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https://mycrazycoding.com/problems/problem_Remove%20duplicate%20element.php

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# Remove duplicate element Problem to remove the duplicate elements in an array. ## Example-1: ```Input : n = [ 30, 20, 10, 20, 30] Output : [10,20,30] Explain : Remove duplicate element (20,30). ``` ## Example-2: ```Input : n = [ 50, 20, 10, 20, 40] Output : [50,20,10,40] Explain : Remove duplicate element (20). ``` # Solution ```import java.util.HashSet; public class Main { public static void main(String [] args) { int a [] = { 30, 20, 10, 20, 30}; HashSet<Integer> number = new HashSet<Integer>(); for(int i=0;i<a.length;i++) { } System.out.print(number); } } ``` ```n = [ 30, 20, 10, 20, 30] a = list(set(n)) print(a) ``` ## Output ```[20,10,30] ```

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CC-MAIN-2023-14

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https://docs.hhvm.com/hack/generics/type-constraints

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Generics: Type Constraints A type constraint on a generic type parameter indicates a requirement that a type must fulfill in order to be accepted as a type argument for that type parameter. (For example, it might have to be a given class type or a subtype of that class type, or it might have to implement a given interface.) A constraint can have one of three forms: • `T as sometype`, meaning that `T` must be a subtype of `sometype` • `T super sometype`, meaning that `T` must be a supertype of `sometype` • `T = sometype`, meaning that `T` must be equivalent to `sometype`. (This is like saying both `T as sometype` and `T super sometype`.) Consider the following example in which function `max_val` has one type parameter, `T`, and that has a constraint, `num`: ``````function max_val<T as num>(T \$p1, T \$p2): T { return \$p1 > \$p2 ? \$p1 : \$p2; } <<__EntryPoint>> function main(): void { echo "max_val(10, 20) = ".max_val(10, 20)."\n"; echo "max_val(15.6, -20.78) = ".max_val(15.6, -20.78)."\n"; } `````` Without the `num` constraint, the expression `\$p1 > \$p2` is ill-formed, as a greater-than operator is not defined for all types. By constraining the type of `T` to `num`, we limit `T` to being an `int` or `float`, both of which do have that operator defined. Unlike an `as` constraint, `T super U` asserts that `T` must be a supertype of `U`. This kind of constraint is rather exotic, but solves an interesting problem encountered when multiple types "collide". Here is an example of how it's used on method `concat` in the library interface type `ConstVector`: ``````interface ConstVector<+T> { public function concat<Tu super T>(ConstVector<Tu> \$x): ConstVector<Tu>; // ... } `````` Consider the case in which we call `concat` to concatenate a `Vector<float>` and a `Vector<int>`. As these have a common supertype, `num`, the `super` constraint allows the checker to determine that `num` is the inferred type of `Tu`. Now, while a type parameter on a class can be annotated to require that it is a subtype or supertype of a particular type, for generic parameters on classes, constraints on the type parameters can be assumed in any method in the class. But sometimes some methods want to use some features of the type parameter, and others want to use some different features, and not all instances of the class will satisfy all constraints. This can be done by specifying constraints that are local to particular methods. For example: ``````class MyWidget<Telem> { public function showIt(): void where Telem as IPrintable { ... } public function countIt(): int where Telem as ICountable { ... } } `````` Constraints can make use of the type parameter itself. They can also make use of generic type parameters on the method. For example: ``````class MyList<T> { public function flatten<Tu>(): MyList<Tu> where T = MyList<Tu> { throw new Exception('unimplemented'); } } `````` Here we might create a list of lists of int, of type `MyList<MyList<int>>`, and then invoke `flatten` on it to get a `MyList<int>`. Here's another example: ``````class MyList<T> { public function compact<Tu>(): MyList<Tu> where T = ?Tu { throw new Exception('unimplemented'); } } `````` A `where` constraint permits multiple constraints supported; just separate the constraints with commas. For example: ``````class SomeClass<T> { function foo(T \$x) where T as MyInterface, T as MyOtherInterface } `````` If a method overrides another method that has declared `where` constraints, it's necessary to redeclare those constraints, but only if they are actually used by the overriding method. (It's valid, and reasonable, to require less of the overriding method.)

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CC-MAIN-2024-38

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https://openspace.infohio.org/curated-collections/38

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# Mathematics 122 affiliated resources # Search Resources View Selected filters: Educational Use Rating In this lesson children will be asked to make a graph, then listen to a story. After the story, they will do a similar graph and then compare the two graphs. Subject: Mathematics Material Type: Lesson Plan Provider: Utah Education Network 02/16/2021 Educational Use Rating In this lesson each student will create a glyph (symbol or icon) which represents them and read the glyphs of others using a legend to understand the data on the glyphs. Subject: Mathematics Material Type: Lesson Plan Provider: Utah Education Network 02/16/2021 Educational Use Rating This lesson will help students understand how to identify, create, and label simple patterns. Subject: Mathematics Material Type: Lesson Plan Provider: Utah Education Network 02/16/2021 Unrestricted Use CC BY Rating This problem helps students practice adding three numbers whose sum are 20 or less. Subject: Mathematics Material Type: Activity/Lab Provider: Illustrative Mathematics Provider Set: Illustrative Mathematics Author: Illustrative Mathematics 09/10/2012 Conditional Remix & Share Permitted CC BY-NC-SA Rating This short video and interactive assessment activity is designed to give fourth graders an overview of 24-hour clocks. Subject: Mathematics Material Type: Assessment Interactive Lecture Provider: CK-12 Foundation Provider Set: CK-12 Elementary Math 11/18/2020 Unrestricted Use CC BY Rating CK-12 Algebra Explorations is a hands-on series of activities that guides students from Pre-K to Grade 7 through algebraic concepts. Subject: Algebra Functions Material Type: Activity/Lab Lesson Plan Textbook Provider: CK-12 Foundation Provider Set: CK-12 FlexBook Author: Mary Cavanagh, Carol Findell, Carole Greenes 10/04/2011 Conditional Remix & Share Permitted CC BY-NC-SA Rating Students develop an understanding of areas as how much two-dimensional space a figure takes up, and relate it to their work with multiplication from Units 2 and 3. Subject: Mathematics Material Type: Unit of Study Provider: Fishtank Learning Provider Set: Mathematics 11/19/2021 Conditional Remix & Share Permitted CC BY-NC Rating You get the general idea of decimal is and what the digits in different places represent (place value). Now you're ready to do something with the decimals. Adding and subtracting is a good place to start. This will allow you to add your family's expenses to figure out if your little brother is laundering money (perhaps literally). Have fun! Common Core Standard: 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider: Provider Set: Author: Khan Salman 11/17/2020 Conditional Remix & Share Permitted CC BY-NC Rating You get the general idea of decimal is and what the digits in different places represent (place value). Now you're ready to do something with the decimals. Adding and subtracting is a good place to start. This will allow you to add your family's expenses to figure out if your little brother is laundering money (perhaps literally). Have fun! Common Core Standard: 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider: Provider Set: Author: Khan Salman 11/17/2020 Conditional Remix & Share Permitted CC BY-NC Rating You get the general idea of decimal is and what the digits in different places represent (place value). Now you're ready to do something with the decimals. Adding and subtracting is a good place to start. This will allow you to add your family's expenses to figure out if your little brother is laundering money (perhaps literally). Have fun! Common Core Standard: 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider: Provider Set: Author: Khan Salman 11/17/2020 Conditional Remix & Share Permitted CC BY-NC Rating The real world is seldom about whole numbers. If you precisely measure anything, you're likely to get a decimal. If you don't know how to multiply these decimals, then you won't be able to do all the powerful things that multiplication can do in the real world (figure out your commission as a robot possum salesperson, determining how much shag carpet you need for your secret lair, etc.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider: Provider Set: Author: Khan Salman 11/17/2020 Conditional Remix & Share Permitted CC BY-NC Rating The real world is seldom about whole numbers. If you precisely measure anything, you're likely to get a decimal. If you don't know how to multiply these decimals, then you won't be able to do all the powerful things that multiplication can do in the real world (figure out your commission as a robot possum salesperson, determining how much shag carpet you need for your secret lair, etc.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider: Provider Set: Author: Khan Salman 11/17/2020 Conditional Remix & Share Permitted CC BY-NC Rating The real world is seldom about whole numbers. If you precisely measure anything, you're likely to get a decimal. If you don't know how to multiply these decimals, then you won't be able to do all the powerful things that multiplication can do in the real world (figure out your commission as a robot possum salesperson, determining how much shag carpet you need for your secret lair, etc.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7 Subject: Number and Quantity Material Type: Activity/Lab Homework/Assignment Lecture Provider:

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# How to Win a Lottery and What the Odds Are of Winning The lottery is a form of gambling that involves drawing numbers at random. Some governments outlaw the lottery while others endorse it, organizing state and national lotteries and regulating the process. Read on to find out how to win a lottery and what the odds are of winning. We will also look at some common numbers and scratch-off games. ## Chances of winning a lottery jackpot While the lottery has become one of the most popular forms of entertainment, chances of winning a lottery jackpot are very low and do not increase with frequency of play. The advertised jackpots are merely the total of annuity payments that will be made over decades, rather than a single lump sum payment. In addition, lottery operators tend to reduce the odds of hitting a jackpot over time to make the jackpot grow larger. ## Odds of winning The formula to calculate the odds of winning the lottery is a little complex. It uses a factorial of n minus r. The n is the total number of numbers in the lottery and r is the number of draws. For example, if you pick five numbers out of the possible seven, you have a 325 percent chance of winning the lottery. However, if you chose four numbers, you have a 66 percent chance of winning. ## Common numbers In recent years, lottery numbers have been appearing at a lower frequency than usual. This is explained by the law of large numbers. This mathematical law states that, after enough drawings, all numbers have an equal chance of occurring. Despite this, lottery enthusiasts still believe that some numbers are more likely to be drawn than others. Regardless of the reason for this phenomenon, there are still some ways you can try to maximize your chances of winning. ## Scratch-off games Scratch-off games in the lottery are a fun and exciting way to win cash prizes. Players can purchase tickets for as little as a dollar or as much as \$30. There are a variety of different games with different jackpots, so it’s important to choose the right one. Also, be sure to research the odds and deadlines before buying a ticket. ## Multistate lotteries Multistate lotteries have a variety of benefits. These benefits include being able to offer more games than a single jurisdiction, and helping to develop multijurisdictional games. These benefits are shared by member lotteries that own and operate the games. In addition, members of the Multistate Lottery Association retain their statutory independence. ## Taxes on winnings If you win the lottery, you’ll probably have to pay taxes on your winnings. As with other income, the federal government takes a cut of the amount you win. You can either pay this tax in a lump sum or over a period of years. The tax you pay will depend on your income and tax bracket.

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http://www.math-only-math.com/worksheet-on-numbers-writing.html

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# Worksheet on Numbers Writing Worksheet on numbers writing for 4th grade math students helps to practice writing big numbers. In this worksheet students need to practice the numbers writing in figures. This numbers printable activity sheets help the fourth grade students to practice changing the numbers in words to figures. ### Write the following numbers in figures: 1. Three hundred forty two 2. Three thousand two hundred three 3. Seventeen thousand five hundred 4. Ten thousand one 5. Eighty thousand forty nine 6. Nine hundred five thousand twenty 7. One hundred fifteen thousand six hundred one 8. Four hundred two thousand sixteen 9. Twelve thousand ninety 10. Six hundred forty thousand two hundred four If students have any queries regarding the questions please fill up the comment box below so that we can help you. However, suggestions for further improvement, from all quarters would be greatly appreciated. Numbers - Worksheets Worksheet on Numbers. Number Worksheets. Worksheets on Comparison of Numbers. Worksheet on Numbers in Words. Worksheet on Numbers Writing. Worksheets Showing Numbers on Spike Abacus. Worksheet on Place Value. Worksheet on Expanded form of a Number. Worksheet on Formation of Numbers. Worksheet on Rounding off Numbers. Worksheet on Place Values. ### New! Comments Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

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CC-MAIN-2017-34

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https://math.stackexchange.com/questions/43276/minimal-turing-machines-are-not-recursively-enumerable-kolmogorov-complexity-i

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# Minimal Turing machines are not recursively enumerable & Kolmogorov complexity is uncomputable ### First problem If $M$ is a Turing machine, then we say that the length of the description of $\langle$M$\rangle$ of M is the number of symbols in the string describing M. We say that M is minimal if there is no Turing machine equivalent to M that has a shorter description. We let $MIN_{TM}$={$\langle$M$\rangle$|M is a minimal TM} Claim: $MIN_{TM}$ is not recursively enumerable. Proof (by Sipser): $C$="On input $w$: 1. Obtain, via the recursion theorem, own description $\langle$C$\rangle$ 2. Run the enumerator $E$ until a machine $D$ appears with a longer description than that of C. 3. Simulate $D$ on input $w$". I am trying to understand the proof above. Since $MIN_{TM}$ is infinite, why does it follow that $E$'s list must contain a TM with a longer description than $C$'s description and is also equivalent to C? Since it follows that $E$'s list must contain a TM with a longer description than $C$'s description, then step $2$ of $C$ must eventually terminate with some TM $D$ that is longer than $C$. Then, $C$ simulates $D$, and is equivalent to it. Since $C$ is shorter than $D$ and is equivalent to it, $D$ cannot be minimal. But $D$ appears on the list that $E$ produces, thus a contradiction. ### Second problem Let $x$ be a binary string. We say that the minimal description of $x$, written as $d(x)$, is the shortest string $\langle$M, w$\rangle$ where TM $M$ on input $w$ halts with $x$ on its tape. So, the Kolmogorov-Chaitin complexity $K(x)$ is written as, $$K(x)=|d(x)|.$$ $K(x)$ is defined to be the length of minimal description of $x$. How can you prove that $K(x)$ is uncomputable? The proofs I read on Wikipedia and other sites are proofs that involve programming arguments. I am not a computer scientist nor do I have any knowledge in programming (just a math student). I would like to see a cohesive (but simple) mathematics proof. I read that the decidability of Kolmogrovo-Chaitin complexity would imply that the halting problem is decidable, how? • Notice that the given proof actually shows that $MIN_{TM}$ contains no infinite c.e. set; i.e. that $MIN_{TM}$ is immune. – Quinn Culver Jun 5 '11 at 4:28 • en.wikipedia.org/wiki/Berry_paradox has a simple explanation of this. – Dan Brumleve Jun 5 '11 at 19:29 Suppose Kolmogorov complexity were computable. One could then write a program that enumerates all strings and outputs a string $s$ of complexity at least $n_0$, where $n_0$ is some constant we choose later. The program has size $C + 2\log n_0$ for some constant $C$ depending on the particular encoding, and so $$C + 2\log n_0 \geq K(s) \geq n_0.$$ For large enough $n_0$, this is contradictory.

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Supplementary Materialssupplemental Supplementary Materialssupplemental. We provide its theoretical properties under the framework of generalized linear models. Powered by an extended Bayesian information criterion as the stopping rule, the method will lead to a final model without the need to choose tuning parameters or threshold parameters. The practical utility of the proposed method is examined via extensive simulations and analysis of a real clinical study on predicting multiple myeloma patients response to treatment based on their genomic profiles. and sequentially recruits more variables into the conditioning set then, and our method is valid even in the absence of the prior information about which variables to condition on. The rest of the paper is organized as follows. In Section 2, 5-R-Rivaroxaban we introduce the proposed sequential conditioning procedure. In Section 3, we establish the sure screening property. Section 4 details the assessment of the finite sample performance of the proposed method and Section 5 illustrates our method by predicting treatment response based on myeloma patients genomic profiles using the aforementioned data example. We conclude the paper with a brief discussion in Section 6 and relegate all the technical details, including lemmas, proofs and conditions, to the online Supporting Information. 2.?Sequentially Conditional Modeling Suppose that there are independent samples (X= 1,, is an outcome, X= (+ 1 predictors for the for all ? 1. We focus on a class of GLMs by assuming that the conditional density of given Xbelongs to the linear exponential family: = ((= 1, , be the mean of ? is on the exponential order of ? ? {0, 1, = {: denotes the collection of covariates for the and to denote the complement of to denote the average log-likelihood of the regression model of on Xfor a given ? {0, 1, to denote the maximizer of the offset evaluated at the ? {0, 1, maximizes and is the estimated intercept without 5-R-Rivaroxaban any other covariates. That is, we start from the null model with only an intercept term. We can also start with a set of given variables according to some Rabbit polyclonal to AP1S1 knowledge, which is in the same spirit as conditional screening (Barut et al., 2016). However, as opposed to Barut et al. (2016), our procedure updates the conditioning set with a sequential selection process dynamically, which is detailed below. First, with such an {1, 5-R-Rivaroxaban on to obtain ? 1, given and for {on to obtain and let ? EBIC(priori known S0. Otherwise, initialize with maximizes ? 1, given and for as a fixed constant which may not vary by datasets. This is analogous to the constant values. 3.?Theoretical Properties Let and denote convergence in distribution and probability, respectively. For a column vector ? 1, denote its satisfying and log = 0 such that denotes the least false value of model 0 is a constant. Let such that the Cramer condition holds for all for all and ? 2. There exist two positive constants 0 , such that and ? {0, 1, 0 and 0 such that and log 0. Condition (A) differs from the Lipschitz assumption in van de Geer (2008), Fan and Song (2010), and Barut et al. (2016). A similar condition is assumed in Bhlmann (2006). The condition log = is an upper bound of the model size, which is required in joint-model-based selection or screening methods with various notation often, such as M in Cheng et al. (2016), and K in Zhang and Huang (2008), Chen and Chen (2008), and Fan and Tang (2013). This condition is weaker than Assumption D in Cheng et al. (2016), which requires log = and is satisfied by a wide range of outcome data, including Gaussian and discrete data (such as binary and count data). Condition (D) has been commonly assumed in literature (Wang, 2009; Zheng et al., 2015; Cheng et al., 2016) and represents the Sparse Riesz Condition (Zhang and Huang, 2008). Compared to those required by joint-model-based sequential screening methods in the literature, the signal condition (E) is not directly imposed on the regression coefficient. Instead, it is imposed on the conditional covariance between a covariate and the response, as 5-R-Rivaroxaban in Barut et al. (2016). The condition can also be reviewed as an strong irrepresentable 5-R-Rivaroxaban condition (Zhao and Yu, 2006) for model identifiability, stipulating that the true model cannot be represented by.

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# Maximize the value of the given expression • Difficulty Level : Easy • Last Updated : 07 Mar, 2022 Given three non-zero integers a, b and c. The task is to find the maximum value possible by putting addition and multiplication signs between them in any order. Note: Rearrangement of integers is allowed but addition and multiplication sign must be used once. Braces can also be placed between equations as per your need. Examples: Input: a = 2, b = 1, c = 4 Output: 12 (1 + 2) * 4 = 3 * 4 = 12 Input: a = 2, b = 2, c = 2 Output: (2 + 2) * 2 = 4 * 2 = 8 Approach: To solve this problem one can opt the method of generating all the possibilities and calculate them to get the maximum value but this approach is not efficient. Take the advantage of given conditions that integers may got rearranged and mandatory use of each mathematical sign (+, *). There are total of four cases to solve which are listed below: 1. All three integers are non-negative: For this simply add two smaller one and multiply their result by largest integer. 2. One integer is negative and rest two positive : Multiply the both positive integer and add their result to negative integer. 3. Two integers are negative and one is positive: As the product of two negative numbers is positive multiply both negative integers and then add their result to positive integer. 4. All three are negative integers: add the two largest integers and multiply them to smallest one. case 3-: (sum – smallest) * smallest Below is the implementation of the above approach: ## C++ `// C++ implementation of the approach``#include ``using` `namespace` `std;` `// Function to return the maximum result``int` `maximumResult(``int` `a, ``int` `b, ``int` `c)``{` `    ``// To store the count of negative integers``    ``int` `countOfNegative = 0;` `    ``// Sum of all the three integers``    ``int` `sum = a + b + c;` `    ``// Product of all the three integers``    ``int` `product = a * b * c;``    ` `    ``// To store the smallest and the largest``    ``// among all the three integers``    ``int` `largest = max(a,max(b,c));``    ``int` `smallest = min(a,min(b,c) );``      ` `    ` `    ``// Calculate the count of negative integers``    ``if` `(a < 0)``        ``countOfNegative++;``    ``if` `(b < 0)``        ``countOfNegative++;``    ``if` `(c < 0)``        ``countOfNegative++;` `    ``// Depending upon count of negatives``    ``switch` `(countOfNegative) {` `    ``// When all three are positive integers``    ``case` `0:``        ``return` `(sum - largest) * largest;` `    ``// For single negative integer``    ``case` `1:``        ``return` `(product / smallest) + smallest;` `    ``// For two negative integers``    ``case` `2:``        ``return` `(product / largest) + largest;` `    ``// For three negative integers``    ``case` `3:``        ``return` `(sum - smallest) * smallest;``    ``}``}` `// Driver Code``int` `main()``{``    ``int` `a=-2,b=-1,c=-4;``    ``cout << maximumResult(a, b, c);` `    ``return` `0;``}``// This code contributed by Nikhil` ## Java `// Java implementation of the approach``class` `GFG``{``    ` `// Function to return the maximum result``static` `int` `maximumResult(``int` `a, ``int` `b, ``int` `c)``{` `    ``// To store the count of negative integers``    ``int` `countOfNegative = ``0``;` `    ``// Sum of all the three integers``    ``int` `sum = a + b + c;` `    ``// Product of all the three integers``    ``int` `product = a * b * c;` `    ``// To store the smallest and the largest``    ``// among all the three integers``    ``int` `largest = (a > b) ? ((a > c) ? a : c) :``                            ``((b > c) ? b : c);``    ``int` `smallest= (a ## Python3 `# Python3 implementation of the approach` `# Function to return the maximum result``# Python3 implementation of the approach` `# Function to return the maximum result``def` `maximumResult(a, b, c):` `    ``# To store the count of negative integers``    ``countOfNegative ``=` `0` `    ``# Sum of all the three integers``    ``Sum` `=` `a ``+` `b ``+` `c` `    ``# Product of all the three integers``    ``product ``=` `a ``*` `b ``*` `c` `    ``# To store the smallest and the``    ``# largest among all the three integers``    ``largest ``=` `max``(a, b, c)``    ``smallest ``=` `min``(a, b, c)` `    ``# Calculate the count of negative integers``    ``if` `a < ``0``:``        ``countOfNegative ``+``=` `1``    ``if` `b < ``0``:``        ``countOfNegative ``+``=` `1``    ``if` `c < ``0``:``        ``countOfNegative ``+``=` `1` `    ``# When all three are positive integers``    ``if` `countOfNegative ``=``=` `0``:``        ``return` `(``Sum` `-` `largest) ``*` `largest` `    ``# For single negative integer``    ``elif` `countOfNegative ``=``=` `1``:``        ``return` `(product ``/``/` `smallest) ``+` `smallest` `    ``# For two negative integers``    ``elif` `countOfNegative ``=``=` `2``:``        ``return` `(product ``/``/` `largest) ``+` `largest` `    ``# For three negative integers``    ``elif` `countOfNegative ``=``=` `3``:``        ``return` `(``Sum` `-` `smallest) ``*` `smallest` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``a, b, c ``=` `-``2``, ``-``1``, ``-``4``    ``print``(maximumResult(a, b, c))` ## C# `// C# implementation of the approach``using` `System;` `class` `GFG``{``    ` `// Function to return the maximum result``static` `int` `maximumResult(``int` `a, ``int` `b, ``int` `c)``{` `    ``// To store the count of negative integers``    ``int` `countOfNegative = 0;` `    ``// Sum of all the three integers``    ``int` `sum = a + b + c;` `    ``// Product of all the three integers``    ``int` `product = a * b * c;` `    ``// To store the smallest and the largest``    ``// among all the three integers``    ``int` `largest = (a > b) ? ((a > c) ? a : c) :``                            ``((b > c) ? b : c);``    ``int` `smallest=(a ## PHP `` ## Javascript `` Output: `12` Time Complexity: O(1) Auxiliary Space: O(1) My Personal Notes arrow_drop_up

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Home My Page Projects Code Snippets Project Openings 3D graphics for Standard ML Summary Activity SCM # SCM Repository [sml3d] View of /src/common/float-onb.sml [sml3d] / src / common / float-onb.sml # View of /src/common/float-onb.sml Mon Apr 7 13:35:25 2008 UTC (11 years, 8 months ago) by jhr File size: 2911 byte(s) ``` Various bug fixes / feature enhancements. ``` ```(* onb-sig.sml * * COPYRIGHT (c) 2008 John Reppy (http://www.cs.uchicago.edu/~jhr) * * Ortho-normal basis and frames. * * NOTE: this code doesn't do any error checking; it probably should! *) structure FloatONB : ONB = struct structure V3 = Vec3f type flt = SML3dTypes.float type vec3 = flt SML3dTypes.vec3 type onb = {u : vec3, v : vec3, w : vec3} type frame = {origin : vec3, onb : onb} (* the standard ONB *) val xyz : onb = {u = V3.e1, v = V3.e2, w = V3.e3} fun fromU u = let val u = V3.normalize u val (vLen, v) = V3.lengthAndDir (V3.cross(u, V3.e1)) val v = if (vLen < 0.01) then V3.cross(u, V3.e2) else v val w = V3.cross(u, v) in {u = u, v = v, w = w} end fun fromV v = let val v = V3.normalize v val (uLen, u) = V3.lengthAndDir (V3.cross(v, V3.e1)) val u = if (uLen < 0.01) then V3.cross(v, V3.e2) else u val w = V3.cross(u, v) in {u = u, v = v, w = w} end fun fromW w = let val w = V3.normalize w val (uLen, u) = V3.lengthAndDir (V3.cross(w, V3.e1)) val u = if (uLen < 0.01) then V3.cross(w, V3.e2) else u val v = V3.cross(w, u) in {u = u, v = v, w = w} end (* construct an ONB where u is parallel to a, v lies in the ab plane * and w is parallel to a cross b. *) fun fromUV (a, b) = let val u = V3.normalize a val w = V3.normalize(V3.cross(a, b)) val v = V3.cross(w, u) in {u = u, v = v, w = w} end (* construct an ONB where v is parallel to a, u lies in the ab plane * and w is parallel to b cross a. *) fun fromVU (a, b) = let val v = V3.normalize a val w = V3.normalize(V3.cross(b, a)) val u = V3.cross(v, w) in {u = u, v = v, w = w} end fun fromUW (a, b) = let val u = V3.normalize a val v = V3.normalize (V3.cross (b, a)) val w = V3.cross (u, v) in {u = u, v = v, w = w} end fun fromWU (a, b) = let val w = V3.normalize a val v = V3.normalize (V3.cross(a, b)) val u = V3.cross(v, w) in {u = u, v = v, w = w} end fun fromVW (a, b) = let val v = V3.normalize a val u = V3.normalize (V3.cross(a, b)) val w = V3.cross(u, v) in {u = u, v = v, w = w} end fun fromWV (a, b) = let val w = V3.normalize a val u = V3.normalize (V3.cross(b, a)) val v = V3.cross(w, u) in {u = u, v = v, w = w} end (* transform a world vector to the ONB *) fun toONB {u, v, w} a = {x = V3.dot(a, u), y = V3.dot(a, v), z = V3.dot(a, w)} (* transform a vector from the ONB to the world coordinates *) fun fromONB {u, v, w} {x, y, z} =

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DISCOVER # How to Read an E-Scale Ruler Updated April 17, 2017 Reading an E-scale, also known as an Engineering scale or tri-scale, can be very confusing to the average person. Having three separate rulers, each with up to four separate scales, it is easy to make a mistake when taking a measurement. After learning how and why the ruler is laid out like it is, your E-scale becomes an invaluable tool and helps decipher engineering and architectural drawings. As full scale drawings would be too big, they are reduced to scale and translated with E-scales. Locate the key on your diagram that contains the scale of the drawing. This information should be listed under the heading of "Legend." Determine if the drawing is an architectural or an engineering type and select the appropriate E-scale. Engineering scales are read left to right, and should have scales of 1:20, 1:25, 1:50, 1:75, 1:100 and 1:125. Architectural scales can be read both left to right, and right to left, and are likely to be labelled with 3/32, 3/16, 1/8, 1/4, 3/8, 1/2, 3/4, 1, 1-1/2, 3 and 16 scales. Select the scale on the ruler to match the scale on the drawing. Engineering scales should match engineering drawings, and architectural scales should match architectural drawings. Ensure the drawing has been printed to size correctly by placing the appropriate scale on the drawing's scale. These two should match. Measure a line in the drawing by placing the appropriate scale at the beginning of the line, matched to the 0 (zero) on the scale, and taking a reading from the ruler where the line ends. If the scale on the drawing states that 1/8 inch equals 1 foot, using your architect scale, select the ruler labelled 1/8. If you measure a line to the 32 mark on the 1/8 scale, the line represents a length of 32 feet in reality. If you accidentally selected the 1/4 scale, the same line would appear to incorrectly represent a 16-foot length. #### Warning Be sure you use the correct scale with the scale of the drawing. Architecture scales read from left to right and from right to left. The numbers on the right to left scales appear lower than the left to right scale and increase in value from right to left. #### Things You'll Need • E-scale ruler

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math what can you use to tie up a spaceship 1. 👍 4 2. 👎 1 3. 👁 1,988 1. Astro Knots 1. 👍 2 2. 👎 1 posted by David 2. ASTRO KNOTS 1. 👍 2 2. 👎 2 posted by poop 3. WITH ASTROKNOTS 1. 👍 2 2. 👎 1 posted by BOB 4. Astro KNots 1. 👍 2 2. 👎 1 posted by Juan 5. Astronauts 1. 👍 1 2. 👎 3 6. What's the written answers to math worksheet what can you use to tie up a spaceship page 156 1. 👍 1 2. 👎 1 7. my nuts 1. 👍 2 2. 👎 0 1. 👍 1 2. 👎 2 posted by leeeee 9. Astro thots 1. 👍 4 2. 👎 1 10. Astro Knots. Btw, what is the math for it? ;-; 1. 👍 0 2. 👎 0 posted by Mel 11. jhklngjdkgndjnb; 1. 👍 0 2. 👎 1 12. Astro Nuts 1. 👍 0 2. 👎 1 13. yes 1. 👍 1 2. 👎 0 14. Roasted and toasted garlic toasted toast 1. 👍 0 2. 👎 1 posted by Hehe 15. Astro knots 1. 👍 0 2. 👎 0 posted by Julie Similar Questions 1. math whats th eanswer to the math page. why do elephants have ivory tusks? and the answer to what can you use to tie up a spaceship? asked by rob on June 6, 2012 2. physics Spaceship 1 and Spaceship 2 have equal masses of 300 kg. Spaceship 1 has a speed of 0 m/s, and Spaceship 2 has a speed of 4 m/s. What is the magnitude of their combined momentum? asked by Anonymous on March 8, 2017 3. physics The length of a moving spaceship is 27.0 m according to an astronaut on the spaceship. If the spaceship is contracted by 14.5 cm according to an Earth observer, what is the speed of the spaceship? I know L=27m and delta L= 0.145 asked by Robby on May 4, 2015 4. College Physics 2 A broken-down spaceship (star Wreck?) flies across an international soccer field (Regulation length = 100 m) at a speed of 0.55c. A.) How long does the spaceship pilot measure the field to be? B.) From the spaceship pilot’s asked by seanmarch58 on August 2, 2014 5. Physics A spaceship in deep space has a velocity of 4000 km/s and an acceleration in the forward direction of 6 m/s2. What is the acceleration of a ball relative to the spaceship after it is released in this spaceship? asked by Kay on February 12, 2015 6. Physics An astronaut with a mass of 90 kg (including spacesuit and equipment) is drifting away from his spaceship at a speed o f.2 m/s with respect to the spaceship. The astronaut is equipped only with a 0.5 kg wrench to help him get back asked by Kelsey on November 13, 2016 7. Science An astronuat who was outside repairing his spaceship with a wrench lost his tie rope. Now he is stranded in space. Unfortunately, you can't swim in space. How can the law of conservation of momentum help the astronaut return to asked by Carolyn on September 28, 2010 8. physics question 1) Let's continue to investigate your space vacation from last week. Your spaceship, which has a proper length of 300 m, passes near a space platform while you are moving at a relative speed of 86% the speed of light (so asked by ronnette on March 4, 2012 9. Physics 1) Let's continue to investigate your space vacation from last week. Your spaceship, which has a proper length of 300 m, passes near a space platform while you are moving at a relative speed of 86% the speed of light (so asked by ronnette on March 4, 2012 10. Physics Bob and Bob Jr. stand at open doorways at opposite ends of an airplane hangar 25 m long. Anna owns a spaceship, 40 m long as it sits on the runway. Anna takes off in her spaceship, then swoops through the hangar at constant asked by Katie on January 29, 2013 More Similar Questions

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or # Simplify each of the following and write as a rational number of the form p/q :\ (-11)/2+7/6+(-5)/8 (ii) (-4)/5+(-7)/(10)+(-8)/(15) Question from  Class 8  Chapter Rational Numbers Apne doubts clear karein ab Whatsapp par bhi. Try it now. Watch 1000+ concepts & tricky questions explained! 106.6 K+ views | 43.9 K+ people like this Share Share

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Category: O Level Here, you will find all the resources for O Level exams. We cover O level Chemistry, Physics and Additional Mathematics . Energy Work and Power – O Level Physics (What’s tested?) In this article, we go through what’s tested in the chapter energy, work and power in O level Physics. This is a application chapter Pressure in O Level Physics – What’s tested? Here, you’ll find what’s tested in the topic pressure in O level Physics. You’ll need to know some formulae, definitions and more Turning Effects of Forces for O Level Physics – What’s tested What’s tested in turning effects of forces for O Level Physics? There are some definitions,, formulae of moments to know and more. Mass, Weight and Density for O Level Physics – What’s tested Let’s look at what’s tested for mass, weight and density in O Level physics. This chapter has many definitions and 2 formulae to remember. Dynamics (Forces) for O Level Physics – What you need to know In this post, let’s look at what you need to know for dynamics (or forces) for O Level Physics. In this chapter, students learn about… Kinematics for O Level Physics – what you need to know What do you need to know for kinematics for O Level Physics? Let’s talk about it in this post. Kinematics is the study of the… What you need to know for O Level Physics Chapter: Physical Quantities, Units and Measurement In O Level Physics, physical quantities, units and measurement is the first chapter students learn. In this post, let’s talk about what you need to… Use of electrostatic charging in a Photocopier In this post, let’s talk about the use of electrostatic charging in a photocopier, tested in O Level Physics (static electricity topic). O Level Physics Formulae Here you’ll get a list of the O level physics formulae to help you study for your exams. I’ve listed them according to the physics chapters. List of O Level Physics Definitions Here you’ll find a list of O Level Physics definitions. In this list, you will find the common terms that you will be asked to define.

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hwset6(2) # hwset6(2) - Carnegie Mellon University Department of... This preview shows page 1. Sign up to view the full content. Carnegie Mellon University Department of Electrical and Computer Engineering 18-100 Fall 2009 Introduction to Electrical and Computer Engineering Homework 6 - Due Tuesday, October 20, 2009 Please do all numbered problems listed below from the Rizzoni text. For our purposes, assume the base-emitter region has a turn-on voltage of 0.6V, and the collector-emitter saturation voltage is 0.2V. 1. Problem 10.11. Don’t use the 2N3904 characteristics, just let β = 100 for this problem. 2. Problem 10.12. Let the input voltage be 8.1V, base resistor be 75k Ω , collector resistor be 2 k Ω , and β = 100. 3. Problem 10.14. Use Vce,sat = 0.2V and β = 100. Let the base resistor be 100k Ω and the collector resistor be 5 k Ω . 4. Problem 10.16. Very similar to problem 10.11 (#1 above). Again, don’t use the 2N3904 characteristics, just let β = 100 for this problem. Now you have both a collector and emitter resistor. 5. Problem 10.6. Change the voltage source to 12V, let β = 100, collector resistor be 2k Ω , and emitter resistor be 1k Ω . This is the end of the preview. Sign up to access the rest of the document. • Fall '07 • Williams • Department of Electrical and Computer Engineering, Thévenin's theorem, small signal model, Emitter Resistor {[ snackBarMessage ]} ### What students are saying • As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students. Kiran Temple University Fox School of Business ‘17, Course Hero Intern • I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero. Dana University of Pennsylvania ‘17, Course Hero Intern • The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time. Jill Tulane University ‘16, Course Hero Intern

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Home Forums Ready-to-use Robots Top 10 EAs Properties & Settings Reply To: Properties & Settings #267430 Marin Stoyanov Keymaster Hi, let me address all the properties one by one. R-Squared is a statistical measure that can be useful to gauge how the strategy’s performance is influenced by broader market movements. An R² of 0 means that none of the performance can be explained by the index used for comparison. An R² of 50 for example, suggests that approximately 50% of the strategy’s performance can be explained by fluctuations in the benchmark. Please check this post and the linked video to learn about R-Squared (this is a complex statistical measure and it’s not that simple to explain so better read the post and watch the video). Return to DD (Return to Drawdown) is a measure that helps evaluate the risk-adjusted return of the trading strategy. This ratio is calculated by dividing the total return by the maximum drawdown experienced during the observed period. For example a ratio of 0.99 suggests that for nearly every 1% of drawdown experienced, the strategy generated an equivalent increase in value. With another example, if Return to DD is 3.57 it indicates that the strategy earned about 3.57 times the maximum percentage decrease experienced in account value. This high ratio is a positive sign, showing that the returns significantly outweigh the risks taken (measured by drawdown). Profit Factor is the ratio of gross profits to gross losses over the specified period. It helps to understand how well the strategy performs in terms of profitability. Regarding what would be good values, we don’t put that much attention to the actual values of the above parameters. The way we use the app is just to pick the top 3/5 for from the weekly/monthly charts and trade with them. Then we check the performance on a daily basis and make amendments to the strategies we trade with if there are new top performers in the top 3/5 for the period we’re looking at. This is the system we follow and it works for us but this is not a recommendation or the only way to do it. Some traders might use the above properties to select their strategies or might have their own way to manage the EAs. Shopping Cart

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## Abstract Syntax ``````Expr ::= 'u' | Expr Expr `````` We write expressions using parentheses where necessary (assuming left-associativity), for example `u u (u u u)`. ## Abstract Semantics An expressions may be reducible to another expression, otherwise (if irreduible) we also call it a value. Expressions are equal (`=`) exactly if they reduce to the same value or reduction does not terminate for either expression. ### Reduction Extend `Expr` with two new abstract expressions `s` and `k`. Apply the following rewrite rules as long as possible. ``````u α ⟶ α s k k α β ⟶ α s α β γ ⟶ α γ (β γ) `````` ### Observability Given an expression `■`, we define `⟦■⟧` as ``````⟦■⟧ = argmin n (n, i, a) ∈ S where S = { (n, i, a) ∈ ℕ × ℕ₀ × ℕ₀ | ∀ α₀, ..., αₙ : ∃ β₁, ..., βₐ : ■ α₀ ... αₙ = αᵢ β₁ ... βₐ } `````` An implementation may assume that it is only asked to compute `⟦■⟧` if it is defined. ## Concrete Syntax The notation used so far is not succinct since multiple copies of identical subexpressions need to be written out. We hence introduce a let binding syntax, like in Haskell or OCaml. In case of name collisions, inner definitions hide outer ones. The expression `u u (u u u)` could be written as `let i = u u in i (i u)`. However, since Lambada is aimed towards ease of implementation, simplicity and minimalism, this will not be our concrete syntax. Instead, we linearize the notation and allow any non-empty sequences of non-whitespace Unicode characters as names. The only predefined (but not reserved!) name is “u”, which represents expression `u`. ### Grammar ``````Terminator ::= ' ' -- space 0x20 Define ::= '\n' -- newline 0x0A Name ::= \S+ Expression ::= α:Name Terminator -- Case: α (name) | β:Expression γ:Expression Terminator -- Case: β γ (application) | β:Expression α:Name Define γ:Expression -- Case: let α = β in γ ``````

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# Natural Response Dive into the fascinating world of physics concepts with a comprehensive exploration of natural response. This inclusive guide sheds light on understanding natural response, its importance in quantum and regular physics, and how it operates in various phenomena. Unpack the concept further with detailed analysis of examples, insightful calculation techniques, and the crucial role it plays in bridging theory and practical physics. The exploration extends to its application in everyday life, with special focus on the natural response in RC circuits. Stay tuned to unlock the significance of this fundamental topic that holds both relevance and application in the modern world of physics. #### Create learning materials about Natural Response with our free learning app! • Flashcards, notes, mock-exams and more • Everything you need to ace your exams ## Understanding Natural Response in Physics ### Fundamental Concepts of Natural Response The term "Natural Response" significantly manifests in the domain of physics. It refers to the reactive behavior of a system when it's left to its own devices, bereft of any external influence. A prime example of this is an oscillating pendulum that gradually comes to a halt. The restful state that the system eventually adopts is known as its natural response. Natural Response: The inherent reaction of a system that surfaces when no external force or influence is acting upon it. Specific systems, for example, an LCR (Inductor-Capacitor-Resistor) circuit will have its bespoke natural response, which can be accurately predicted using equations of motion. These equations are generally second order differential equations, represented as: $a\frac{d^2x}{dt^2} + b\frac{dx}{dt} + cx = 0$ To decipher the natural response, two constituents are integral: the Initial Condition and Damping Ratio. • Initial Condition: Describes the state of the system at time $$t = 0$$. • Damping Ratio: A dimensionless quantity which describes how oscillations in a system decay after a disturbance. ### The Importance of Natural Response in Regular and Quantum Physics The concept of natural response isn't confined to regular physics, but also extends its relevance to the field of quantum physics. In both domains, the ability to predict a system's natural response proves indispensable for multiple reasons: • Understanding System Behaviour: Knowing the natural response helps describe how a system unearths its equilibrium condition. • Designing Efficient Systems: Engineers often design systems to have the desired natural response - such as designing a car suspension system to cushion the bumps on a road. • Manipulating Quantum Systems: In quantum physics, understanding natural responses can govern quantum entanglement and other fascinating phenomena. In quantum physics, every system riddled with energy levels beholds its unique natural response. In a state of non-equilibrium, these systems tend to strive toward equilibrium - their natural response. This non-equilibrium scenario can be attained via quantum quenching, where parameters in the system change suddenly. Understanding the natural responses here can lead to quantum information protocol enhancements and paving the way for quantum computers. ### How Natural Response Works in Various Phenomena Applying the concept of natural response implies a myriad of phenomena, be it in electronics or mechanics. Let's look at some examples: RLC Circuit: In an electric circuit consisting of a Resistor (R), an Inductor (L), and a Capacitor (C), the natural response describes how current changes over time when an external power supply is disconnected. Damped Harmonic Oscillator: In a simple pendulum, once set into motion, it will eventually stop due to air resistance, which acts as a damping force. This eventual halt is the result of the system's natural response, which is, in this case, rest. In both examples, captivatingly, natural responses are observed when these systems are left alone, undeterred by any external influence. Hence, natural responses help understand the inherent behaviours of systems, as reflected in the diverse array of phenomena across physics. ## Natural Response of RC Circuit ### RC Circuit: An Overview and its Connection to Natural Response In the broad spectrum of electric circuits, Resistor-Capacitor or RC circuits secure a pivotal place. They consist of a resistor and capacitor connected in series or parallel. The values of resistance and capacitance define the circuit's behaviour, dictating its frequency and phase response. Your television tuner or an audio amplifier might be utilising this very principle! RC Circuit: An electric circuit that includes a resistor (R) and a capacitor (C) connected in series or parallel. The ingrained relation between an RC circuit and its natural response is consequential. An RC circuit's natural response manifests when the external supply is severed. The electrical energy stored in the capacitor commences to discharge through the resistor, forming the free, unprovoked or "natural" response of the circuit over a distinct period, essentially called the time constant. Time Constant: Denoted by $$\tau$$ in the context of an RC circuit, the time constant refers to the time required for the charge or voltage in the circuit to rise or fall approximately 63.2% of its final value. It can be calculated as $$\tau = RC$$. ### Calculating the Natural Response in RC Circuits For the natural response of an RC circuit, one could focus on two potential scenarios: charging and discharging of the capacitor. Both circumstances procuring distinctive differential equations and subsequent solutions. 1. Charging: As the capacitor charges from a power source through a resistor, we can use Kirchhoff's voltage law to write an equation for the loop as $$V = V_R + V_C$$. Using Ohm's law ($$V_R = IR$$) and the definition of a capacitor ($$V_C = Q/C$$) gives $$V = IR + Q/C$$. With I defined as $$dq/dt$$ (rate of change of charge), this transforms into a differential equation: $$V = R\frac{dq}{dt} + \frac{Q}{C}$$. Given a 10 V external supply with R=2 $$\Omega$$ and C=0.01 F, to calculate the charge at any time $$t$$ and the circuit's natural response, use the differential equation and the boundary condition, $$Q(0) = 0$$. Solving this yields a charging equation, $$Q(t) = CV(1 - e^{-t/RC})$$, which describes the natural response during charging. 2. Discharging: With the external supply removed, the capacitor discharges via the resistor. Applying similar principles as during charging, a similar differential equation is established for the discharge case, except for the absence of the external voltage $$V$$. The boundary condition changes to Q at $$t = 0$$ being the maximum charge stored, CV. Solving this yields a discharging equation, $$Q(t) = CVe^{-t/RC}$$, which describes the natural response during discharging. ### Practical Instances of Natural Response of RC Circuits in Everyday Life The understanding of the RC circuit's natural response isn't just a theoretical exercise, but also underpins various real-life applications. Here are a few examples: • Photoflash Capacitor: Cameras often use capacitors in their flash circuits. When you capture a picture, the capacitor discharges quickly, lighting up the flash - a practical application of the natural response of an RC circuit during discharge. • Tuning Circuits: An RC circuit can act as a signal filter in radio tuners. Different RC combinations tune to specific frequencies, making choosing radio stations on your car ride possible! • Timing and Control: In computers, RC circuits are key in creating time delays and controlling waveforms like clock signals, critical for synchronisation of processes; hence their natural response controls the overall system timing. Each integration of the natural response of RC circuits into our everyday life embodies the quintessence of physics - taking abstract theoretical concepts right into tangible, familiar applications. ## Techniques for Natural Response Calculation ### Essential Formulas in Natural Response Calculation Rendering the natural response calculations effective invariably requires a toolbox of essential formulas. Each corresponds to a particular situation or scenario in the respective physical system. Time Constant ($$\tau$$): In the context of an RC or RL circuit, the time constant refers to the span required to charge or discharge approximately 63.2% of its maximum value. It can be computed as $$\tau = RC$$ in an RC circuit or $$\tau = L/R$$ in an RL circuit. Damping Ratio ($$\zeta$$): A dimensionless measure involving the time constant, damping coefficient, and mass (or resistance and inductance) of a system, denoted by the formula $$\zeta= \frac{c}{2 \sqrt{mk}}$$ for mechanical systems or $$\zeta = \frac{R}{2\sqrt{L/C}}$$ for electrical circuits. Each system waveforms are predominantly dictated by the damping ratio $$\zeta$$. • If $$\zeta > 1$$, the system is over-damped. • If $$\zeta = 1$$, the system is critically damped. • If $$\zeta < 1$$, the system is under-damped. • If $$\zeta = 0$$, the system is undamped. Natural Frequencies: They are the frequencies at which the system freely oscillates after an initial disturbance. Un-damped natural frequency ($$\omega_n$$) for a mass-spring system is defined as $$\omega_n = \sqrt{\frac{k}{m}}$$ and for an L-C circuit, it's $$\omega_n = \frac{1}{\sqrt{LC}}$$. ### Steps and Methods for Calculating Natural Response To successfully calculate a system's natural response, a methodical step-by-step approach can be adhered to: 1. Identify System Parameters: Primarily, identify system characteristics such as resistance, capacitance, mass, or spring constant, depending on the system under consideration. 2. Define the Equation of Motion: Characterising the differential equation that models your system behaviour is crucial. Systems like oscillating springs or RC circuits can be represented by second-order differential equations, like $$a\frac{d^2x}{dt^2} + b\frac{dx}{dt} + cx = 0$$. 3. Compute the Time Constant and Damping Ratio: Employing the formulas mentioned earlier, determine the time constant and damping ratio, which would narrow down the type of response the system exhibits. 4. Determine the Natural Frequencies: Capitalising on the system parameters evaluated above, calculate the natural frequencies. 5. Solve the Differential Equation: Utilise the initial conditions to solve the differential equation, using methods such as characteristic equations or Laplace transform. 6. Graph the Response: Finally, graph the response as a function of time that elucidates how the system evolves with the passage of time. ### Tips for Accurate Natural Response Calculations To ensure accuracy while calculating the natural response of a system, heed to these pivotal tips: 1. Identify the System Correctly: Correctly identifying the type of system (be it RC, RL, RLC circuits, or mass-spring-damper systems) is the first stride towards the accurate calculation of the natural response. The system type influences which formulas are to be employed. 2. Verify the Units: Ensuring congruity and consistency of units across all parameters can ward off calculative errors. 3. Confirm the Initial Conditions: The initial conditions can drastically sway the response calculations, rendering them essential in accurately solving the system's differential equation. 4. Utilise Appropriate Mathematical Methods: Depending on the complexity of the differential equation, properly employ methods for solving them, such as the characteristic equations method for second-order homogeneous equations or Laplace transforms for more complex systems. 5. Double-Check Calculations: When dealing with mathematical operations or when substituting values into formulas, it's always salient to double-check calculations to avoid any discrepancies. ## Examples of Natural Response ### Simplified Examples of Natural Response in Physics To acquaint you with the concept of natural response, let's delve into the realm of physics with some simplified examples: 1. A Simple Pendulum: Imagine a pendulum suspended from a rigid support. You jolt it from its equilibrium position, then the pendulum starts oscillating back and forth. Thus, it displays a natural response with its inherent natural frequency, governed by the length of the pendulum and gravity ($$T=2\pi \sqrt{\frac{l}{g}}$$). 2. Spring-Mass-Damper System: Envisage a system with a mass attached to a spring and dashpot (damper). An external force displaces the mass from its equilibrium position, and upon removal of the force, the system starts oscillating at its natural frequency, exhibiting natural response. The natural frequency of the system is $$\omega_n=\sqrt{\frac{k}{m}}$$, where $$k$$ is spring constant and $$m$$ is the mass. 3. LC Circuit: An LC (inductor-capacitor) circuit also stands as an example. Disconnected from the voltage source, the energy shifts back and forth between the inductor and capacitor at the natural frequency $$\omega= \frac{1}{\sqrt{LC}}$$, elucidating the natural response of the LC circuit. ### Detailed Example: Natural Response Scenario Let's delve into an in-depth example to illustrate the natural response of a system under consideration: RC Circuit: Consider a series RC circuit where the resistor's resistance is 1kΩ, and the capacitor's capacitance is 1μF. An external voltage of 5V charges the capacitor. After the capacitor charges, the voltage source is disconnected, and the circuit exhibits a natural response as the capacitor discharges through the resistor. The time constant, which represents how quickly the capacitor discharges, is calculated by $$\tau=RC$$ as '1ms'. The voltage across the capacitor as a function of time during discharging can be depicted by the equation: $$V(t) = V_{max}e^{-t/\tau}$$. Hence, the natural response can be graphically constructed, illustrating an exponential decay. ### Natural Response in Critical, Over-damped and Under-damped Circuits Circuits exhibit different natural responses based on their damping situations. Here's how circuits behave in the case of critical damping, over-damping, and under-damping: 1. Critical Damping ($$\zeta = 1$$): In a critically damped response, the system returns to its steady state in the quickest time possible without oscillating back and forth. This swift, non-oscillatory response is hugely beneficial for many practical applications. 2. Over-damping ($$\zeta > 1$$): Over-damping is a situation where the damping is higher than necessary. The high resistance or small inductance/capacitance in the system result in an over-damped state. Although the system returns to a steady state eventually, the response is sluggish and potentially undesirable for time-sensitive applications. 3. Under-damping ($$\zeta < 1$$): An under-damped system oscillates around its resting position or voltage. It goes beyond its steady state back and forth with the amplitude of oscillation gradually decreasing, showcasing a slightly fluctuating return to stability. The system eventually becomes steady, but this oscillatory transition is a characteristic of under-damped systems. Each of these damping scenarios presents distinct natural responses, constituting the cornerstone of designing suitable physical systems for desired applications. ## Role of Natural Response in Physics In the vast landscape of Physics, the natural response plays a central role, working as a marker for physical systems' inherent attributes, be it electrical circuits or mechanical oscillators. Understanding how a system responds to an external stimulus when left to its own attributes gives an insight into its inherent characteristics. It helps to delineate the inherent frequencies, damping characteristics, and the temporal evolution of a system, ultimately forging a path from abstract Physics laws and formulas towards concrete physical phenomena. ### Significance of Natural Response in Bridging Theory and Practical Physics The natural response stands as a compelling bridge between theoretical and practical physics. Theoretical models build strong foundations, but practical manifestations of these theories bring them to life. In the realm of this interplay, enter the natural response. From the theoretical perspective, the natural response of an electrical or mechanical system is a direct result of the principles articulated by fundamental laws such as Newton's laws of motion for mechanical systems or Ohm’s and Faraday’s laws for electrical circuits. The collective interaction of system's individual elements through these laws manifests as the system's natural response. For example, in oscillatory systems like an LC circuit or a mass-spring system, the natural frequency is a function of system's parameters (inductance and capacitance in an LC circuit, or mass and spring constant in a mass-spring system). Applying these relationships theoretically enables you to discern the natural frequencies, which in turn determine their response to external disturbances. The practical side of the natural response unravels when you observe real-world systems. The natural response isn't just confined to numbers and equations, but acts as the key which unlocks the practical behaviour of components around us. Imagine switching off your television, the slow dimming is your system’s natural response to the cessation of the electric signal. Similarly, the oscillations of a suspension bridge in the wind, or the humming of a tuning fork when struck, all are real-world instances of a system's natural response. Thus, such practical instances of natural response deliver an understanding of Physics that neither theory nor practice can offer individually: a coherent, tangible manifestation of abstract principles. ### How Natural Response Affects Other Aspects of Physics With natural response holding importance in its own right, it interlaces with several other aspects of Physics, exerting substantial influence on them. Firstly, the study of natural response aids in interpreting the $$Stability$$ of systems. The concepts of critical damping, over-damping, and under-damping are tethered to how a system behaves post an external disturbance, leading us to understand whether a system is stable, marginally stable or unstable. Similarly, in the domain of $$Resonance$$, one of the critical factors is the system's natural frequency. When a system is subjected to a periodically varying external force whose frequency matches with the system's natural frequency, it leads to resonance, exhibiting an amplified response. The bridges shaking in sync with the wind, or glass shattering at a specific sound frequency, are examples of resonance, which are fundamentally woven with the concept of a system's natural response. Beyond these, natural response also impacts the design and analysis of $$Control systems$$. Here, the natural response, along with forced response, dictates how a system responds over time. It helps in predicting system behaviour, analysing error, system stability and designing control strategies accordingly. Furthermore, in $$Wave theory$$, the concept of the natural response can be seen through the phenomenon of 'beat frequencies'. When two waveforms of slightly different frequencies interact, they produce a waveform that oscillates at a frequency equal to the difference of the two; essentially the natural response of the system. Thus, the ripple effect of the natural response is evident in disparate aspects of Physics, underlying its central role. ### Applications and Implications of Natural Response in Modern Physics In the arena of modern physics, the importance, the applications, and the implications of the 'natural response' concept have only magnified, establishing a significant foothold in numerous fields. In $$Signal Processing$$ and $$Communications$$, the concept of natural response plays a crucial role. For example, in radio receivers, the LC circuits are tuned to resonate at the desired transmission frequency, demonstrating an application of the natural response concepts. Additionally, in $$Mechanical Engineering$$, buildings and bridges are designed keeping in mind the natural frequencies to prevent catastrophic failures due to resonance. The horrifying collapse of the Tacoma Narrows Bridge in 1940 due to wind-induced resonant oscillations serves as a grim reminder of how pivotal understanding of natural response is in engineering design. Moreover, in $$Electrical Engineering$$, natural response forms an integral part of the controller design for systems. Whether it's designing an automobile's cruise control or stabilising an aircraft's flight, the natural response, along with the forced response, aids in shaping the required system response. Even computer systems, particularly in the design of computer networks, the packets of data sent across the network can be interpreted as external disturbances. The waveform or data packet transmission then takes the form of system's natural response. Hence, deciphering this response is fundamental to achieving efficient data transmission. In essence, understating the concept of natural response in Physics forms the bedrock for designing, interpreting and improving many modern technologies, punctuating the nothing-less-of-amazing applications it has, along with the far-reaching implications it kindles. ## Natural Response - Key takeaways • RC Circuit: An electric circuit with a resistor (R) and a capacitor (C) connected in series or parallel. The behavior of the circuit, including its frequency and phase response, is determined by the values of resistance and capacitance. Found in common appliances such as television tuners and audio amplifiers. • Natural Response of RC Circuit: Occurs when the external power supply to the circuit is cut off. The electrical energy stored in the capacitor begins to discharge through the resistor, triggering the circuit's "natural" response over a distinct time period known as the time constant. • Time Constant: Symbolized as $$\tau$$, it is the time required for the charge or voltage in a circuit to rise or fall about 63.2% of its final value. In the context of an RC circuit, it can be calculated as $$\tau = RC$$. • Natural Response Calculation Techniques: Different scenarios, such as the charging and discharging of the capacitor, lead to different types of equations. Understanding these equations and their solutions supports the calculation of the circuit's natural response. • Practical Applications of Natural Response: The concept of natural response in RC circuits is not just theoretical, but also has various real-life applications. Examples include photoflash capacitors in cameras, tuning circuits in radios, and timing and control in computers. #### Flashcards inNatural Response 15 ###### Learn with 15 Natural Response flashcards in the free StudySmarter app We have 14,000 flashcards about Dynamic Landscapes. What is the definition and importance of Natural Response in Physics? Natural response in physics refers to a system's reaction to changes internally, such as a circuit reacting to a change in current. It's vital as it helps understand systems' inherent behaviours, allowing us to predict and manipulate outcomes. How is the Natural Response of a system determined in Physics? The Natural Response of a system in Physics is determined by studying the characteristic properties of the system (like mass, spring, or capacitance) when it is not subjected to any external forces or excitations. The rules of physics governing the system are then used to predict its behaviour. What factors can influence the Natural Response of a system in Physics? The Natural Response of a system in Physics can be influenced by factors such as the initial conditions of the system, the inherent characteristics of the system, external forces acting on the system, and inputs to the system. What is the correlation between Natural Response and resonance in Physics? In physics, the natural response of a system corresponds to its inherent behaviour after an excitation or external influence is removed. When this natural frequency matches the frequency of a new, external influence, the system is said to be in 'resonance', resulting in amplified oscillations. What are the real-life applications of Natural Response in Physics? Natural response in physics is integral to engineering and technology. It's used in audio speaker design, mobile signal transmission, and circuit design for electronics. It's also critical in designing suspension for cars to absorb shocks and vibrations. ## Test your knowledge with multiple choice flashcards What is the meaning of 'Natural Response' in physics? What are the two integral constituents needed to decipher a system's natural response? Why is understanding a system's natural response important in regular and quantum physics? StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance. ##### StudySmarter Editorial Team Team Natural Response Teachers • Checked by StudySmarter Editorial Team

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# How Do I Calculate Internal Heat Transfer Rates in a Plate Chiller System? • johnp909 In summary, a person is working on a design project for school involving a plate chiller that cools hot liquid. They have researched and calculated various factors, but are struggling with determining the heat transfer rate inside the plate due to changing temperatures and fluid flow. They are considering using the LMTD method, but are unsure if it is applicable in this situation. They are seeking advice or guidance. johnp909 Hi There. I'm currently working on a design project for school. Basically its a plate chiller which is lowered into a pot of hot liquid. The plates have channels in them thru which water will flow, the result being cooling of the hot liquid in the pot. I've done a lot of research and reading in my fluid science book and I can't figure out the direction to go in my calcuations. Here's what I know so far... I have the specifics for the hot liquid. Thermal properties, mass volume, intended starting temperature and final temperature for the liquid etc. As well as a goal for the time that this process will take. I know the total heat to be removed. Calculated that. I know the heat transfer rate on the outside of the plates by convection. I calculated that using lumped system analysis and calculating coefficient of convection for a vertical plates. etc. The thing that's got me crossed up is figuring out how to calculate the heat transfer rate inside the plate. I know the dimensions of the channels. At least as a starting point. I understand that... Heat Transfer Rate = mass flow rate x Cp x Difference in Temperature and Heat Transfer Rate = convection coefficient x cooling area x Difference in Temperature The Thing that confuses me is that in this system the temperature of the hot liquid is changing over time as it's cooled this will slow the heat transfer rate. In addition the cooling fluid is raising temperature as it flows thru, therefore removing less heat at the end then at the begining. It seems like LMTD only deals with constant inlet and outlet temperature situations. This seems different to me. If the fluid residence time is short in the heat exchange is short, you can still assume quasi steady state and employ the LMTD. ## 1. What is heat transfer and why is it important? Heat transfer is the movement of thermal energy from one object to another. It is important because it helps regulate temperature and allows for the transfer of energy in various systems, such as in the human body, in buildings, and in industrial processes. ## 2. What are the three modes of heat transfer? The three modes of heat transfer are conduction, convection, and radiation. Conduction is the transfer of heat through direct contact between two objects, convection is the movement of heat through a fluid or gas, and radiation is the transfer of heat through electromagnetic waves. ## 3. How is heat transfer calculated? The rate of heat transfer can be calculated using the equation Q=mcΔT, where Q is the heat transferred, m is the mass of the object, c is the specific heat capacity of the object, and ΔT is the change in temperature. ## 4. What factors affect heat transfer? The rate of heat transfer is affected by several factors, including the temperature difference between the objects, the thermal conductivity of the materials, the surface area of the objects, and the distance between them. ## 5. How can heat transfer be controlled or manipulated? Heat transfer can be controlled or manipulated by using insulating materials to reduce the rate of heat transfer, by increasing or decreasing the surface area of the objects, and by adjusting the temperature difference between the objects. Additionally, heat transfer can be enhanced by using materials with high thermal conductivity and by increasing the surface area or temperature difference. • Thermodynamics Replies 14 Views 1K • Thermodynamics Replies 7 Views 906 • Thermodynamics Replies 2 Views 1K • Thermodynamics Replies 5 Views 1K • Thermodynamics Replies 8 Views 1K • Thermodynamics Replies 11 Views 1K • Thermodynamics Replies 15 Views 1K • Mechanical Engineering Replies 15 Views 2K • Thermodynamics Replies 4 Views 4K • Engineering and Comp Sci Homework Help Replies 22 Views 2K

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# Barney is the Hitler of my Generation Barney is The Hitler of My Generation 1992 marked a dark year in our nation's television history. That was the year a small production company brought to life an over-sized, giggling buffoon on the Public Broadcasting System (PBS). Barney & Friends was born. Oh, if only we had known better. Barney rode the crest of a dinosaur wave that swept the country in the early 1990's. Since then, researchers have proven Barney to be the spawn of Satan, and even educational leaders ponder the validity of his Nazi-like teachings. Barney & Friends is a show that PBS should definitely remove from its television line-up. Following the film Jurassic Park and TV shows such as Dinosaurs, Sheryl Leach, a Texas school teacher frustrated by the lack of "quality interactive and educational entertainment," created a fat purple dinosaur named Barney, and a dragon sidekick named Baby Bop for her two years old son(Tolentino 2). Thus, the Barney phenomenon was born. Conceived as a show that would help little children celebrate childhood and understand the complex business of growing up in a world where sixth graders carry guns to school, Barney & Friends has achieved a cult-like following among toddlers who swear upon their mother's graves that he is God himself. Barney has become a marketing win-fall. With an international fan club of more than six hundred thousand and video sales that outnumber copies of Cannonball Run and Playboy's Lingerie Video combined, Barney is slowly taking over the world. Barney has brainwashed the world's children into thinking he is a god, when truly he is the spawn of Satan himself. Recently, a student at Duke University published a theorem about Barney being the spawn of Satan: Given that Barney is a CUTE PURPLE DINOSAUR, Barney can be shown to be Satan. The Romans had no letter 'U', and used 'V' instead for printing, meaning the roman representation for Barney would be CVTE PVRPLE DINOSAVR. Extracting the roman numerals, we have: C, V, V, L, D, I, and V. Their decimal equivalents are 100, 5, 5, 50, 500, 1, 5. Adding those numbers produces 666. 666 is the number of the Beast. Barney is Satan. ("More" 1) Another example comes from the Bible: "And the beast which I saw was like unto a leopard, his feet were the feet of a bear, and his mouth as the mouth of a lion: and the dragon gave him power, and his seat great authority" (Revelation 13:2). Based on my interpretation, that passage says that Barney has spots, not unlike a leopard, his feet are large and broad, just like a bear, and his mouth is large and protruding creating senseless noise, like a lion ("Biblical" 1). As we know Baby Bop, his sidekick, does take on the form of a dragon, and toddlers will scream and cry, forcing their parents to travel many miles just to watch his purple majesty sing at any given shopping mall. Barney is now coming under scrutiny from researchers asking whether the overstuffed purple menace makes the grade as a teacher. A study by Yale University suggests that "The show contains many positive educational elements that aid young children preparing for school" (Walsh 1). Barney teaches ideas such as sharing, respect for different cultures, and washing your hands after you go potty. However, other observers argue that the show suffers from unfocused educational goals. Gerald S. Lesser, a professor of education and developmental psychology at Harvard University, said, "I'm not sure what the strong attraction is, because it does not make much use of the television medium. If there is an organized curriculum, I haven't figured it out" (Walsh 2). Some more reasons that Barney hurts, rather than helps, children are as follows: Barney presents a candy-coated, unrealistically nice view of the world. Barney does not promote individual thought. Rather, you are condemned for going against the wishes of the majority. (Just like a Barney replaces parental figures with himself in the 'I luv you' song. Barney emotionally cripples children by forcing them to repress negative emotions. Human beings have to be able to express all of their emotions to be functional in today's society. Barney tells children "A stranger is a friend you've never met" when the rates of kidnapping and molestation are rising. Barney encourages children to place boxes, and other objects (such as bags), over their heads, despite child mortality rates from suffocation. ("Hate" 3) Barney is turning our children into sissies, raising them with a false sense of security and universal harmony that bears little resemblance to reality. What is going to happen when those children grow up and find out what it is like in the real world? Will Barney be around to pay for all those psychiatric bills? I would venture to say not. We see quite clearly that the show Barney & Friends is detrimental to today's children. Parents should use the television more as a tool, and less as a babysitter. Barney & Friendsclaims to be educational when the producers admit it is escapism for a child. It teaches children to repress their emotions and conform to his will. Barney has formed his own legions of toddlers and is planning on domination of the earth. It is time that we must unite against a force that is covering our planet like a dark cloud. Barney is evil, and if you see him do not make eye contact, or listen to his songs. For if you do, you too will become one of his minions. He must be stopped, not just for the collective sanity, but also for the children. Laugh if you must, but when your brain turns to sponge, don't come crying to me. You have been forewarned. Works Cited "Biblical Proof Barney is Satan" Online. Internet. March 1, 1994 Available http://www.geocities.com/TelevisionCity/3194/biblel.html Accessed 21 Oct, 1997 "Hate Barney" ABL's Barney Must Die Newsletter. Online. Internet Available http://www.visi.com/~nathan/humor/general/hate.barney.html Accessed 21 Oct, 1997 "Mor reasons to hate Barney" Online. Internet Available http://www.duke.edu/~jlg1/humor/general/barney2.html Accessed 22 Oct, 1997 Tolentino, Nichole "Shove it up jurassic" Online. Internet. July 1995 Available http://www.bunnyhop.com/WA4/barney.html Accessed 22 Oct, 1997 Walsh, Mark "Experts Ponder Academic Value of 'Barney'" Education Week Online. Internet September 22, 1993 Available http://www.edweek.org/htbin/fastweb...26OR%26%28barney%26and%26friends%29 Accessed 22 Oct, 1997

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The Fool FAQ # Bonds The Fool FAQ ### I'm totally new to this whole thing, could you possibly give an example of how bonds work? When you buy a bond, you are actually loaning your money to the organization that issued the bond. That is why bonds are often called "debt instruments." The principal (the "face value" of the bond) is repaid on the maturity date. In the meantime, you are paid a set amount of interest, usually every six months. This interest is called the "coupon" or "coupon rate." It's called that because bonds used to come with little coupons attached that you would cut off and send in twice a year to receive the interest payment. How quaint. Nowadays, the coupon rate is nothing more than the annual interest rate. When bonds are originally issued they are issued in even denominations, usually \$1,000 and occasionally \$5,000, with some government bonds at \$10,000. Let's look at how bonds are described. For example, you could buy a New York State G.O. (Government obligation) with a maturity of 5-1-06 and a coupon of 5.25% priced at \$96.425. Now, before we get confused, let's look at each of those components. Maturity = the day that you get all of your original investment back. In this case, you would get back \$1,000 per bond on May 1, 2006. Coupon of 5.25% = This is the annual interest rate. Every six months you will receive an interest payment on your initial investment. This payment will be equal to the coupon rate (divided by 2 since each payment is for only half a year) times the denomination of the bond. So in this case you will receive \$26.25 (\$1,000 x .0525/2) every six months. In the case of tax-free bonds, the interest is tax-free. Priced at \$96.425 = All bonds are priced in units of \$100 even though you can't buy them for that. Our example is the most common kind of bond -- a \$1,000 denomination. This bond is selling at a discount to its original price. If a \$1,000 bond is priced at 96.425, then your price per bond will be \$964.25, not \$1,000.00. The original price was \$100.00 (meaning that it actually sold for \$1,000), but once a bond enters the secondary market (you can buy and sell them at any time), the price fluctuates based on how the bond compares to the terms that new bonds are being offered under. This bond's coupon rate is lower than the current interest rate for a comparable newly issued bond, so it is selling for less than its original value. Bond pricing gets complicated so hold on... Say you have an old 20-year bond that expires in 5 years and has a coupon interest rate of 10%. Wow. New 5-year bonds are only paying 5%, so obviously your bond is worth more -- the question is, what price would someone pay for your bond with its premium interest rate? Well, the market tells us that people are willing to pay \$1,000 for a 5-year bond that pays them \$50 a year in interest, but your bond pays \$100 a year in interest, so you might think that someone would be willing to pay a lot more for it, based on the interest it pays. That's true, but there are other factors -- both bonds pay \$1,000 in cash at the end of 5 years -- so maybe the price should be the same. Then you have to look at the entities that issued the bonds -- are they equally likely to still be in business and be able to repay the bond's principal? There is a complicated formula that takes into account the length of time to maturity, the coupon rate vs. the current equivalent interest rate, the relative possibility of default between the two issuing companies, and comes up with a price for your bond that is supposed to make it the equivalent of a newly issued, comparable bond. Oh, if you have something called a 'bearer bond', which means that the issuer of the bond doesn't have your name on the certificate, make sure that you register them in order to get your interest and to be notified of any changes. Here are two books you may want to look at. The first is The Bond Book by Annette Thau (Probus Publishing, 1992). It was written for the individual investor and covers all areas of the bond market, including governments, municipals, and corporates. On a more professional level is Frank J. Fabozzi, editor, The Handbook of Fixed Income Securities (4th edition, 1994, Irwin). This is loaded with more information than most people want or need to know. However, it is a great resource. ### So what are tax-free bonds? Those sound good! Tax-free municipal bonds (often called munis) are debt securities issued by various cities throughout the United States. As such, they are given special tax-free status. If you live in, say, Indiana, and buy an Indianapolis muni, you will pay no federal, state, and, in many cases, local taxes on the interest you receive. But before you start dreaming of an IRS-free life, let's take a closer look. First, tax-free bonds usually come with a lower interest rate -- after all, you get to keep all of it, right? But how do you know that you won't come out ahead buying a higher interest rate bond and paying the taxes? ### So how do I decide between a tax-free or a regular bond? It's not that hard a decision. You simply convert the regular bond's interest rate to its equivalent "after-tax rate" and compare that with the tax-free bond's interest rate. Whichever is larger wins. To find the after-tax rate, first you add up the your marginal income tax rates. If you are in the 15% bracket for federal income taxes and pay 4% in state taxes, your marginal tax rate is 19%. If you are in the 28% federal bracket and your state taxes are 7%, your marginal tax rate is 35%. Now you have to find the reciprocal of your marginal tax rate. Just subtract from 1. The reciprocal of 19% (0.19) would be 0.81, the reciprocal of 35% (0.35) would be 0.65. Now you can convert any interest rate to its after-tax equivalent. Say you were considering buying a new issue corporate (taxable) bond with a coupon rate of 8%. The after-tax equivalent for someone in the 19% bracket is 6.48% (8% x 0.81), but for someone in the 35% tax bracket, the after-tax equivalent rate is 5.2% (8% x 0.65). So for someone in the lower tax bracket, the tax-free bond would have to beat 6.48% to be a good deal, but for someone in the higher bracket, the tax-free bond would only have to beat 5.2%. It works in reverse, too. To see how a tax-free bond compares to a taxable bond, divide the tax-free interest rate by your tax rate. A tax-free bond paying 5% would be equivalent to a taxable interest rate of 7.69% for someone with a marginal tax rate of 35% (5/.65= 7.69), but only equivalent to 6.17% for someone with a marginal rate of 19% (5/.81=6.17). In general, tax-free bonds are usually only attractive to persons in the upper income brackets. ### What percentage of bonds should I have in my portfolio? Hold on there. You are assuming you should have bonds in your portfolio? There is a whole industry out there dedicated to convincing you of that, and people who make a living issuing pronouncements such as "Due to market machinations attributable to phase inducted waves in the upper troposphere we are instructing our clients to move from a 35% position in bonds to a 40% position until the deep geothermal wave pattern returns to an asynchronous mode." Right. The whole portfolio allocation rap is just another facet of market timing. One can construct wonderful scenarios showing how much money one would have made by switching from stocks to bonds at strategic points in the past, but we have yet to find someone who can accurately and consistently call the shots for the future. Until we locate such a prognosticator, we suggest staying fully invested in stocks with all money that you won't need for the next 5 years. Gonna buy a house in three years? Put the down payment in some nice bonds, CDs, or even a money market account. But planning for retirement in 20 years? History has spoken pretty firmly on that one -- stocks have always done better over the long run. Here at the Motley Fool we don't spend a lot of time on bonds because we are focused on your long-term investments and we don't think bonds are good for that -- but, hey, we also don't tell you what to do with your money, so if if it makes you feel better to have a few bonds in your long-term portfolio, be our guest. Just don't expect a lot from them. :) For more on this topic see our Tax Q&A area.

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# Using Runge-Kutta integration to increase the speed and stability of gradient descent? For a gradient descent problem with $$\mathbf{x}\in \mathbb{R}^N$$ I can evaluate the gradient $$\mathbf{\nabla}_\mathbf{x} \in \mathbb{R}^N$$ that reduces the least squares error, $$y$$. However, simply updating the position using $$\mathbf{x'} = \mathbf{x} + \mathbf{\nabla}_\mathbf{x}$$ converges very slowly to the global minimum of the least squares error (which is also the global minimum of the gradient magnitude, where the gradient is zero). I tried simply scaling up the step, i.e. $$\mathbf{x'} = \mathbf{x} + h\mathbf{\nabla}_\mathbf{x}$$, however while this dramatically improves convergence times in some cases, it can become unstable in others (particularly when some of the components of $$\mathbf{\nabla}_\mathbf{x}$$ are much larger than others -- scaling up all components of the gradient can cause the gradient descent method to "climb up the side of a canyon" rather than descending the canyon, and the system can either oscillate or explode). I would like to use the 3rd order Runge-Kutta method to follow the curvature of the gradient space, so that I can take larger steps without the system blowing up. I have applied this to simulating mass-spring systems before (using Runge-Kutta integration to integrate acceleration to find velocity, and velocity to find position) -- however I can't figure out how to apply it to this gradient descent problem. I think I have some fundamental misunderstanding about how the Runge-Kutta methods work. They requires a function $$f=(x, y)$$ to be defined, which I believe computes the gradient of the curve at $$x$$. However I don't understand why $$y$$ needs to be supplied to the function -- isn't $$y$$ a function of $$x$$? Can Runge-Kutta even be applied to the gradient descent problem? It seems like there should be a way to adapt Runge-Kutta to gradient descent, since each update step $$\mathbf{x'} = \mathbf{x} + \mathbf{\nabla}_\mathbf{x}$$ is basically an integration step. Is the step size $$h$$ simply the magnitude of the gradient, i.e. $$h_i = |{\mathbf{\nabla}_{\mathbf{x}_i}}|$$ and $$\mathbf{y}_i = {\mathbf{\nabla}_{\mathbf{x}_i}} / h_i$$? If Runge-Kutta is not applicable here, can somebody please suggest a robust and fast gradient descent algorithm to try? Some more detail: in the case of this problem, the gradient surface is fairly smooth, and quite strongly convex (there are few if any local minima that are not global minima), but the error surface is less convex. In other words, sometimes gradient descent will continue walking down the gradient slope in the direction of the global minimum of gradient, and the least squares error will increase temporarily before decreasing to the global minimum of least squares error. (The gradient is not computed from the least squares error measure itself, but using a different method that directly identifies the locally-best least squares solution, which moves the system closer to the globally-optimal least squares solution.) The gradient is therefore more reliable for gradient descent than the slope of the least squares error surface. • @RodrigodeAzevedo thanks, fixed Jul 26, 2020 at 1:35 • @RodrigodeAzevedo I honestly don't know how to clarify my question any further. Let me ask a more direct question: Is it possible to use Runge-Kutta as a gradient descent algorithm, i.e. can an integration algorithm be turned into a gradient descent algorithm? My gut says it can, but I am having trouble figuring out how to do this. Jul 26, 2020 at 13:01 • Do you have something like this in mind? Jul 26, 2020 at 13:08 • I'm trying to implement Iterative Closest Point, which requires gradient descent in the rotation and translation space to align two point clouds. However, I'm also interested in speeding up training of deep nets by using Runge-Kutta to predict the curvature of the error surface, which would allow for larger gradient descent step sizes. Jul 27, 2020 at 7:04 • That did not quite answer my question. Jul 27, 2020 at 10:07 First, gradient descent and Runge-Kutta methods solve different problems. 1. Gradient descent is a method to find an extremum of $$f(\mathbf x)$$ by solving $$\mathbf g(\mathbf x) = \nabla f(\mathbf x) = 0$$. Gradient descent simply does $$\mathbf x_{n+1} = \mathbf x_{n} + \alpha_n \mathbf g(\mathbf x_n)$$ with $$\alpha_n$$ being fixed or chosen smartly. 2. Runge-Kutta methods are used to solve ODEs, that is solve an initial value problem $$\mathbf x'(t) = \mathbf F(t, \mathbf x(t))\\ \mathbf x(0) = \mathbf x_0.$$ The simplest RK method is Euler's method which has quite similar to GD form $$\mathbf x_{n+1} = \mathbf x_n + (t_{n+1} - t_n) \mathbf F(t_n, \mathbf x_n)$$ In other words, GD may be treated as Euler's method applied to an ODE $$\mathbf x'(t) = \pm \mathbf g(\mathbf x)\\ \tag{*} \mathbf x(0) = \mathbf x_0.$$ I used $$\pm$$ since $$\alpha_n$$ may be positive or negative (depending whether you're searching for a minimum or a maximum). ODE's are usually solved forward in time, so $$t_{n+1} - t_n$$ is positive. The solution you're searching is the steady state $$\mathbf x(\infty)$$ at which the left hand side (and, consequently the right side) becomes zero. The correct sign also ensures that $$\mathbf x(t)$$ really tends to the steady state and not away from it. Further I will assume that the correct sign is $$+$$. You may use higher order RK methods for (*) problem. For example, the midpoint rule $$\mathbf x_{n+1/2} = \mathbf x_{n} + \frac{\Delta t_n}{2} \mathbf g(\mathbf x_n)\\ \mathbf x_{n+1} = \mathbf x_{n} + \Delta t_n \mathbf g(\mathbf x_{n+1/2})$$ Higher order RK methods are known to be more accurate than the Euler's method. That is the numerical trajectory (formed by $$\mathbf x_n$$ sequence) is much closer to the true trajectory $$\mathbf x(t)$$, which is the true solution of (*). Unfortunately, you don't need this property. In fact you don't care how close your $$\mathbf x_n$$ are to the true trajectory, instead you're interested in how close are your $$\mathbf x_n$$ to $$\mathbf x(\infty)$$. It is attractive to choose $$\Delta t_n$$ large, so one faster approaches to the $$t = \infty$$. Unfortunately, it does not work that way, because all explicit methods for ODEs (and any RK method is one of them) have a stability condition that restricts the largest step $$\Delta t$$. In fact even choosing $$\Delta t$$ close to that bound won't work either since the method will be oscillating forward and backward (exactly like GD does). Choosing $$\Delta t$$ which maximizes the convergence speed is quite nontrivial. Another disappointing fact is the stiffness phenomenon. You probably know that there are pathological functions $$f(\mathbf x)$$ for which GD converges very slowly. Usually it happens when Hessian matrix of $$f$$ is badly conditioned. For these cases the corresponding systems (*) are (infamously) known in numerical integration as stiff problems. For these problems all explicit methods perform roughly the same - the limit for $$\Delta t$$ and the convergence speed is believed to be practically the same. Stiff problems are often solved by implicit methods. Those methods cannot be converted to a GD-like method, since they require solving a nonlinear problem for each iteration. And this problem is roughly equivalent to the minimization problem itself. For example implicit Euler method has the form $$\mathbf x_{n+1} = \mathbf x_{n} + \Delta t_n \mathbf g(\mathbf x_{n+1}).$$ Separating known $$\mathbf x_n$$ and unknown $$\mathbf x_{n+1}$$ gives a nonlinear problem for $$\mathbf x_{n+1}$$ $$\mathbf G(\mathbf x_{n+1}) \equiv \mathbf x_{n+1} - \Delta t_n \mathbf g(\mathbf x_{n+1}) = \mathbf x_{n}$$ This problem is only slightly simpler to solve than the original $$\mathbf g(\mathbf x) = 0$$. Summarizing all above: using more precise methods for (*) won't get you to the solution faster. Instead you may want to use conjugated gradients method or other methods that are specialized for minimization problems, possibly involving more information about the function. • Thank you for the exceptionally great answer! Jul 30, 2020 at 5:36 • This answer is wrong. Runge-Kutta is not specifically a method to solve ODE's. It is a method of numerical differentiation, which in turn means it can be used to solve ODE's. If you have to use numerical differentiation for a gradient descent problem, Runge-Kutta is definitely the best way to do it. – Paul May 8, 2023 at 13:14 • @Paul Feel free to add an answer with details on how to use R-K for numerical differentiation and gradient descent. To my knowledge, R-K are for numerical integration (in wide sense) and in no way for differentiation. May 9, 2023 at 21:16

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# On the coefficients of the polynomial in the number field sieve Save this PDF as: Size: px Start display at page: Download "On the coefficients of the polynomial in the number field sieve" ## Transcription 1 On the coefficients of the polynomial in the number field sieve Yang Min a, Meng Qingshu b,, Wang Zhangyi b, Li Li a, Zhang Huanguo b a International School of Software, Wuhan University, Hubei, China, b Computer School, Wuhan University, Hubei, China, Abstract Polynomial selection is very important in number field sieve. If the yield of a pair of polynomials is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first. This can speed up the selection of good polynomials. In this paper, we aim to study the correlation between the polynomial coefficients and the yield of the polynomials. By theoretical analysis and experiments, we find that a polynomial with the ending coefficient containing more small primes is usually better in yield than the one whose ending coefficient contains less. One advantage of the ending coefficient over the leading coefficient is that the ending coefficient is bigger and can contain more small primes in root optimizing stage. Using the complete discrimination system, we also analyze the condition on coefficients to obtain more real roots. Key words: cryptography, integer factorization, number field sieve, polynomial selection, coefficients 1 Introduction The general number field sieve[1,2] is known as the asymtotically fastest algorithm for factoring large integers. It is based on the observation that if a 2 = b 2 mod N and a b, gcd(a b, N) will give a proper factor of N with at This work is supported in part by the National Natural Science Foundation of China(No , No , No ) and the Fundamental Research Funds of Central Universities Corresponding author address: (Meng Qingshu ). 2 least a half chance. The number field sieve starts by choosing two irreducible and coprime polynomials f(x) and g(x) over Z which share a common root m modulo N. Let F (x, y) = y d 1 f(x/y) and G(x, y) = y d 2 g(x/y) be the homogenized polynomials corresponding to f(x) and g(x) respectively, where d 1 and d 2 are the degree of f(x) and g(x) respectively. We want to find many coprime pairs (a, b) Z 2 such that the polynomials values F (a, b) and G(a, b) are simultaneously smooth with respect to some upper bound B and the pair (a, b) is called a relation. An integer is smooth with respect to bound B (or B-smooth) if none of its prime factors are larger than B. If we find enough number of relations, by finding linear dependency[3,4] we can construct: (a,b) S (a bα 1 ) = β 2 1, where f(α 1 ) = 0, β 1 Z[α 1 ] (a,b) S (a bα 2 ) = β 2 2, where g(α 2 ) = 0, β 2 Z[α 2 ]. As there exist maps such that ϕ 1 (α 1 ) = m mod N and ϕ 2 (α 2 ) = m mod N, we have ϕ 1 (β 2 1) = ϕ 2 (β 2 2). We can obtain the square root β 1 and β 2 from β 2 1 and β 2 2 respectively using method in [5]. If we let ϕ 1 (β 1 ) = x and ϕ 2 (β 2 ) = y, then y 2 = x 2 mod N, and we have constructed a congruent squares and so may attempt to factor N by computing gcd(x y, N). In order to obtain enough relations, selecting a polynomial with high probability of being smooth is very important. A good polynomial not only can decrease sieving time, but also can reduce the expected matrix size[6]. The polynomial selection is now a hot research area. Based on base-m method and with translation and rotation technique[6], non-skewed or skewed polynomial can be constructed, where one polynomial f(x) is nonlinear and the other g(x) is monic and linear. If the linear polynomial is nonmonic, the size of nonlinear polynomial can be greatly reduced[7,1]. The two methods above are called linear method. Montgomery[8] proposed the nonlinear method, where the two polynomials are both nonlinear. Recently several papers[9 11] address nonlinear polynomials construction problem. The concept of evolutionary cryptography was proposed in [12] and its idea was used to design cryptographic functions[13,14] or cryptographic algorithm[15]. The capabilities of evolutionary cryptosystem against linear and differential cryptanalysis were given in [16,17] respectively. To search for good polynomials for number field sieve with evolutionary computing, we need to solve the difficulty that the space of candidate polynomials is too huge. With the idea similar to [13], we study if there exists any correlation between the polynomial coefficients and the yield of the polynomials. If they are closely related, we can search for polynomials by checking the coefficients first before calculating its alpha value[6]. Therefore we can search for good polynomials in a 2 3 smaller space. This would speed up the selection of polynomials. By theoretical analysis and experiments, we find that the yield of polynomial is closely related to the coefficients of the polynomial. The polynomial with ending coefficient containing more small primes usually have better yields. We also find that the number of real roots can be determined by partial coefficients of the polynomial if it is skewed. The rest of the paper is organized as follows. In Section 2 we review elements related to the yield of a polynomial. In Section 3 we recite the number of real roots of a rational polynomial. In Section 4 we analyze the effect of the ending coefficient and leading coefficient on the yield respectively. In Section 5 we analyze the effects of coefficients on the number of real roots and on the yield. Finally we make a conclusion in Section 6. 2 Elements related to smoothness of a polynomial An integer is said to be B-smooth if the integer can be factored into factors bounded by B. By Dickman function, given the smooth bound B, the less the integer is, the more likely the integer is B-smooth. In number field sieve, we want the homogenous form F (x, y) = a d x d + + a 1 xy d 1 + a 0 y d of the polynomial f(x) = a d x d + + a 1 x + a 0 to be small. In [6], the size and root property are used to describe the quantity. By size we refer to the magnitude of the values taken by F (x, y). By root property we refer to the distribution of the roots of F (x, y) modulo small p k, for p prime and k 1. If F (x, y) has many roots modulo small p k, values taken by F (x, y) behave as if they are smaller than they actually are. That is, on average, the likelihood of F (x, y) values being smooth is increased. It has always been well understood that size affects the yield of F (x, y). In [18], the number of real roots, the order of Galois group of f 1 (x)f 2 (x) were taken into account. By the number of real roots, if a/b is near a real root, the value F (a, b) will be small and will be smooth with high chance. By the order of Galois group of f 1 f 2, it is better to chose polynomial for which the order of Galois group of f 1 f 2 are small, because they provide more free relations. Obviously, if the coefficients of f(x) are small, F (x, y) would have good size property. In order to obtain polynomial with small coefficients, we can search extensively, or let the linear polynomial be nonmonic as suggested in [1,7]. In order to obtain good root property, usually it is required that the leading coefficient contains many small prime as its factors[6]. The paper[19,20] discussed other ways to improve root property. As for the number of the real roots, it is left as random. 3 4 3 The number of real roots of a polynomial In [21,22],the number of real roots or roots distribution of a rational polynomial is given by CDS(complete discrimination system). In degree 3, take polynomial f(x) = ax 3 + bx 2 + cx + d as example. The CDS is = 18abcd 4b 3 d + b 2 c 2 4ac 3 27a 2 d 2. The root distribution is as follows. (1) If > 0, the equation has three distinct real roots. (2) If = 0, the equation has a multiple root and all its roots are real. (3) If < 0, the equation has one real root and two nonreal complex conjugate roots. In degree 4, take f(x) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0, (a 4 0) as example. Its CDS is as follows: D 2 = 3a 2 3 8a 2 a 4, D 3 = 16a 2 4a 0 a 2 18a 2 4a 2 1 4a 4 a a 4 a 1 a 3 a 2 6a 4 a 0 a a 2 2a 2 3 3a 1 a 3 3, D 4 = 256a 3 4a a 2 4a a 2 4a 1 a 2 0a 3 27a 4 3a 2 0 6a 4 a 2 3a 0 a 2 1+a 2 2a 2 1a 2 3 4a 4 a 3 2a a 2 a 0 a 3 3a a 4 a 2 a 2 0a a 4 a 2 2a 0 a 3 a 1 +18a 4 a 2 a 3 1a 3 4a 3 2a 0 a 2 3 4a 3 3a a 4 a 4 2a 0 128a 2 4a 2 2a a 2 4a 2 a 0 a 2 1, E = 8a 2 4a 1 + a 3 3 4a 4 a 3 a 2. The numbers of real and imaginary roots and multiplicities of repeated roots in all cases are given as follows: (1) D 4 > 0 D 3 > 0 D 2 > 0 {1, 1, 1, 1}, (2) D 4 > 0 (D 3 0 D 2 0) {}, (3) D 4 < 0 {1, 1}, (4) D 4 = 0 D 3 > 0 {2, 1, 1}, (5) D 4 = 0 D 3 < 0 {2}, (6) D 4 = 0 D 3 = 0 D 2 > 0 E = 0 {2, 2}, (7) D 4 = 0 D 3 = 0 D 2 > 0 E 0 {3, 1}, (8) D 4 = 0 D 3 = 0 D 2 < 0 {}, (9) D 4 = 0 D 3 = 0 D 2 = 0 {4}, where the column on the right side describes the situations of the roots. For example, {1, 1, 1, 1} means four real simple roots and {2, 1, 1} means one real double root plus two real simple roots. In degree 5, take f(x) = x 5 + px 3 + qx 2 + rx + s as example. Its CDS is as follows: D 2 = p 4 5 D 3 = 40rp 12p 3 45q 2 D 4 = 12p 4 r 4p 3 q prq 2 88r 2 p 2 40qp 2 s+125ps 2 27q 4 300qrs+160r 3 D 5 = 1600qsr ps 3 q+2000ps 2 r 2 4p 3 q 2 r 2 +16p 3 q 3 s 900rs 2 p q 2 p 2 s pq 2 r q 2 rs 2 +16p 4 r p 5 s 2 128r 4 p 2 27q 4 r q 5 s+256r s 4 72p 4 rsq + 560r 2 p 2 sq 630prq 3 s E 2 = 160r 2 p q 2 r 2 48rp 5 +60q 2 p 2 r+1500rpsq+16q 2 p qp 3 s+625s 2 p q 3 s F 2 = 3q 2 8rp The numbers of real and imaginary roots and multiplicities of repeated roots of polynomial in all cases are given as follows: (1) D 5 > 0 D 4 > 0 D 3 > 0 D2 > 0 {1, 1, 1, 1, 1} (2) D 5 > 0 (D 4 0 D 3 0 D 2 0) {1} (3) D 5 < 0 {1, 1, 1} (4) D 5 = 0 D 4 > 0 {2, 1, 1, 1} (5) D 5 = 0 D 4 < 0 {2, 1} (6) D 5 = 0 D 4 = 0 D 3 > 0 E 2 0 {2, 2, 1} (7) D 5 = 0 D 4 = 0 D 3 > 0 E 2 = 0 {3, 1, 1} (8) D 5 = 0 D 4 = 0 D 3 < 0 E 2 0 {1} (9) D 5 = 0 D 4 = 0 D 3 < 0 E 2 = 0 {3} (10) D 5 = 0 D 4 = 0 D 3 = 0 D 2 0 F 2 0 {3, 2} (11) D 5 = 0 D 4 = 0 D 3 = 0 D 2 0 F 2 = 0 {4, 1} (12) D 5 = 0 D 4 = 0 D 3 = 0 D 2 = 0 {5} 4 The yield and the ending coefficient Let f(x) = a d x d + + a 1 x + a 0 be a nonlinear polynomial. Let F (x, y) = a d x d + + a 1 xy d 1 + a 0 y d be the homogenous form of polynomial f(x). Let p q be the number of roots of the homogeneous polynomial F modulo p and let α(f ) = small prime p p (1 p q p + 1 ) log p p 1. In order to make α(f ) small, we can increase the value of p q. Equation F (x, y) = 0 mod p has three kind of roots. (1) If p b and p a d, the pair (a,b) is called the projective root. (2) If p a and p a 0, the pair (a,b) is called the zero root. (3) The rest of pairs (a,b) satisfying F (a, b) = 0 mod p are called ordinary roots, or simply roots. 5 6 Correspondingly, there are three methods to increase the value of p q. The first method is already known that the leading coefficient a d containing many small primes can increase the number of projective roots. For example, the leading coefficient usually is the multiple of 60[7]. As for the ordinary roots,we refer to [19,20].We propose the third method that if the ending coefficient contains many small primes, the number of zero roots can be increased. As f(x) is skewed with a d << a 0, a 0 can contain more small primes than a d can. The number of pair (a,b) satisfying second case would be much more than the one in the first case, especially after the optimization of root properties. In order to check whether the above analysis is right, we do many experiments. In our experiments, we let N be an integer of about 30 digits. In Experiment 1, the polynomials are generated by base-m method as described in [6], but without the optimization step. Experiment 1: (1) Generate polynomial as [6]. For each leading coefficient a d below a bound, we examine m ( N a d ) 1 d. Check the magnitude of a d 1, and of a d 2 compared to m, by computing the integral and non-integral parts of N a d m d m d 1 = a d 1 + a d 2 m + O(m 2 ). If these are sufficiently small, accept a d and m, and we get a polynomial f(x) by the expansion of N in base m and with leading coefficient a d. (2) Collect relations. For each above polynomial, skew the sieving area with skewness= d a0 a d. Randomly choose enough pair of coprime (a,b) in sieving area and check if they form relations. For each polynomial,we denote the number of relations by num rel. (3) For each polynomial, there is a row corresponding to it. It includes the following items: the number of relations num rel, the number of small primes contained in a d below a predefined bound, denoted by num ad, the number of small primes in a 0 below a predefined bound, denoted by num a0. A file is formed. (4) Sort the above file in ascending order with num rel as key word. From the sorted file, we can find the parameter num a0 and num ad both are in ascending order, but not strictly. In order to obtain an obvious impression, we divide the sorted file by rows into many length-equal parts, each of which contains equal number of rows and calculate the sum of the parameters num ad and num a0 in each part respectively. 6 7 Table 1 The trend of three parameter(256 rows/block) Block num num ad num a num root Table 1 lists the sums of num ad and num a0 respectively, where the parameters are as follows. N = ( an integer in example 3 [9] ). The degree of the nonlinear polynomial is 3. Sieving area is 2A A, where A = 4000, coprime pair (a, b) are chosen randomly from sieving area in a way like for(a=-a s;a< A s;a+=rand()%6+1) for(b=1;b< A/s;b+=rand()%6+1), where s = 6 a 0 /a 3. From Table 1 we find that the ending coefficients correlate with the yields of polynomials as the leading coefficient does. For polynomial of degree 4 or 5, we get similar results. For nonmonic linear polynomial generated as suggested in [1,7], we can get similar results. For nonlinear polynomials as suggested in[9], we don t do the experiments. We conjecture the results should be similar. Remark 1: If the size of N, the integer to be factored, gets larger, the skewness of the polynomial would become larger, and a 0 can contain more small primes than a d can. This can be used in optimization of root properties. The advantage of ending coefficient over leading coefficient will be more obvious in term of relation yield. 5 The number of real roots and the coefficients A polynomial with more real roots are preferable in number field sieve because if a/b is near a real root, the value F (a, b) will be small and will be smooth with high possibility. Usually the number of real roots is left as random in polynomial generation. In [21,22],the number of real roots or roots distribution of a rational polynomial is given by CDS. From CDS, the number of real roots should depends on all coefficients of the polynomial. However, the polynomial for NFS is not randomly generated. It has many special properties. For example, it is skewed and usually its first 3 coefficients a d, a d 1 and a d 2 are small compared to m while the size of the rest coefficients is similar to or bigger than that of m. From this, the ratio abs(a d 3 /a d 2 ) will bigger than d a 0 /a d while the ratio abs(a 0 /a 1 ) is smaller than d a 0 /a d. Based on all these features and CDS we can analyze the correlation between the number of real roots and the coefficients. 7 8 In degree 3, from the expression of, the sign of should depend on the item 4ac 3 or 27a 2 d 2, but not depend on 4b 3 d since b is usually small. If the ratio abs(d/c) is not big, the sign of mainly be determined by 4ac 3. If c is small enough, > 0, which means the polynomial f(x) has 3 real roots. Similarly if the ratio abs(d/c) is big, < 0, which means the polynomial f(x) has only one real root. In Experiment 2, on about more than a half cases the polynomial has 1 real root and on less than a half cases the polynomial has 3 real roots, where the ratio abs(d/c) is small and c is negative and small enough. In degree 4, from the expression of D 4, each monomial s exponent sum is equal to 6 and highest exponent of a 3, a 2, a 1 are 4 and the exponent of a 0 is 3. If the polynomial is skewed, the coefficients a 1, a 0 or the ratio abs(a 0 /a 1 ) determine the sign of D 4. As the ratio abs(a 0 /a 1 ) is usually smaller than 4 a 0 /a 4, we have abs(27a 2 4a 4 1) > abs(256a 3 4a 3 0) and D 4 < 0 with large chance. That is, in most case the polynomial has 2 real roots. To obtain 4 real roots, the coefficient a 2 should be negative and small enough and the absolute value of a 1 should be of similar size with that of a 2. In Experiment 2, on about 80 percent of cases the polynomial has 2 real roots and on about 20 percent of cases the polynomial has 4 real roots, where the absolute values of a 2 and a 1 are of similar size. The case with zero real roots should be avoided. For degree =5, from the expression of D 5, the monomials exponent sums are not equal. It is not obvious to determine its sign. However, the sign of D 5 should mainly depend on the sum of 4p 3 q 2 r p 3 q 3 s + 16p 4 r p 5 s 2. To obtain 5 real roots, first the variable p should be negative and small enough such that D 2 > 0, D 3 > 0 and D 4 > 0. Secondly, let p < 0, q < 0, r > 0, s > 0, or let p < 0, s < 0, q > 0, r > 0 such that D 5 > 0. The analysis above in case of degree 3 should be useful in choosing polynomial in nonlinear method, where a polynomial with degree 3 is already enough for practical purpose. Experiment 2: (1) Generate polynomial as step 1 of experiment 1. (2) Collect relation as step 2 of experiment 1. Denote the number of relation by num rel. (3) For each polynomial, there is a row corresponding to it. The row has num rel, the number of real roots num root, and all coefficients as its items. We form a file now. (4) Sort the above file in ascending order with num root as the key word. From the sorted file, we observe the correlation between num root and the polynomial coefficients. In case degree =3, if the coefficients of degree 1 is below some value, there will be 3 real roots for most cases. In case 8 9 degree=4, if coefficient of degree 2 is below some value, there will be 4 real roots for most cases. In case degree=5, only a few polynomials have 5 real roots. (5) Sort the file above in ascending order with num rel as the key word. From the sorted file, we observe the correlation between num root and num rel. We can find num root is also in ascending order, but not strictly. In order to obtain an obvious impression, we divide the sorted file by rows into many length-equal parts, each of which contains equal number of rows and calculate the sum of the parameters num root in each part. Table 1 lists the sum of num root, where the parameters are the same as in experiment 1. From Table 1 we find that increasing the number of roots can increase the yield in degree 3. For degree 4 or 5, we get similar results, but not strong as the case in degree 3. For polynomial generated as suggested by Kleinjung in [7], where the linear polynomial is nonmonic, the results is similar. As for the nonlinear polynomial, we don t do the experiments, but we conjecture results should be similar if the polynomials are skewed. Remark 2: Usually the number of real roots is left as random. However, based on our analysis, we can adjust the value of the related coefficients in polynomial optimizing stage such that the polynomial have more real roots. This is useful because it was stated that increasing the number of real roots could increase the yield[18]. The relationship between the size of real roots and the coefficients is another good research topic because if the size of real roots is near the size of skewness the yield will be high. 6 Conclusion Studying the correlation between the yield of a polynomial and its coefficients is important because it take less computation if we can choose polynomial by checking its coefficients first. In this paper, we study the correlation between the yield of a polynomial and its coefficients. The theoretical analysis and the experiments both show that the ending coefficient containing more small primes will increase the yield of the polynomial. As the ending coefficient is much bigger than the leading coefficient, the ending coefficient can contain more small primes in root optimization step. This is one advantage of ending coefficient over leading coefficient. Using CDS we analyze the coefficients condition to obtain more real roots. This is a necessary consideration in choose polynomial as increasing the number of real roots can increase the yield of the polynomial pair. As the yield will be high when the size of real roots is near the size of the skewness, the relation between the size of real roots and the 9 10 coefficients should be under research. References [1] J. P. Buhler, H.W. Lenstra, JR., C. Pomerance, Factoring Integers With The Number Field Sieve, in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, LNCS 1554, 50-94, [2] C. Pomerance, The Number Field Sieve, Proceedings of Symposia in Applied Mathematics, Vol.48, , [3] P. L. Montgomery, A Block Lanczos Algorithm for Finding Dependencies over GF(2), Eurocrypt 95, LNCS921, , [4] D. Coppersmith, Solving Homogeneous Linear Equations over GF(2) via Block Wiedemann Algorithm, Mathematics of Computation. 62, , [5] P. Nguyen, A Montgomery-like Square Root for the Number Field Sieve, Proceedings ANTS III, Springer-Verlag, LNCS 1423,151-68, [6] B. Murthy, Polynomial Selection for the Number Field Sieve Integer Factorisation Algorithm, Ph.D. thesis, The Australian National University, [7] T. Kleinjung, On Polynomial Selection For The General Number Field Sieve, Mathematics of Computation, Vol.75, No.256, , [8] P. L. Montgomery, Small Geometric Progressions Modulo n, manuscript (1995). [9] N. Koo, G.H. Jo, and S. Kwon, On Nonlinear Polynomial Selection and Geometric Progression (mod N) for Number Field Sieve. [10] T. Prest, P. Zimmermann, Non-linear Polynomial Selection For The Number Field Sieve, Journal of Symbolic Computation, Vol.47, Issue 4, , [11] R. S. Williams, Cubic Polynomials in the Number Field Sieve, Master Thesis, Texas Tech University, [12] Huanguo Zhang, Xiutao Feng, Zhongpin Qin, Yuzhen Liu, Research on Evolutionary Cryptosystems and Evolutionary DES, Journal of computer, Vol.26,No.12, , [13] Qingshu Meng, Huanguo Zhang, Zhangyi Wang, Zhongpin Qin, Wenling Peng, Designing Bent Functions Using Evolving Method, Acta Electronica Sinica, Vol.32, No.11, , [14] Min Yang, Qingshu Meng, Huanguo Zhang, Evolutionary Design of Trace Form Bent Functions in Cryptography, International Journal of Information and Computer Security, Vol.3, NO.1, 47-59, 11 [15] Huanguo Zhang, Qingshu Meng, Sequence Generator Based on Time-Varying Binary Combiner, Acta Electronica Sinica, Vol.32,No.4, , [16] Huanguo Zhang, Chunlei Li, Ming Tang, Evolutionary Cryptography Against Multidimensional Linear Cryptanalysis, Science China: Information Science, Vol.54, No.12, , [17] Huanguo Zhang, Chunlei Li, Ming Tang, Capability of Evolutionary Cryptosystem Against Differentil Cryptanalysis. Science China: Information Science, Vol. 54,No.10, , [18] M. Elkenbracht-Huizing,An Implementation of the Number Field Sieve, Experimental Mathematics, Vol.5, No.3, , [19] J.E. Gower,Rotations and Translations of Number Field Sieve Polynomials, Advances in Cryptology - ASIACRYPT 2003, LNCS2894, , [20] Shi Bai, Polynomial Selection for the Number Field Sieve, Ph.D thesis, the Australian National University,2011. [21] Lu. YANG, Recent Advances on Determining the Number of Real Roots of Parametric Polynomials, Journal of Symbolic Computation, Vol. 28, , [22] function. 11 ### Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give ### Factoring Algorithms Institutionen för Informationsteknologi Lunds Tekniska Högskola Department of Information Technology Lund University Cryptology - Project 1 Factoring Algorithms The purpose of this project is to understand ### An Overview of Integer Factoring Algorithms. The Problem An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm ### 3 1. 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Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer ### Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c ### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical ### STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION. Takayuki Yato. A Senior Thesis. Submitted to STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION by Takayuki Yato A Senior Thesis Submitted to Department of Information Science Faculty of Science The University of Tokyo on ### Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression ### The Quadratic Sieve Factoring Algorithm The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 Introduction Mathematicians have been attempting to find better and faster ways to factor composite ### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role ### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include ### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list ### Cryptography. Course 2: attacks against RSA. Jean-Sébastien Coron. September 26, Université du Luxembourg Course 2: attacks against RSA Université du Luxembourg September 26, 2010 Attacks against RSA Factoring Equivalence between factoring and breaking RSA? Mathematical attacks Attacks against plain RSA encryption ### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing ### Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS The following is a list of terms and properties which are necessary for success in Math Concepts and College Prep math. You will ### Basics of Polynomial Theory 3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where ### ( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated ### 1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable. ### Polynomials Classwork Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable ### Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose ### Factorization Algorithms for Polynomials over Finite Fields Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is ### Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For ### ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for ### Factoring Polynomials Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent ### SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra ### 1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts, ### x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense ### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic ### An Introduction to Galois Fields and Reed-Solomon Coding An Introduction to Galois Fields and Reed-Solomon Coding James Westall James Martin School of Computing Clemson University Clemson, SC 29634-1906 October 4, 2010 1 Fields A field is a set of elements on ### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public ### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear ### Factoring. Factoring 1 Factoring Factoring 1 Factoring Security of RSA algorithm depends on (presumed) difficulty of factoring o Given N = pq, find p or q and RSA is broken o Rabin cipher also based on factoring Factoring like ### Computer Algebra for Computer Engineers p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions ### 6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic ### calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0, Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials ### RESULTANT AND DISCRIMINANT OF POLYNOMIALS RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results ### Factoring Cubic Polynomials Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than ### Advanced Higher Mathematics Course Assessment Specification (C747 77) Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational ### Solving Cubic Polynomials Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial ### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of. 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We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational ### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called. Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate ### ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS. Carl Pomerance ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS Carl Pomerance Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of computing an integer l with g l ### ECE 842 Report Implementation of Elliptic Curve Cryptography ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic ### Cubic Polynomials in the Number Field Sieve. Ronnie Scott Williams, Jr., B.S. A Thesis. Mathematics and Statistics Cubic Polynomials in the Number Field Sieve by Ronnie Scott Williams, Jr., B.S. A Thesis In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment ### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map 22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find ### 0.4 FACTORING POLYNOMIALS 36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use ### Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x) ### STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an ### Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor ### The Sieve Re-Imagined: Integer Factorization Methods The Sieve Re-Imagined: Integer Factorization Methods by Jennifer Smith A research paper presented to the University of Waterloo in partial fulfillment of the requirement for the degree of Master of Mathematics ### In this lesson you will learn to find zeros of polynomial functions that are not factorable. 2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. 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Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory ### Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we ### The Method of Least Squares The Method of Least Squares Steven J. 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Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)). ### it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore ### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize ### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module ### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra ### Smooth numbers and the quadratic sieve Algorithmic Number Theory MSRI Publications Volume 44, 2008 Smooth numbers and the quadratic sieve CARL POMERANCE ABSTRACT. This article gives a gentle introduction to factoring large integers via the ### 3.7 Complex Zeros; Fundamental Theorem of Algebra SECTION.7 Complex Zeros; Fundamental Theorem of Algebra 2.7 Complex Zeros; Fundamental Theorem of Algebra PREPARING FOR THIS SECTION Before getting started, review the following: Complex Numbers (Appendix, ### SMT 2014 Algebra Test Solutions February 15, 2014 1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the ### The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method ### Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F. ### A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers Johannes Blömer, Alexander May Faculty of Computer Science, Electrical Engineering and Mathematics University of Paderborn ### 8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph ### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables. ### Factoring Special Polynomials 6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These ### minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We ### Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression ### I. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F. F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 3/22 F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 4/22 MPRI Cours 2-12-2 I. Introduction Input: an integer N; logox F. Morain logocnrs ### Short Programs for functions on Curves Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function ### Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms ### FACTORING. n = 2 25 + 1. fall in the arithmetic sequence FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard, ### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise ### On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University ### COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, ### Portable Assisted Study Sequence ALGEBRA IIA SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

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Today I am sharing the TCS aptitude questions which will help you to get the placement in the TCS, Wipro, and other reputed companies. if you are going to give the TCS fresher placement exam then I highly recommend you to solve these TCS aptitude questions. if you are able to solve these questions then you can crack the TCS NQT exam or any Placement exam. These Aptitude questions are designed by the already placed candidates in the reputed company. these questions are based on the previous year question paper of TCS Company Name: TCS Type: Fresher #### 1. (1/3) of a number is 3 more than the (1/6) of the same number? a) 6 b)16 c)18 d)21 Ans: 18 Related: TCS NQT All Previous Year Question & Sample paper with Solution a) 12 b) 25 c) 05 d) 27 Ans: 12 #### 3. Samita was making a cube with dimensions 5*5*5 using 1*1*1 cubes. What is the numberof cubes needed to make a hollow cube looking of the same shape? If we are painting only 2face of each cube then how many faces will remain unpaint??? a) 98 b) 104 c) 538 d) 650 Ans: 538 Sol: (5*5*5-3*3*3)=125-27=98*no of faces=98*6=588-(no of sides painted)=588-50=538 #### 4. Middle- earth is a fictional land inhabited by hobbits, elves, dwarves and men. The hobbitsand elves are peaceful creatures that prefer slow, silent lives and appreciate nature and art.The dwarves and the men engage in physical games. The game is as follows. A tournament isone where out of the two teams that play a match, the one that loses get eliminated. Thematches are played in different rounds, where in every round; half of the teams get eliminatedfrom the tournament. If there are 8 rounds played in knock out tournament, how manymatches were played? a) 257 b) 256 c) 72 d) 255 Ans: 255 Sol: 2^n-1=2^8-1=255 5. Mr. bean having magical balls 25 pink, 10 green, 31 red, 31 yellow, 30 purple. He drenched in rain red, green, and yellow turn into white what is the maximum probability of a pair of same color ? Ans : 31+31+ 2(worst case probability)= 64 #### How many handshakes possible? a) 6 b) 21 c) 28 d) 7 Ans: 21 since cycle will not form. #### 7. we are having 54 men doing handshake in set what will be the minimum required hand shakesfor a minimum of 1 handshake? Ans : {1,2,3,4,5,6,.....54} Set= { 2,5,8,...........53} So1 set will be ans 18 a) 17 b) 21.25 c) 12.25 d) 14.05 a) 20.72 b) 3.5 c) 238.25 d) 6.18 #### 10. A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70.Statement n says ' At least n of the statements on this sheet are false. ‘Which statements aretrue and which are false? a) The even-numbered statements are true and the odd-numbered are false. b) The odd-numbered statements are true and the even-numbered are false. c) The first 35 statements are true and the last 35 are false. d) The first 35 statements are false and the last 35 are false. Sol: when Rule 1: exact ( n-1)th will be true and other will be false Rule2: At least (first half will be true) Rule 3: At most (all true) Example : exactly 40 statement 39th will be true other than it false #### 11. If there are 254 barrels out of them one is poisoned if a person tastes very little he will diewithin 14 hours so if there are mice to test and 24 hours to test, how many mices are requiredto find the poisoned can? a) 3 b) 2 c) 6 d) 8 Ans : 2^n > no of barrels Then n=will be required mice N=8 #### 12. Consider two tumblers, the first containing Water and next contains coffee. Suppose youtake one spoon of water out of the first tumbler and pour it into the second tumbler. Aftermoving you take one spoon of the mixture from the second tumbler and pour it back into thefirst tumbler . Which one of the following statement holds now? a) There is less coffee in the first tumbler than water in the second tumblers b) There is more coffee in the firs tumbler than water in the second tumbler c) There is as much coffee in the first tumbler as there is water in the second tumbler d) None of the statements holds true Ans: both will be equal #### 13. Given a collection of points P in the plane, a 1-set is a point in P that can be separatedfrom the rest by a line, that is the point lies on one side of the line while the others lie on theother side. The number of 1-sets of P is denoted by n1(P). The minimum value of n1(P) overall configurations P of 5 points in the plane in general position (that is no three points in P lieon a line) is a)3 b)5 c) 2 d)1 Ans: 5 same as given no of points #### 14. The citizens of planet nigiet are 8 fingered and have thus developed their decimal systemin base 8. A certain street in nigiet contains 1000 (in base 8) buildings numbered 1 to 1000.How many 3s are used in numbering these buildings? a) 54 b) 64 c) 265 d) 192 Ans: 192 Some times base value is change like: 9finger, 1 to 100(base 9) For 1..100 = 2x For 1....1000 = 3*x^2 a)1 b)3 c)4 d)0 Ans: 4 a) 30.33 b)8 c) 40 d) 5 Ans: 30.33 Sol: 1/3, 1/8 3*8=24 (24-3)=21 (21-8)=13 (21*13)/3^2 #### 17. Here 10 programmers, type 10 lines with in 10 minutes then 60lines can type within 60minutes. How many programmers are needed? a) 16 b) 6 c) 10 d) 60 Solution: (men*time)/work) Ans: 10 This type of Q's repeated 4times for me but the values are different. #### 18. Alok and Bhanu play the following min-max game. Given the expression N = 9 + X + Y -Z Where X, Y and Z are variables representing single digits (0 to 9) Alok would like tomaximize N while Bhanu would like to minimize it. Towards this end, Alok chooses a singledigit number and Bhanu substitutes this for a variable of her choice (X, Y or Z). Alok thenchooses the next value and Bhanu, the variable to substitute the value. Finally Alok proposesthe value for the remaining variable. Assuming both play to their optimal strategies, the valueof N at the end of the game would be. a) 0 b) 27 c) 18 d) 20 The Q's concept is same but the equation of N's is changing. #### 19. Alice and Bob play the following coins-on-a-stack game. 20 coins are stacked one abovethe other. One of them is a special (gold) coin and the rest are ordinary coins. The goal is tobring the gold coin to the top by repeatedly moving the topmost coin to another position inthe stack.Alice starts and the players take turns. A turn consists of moving the coin on the topto a position i below the top coin (0 = i = 20). We will call this an i-move (thus a 0-moveimplies doing nothing). The proviso is that an i-move cannot be repeated; for example once aplayer makes a 2-move, on subsequent turns neither player can make a 2-move. If the goldcoin happens to be on top when it's a player's turn then the player wins the game. Initially, thegold coins the third coin from the top. Then a) In order to win, Alice's first move should be a 1-move. b) In order to win, Alice's first move should be a 0-move. c) In order to win, Alice's first move can be a 0-move or a 1-move. d) Alice has no winning strategy. Ans: d a)1/9 b)4/9 c)5/9 d)2/3 Ans: 5/9 a)12 b)11 c)13 d)18 Ans: 12 a)1/12 b)0 c)12/212 d)11/12 Ans: b #### 23. A sheet of paper has statements numbered from 1 to 40. For each value of n from 1 to 40,statement n says "At least and of the statements on this sheet are true." Which statements aretrue and which are false? a) The even-numbered statements are true and the odd-numbered are false. b) The first 26 statements are false and the rest are true. c) The first 13 statements are true and the rest are false. d) The odd-numbered statements are true and the even-numbered are false. Ans: c Related: TCS NQT All Previous Year Question & Sample paper with Solution a)1/2 b)14/19 c)37/38 d)3/4 Ans: 14/19 a) 0.75 b) 1 c) 0.5 d) 0.25 Ans: d a) 8 b) 16 c) 25 d) 21 Ans: 16 #### 27. A sheet of paper has statements numbered from 1 to 40. For all values of n from 1 to 40,statement n says: 'Exactly n of the statements on this sheet are false.' Which statements aretrue and which are false? a) The even numbered statements are true and the odd numbered statements are false. b) The odd numbered statements are true and the even numbered statements are false. c) All the statements are false. d) The 39th statement is true and the rest are false. Ans: d #### 28. Alok and Bhanu play the following coins in a circle game. 99 coins are arranged in acircle with each coin touching two other coin. Two of the coins are special and the rest areordinary. Alok starts and the players take turns removing an ordinary coin of their choicefrom the circle and bringing the other coins closer until they again form a (smaller) circle.The goal is to bring the special coins adjacent to each other and the first player to do so winsthe game. Initially the special coins are separated by two ordinary coins O1 and O2. Which ofthe following is true? a) In order to win, Alok should remove O1 on his first turn. b) In order to win, Alok should remove one of the coins different from O1 and O2 on his first turn. c) In order to win, Alok should remove O2 on his first turn. d) Alok has no winning strategy. Ans: a if the gold coin in 3rd position then mark it otherwise leave it Ans: 23 #### 30. One day Alice meets pal and byte in fairyland. She knows that pal lies on Mondays,Tuesdays and Wednesdays and tells the truth on the other days of the week byte, on the otherhand, lies on Thursdays, Fridays and Saturdays, but tells the truth on the other days of theweek. Now they make the following statements to Alice – pal. Yesterday was one of thosedays when I lie byte. Yesterday was one of those days when I lie too. What day is it? a) Thursday b) Tuesday c) Monday d) Sunday Ans: a . #### 31. Sudha Patel +> perfume factory Ans : 2 more cedar #### 32. A toy train can make 10 sounds sound changes after every 4 minute now train is defectiveand can make only 2 sounds find probability that same sound is repeated 4 timesconsecutively (1 OUT OF__)? a 16 b 8 c 12 d 4 Ans: (1/2)*(1/2)*(1/2)*(1/2)+ (1/2)*(1/2)*(1/2)*(1/2)=(1/8) thus 1 out of 8 ans a) 29 b) 12 c) 67 d) 98 Ans :29 #### 34. In a country x we are having diff types of coins ranging from 64...512. All coins havingdifferent integral value the difference between two coins comes out to be 50% more than theformer coin. Then how many coins can be made? Ans : 6 coins Sol.64 64*1.5=96 96*1.5=144 144*1.5=216 216*1.5=324 324*1.5=486 Then total coin will be 6 #### 35. There is a relation is given which is n/P=195 Where n- no of steps in meters P- pacelength These Is a man shivam he know his pace length =185cm then what will be the speedof shivam kmph? Ans=195*1.85*1.85*60/1000=40.04kmph Thank You.

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# Capacity Worksheets Third Grade 👤 will chen 🗓 May 17, 2021, 10:27 am ( Last Modified ) Metric capacity worksheets: liters and milliliters. Below are three grade 3 math worksheets on using metric units of volume or capacity, and in particular understanding the relationship between milliliters and liters in a real world context. These worksheets are pdf files..Learn third grade math online for free. Check 3rd Grade Math Games and Fun Math Worksheets Curriculum Interactive Practice Learning. SplashLearn is an award winning math learning program used by more than 40 Million kids for fun math practice..Our grade 3 math worksheets are free and printable in PDF format. Based on the Singaporean math curriculum grade level 3, these worksheets are made for students in third grade level and cover math topics such as: place value, spelling, addition, subtraction, division, multiplication, fractions, graphing, measurement, mixed operations, geometry, area and perimeter, and time..All the free 3rd Grade Math Worksheets in this section follow the Elementary Math Benchmarks for Third Grade. Adding on 1, 10, 100 and 1000 These sheets will help you learn to add on 1, 10, 100 or 1000 to any 3 or 4 digit numbers.. The following worksheets contain a mix of grade 3 addition, subtraction, multiplication and division word problems. Mixing math word problems is the ultimate test of understanding mathematical concepts, as it forces students to analyze the situation rather than mechanically apply a solution..Take your students' geometry skills to the next level with our second grade geometry worksheets and printables. Begin by reviewing 2D shapes and advance to introducing more complex 3D shapes and rare polygons. Explore concepts of angles, lines, and symmetry, and use visual guides to practice fractions..Make 1000s of FREE Math Worksheets! No registration needed! Just print and go! The Math Worksheet Wizard is a FREE resource for teachers and homeschooling moms and dads. You can make an unlimited number of printable math worksheets for children, for the classroom or for homework, simply by clicking a button. What makes this site unique is that every time you create a worksheet, you get .. This is the first week of the 4th grade math buzz series. This file contains 5 worksheets, reviewing basic skills from the previous grade. Skills covered include: place value, linear measurement, multiplication facts, and comparing fractions..First Grade Math Curriculum: What Students Will Learn. Common Core Standards for first-grade math include representing and solving problems that involve addition and subtraction; understanding and applying properties of operations and the relationship between addition and subtraction; adding and subtracting within 20; working with addition and subtraction equations; extending the counting ..Dive into these printable worksheets on converting between milliliters and liters, and help grade 3, grade 4, and grade 5 kids master conversions. Begin with converting from liters to milliliters, followed by milliliters to liters, and find an enormous collection to recapitulate capacity unit conversions, we sure have something for everyone in ... Related to "Capacity Worksheets Third Grade" ⤵ Name : __________________ Seat Num. : __________________ Date : __________________ 33 + 94 = ... 62 + 70 = ... 56 + 49 = ... 62 + 98 = ... 41 + 97 = ... 68 + 25 = ... 96 + 15 = ... 45 + 69 = ... 37 + 67 = ... 69 + 57 = ... 35 + 24 = ... 40 + 88 = ... 73 + 53 = ... 39 + 77 = ... 46 + 59 = ... 28 + 82 = ... 19 + 86 = ... 32 + 55 = ... 14 + 31 = ... 17 + 92 = ... 21 + 83 = ... 32 + 100 = ... 39 + 51 = ... 75 + 35 = ... 62 + 54 = ... 38 + 98 = ... 63 + 11 = ... 10 + 95 = ... 32 + 30 = ... 35 + 69 = ... 50 + 47 = ... 24 + 19 = ... 64 + 83 = ... 45 + 17 = ... 93 + 70 = ... 15 + 67 = ... 50 + 14 = ... 41 + 90 = ... 92 + 25 = ... 50 + 77 = ... 44 + 44 = ... 63 + 93 = ... 56 + 98 = ... 54 + 21 = ... 11 + 46 = ... 50 + 98 = ... 15 + 14 = ... 12 + 98 = ... 91 + 63 = ... 33 + 12 = ... 27 + 23 = ... 11 + 64 = ... 41 + 58 = ... 59 + 89 = ... 95 + 49 = ... 22 + 97 = ... 27 + 76 = ... 87 + 70 = ... 87 + 30 = ... 43 + 16 = ... 83 + 83 = ... 68 + 82 = ... 23 + 60 = ... 82 + 63 = ... 55 + 51 = ... 11 + 13 = ... 38 + 51 = ... 76 + 91 = ... 99 + 57 = ... 27 + 91 = ... 24 + 73 = ... 26 + 86 = ... 12 + 94 = ... 97 + 42 = ... 29 + 53 = ... 63 + 22 = ... 13 + 98 = ... 35 + 92 = ... 24 + 96 = ... 97 + 86 = ... 51 + 89 = ... 43 + 44 = ... 62 + 95 = ... 16 + 43 = ... 22 + 26 = ... 64 + 12 = ... 53 + 84 = ... 26 + 41 = ... 41 + 70 = ... 75 + 81 = ... 69 + 35 = ... 55 + 77 = ... 88 + 59 = ... 18 + 84 = ... 10 + 72 = ... 25 + 93 = ... 67 + 68 = ... 30 + 20 = ... 44 + 67 = ... 47 + 35 = ... 66 + 41 = ... 25 + 83 = ... 80 + 45 = ... 54 + 97 = ... 10 + 26 = ... 79 + 53 = ... 68 + 48 = ... 51 + 86 = ... 73 + 14 = ... 72 + 58 = ... 51 + 50 = ... 55 + 46 = ... 81 + 22 = ... 46 + 70 = ... 90 + 90 = ... 63 + 79 = ... 100 + 16 = ... 65 + 62 = ... 66 + 21 = ... 19 + 34 = ... 83 + 16 = ... 16 + 50 = ... 11 + 93 = ... 60 + 94 = ... 33 + 87 = ... 50 + 83 = ... 23 + 91 = ... 36 + 47 = ... 19 + 52 = ... 49 + 89 = ... 99 + 97 = ... 53 + 24 = ... 15 + 67 = ... 68 + 12 = ... 39 + 30 = ... 61 + 60 = ... 87 + 85 = ... 14 + 75 = ... 53 + 81 = ... 95 + 58 = ... 35 + 95 = ... 63 + 99 = ... 74 + 62 = ... 10 + 88 = ... 58 + 69 = ... 58 + 19 = ... 74 + 11 = ... 23 + 12 = ... 51 + 25 = ... 52 + 24 = ... 24 + 38 = ... 28 + 84 = ... 27 + 36 = ... 26 + 26 = ... 83 + 39 = ... 49 + 97 = ... 59 + 26 = ... 53 + 81 = ... 79 + 27 = ... 10 + 66 = ... 61 + 34 = ... 100 + 69 = ... 78 + 17 = ... 34 + 59 = ... 37 + 98 = ... 66 + 81 = ... 52 + 15 = ... 91 + 89 = ... 95 + 54 = ... 28 + 48 = ... 38 + 14 = ... 52 + 86 = ... 79 + 37 = ... 50 + 41 = ... 97 + 46 = ... 45 + 30 = ... 37 + 53 = ... 14 + 78 = ... 20 + 58 = ... 50 + 100 = ... 35 + 13 = ... 56 + 28 = ... 91 + 74 = ... 66 + 18 = ... 45 + 94 = ... 89 + 18 = ... 61 + 36 = ... 73 + 58 = ... 85 + 19 = ... 57 + 99 = ... 84 + 24 = ... 14 + 79 = ... 18 + 22 = ... 21 + 78 = ... 69 + 56 = ... 28 + 90 = ... 76 + 78 = ... 64 + 28 = ... 84 + 65 = ... 95 + 31 = ... show printable version !!!hide the show Free-3rd-grade-math-worksheets-reading-scales-3b.gif (1000×1294) Measurement Worksheets 3rd Grade Measurement Worksheets 3rd Grade Measurement Worksheets Exploring Capacity The Hands-on Way! 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# Engineering Mechanics-Pune-Winter-2016 ## All Branches (C.B.S.) Time: 2 hours Maximum marks: 50 Notes : 1. Attempt Q. No. 1 or Q. No. 2, Q. No. 3 or Q. No. 4 and Q. No. 5 or Q. No. 6. 2. Neat diagram must be drawn wherever necessary. 3. Figures to the right indicate full marks. 4. Assume suitable data, if necessary and clearly state. 5. Use of cell phone is prohibited in the examination hall. 6. Use of electronic pocket calculator is allowed. 1. (a) The resultant of two forces P and Q is 1400 N vertical. Determine the force Q and the corresponding angle θ for the system of forcess as shown in Fig. 1A. [4M] (b) The system shown in Fig. 1b is initially at rest. Neglecting axle friction and mass of pulley, determine the acceleration of block. [4M] (c) A cricket ball shot by a batsman from a height of 2.0 m at an angle of 30° with the horiziontal with a velocity of 20 m/s is caught by a fielder at a height of 0.8 m from the ground. Determine the distance between the batsman and fielder. [4M] (d) A ball has a mass of 30 kg and is thrown upward with a speed of 15 m/s. Determine the time to attain maximum height using impulse momentum principle. Also find the maximum height. [4M] OR 2. (a) Determine the y coordinate of centroid of the shaded area as shown in Fig. 2a. [4M] (b) If the crest of the hill has a radius of curvature ρ = 60m, determine the maximum constant speed at which the car can travel over it without leaving the surface of the road. The car has a weight of 17.5kN. (Refer Fig. 2b) [4M] (c) A particle moves along a straight line with an acceleration α=(4t3 – 2t), where α is in m/s2 and t is in s. When t = 0, the practicle is at 2 m to the left of origin and when t = 2s the praticle is at 20 m to left of origin. Determine the position of particle at t = 4s. [4M] (d) A woman having a mass of 70 kg stands in an elevator which has a downward acceleration of 4 m/s2 starting from rest. Determine work done by her weight and the work of the normal force which the floor exerts on her when the elevator descends 6 m. [4M] 3. (a) Two spheres A and B a diameter 80 mm and 120 mm respectively are held in equilibrium by separate strings as shown in Fig. 3a. Sphere B rests against vertical wall. If masses of spheres A and B are 10 kg and 20 kg, determine the tension in the string and reactions at point of contact. [6M] (b) Four parallel bolting forces act on the rim of the circular cover plate as shown in Fig. 3b. If the resultant force 750 N is passing through (0.15m, 0.1m) from the origin O, determine the magnitude of force P1 and P2 [6M] (c) Determine the support reaction for the beam loaded and supported as shown in Fig. 3c. [5M] OR 4. (a) Three cables are joined at the junction C as shown in Fig. 4a. Determine the tension in cable AC and BC caused by the weight of the 30 kg cylinder. [6M] (b) The square steel plate has a mass of 1800 kg with mass center G as shown in Fig. 4b. Determine the tension in each cable so that the plate remains horiziontal. [6M] (c) Determine the component of reaction at hinge A and tension in the cable BC as shown in Fig. 4c. [5M] 5. (a) A block of mass 10 kg rests on an inclined plane as shown in Fig. 5a. If the coefficient of static friction between the block and plane is μs= 0.25, determine the maximum force P to maintain equalibrium. [6M] (b) Determine the components of reactions at supports A and B for the frame loaded and supported as shown in Fig. 5b. [6M] (c) The 15 m ladder has uniform weight of 80N. It rests against smooth vertical wall at B and horiziontal floor at A. If the coefficient of static friction between ladder and floor at A is μs=0.4, determine the smallest angle θ with vertical wall at which the ladder will slip. [5M] OR 6. (a) The cable segment support the loading as shown in Fig. 6a. Determine the support reaction and maximum tension in segment of cable. [6M] (b) A cable is passing over the disc of belt friction apparatus at a lap angle 180° as shown in Fig. 6b. If the weight of block is 500 N, determine the range of force P to maintain equilibrium. [6M] (c) Determine the forces in the members of the truss loaded and supported as shown in the Fig. 6c. Tabulate the result with magnitude and nature of force in the members. [5M] Scroll to Top error: Alert: Content selection is disabled!!

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GeeksforGeeks App Open App Browser Continue # Print path between any two nodes in a Binary Tree Given a Binary Tree of distinct nodes and a pair of nodes. The task is to find and print the path between the two given nodes in the binary tree. For Example, in the above binary tree the path between the nodes 7 and 4 is 7 -> 3 -> 1 -> 4 The idea is to find paths from root nodes to the two nodes and store them in two separate vectors or arrays say path1 and path2. Now, there arises two different cases: 1. If the two nodes are in different subtrees of root nodes. That is one in the left subtree and the other in the right subtree. In this case it is clear that root node will lie in between the path from node1 to node2. So, print path1 in reverse order and then path 2. 2. If the nodes are in the same subtree. That is either in the left subtree or in the right subtree. In this case you need to observe that path from root to the two nodes will have an intersection point before which the path is common for the two nodes from the root node. Find that intersection point and print nodes from that point in a similar fashion of the above case. Below is the implementation of the above approach: ## C++ `// C++ program to print path between any``// two nodes in a Binary Tree``#include ``using` `namespace` `std;` `// structure of a node of binary tree``struct` `Node {``    ``int` `data;``    ``Node *left, *right;``};` `/* Helper function that allocates a new node with the``given data and NULL left and right pointers. */``struct` `Node* getNode(``int` `data)``{``    ``struct` `Node* newNode = ``new` `Node;``    ``newNode->data = data;``    ``newNode->left = newNode->right = NULL;``    ``return` `newNode;``}` `// Function to check if there is a path from root``// to the given node. It also populates``// 'arr' with the given path``bool` `getPath(Node* root, vector<``int``>& arr, ``int` `x)``{``    ``// if root is NULL``    ``// there is no path``    ``if` `(!root)``        ``return` `false``;` `    ``// push the node's value in 'arr'``    ``arr.push_back(root->data);` `    ``// if it is the required node``    ``// return true``    ``if` `(root->data == x)``        ``return` `true``;` `    ``// else check whether the required node lies``    ``// in the left subtree or right subtree of``    ``// the current node``    ``if` `(getPath(root->left, arr, x) || getPath(root->right, arr, x))``        ``return` `true``;` `    ``// required node does not lie either in the``    ``// left or right subtree of the current node``    ``// Thus, remove current node's value from``    ``// 'arr'and then return false``    ``arr.pop_back();``    ``return` `false``;``}` `// Function to print the path between``// any two nodes in a binary tree``void` `printPathBetweenNodes(Node* root, ``int` `n1, ``int` `n2)``{``    ``// vector to store the path of``    ``// first node n1 from root``    ``vector<``int``> path1;` `    ``// vector to store the path of``    ``// second node n2 from root``    ``vector<``int``> path2;` `    ``getPath(root, path1, n1);``    ``getPath(root, path2, n2);` `    ``int` `intersection = -1;` `    ``// Get intersection point``    ``int` `i = 0, j = 0;``    ``while` `(i != path1.size() || j != path2.size()) {` `        ``// Keep moving forward until no intersection``        ``// is found``        ``if` `(i == j && path1[i] == path2[j]) {``            ``i++;``            ``j++;``        ``}``        ``else` `{``            ``intersection = j - 1;``            ``break``;``        ``}``    ``}` `    ``// Print the required path``    ``for` `(``int` `i = path1.size() - 1; i > intersection; i--)``        ``cout << path1[i] << ``" "``;` `    ``for` `(``int` `i = intersection; i < path2.size(); i++)``        ``cout << path2[i] << ``" "``;``}` `// Driver program``int` `main()``{``    ``// binary tree formation``    ``struct` `Node* root = getNode(0);``    ``root->left = getNode(1);``    ``root->left->left = getNode(3);``    ``root->left->left->left = getNode(7);``    ``root->left->right = getNode(4);``    ``root->left->right->left = getNode(8);``    ``root->left->right->right = getNode(9);``    ``root->right = getNode(2);``    ``root->right->left = getNode(5);``    ``root->right->right = getNode(6);` `    ``int` `node1 = 7;``    ``int` `node2 = 4;``    ``printPathBetweenNodes(root, node1, node2);` `    ``return` `0;``}` ## Java `// Java program to print path between any``// two nodes in a Binary Tree``import` `java.util.*;``class` `Solution``{` `// structure of a node of binary tree``static` `class` `Node {``    ``int` `data;``    ``Node left, right;``}` `/* Helper function that allocates a new node with the``given data and null left and right pointers. */`` ``static` `Node getNode(``int` `data)``{``     ``Node newNode = ``new` `Node();``    ``newNode.data = data;``    ``newNode.left = newNode.right = ``null``;``    ``return` `newNode;``}` `// Function to check if there is a path from root``// to the given node. It also populates``// 'arr' with the given path``static` `boolean` `getPath(Node root, Vector arr, ``int` `x)``{``    ``// if root is null``    ``// there is no path``    ``if` `(root==``null``)``        ``return` `false``;` `    ``// push the node's value in 'arr'``    ``arr.add(root.data);` `    ``// if it is the required node``    ``// return true``    ``if` `(root.data == x)``        ``return` `true``;` `    ``// else check whether the required node lies``    ``// in the left subtree or right subtree of``    ``// the current node``    ``if` `(getPath(root.left, arr, x) || getPath(root.right, arr, x))``        ``return` `true``;` `    ``// required node does not lie either in the``    ``// left or right subtree of the current node``    ``// Thus, remove current node's value from``    ``// 'arr'and then return false``    ``arr.remove(arr.size()-``1``);``    ``return` `false``;``}` `// Function to print the path between``// any two nodes in a binary tree``static` `void` `printPathBetweenNodes(Node root, ``int` `n1, ``int` `n2)``{``    ``// vector to store the path of``    ``// first node n1 from root``    ``Vector path1= ``new` `Vector();` `    ``// vector to store the path of``    ``// second node n2 from root``    ``Vector path2=``new` `Vector();` `    ``getPath(root, path1, n1);``    ``getPath(root, path2, n2);` `    ``int` `intersection = -``1``;` `    ``// Get intersection point``    ``int` `i = ``0``, j = ``0``;``    ``while` `(i != path1.size() || j != path2.size()) {` `        ``// Keep moving forward until no intersection``        ``// is found``        ``if` `(i == j && path1.get(i) == path2.get(i)) {``            ``i++;``            ``j++;``        ``}``        ``else` `{``            ``intersection = j - ``1``;``            ``break``;``        ``}``    ``}` `    ``// Print the required path``    ``for` `( i = path1.size() - ``1``; i > intersection; i--)``        ``System.out.print( path1.get(i) + ``" "``);` `    ``for` `( i = intersection; i < path2.size(); i++)``        ``System.out.print( path2.get(i) + ``" "``);``}` `// Driver program``public` `static` `void` `main(String[] args)``{``    ``// binary tree formation``     ``Node root = getNode(``0``);``    ``root.left = getNode(``1``);``    ``root.left.left = getNode(``3``);``    ``root.left.left.left = getNode(``7``);``    ``root.left.right = getNode(``4``);``    ``root.left.right.left = getNode(``8``);``    ``root.left.right.right = getNode(``9``);``    ``root.right = getNode(``2``);``    ``root.right.left = getNode(``5``);``    ``root.right.right = getNode(``6``);` `    ``int` `node1 = ``7``;``    ``int` `node2 = ``4``;``    ``printPathBetweenNodes(root, node1, node2);` `}``}``// This code is contributed by Arnab Kundu` ## Python `# Python3 program to print path between any``# two nodes in a Binary Tree` `import` `sys``import` `math` `# structure of a node of binary tree``class` `Node:``    ``def` `__init__(``self``,data):``        ``self``.data ``=` `data``        ``self``.left ``=` `None``        ``self``.right ``=` `None` `# Helper function that allocates a new node with the``#given data and NULL left and right pointers.``def` `getNode(data):``        ``return` `Node(data)` `# Function to check if there is a path from root``# to the given node. It also populates``# 'arr' with the given path``def` `getPath(root, rarr, x):` `    ``# if root is NULL``    ``# there is no path``    ``if` `not` `root:``        ``return` `False``    ` `    ``# push the node's value in 'arr'``    ``rarr.append(root.data)` `    ``# if it is the required node``    ``# return true``    ``if` `root.data ``=``=` `x:``        ``return` `True``    ` `    ``# else check whether the required node lies``    ``# in the left subtree or right subtree of``    ``# the current node``    ``if` `getPath(root.left, rarr, x) ``or` `getPath(root.right, rarr, x):``        ``return` `True``    ` `    ``# required node does not lie either in the``    ``# left or right subtree of the current node``    ``# Thus, remove current node's value from``    ``# 'arr'and then return false``    ``rarr.pop()``    ``return` `False` `# Function to print the path between``# any two nodes in a binary tree``def` `printPathBetweenNodes(root, n1, n2):` `    ``# vector to store the path of``    ``# first node n1 from root``    ``path1 ``=` `[]` `    ``# vector to store the path of``    ``# second node n2 from root``    ``path2 ``=` `[]``    ``getPath(root, path1, n1)``    ``getPath(root, path2, n2)` `    ``# Get intersection point``    ``i, j ``=` `0``, ``0``    ``intersection``=``-``1``    ``while``(i !``=` `len``(path1) ``or` `j !``=` `len``(path2)):` `        ``# Keep moving forward until no intersection``        ``# is found``        ``if` `(i ``=``=` `j ``and` `path1[i] ``=``=` `path2[j]):``            ``i ``+``=` `1``            ``j ``+``=` `1``        ``else``:``            ``intersection ``=` `j ``-` `1``            ``break` `    ``# Print the required path``    ``for` `i ``in` `range``(``len``(path1) ``-` `1``, intersection ``-` `1``, ``-``1``):``        ``print``(``"{} "``.``format``(path1[i]), end ``=` `"")``    ``for` `j ``in` `range``(intersection ``+` `1``, ``len``(path2)):``        ``print``(``"{} "``.``format``(path2[j]), end ``=` `"")``    ` `# Driver program``if` `__name__``=``=``'__main__'``:` `    ``# binary tree formation``    ``root ``=` `getNode(``0``)``    ``root.left ``=` `getNode(``1``)``    ``root.left.left ``=` `getNode(``3``)``    ``root.left.left.left ``=` `getNode(``7``)``    ``root.left.right ``=` `getNode(``4``)``    ``root.left.right.left ``=` `getNode(``8``)``    ``root.left.right.right ``=` `getNode(``9``)``    ``root.right ``=` `getNode(``2``)``    ``root.right.left ``=` `getNode(``5``)``    ``root.right.right ``=` `getNode(``6``)``    ``node1``=``7``    ``node2``=``4``    ``printPathBetweenNodes(root,node1,node2)` `# This Code is Contributed By Vikash Kumar 37` ## C# `// C# program to print path between any``// two nodes in a Binary Tree``using` `System;``using` `System.Collections.Generic;` `class` `Solution``{` `// structure of a node of binary tree``public` `class` `Node``{``    ``public` `int` `data;``    ``public` `Node left, right;``}` `/* Helper function that allocates a new node with the``given data and null left and right pointers. */``static` `Node getNode(``int` `data)``{``    ``Node newNode = ``new` `Node();``    ``newNode.data = data;``    ``newNode.left = newNode.right = ``null``;``    ``return` `newNode;``}` `// Function to check if there is a path from root``// to the given node. It also populates``// 'arr' with the given path``static` `Boolean getPath(Node root, List<``int``> arr, ``int` `x)``{``    ``// if root is null``    ``// there is no path``    ``if` `(root == ``null``)``        ``return` `false``;` `    ``// push the node's value in 'arr'``    ``arr.Add(root.data);` `    ``// if it is the required node``    ``// return true``    ``if` `(root.data == x)``        ``return` `true``;` `    ``// else check whether the required node lies``    ``// in the left subtree or right subtree of``    ``// the current node``    ``if` `(getPath(root.left, arr, x) || getPath(root.right, arr, x))``        ``return` `true``;` `    ``// required node does not lie either in the``    ``// left or right subtree of the current node``    ``// Thus, remove current node's value from``    ``// 'arr'and then return false``    ``arr.RemoveAt(arr.Count-1);``    ``return` `false``;``}` `// Function to print the path between``// any two nodes in a binary tree``static` `void` `printPathBetweenNodes(Node root, ``int` `n1, ``int` `n2)``{``    ``// vector to store the path of``    ``// first node n1 from root``    ``List<``int``> path1 = ``new` `List<``int``>();` `    ``// vector to store the path of``    ``// second node n2 from root``    ``List<``int``> path2 = ``new` `List<``int``>();` `    ``getPath(root, path1, n1);``    ``getPath(root, path2, n2);` `    ``int` `intersection = -1;` `    ``// Get intersection point``    ``int` `i = 0, j = 0;``    ``while` `(i != path1.Count || j != path2.Count)``    ``{` `        ``// Keep moving forward until no intersection``        ``// is found``        ``if` `(i == j && path1[i] == path2[i])``        ``{``            ``i++;``            ``j++;``        ``}``        ``else``        ``{``            ``intersection = j - 1;``            ``break``;``        ``}``    ``}` `    ``// Print the required path``    ``for` `( i = path1.Count - 1; i > intersection; i--)``        ``Console.Write( path1[i] + ``" "``);` `    ``for` `( i = intersection; i < path2.Count; i++)``        ``Console.Write( path2[i] + ``" "``);``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``// binary tree formation``    ``Node root = getNode(0);``    ``root.left = getNode(1);``    ``root.left.left = getNode(3);``    ``root.left.left.left = getNode(7);``    ``root.left.right = getNode(4);``    ``root.left.right.left = getNode(8);``    ``root.left.right.right = getNode(9);``    ``root.right = getNode(2);``    ``root.right.left = getNode(5);``    ``root.right.right = getNode(6);` `    ``int` `node1 = 7;``    ``int` `node2 = 4;``    ``printPathBetweenNodes(root, node1, node2);` `}``}` `// This code is contributed by Princi Singh` ## Javascript `` Output `7 3 1 4 ` Complexity Analysis: • Time Complexity: O(N), as we are using recursion for traversing the tree. Where N is the number of nodes in the tree. • Auxiliary Space: O(N), as we are using extra space for storing the paths of the trees. Where N is the number of nodes in the tree. My Personal Notes arrow_drop_up

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# Is the circuit breaker amperage the rms amperage • Puglife #### Puglife The circuit breakers of my house say that they are 15a (the majority of them). The rms voltage is 120v for the outlets, and so the peak voltage is about 170v. I was wondering, if I hypthetically connected the terminals together with a 10 ohm resistor, would the breaker trip? I don't know if the amperage drawn is total amps, or amps at the rms voltage, and so that is what I am asking. I know its not safe to do, and I am not going to do it, but I would like to know if it is 15a in general or at the rms, allowing for more amperage at its peak voltages. Thank You. http://static.schneider-electric.us/docs/Circuit Protection/Molded Case Circuit Breakers/0100-400 A Frame FA-LA/FA-FC-FH/0600DB0105.pdf Look at page 2, it has a curve of time at rated current vs trip time. A circuit breaker will take approx 1000 seconds to trip if the current is right at the rated value, according to this graph at least. Im not exactly sure what it is saying, because no current will be consistantly be drawn out of the outlet, because many times a second, the voltage reaches zero, and thus so is the current. Im not exactly sure what it is saying, because no current will be consistantly be drawn out of the outlet, because many times a second, the voltage reaches zero, and thus so is the current. I don't know if the amperage drawn is total amps, or amps at the rms voltage, and so that is what I am asking. The datasheet Grinkle linked describes RMS symetrical current. Look up "RMS current" then think about it - fuses amd thermal breakers work by heating an element. Not surprising they'd be rated in terms of "heating value" of current, would you think ? Thanks, Jim - yes, RMS. Thanks, Jim - yes, RMS. Thank You Both!

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```ENSC 461 Assignment #2 (Cycles) Assignment date: Tuesday Jan 23, 2011 Problem 1: (the Stirling cycle) Show that for an idealized Stirling cycle, the thermal efficiency is:  th  1  TL TH Problem 2: (Otto cycle) An open, ideal Otto-cycle engine has a compression ratio of 10:1. The air just prior to the compression stroke is at 20C and 100 kPa. The maximum cycle temperature is 2000 C. The thermal efficiency of the ideal Otto cycle is 0.60. Rather than simply discharging the air to the atmosphere after expansion in the cylinder, an isentropic turbine is installed in the exhaust to produce additional work. Assume constant specific heats, the mass flow rate through the turbine is steady and the pressure at the inlet to the cylinder is identical to the pressure at the discharge of the turbine. i) draw a T-s diagram process for the compound engine ii) determine the work output of the turbine, (kJ/kg) iii) determine the overall thermal efficiency of the compound engine. Otto Cycle Turbine in Wturbine out M. Bahrami ENSC 461 (S 11) Assignment 2 1 Problem 1 Solution: For idealized Stirling cycle with perfect regeneration, one can write: QH  TH s 2  s1  QL  TL s 3  s 4  Thermal efficiency is defined:  th  Wnet QH  QL T s  s 4  Q (1)   1 L  1 L 3 QH QH QH TH s 2  s1  From Gibb’s equation: Tds  du  Pdv  c v dT  Pdv If T = const. Tds  Pdv Rdv ds  v using ideal gas equation of state Integrating gives: v s 3  s 4  R ln 3  v4 v    R ln 2  v1     s 2  s1  Therefore, s3 - s4 = s2 – s1; and Eq. (1) gives:  th  1  TL TH Problem 2 Solution: T 3 4 2 1 5 T1 = 20 C s Part i) For an isentropic process M. Bahrami ENSC 461 (S 11) Assignment 2 2 T2  v1    T1  v 2   k 1 T4  v3  and   T3  v 4   k 1 P   4  P3     k 1 / k Therefore; T2 = 293 K (10) 1.4-1 = 735.98 K and v T4  T3  3  v4     k 1 v  T3  4  v3    v  T3  1  v2      k 1 For an IC engine, v 4 v1  v3 v 2 Therefore v T4  T3  3  v4     k 1   k 1  2000  273K  10   1.4 1  904.9 K The net work output from the Otto cycle is wnet, Otto = cp (T3 – T4) – cp (T2 – T1) = 1.005 kJ/kg.K (2273 - 904.9 – 735.98 + 293) K = 929.75 kJ/kg We have an isentropic process between 3 and 5, we can write: T5  P5    T3  P3   k 1 / k Since the air behaves as an ideal gas, and we know that P1 = P5 = Patm, we can write P3 = R T3 / v3 P1 = P5 = R T1 / v1 Therefore T5  P5    T3  P3   k 1 / k  R.T v    1 3   v1 R.T3   k 1 / k But for an Otto cycle, we have constant volume heat addition between 2 and 3 and v2 = v3 T5  T1 v 2    T3  v1 T3   k 1 / k M. Bahrami T v    1 2   T3 v1   k 1 / k  473K 1   2273K    2273K 10  ENSC 461 (S 11) 0.4 / 1.4  655.7 K Assignment 2 3 Part ii) The work output of the turbine is w turbine = (h4 – h5) = cp (T4 – T5) = 1.005 kJ/kg.K (904.9 – 655.7) K = 250.4 kJ/kg Part iii) The thermal efficiency of the compound engine is given by  th  wnet ,Otto  wturbine qh q h  wnet ,Otto /  Otto  929.75kJ / kg  / 0.6  1549.58kJ / kg  thus  th  929.75kJ / kg  250.4kJ / kg  / 1549.58kJ / kg  0.762 M. Bahrami ENSC 461 (S 11) Assignment 2 4 ```

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# A very different property of primitive Pythagorean triplets: Can number be in more than two of them? While playing with numbers, I thought about squares of numbers, and then the first thing that came to mind was Pythagorean triplets. I observed a very interesting fact that any $x\in\mathbb N$ can never be a member of more than two Pythagorean triplets of pairwise coprime numbers, like $(3,4,5)$ and $(8,15,17)$. For example $$16^2+63^2=65^2$$$$33^2+56^2=65^2$$ are the possible triplets for $x=65$, and $65$ cannot exist in any other triplet of co-primes. Now I need to prove this conjecture. So I thought that in a Pythagorean triplet, the three numbers are of the form $(2mn, m^2-n^2, m^2+n^2)$. Let $x$ be a number . Then I have to show that $$x=2mn$$$$x=a^2+b^2$$$$x=y^2-z^2$$ are not simultaneously possible. But I am stuck and don't know where to go from here. • The pythagorean triples are uniquely generated by $2kmn, k(m^2-n^2),k( m^2+n^2)$ where $\gcd(m,n)=1, m \gt n\ge 1,k\ge1$ Commented Dec 9, 2016 at 10:41 • maybe you should post another question with the requirement $(x,y)=(y,z)=(z,x)=1$ Commented Dec 9, 2016 at 10:46 • $1105^2 = 1104^2 + 47^2 = 1073^2 + 264^2 = 943^2 + 576^2 = 817^2 + 744^2$ Commented Dec 9, 2016 at 11:08 • This is a nice question +1 Commented Dec 9, 2016 at 11:57 • "solve it again" is not a fair approach. As I already suggested you should open another question if your question is already answered but you you want to change it substantially Commented Dec 9, 2016 at 16:56 Start from $1105 = 5\cdot 13\cdot 17$ whose prime factors all has the form $4k+1$. Decompose the prime factors in $\mathbb{Z}$ over gaussian integers $\mathbb{Z}[i]$ as $$5 = (2+i)(2-i),\quad 13 = (3+2i)(3-2i)\quad\text{ and }\quad17 = (4+i)(4-i)$$ Recombine the factors of $1105^2$ over $\mathbb{Z}[i]$ in different order and then turn them to sum of two squares. We get: $$\begin{array}{rc:rr} 1105^2 = & 943^2 + 576^2 & ((2+i)(3+2i)(4+i))^2 = & -943 + 576i\\ = & 817^2 + 744^2 & ((2-i)(3+2i)(4+i))^2 = & 817 + 744i\\ = & 1073^2 + 264^2 & ((2+i)(3-2i)(4+i))^2 = & 1073 + 264i\\ = & 1104^2 + 47^2 & ((2-i)(3-2i)(4+i))^2 = & -47 - 1104i\\ \end{array}$$ A counter-example for the speculation that an integer can appear in at most two primitive Pythagorean triples. • shouldn't it be 13 in the place of 11@achille hui? Commented Dec 9, 2016 at 11:41 • Ahhh, you are right. Commented Dec 9, 2016 at 11:42 • More examples here Commented Dec 9, 2016 at 23:18 The "smallest" counterexample is \begin{gather*} 5^2 + 12^2 = 13^2 \\ 9^2 + 12^2 = 15^2 \\ 12^2 + 16^2 = 20^2 \end{gather*} ($12$ also satisfies $12^2 + 35^2 = 37^2$.) Edit: now that the OP has stipulated that the elements be pairwise coprime, the smallest counterexample (in the sense that the repeated number is minimal) is \begin{gather*} 11^2 + 60^2 = 61^2 \\ 60^2 + 91^2 = 109^2 \\ 60^2 + 221^2 = 229^2 \end{gather*} $60$ also satisfies $60^2 + 899^2 = 901^2$. • I have added something to the question , please solve it again @lokodiz Commented Dec 9, 2016 at 11:00 $120,160,200$ and $90,120,150$ and $72,96,120$ How I arrived at it: It is a common knowledge that if we scale the triplet $3,4,5$ by any constant, we get another triplet. So I found out a common multiple of $3,4,5$, which is $120$ and then scaled the triplet one by one with the constants $\frac{120}{3}$, $\frac{120}{4}$ and $\frac{120}{5}$. • Note that the question has now been edited to allow only triplets whose numbers don't share a common factor. By construction, your examples don't pass that added requirement. Commented Dec 9, 2016 at 14:39 • Yes, I saw that. but unfortunately, I am unable to prove/disprove it. But I have a hunch that a counter example can be found. I should maybe start thinking about a proof rather than search for counter example. Commented Dec 9, 2016 at 14:51 • My answer now provides a counterexample where the numbers are pairwise coprime. Commented Dec 9, 2016 at 15:04 $16, 63, 65$ $25, 60, 65$ $33, 56 , 65$ Observed from: http://www.tsm-resources.com/alists/trip.html Possible interesting reasearch question: Are there numbers that appears in infinitely many Pythagorean triples? • @yoyostein, suppose such a number exists. Call it $n$. Then at least one of $a^2+b^2=n$, $2ab=n$, $a^2-b^2=n$ has infinitely many solutions. The first two obviously don't, and the third has finitely many because $a+b$ and $a-b$ have to be factors of $n$ (a finite set). Commented Dec 9, 2016 at 14:40 • @Sophie Nice analysis Commented Dec 9, 2016 at 14:43 Any prime which is congruent to $1$ modulo $4$ is the largest member of a Pythagoren triplet. (That is not obvious.) If $P_1,..., P_n$ are $n$ distinct primes, each congruent to $1$ modulo 4, then $\prod_{j=1}^nP_j$ is the largest member of $2^{n-1}$ different primitive Pythagorean triplets. ( It is not obvious that the method, below, always yields that many.) Use $(aa'+bb')^2+(ab'-ba')^2=(aa'-bb')^2+(ab'+ba')^2=(a^2+b^2)(a'^2+b'^2).$ Example : (For ease of typing let "$.$" denote "$\times$"). From $3^2+4^2=5^2$ and $5^2+12^2=13^2$ we have $$(3.5\pm 4.12)^2+(3.12\mp 4.5)^2=5^2.13^2.$$ That is, $63^2+16^2=33^2+56^2=65^2.$...... Since $17=1^2+4^2$ we have $$(63.1\pm 16.4)^2+(63.4\mp 16.1)^2=17^2.65^2=$$ $$= (33.1\pm 56.4)^2+(33.4\mp 56.1)^2=17^2.65^2.$$ That is, $$127^2+236^2=1^2+268^2=257^2+76^2=191^2+188^2=1105^2.$$ • $127^2 + 236^2 = 1^2 + 268^2 = \cdots = 17\times 65^2 \ne 1105^2$. Commented Dec 9, 2016 at 17:46 • @achillehui. Right. I should have proceeded from $17^2=15^2+8^2$, not from $17=4^2+1^2$. I'll fix it tomorrow. Commented Dec 9, 2016 at 18:27 There are $$2^{n-1}$$ primitive Pythagorean triples for every valid $$C$$-value where $$n$$ is the number of "distinct" prime factors of $$C$$. For $$C=5*13*17*29=32045 ,\quad$$ there are $$2^{4-1}=8$$ primitives $$f(131,122)=(2277,31964,32045)\\ f(142,109)=(8283,30956,32045)\\ f(157,86)=(17253,27004,32045)\\ f(163,74)=(21093,24124,32045)\\ f(166,67)=(23067,22244,32045)\\ f(173,46)=(27813,15916,32045)\\ f(178,19)=(31323,6764,32045)\\ f(179,2)=(32037,716,32045)\\$$ All Pythagorean triples take the form $$\quad A=4x^2+1\quad B=4x\quad C=4x+1$$ $$LCM\big((4x^2+1),\space 4x,\space (4x+1)\big)=100$$ For $$A=B=C=100,\quad$$ there are $$3$$ triples \begin{align*} F(26,24)\quad &\rightarrow A= 26^2 - 24^2=100 &\rightarrow (100,1248,1252)&\\ F(10,5)\quad & \rightarrow B= 2(10)(5) = 100 &\rightarrow (75,100,125)&\\ F(8,6)\quad & \rightarrow C= 8^2 + 6^2 = 100 &\rightarrow ( 28,96,100)& \end{align*}

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Instantly share code, notes, and snippets. # gmalecha/member_heq_eq.v Last active Feb 22, 2017 Solution to member_heq_eq Require Import Coq.Lists.List. Require Import ExtLib.Data.HList. Require Import ExtLib.Data.Member. Set Implicit Arguments. Set Strict Implicit. Arguments MZ {_ _ _}. Arguments MN {_ _ _ _} _. Section del_val. Context {T : Type}. Variable ku : T. Fixpoint del_member (ls : list T) (m : member ku ls) : list T := match m with | @MZ _ _ l => l | @MN _ _ ku' _ m => ku' :: del_member m end. End del_val. Section member_heq. Context {T : Type}. Fixpoint member_heq (l r : T) (ls : list T) (m : member l ls) : member r ls -> member r (del_member m) + (l = r) := match m as m in member _ ls return member r ls -> member r (del_member m) + (l = r) with | @MZ _ _ ls => fun b : member r (l :: ls) => match b in member _ (z :: ls) return l = z -> member r (del_member (@MZ _ _ ls)) + (l = r) with | MZ => @inr _ _ | MN m' => fun pf => inl m' end eq_refl | @MN _ _ l' ls' mx => fun b : member r (l' :: ls') => match b in member _ (z :: ls) return (member _ ls -> member _ (del_member mx) + (_ = r)) -> member r (del_member (@MN _ _ _ _ mx)) + (_ = r) with | MZ => fun _ => inl MZ | MN m' => fun f => match f m' with | inl m => inl (MN m) | inr pf => inr pf end end (fun x => @member_heq _ _ _ mx x) end. (* This function sets up a `match` on a value of type * `member t (t' :: ts)` that extracts a maximal amount of * information from the unification. *) Definition member_match {t t' : T} {ts} (P : forall t t' ts, member t (t' :: ts) -> Type) (Hz : P t' t' ts (MZ)) (Hn : forall m : member t ts, P t t' ts (MN m)) : forall m, P t t' ts m := fun m => match m in member _ (x :: y) return P x x y (@MZ _ x y) -> (forall m0 : member t y, P t x y (MN _)) -> P t x y m with | MZ => fun X _ => X | MN m => fun _ X => X m end Hz Hn. Definition inj_inr {T U a b} (pf : @inr T U a = inr b) : a = b := match pf in _ = X return match X with | inl _ => True | inr X => a = X end with | eq_refl => eq_refl end. Lemma member_heq_eq : forall {l l' ls} (m1 : member l ls) (m2 : member l' ls) pf, member_heq m1 m2 = inr pf -> match pf in _ = X return member X ls with | eq_refl => m1 end = m2. Proof. induction m1. { refine (@member_match _ _ _ (fun l' a ls m2 => forall (pf : a = l'), member_heq (@MZ _ a ls) m2 = inr pf -> match pf in (_ = X) return (member X (a :: ls)) with | eq_refl => MZ end = m2) _ _). { simpl. intros. refine match H in _ = X return match X with | inl _ => True | inr X => match X in (_ = X) return (member _ (l :: ls)) with | eq_refl => MZ end = MZ end with | eq_refl => eq_refl end. } { simpl. inversion 1. } } { intro. revert IHm1. revert m1. revert m2. refine (@member_match _ _ _ (fun l' l0 ls m2 => forall m1 : member l ls, (forall (m3 : member l' ls) (pf : l = l'), member_heq m1 m3 = inr pf -> match pf in (_ = X) return (member X ls) with | eq_refl => m1 end = m3) -> forall pf : l = l', member_heq (MN m1) m2 = inr pf -> match pf in (_ = X) return (member X (l0 :: ls)) with | eq_refl => MN m1 end = m2) _ _). { inversion 2. } { intros. specialize (H m pf). simpl in *. destruct (member_heq m1 m). { inversion H0. } { specialize (H (f_equal _ (inj_inr H0))). subst. reflexivity. } } } Defined. End member_heq. Arguments member_heq_eq [_ _ _ _] _ _ _ _ : clear implicits. Eval compute in member_heq_eq (@MZ _ 1 nil) MZ eq_refl eq_refl. Eval compute in member_heq_eq (@MN _ 2 1 (2 :: 3 :: nil) MZ) (MN MZ) eq_refl eq_refl. to join this conversation on GitHub. Already have an account? Sign in to comment

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Contents Previous Next Subchapters Current Chapters-> fourier kalman_bucy filter mlmode_filter arcov fspec dpss wavelet iwavelet fnbut fncheb fnbpole fncpole fn2cbp fn2clp fc2dig fn2dbp fn2dlp Parent Chapters-> Omatrix6 filtering fn2clp Search Tools-> contents reference index search Converting From A Normalized To Continuous Lowpass Filter Syntax `fn2clp(`cutoff`, `numin`, `denin`, `numout`, `denout`)` See Also fnbut , fncheb , fn2dlp Description Converts a normalized filter to a continuous lowpass filter. The real or double-precision scalar cutoff specifies the cutoff frequency for the lowpass filter. The column vector numin has the same type as cutoff and specifies the numerator polynomial for the normalized filter. The column vector denin has the same type as cutoff and specifies the denominator polynomial for the normalized filter. ``` ```The input values of numout and denout have no effect. The output value of numout is set to the column vector representing the numerator polynomial for the continuous lowpass filter. The output value of denout is set to the column vector representing the denominator polynomial for the continuous lowpass filter. The output values of numout and denout have the same type as cutoff. ``` ```The response of a continuous filter is ```              2      |num[s]|      |------|      |den[s]| ```where `num[s]` is the numerator polynomial and `den[s]` is the denominator polynomial corresponding to the filter. The response of a normalized filter is near 1 for `s` in the interval ```            __      [0, \/-1] ```and near 0 for the rest of the positive imaginary axis. The response of the continuous lowpass filter is near 1 for `s` in the interval ```            __      [0, \/-1 cutoff] ```and near 0 for the rest of the positive imaginary axis. Example ```      numin = 1.      denin = {1., sqrt(2.), 1.}      numout = novalue      denout = novalue      cutoff = 2.      fn2clp(cutoff, numin, denin, numout, denout)            xmin = 1e-1      xmax = 1e+1      ymin = 1e-4      ymax = 1e+2      fcplot(xmin, xmax, ymin, ymax, cutoff, numout, denout)       ``` returns the following plot: ``` ```

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# [FOM] origins of completeness in modal logic Dana Scott dana.scott at cs.cmu.edu Tue Mar 3 18:10:03 EST 2009 ```On Mon, 2 Mar 2009 Max Weiss <30f0fn at gmail.com> wrote: > For first-order logic, there is some intuitive notion of (classical) > validity that is sufficiently stable to allow axiomatization and even > the proof of a completeness theorem before the relevant semantical > notions were completely formalized. However, for modal logic this > seems not clearly to be the case: the question of an arbitrary modal > formula "is it valid?" tout court, seems simply ill-posed. > > But moreover, it is not just any mathematically precise method of > "interpretation" that delivers a notion of validity nor hence of > completeness. Consider, for example, the result of McKinsey and > Tarski (1944) that a formula is a theorem of S4 iff it is always > assigned the top element in all closure algebras. My impression is > that this would not be considered a completeness theorem, since > "always denotes the top element" is not an intuitively plausible > notion of validity. (For whatever reasons, the authors don't call it > a completeness theorem.) > > Roughly speaking, what I'm wondering is what sort of basis there might > be for considering, say, Kripke (1959), as opposed to such earlier > work, indeed to contain "a completeness theorem in modal logic". The > motivation is not to try to establish relationships of historical > priority, but rather to try to understand how the notion of > completeness in modal logic arose in the first place. > > Perhaps understanding the development of the relevant notion of > completeness would help to explain why Jonsson and Tarski don't, in > their (1950), draw the retrospectively obvious connections to logic > for their representation theorem. > > Any references, hypotheses, critical remarks, etc. would be much > appreciated. I think it is fair to say that Tarski & Co. were more interested in Algebraic Logic than in completeness theorems. Just as Boolean Algebra is just a reformulation of Propositional Calculus (one an equational theory, the other an assertional theory). So Closure Algebra is just an equational form of Lewis' S4 Modal Logic, and Heyting Algebra is just an equational form of Intuitionistic Propositional Calculus. Equational theories have free algebras, and in finitary cases, the free algebra on countably many generators satisfies EXACTLY the equations provable in the theory. Some, might call this a completeness theorem, but it is a very lazy one. The so-called Lindenbaum algebras of the propositional calculi mentioned give free "interpretations" (and free algebras); and saying "phi = 1" is just another way of saying "phi is provable". In the Boolean case, the two-element algebra can be considered "canonical" -- especially when we take 0 as standing for False and 1 for True. (Tarski actually liked it the other way around!) So the question comes up: If an equation holds in the two-element algebra, does it hold in the free algebra? The well-known proof uses homomorphisms to the two-element algebra. The facts that only finitary operations are used, and that a finitely generated Boolean algebra is finite, make the existence of homomorphisms easy to prove. Another, more formal proof argues that if an equation is not provable, then its adjunction to the axioms of Boolean algebra allows a proof of the equation "x = y". The other two logics are not so easy to deal with. Is there a canonical Heyting algebra or a canonical Lewis algebra? I don't think so. Thus there is no way to argue that the axioms have "completely captured" all the valid equations of a particular "God-given" structure. There may indeed be single structures that do have just the equations of the theory, but we may not know of them in advance of setting out the formal theory. The Boolean situation is very special. In High-School Algebra we are very lucky indeed: The equations in (+, x, -, 0, 1) true in the signed integers Z are exactly the same as those true in the rationals, the reals, and the complex numbers. So one formal system of algebra does for all these interpretations. (We speak here of equations universally valid, not whether certain equations have solutions.) It was not until we came to the quaternions that we had to review the rules of algebra. And finite fields need MORE equations. With the logics mentioned it did turn out that equations true in all finite Heyting algebras were generally valid, and the same for Lewis algebras. But as Godel showed, no finite model can be canonical. What to do? Perhaps there is a canonical CLASS of models short of the class of all models? Maybe. Maybe not. Tarski (and many others) showed that topological models suffice. In the intuitionistic case, using the the lattices of open subsets of topological spaces suffices; in the modal case topological closure algebras suffice. And there are even FIXED topological spaces that can be used. This is not philosophically satisfactory, however, because the the discovered models were not the primary motivation for the study of the logics in the first place. Kripke models (also discovered by Tarski & Co.) are specialized topological spaces which also suffice, and the "possible worlds" story makes them somewhat attractive philosophically. But where do possible worlds come from and what governs choices of alternative relations? Things become even less clear (in the opinion of this writer) when one moves on to predicate logic and the need to interpret structures of individuals. The topological models work somewhat better for first- and higher-order logic, and this has been studied in new detail since the discovery of Topos Theory -- especially for intuitionistic logic. In a completely different direction realizability and "provability" interpretations of intuitionistic and modal logics have more philosophical interest, but completeness theorems are quite hard to formulate and work out. I hope these opinions do not raise more questions than they DANA S. SCOTT University Professor, Emeritus Carnegie Mellon University Visiting Scholar in Logic and the Methodology of Science University of California, Berkeley ```

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24 7 payday loans # Computation From Active Rate of interest And you may Mortgage AMORTIZATION Computation From Active Rate of interest And you may Mortgage AMORTIZATION ## Dealing with Consultant This new effective rate of interest is actually determined compliment of a straightforward algorithm: roentgen = (1 + i/letter)^letter – 1. Within formula, roentgen means the energetic rate www.paydayloanservice.org/payday-loans-il of interest, i signifies new stated interest, and you will n is short for the number of compounding attacks annually. ## Joseph Ezenwa Whenever examining a loan or a good investment, it could be hard to find a clear picture of the brand new loan’s true prices or even the investment’s real yield. There are some more conditions accustomed identify the pace or produce for the that loan, along with yearly fee yield, annual percentage rate, productive speed, moderate price, plus. Of them, the energetic rate of interest could very well be probably the most of use, offering a relatively done image of the true price of credit. To estimate brand new effective interest into financing, just be sure to comprehend the loan’s said words and carry out an easy formula. The latest said interest is often the “headline” interest rate. It’s the count the bank usually promotes as the desire speed. Dictate how many compounding episodes on financing. The compounding periods will normally feel month-to-month, quarterly, annually, or consistently. That it makes reference to how often the eye are applied. Like, envision that loan which have a reported rate of interest of five % that is combined monthly. Making use of the algorithm returns: roentgen = (step 1 + .)^12 – step 1, otherwise r = 5.several per cent. An identical loan compounded daily do yield: r = (step one + .)^365 – step 1, or r = 5.13 percent. Observe that the brand new effective interest rate will still be higher than brand new said rates. Get acquainted with the brand new algorithm used in matter-of consistently compounding appeal. If the focus try compounded continuously, you will want to assess the new active rate of interest using a unique algorithm: r = e^i – step 1. Inside formula, roentgen ‘s the energetic interest rate, i ‘s the mentioned interest, and age ‘s the lingering 2.718. Including, imagine that loan which have an affordable rate of interest away from nine per cent compounded consistently. The fresh new algorithm significantly more than output: roentgen = dos.718^.09 – step 1, otherwise 9.417 percent. The fresh formulas used for amortization computation shall be type of confusing. Very, let us first start by the explaining amortization, basically, because process of reducing the property value a valuable asset or the bill regarding that loan of the a periodic count . Every time you make a fees to the that loan you have to pay certain appeal along with a part of the main. The principal is the unique loan amount, or the harmony you need to pay off. By simply making regular occasional repayments, the primary slowly decreases, of course, if they is at zero, you have entirely paid back the debt. Constantly, if you can afford financing hinges on if or not you might pay the occasional fee (aren’t a payment months). So, the very first amortization algorithm is amongst the calculation of your own commission amount per period. Example: What can the latest payment be on a 5-season, \$20,100 car loan with a moderate 7.5% yearly interest rate?.I am able to follow this new formular I normally use in financing amortization Where Good = Comparable to annual payment necessary to repay or amortise the newest loan, PVA =present worth of annuity at the K% interesting.We should instead note that since the loan is paid down into equivalent monthly payments, it’s intra-period compounding . Which ,we should instead divide the rate by twelve. A= 20,=\$ .Please, keep in mind that the clear answer they got making use of the very first formular offered is the same as the thing i had. A= 20,=\$ .Delight, note that the answer they got making use of the basic formular provided is the same as the thing i had.

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# Blog ## Is average abbreviation Ave or AVG? There are two main forms of abbreviating average and those are: avg. av.Sep 9, 2020 ## What are the 3 types of averages? There are three main types of average: mean, median and mode. ## What is the full form of Ave? Ave. is a written abbreviation for avenue. ... ## What is AVG average? Avg calculates the sum of a single value list and divides the result by the number of values in the list. This returns the average (arithmetic mean) of the listed values. Avg is often used to create subtotals and metrics based on fact data. ## What is the symbol for average? The mathematical symbol or notation for mean is 'x-bar'. This symbol appears on scientific calculators and in mathematical and statistical notations. The 'mean' or 'arithmetic mean' is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data set). ## What is the full form of MISC? misc. adjective. us. written abbreviation for miscellaneous : There was a folder labelled "misc.4 days ago ## What is the abbreviation of Dr? Dr is a written abbreviation for doctor. ... Dr John Hardy of St Mary's Medical School in London. 2. Dr is used as a written abbreviation for drive when it is part of a street name. ## What is the abbreviation of St? St is a written abbreviation for street. ... 116 Princess St. 2. St is a written abbreviation for saint. ## How do you write an average? Average This is the arithmetic mean, and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5. ## How do you get an average? Average equals the sum of a set of numbers divided by the count which is the number of the values being added. For example, say you want the average of 13, 54, 88, 27 and 104. Find the sum of the numbers: 13 + 54 + 88+ 27 + 104 = 286. There are five numbers in our data set, so divide 286 by 5 to get 57.2.Feb 22, 2021 ### What is the abbreviation for usual? • The abbreviation "usu." is the standard abbreviation for the adverb "usually". If we want to abbreviate the adjective "usual", I think "usu." would be the best choice. ### What is the correct abbreviation for average? • Of these two abbreviations, avg . is by far the more common abbreviation. The plural abbreviation of average is avgs. or avs. This abbreviation is usually found in mathematics, weather reports, and Consumer Price Indexes. You might abbreviate the word average to avg. on a math report or weather report temperatures. ### What is the abbreviation for batting average? • AVG is an abbreviation for batting average, which is simply the number of hits a batter had, divided by the number of at-bats. So a .300 hitter is someone who had three hits for every 10 at-bats. ### What is the abbreviation for typical? • How is Typical abbreviated? TYP stands for Typical. TYP is defined as Typical very frequently. ### What does IE mean? I.e. is an abbreviation for the phrase id est, which means "that is." I.e. is used to restate something said previously in order to clarify its meaning. E.g. is short for exempli gratia, which means "for example." E.g. is used before an item or list of items that serve as examples for the previous statement. ### What is the abbreviation of Ave? Ave. is a written abbreviation for avenue. ... ### What is number shortened? No. —This is the most common abbreviation of the word number. #—This is a number sign, sometimes called a hash or a pound sign.Feb 19, 2020 ### Is eg a Scrabble word? "EG" is not a Scrabble word. It is an abbreviation (e.g. = exempli gratia). But if you did have to play those letters: List of words with E, G and one more letter. ### How do you shorten Reverend? Reverend is a title used before the name or rank of an officially appointed religious leader. The abbreviation Rev. or , Revd is also used. ### What does TV stand for? TV is an abbreviation for `television'. ### How do you shorten large numbers? Depending on how large the number is, it's shortened by using a locale-specific abbreviation. In English, it's K for thousands, M for millions, B for billions, and T for trillions. For example, the value of 1500 is shown as 1.5K, and the value of 1500000000 is shown as 1.5B. ### What does Max stand for? Max. is the abbreviation for maximum. ### Does ST stand for Saint? St. is a written abbreviation for Saint. ### Do you put a period after Blvd? Locations (no ZIP code present) Spell them out and capitalize when part of a formal street name without a number: Wilbur Avenue. ... Do not abbreviate if the number is omitted: West Michigan Avenue. Do not use periods in quadrant abbreviations—NW, SE: 2333 E. Beltline Ave. ### What is the abbreviation for average daily quantity? • How is Average Daily Quantity abbreviated? ADQ stands for Average Daily Quantity. ADQ is defined as Average Daily Quantity rarely. ### What is the abbreviation for average total cost? • How is Average Total Unit Cost abbreviated? ATUC stands for Average Total Unit Cost. ATUC is defined as Average Total Unit Cost very rarely. ### What is the abbreviation for three times daily? • BID, also b.i.d. or BD, is a medical abbreviation for bis in die, meaning to give medication twice a day. TID, also t.i.d., is a medical abbreviation for ter in die, meaning to give medication three times a day.

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Full text of "Treatise On Analysis Vol-Ii" See other formats ```11 THE SPACES L1 AND L2 167 10. Suppose that the measure ju. is bounded and that /x(X) = 1. Let u be a £i-measur- able mapping of X into itself which leaves invariant the measure p, (Section 13.9, Problem 24). Then for every function/e J5f1(X, /x) the class U -f=f° u depends only on the class of/, hence defines an endomorphism of L1, also denoted by /t—> U •/, such that NI(£/ •/) = N!(/). The restriction of U to L2 is a unitary operator on L2. (a) If P is the orthogonal projection of L2 on the subspace of vectors which are invariant under £/, denned in Problem 9, show that N^/)2 <; (/| P •/) for all/e L2. (b) Show that for each/e L1 and each e > 0 there exists an integer n such that, for all integers m > 0, (1 m+n-l \ p./_i E £/*•/<*. rt * = m / (c) Deduce that, for each measurable subset A of X and each e > 0, there exists an integer n > 0 with the property that for each integer m > 0, there exists an integer k such that m^k <zm-{- n — 1 and p,(A n u~k(AJ) ^ (p,(A))2 — e ("Khintchine's statistical recurrence theorem "). (d) For each/e 3?^ and each integer n, put Show that the limit I m-l m-*oo m, fcasO (where P -/denotes a function in the class P •/) exists almost everywhere and that lim Fw dp, = 0. (Use Birkhoffs ergodic theorem.) (e) Let/e^(X, fM) be such that/(x)^ 0 almost everywhere. Show that, for almost all x e X, either f(x) = 0 or (P •/)(*) > 0. (Consider the set N of points x e X at which (P •/)(•*) is either undefined or equal to zero.) 1. Let X be a metrizable compact space and let u: X-*X be a homeomorphism. The set I of measures ^0 on X of total mass 1 and invariant under u is then a nonempty vaguely compact subset of M(X) (Section 13.4, Problem 8). A point x e X is said to be quasi-regular (relative to u) if the sequence of measures converges vaguely as n -> + °o (necessarily to a measure />tx e I). Let Q denote the set of quasi-regular points of X. A point x e Q is said to be ergodic (relative to u) if the measure p,x is ergodic (Section 13.9, Problem 13). Let E be the set of ergodic points x e Q. A point x e Q is said to be dense ifx belongs to the support of p,x. Let D be the set of dense points xeQ, The points belonging to R Ğ E n D are said to be regular (relative to p). (a) Show that the complement of R (and hence also the complement of Q, E and D) is negligible with respect to any invariant measure vel. (To show that Q has measure 1 for any measure v e I, apply BirkhofFs ergodic theorem to the functions belonging to a dense sequence in ^(X), To show that D has measure 1, consider a denumerable basis (UĞ) for the topology of X, and for each pair (m, n) such that Om c Un, a continuousfore the series with general term aHfn(x) converges absolutely ```

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cancel Showing results for Did you mean: Find everything you need to get certified on Fabric—skills challenges, live sessions, exam prep, role guidance, and more. Get started Frequent Visitor ## Compare Sales to Last Month by Exact Same Sales Days Hello Everyone and @ImkeF, I want to ask about Sales Data. Recently, i watched the youtube video about using DAX formula related to Time Intelligence (such as DATESMTD, DATEADD, PARALLELPERIOD, etc). However, no such DAX that fulfill my needs, since i need to do indexing the sales date manually. The reason i need this is to compare the sales date month-to-month within the exact same working days. So that the comparison will be precise and apple-to-apple each month. As Is SalesDate PrincipalAmt DatesMTD Indexing in Day() 01-Oct-22 18,726,838 18,726,838 1 03-Oct-22 32,035,711 50,762,549 3 04-Oct-22 33,111,669 83,874,219 4 01-Nov-22 24,011,148 24,011,148 1 02-Nov-22 25,004,098 49,015,246 2 03-Nov-22 21,573,716 70,588,962 3 As Is Summary As Is Calendar Days 01-03 November (3 Calendar Days) 01-03 November (3 Calendar Days) 3 70,588,962 50,762,549 To Be: SalesDate PrincipalAmt DatesMTD Indexing in Excel 01-Oct-22 18,726,838 18,726,838 1 03-Oct-22 32,035,711 50,762,549 2 04-Oct-22 33,111,669 83,874,219 3 01-Nov-22 24,011,148 24,011,148 1 02-Nov-22 25,004,098 49,015,246 2 03-Nov-22 21,573,716 70,588,962 3 To Be Summary To Be Sales Days 01-03 November (3 Sales Days) 01-04 October (3 Sales Days) 3 70,588,962 83,874,219 I can do this in excel (attached) by using Pivot and then i do a little bit manual by typing each row or applying Excel Formula, but i don't know how to apply it in powerBI. I believe i need the M-Code rather than DAX because i will run the script directly (using Direct Query Mode) into the Database and generate index like in excel. SalesData.xlsx Best Regards, Eddy W. 2 ACCEPTED SOLUTIONS Community Support Hi @EddyW You can add a calculated column with below DAX to have an Index column. ``Index = COUNTROWS(FILTER('Table','Table'[YearMonth]=EARLIER('Table'[YearMonth])&&'Table'[SalesDate]<=EARLIER('Table'[SalesDate])))`` Then get the expected result in a matrix visual. You can use the "DatesMTD" column from the current table, or create a MTD measure with "PrincipalAmt" and "Index". I have attached the sample file at bottom. Hope it helps. Best Regards, Community Support Team _ Jing If this post helps, please Accept it as Solution to help other members find it. Community Support Hi @EddyW Sorry I cannot think of a solution under DirectQuery mode. Power Query can create this indexing in every month group but it isn't supported under DirectQuery either. To use DirectQuery, it seems the indexing column needs to be added in the data source. Best Regards, Jing 4 REPLIES 4 Frequent Visitor Hi @v-jingzhang, Ok then, i will eitheir create the DB under the import mode or discuss with my IT team. Thank you so much for the solution! Best Regards, Eddy W. Community Support Hi @EddyW You can add a calculated column with below DAX to have an Index column. ``Index = COUNTROWS(FILTER('Table','Table'[YearMonth]=EARLIER('Table'[YearMonth])&&'Table'[SalesDate]<=EARLIER('Table'[SalesDate])))`` Then get the expected result in a matrix visual. You can use the "DatesMTD" column from the current table, or create a MTD measure with "PrincipalAmt" and "Index". I have attached the sample file at bottom. Hope it helps. Best Regards, Community Support Team _ Jing If this post helps, please Accept it as Solution to help other members find it. Frequent Visitor Hi @v-jingzhang, Thankyou for the helps, i believe it works. However, when i tried to apply it, it didn't work because i was using DirectQuery mode. Is there another way to do the indexing under DirectQuery mode instead of Import mode? Best Regards, Eddy W. Community Support Hi @EddyW Sorry I cannot think of a solution under DirectQuery mode. Power Query can create this indexing in every month group but it isn't supported under DirectQuery either. To use DirectQuery, it seems the indexing column needs to be added in the data source. Best Regards, Jing Announcements #### Europe’s largest Microsoft Fabric Community Conference Join the community in Stockholm for expert Microsoft Fabric learning including a very exciting keynote from Arun Ulag, Corporate Vice President, Azure Data. #### Power BI Monthly Update - August 2024 Check out the August 2024 Power BI update to learn about new features. #### Fabric Community Update - August 2024 Find out what's new and trending in the Fabric Community. Top Solution Authors Top Kudoed Authors

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# How to declare 2D array in bash I'm wondering how to declare a 2D array in bash and then initialize to 0. In C it looks like this: ``````int a[4][5] = {0}; `````` And how do I assign a value to an element? As in C: ``````a[2][3] = 3; `````` • Related: multi-dimensional arrays in BASH – Izzy Commented Nov 6, 2014 at 15:14 • Btw a multi-dimensional array is actually (deep down) a one dimensional array, which is handled a little bit different especially when it comes to accessing its elements. For example a 3x4 matrix has 12 cells. The "rows" you traverse with an outer loop with a step of 3 and the "columns" you traverse with an inner loop with a step of 1. Commented Jun 7, 2015 at 20:49 You can simulate them for example with hashes, but need care about the leading zeroes and many other things. The next demonstration works, but it is far from optimal solution. ``````#!/bin/bash declare -A matrix num_rows=4 num_columns=5 for ((i=1;i<=num_rows;i++)) do for ((j=1;j<=num_columns;j++)) do matrix[\$i,\$j]=\$RANDOM done done f1="%\$((\${#num_rows}+1))s" f2=" %9s" printf "\$f1" '' for ((i=1;i<=num_rows;i++)) do printf "\$f2" \$i done echo for ((j=1;j<=num_columns;j++)) do printf "\$f1" \$j for ((i=1;i<=num_rows;i++)) do printf "\$f2" \${matrix[\$i,\$j]} done echo done `````` the above example creates a 4x5 matrix with random numbers and print it transposed, with the example result `````` 1 2 3 4 1 18006 31193 16110 23297 2 26229 19869 1140 19837 3 8192 2181 25512 2318 4 3269 25516 18701 7977 5 31775 17358 4468 30345 `````` The principle is: Creating one associative array where the index is an string like `3,4`. The benefits: • it's possible to use for any-dimension arrays ;) like: `30,40,2` for 3 dimensional. • the syntax is close to "C" like arrays `\${matrix[2,3]}` • the obvious disadvantage of this method is that the length of a dimension cannot be known. Nonetheless, it works great in most other scenarios! Thanks!! Commented Dec 13, 2013 at 9:37 • Please can you explain what `f1` and `f2` do? Commented Oct 6, 2016 at 10:03 • @Jodes the `f1` and `f2` contains the `format` for the `printf` for the nice aligned printing. It could be hardcoded, for example `printf "%2s"` but using variables are more flexible - as in the above `f1`. The `width` of the row number is calculated as the length of `\$num_rows` variable - e.g. if the number of rows `\$num_rows` is 9, its length is `1` the format will be `1+1` so `%2s`. For the `\$num_rows` 2500, its length is `4` so the format will be `%5s` - and so on... Commented Oct 6, 2016 at 13:51 • Faced a big issue implementing this mehtod when it came to modifying array elements. E.g. for `matrix[1,1]=0; matrix[2,1]=0; matrix[1,1]=1;` you expect element `matrix[2,1]` keep `0` but in fact it comes `1`. In other words, attempt to change element by one column index you modify all rows on that column index which is frustrating Commented Jul 14, 2021 at 21:13 • @nakli_batman strange, because with `-a` it should not work. The lowercase `-a` declares an indexed array and for my solution is NEEDED tha associative array (which is declared by uppercase `-A`). Strange (old?)`bash` if it doesn't knows the `-A` and even more strange if the lowercase "working fine" :) :) Commented Oct 28, 2022 at 10:29 Bash doesn't have multi-dimensional array. But you can simulate a somewhat similar effect with associative arrays. The following is an example of associative array pretending to be used as multi-dimensional array: ``````declare -A arr arr[0,0]=0 arr[0,1]=1 arr[1,0]=2 arr[1,1]=3 echo "\${arr[0,0]} \${arr[0,1]}" # will print 0 1 `````` If you don't declare the array as associative (with `-A`), the above won't work. For example, if you omit the `declare -A arr` line, the `echo` will print `2 3` instead of `0 1`, because `0,0`, `1,0` and such will be taken as arithmetic expression and evaluated to `0` (the value to the right of the comma operator). Bash does not support multidimensional arrays. You can simulate it though by using indirect expansion: ``````#!/bin/bash declare -a a0=(1 2 3 4) declare -a a1=(5 6 7 8) var="a1[1]" echo \${!var} # outputs 6 `````` Assignments are also possible with this method: ``````let \$var=55 echo \${a1[1]} # outputs 55 `````` Edit 1: To read such an array from a file, with each row on a line, and values delimited by space, use this: ``````idx=0 let idx++; done </tmp/some_file `````` Edit 2: To declare and initialize `a0..a3[0..4]` to `0`, you could run: ``````for i in {0..3}; do eval "declare -a a\$i=( \$(for j in {0..4}; do echo 0; done) )" done `````` • Can you please demonstrate how to fill the above "2d array simulation" from a file-table? e.g. having a file with random number of rows and in each row containing 5 space delimited numbers. Commented May 10, 2013 at 18:29 • @kobame: I edited the answer to provide a solution for what you're asking. It will read a 2d array with a variable number of rows and variable number of columns, into a0, a1 and so on. Commented May 10, 2013 at 22:40 • How would you use another delimiter such as a comma or tab? Commented Jul 19, 2016 at 21:30 Another approach is you can represent each row as a string, i.e. mapping the 2D array into an 1D array. Then, all you need to do is unpack and repack the row's string whenever you make an edit: ``````# Init a 4x5 matrix a=("00 01 02 03 04" "10 11 12 13 14" "20 21 22 23 24" "30 31 32 33 34") aset() { row=\$1 col=\$2 value=\$3 IFS=' ' read -r -a rowdata <<< "\${a[\$row]}" rowdata[\$col]=\$value a[\$row]="\${rowdata[@]}" } aget() { row=\$1 col=\$2 IFS=' ' read -r -a rowdata <<< "\${a[\$row]}" echo \${rowdata[\$col]} } aprint() { for rowdata in "\${a[@]}"; do echo \$rowdata done } echo "Matrix before change" aprint # Outputs: a[2][3] == 23 echo "a[2][3] == \$( aget 2 3 )" echo "a[2][3] = 9999" aset 2 3 9999 # Show result echo "Matrix after change" aprint `````` Outputs: ``````Matrix before change 00 01 02 03 04 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 a[2][3] == 23 a[2][3] = 9999 Matrix after change 00 01 02 03 04 10 11 12 13 14 20 21 22 9999 24 30 31 32 33 34 `````` • and how to easily unpack a row into multiple values? Commented Mar 9, 2022 at 9:07 • @noisy `IFS=' ' read -r -a tmp <<< "\${a[\$row]}"` unpacks a row into multiple values. Commented Mar 10, 2022 at 7:21 2D array can be achieved in bash by declaring 1D array and then elements can be accessed using `(r * col_size) + c)`. Below logic delcares 1D array (`str_2d_arr`) and prints as 2D array. ``````col_size=3 str_2d_arr=() str_2d_arr+=('abc' '200' 'xyz') str_2d_arr+=('def' '300' 'ccc') str_2d_arr+=('aaa' '400' 'ddd') echo "Print 2D array" col_count=0 for elem in \${str_2d_arr[@]}; do if [ \${col_count} -eq \${col_size} ]; then echo "" col_count=0 fi echo -e "\$elem \c" ((col_count++)) done echo "" `````` Output is ``````Print 2D array abc 200 xyz def 300 ccc aaa 400 ddd `````` Below logic is very useful to get each row from the above declared 1D array `str_2d_arr`. ``````# Get nth row and update to 2nd arg get_row_n() { row=\$1 local -n a=\$2 start_idx=\$((row * col_size)) for ((i = 0; i < \${col_size}; i++)); do idx=\$((start_idx + i)) a+=(\${str_2d_arr[\${idx}]}) done } arr=() get_row_n 0 arr echo "Row 0" for e in \${arr[@]}; do echo -e "\$e \c" done echo "" `````` Output is ``````Row 0 abc 200 xyz `````` You can also approach this in a much less smarter fashion ``````q=() q+=( 1-2 ) q+=( a-b ) for set in \${q[@]}; do echo \${set%%-*} echo \${set##*-} done `````` of course a 22 line solution or indirection is probably the better way to go and why not sprinkle eval every where to . • Where does the 22 line solution use indirection? For your solution, what are you going to do when writing a script that requires i/o and a user wants to input a `-` into the 'array'. Also if you want to simulate an array probably makes more sense to `echo \${set//-/ }` instead of your two. Commented Jun 25, 2014 at 1:23 • That was my mistake i missed an or . I think that \${set//-/} is probably a better way to go ( I don`t know about the portability issues of %% and ## though I believe you ) . What if is a very dangerous question , if you ask it to many times you'll find you need A.I. for your option parser :{p Commented Jun 25, 2014 at 1:27 • I don't understand how that is relevant or helps answer the question. \${set//-/ } would eliminate the '-', merging the values together. i.e., the echo results in 'ab' whereas the original code returns either 'a' or 'b' depending on which side of the '-' you want. Commented Feb 2, 2019 at 14:31 If each row of the matrix is the same size, then you can simply use a linear array and multiplication. That is, ``````a=() for (( i=0; i<4; ++i )); do for (( j=0; j<5; ++j )); do a[i*5+j]=0 done done `````` Then your `a[2][3] = 3` becomes ``````a[2*5+3] = 3 `````` This approach might be worth turning into a set of functions, but since you can't pass arrays to or return arrays from functions, you would have to use pass-by-name and sometimes `eval`. So I tend to file multidimensional arrays under "things bash is simply Not Meant To Do". A way to simulate arrays in bash (it can be adapted for any number of dimensions of an array): ``````#!/bin/bash ## The following functions implement vectors (arrays) operations in bash: ## Definition of a vector <v>: ## v_0 - variable that stores the number of elements of the vector ## v_1..v_n, where n=v_0 - variables that store the values of the vector elements # Adds the string contained in variable \$2 in the next element position (vector length + 1) in vector \$1 local elem_value local vector_length local elem_name eval elem_value=\"\\$\$2\" eval vector_length=\\$\$1\_0 if [ -z "\$vector_length" ]; then vector_length=\$((0)) fi vector_length=\$(( vector_length + 1 )) elem_name=\$1_\$vector_length eval \$elem_name=\"\\$elem_value\" eval \$1_0=\$vector_length } # Vector Add Element Direct Value Next # Adds the string \$2 in the next element position (vector length + 1) in vector \$1 local elem_value local vector_length local elem_name eval elem_value="\$2" eval vector_length=\\$\$1\_0 if [ -z "\$vector_length" ]; then vector_length=\$((0)) fi vector_length=\$(( vector_length + 1 )) elem_name=\$1_\$vector_length eval \$elem_name=\"\\$elem_value\" eval \$1_0=\$vector_length } # Adds the string contained in the variable \$3 in the position contained in \$2 (variable or direct value) in the vector \$1 local elem_value local elem_position local vector_length local elem_name eval elem_value=\"\\$\$3\" elem_position=\$((\$2)) eval vector_length=\\$\$1\_0 if [ -z "\$vector_length" ]; then vector_length=\$((0)) fi if [ \$elem_position -ge \$vector_length ]; then vector_length=\$elem_position fi elem_name=\$1_\$elem_position eval \$elem_name=\"\\$elem_value\" if [ ! \$elem_position -eq 0 ]; then eval \$1_0=\$vector_length fi } # Adds the string \$3 in the position \$2 (variable or direct value) in the vector \$1 local elem_value local elem_position local vector_length local elem_name eval elem_value="\$3" elem_position=\$((\$2)) eval vector_length=\\$\$1\_0 if [ -z "\$vector_length" ]; then vector_length=\$((0)) fi if [ \$elem_position -ge \$vector_length ]; then vector_length=\$elem_position fi elem_name=\$1_\$elem_position eval \$elem_name=\"\\$elem_value\" if [ ! \$elem_position -eq 0 ]; then eval \$1_0=\$vector_length fi } VectorPrint () { # Vector Print # Prints all the elements names and values of the vector \$1 on separate lines local vector_length vector_length=\$((\$1_0)) if [ "\$vector_length" = "0" ]; then echo "Vector \"\$1\" is empty!" else echo "Vector \"\$1\":" for ((i=1; i<=\$vector_length; i++)); do eval echo \"[\$i]: \\\"\\$\$1\_\$i\\\"\" ###OR: eval printf \'\%s\\\n\' \"[\\$i]: \\\"\\$\$1\_\$i\\\"\" done fi } VectorDestroy () { # Vector Destroy # Empties all the elements values of the vector \$1 local vector_length vector_length=\$((\$1_0)) if [ ! "\$vector_length" = "0" ]; then for ((i=1; i<=\$vector_length; i++)); do unset \$1_\$i done unset \$1_0 fi } ################## ### MAIN START ### ################## ## Setting vector 'params' with all the parameters received by the script: for ((i=1; i<=\$#; i++)); do eval param="\\${\$i}" done # Printing the vector 'params': VectorPrint params ## Setting vector 'params2' with the elements of the vector 'params' in reversed order: if [ -n "\$params_0" ]; then for ((i=1; i<=\$params_0; i++)); do count=\$((params_0-i+1)) done fi # Printing the vector 'params2': VectorPrint params2 ## Getting the values of 'params2'`s elements and printing them: if [ -n "\$params2_0" ]; then echo "Printing the elements of the vector 'params2':" for ((i=1; i<=\$params2_0; i++)); do eval current_elem_value=\"\\$params2\_\$i\" echo "params2_\$i=\"\$current_elem_value\"" done else echo "Vector 'params2' is empty!" fi ## Creating a two dimensional array ('a'): for ((i=1; i<=10; i++)); do for ((j=1; j<=8; j++)); do value=\$(( 8 * ( i - 1 ) + j )) done done ## Manually printing the two dimensional array ('a'): echo "Printing the two-dimensional array 'a':" if [ -n "\$a_0" ]; then for ((i=1; i<=\$a_0; i++)); do eval current_vector_lenght=\\$a\_\$i\_0 if [ -n "\$current_vector_lenght" ]; then for ((j=1; j<=\$current_vector_lenght; j++)); do eval value=\"\\$a\_\$i\_\$j\" printf "\$value " done fi printf "\n" done fi ################ ### MAIN END ### ################ `````` One can simply define two functions to write (\$4 is the assigned value) and read a matrix with arbitrary name (\$1) and indexes (\$2 and \$3) exploiting eval and indirect referencing. ``````#!/bin/bash matrix_write () { eval \$1"_"\$2"_"\$3=\$4 # aux=\$1"_"\$2"_"\$3 # Alternative way # let \$aux=\$4 # --- } aux=\$1"_"\$2"_"\$3 echo \${!aux} } for ((i=1;i<10;i=i+1)); do for ((j=1;j<10;j=j+1)); do matrix_write a \$i \$j \$[\$i*10+\$j] done done for ((i=1;i<10;i=i+1)); do for ((j=1;j<10;j=j+1)); do done done `````` • Hi, do add a bit of explanation along with the code as it helps to understand your code. Code only answers are frowned upon. Commented Sep 10, 2016 at 18:17 Mark Reed suggested a very good solution for 2D arrays (matrix)! They always can be converted in a 1D array (vector). Although Bash doesn't have a native support for 2D arrays, it's not that hard to create a simple ADT around the mentioned principle. Here is a barebone example with no argument checks, etc, just to keep the solution clear: the array's size is set as two first elements in the instance (documentation for the Bash module that implements a matrix ADT, https://github.com/vorakl/bash-libs/blob/master/src.docs/content/pages/matrix.rst ) ``````#!/bin/bash matrix_init() { # matrix_init instance x y data ... declare -n self=\$1 declare -i width=\$2 height=\$3 shift 3; self=(\${width} \${height} "\$@") } matrix_get() { # matrix_get instance x y declare -n self=\$1 declare -i x=\$2 y=\$3 declare -i width=\${self[0]} height=\${self[1]} echo "\${self[2+y*width+x]}" } matrix_set() { # matrix_set instance x y data declare -n self=\$1 declare -i x=\$2 y=\$3 declare data="\$4" declare -i width=\${self[0]} height=\${self[1]} self[2+y*width+x]="\${data}" } matrix_destroy() { # matrix_destroy instance declare -n self=\$1 unset self } # my_matrix[3][2]=( (one, two, three), ("1 1" "2 2" "3 3") ) matrix_init my_matrix \ 3 2 \ one two three \ "1 1" "2 2" "3 3" # print my_matrix[2][0] matrix_get my_matrix 2 0 # print my_matrix[1][1] matrix_get my_matrix 1 1 # my_matrix[1][1]="4 4 4" matrix_set my_matrix 1 1 "4 4 4" # print my_matrix[1][1] matrix_get my_matrix 1 1 # remove my_matrix matrix_destroy my_matrix `````` For simulating a 2-dimensional array, I first load the first n-elements (the elements of the first column) ``````local pano_array=() i=0 for line in \$(grep "filename" "\$file") do url=\$(extract_url_from_xml \$line) pano_array[i]="\$url" i=\$((i+1)) done `````` To add the second column, I define the size of the first column and calculate the values in an offset variable ``````array_len="\${#pano_array[@]}" i=0 while [[ \$i -lt \$array_len ]] do url="\${pano_array[\$i]}" offset=\$((\$array_len+i)) found_file=\$(get_file \$url) pano_array[\$offset]=\$found_file i=\$((i+1)) done `````` The below code will definitely work provided if you are working on a Mac you have bash version 4. Not only can you declare 0 but this is more of a universal approach to dynamically accepting values. # 2D Array ``````declare -A arr echo "Enter the row" echo "Enter the column" i=0 j=0 echo "Enter the elements" while [ \$i -lt \$r ] do j=0 while [ \$j -lt \$c ] do echo \$i \$j arr[\${i},\${j}]=\$m j=`expr \$j + 1` done i=`expr \$i + 1` done i=0 j=0 while [ \$i -lt \$r ] do j=0 while [ \$j -lt \$c ] do echo -n \${arr[\${i},\${j}]} " " j=`expr \$j + 1` done echo "" i=`expr \$i + 1` done ``````

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# Geometry Math Worksheets Magnolia Amaya July 1, 2018 worksheets Tip #2 – Neat & Professional. Because there are so many sources of worksheets on the internet, you’re never sure what you’re going to find at a particular site. Choose worksheets that are neat and organized without too many problems jumbled on to each page. The concept of neatness needs to be taught to your child as they do math. If your child doesn’t learn this, be prepared for many careless mistakes later on in his math work. Boys in particular don’t take the time to be neat and careful. If you give him multiplication worksheets that are crowded on to each page without room to write the answers, this in encouraging messiness. Crowded problems also confuse kids. When a child is first learning a new concept in math and they lack confidence, being faced with an overcrowded worksheet can cause instant panic. Avoid this with neat and professional worksheets. Another one of the many ways that you can go about saving money with preschool worksheets is by using photo protectors. These are pages that you can slip photographs into to protect them. It is possible to find photo protectors that are designed for full pages, like preschool worksheets. You can simply slip each preschool worksheet into a page protector and give your child a dry erase marker. Each time that they are finished, you can wipe off the marker and the worksheet is good to go, again and again! The above mentioned methods are just a few of the many ways that you can go about saving money with preschool worksheets for your child. Of course, these steps are optional, but they can help you save money, as well as prolong the life of your child’s preschool worksheets. Rather than using worksheets, a better method is to use individual size white boards and have the child writing entire facts many times. Having a child writing 9 x 7 = 7 x 9 = 63 while saying ”nine times seven is the same as seven times nine and is equal to sixty-three” is many times more successful than a worksheet with 9 x 7 = ___ and the student just thinks the answer once and then writes that answer on the duplicate problems. I will admit that there is one type of worksheet that I used in the past and found relatively beneficial, although it had a different kind of flaw. For my Basic Math, Pre-Algebra, and Algebra classes, I had several books of ”self-checking” worksheets. These worksheets had puns or puzzle questions at the top, and as the students worked the problems they were given some kind of code for choosing a letter to match that answer. If they worked the problems correctly, the letters eventually answered the pun or riddle. Students enjoyed these worksheets, but there are a couple problem areas even with these worksheets. Some students would get the answer to the riddle early and then work backward from letter to problem answer, so they weren’t learning or practicing anything. Homeschool worksheets have pros and cons that depend on the type of material the worksheet deals with. One advantage is that worksheets are very handy if you want to give your child something to do. Some types of worksheets are very easy to grade and can be completed without much input from you. Worksheets can also give you a good idea of how much your child was able to understand of the subject matter. While worksheets for homeschool can assist in home schooling, they cannot take the place of a proper homeschool curriculum. One disadvantage they have is that they often focus on one subject area only, without integrating the whole curriculum. They can also be simplistic and give the impression that the student understands more than he actually does. Free homeschool worksheets that you can print are available online. Some of them may be excellent, but you will have to make sure that they are accurate and suitable for your curriculum or child. Planning Worksheets for Kids. Before creating the worksheet for children, it is important to understand why the worksheet is being made. Is there a message to be conveyed? Can students record information that can be understood later? Is it being created to just teach a basic concept to little children? A well designed worksheet will make its objective clear. The different aspects that should influence the design of the worksheet are the age, ability and motivation of the students. A young child may not be able to write or read more than a few words. Worksheets should be created keeping these factors in mind. When you buy worksheets for your children, look for how the concept is explained. Is it pictorial or is it just a collection of words? A pictorial worksheet will hold the attention of a child more than just a combination of words. Another thing to look out for is what the pupil will need to solve the worksheets? Does the worksheet require the use of crayons? Does it require other things like a pair of scissors, glue and so on? Before you buy worksheets, make sure to check if they have been created to suit the geographical location that you reside in. The language and usage of words differs from country to country. It is no point buying a worksheet which is designed for children in the US for children residing in India. Also see if the worksheets involve just one way of teaching or multiple ways. Do the worksheets involve short assessments? Does it have some activity built in; does it involve elements from the child’s surroundings? Feb 22, 2021 Feb 22, 2021 Feb 20, 2021 Feb 22, 2021 Feb 20, 2021 Feb 22, 2021 Feb 22, 2021 Feb 22, 2021 ### Photos of Geometry Math Worksheets Rate This Geometry Math Worksheets Reviews are public and editable. Past edits are visible to the developer and users unless you delete your review altogether. Most helpful reviews have 100 words or more Feb 22, 2021 Feb 22, 2021 Feb 22, 2021 Feb 20, 2021 Static Pages Categories Most Popular Feb 22, 2021 Feb 20, 2021 Feb 22, 2021 Feb 22, 2021 Feb 22, 2021 Latest Review Feb 20, 2021 Feb 22, 2021 Feb 22, 2021 Latest News Feb 20, 2021 Feb 22, 2021 Feb 22, 2021

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Upcoming database upgrades. Airtable functionality will be reduced for ~15 minutes at 06:00 UTC on Feb. 4 / 10:00 pm PT on Feb. 3. Learn more here # My multiplication keeps giving an error Topic Labels: Formulas Solved 493 4 cancel Showing results for Did you mean: 4 - Data Explorer Hi, I am trying to figure out why I keep getting this error message: The Impact Value is the average of the Confidentiality, Integrity and Availability column. (Confidentiality+Integrity+Availability)/3 The Risk Value should be multiplying the Impact Value with the Likelihood Value. But I am getting an error message with this basic formula {Impact Value}*{Likelihood Value} 1 Solution Accepted Solutions 9 - Sun Hi and welcome to the Airtable community :slightly_smiling_face: The output of a single select is not a number but a string. So you have to convert that string to a number first which you do with ` Value({Likelihood Value})`. Cheers, Rupert 4 Replies 4 9 - Sun Hi and welcome to the Airtable community :slightly_smiling_face: The output of a single select is not a number but a string. So you have to convert that string to a number first which you do with ` Value({Likelihood Value})`. Cheers, Rupert 4 - Data Explorer Amazing! This worked. Thank you so much! 5 - Automation Enthusiast That helped me, too. I’m trying to average scores from a survey, but my survey has 19 questions. Worse yet, my fields are more like “My position is a good fit for someone with my knowledge, skills, and personality.” Do I need to apply the VALUE function to each field (question)? In the example above, would I need to apply VALUE({Confidentiality}) and VALUE({Integrity}) and VALUE ({Availability}) and so forth or is there a better way? Is there a Value version of Single Select? I found the number field type, which might do the trick. Is there a way to limit number responses to within an acceptable range? 18 - Pluto There is not a way to limit a number field to a range in a form. Is your maximum number is 10 or less, you could use a rating field, which can have a maximum value of 10 or less. If you want to keep your answer fields as single select values with text values, you can use a `SWITCH` formula to convert the text to a number. ``````SWITCH( {singe select field name}, "High", 5, "Medium", 3, "Low", 1 ) ``````

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Two irrational functions that seem equivalent to me have different positivity I was analysing the following function: $f(x)=\sqrt{\dfrac{x^2-4}{x^2-1}}$ And when I had to differentiate I found the first derivative as: $f'(x)=\dfrac{3x}{\left(x^2-1\right)\sqrt{\left(x^2-4\right)\left(x^2-1\right)}}$ I studied its positivity and I found it was positive for -1 < x < 0 or x > 2. This went in contrast with what the function should be like because the original is increasing for 0 < x < 1 or x > 2. I asked my friend for a hand and he calculated the derivative as: $f'(x) = \dfrac{3x}{\left(x^2-1\right)^2\sqrt{\frac{x^2-4}{x^2-1}}}$ Which to me seems equivalent because you could take $(x^2-1)$ and bring it inside the square root. By simplifying I get my own derivative, but the positivity of this function is the correct one (positive for 0 < x < 1 or x > 2) while mine is wrong. Both derivatives have the same domain of f(x) if I'm not mistaken. Are these two functions not equivalent? Or perhaps I'm missing a facepalm-worthy basic algebraic rule? No, they are not equivalent. This a simpler example of what is happening: if $x<0$ then $$x\sqrt {\strut y\,x} \text{ is not equivalent to } x^2\sqrt{\strut y/x}.$$ • But would $$x\sqrt x \text = x^2\sqrt{x/x^2}.$$ ? That's the actual question that I'm asking. Also if the problem would be in the domain, I actually find that both my derivatives have the same domain. – user3624242 Mar 20 '17 at 17:24 • Not exactly. What you have is $x\,\sqrt {y\,x}$ and $x^2\sqrt{y/x}$. $y\,x$ and $x/y$ have the same sign, but $x$ and $x^2$ have different sign if $x<0$. – Julián Aguirre Mar 20 '17 at 18:27 • Yes, exactly! That's the point I'm trying to understand: which algebraic rules did I break while doing that? Because to me it seems correct, but I see that the result is wrong. Even while differentiating I didn't notice any evident rule that would block me from getting to the result I had before. Perhaps later I can send all the calculations I did for the derivative. – user3624242 Mar 20 '17 at 19:47 $$x\sqrt{x} = \sqrt{x^2 x} = \sqrt{\frac{x^4}{x^2} x} = x^2 \sqrt{\frac{x}{x^2}}$$ EDIT: Note that this only works for $x \geq 0$, as pointed out in the comment below. • But your first equality holds only when $x\ge0$. It’s not true that $\sqrt{x^2}=x$. Rather it’s the case that $\sqrt{x^2}=|x|$. – Lubin Mar 20 '17 at 20:17

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{[ promptMessage ]} Bookmark it {[ promptMessage ]} # ch22_p40 - 40(a The initial direction of motion is taken to... This preview shows page 1. Sign up to view the full content. 40. (a) The initial direction of motion is taken to be the + x direction (this is also the direction of G E ). We use vv a x fi 22 2 −=∆ with v f = 0 and G G G aFm e Em e == to solve for distance x : x v a mv eE ie i = = = −× × × 31 19 2 911 10 2 160 10 712 10 . . . kg 5.00 10 m s C 1.00 10 N C m. 6 2 3 c hc h c h (b) Eq. 2-17 leads to t x v x v i = × × ∆∆ avg m ms s. 2 2 712 10 500 10 285 10 2 6 8 . . . c h (c) Using v 2 = 2 a x with the new value of x , we find This is the end of the preview. Sign up to access the rest of the document. {[ snackBarMessage ]} Ask a homework question - tutors are online

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GIL .. BIOSCI/Bionet News .. Biosequences .. Software .. FTP # PDB coordinate and psi, phi, omega angles Dr. Mrigank mrigank at imtech.ernet.in Tue Sep 28 09:45:59 EST 1993 ```In article <9309211522.AA04663 at conch.senod.uwf.edu>, hhuang at CONCH.SENOD.UWF.EDU (Hong Huang) writes: > Dear Netters: > > Could anybody be so kind to show my how to display the 3D protein structures > which are in PDB's orthogonal coordinate system. My problems are: > > (1) How to map 3D coordinates to our 2D screen, is it done by Ray Tracing? > (2) What are psi, phi and omega angles? > > I know there are many excellent packages out there but the point are, I have to > do it myself and I do not have biology background. > Hi Ans: (1) It is done by taking the xy projection of the system. For depth effect depth cueing , Elimination of hidden line - Z buffering etc. are done. Read any std. Text book on computer graphics it will have the details. If you can't find one in lib. I can send the details[ only brief]. Ans: (2) Phi psi and omega are 3 torsion angles in peptide bond, which can define entire 3-D structure of backbone. See below: +-------------------+ | | R | H O | H | | | || | | N----Ca----C=-|=-N-----Ca-----C-=-|=-N-- || ^ | ^ | ^ ^| O | | | R | || | | | | || Peptide | | || Bond| Phi Psi Omega | | +-------------------+ Here Phi and Psi are flexible and can rotate freely (In reality no so freely) and omega has partial double bond so is fixed around 180 degrees. Refer to any text on protein structure to get the details. Or again if you want I can post the details to you. -- Dr. Mrigank \/Phone +91 172 45004 x216 Institute of Microbial Technology /\Email: mrigank at imtech.ernet.in P O Box 1304, Sector 39A \/UUCP:...!uunet!sangam!vikram!imtech!mrigank Chandigarh 160 014 India. /\ ==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+==+ -- When I feed the poor, they call me saint. When I ask why the poors do not have food, they call me communist - Archbishop Camaran ```

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New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Measurement Conversion Measurement Converter (Help) Convert From: (required) Click here to Convert To: (optional) Examples: 5 kilometers, 12 feet/sec^2, 1/5 gallon, 9.5 Joules, or 0 dF. Help, Frequently Asked Questions, Use Currencies in Conversions, Measurements & Currencies Recognized Examples: miles, meters/s^2, liters, kilowatt*hours, or dC. Conversion Result: ```Canadian gallon = 0.00454609 volume (volume) ``` Related Measurements: Try converting from "Canadian gallon" to acre foot, barrel, cord foot (of wood), cup, dram fluid (fluid dram), drop, dry pint, gill, jeroboam, kilderkin, last, noggin, oil arroba (Spanish oil arroba), rehoboam, salmanazar, teaspoon, tou (Chinese tou), tun (English tun), UK bushel (British bushel), UK quart (British quart), or any combination of units which equate to "length cubed" and represent capacity, section modulus, static moment of area, or volume. Sample Conversions: Canadian gallon = .00000369 acre foot, .11659708 amphora (Greek amphora), 123.3 bath (Israeli bath), 4,546.09 cc (cubic centimeters), .11467297 ephah (Israeli ephah), 6 fifth, .13343888 firkin, 38.43 gill, .95333138 hekat (Israeli hekat), 102.48 jigger, .06671944 kilderkin, .02523004 koku (Japanese koku), .60047496 methuselah, .24018999 nebuchadnezzar, 9.61 pint (fluid pint), .17557747 Roman amphora, .04300236 sack, 307.44 tablespoon, 922.33 teaspoon, .27929068 wine arroba (Spanish wine arroba). Feedback, suggestions, or additional measurement definitions? Please read our Help Page and FAQ Page then post a message or send e-mail. Thanks!

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At the behest of a friend of mine, Alison;  whom desperately wishes to learn how to read a map, and use a compass, I will begin a few posts regarding Navigation. I want to focus first on reading a topographic map; mostly because you can navigate a lot easier with just a map, than you can with just a compass. (GPS is just plain cheating!) ## Map Features The first order of business is to be able to identify the features on a map, once you can identify the features, and what they mean, a map begins to become useful. Contour lines, river markings and other similar features allow you to plan a trip that will allow you to cover distance easier, make sure you achieve the highest possible points in an area, avoid these heights or make sure you detour via water sources. Knowing what surrounds you, allows you to get the most out of a trip. ### Contour Lines Contour lines are always subject to the legend of the map, but the concepts stay the same, for every line, there is an associated ascent. ergo, should you have a lot of lines close together, this means a steep hill, versus long distances between lines which means a slower ascent. In the example below the walker who takes the blue line, has a steeper initial walk toward the peak, before mounting a very steep face where there is an almost 20M climb, then travelling along the razorback it is quite flat, less than 20M up or down before walking to the top of the peak at 360. A walker on the red line, would have a very flat initial walk, then gradually stepper before a consistent angle for the latter part of the walk, getting slowly steeper. Every time either walker travels over one of those lines, they have covered 20M of vertical distance between that line and the last. ### Scale Scale too is important, what is depicted above could be very different walk depending on the scale of the map, it its a 1:30,000 map, one centimeter on the map is equivalent to 30,000cm, 300M. If the scale was 1:50,000 the distance becomes 500M per centimeter on the map. the decisions that are made by the hike planner could be very different. on a 1:30,000 map, the Blue hikers walk would be very difficult, if not impossible, but on a 1:50,000 it would much more likely be possible. ### Lat/Long Latitude and Longitude lines, (not depicted on the map above) allow for co-ordinates to be designated to all locations on earth on a unique basis. Latitude lines run east to west, while longitude run north to south. In lat/long coordinates; the lattitude comes first followed by the longitude. Latitude is measured from the equator with a directional symbol, the Equator is 0 degrees. Latitude is measured east and west with 0 degrees at Greenwich in England (the same place that GMT or Greenwich mean time starts and counts to 180 degrees). After the first measurements, both are broken into minutes and seconds in 60 unit intervals to increase accuracy even further. For example, Melbourne, Australia is 37 degrees 47 seconds south, 144 degrees 58 seconds east Latitude and longitude can also be depicted as decimal, most GPS in todays market can give you both types of co-ordinates. Most maps will allow you to get your accuracy down to accurate minutes, with a ‘guestamation’ on seconds. A latitude and Longitude grid over Cathedral ranges; with measurements down to seconds ### Orientation Orienting a map is quite useful and actually quite straight forward. It can be done using a compass if you have one, you simply find the magnetic north line, orient your compass to north and hurrah you have a correctly aligned map. Without the compass however, using large and distinctive features such as peaks or rivers can help you align the map; i usually try to triangulate my position using three features roughly the same distance apart(in a 360 degrees sense) , once aligned, you can use the map to figure out where you are and where you plan on going. ### Magnetic Declination Magnetic declination is the difference between magnetic north and true north. I will cover more in compass navigation as it doesnt affect reading a map; however it is very important when using a compass to align a map as it give us the difference between magnetic north and true north. Suffice to say for now, that due to the way the magnetic fields of earth work, a compass doesnt always point due north . With notable exceptions of course, such as heavy iron or other ferrus metal deposits in an area which can seriously affect the reliability of a compass. ### Map Number & Print date Relevant only in the way that an expiry date is relevent on food, very. If the print date is a long time ago, while the general landscape may not have changed, paths and river locations as well as drinkable water sources may have varied significantly. Planning a hike with an older map is fine, and using the map on a walk is fine too, just be aware that geo-spacial data is subject to change and because of shifting magnetic fields around earth magnetic declination in an area can change significantly in as little as 10 years. When taking survival and navigation into consideration \$10 it takes to buy a new map might well be worth the effort. ### Trail info / suggesting trails Many maps come with several suggested trails, some even going to the level of detail that shows elevation profiles, water stops and all the scenic excitement you can find. They can also be very helpful for noting logging truck trails, trails shared with motorbikes and other vehicles. ### Distinct features Why do we go walking? Most people go to see something, or conquer a mountain, either way, these kind of distinct features are readily depicted on most maps.

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Re: [eigen] How to raise double to array powers in Eigen • To: eigen@xxxxxxxxxxxxxxxxxxx • Subject: Re: [eigen] How to raise double to array powers in Eigen • From: Ian Bell <ian.h.bell@xxxxxxxxx> • Date: Mon, 18 Aug 2014 19:30:54 +0200 • Dkim-signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20120113; h=mime-version:in-reply-to:references:from:date:message-id:subject:to :content-type; bh=0bxtkrSXE4c4HVTR5cQyp7WzdGnVLPgjxnePuzDTuL0=; b=lz0d2skAgX86IxYSHfZrC/6zifKuP0aY4JF7mkGC7ebJQvRpcsd8vJE4BVbXdbMlT6 VimoVJgG6TCYx6vQmIDwgKxPGfjTgMSiUQLLexYVdxbKUbCVMjQGtk4ERaqG9ynebub6 9Hduq41+Kjy/vIDT4hIOzy8kU5CA5GOjjdFJghZDq5mIGCZb96jW6HZ9TRaqE07uLjC7 i/rtZU4R3S37hTNbtdA9lt0JZ97sxMb8IwHBbCss/iOXRpE9WEZkV4wEsDPHtsHWOeJX thRIrcBPUY/AkDhu37C5zLZwN6h30JEaf+e0TpPin8L2JXRllb2uTvjOTCBehh7sEEvb 8YBg== Chen-Pang, Can you provide a bit more detail on the exponentiation by squaring?  How would I re-write my simple c++ loop above using your method? Thanks, Ian On Mon, Aug 18, 2014 at 2:30 PM, Chen-Pang He wrote: If you need no optimization and want functional programming, ArrayBase::unaryExpr with a homemade functor is enough. If you need optimization, because all exponents are integers, binary powering (aka exponentiation by squaring) is *the* algorithm.   We can reuse `pow(delta, (1 << k))`. Cheers, Chen-Pang > On Mon, Aug 18, 2014 at 1:28 PM, Christoph Hertzberg < > chtz@xxxxxxxxxxxxxxxxxxxxxxxx> wrote: > > > On 18.08.2014 00:23, Ian Bell wrote: > > > >> cross-posted to stack overflow... > >> > > > > If you cross-post, could you post a link as well? > > http://stackoverflow.com/questions/25354205/how-to-raise-double-to-array-powers-in-eigen > > > > > > >  I need to raise a double value to an array of integer powers, in c++ this > >> would look something like > >> > > > > Could you be a bit more specific about the "array of integer powers" (I > > guess 'exponents')? > > If they are a sequence 0,1,2,3,... your proposed solution is definitely > > very inefficient. If they are of great range, with no apparent pattern, I > > would say the exp(log(delta)*exponent) approach is basically the most > > efficient solution (maybe using log2 and exp2 would be more efficient, if > > they were available). > > > > The exponents are typically in the range [0, 10), and can be sorted if that > is of use.  They are not necessarily linearly increasing, we might have > exponents 0,0,0,1,1,1,1,2,2 for instance.  Obviously it would be better to > short-circuit the 0 powers as well since they yield 1 again (x^0=1).  This > method with the exp(log(delta)*exponent) also doesn't allow you to skip > those powers.  I think that the >0 comparison should be cheaper than doing > pow(x, 0) when 0 is integer.  Especially this will be true when using exp(). -- Mail converted by MHonArc 2.6.19+ http://listengine.tuxfamily.org/

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Re: Rolling up some integers • To: mathgroup at smc.vnet.net • Subject: [mg33455] Re: [mg33404] Rolling up some integers • From: Ken Levasseur <Kenneth_Levasseur at uml.edu> • Date: Thu, 21 Mar 2002 09:28:57 -0500 (EST) • References: <200203200653.BAA07992@smc.vnet.net> • Sender: owner-wri-mathgroup at wolfram.com ```Mark: Here is a non-snaking solution, but it looks sort of like what you describe. In[1]:= data[n_] := Range[n^2] In[2]:= order[n_] := Sort[Flatten[Outer[List, Range[n], Range[n]], 1], Plus @@ #1 <= Plus @@ #2 & ]; In[3]:= Transpose[{order[4],data[4]}]//Sort[#,(First[First[#1]]?First[First[#2]])&]&//Transpose//Last//Partition[#,4]& Out[3]= {{1, 2, 4, 7}, {3, 5, 8, 11}, {6, 9, 12, 14}, {10, 13, 15, 16}} Ken Levasseur UMass Lowell "DIAMOND Mark R." wrote: > Can anyone see an easy way of putting a fiven list of n^2 integers into an > n*n matrix along the diagonals---could be either not-snaking (preferred) > snaking. The *ordering*, but not necessarily the integers themselves, would > be > > 1 3 6 > 2 5 8 > 4 7 9 > > I would call this non-snaking, or the following, snaking. > > 1 2 6 > 3 5 7 > 4 8 9 > > > It occurred to me that this might be possible by first partitioning into > ascending and descending size lists representing each diagonal, but aside > from constructing an explicit secondary matrix of indices, I can't see > another way of doing it. > > Cheers, > > Mark > -- > Mark R. Diamond > No spam email ROT13: znexq at cfl.hjn.rqh.nh > No crawler web page ROT13 uggc://jjj.cfl.hjn.rqh.nh/hfre/znexq ``` • Prev by Date: Re: Export Directory • Next by Date: Re: newbie question - printing Pi • Previous by thread: Rolling up some integers • Next by thread: Re: Rolling up some integers

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All India Management Association Management Trainee model question papers - 2018 2019 Student Forum 2018 2019 Student Forum All India Management Association Management Trainee model question papers #1 25th July 2014, 01:54 PM Unregistered Guest Posts: n/a All India Management Association Management Trainee model question papers Will you please share with me the All India Management Association Management Trainee model question papers? #2 25th July 2014, 02:11 PM Super Moderator Join Date: Apr 2013 Posts: 23,085 Re: All India Management Association Management Trainee model question papers As you want to get the All India Management Association Management Trainee model question papers so here it is for you: Q1 There are 4 boys and 3 girls. What is the probability the boys and girls sit alternately? Ans 1/35 Q2 Two trains are 2 kms apart. Speed of one train is 20m/s and the other train is running at 30 m/s . Lengths of the trains are 200 and 300m. In how much time do the trains cross each other? Ans 50 seconds Q3 A& B are two players. They need to select one number from 1 to 25. If both the players select the same numbers they will win the prize. What is the probability of not winning in a single trial? Ans 24/25 Q4 Four different integers are in increasing AP such that one number is equal to sum of the squares of the other three numbers. Ans –1,0,1,2 Q5 A train runs first half of the distance at 40 km/hr and the remaining half at 60 km/hr. What is the average speed for the entire journey? Ans 48km/hr Q6 In a company there were 75 % skilled employees and the remaining unskilled. 80% of the skilled and 20% of the unskilled were permanent. If the temporary employees were 126 find the total number of employees. Ans 360 Q7 A person sells a horse at 12.5% loss. If he sells for 92.5 more, he will have a profit of 6%. What is the CP? Ans 500 Q8 One tap takes 15 min to fill, another tap takes 12 min to fill and the third tap can empty in 20 min. In how much time the tank would be full. Ans 10 min Q9 Three mixtures of water and milk are in the ratio 6:7, 5:9, 8:7. What would be the ratio if they were mixed together? Ans: None of the options were correct. The right answer would be data insufficient. Q10. Two trains are separated by 200km. One leaves at 6:00 am from Delhi and reaches Merrut at 10:00 am. Another train leaves from Merrut at 8:00 am and reaches Delhi at 11:30 am. At what time two trains meet each other? Ans 8:56 am Q11 If the height of the triangle decreases by 40% and the breadth increases by 40%. Then the effect on the area is Ans 16% decrease. Q12 In how many ways can you go from one corner to the diagonally opposite corner if the movement is through the shortest route.. Ans: 6 Q13 There are only two candidates. 10% of the voter did not vote. 60 votes were invalid. If elected got 308 votes more than the opponent. The elected person got 47% of the total votes. How many Ans: Option 4 ( only one option is having a difference of 308 votes.) Q14. If area of circle is equal to area of triangle is equal to area of square then which is having the larger perimeter. Ans: Triangle. Q15. If log1227= a then find the value of log616 Ans. (4(3-a)) / (3+a) Q16 A grandfather has 5 sons and daughters and 8 grandchildren.. They have to be arranged in arrow such that the first 4 seats and last four seats are to be taken by grandchildren and the grandfather would not sit adjacent to any of the grandchildren. Ans. 4 * 8! * 5! Q17 A parellopiped of length = 5cm, breadth = 3cm and height 4 cm. Cube of side 4 cm, cylinder of radius 3 cm and length = 3cm and sphere of radius 3cm. Arrange them in descending order of volume Ans: Sphere>Cylinder>Cube>Parellopiped. Q18 Which of the following statements is/are correct. Ans: The correct Statements were (1) 99/101 > 97/99 > 95/97 , (2) 992/1012 > 972/992 > 952/972 Q19. A farmer has a rectangular plot. He wants to do fencing along one of the side with the help of the posts. Two posts being on two corners. He brings 5 post less than what he has initially plan because of which the distance between two consecutive post became 8 m instead of 6 m.. What is the length of the side and no of post? Ans. 120 , 16 Q20: A circular ground of circumference of 88m. A strip of land 3m wide inside and along the circumference is to be leveled. What is the expenditure if leveling cost is Rs. 7 per metre square. Ans. 1650 Q21: Four horses are tethered at the four corners of a square of side 14cm such that two horses along the same side can just reach each other. They were able to graze the area in 11 days. How many days will they take in order to graze the left out area? Ans. 3 Q22: Two clocks are synchronized at 12 midnights. One clock gains 2 min per hour and the other clock loses 1min per hour. How many minutes apart its minute hand will be at 11.am Ans. None of these. Sample Questions and Answers in GK Section: Ques1. Bull and bear refer to Ans: Stock exchange Ques2. Bhakra Nangal dam was built on river Ans. Sutlej Ques3. Tetanus is caused by Ans. Bacteria Ques4. De beers manufacture Ans: Diamonds Ques5. Commercial vehicles are not manufactured by which of the following Ans: Birla Yamaha Ques6. Wings of fire autobiography by Ans: President A.P.J Abdul Kalam. Ques7. Which is not a courier company. Ans: Essar Ques8. Software and Services Company. Ans: NASSCOM Ques9. Panchayti Raj started from which state in India. Ans: Rajastan Ques10. Duration of First five yr plan Ans 1950-1951 to 1955-1956 Ques11.Which one of following is Banker’s Bank Ans: Reserve Bank of India Ques12.Economic growth is indicated by Ans: GDP Ques13. All India service is created by Ans: Parliament. Ques14. Procedure for removal of which two are same out of the following… Ans. CEC and Judge of the supreme court. Ques15. Arrange PMs of India in chronological order…… Ans: Morarjee Desai, Charan Singh, VP Singh, Chanderashekhar. Ques16. Volkswagon is from which country? Ans Germany Ques17. First woman Prime minister in the world Ans: Seremavo Bandarnaike Ques18. Which is associated with football Ans: Merdeka Ques 19. Venue of XXI winter Olympics 2010 Ques 20. Crossed cheque can be Encashed ….. Ans: Through bank Ques21. Standard charterd bank merged with…… Ans: ANZ Grindlay’s bank Ques 22. Which disease is mentioned along with national policy with AIDS Ans. Tuberculosis Ques 23. National income in India compilation is done by… Ans: CSO Ques 24. Rajdhani Express runs between Ans: Between Delhi and few state capitals Ques 25. Maple leaf represents which one of the foolowing countr Ques 26. PAN is required for…… Ans: D-Mat account Ques27. Which one of following is not matched correctly…. #3 26th November 2015, 02:33 PM Unregistered Guest Posts: n/a Re: All India Management Association Management Trainee model question papers I am going to appear in All India Management Association Management Trainee exam. So I need a sample paper. So here can you provide me a sample paper of this exam? #4 26th November 2015, 02:33 PM Super Moderator Join Date: Apr 2013 Posts: 36,164 Re: All India Management Association Management Trainee model question papers I have a sample paper of All India Management Association Management Trainee exam. So here I am providing you as you want. Sample Questions and Answers in GK Section: Ques1. Bull and bear refer to Ans: Stock exchange Ques2. Bhakra Nangal dam was built on river Ans. Sutlej Q3 A& B are two players. They need to select one number from 1 to 25. If both the players select the same numbers they will win the prize. What is the probability of not winning in a single trial? Ans 24/25 Q4 Four different integers are in increasing AP such that one number is equal to sum of the squares of the other three numbers. Ans –1,0,1,2 Q5 A train runs first half of the distance at 40 km/hr and the remaining half at 60 km/hr. What is the average speed for the entire journey? Ans 48km/hr Q6 In a company there were 75 % skilled employees and the remaining unskilled. 80% of the skilled and 20% of the unskilled were permanent. If the temporary employees were 126 find the total number of employees. Ans 360 Ques13. All India service is created by Ans: Parliament. Ques14. Procedure for removal of which two are same out of the following… Ans. CEC and Judge of the supreme court. Ques15. Arrange PMs of India in chronological order…… Ans: Morarjee Desai, Charan Singh, VP Singh, Chanderashekhar. Ques16. Volkswagon is from which country? Ans Germany Ques10. Duration of First five yr plan Ans 1950-1951 to 1955-1956 Ques11.Which one of following is Banker’s Bank Ans: Reserve Bank of India Ques12.Economic growth is indicated by Ans: GDP Ques17. First woman Prime minister in the world Ans: Seremavo Bandarnaike Q18 Which of the following statements is/are correct. Ans: The correct Statements were (1) 99/101 > 97/99 > 95/97 , (2) 992/1012 > 972/992 > 952/972 Q19. A farmer has a rectangular plot. He wants to do fencing along one of the side with the help of the posts. Two posts being on two corners. He brings 5 post less than what he has initially plan because of which the distance between two consecutive post became 8 m instead of 6 m.. What is the length of the side and no of post? Ans. 120 , 16 Ques18. Which is associated with football Ans: Merdeka Ques 19. Venue of XXI winter Olympics 2010 Ques 20. Crossed cheque can be Encashed ….. Ans: Through bank Ques21. Standard charterd bank merged with…… Ques 26. PAN is required for…… Ans: D-Mat account Ques27. Which one of following is not matched correctly…. Ques3. Tetanus is caused by Ans. Bacteria Ques4. De beers manufacture Ans: Diamonds Ques5. Commercial vehicles are not manufactured by which of the following Ans: Birla Yamaha Ques6. Wings of fire autobiography by Ans: President A.P.J Abdul Kalam. Ques7. Which is not a courier company. Ans: Essar Ques8. Software and Services Company. Ans: NASSCOM Ques9. Panchayti Raj started from which state in India. Ans: Rajastan Q21: Four horses are tethered at the four corners of a square of side 14cm such that two horses along the same side can just reach each other. They were able to graze the area in 11 days. How many days will they take in order to graze the left out area? Ans. 3 Q7 A person sells a horse at 12.5% loss. If he sells for 92.5 more, he will have a profit of 6%. What is the CP? Ans 500 Q8 One tap takes 15 min to fill, another tap takes 12 min to fill and the third tap can empty in 20 min. In how much time the tank would be full. Ans 10 min Q9 Three mixtures of water and milk are in the ratio 6:7, 5:9, 8:7. What would be the ratio if they were mixed together? Ans: None of the options were correct. The right answer would be data insufficient. Q10. Two trains are separated by 200km. One leaves at 6:00 am from Delhi and reaches Merrut at 10:00 am. Another train leaves from Merrut at 8:00 am and reaches Delhi at 11:30 am. At what time two trains meet each other? Ans 8:56 am Q13 There are only two candidates. 10% of the voter did not vote. 60 votes were invalid. If elected got 308 votes more than the opponent. The elected person got 47% of the total votes. How many Ans: Option 4 ( only one option is having a difference of 308 votes.) Q14. If area of circle is equal to area of triangle is equal to area of square then which is having the larger perimeter. Ans: Triangle. Q15. If log1227= a then find the value of log616 Ans. (4(3-a)) / (3+a) Q22: Two clocks are synchronized at 12 midnights. One clock gains 2 min per hour and the other clock loses 1min per hour. How many minutes apart its minute hand will be at 11.am Ans. None of these. 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by ## Where communities thrive • Join over 1.5M+ people • Join over 100K+ communities • Free without limits ##### Activity Sheehan Olver @dlfivefifty I don't think so: that needs to be refactored. FullPDETest.jl suffered a bit of version control neglect. On Tuesday let's discuss what an FEM analogue would be. That's a very different viewpoint from ApproxFun as one works with weak formulations, not operators. Stefanos Carlström @jagot Yes, I realized I have too many questions to work efficiently at the moment Sheehan Olver @dlfivefifty Domains.jl originated with @daanhb so some of the design is different. It also uses the term "Space" as things like R are Euclidean spaces. The analogue of AnyDomain() would be FullSpace{Any}(). Some more thought should be put in to how to unify this notion of Space with ApproxFuns. Stefanos Carlström @jagot I see. I was also wondering, how I, if I knew a set of points x,y, would use transform to get the expansion coefficents, i.e. the best possible approximation/interpolation. Some notion on how to perform eigendecompositions of operators acting on a certain basis, should also be encoded. For e.g. BSplines, since the basis functions are not orthogonal, you have an overlap matrix and hence you have a generalized eigenvalue problem. Sheehan Olver @dlfivefifty Stefanos Carlström @jagot Maybe I do, but it seems that points then need to be attached to an existing Space. Ok, Vandermonde seems to be what I want Kirill Ignatiev @ikirill Is this supposed to work? julia> dot(Fun(), Fun()) ERROR: StackOverflowError: Stacktrace: [1] dot(::Function, ::Function) at /Users/osx/buildbot/slave/package_osx64/build/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/generic.jl:665 (repeats 39122 times) [2] top-level scope at none:0 Sheehan Olver @dlfivefifty Can you file an issue? Kirill Ignatiev @ikirill Am I going crazy, or is there something wrong with transposes? JuliaApproximation/ApproxFun.jl#624 Up there when I was solving the associated Legendre ODE: is there a way to get the diagonal operator whose entries are 1/sqrt(dot(psi[j], psi[k])), psi being the basis of a space? It would make the self-adjoint ODE actually have a symmetric matrix when solving the generalized eigenvalue problem, but I couldn't find this. Sheehan Olver @dlfivefifty It's just an operator that hasn't been added yet. Probably there should be a Normalized(Jacobi(1,1)) space and this done as a conversion operator. (Before you ask "why not always use normalized", its because the unnormalized versions have rational formulae which are much faster.) Mikael Slevinsky @MikaelSlevinsky I showed how to get a symmetric-definite + banded GEP for associated Legendre functions here (section 3.1 https://arxiv.org/abs/1711.07866). I have some ApproxFun-like code that I wrote to verify the correctness of the formulas, (but not the FMM speedup) Mikael Slevinsky @MikaelSlevinsky Sorry, that should say "one way" to get banded symmetry if I have 2D cartesian grid data at a certain resolution, is there any way to make a high precision Chebychev interpolant using ApproxFun? ah, I see a comment above that answers this! Soham Mukherjee @soham1112 Hi, I was just trying out an example from https://github.com/JuliaApproximation/ApproxFunExamples/blob/master/PDEs/Rectangle%20PDEs.ipynb. When trying the example for solving the convection equation (see In[10]), I end up getting an error if I end up changing the interval for t, say, for example, from 0 .. 2 to 2 .. 3. ERROR: LoadError: LoadError: Cannot convert coefficients from Chebyshev(-1.0..1.0) to Chebyshev(-1.0..1.0)⊗ConstantSpace(Point(2.0)) I don't understand why this should be giving an error, since I should be able to choose an arbitrary time-interval. Am I missing something? Sheehan Olver @dlfivefifty It’s using an embedding of a 1D interval in the 2D plane where the interval is assumed to be at t = 0. You can use a fully 2D version: u0 = Fun((x,_) -> ..., Chebyshev(-1.0..1.0)⊗ConstantSpace(Point(2.0))) Soham Mukherjee @soham1112 Ah! I see. Thanks--that worked like a charm! However, I'm curious if there's a reason for this default behaviour? Mitko Georgiev @mitkoge Hi, I tried tzhe following using ApproxFun sp1= Chebyshev(0.0..50.0) ∪ Chebyshev(50.0..60.0) x1=Fun(sp1); g1 = Fun(exp(-x1^2),sp1) s1= sum(g1) #NaN sp2= Chebyshev(0.0..5.0) ∪ Chebyshev(5.0..60.0) x2=Fun(sp2); g2 = Fun(exp(-x2^2),sp2) s2= sum(g2) #0.8862269254527582 i expected s1~= s2. but it seems it is not. Mikael Slevinsky @MikaelSlevinsky I pushed a fix for this on this branch JuliaApproximation/ApproxFun.jl@07f50d0. If the tests pass, then it could be merged. Mitko Georgiev @mitkoge Wow i am impressed by your reactivity! Fixed, pushed and merged in a day! Thank you! Mitko Georgiev @mitkoge I try to reproduce an example from this chat given by @cortner using ApproxFun dom = Interval(0.001, 1) * PeriodicInterval(-pi, pi) space = Space(dom) Dr = Derivative(space, [1,0]) Dθ = Derivative(space, [0,1]) Mr = Multiplication(Fun( (r, θ) -> r, space ), space) rDr = Mr * Dr L = rDr * rDr + Dθ * Dθ but PeriodicInterval(-pi, pi) is no more recognized. What alternative can be used? Circle? Sheehan Olver @dlfivefifty It’s been renamed PeriodicSegment as it’s really an oriented line segment Mitko Georgiev @mitkoge Thank you, i tried julia> dom = Interval(0.001, 1) * PeriodicSegment(-pi, pi) ERROR: MethodError: no method matching *(::Interval{:closed,:closed,Float64}, ::PeriodicSegment{Floa t64}) and julia> dom = Segment(0.001, 1) * PeriodicSegment(-pi, pi) ERROR: MethodError: no method matching *(::Segment{Float64}, ::PeriodicSegment{Float64}) Sheehan Olver @dlfivefifty Ah and * has changed to imes imes \times Mitko Georgiev @mitkoge thank you. using ApproxFun,LinearAlgebra dm = Segment(0.001,1) × PeriodicSegment(-pi,pi) sp = Space(dm) Dr = Derivative(sp, [1,0]); Dθ = Derivative(sp, [0,1]) Mr = Multiplication(Fun( (r, θ) -> r, sp ), sp) rDr = Mr * Dr Lr = rDr * rDr; L = Lr + Dθ * Dθ i modified it and now can play with. Simon Etter @ettersi Not sure whether this is the right place to ask since technically my question concerns SingularIntegralEquations.jl, but that package seems to be related enough to ApproxFun.jl that I'll go ahead anyway. In the code of SingularIntegralEquations it says # stieltjesintegral is an indefinite integral of stieltjes # normalized so that there is no constant term Can someone elaborate on that normalisation condition? Sheehan Olver @dlfivefifty Sure, fine to ask here. This normalisation condition comes from the fact that Jacobi polynomials have nice integration formulae coming from the weighted differentiation d/dx[(1-x)^a *(1+x)^b P_n^(a,b)] = C (1-x)^{a-1} (1+x)^{b-1} P_{n+1}^(a-1,b-1). Roughly speaking, for functions that vanish at ±1, derivatives and Stieltjes transforms commute, so provided a,b > 0 we can use this to write the integral of a Stieltjes transforms in terms of the integral of the integrand. Note that the n = 0 case is special, and dictates the normalisation constant (as the construction for other n will automatically decay something like O(z^(-n)) ) For the cases actually implemented, I think its normalized to go to zero at ∞. Sorry, that's not right. You always have logarithmic growth (unless the integrand integrates to zero). Sheehan Olver @dlfivefifty On -1..1 it just vanishes apart from the log term: julia> x = Fun(); julia> f = exp(x)/sqrt(1-x^2); julia> z = 10_000; stieltjesintegral(f,z) - log(z) * sum(f) -0.00017756097918919522 julia> f = exp(x) * sqrt(1-x^2); julia> z = 10_000; stieltjesintegral(f,z) - log(z) * sum(f) -4.264886882765495e-5 Oh, wait, that's true for other intervals too: julia> x = Fun(1..2); julia> f = exp(x) * sqrt((x-1)*(2-x)); julia> z = 100_000; stieltjesintegral(f,z) - log(z) * sum(f) -2.8356239955229512e-5 Sorry for making it seem more complicated than it actually it is. It's really simple: it's normalized so that stieltjes(f,z) = log(z) *sum(f) + o(1).

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# What were Nathaniel Hawthorne feelings towards Puritan New England? 2 answers ###### Question: what were Nathaniel Hawthorne feelings towards Puritan New England? ## Answers 1 answer ### What is health called wealth?​ what is health called wealth?​... 1 answer ### Which is true about inelastic collisions: a. An inelastic collision does not obey conservation of energy. b. An inelastic collision conserves kinetic energy. c. Objects will stick together upon collision. d. Momentum is not conserved in inelastic collisions.. Which is true about inelastic collisions: a. An inelastic collision does not obey conservation of energy. b. An inelastic collision conserves kinetic energy. c. Objects will stick together upon collision. d. Momentum is not conserved in inelastic collisions..... 1 answer ### Find an equation with an x intercept of (-1, 0) Find an equation with an x intercept of (-1, 0)... 1 answer ### State under which condition the original and the simplified expressions are equivalent. The expressions are equivalent if... a) a≠0 b) a≠x c) a≠-x $\frac{a^2}{ax-x^2} +\frac{x}{x-a}$ State under which condition the original and the simplified expressions are equivalent. The expressions are equivalent if... a) a≠0 b) a≠x c) a≠-x $\frac{a^2}{ax-x^2} +\frac{x}{x-a}$... 1 answer ### Identify the domain and range of the function graphed below Identify the domain and range of the function graphed below... 1 answer ### Can someone help me... What is the answer Can someone help me... What is the answer... 1 answer ### A skateboard is moving to the right with a velocity of 8 m/s after a steady Gust of wind that last 5 seconds the skateboarder is moving to the right with a velocity of 5 m/s assuming the acceleration is constant what is the acceleration of the skateboard during the five second time. A skateboard is moving to the right with a velocity of 8 m/s after a steady Gust of wind that last 5 seconds the skateboarder is moving to the right with a velocity of 5 m/s assuming the acceleration is constant what is the acceleration of the skateboard during the five second time.... 2 answers ### Where does the email address of the recipient go?​ where does the email address of the recipient go?​... 1 answer ### Let f(x) = 4x + 3 and g(x) = -2x + 5 find (g•f)(5) • -5 •23 •-17 •-41 let f(x) = 4x + 3 and g(x) = -2x + 5 find (g•f)(5) • -5 •23 •-17 •-41... 1 answer ### Which of the four spheres of the Earth encompasses all of the gases at and above the Earth's surface? 1. Biosphere 2. Atmosphere 3. Lithosphere 4. Hydrosphere Which of the four spheres of the Earth encompasses all of the gases at and above the Earth's surface? 1. Biosphere 2. Atmosphere 3. Lithosphere 4. Hydrosphere... 2 answers ### Pythagorean theorem , HELPPP Pythagorean theorem , HELPPP... 2 answers ### According to Thomas Hobbes, what is the purpose of government control over citizens? to provide safety and security to erase natural human tendencies to create a balanced society to change the corrupt nature of humans. According to Thomas Hobbes, what is the purpose of government control over citizens? to provide safety and security to erase natural human tendencies to create a balanced society to change the corrupt nature of humans.... 1 answer ### How are forces able to act at a distance ? How are forces able to act at a distance ?... 1 answer ### Colton has 888 comedy movies, 666 action movies, 555 fantasy movies, and 111 drama movie on his list of favorite movies. For every 555 movies on the list, there are 222 \_\_\_\_____\_, \_, \_, \_ movies. Colton has 888 comedy movies, 666 action movies, 555 fantasy movies, and 111 drama movie on his list of favorite movies. For every 555 movies on the list, there are 222 \_\_\_\_____\_, \_, \_, \_ movies.... 1 answer ### The result of the investiture controversy between pope Gregory VII and emperor Henry IV was The result of the investiture controversy between pope Gregory VII and emperor Henry IV was... 1 answer ### How many solutions are there to the equation below? 9(x - 4) = 9x - 33 A. Infinitely many B. 1 C. 0 How many solutions are there to the equation below? 9(x - 4) = 9x - 33 A. Infinitely many B. 1 C. 0... 1 answer ### Corruption is one of the main obstacles of Nepal.Justify​ Corruption is one of the main obstacles of Nepal.Justify​... 1 answer ### Q1 Give a combination of four quantum numbers that could be assigned to an electron occupying a 5p orbital. Express your answers using one significant figures. Enter your answers separated by commas. The answer n, l, ml, ms = 5,1,-1,0,or1,-1/2or+1/2 mastring chemistry says "Incorrect; Try Again; 6 attempts remaining; no points deducted Your answer does not have the correct number of comma-separated terms." the same for q2 Do the same for an electron occupying a 6d orbital. Express your Q#1 Give a combination of four quantum numbers that could be assigned to an electron occupying a 5p orbital. Express your answers using one significant figures. Enter your answers separated by commas. The answer n, l, ml, ms = 5,1,-1,0,or1,-1/2or+1/2 mastring chemistry says "Incorrect; Try Again... 1 answer ### WILL GIVE BRAINLIST!!! The circumference of a circle is 62.8 and the diameter of the circle is 20. Which best represents the value of π? 1062.8 2062.8 62.820 62.810 WILL GIVE BRAINLIST!!! The circumference of a circle is 62.8 and the diameter of the circle is 20. Which best represents the value of π? 1062.8 2062.8 62.820 62.810... 2 answers ### What did the RMS Titanic have that showed it was a luxury cruise-liner? What did the RMS Titanic have that showed it was a luxury cruise-liner?... -- 0.043019--

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## Effect of Magnet on Current-Carrying Wire Updated on 31-Jan-2023 10:39:40 Introduction Generally, we see when we put iron nails at some distance to the magnet, it gets attracted towards it. Why is that so? We see that certain materials have the capability to attract materials towards themselves. The materials are attracted to the field around the magnet known as the magnetic field. Like due to electric charge producing an electric field also a magnet produces a magnetic field around it. The magnetic field is also created around the wire-carrying current. What is a Magnet? This is a piece of metal that attracts certain kinds of materials towards itself. There ... Read More ## Difference Between Earthing and Grounding Updated on 30-Jan-2023 17:33:15 Introduction Electricity is one of the most used things by humans. It is transported from the powerhouse through the wires to the various houses and industrial locations. Electric current has two main types - Direct current and Alternating current. Alternating current mainly flows in the wires which see around us. Electricity has given new dimensions to power systems and technology, however, it can be dangerous also. Spark in electrical wires can lead to a fire. If touched directly, live wires can also give shocks. Keeping these things in mind, for the safety of humans and the powerhouses we used several ... Read More ## Joules Law Updated on 30-Jan-2023 17:11:28 Introduction Energy and work are measured in joules. By applying a force of one newton over a distance of one meter, an object is made to work (or to be energized). One joule is produced when an object is subjected to a force of 1 N and moves 1 m. Newton meters are used to measure it. The exact equivalent of heat is produced each time mechanical force is applied. The heat generated in any electrical component varies directly to the squared current, the resistance, and the duration of the current's flow, as per Joule's law of heating. What ... Read More ## Electrical Energy and Power Updated on 24-Jan-2023 15:49:51 Introduction We do many works in our daily life. We do many of them through our physical activities. Some work is done with tools and other devices. But they also require energy. There are many types of energy like electrical energy, mechanical energy, thermal energy, light energy, and wind energy. In our daily life, we get electrical energy from the battery. Electricity is also produced from nuclear power plants, hydroelectric plants, wind farms, and sunlight. Eels produce electricity. They use this energy to protect themselves from their enemies. Fans, lights, televisions, washing machines, and refrigerators all require electricity to operate. ... Read More ## Electric Currents In Conductors Updated on 24-Jan-2023 15:19:39 Introduction Electric current and conductors are essential for each other because the flow of current is only possible in conductors without any loss. The current is a flow of charges in a particular direction. The flow of current through a conductor is possible due to the difference in potential at both ends. A conductor also charges particles whether it is connected to a battery or not. However, when it is not related to any source of voltage then charges are just neutral and moving in random directions which results in net-zero charge flow and thermal velocity. So, to get a ... Read More ## Electrical Force Updated on 24-Jan-2023 16:50:14 Introduction Electromagnetism is one of the most important branches of physics. Among the forces we see in our daily life, all the forces except gravity are electromagnetic. All the forces we encounter in life except gravity (including the tension of a wire, the vertical force of a surface, and the force of friction) are electromagnetic forces that appear between atoms. Any neutral substance has equal number of electrons and protons. If the outermost electrons leave the atom, it becomes free electron and causes an electric current. An atom that has lost an outer electron has a higher positive charge. ... Read More ## Electric Car Updated on 24-Jan-2023 14:52:33 Introduction The electric car is the best achievement by the automobile sector to make the earth pollution-free. Also, it reduces transportation costs due to no fuel charges or lowmaintains costs. If we have a look at the history of the electric car, the first electric car was invented by Gustave Trouve in 1881. The car made by Gustave is a personal fullscale electric car. However, Gustave’s car didn’t get any admiration or success. Gustave Trouvé's electric tricycle, the first electric vehicle in history to be displayed to the public Jacques CATTELIN, Capture d’écran 2016-10-14 à 21.26.28, CC BY-SA 4.0 ... Read More ## Displacement Current Updated on 24-Jan-2023 14:11:28 Introduction Initially, electricity and magnetism were treated as separate subjects. In a later period due to the contribution of Oersted, Faraday, Maxwell, etc., it evolved as a unified subject. The current-carrying wire creates a magnetic field around it. Faraday through his experiments showed that current can be produced even if there are no batteries in the circuit. The change in the magnetic field can produce a current in the circuit. This result is known as electromagnetic induction. Maxwell tried to write all the main equations of electricity and magnetism in a unified and compact manner. These equations are known as ... Read More ## Electronics in Daily Life Updated on 17-Jan-2023 17:38:15 Introduction You are use electronics in your daily life from every small object to a big machine. The device by which you are reading this article is also a part of electronics. Electronics is a vast field that contains a huge amount of components like conductors, switches, circuits, diodes, processors, inductors, resistors, etc. All these components have important features to develop this field to more extant. Simply, when we study charges, current, electric or magnetic field, etc. actually we are working with electronics. What are Electronics? Electronics is a part or branch of science in physics and technology portion. ... Read More ## Temperature Dependence Resistance Updated on 09-Jan-2023 15:14:43 Introduction Resistance is an obstacle that is present in an object with the flow of electrons. Objects with a lower amount of resistance can flow a higher amount of electrons. The significance of the current that is provoked strictly counts on the aspect of the item. Any item resists the flow of electric control which is named resistance. The dependency resistance is followed by the conductor’s geometry and it is as well as what the conductor is constructed from. However, it also relies on temperature. The Concept of Electrical Resistance The application of resistance can be found with the application ... Read More

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1 You visited us 1 times! Enjoying our articles? Unlock Full Access! Question # A block of mass 10 kg is kept on a fixed rough (μ=0.8) inclined plane of angle of inclination 30. The frictional force acting on the block is A 50 N Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses B 503 N No worries! We‘ve got your back. Try BYJU‘S free classes today! C 52 N No worries! We‘ve got your back. Try BYJU‘S free classes today! D 54 N No worries! We‘ve got your back. Try BYJU‘S free classes today! Open in App Solution ## The correct option is A 50 NWhen a block of 10kg is moving n a rough surface and the angle of inclination is 30.the force of static friction is 5kg .wt1kg.wt means a force that is required to accelerate a mass of 1kg with an acceleration of approximately 9.8 m/s2.here the friction force to stop the block is mgsin30\$Since mass is equal to 10 kgforce required =10×sin30 newton=5×gnewton=5kg.wtHence, Option A is correct. Suggest Corrections 0 Join BYJU'S Learning Program Related Videos Rubbing It In: The Basics of Friction PHYSICS Watch in App Explore more Join BYJU'S Learning Program

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# physics posted by . A car with a 65cm diameter wheel travels 3.0km. How many revolutions does the wheel make in this distance? • physics - find the circumference C = pi D = pi * 0.65 meters per revolution C * n = 3,000 meters so n = 3,000/C ## Similar Questions 1. ### trigonometry these two are the same type of questions but i have no clue how to figure out the answer. i have tried drawing it out but don't know what to do after that. 1. a bicycle with a 28-inch wheel(diameter) travels a distance of 800 feet. … 2. ### maths a car travels 1 km in which each wheel makes 450 revolutions. Find the radius of its wheel 3. ### Math A bicycle travels 42 meters for every 20 revolutions of the wheel. How many revolutions would the wheel make in a kilometer? 4. ### Math A bicycle travels 42 meters for every 20 revolutions of the wheel. How many revolutions would the wheel make in a kilometer? 5. ### Trigonometry A bicycle with a 26-inch wheel(diameter) travels a distance of 700 feet. How many revolutions does the wheel make? 6. ### math One wheel has a diameter of 30 inches and a second wheel has a diameter of 20 inches. The first wheel traveled a certain distance in 240 revolutions. In how many revolutions did the second wheel travel the same distance? 7. ### math The diameter of a bicycle wheel is 65cm. How far does the bicycle go in 20 revolutions of the wheel 8. ### MATHS "A DIAMETER OF THE WHEEL OF A CAR IS 84 CM AND THE SPEED OF A CAR IS 72KM/H. THE NO. OF REVOLUTIONS MADE BY THE WHEEL IN 11 MINUTES IS."IS THIS QUESTION EXAMPLE OF CIRCLE OR CYLINDER? 9. ### physics A car with a 65cm diameter wheel travels 3.0km. How many revolutions does the wheel make in this distance? 10. ### physics A car with a 65cm diameter wheel travels 3.0km. How many revolutions does the wheel make in this distance? More Similar Questions

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# 【换根dp】9.22小偷 换根都不会了 ## 题目大意 $n,m\le 30000,LIM\le 30000$ ## 题目分析 1 #include<bits/stdc++.h> 2 const int maxn = 30035; 3 const int maxm = 60035; 4 5 int n,m,lim,ans,sum,p[maxn],mx[maxn],dx[maxn]; 7 bool tag[maxn]; 8 10 { 11 char ch = getchar(); 12 int num = 0, fl = 1; 13 for (; !isdigit(ch); ch=getchar()) 14 if (ch=='-') fl = -1; 15 for (; isdigit(ch); ch=getchar()) 16 num = (num<<1)+(num<<3)+ch-48; 17 return num*fl; 18 } 19 void addedge(int u, int v) 20 { 23 } 24 void dfs1(int x, int fa) 25 { 26 mx[x] = dx[x] = -1; 27 if (tag[x]) mx[x] = 0; 28 for (int i=head[x]; i!=-1; i=nxt[i]) 29 { 30 int v = edges[i]; 31 if (v==fa) continue; 32 dfs1(v, x); 33 if (mx[v]!=-1&&mx[v]+1 >= mx[x]) dx[x] = mx[x], mx[x] = mx[v]+1; 34 else if (mx[v]!=-1&&mx[v]+1 > dx[x]) dx[x] = mx[v]+1; 35 } 36 } 37 void dfs2(int x, int fa) 38 { 39 for (int i=head[x]; i!=-1; i=nxt[i]) 40 { 41 int v = edges[i], val = 0; 42 if (v==fa) continue; 43 if (mx[x]==mx[v]+1&&mx[v]!=-1) val = dx[x]+1; 44 else val = mx[x]+1; 45 if (val&&val >= mx[v]) dx[v] = mx[v], mx[v] = val; 46 else if (val&&val > dx[v]) dx[v] = val; 47 dfs2(v, x); 48 } 49 if (mx[x] <= lim) ++ans; 50 } 51 int main() 52 { 55 for (int i=1; i<=m; i++) tag[read()] = true; 57 dfs1(1, 0); 58 dfs2(1, 0); 59 printf("%d\n",ans); 60 return 0; 61 } END posted @ 2019-09-22 21:02  AntiQuality  阅读(186)  评论(0编辑  收藏  举报

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Cody # Problem 43144. BASICS - sum part of vector Solution 2084927 Submitted on 9 Jan 2020 by Ana Maria Alzate This solution is locked. To view this solution, you need to provide a solution of the same size or smaller. ### Test Suite Test Status Code Input and Output 1   Pass x = [-1 2 3 4 7 9]; c=[1 4 6]; y_correct = 12; assert(isequal(sumvec(x,c),y_correct)) 2   Pass x = [-50 -24 0 4 10 14 19 18]; c=[2 3 6 7]; y_correct = 9; assert(isequal(sumvec(x,c),y_correct))

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Cody # Problem 2016. Area of an equilateral triangle Solution 1999326 Submitted on 2 Nov 2019 by Son Pham This solution is locked. To view this solution, you need to provide a solution of the same size or smaller. ### Test Suite Test Status Code Input and Output 1   Pass x = 1; y_correct = sqrt(3)/4; tolerance = 1e-12; assert(abs(equilateral_area(x)-y_correct)<tolerance) y = 0.4330 2   Pass x = 2; y_correct = sqrt(3); tolerance = 1e-12; assert(abs(equilateral_area(x)-y_correct)<tolerance) y = 1.7321 3   Pass x = 3; y_correct = sqrt(3)*9/4; tolerance =1e-12; assert(abs(equilateral_area(x)-y_correct)<tolerance) y = 3.8971 ### Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting!

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# Is EMF and Voltage the Same Thing? Last Updated on 1 year by Francis Electromotive force (EMF) and voltage are terms that are often used interchangeably but are not necessarily the same thing. In the field of physics and electrical engineering, it is important to understand the distinctions between these terms as they play a fundamental role in many electrical systems. This article will explore the differences between EMF and voltage, and provide a clear understanding of each concept. Contents ## Understanding EMF Electromagnetic fields (EMF) are invisible forces that surround us. They are produced by anything that has an electric current flowing through it, such as power lines, appliances, and electronics. EMF can also come from natural sources, such as the sun and the Earth’s magnetic field. ### Types of EMF There are two types of EMF: ionizing and non-ionizing. Ionizing EMF has enough energy to remove electrons from atoms, which can damage DNA and potentially lead to cancer. Non-ionizing EMF, on the other hand, does not have enough energy to remove electrons from atoms and is considered safe. ## Understanding Voltage Voltage is a measure of the electric potential difference between two points. In other words, it measures the force that drives an electric current. Voltage is measured in volts (V) and can be thought of as the pressure that pushes electrons through a circuit. One key takeaway from this article is that EMF and voltage are not the same thing, although they are related. EMF is the force that causes electric current to flow, while voltage is the measure of that force. It’s important to understand the differences between these two concepts in order to fully understand the potential health effects of exposure to EMF. While the evidence linking EMF to health effects is inconclusive, it’s still a good idea to take precautions to reduce exposure, such as using a headset with your cell phone or avoiding prolonged exposure to high-voltage power lines. ### Voltage in Daily Life We encounter voltage in our daily lives whenever we use electronics or appliances. For example, the voltage of a standard electrical outlet in the United States is 120 volts. ## Comparing EMF and Voltage Although EMF and voltage are related, they are not the same thing. EMF is the force that causes electric current to flow, while voltage is the measure of that force. In other words, voltage is the cause and EMF is the effect. One key takeaway from this text is that electromagnetic fields (EMF) and voltage are not the same thing. While voltage is a measure of the electric potential difference between two points, EMF is the force that causes electric current to flow. It’s important to differentiate between these two concepts in order to fully understand the potential health effects of exposure to EMF. Although research on the health effects of EMF exposure is inconclusive, it’s still recommended to take precautions to reduce exposure, such as using a headset with your cell phone and avoiding prolonged exposure to high-voltage power lines. ### EMF in Relation to Voltage EMF can be thought of as the voltage that is generated by a changing magnetic field. When a magnetic field changes, it creates an electric field, which in turn produces an EMF. ### Misconceptions About EMF and Voltage There are many misconceptions about EMF and voltage, including the belief that they are the same thing. It’s important to understand the differences between these two concepts in order to fully understand the potential health effects of exposure to EMF. ## Understanding the Health Effects of EMF Exposure to EMF has been linked to a variety of health effects, including cancer, neurological disorders, and reproductive issues. However, the evidence linking EMF to these health effects is still inconclusive. ### Research on EMF and Health Many studies have been conducted on the potential health effects of EMF, but the results have been mixed. Some studies have suggested a link between EMF exposure and health problems, while others have not found any significant association. ### Precautions for Reducing EMF Exposure Although the evidence is inconclusive, it’s still a good idea to take precautions to reduce your exposure to EMF. This can include things like using a headset with your cell phone, avoiding prolonged exposure to high-voltage power lines, and using EMF shielding products. ## FAQs – Is EMF and Voltage the Same Thing? ### What is EMF? EMF stands for Electromotive Force. It can be defined as the potential difference between two points in a circuit or a device. It is usually measured in volts and is the force that pushes the electrons to move through the circuit. EMF is a source of energy that is created by a battery or generator and it is the force that makes charges move along conductive material. ### What is Voltage? Voltage is a measure of electrical potential energy per unit of charge. It is the difference in electric potential between two points in an electric circuit, and it is measured in volts. Voltage can also be defined as the force that moves electric current through a circuit. In simple terms, voltage is the pressure that causes the electrons to move from one point to another. ### Are EMF and Voltage the same thing? No, EMF and voltage are not the same thing. EMF is the force that drives the electric current in a circuit, while voltage is the potential difference between two points in that circuit. EMF is the energy that is supplied to the circuit to make electric charges move, while voltage is the amount of electrical potential energy that each charge has when it is passing through the circuit.

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204 Answered Questions for the topic integral 05/12/18 #### How do you find the upper bound of an integral 12000-inte(0 to A) r(t)dt= 9000   r(t)= 600t/t+3 for 0 less than or equal to t less than or equal 5 r(t)= t>5 (Piecewise) 03/26/18 #### Assume s,v, and a are continuous functions and s is twice differentiable, where s(t) represents the position, v(t) is velocity, a(t) is acceleration Find the following:   d/dx of integral from t to t^2 of v(w)dw   integral from t to t^2 of v(w)dw   second derivative of integral from t to t^2 of v(w)dw    Calculus question I am stuck on Integral 07/26/17 #### Integral of Sin-1 logx Sin-1logx find integral 04/29/17 #### Stuck on pratice problem reguarding integrals a. Explain why ∫02pi sin(x)dx = 0   b. Can you set up another integral, or perhaps multiple integrals, to compute the total (unsigned) area between sin(x) and the x-axis on the interval [0,2pi] 04/02/17 Find the area bounded by the curve: y = x(x-1)^2, the line y = 2, and the y - axis Integral Calculus 3 11/15/16 #### I=R⌠⌠⌠f(x,y,z)dv Set up an iterated integral to evaluate I=R⌠⌠⌠f(x,y,z) dv, where R is the region in the first octant bounded by the surface z=x+y^2, the cylinders y=2sqrt(x), x=2sqrt(y), and the... more 04/28/16 #### Integration question involving cosh^-1??? Evaluate the integral (1/sqrt(x^2 - 5))dx on limits of 3 to 4, using the formula: integral of (1/sqrt(x^2 - a^2))dx. Recall that this formula was found by using the substitution x = a cosh t, and... more Integral Calculus 04/18/16 #### Find an anti-derivative F(x) with F'(x) = f(x) and F(0)=0 Find an anti-derivative F(x) with F'(x) = f(x) and F(0)=0   f(x)=(1/5)x 03/13/16 #### Evaluate the integral below given R. (Hint: what is the upper boundary of x?) Evaluate the integral below given R. (Hint: what is the upper boundary of x?)     ∫∫1_R^  xy daR bounded byx=0y=0y=9−x^2 03/13/16 #### Find the integral below. Find the integral below.       ∫∫1_R^ (x+2y) dar={(x, y):0≤x≤3, 1≤y≤4} Integral Calculus 03/13/16 #### Integral of the equation ∫1/(√x√(1-x)) dx Integral Calculus 03/13/16 #### How to find this integral? ∫(x+5)/(√9-(x-3)2)   This problem is stumping me! Also the square root is over the whole denominator. 02/02/16 #### Cubic curve integral - area enclosing curve and x-axis The curve is x^3-6x^2+11x-6. I have to prove that the two regions enclosed between the x-axis and the curve are equal. Given is, that the curve cuts the x-axis at x=1,x=2 and x=3. I tried finding... more 08/15/15 #### solution of indefinite integral What is a particular solution of the indefinite integral ∫(2)/(1-x2)dx that passes through (√3 , (Π/2)?   a) f(x)=√(3) tan-1x-(Π/6) b) f(x)=2tan-1x-(Π/6) c) f(x)=2tan-1x+√(3) Integral Math Calculus Ap Calc 08/13/15 #### substituting in an integral If the substitution √(X)=sin y is made in the integrand of: (top of the integral sign is (1/2), bottom of the sign is 0) ∫ ( √(X) )/( √(1-X) )dx, the resulting integral is_____________.   A.)... more 08/12/15 #### chain rule is also know as The Chain Rule for antiderivatives is also known as the ___________________ rule.   -derivative -substitution -product -integral 08/12/15 #### integrals for area between curves what integral formula would represent the area between the curves f(x)=-x2+2x+3 and g(x)=x+1 on the interval [-1,2]? 08/12/15 #### solving integrals ∫ ( (dx)/√(x2+36) )=_____________     a. ( (√(x2+36)+x)/6 )+C   b. ln abs( (x+6)/(√(x2+36)) )+C   c. ln ( (√(x2+36)+x)/6 )+C 08/12/15 #### trigonometric substitution to evaluate integral Which trigonometric substitution can be used  to evaluate the integral: ∫ ( (dx)/(√(a2+x2)) ) ...?   x= a tanθ   x=a sinθ   x=a secθ 08/11/15 #### substituting a value into an integrand if the substitution √(x)=sin(y) is made into the integrand of (with the upper limit of: 1/2  and the lower limit of: 0) ∫ ( (√(x) )/( (√(1-x) )dx , the resulting integral is _______________. 06/29/15 #### Area bounded by two curves What is the area of the region bounded by the curves y=2(x^2)-7 and y=2/x between x=1 and x=3? Please and thank you! 03/08/15 #### Evaluate the integral Evaluate the integral  ∫(-9/3√(x-9))dx with upper bound 9 and lower bound 1 Integral Derivative 01/29/15 #### Find the derivative of Find the derivative of  g'(x)=∫(u+4/u-5)du with upper bound 3x and lower bound 9x Integral Derivative Function 01/26/15 #### Find the derivative of the following integral find the derivative of    g(x)= ∫(u+4/u-5)du with upper bound 3x and lower bound 9x g'(x)=? Integral Calculus 01/23/15 #### At what value of t does the local max occur? f(t)= integral (x^2+11x+24)/(1+cos^2(x)) from 0 to t  At what value of t does the local max of f(t) occur? ## Still looking for help? Get the right answer, fast. Get a free answer to a quick problem. Most questions answered within 4 hours. #### OR Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.

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CC-MAIN-2021-25

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https://casaplorer.com/fha-mip-calculator

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# The FHA Mortgage Insurance Calculator CASAPLORERTrusted & Transparent ### What You Should Know • This FHA Mortgage Insurance Premium Calculator allows you to estimate your upfront and annual mortgage insurance premiums. • FHA MIP is mortgage insurance that is paid by all FHA loan holders regardless of the loan-to-value ratio of the loan. • FHA mortgage insurance premium includes two fees: an upfront fee and an annual fee. • An upfront fee is paid once at the time of closing on the loan while an annual fee is charged annually and paid in monthly installments. FHA MIP Calculator Inputs \$ % = \$ Results \$ 7,000 \$ 150 You Will Pay FHA Mortgage Insurance Premium for 11 years. This FHA PMI Calculator allows you to estimate your upfront MIP and your annual MIP payments. It also provides you with information about how long you will have to pay your mortgage insurance premium. In addition to that, this calculator also estimates your annual FHA mortgage insurance premium rate. To estimate the results, you need to input a home purchase price, your down payment, and your loan term. Once you adjust your inputs, the calculator will yield precise results related to the mortgage insurance premium on your FHA loan. This calculator does not require your credit score or any other information regarding the home loan because it only estimates the mortgage insurance premium charged by the Federal Housing Administration. It does not estimate the monthly payments on your FHA loan. If you want to estimate your monthly mortgage payments, you should use the FHA mortgage calculator instead. ## How to Calculate PMI on FHA Loan It is easy to estimate your mortgage insurance on an FHA loan because FHA MIP is fixed, and it does not depend on your credit report. FHA MIP has two separate fees: an upfront fee and an annual fee. An upfront fee is paid once at the time of closing on the mortgage. An annual fee is charged annually and paid in monthly installments for at least 11 years of the loan term. Upfront Fee (UFMIP): 1.75% of the original mortgage principal. Annual Fee (MIP): 0.45% to 1.05% of the mortgage principal. The rate depends on the loan term, original loan amount, and loan-to-value (LTV) ratio. FHA MIP Rates Chart Base Loan AmountLTV RatioMIPDuration Mortgage Term Up to 15 years Less than or equal to \$625,000≤ 90%0.004511 years > 90%0.007Mortgage Term Greater than \$625,000≤ 78%0.004511 years > 78% but ≤ 90%0.00711 years > 90%0.0095Mortgage Term Mortgage Term More Than 15 years Less than or equal to \$625,000≤ 90%0.00811 years > 90% but ≤ 95%0.008Mortgage Term > 95%0.0085Mortgage Term Less than or equal to \$625,000≤ 90%0.00111 years > 90% but ≤ 95%0.001Mortgage Term > 95%0.0105Mortgage Term Any calculators or content on this page is provided for general information purposes only. Casaplorer does not guarantee the accuracy of information shown and is not responsible for any consequences of its use.

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https://docs.taichi.graphics/lang/articles/basic/external

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# Interacting with external arrays Although Taichi fields are mainly used in Taichi-scope, in some cases efficiently manipulating Taichi field data in Python-scope could also be helpful. We provide various interfaces to copy the data between Taichi fields and external arrays. External arrays refer to NumPy arrays or PyTorch tensors. Let's take a look at the most common usage: interacting with NumPy arrays. Export data in Taichi fields to NumPy arrays via `to_numpy()`. This allows us to export computation results to other Python packages that support NumPy, e.g. `matplotlib`. ``````@ti.kerneldef my_kernel(): for i in x: x[i] = i * 2 x = ti.field(ti.f32, 4)my_kernel()x_np = x.to_numpy()print(x_np) # np.array([0, 2, 4, 6])`````` Import data from NumPy arrays to Taichi fields via `from_numpy()`. This allows us to initialize Taichi fields via NumPy arrays: ``x = ti.field(ti.f32, 4)x_np = np.array([1, 7, 3, 5])x.from_numpy(x_np)print(x[0]) # 1print(x[1]) # 7print(x[2]) # 3print(x[3]) # 5`` ## External array shapes# Shapes of Taichi fields and those of corresponding NumPy arrays are closely connected via the following rules: • For scalar fields, the shape of NumPy array is exactly the same as the Taichi field: ``````field = ti.field(ti.i32, shape=(256, 512))field.shape # (256, 512) array = field.to_numpy()array.shape # (256, 512) field.from_numpy(array) # the input array must be of shape (256, 512)`````` • For vector fields, if the vector is `n`-D, then the shape of NumPy array should be `(*field_shape, vector_n)`: ``````field = ti.Vector.field(3, ti.i32, shape=(256, 512))field.shape # (256, 512)field.n # 3 array = field.to_numpy()array.shape # (256, 512, 3) field.from_numpy(array) # the input array must be of shape (256, 512, 3)`````` • For matrix fields, if the matrix is `n`-by-`m` (`n x m`), then the shape of NumPy array should be `(*field_shape, matrix_n, matrix_m)`: ``````field = ti.Matrix.field(3, 4, ti.i32, shape=(256, 512))field.shape # (256, 512)field.n # 3field.m # 4 array = field.to_numpy()array.shape # (256, 512, 3, 4) field.from_numpy(array) # the input array must be of shape (256, 512, 3, 4)`````` • For struct fields, the external array will be exported as a dictionary of arrays with the keys being struct member names and values being struct member arrays. Nested structs will be exported as nested dictionaries: ``````field = ti.Struct.field({'a': ti.i32, 'b': ti.types.vector(float, 3)} shape=(256, 512))field.shape # (256, 512) array_dict = field.to_numpy()array_dict.keys() # dict_keys(['a', 'b'])array_dict['a'].shape # (256, 512)array_dict['b'].shape # (256, 512, 3) field.from_numpy(array_dict) # the input array must have the same keys as the field`````` ## Using external arrays as Taichi kernel arguments# Use the type hint `ti.ext_arr()` for passing external arrays as kernel arguments. For example: ``````import taichi as tiimport numpy as np ti.init() n, m = 4, 7 @ti.kerneldef test_numpy(arr: ti.ext_arr()): for i in range(n): for j in range(m): arr[i, j] += i + j a = np.empty(shape=(n, m), dtype=np.int32) for i in range(n): for j in range(m): a[i, j] = i * j test_numpy(a) for i in range(n): for j in range(m): assert a[i, j] == i * j + i + j``````

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CC-MAIN-2022-05

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https://www.askmehelpdesk.com/online-education/student-recorded-length-piece-wire-measured-meter-scale-98-a-801626.html

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Virojosh Posts: 1, Reputation: 1 New Member #1 Sep 20, 2014, 01:46 PM A student recorded the length of a piece of wire measured with the meter scale as 98? A student recorded the length of a piece of wire measured with the meter scale as 98.6cm. What is the relative error in the measurement? teacherjenn4 Posts: 3,998, Reputation: 468 Education Expert #2 Sep 20, 2014, 02:41 PM ma0641 Posts: 15,681, Reputation: 1012 Uber Member #3 Sep 21, 2014, 05:33 PM Depends on the instrument you are measuring with. But as a general rule: The degree of accuracy is half a unit each side of the unit of measure. ballengerb1 Posts: 27,379, Reputation: 2280 Home Repair & Remodeling Expert #4 Sep 21, 2014, 07:14 PM A meter scale could be a stick like a yard stick or a steel bar. You need to give more specifics or just go with ma0641 because its should get you past this quiz. paraclete Posts: 2,704, Reputation: 171 Ultra Member #5 Sep 21, 2014, 08:41 PM 14 mm is not much in a meter measurement Question Tools Search this Question Search this Question: Advanced Search ## Check out some similar questions! In an experiment with a simple pendulum a student recorded 10 complete oscillations in 25 seconds. The effective length of the pendulum is a)10 b)25 c)2.5 d)0.4 220V wire size (80 ft length) [ 1 Answers ] I've been reading these thread and already have a wealth of information. Thank you Basic Info: I have a bridgeport mill, and DoAll band saw. Both of which are 220 3phase. I'm going to have a rotary phase converter to supply the 3 phase. The Saw has 3hp & Instant Amps 28 (prob for he blade... What is the different between 3phase 3 wire and 3 phase 4 wire energy meter? I have a old 60 AMP feeder line from the top of my house to the meter. From the meter is 200 AMP new wire. I was wondering if this is even safe and what would the cost be of updating the feeder line? How to wire a ct meter pan [ 1 Answers ] I want to know which side of a ct is the line and which side is the load on a 400 amp meter pan

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# 1. The sum of present ages of Anil and his father is 54 years. Six years before, Anil's father was 6 times as old as Anil. Find their present ages. 2. Rishabhscored thrice as many runs as Varad. Together, their runs fell 8 short of a double century. How many runs did each one score? . Respected Teachers , Please find out the solution for three problems. Friday I have class test in this portion. Dear Student , If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible . Regards • 0 let Anil's present age=x and age of Anil's father=y then. x+y=54____1 and. y-6=6(x-6)=6x-y=30___2 x+y=54 6x-y=30 ________ 7x=84 x=12 y=42 • 3 What are you looking for?

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https://openlab.citytech.cuny.edu/halleckmat1275sp2017/

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# Overall Class Experience The class started off easy than gradually became more difficult as it continued. I feel like I learned many new skills from this class that I didn’t know before. My professor was always there when I couldn’t understand the material or if I was confused. Being said that My experience during this class was good. This exemple of a parabola with test tube. So when the Y =2, what what are the two point for X going to be ? (__, 2) (__,2) This is a perfect exemple of a parabola upside down where the the is propulsed from the top , go up to pick of 2 and return . Here is the équation y= -(x-2)^2+2 # overall experience for this class Math 1275 is the first math course I take in this college. Overall this course is better than what I expected. What I mean is everybody can do good in this class as long as you spend time on doing all the homework and practice questions.  Math is one of tough subject for me, but throught out this semester, it make me feel I can be good with this subject with my professors help. My major is Radiology, for this major I need to know basic algebra and trigonometry to work and now I am another step closer for career.  And math is needed thoughout our daily routine. # My Class Experience I had a both challenging and great experience in this class. Challenging because there were times when the class materials were tough and overwhelming; and great because actually learning the material and managing to do well helped me realize that I can do whatever I put my mind to. Also, algebra and trigonometry is math that I can use throughout both my life and future career as a nurse. In addition, we had a Professor that was there to assist us with any questions and concerns we may have had, and gave us many opportunities to increase our grade. # Parabola Equation A satellite has a parabolic dish with a diameter of -1 cm. The radio signals are reflected on a focal point, being the focus of the parabola. If the focal length is 1.4 cm, find the depth of the dish. # Real Life Parabola The real life parabola that i have chosen is The St.Louis Arch. It is a parabola that goes upward then ends up curving back downwards. Parabolas like these are common in many places. # My Overall experience My overall experience in this class was interesting. As any class it is easy and understandable in the beginning and as class goes on it becomes difficult. I am not strong in math at all, it is very hard for me. But I really hope I get a decent enough grade to help me pursue my goals and dreams in the nursing field. Thank you so much, the professor was a good professor and gave us many alternatives to get extra points or to boost our grades. There`s not a lot of professors like that, so I am grateful. Math is present in the real world esp, in hospital fields. so math relates to nursing in many ways. We do math in our everyday lives. # Surveys On Thursday, we will reserve the last 10 minutes of class for the standard college course evaluation. You will have an opportunity to rate the course, instruction, etc. In addition, this course was part of a grant. Development of certain aspects of the course, webwork for example, received funding from the grant. As part of the evaluation of the grant, we ask that you do an online survey. If we have time on Tuesday next week at the end of class, I may have you do it using your phones. Those who have already done it may be allowed to leave early. There is a lot of legalese that you have agree to, but be assured that I will never have access to any of the individual responses and even the overall responses will be kept from me until the grades have been submitted.

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CC-MAIN-2023-23

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http://stackoverflow.com/questions/8961861/find-the-value-in-the-minimum-number-of-trials

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# Find the value in the minimum number of trials I have an array of 52 different values that I can pass through a class to get a number in return. ``````\$array = array("A","B","C","D"...); `````` Each value passed through the class gives a different number that can be either positive or negative. The numbers are not equally distributed but are sorted in natural order. E.g. ``````\$myclass->calculate("A"); // 2.3 \$myclass->calculate("B"); // 0.25 \$myclass->calculate("C"); // -1.3 \$myclass->calculate("D"); // -6 `````` I want to get the last value that return a number >= 0.20 (in the example would be "B"). This should be done in the minimum number of "class invocation" to avoid time wasting. I thought something like: divide \$array in 2 pieces and calculate the number I get, if it is >= 20, then split the last part of \$array in other 2 smaller pieces and so on. But I don't know if this would work. How would you solve this? - This is called a binary search. – Dmitri Budnikov Jan 22 '12 at 14:43 Or Divide and conquer. It would definitely work splitting your array. – Loïs Di Qual Jan 22 '12 at 15:02 What do you mean for "Divide and conquer"? – Giorgio Jan 22 '12 at 16:56 What you're describing is called a binary search, but it won't really work for this use case, because you aren't searching for a known value. Rather, you're searching for the value that is the lowest number >= 0.2 in a set where the exact value 0.2 may not exist (if it were guaranteed to exist, then you could do a binary search for 0.2, and then your letter would simply be `n - 1; n != 0`).

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http://export.arxiv.org/abs/2203.00816

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math.CO (what is this?) # Title: Mutually orthogonal cycle systems Abstract: An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\ell$-cycle systems of $\Gamma$ is said to be a set of mutually orthogonal cycle systems of $\Gamma$. Let $\mu(\ell,n)$ (respectively, $\mu'(\ell,n)$) be the maximum integer $\mu$ such that there exists a set of $\mu$ mutually orthogonal (cyclic) $\ell$-cycle systems of the complete graph $K_n$. We show that if $\ell\geq 4$ is even and $n\equiv 1\pmod{2\ell}$, then $\mu'(\ell,n)$, and hence $\mu(\ell,n)$, is bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the following upper bounds: $\mu(\ell,n)\leq n-2$; $\mu(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $\mu(\ell,n)\leq 1$ when $\ell>n/\sqrt{2}$; and $\mu'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain computational results for small values of $n$ and $\ell$. Subjects: Combinatorics (math.CO) MSC classes: 05B30, 05C51 Cite as: arXiv:2203.00816 [math.CO] (or arXiv:2203.00816v1 [math.CO] for this version) ## Submission history From: Andrea Burgess [view email] [v1] Wed, 2 Mar 2022 01:44:33 GMT (21kb) Link back to: arXiv, form interface, contact.

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https://www.physicsforums.com/threads/force-on-an-iron-ball-due-to-a-dipole-magnet.595646/

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# Force on an iron ball due to a dipole magnet ## Homework Statement A soft iron ball is fixed a distance d above the pole of a rectangular dipole magnet which is permanently magnetized. What is the force the iron ball feels due to the magnetic field? The dimensions of the dipole magnet are a x a x b, where a < b ## Homework Equations $$B= \frac{μ_0}{4π}\frac{m}{d^3}$$ Dipole Moment: $$m = pl$$ p = magnetic dipole strength (how is this even calculated?) l = displacement vector between poles ## The Attempt at a Solution I know the following: $$F = qv x B$$ But I don't think I can use that because there's no velocity as the iron ball is fixed. The ball has to feel some kind of force, though that formula suggests it isn't possible. It seems to me that if I hung a ball from a string, the tension in the string would increase if a magnet was placed below the iron ball. Is there some concept I'm not understanding, here? $$F_{mag}=\frac{-3μ_0m_1m_2}{4πr^4}$$

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https://www.justcrackinterview.com/interviews/ific-bank-limited/

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# Ific bank limited Interview Questions, Process, and Tips Ques:- In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs. 16.50 per kg? A. 3 : 7 B. 5 : 7 C. 7 : 3 D. 7 : 5 7 : 3 Ques:- Do you apply for another job? Ques:- Find the odd man out: 16, 25, 36, 72, 144, 196, 225 A. 225 B. 196 C. 72 D. 36 72 Because it is not a square number Ques:- Which colour do u like? Ques:- Tell me something abt urself Ques:- What amount does Kiran get if he invests Rs. 18000 at 15% p.a. simple interest for four years? Ques:- What is role of store manegment Ques:- A student scored an average of 80 marks in 3 subjects: Physics, Chemistry and Mathematics. If the average marks in Physics and Mathematics is 90 and that in Physics and Chemistry is 70, what are the marks in Physics? P+C+M=80 P+C=70 Therefore M=10 Given , P+M=90 If M=10 P=80 Ques:- 361 -> 22121 -> 1481 -> 1225 -> XWhat is the value of X? Ques:- The egg vendor calls on his first customer and sells half his eggs and half an egg. To the second customer, he sells half of what he had left and half an egg and to the third customer he sells half of what he had then left and half an egg. By the way he did not break any eggs. In the end three eggs were remaining . How many total eggs he was having ? 31 Ques:- ABOUT my drawings and my personal life and about few technical terms and about any criminal records and etc etc tests Ques:- How will you handle confidential company matters? Ques:- Mr. Rahul spends 30% of his monthly salary on domestic expenses. He spends respectively 20% and 10% of the remaining salary on education of children and conveyance. Of the remaining amounts now he spends respectively 20% and 30%on entertainment and maintenance of house. He saves Rs.5512.50. what is the monthly salary of Mr. Rahul? A. Rs. 22,500 B. Rs. 20,000 C. Rs. 25,000 D. Rs. 24,500 E. None of these How it is possible ? Please write in detail !!! This answer s not a use !!! Ques:- Co-ordination should be an individual responsibility rather than the other insisting the co-ordination. Justify your answer. Ques:- You have changed 4 jobs in 5years, why you changed so many companies? Ques:- What is your father’s name ? Ques:- What is your first reaction when your senior manager assigns a task that you think is impossible? Ques:- A family I know has several children. Each boy in this family has as many sisters as brothers but each girl has twice as many brothers as sisters. How many brothers and sisters are there? 4 boy 3 sis Ques:- The film that won the Best Film Award at 39th International Film Festival of India (IFFI) is? Ques:- A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is: Ques:- P is heavier than Q but lighter than R. Q is heavier than T. S is heavier than P but lighter than V. Who among them is the lightest? (a) V (b) S (c) T (d) R (e) None of these R>P>Q>T V>S>P so the lightest is T. Ques:- What’s the square root of 2000? Ques:- Why do you want to work for STG after working for Retail Industry? Ques:- A train crosses a pole in 9 seconds at the speed of 60 km/hr ?Evaluate the lengthy of the train? 150 m Ques:- You have been given sand time clocks of 4 and 7 minutes. How to measure 9 minutes? 7 min clock:|——-7——-| 4 min clock:|—–4—-|—-4—–|—–4—-|—–4—-| you got 9 min: |—————9————–| Ques:- Checked communication skills Ques:- Why are you changing your industry, I think this industry is not right option for you? Ques:- What is a Crinoline? A. Skirt B. Shirt C. Thong D. Petticoat

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http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=1607

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Welcome to ZOJ Problem Sets Information Select Problem Runs Ranklist ZOJ Problem Set - 1607 Varacious Steve Time Limit: 2 Seconds      Memory Limit: 65536 KB Steve and Digit bought a box containing a number of donuts. In order to divide them between themselves they play a special game that they created. The players alternately take a certain, positive number of donuts from the box, but no more than some fixed integer. Each player's donuts are gathered on the player's side. The player that empties the box eats his donuts while the other one puts his donuts back into the box and the game continues with the "looser" player starting. The game goes on until all the donuts are eaten. The goal of the game is to eat the most donuts. How many donuts can Steve, who starts the game, count on, assuming the best strategy for both players? Write a program that: > reads the parameters of the game from the standard input, > computes the number of donuts Steve can count on, > writes the result to the standard output. Input The rst and only line of the input contains exactly two integers n and m separated by a single space, 1 <= m <= n <= 100 - the parameters of the game, where n is the number of donuts in the box at the beginning of the game and m is the upper limit on the number of donuts to be taken by one player in one move. Process to the end of file. Output The output contains exactly one integer equal to the number of donuts Steve can count on. Sample Input 5 2 Sample Output 3 Source: Central Europe 2002 Submit    Status

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# The length of an arc and the perimeter of a sector ## Home learning focus Learn how to find the length of an arc and the perimeter of a sector. This lesson includes: • learning summary • two activity sheets # Learn Students looking to achieve grade 4 in GCSE Maths must understand how to calculate the length of an arc and the perimeter of a sector. Read page 5 of our 'Circles, sectors and arcs' Bitesize revision guide and page 3 from our 'Sector, segment and arc' Bitesize revision guide to understand: ## Arc length An arc is a portion of the circumference of a circle. A chord separates the circumference of a circle into two sections - the major arc and the minor arc. It also separates the area into two segments - the major segment and the minor segment. Worked example Calculate the arc length to 2 decimal places (dp). First calculate what fraction of a full turn the angle is. The angle is 90°, so it is one quarter of the whole circle (360°). The arc length is ¼ of the full circumference. The circumference of a circle is πd (or π × diameter) and the diameter is = 2 × radius. Remember that π is = 3.14 (to 2 dp) The formula to calculate the arc length is: Arc length = angle ÷ 360° × π × d The arc length is: ¼ × π × 8 = 2π is 2 × 3.14 The arc length is 6.28 cm (to 2 dp). ## Perimeter of a sector The perimeter is the distance all around the outside of a shape. We can find the perimeter of a sector using what we know about finding the length of an arc. A sector is formed between two radii and an arc. To find the perimeter, we need to add these values together. Perimeter = Arc length + 2r Worked example In this diagram, the arc length (27.5 cm) and the radius (45 cm) are shown. From this the perimeter can be calculated: Perimeter = 27.5 + (2 × 45) 27.5 + 90 = 117.5 cm The perimeter is 117.5 cm. # Practise ## Activity 1 Finding the length of an arc Complete the activity sheet from TES on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper. ## Activity 2 The length of an arc Complete the activity sheet from White Rose Maths on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper.

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Qwika Toolbar for IE and Firefox users! ## Home > English Searching 21,964,380 articles in 1,158 wikis. Beta release. Any comments please contact us Press release (Feb 17): New search engine helps bridge the language gap in Wikipedia Press release (Apr 4): Qwika search engine now indexes 1158 wikis in 12 languages Search wikis: Results for Applied statistics   1 to 10 of 3294 Applied statistics Applied statistics Applied statistics is the use of statistics and statistical theory in real-life ... http://en.wikipedia.org/wiki/Applied_statistics - 1k - Cached - Similar pages Applied statistics (Psychology wiki) Applied statistics Applied statistics is the use of statistics and statistical theory in real-life ... http://psychology.wikia.com/wiki/Applied_statistics - 1k - Cached - Similar pages Talk:Applied statistics Talk:Applied statistics How is statistics a science of appropriation Why does this ... here that wouldn't go on the statistics page? -- Walt Pohl 01:34, 7 October ... http://en.wikipedia.org/wiki/Talk:Applied_statistics - 0k - Cached - Similar pages Statistics Statistics This article is about the field of statistics. For statistics of Wikipedia, see Special:Statistics. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various ... http://en.wikipedia.org/wiki/Statistics - 33k - Cached - Similar pages Statistics (Psychology wiki) Statistics Home Support Help Site Support Orientation Background Policies To Do Areas Philosophy Psychometrics Statistics Experimental Comparative Biological Language Developmental Soc. processes ... Professional Education Organisational Other fields Transpersonal World Statistics Research methods Exp. design Stats. tests Stats ... bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading ... scores, standard nine, and percent in stanine. Statistics refers to the branch of mathematics ... http://psychology.wikia.com/wiki/Statistics - 31k - Cached - Similar pages Statistics   (translated from German) Statistics As one Statistics designates one: name-giving (v. lat.: status ... barrier around 1749); those today as official statistics away-lived; in addition, independently of the ... already for over 5000 years as Population statistics and Business statistics existed; of it generalizing quantitative Collections ... http://de.wikipedia.org/wiki/Statistik - 19k - Cached (German) - Wikipedia (German) - Similar pages Statistics   (translated from French) Statistics statistics is the science and the practice of ... theory who is a branch of mathematics applied. One can bind it to decision theory ... not interdict to use them jointly. descriptive statistics mitigate simply a weakness of the human ... values, values of dispersion, histograms, etc. mathematical statistics has a more ambitious objective: to ... http://fr.wikipedia.org/wiki/Statistiques - 26k - Cached (French) - Wikipedia (French) - Similar pages Statistics   (translated from Greek) Statistics Statistics it is science that attempts it exports ... the use of statistical theory, a sector applied mathematics. In the statistics, tyhajo'tita and vagueness they are fixed ... frames of theory probabilities. The practice of statistics includes the designing, collection and interpretation ... http://el.wikipedia.org/wiki/Στατιστική - 5k - Cached (Greek) - Wikipedia (Greek) - Similar pages Statistics   (translated from Portuguese) Statistics Statistics it is a probabilisticas science that uses ... forecast and organization of the future. The Statistics it is also a practical science and ... knowledge through the use of empirical data. Statistics is based on the theory, a branch of the applied mathematics. In the theory statistics, the ... http://pt.wikipedia.org/wiki/Estatística - 12k - Cached (Portuguese) - Wikipedia (Portuguese) - Similar pages Applied   (translated from German) Applied Applied a city is in Sacramento County in ... persons per household with 3,23. Age statistics Distribution of the population on age groups ... on the average 98 men come into applied. If one regards the adult part of ... the families of the city live in applied 10.6 for % of the population ... http://de.wikipedia.org/wiki/Galt - 4k - Cached (German) - Wikipedia (German) - Similar pages Page:1 2 3 4 5 6 7 8 9 10 Next >> Search wikis: Search: Try your search on: FactBites (sentence-based)

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https://www.physicsforums.com/threads/i-keep-getting-negative-volumes-volume-of-sin-x-around-y-c-c-is-in-0-1.465870/

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# I keep getting negative volumes (volume of sin(x) around y=c, c is in [0,1]) ## Homework Statement The arch y= sin(x), x is in [0, pi], is revolved around the line y=c, where c is a constant in [0, 1], to generate a solid... Anyway, then I have to represent the volume of the solid as a function of c and other stuff. ## The Attempt at a Solution I remember asking in class and the proff said not to find the points of intersection with c. The volume of the solid would be the integral of sin(x)^2 - c^2 from x=0 to x=1. OR the integral of c^2 - sin(x)^2, x=0--->1. Either way, there are values of x for which sin(x) is larger than c, and some where it is smaller. And either way, there are values of c for which I get a negative number, which makes no sense. What am I doing wrong? I can't think of any way of doing this without finding the intercepts, which is said specifically not to do. Related Calculus and Beyond Homework Help News on Phys.org LCKurtz Homework Helper Gold Member I think this problem is either not well posed or well understood. You don't rotate "the arch". You rotate an area, presumably the area under the arch, to create a volume. As you have noticed, if c is in (0,1) the area wraps on itself. Your teacher needs to clarify what he wants you to do about that. Does he not want to count the doubled part? It is a poorly phrased problem as you give it. Also, did the question ask to rotate the area between the curve and the line? Last edited: I think this problem is either not well posed or well understood. You don't rotate "the arch". You rotate an area, presumably the area under the arch, to create a volume. As you have noticed, if c is in (0,1) the area wraps on itself. Your teacher needs to clarify what he wants you to do about that. Does he not want to count the doubled part? It is a poorly phrased problem as you give it. I copied that from my assignment. When I asked him, he said the volume will look like a candy wrapper. Which is what happens if I find the sin(x) intercepts with c, and take the areas either under c when c > sin(x), or under sin(x) when sin(x) > c. But he said that this is not to be done, that it should be a single integral, (he added that it should be the area under the curve). This thing is giving me a headache because it looks completely contradictory. How can I take the area under sin(x) without finding the intercepts and not get negative volumes at some values of c? LCKurtz Homework Helper Gold Member I think this problem is either not well posed or well understood. You don't rotate "the arch". You rotate an area, presumably the area under the arch, to create a volume. As you have noticed, if c is in (0,1) the area wraps on itself. Your teacher needs to clarify what he wants you to do about that. Does he not want to count the doubled part? It is a poorly phrased problem as you give it. Also, did the question ask to rotate the area between the curve and the line? I copied that from my assignment. When I asked him, he said the volume will look like a candy wrapper. Which is what happens if I find the sin(x) intercepts with c, and take the areas either under c when c > sin(x), or under sin(x) when sin(x) > c. But he said that this is not to be done, that it should be a single integral, (he added that it should be the area under the curve). This thing is giving me a headache because it looks completely contradictory. How can I take the area under sin(x) without finding the intercepts and not get negative volumes at some values of c? From what you have written I think you understand very well that there is something wrong with the statement of the problem. The only interpretation that makes any sense to me is to rotate the area between the line and the curve about the line. So the integrand on the ends would be π(c2 - sin2(x)) and on the middle part would be π(sin2(x) - c2). And I agree with you, you need the values where sin(x) = c to do the limits. Last edited: LCKurtz Homework Helper Gold Member And, for extra credit, find the value of c for which the volume generated between the arch and the line is a minimum. And, for extra credit, find the value of c for which the volume generated between the arch and the line is a minimum. That's part (a) (and the max) I guess I'll go head and do it with the intercepts. An answer is an answer right? I think I know what I have to do. The radius of the figure will essentially be the absolute value of c-sin(x), so the integral is just pi*(c-sin(x) )^2. That's how you get the shape he drew on the board when I asked. ... I think the question could have been worded differently, or at least included the figure he drew, because without it it can be interpreted in two different ways :/ LCKurtz Homework Helper Gold Member I think I know what I have to do. The radius of the figure will essentially be the absolute value of c-sin(x), so the integral is just pi*(c-sin(x) )^2. That's how you get the shape he drew on the board when I asked. ... I think the question could have been worded differently, or at least included the figure he drew, because without it it can be interpreted in two different ways :/ Careful, you mean π|c2-sin2(x)|, not what you wrote. And you still need those sin-1(c) limits to evaluate it. Careful, you mean π|c2-sin2(x)|, not what you wrote. And you still need those sin-1(c) limits to evaluate it. Wouldn't pi*(c-sin(x))^2 work in this case? -- the area is being revolved around c, and the distance from c to the curve of sin(x) is |c-sin(x)|. So choosing |c-sin(x)| to be R, would make the integral pi*R^2. |c^2-Sin(x)^2| would be the area under sin(x). This is the figure he drew in class when I asked: |>O<| like a candy wrapper. From the figure, I think he wants the area between c and sin(x). LCKurtz Homework Helper Gold Member Wouldn't pi*(c-sin(x))^2 work in this case? -- the area is being revolved around c, and the distance from c to the curve of sin(x) is |c-sin(x)|. So choosing |c-sin(x)| to be R, would make the integral pi*R^2. |c^2-Sin(x)^2| would be the area under sin(x). This is the figure he drew in class when I asked: |>O<| like a candy wrapper. From the figure, I think he wants the area between c and sin(x). Yes, you are correct. I responded too quickly this morning and was thinking revolving about the x axis when I wrote that formula.

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# i want code for the below given task 9 views (last 30 days) kalyani chava on 4 Apr 2020 Edited: DGM on 29 Apr 2022 Load an ECG signal. Add a 50 Hz, 100 Hz and 150 Hz frequency sinusoidal signals to this ECG signals. Plot the original ECG signal, corrupted ECG signal and their spectrum. Ameer Hamza on 4 Apr 2020 Exactly, this is clearly an assignment or a project. I hope that you don't expect us to complete it for you. As Peng mentioned, you need to show some code that you have already tried. This forum is focused on MATLAB related questions, e.g., errors, syntax issues, suggestions, etc. Your problem is not specifically related to MATLAB, and it is still very kind of Peng to solve Task 1 of the assignment for you. Now you can extend it further to solve the other two tasks. Peng Li on 4 Apr 2020 close all; clear; %% this is an examplary ECG recording with sampling frequency 200 hz fs = 200; N = length(y); % show data hfd = figure('Name', 'Orignal data and contaminated data'); t = (0:1/fs:(N-1)/fs)'; % zoom in to better visualize data title('Original ECG'); xlabel('time (s)'); % add 50 Hz noise, SNR 1:1 fn1 = 50; n50 = std(y).*sin(2*pi*fn1*t); yn50 = y + n50; title('Original ECG contaminated by 50 Hz noise'); xlabel('time (s)'); %% fft Fy = fft(y); f = fs/N .* ((-N/2+1):N/2); % show fft hff = figure('Name', 'FFT'); haf1 = subplot(211); plot(f, fftshift(abs(Fy)./N)); haf1.XLim = [0 fs/2]; title('Amplitude spectrum of orignal ECG'); xlabel('Frequency (Hz)'); ylabel('Amplitude spectrum'); % fft of contaminated data Fyn50 = fft(yn50); % show fft haf2 = subplot(212); plot(f, fftshift(abs(Fyn50)./N)); haf2.XLim = [0 fs/2]; title('Amplitude spectrum of ECG + 50 Hz noise'); xlabel('Frequency (Hz)'); ylabel('Amplitude spectrum'); ### Categories Find more on Signal Operations in Help Center and File Exchange ### Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting! Translated by

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• A • B # Friday 19th May Friday 19th May REMINDER SPORT'S DAY - It will be held on Tuesday 23rd May 9am - 11 am (weather permitting). Should it need to be cancelled there will be posters displayed around school and text messages will be sent out. CHOIR, CAKE and a CUPPA - Why not come along to school if you're free on Wednesday 24th May at 9am and watch Westfield's OUTSTANDING choir perform a selection of songs? Tickets are on sale at the Front Office for £1 each. All the proceeds will go towards the School Christmas Party Fund for the children. Welcome to this week’s blog from Year 5F. On Thursday, both Year 5 classes visited Liverpool as part of our topic on the Ancient Greeks. The children were fantastic and had a fabulous day. Thankfully, the weather stayed dry for us all, which was a massive bonus! In the morning, we visited Liverpool Museum where the children had chance to explore all five floors! They visited the aquarium, the bug house, the Ancient Egyptian display which had REAL mummies lying in their sarcophaguses, the dinosaur exhibition with its impressive display of skeletons and the top floor which was all about space. After lunch, we moved on to the Walker Art Gallery. The children had time to view all the MAGNIFICENT paintings and sculptures throughout the building. Some of the artwork dated back to the 1400s! After that, we gathered together and were taken to view a painting depicting a scene from a Greek Myth. The painting is called Atalanta and Meleager by Charles LeBrun. Take a look for yourself at this simply stunning painting... The children sat mesmerised and spellbound as they heard the tragic story behind the painting. Unhappy with the terribly sad ending, they rewrote it and then performed their new (much happier) ending with their partners. Here are a few photos from the day… Maths – The children have continued their work investigating angles. They have been learning about reflex angles and how to measure them. A reflex angle is an angle that measures more than 180°. Also, they have been working out missing angle measurements along a straight line. A straight line measures 180° and if you are given one of the angle measurements it is just a case of subtracting that from 180 and you have the solution to the missing angle measurement! Take a look at this website which will give you the chance to use what you know about angles to complete the mission… English – The children have been learning about Ancient Greek myths. A definition of a myth is a story without an author that is passed along by word of mouth, from generation to generation, and is usually intended to teach a lesson. They have read some the stories about Perseus and his bravery. They have learned about Theseus and the Minotaur, Athena and Medusa and later how Perseus slayed Medusa, Arachne the Spinner as well as Persephone and her mother Demeter. The children are enjoying reading these action-packed, magical stories from the past. Take a look at this website where you can either read or watch these amazing Greek myths (as well as some from other countries around the world including England)… Topic – The children have been finding out about Greek gods and goddesses. Did you know they lived at the top of Mount Olympus? Also, Zeus was the King of all the gods and was married to his sister Hera! The children have been fascinated by this mysterious world of gods, goddesses and demi-gods! They have begun to design their very own god or goddess. Science – The children have been investigating air resistance on different objects. They have looked carefully at the shape of the object and asked the question…will the shape affect the amount of air resistance on the object? Here is our CERTIFICATE WINNER for this week. He won for working super hard to improve his attitude over the last few weeks. His mature attitude shone during yesterday’s trip to Liverpool. All that remains to say is have a fabulous weekend and we’ll see you again on Monday. Please try to support our School Choir Coffee Morning on Wednesday.  Check out the details below. Finally, have your say.  Each week you will have a chance to decide on which new song we will learn in next week's whole school assembly.  Click on the link below to take part in an online vote. (Note -this is carried from last week's choices.) Welcome to our website. ........ .Attendance for w/c 18th June 2018... RW 93%, RF 90%, 1W 97%, 1F 96%, 2W 96%, 2F 90%, 2WF 93% 3W 94%, 3F 92%, 4W 97%, 4F 93%, 5W 98%, 5F 96%, 6W 92%, 6F 98% ** ** Congratulations to 5W and 6F for achieving the highest attendance. Well Done!**.... Top

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Sep 17, 2016 Here's what I got. #### Explanation: The thing to remember here is that • a single bond $\to$ contains one sigma bond • a double bond $\to$ contains one sigma bond and one pi bond • a triple bond $\to$ contains one sigma bond and two pi bonds In order to find the number of sigma and pi bonds present in a molecule of cyanogen, ${\text{C"_2"N}}_{2}$, you must examine its Lewis structure. As you can see, the cyanogen molecule contains one $\text{C"-"C}$ single bond and two $\text{C"-="N}$ triple bonds. This means that you will have • a total of three sigma bonds Here you have one sigma bond from the $\text{C"-"C}$ single bond and one sigma bond from each of the two $\text{C"-="N}$triple bonds • a total of four pi bonds Here you get two pi bonds from each of the two $\text{C" -= "N}$ triple bonds

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Home Hostname: page-component-dc8c957cd-rqfrn Total loading time: 0.4 Render date: 2022-01-28T00:45:10.480Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true } # Part V - Energy-Based Numerical Solutions Published online by Cambridge University Press:  05 June 2012 ## Summary Preface In Chapters 16–2, the focus shifts from solving differential equations to employing the Principle of Virtual Work, or the Principle of Complementary Virtual Work, or, for those who prefer them, the corresponding energy principles. The goal remains the same: to solve larger and more complicated structural analysis problems. The shift in focus is more stylistic than fundamental. As the last chapter's endnotes demonstrate, the beam differential equations follow from either the Principle of Virtual Work, in the case of bending and extension, or the Principle of Complementary Virtual Work, in the case of twisting. Although not demonstrated here, the reverse path from differential equations to a work or energy principle is also possible when the differential equation meets certain requirements as described by Ref. [3], p. 158. Hence it is essentially a matter of convenience whether a differential equation or a work principle is the starting point of an analysis. If the structure contains more than a couple of structural elements, it is usually, if not always, the work or energy principles that are most convenient. Indeed, one particular application of the Principle of Virtual Work – the finite element method – coupled with modern digital computers, permits the routine analysis of structures with many thousands of structural elements. The finite element method is a numerical method that is unperturbed by geometric or material complexity, and it allows the analyst to minutely model (and thus analyze) one part of a structure while getting by with a crude model of other parts of the structure. Type Chapter Information Analysis of Aircraft Structures An Introduction , pp. 521 - 522 Publisher: Cambridge University Press Print publication year: 2008 ## Access options Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.) ### Purchase Buy print or eBook[Opens in a new window] # Send book to Kindle Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service. Available formats × # Send book to Dropbox To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Available formats × # Send book to Google Drive To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Available formats ×

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# Free Energy | searching for free energy and discussing free energy ## New theories about free energy systems => Theory of overunity and free energy => Topic started by: rushi95 on February 05, 2018, 03:15:54 AM Title: Mechanical Lever to get pass sticky point Post by: rushi95 on February 05, 2018, 03:15:54 AM What is the issue with a mechanical lever to move stator magnets and get pass the sticky point? something similar to (https://www.youtube.com/watch?v=bssBAb6EzM4&t=48s) V gate motor. All my understanding supports in favour of V gate motor design. But, I am not sure why there are failed attempts to replicate. Can you help me in knowing what might be the possible issues with this idea. OR in general will a mechanical lever help in avoiding sticky point? What I do hear it takes more mechanical energy than what can be produced in one revolution. I have no means to verify that. What are your thoughts on this? Title: Re: Mechanical Lever to get pass sticky point Post by: Low-Q on February 07, 2018, 10:11:11 AM What is the issue with a mechanical lever to move stator magnets and get pass the sticky point? All my understanding supports in favour of V gate motor design. But, I am not sure why there are failed attempts to replicate. Can you help me in knowing what might be the possible issues with this idea. OR in general will a mechanical lever help in avoiding sticky point? What I do hear it takes more mechanical energy than what can be produced in one revolution. I have no means to verify that. What are your thoughts on this? A lever is doing movement as any other movements. When you avoid the sticky spot, you need some input force. And with that force you need desplacement to actually move the object away from the sticky spot. Now, the sticky spot is the reason why magnets attract eachother. This force is what we want in a magnet motor, but when the driving forces increase, the closer the magnets are. At some point, the force is not longer perpendicular but purely radial. A radial force just wants to pull the rotor off the hub and not contribute to rotational motion. This is where the sticky spot is. By using a mechanism to avoid the sticky spot, at the same time you remove the force which drives the rotation, because that mechanism is driven by the same closed system. You need a separate mechanism that is not connected to the rotor, but that means you have to apply energy to that mechanism. The V-gate is nothing different from lettig two magnets approach eachother. The magnetic field at the end of the track is where the field is strongest, working perpendicular to the wanted direction. Pass that point you still have a strong local magnetic field that wants to pull the moving object in reverse. Theerfor V-gates can't work. Apologies for typing errors. My cellphone have very small "keys" and my figer is large :-) Br. Vidar Title: Re: Mechanical Lever to get pass sticky point Post by: Belfior on February 07, 2018, 11:30:19 AM V-gates can totally work. Just have a track between the 2 strips of magnets and a hole in the end to the track. Before a ball gets to the end of the track it drops through the hole. Under the hole is another V-gate. You tell me how this cannot work? I think lack of imagination and intuition is the only thing that does not work. Title: Re: Mechanical Lever to get pass sticky point Post by: Low-Q on February 07, 2018, 01:21:09 PM V-gates can totally work. Just have a track between the 2 strips of magnets and a hole in the end to the track. Before a ball gets to the end of the track it drops through the hole. Under the hole is another V-gate. You tell me how this cannot work? I think lack of imagination and intuition is the only thing that does not work. You are right about the hole. The ball will be able to drop, but, that is becaus the initial position of the ball is inside attraction area. Do you know what happen if to put the ball too far away from the V-track input? It will be repelled. Yes, repelled - believe it or not. So when you place a ball at the very entrance of a V-track, you have already applied the energy required to enter the V-track. That is why the ball can continue along the track, and finally drop through that hole. Second, the track must have an incline, then you have a SMOT. But the next V-track will repel the ball slightly before it finish the first track, AND the end of the first track will hold back the drop a little bit due to the strong magnetic field, so the ball cannot fall with 9.81m/s^2. The total magnetic gain is perfectly zero in a closed loop. Friction and eddy currents is two of the factors that prevents the ball to complete a cycle and accelerate. Imagination and intuition does not help. Physics do things we do not want it to do, and therefor, you cannot force a device like this to work continously, or absolutely not accelerate, just by using imagination and intuition. Quite frakly, nature does not care about our imaginations. It does what it is created to do. Conserve energy ;-) Vidar Title: Re: Mechanical Lever to get pass sticky point Post by: vineet_kiran on February 07, 2018, 02:19:04 PM If magnets are correctly arranged, it should be possible to jump the sticky spot.  See this video: Can somebody 'refine' it? Title: Re: Mechanical Lever to get pass sticky point Post by: Low-Q on February 07, 2018, 02:49:10 PM If magnets are correctly arranged, it should be possible to jump the sticky spot.  See this video: Can somebody 'refine' it? This visio is fake. Busted many years ago. It loops approx every 1 second for a while before the video continue as normal when the magnet stops. Vidar Title: Re: Mechanical Lever to get pass sticky point Post by: vineet_kiran on February 08, 2018, 09:56:16 AM This video is fake. Busted many years ago. Vidar This one ? Title: Re: Mechanical Lever to get pass sticky point Post by: Low-Q on February 08, 2018, 10:28:40 AM This one ? I don't know what drives this. With no back emf this runs surprizingly smooth. Why is this motor located so close to the edge? Why doesn't it bounce around? Why not lift it up to prove nothing under the table runs it with magnetic coupling? Vidar Title: Re: Mechanical Lever to get pass sticky point Post by: sm0ky2 on February 08, 2018, 12:01:11 PM Using gravity to drive a magnetic assembly in this manner IS possible. It, however, requires a lot of precision and an appropriate magnetic-gravitational proportionality. To understand this, you have to look at it in terms of vertical lift, vs horizontal displacement. When repulsion is primarily in the vertical domain, ‘lift’ occurs, as the magnetism is counteracting against gravity. When the repulsion is more horizontal, downward gravitational force can create a horizontal translation. The difference between these two actions is where the energy comes from. How much energy? In an ideal state, it is equal to the combined magnetic repulsion force -9.8m/s/s In actuality, our designs operate at less than this value. why? Because in a vertical-lift scenario gravity subtracts from the magnetic repulsion. In a horizontal translation, we have the full repulsion force. gravity in the horizontal case pushes the magnet, which repels the rotor. The test-bed i designed for adjusting the parameters of the Archer Quinn device is a good tool for determining the magnetic repulsion vs gravity proportion. Basically a vertical slide (rod/tube/linear bearing/etc) with one of the magnets attached to the bottom. The other magnet of the repelling pair is placed under it. You can then adjust the two magnets by distance and add/subtract weight to the magnet being lifted. From this you can develop a mass-vs-distance scale for your magnet pair. Every pair of magnets is slightly different, so if you require similar forces in your design, you may need to test a lot of magnets and select the ones that are the closest Two important things to reiterate When the magnets do work against gravity, the repulsion should be mostly vertical When gravity does work against the magnets, the repulsion should be mostly horizontal. And a word of warning: use caution with this, because when you have the parameters correct for it to run, using the difference between gravity and magnetism, it will not run at a constant rpm, but continue to increase until it destroys itself. Which is very dangerous, in fact, becomes increasingly dangerous the stronger you construct the device. It is best to start with low-strength materials, that will break apart with little force. the slower it is spinning when it breaks, the less chance is seriously injuring yourself or others, and causing damage to your home. Title: Re: Mechanical Lever to get pass sticky point Post by: Belfior on February 08, 2018, 12:23:29 PM Imagination and intuition does not help. Physics do things we do not want it to do, and therefor, you cannot force a device like this to work continously, or absolutely not accelerate, just by using imagination and intuition. Quite frakly, nature does not care about our imaginations. It does what it is created to do. Conserve energy ;-) Vidar I don't see any evidence, that nature would like to conserve energy. Nature wants to have things in equilibrium. If you disturb that, nature will go to extreme lengths to equalize that. Conservation of energy is just a property of energy. Energy conservation is used by free energy debunkers to "prove" that free energy does not exist. They want to steer the debate into an area where energy is created from nothing. In reality we are just trying to dip a bucket to an ocean. Nature will fill the hole that the bucket made. What I mean with the use of intuition and imagination is that do not take anything for a "Natural Law" that some white dude wrote 200 years ago. We have moved from experimentation to theoretical physics and proving things just by solving mathematical equations. We can see how this is going towards total bullshit just because people can't admit that they might be wrong OR that their demi-god master jedi might have been wrong. Now we are adding Dark Mater AND the new addition Dark Energy into equations, because we can't explain what is happening otherwise. We are using virtual particles and "cosmological constants" and the reason is "I had to add that or it does n't make any sense". I am not saying that Einstein was wrong in everything or that Newton was just crazy. What I'm saying is that we need to take few steps back every time we find out that our equations need a new virtual element or reality does not make sense. Science is turning into archaeology where professors hide new evidence or deny any new findings. The Sphinx is 4511y old and there is no water erosion! It is the tomb of Khufu! Title: Re: Mechanical Lever to get pass sticky point Post by: Low-Q on February 08, 2018, 01:37:43 PM @ sm0key: Gravity is a constant, and cannot provide help. Permanent magnetism cannot provide help. As long the devices we build is based on elementary physics, there cannot be any way anyone can calculate the precision required to make these things work - creating energy from nothing, because there are no prerequisites available for making this happen. @ Belfior: Equilibrium is what conservation is much about. Though, the sun is providing energy all the time, so this source is the only source we actually can harness through power plants of different sorts. The sun has for a billion years, and in billions of years to come, worked/working towards equilibrium. As this happens, energy is released due to the nuclear potential energy that already exist in the hydrogen atoms. A small motor at 200 grams cannot crush atoms. The thing is, by my perspective, is to harness potential energy which is just waiting for being released. As long we have potential energy in mass or in magnetism, we cannot harness these potentials without destroying the mass or weaken the magnets. Our physics is limited to the very elementary ones. We cannot make things work just because we cannot explain dark matter or other exotic properties of the universe. The 200 years old discoveries are still valid today - with remarkable precision. Vidar Title: Re: Mechanical Lever to get pass sticky point Post by: Belfior on February 08, 2018, 02:19:02 PM "Equilibrium is what conservation is much about." In a way yes. Nature can use a lot of energy to gain equilibrium, but since you can only transform energy that did not destroy any energy. It just changed form or potential. All we need to do is create unequilibrium and provide a path for nature to correct this. "Though, the sun is providing energy all the time, so this source is the only source we actually can harness through power plants of different sorts." Sure we can use the Sun, any other radiation from space or something that has potential already on this planet because of those radiations. Some experiments say that the magnetic field does not rotate with the Earth. We could just have 500m long coils on the ground to pick up electricity. "The sun has for a billion years, and in billions of years to come, worked/working towards equilibrium. As this happens, energy is released due to the nuclear potential energy that already exist in the hydrogen atoms. A small motor at 200 grams cannot crush atoms." I think splitting atoms is the last thing we should be doing. It is the worst thing that we have developed. I don't think Moray had a fission plant in his wooden box. "The thing is, by my perspective, is to harness potential energy which is just waiting for being released. As long we have potential energy in mass or in magnetism, we cannot harness these potentials without destroying the mass or weaken the magnets. Our physics is limited to the very elementary ones. We cannot make things work just because we cannot explain dark matter or other exotic properties of the universe. The 200 years old discoveries are still valid today - with remarkable precision." We also cannot make things work, if we just take theory for the truth. There are older discoveries that are still valid and younger that are just dead in the water. I have serious doubts about some "natural laws" that some men have come up with. I find that with nature it is always about circumstances and not so much that something is illegal. These artificial rules are just holding us back. You might not be able to push a rocket to light speed, but does that also mean that you cannot travel from A to B faster than light? There might be some way to do this, but we will never find it, because "it is illegal to go faster than light". Title: Re: Mechanical Lever to get pass sticky point Post by: sm0ky2 on February 11, 2018, 05:32:47 AM @ LowQ Gravity alone does not provide energy. Nor does magnetism, by itself. But the difference between the two can be made to perform work. As can the difference between two magnetic forces. Field symmetry is the wrong approach A conservative field, by itself, is exactly that. A difference in fields results in a vectored potential. reverse the conditions and the potential is in the opposite direction. this is the key. For example, magnetism opposing a gravitational vector, increases gravitational potential. gravity opposing a magnetic vector increases magnetic potential. A system that utilizes both of these functions, can constantly increase it’s own kinetic energy.

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# Multiplying whole numbers worksheets ideas » » Multiplying whole numbers worksheets ideas Your Multiplying whole numbers worksheets images are available in this site. Multiplying whole numbers worksheets are a topic that is being searched for and liked by netizens today. You can Download the Multiplying whole numbers worksheets files here. Get all royalty-free photos and vectors. If you’re searching for multiplying whole numbers worksheets pictures information linked to the multiplying whole numbers worksheets keyword, you have come to the ideal site. Our site frequently gives you suggestions for seeing the maximum quality video and picture content, please kindly search and locate more informative video content and graphics that match your interests. Multiplying Whole Numbers Worksheets. English School subject. Multiplying Mixed Numbers by Mixed Numbers. Some of the worksheets for this concept are Mixed whole number s1 Mixed number multiplication l1s1 Multiplying mixed numbers Multiplying mixed numbers Multiplication of mixed numbers Fractions work Multiplyingdividing fractions and mixed numbers. In these worksheets one of the multiplicands is a whole number between 1 and 10. The Multiplying 3 Digit By 2 Digit Numbers With Various Decimal Places A Math Wor Multiplying Decimals Decimal Multiplication Multiplying Decimals Worksheets From pinterest.com Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions. Change the mixed numbers to improper fractions cross-cancel to reduce them to the lowest terms multiply the numerators together and the denominators together and convert them to mixed numbers if improper fractions. To downloadprint click on pop-out icon or print icon to worksheet to print or download. In these worksheets one of the multiplicands is a whole number between 1 and 10. Multiplying Fractions by Whole Numbers Engage grade 4 and grade 5 kids with this bunch of printable multiplying fractions by whole numbers worksheets and reinforce skills in multiplying fractions and mixed numbers by single and two-digit whole numbers. Decimal and Whole - 2. ### Includes whole numbers decimal numbers and negative numbers. These worksheets are pdf files. Missing factor questions are also included. Multiply Decimals by Whole Numbers. Multiply Decimals by Whole Numbers Find the product for each exercise. Decimals Tenths and Hundredths by mrskrause. Simply multiply the numerators and the denominators by each other and then simplify. Source: pinterest.com Change the mixed numbers to improper fractions cross-cancel to reduce them to the lowest terms multiply the numerators together and the denominators together and convert them to mixed numbers if improper fractions. Decimal and Whole - 1. With oodles of practice in using repeated addition to multiply fractions finding the product of fractions and whole numbers our. These decimals worksheets are pdf files. Multiplying Decimal with Whole Number. Source: pinterest.com Journey through this ensemble of printable multiplying fractions with whole numbers worksheets and be au fait with the steps followed in multiplying proper improper and unit fractions and mixed numbers by whole number multipliers. Decimal and Whole - 1. Multiplying Whole Numbers -2 Multiplying With and Without Regrouping ID. Multiplying Tenths - Moderate. Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions. Source: pinterest.com Multiplying Whole Numbers -2 Multiplying With and Without Regrouping ID. Apply to the whole worksheet. To downloadprint click on pop-out icon or print icon to worksheet to print or download. More Multiplication without Regrouping interactive worksheets. Multiply Decimals by Whole Numbers. Source: pinterest.com Multiplying Decimal with Whole Number. The html format is even editable. More Decimals interactive worksheets. Multiply decimals by whole numbers or other decimals These practice exercises range from multiplying one digit decimals by whole numbers to general multiplication of multi-digit decimals in columns. Includes whole numbers decimal numbers and negative numbers. Source: pinterest.com English School subject. Simply multiply the numerators and the denominators by each other and then simplify. Apply to the whole worksheet. Similar sets of ordering numbers worksheets are presented in both horizontal and vertical formats. Practice ordering numbers worksheets for with multiple numbers in ascending greatest to least and descending least to greatest orders. Source: pinterest.com Multiply Decimals by Whole Numbers Find the product for each exercise. Simply multiply the numerators and the denominators by each other and then simplify. 08102020 To download the above worksheet as a PDF click here. Multiplying decimals by whole numbers Math worksheets. Includes whole numbers decimal numbers and negative numbers. Source: pinterest.com Decimal and Whole - 1. Free Multiplying Fractions by Whole Numbers flash cards. You can also customize them using the generator below. To downloadprint click on pop-out icon or print icon to worksheet to print or download. Multiplying fractions with a whole number worksheet game quiz and flash card. Source: pinterest.com Decimal and Whole - 2. Missing factor questions are also included. Kindergarten 1st Grade 2nd Grade 3rd. When youre not multiplying fractions by a whole number the same methodology applies. These decimals worksheets are pdf files. Source: pinterest.com Multiplying Decimal with Whole Number. More Multiplication without Regrouping interactive worksheets. Multiplying decimals by whole numbers Math worksheets. Includes whole numbers decimal numbers and negative numbers. When youre not multiplying fractions by a whole number the same methodology applies. Source: pinterest.com To downloadprint click on pop-out icon or print icon to worksheet to print or download. Multiplying Tenths - Easy. Sample Grade 5. If you need advice on how to do this check out this post. Make lightning-fast progress with these multiplying mixed fractions worksheet pdfs. Source: pinterest.com Multiplying whole numbers multiply whole numbers by one two and three digit multipliers and powers of 10. When youre not multiplying fractions by a whole number the same methodology applies. With oodles of practice in using repeated addition to multiply fractions finding the product of fractions and whole numbers our. Multiplying fractions with a whole number worksheet game quiz and flash card. You can also customize them using the generator below. Source: pinterest.com These decimals worksheets are pdf files. You can also customize them using the generator below. Decimal and Whole - 2. Decimal and Whole - 1. More Decimals interactive worksheets. Source: pinterest.com Apply to the whole worksheet. Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions. Adding and Subtracting. Multiplying whole numbers multiply whole numbers by one two and three digit multipliers and powers of 10. Multiply decimals by whole numbers or other decimals These practice exercises range from multiplying one digit decimals by whole numbers to general multiplication of multi-digit decimals in columns. Source: pinterest.com Multiplying Mixed Numbers by Mixed Numbers. Suitable for kids in 3rd 4th 5th and 6th grades. Some of the worksheets for this concept are Mixed whole number s1 Mixed number multiplication l1s1 Multiplying mixed numbers Multiplying mixed numbers Multiplication of mixed numbers Fractions work Multiplyingdividing fractions and mixed numbers. Children begin their study of fraction multiplication by learning how to multiply a fraction by a whole number such as 5. Multiplying Mixed Numbers by Mixed Numbers. Source: pinterest.com Multiplying decimals by whole numbers Math worksheets. Similar sets of ordering numbers worksheets are presented in both horizontal and vertical formats. Multiplying fractions with a whole number worksheet game quiz and flash card. Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions. Kindergarten 1st Grade 2nd Grade 3rd. Source: pinterest.com Decimal and Whole - 2. Suitable for kids in 3rd 4th 5th and 6th grades. Kindergarten 1st Grade 2nd Grade 3rd. Multiplying decimals by whole numbers in columns Below are six versions of our grade 5 math worksheet on multiplying decimals using column form multiplication. Multiplying Tenths - Moderate. Source: pinterest.com Apply to the whole worksheet. Missing factor questions are also included. Sample Grade 5. Some of the worksheets for this concept are Mixed whole number s1 Mixed number multiplication l1s1 Multiplying mixed numbers Multiplying mixed numbers Multiplication of mixed numbers Fractions work Multiplyingdividing fractions and mixed numbers. Multiplying Decimal with Whole Number. Source: pinterest.com Multiplying Whole Numbers -2 Multiplying With and Without Regrouping ID. Simply multiply the numerators and the denominators by each other and then simplify. Multiplying Whole Numbers -2 Multiplying With and Without Regrouping ID. To downloadprint click on pop-out icon or print icon to worksheet to print or download. Decimal and Whole - 2. 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# Problem You are given an M by N matrix consisting of booleans that represents a board. Each True boolean represents a wall. Each False boolean represents a tile you can walk on. Given this matrix, a start coordinate, and an end coordinate, return the minimum number of steps required to reach the end coordinate from the start. If there is no possible path, then return null. You can move up, left, down, and right. You cannot move through walls. You cannot wrap around the edges of the board. For example, given the following board: ``````[ [f, f, f, f], [t, t, f, t], [f, f, f, f], [f, f, f, f], ] `````` and start = (3, 0) (bottom left) and end = (0, 0) (top left), the minimum number of steps required to reach the end is 7, since we would need to go through, (1, 2) because there is a wall everywhere else on the second row. # Solution It is a new type of problems I faced. I remember, I solved some during university, but it was pretty hard to come up with the solution right away. I googled basic types of maze solving algorithms, and it looks like Lee algorithm will be a pretty good choice in most of the `shortest path` problems since at the end of the day a number of different paths in a maze makes a tree. 1. go to start cell, mark it `0`. 2. mark all neighbors as `+1`. It is a distance to the starting cell 3. make the same for each of the neighbors By running this algorithm for each cell, we will get the number of steps it takes to get to any other point from the start. Of course, we should ignore walls and previously marked cells on each iteration. This is a basic solution and can be optimized for a given problem. For example, we can stop our recursive calls once we meet finish cell. # Code ``````// the maze is a matrix that represents a maze. // all cells have value 0, and all walls have value 1. // start and finish are arrays of 2 elements, [i,j] of the cells. func solution(maze [][]int, start []int, finish []int) int { // mark start cell is -1 maze[start[0]][start[1]] = -1 // mark all cells starting from start recursively mark(maze, start, 0) // return value of a finish cell return maze[finish[0]][finish[1]] } // pos is a structure to hold cell coordinates, // because []int can't ba used as a map key ¯\_(ツ)_/¯ type pos struct { i int j int } // mark marks all neighbors of a given cell with n+1 func mark(maze [][]int, point []int, n int) { i, j := point[0], point[1] neighbors := map[pos]bool{ pos{i + 1, j}: false, pos{i - 1, j}: false, pos{i, j - 1}: false, pos{i, j + 1}: false, } for p := range neighbors { neighbors[p] = markP(maze, p.i, p.j, n+1) } for p, ok := range neighbors { if ok { mark(maze, []int{p.i, p.j}, n+1) } } } // markP used to mark maze[i][j] with given n if exists and not marked. // returns true if it was marked, otherwise false. func markP(maze [][]int, i, j, n int) bool { if i >= len(maze) || j >= len(maze[0]) { return false } if i < 0 || j < 0 { return false } if maze[i][j] != 0 { return false } maze[i][j] = n return true }``````

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# 97970 (number) 97,970 (ninety-seven thousand nine hundred seventy) is an even five-digits composite number following 97969 and preceding 97971. In scientific notation, it is written as 9.797 × 104. The sum of its digits is 32. It has a total of 4 prime factors and 16 positive divisors. There are 38,400 positive integers (up to 97970) that are relatively prime to 97970. ## Basic properties • Is Prime? No • Number parity Even • Number length 5 • Sum of Digits 32 • Digital Root 5 ## Name Short name 97 thousand 970 ninety-seven thousand nine hundred seventy ## Notation Scientific notation 9.797 × 104 97.97 × 103 ## Prime Factorization of 97970 Prime Factorization 2 × 5 × 97 × 101 Composite number Distinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 97970 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0 The prime factorization of 97,970 is 2 × 5 × 97 × 101. Since it has a total of 4 prime factors, 97,970 is a composite number. ## Divisors of 97970 1, 2, 5, 10, 97, 101, 194, 202, 485, 505, 970, 1010, 9797, 19594, 48985, 97970 16 divisors Even divisors 8 8 8 0 Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 179928 Sum of all the positive divisors of n s(n) 81958 Sum of the proper positive divisors of n A(n) 11245.5 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 313.002 Returns the nth root of the product of n divisors H(n) 8.71193 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors The number 97,970 can be divided by 16 positive divisors (out of which 8 are even, and 8 are odd). The sum of these divisors (counting 97,970) is 179,928, the average is 1,124,5.5. ## Other Arithmetic Functions (n = 97970) 1 φ(n) n Euler Totient Carmichael Lambda Prime Pi φ(n) 38400 Total number of positive integers not greater than n that are coprime to n λ(n) 2400 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 9406 Total number of primes less than or equal to n r2(n) 32 The number of ways n can be represented as the sum of 2 squares There are 38,400 positive integers (less than 97,970) that are coprime with 97,970. And there are approximately 9,406 prime numbers less than or equal to 97,970. ## Divisibility of 97970 m n mod m 2 3 4 5 6 7 8 9 0 2 2 0 2 5 2 5 The number 97,970 is divisible by 2 and 5. • Deficient • Polite • Square Free ## Base conversion (97970) Base System Value 2 Binary 10111111010110010 3 Ternary 11222101112 4 Quaternary 113322302 5 Quinary 11113340 6 Senary 2033322 8 Octal 277262 10 Decimal 97970 12 Duodecimal 48842 20 Vigesimal c4ia 36 Base36 23le ## Basic calculations (n = 97970) ### Multiplication n×y n×2 195940 293910 391880 489850 ### Division n÷y n÷2 48985 32656.7 24492.5 19594 ### Exponentiation ny n2 9598120900 940327904573000 92123924811016810000 9025380913735316875700000 ### Nth Root y√n 2√n 313.002 46.0997 17.6919 9.95907 ## 97970 as geometric shapes ### Circle Diameter 195940 615564 3.01534e+10 ### Sphere Volume 3.93884e+15 1.20614e+11 615564 ### Square Length = n Perimeter 391880 9.59812e+09 138551 ### Cube Length = n Surface area 5.75887e+10 9.40328e+14 169689 ### Equilateral Triangle Length = n Perimeter 293910 4.15611e+09 84844.5 ### Triangular Pyramid Length = n Surface area 1.66244e+10 1.10819e+14 79992.2 ## Cryptographic Hash Functions md5 89f487d1a0ece62b5feb443b4e8d5de2 28718dd35e08817964387049838192a3b36e04c1 f5ee468f3cf57544ca8ae97f6369e2e3bd5a078af0ee2e02793bf7207a367fb4 486b5dd7540aeb1a4295d23faf231ed3bf54e8e23e971bb16f72a1bb6fd18243ce7ae8654c9cf6013e90e56cb98ef04440dcb2a247508b1a2159204c41d72260 958af20663261bb85c9ff6a81c88929411b5d2a2

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Presentation is loading. Please wait. # Subtracting Integers with Tiles ## Presentation on theme: "Subtracting Integers with Tiles"— Presentation transcript: Subtracting Integers with Tiles Student Expectation 7th Grade: 7.1.2C Use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms. SUBTRACTING INTEGERS +3 +3 We often think of subtraction as a “take away” operation. Which diagram could be used to compute = ? +3 +3 The University of Texas at Dallas SUBTRACTING INTEGERS This diagram also represents +3, and we can take away +5. When we take 5 yellow tiles away, we have 2 red tiles left. We can’t take away 5 yellow tiles from this diagram. There is not enough tiles to take away!! The University of Texas at Dallas SUBTRACTING INTEGERS -2 - -4 = ? Use your red and yellow tiles to model each subtraction problem. = ? ANSWER The University of Texas at Dallas -2 - -4 = +2 SUBTRACTING INTEGERS Now you can take away 4 red tiles. 2 yellow tiles are left, so the answer is… This representation of -2 doesn’t have enough tiles to take away -4. Now if you add 2 more reds tiles and 2 more yellow tiles (adding zero) you would have a total of 4 red tiles and the tiles still represent -2. = +2 The University of Texas at Dallas SUBTRACTING INTEGERS Work this problem. +3 - -5 = ? ANSWER The University of Texas at Dallas +3 - -5 = +8 SUBTRACTING INTEGERS Add enough red and yellow pairs so you can take away 5 red tiles. Take away 5 red tiles, you have 8 yellow tiles left. = +8 The University of Texas at Dallas SUBTRACTING INTEGERS Work this problem. -3 - +2 = ? ANSWER The University of Texas at Dallas -3 - +2 = -5 SUBTRACTING INTEGERS Add two pairs of red and yellow tiles so you can take away 2 yellow tiles. Take away 2 yellow tiles, you have 5 red tiles left. = -5 The University of Texas at Dallas SUBTRACTING INTEGERS A fact family gives 4 true equations using the same 3 numbers. For example: 7 + 8 = 15 8 + 7 = 15 15 – 7 = 8 15 – 8 = 7 All of these statements are true. The University of Texas at Dallas SUBTRACTING INTEGERS We can also use fact family with integers. Use your red and yellow tiles to verify this fact family: = +5 = +5 = -3 = +8 The University of Texas at Dallas Download ppt "Subtracting Integers with Tiles" Similar presentations Ads by Google

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Community Profile # Benson Gou ### Smart Electric Grid Last seen: 23 days ago Active since 2017 Founder, Smart Electric Grid #### Content Feed View by Question How can I get the alternatives for the functions Not supported by Matlab Coder? Dear All, I want to convert my Matlab code into C language using Matlab Coder. But I got the following functions which are Not ... 5 months ago | 1 answer | 0 ### 1 Question How to quickly calculate the sum of the transpose of a sparse matrix? Dear All, I have a very big sparse matrix A. I want to obtain the sum of its transpose of the selected columns in A. Here is my... 5 months ago | 2 answers | 0 ### 2 Question How to quickly do the calculation? Dear All, I have a code to calculate an array using several arrays. The code is as follows: Ieq1 = Haltmr(mmm(jbb2),IndBus(sol... 5 months ago | 1 answer | 0 ### 1 Question How to systematically handle a cell? Dear All, I have an array Measinj. I want to find out the indecies of Lineon whose 2nd or 3rd columns equal to the entry in Mea... 5 months ago | 0 answers | 0 ### 0 Question How to deal with the structure? Dear All, I have a struct which saves a number of records. I have the following codes which takes more time than expected. Re... 5 months ago | 1 answer | 0 ### 1 Question How can I find a faster way to calculate the transpose of a matrix? Dear All, I have a matrix A and I want to select some columns from A to form another matrix B, and calculate the sum of each co... 5 months ago | 1 answer | 0 ### 1 Question How to run find function faster? Dear All, I have an array A with 6000 elements in integers, for example, "10", "12", "5" and others. Actually array A saves the... 6 months ago | 1 answer | 0 ### 1 Question Can I calculate the inverse of a matrix using arrayfun? Dear All, I have a sparse matrix A. I want to calculate its inverse. I used the following way to calculate invA. invA = A \ s... 6 months ago | 2 answers | 0 ### 2 Question How to quickly save a matrix by adding rows? Dear All, I calculate a row Ai and save it in my defined matrix A using the following way: A = [A; Ai]; But I foud it took a ... 6 months ago | 1 answer | 0 ### 1 Question How can I quickly obtain the inverse of a square sparse matrix? Dear All, I have a square sparse matrix A. I use the following way to obtain its inverse: invA = A \ eye(size(A)); I am wonde... 6 months ago | 1 answer | 0 ### 1 Question How to quickly find the indecis of an array in another array? Dear All, I have arrays A and B. I want to quickly find out the repeat entries of A in B without using intersect. For example, ... 6 months ago | 2 answers | 0 ### 2 Question Looking for a way faster than find? Dear All, I have an array A. I want to find out the index of those "1" entries in the array A. 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I want to extract the first 20 values from ... 6 months ago | 1 answer | 0 ### 1 Question how to form a two-column array for a given but dimmension-unknown array? Dear All, I have two array with the same length A and B. I want to form another array C which is formed by A and B. But the dim... 6 months ago | 2 answers | 0 ### 2 Question How to apply a function independently for a set of data? Dear All, I have a set of data {Ai, bi}, i = 1, ..., N. I want to apply a function which solves Ai(xi) = bi, a nonliear problem... 6 months ago | 0 answers | 0 ### 0 Question How to quickly find out the repeat integers in an array? Dear All, I have an array A which contains repeat intergers (entries). For example, A = [4 20 5 4 7 5 9 5 31]. I want to find o... 6 months ago | 2 answers | 0 ### 2 Question How to assign the values to a matrix? Dear All, I have a zero matrix A whcih needs to be assigned values from an column vector B. The indecies of those elements are ... 7 months ago | 1 answer | 0 ### 1 Question How to generate a sparse matrix for a given array? Dear All, I want to generate a sparse matrix B for a given array A. A contains two columns of indecis. For example, A = [1 3;... 7 months ago | 2 answers | 0 ### 2 Question How to quickly assign the values of a matrix using a given array? Dear All, I have an array A which contains two columns of integers. I want to build a matrix B in the following way. For exam... 7 months ago | 2 answers | 0 ### 2 Question How to quickly find the indecies of elements in an array? Dear All, I have an array A which contains integers. I have another array B. I want to find out the indecis of B in A. For ex... 7 months ago | 2 answers | 0 ### 2 Question Can Ridge Regression solve my problem? Dear All, I plan to buy Statistics and Machine Learning Toolbox to apply Ridge Regression to solve my problem. But I do not kno... 7 months ago | 1 answer | 0 ### 1 Question How to find the CUP time used by each function in my code? Dear All, I want to know the CPU time used by each function of my code. I donot know if it is possible. I donot want to use tic... 7 months ago | 1 answer | 0 ### 1 Question Is the C code generated by Matlab Coder Faster than my Matlab Code? Dear All, I am thinking to buy a Matlab Coder to convert mt Matlab code into C code. My purpose is to largely reduce the CPU ti... 9 months ago | 1 answer | 1 ### 1 Question How to quickly obtain the row indices in the original matrix A for a sub-matrix B? Dear All, I have an original matrix A and a sub-matrix which is formed by selected rows from A. Now given A and B, how can I fi... 10 months ago | 1 answer | 0 ### 1 Question How to efficiently solve a matrix eqation when the matrix is updated? Dear All, I need to solve a matrix linear equation: Ax = b. I know the most fast way to solve x is x = A\b. My question is: for... 10 months ago | 1 answer | 0 ### 1 Question How to find out a smallest sub-matrix B from a sparse matrix A which has the equal rank and # of non-zero columns? Dear All, I have a very sparse matrix A. I need to find out a number of rows (smallest #) of A which satisfies the following co... 11 months ago | 2 answers | 0

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Teaching Math with the KDE Interactive Geometry Program I've written quite a bit about using Linux to help educate people. In the past, I've discussed using Linux to teach astronomy, programming and computer logic design. So today, I'm writing about using the KDE Interactive Geometry (Kig) program to teach mathematics. Kig allows you to use various tools to diagram and demonstrate different mathematical concepts. With Kig, you can draw points, lines, line segments, half lines, vectors, circles and various other conic sections. When Kig refers to a “half line”, it means what I was taught was a ray—essentially a line with one endpoint. Drawing hyperbolic curves on the computer sure beats getting dry-erase marker all over yourself or sneezing because of chalk dust. Even more important though, Kig diagrams are interactive, which means that once you create a diagram, you can move various elements around and see how they behave (more on that later). Kig's user interface can be a bit deceptive at times. When you first start the program, you are presented with a grid and a group of tools used to create various diagram elements. At this point, you begin to think that the interface is intuitive and that you already “know” how to use it. Then, for a brief moment, you run into trouble. For example, if you try to use a tool to construct a circle given a center point and a line segment as a radius, the program is expecting that the line segment already has been created; you can't create the line segment as part of constructing the circle. After you've used the program for a few minutes, you begin to understand how it “thinks” and things go pretty smoothly. Thankfully I read the documentation that came with Kig, otherwise, I would have missed out on some of its more powerful features. For example, if you select a curve, say a parabola, from your diagram, you can use the point tool to create a point on that curve. Later, you can drag that point around with the mouse, and it won't leave the curve to which it was constrained. Then, you can use that point to construct other curves, such as a tangent to the curve. Without reading the documentation, I would have completely overlooked the Add Text Label function that is available by right-clicking on a curve. This function doesn't merely add text to your diagram; there's a text tool for that. The Add Text Label function lets you display information about a curve, such as slope, equation, focus and so on. Once the label has been added to the diagram, you can change various parts of the curve, and the label will reflect those changes. For example, if you created a parabola through three points, you can add a label that displays the equation of that curve. You also can create a label that displays the coordinates of the points. Then, you can move the points around with your mouse, and see the labels change. So, what can you do with Kig? Is it just a geometry teaching tool? Kig would be interesting if it were only for teaching geometry, but it can be used for much more. I can easily see how to use Kig to teach algebra, geometry, trigonometry, physics, analysis and calculus. Let's start with algebra. Figure 1 shows two lines on the Cartesian (X,Y) coordinate plane. Each line is defined by two points, and the coordinates of those points are displayed nearby. The red point in the center is the intersection of the two lines. The equations of the lines also are displayed. By dragging the points around, you can change the lines and explore concepts such as slope, Y-intercept, the solution of an equation and the solution of a system of two equations. I vaguely remember something from high-school geometry that said “the opposite interior angles formed by two parallel lines crossed by a third, are equivalent.” Figure 2 demonstrates this statement for the case where the angles happen to be 90 degrees. Kig makes it easy to construct two parallel lines. Then, you create a third line that crosses the other two. Finally, you tell Kig to label the angle formed by the various lines. Once this is done, you or your student can change the angles, the distance between the parallel lines and so on—the theorem still holds. I also remember from when I was in school how obtuse mathematical statements tend to be (particularly those statements related to triangles and the size of their sides and angles), but quite often a picture easily much explains them. Kig would allow you to create an interactive demonstration of each the Euclidean geometry theorems. Measurements of 30-60-90 do not make a very attractive super model, but they do make a great triangle (at least in Euclidean space, but let's not warp things too much here). Figure 3 demonstrates that the sum of all angles in a triangle is 180 degrees. Once this diagram is constructed, students can drag the angles around and explore any triangle they please. This also would be a great way to demonstrate the various trigonometric ratios, such as sin, cos and tan. In this case, you simply would construct a right angle, and let the student manipulate the lengths of the sides. Kig could be asked to display the angles and lengths of the sides, and the student then could calculate and verify the various ratios. But, this is where I found what I think to be one of the weaknesses of Kig. Perhaps I just don't know how, but I was unable to create a triangle and explicitly configure two of the angles. I could drag points to the approximate position I wanted, but I was unable to construct an exact 30-60-90 triangle. It seemed like Kig was keeping the lengths of the sides constant, and thus, when I tried to change the angles, they just didn't “fit”. I think this is important, so if it can be done, please let me know how. Many concepts in physics can be expressed with what's known as a “vector”. Such things as position, velocity and acceleration can all be expressed as a vector. Kig has rudimentary support for vectors, including vector addition. Figure 4 shows a hypothetical physics problem expressed as the sum of two vectors. Here we have an object moving along a given vector. This object also is being affected by a force, expressed as another vector—gravity in this case. The resulting motion is found by adding the two vectors, as shown. In analysis, the student begins to learn the features of various curves. Figure 5 gives an example of a hyperbolic curve. I've also included the calculated asymptotes as well as the equation of the curve in both the Cartesian and Polar coordinate systems. Of course, this diagram is completely interactive. Finally, Figure 6 shows a parabola and a tangent line. You also can see the equation of the tangent and watch the equation change as you move the tangent point along the parabola. This could be a nice way to introduce the concept of the differential and, eventually, the derivative in calculus. Despite a few weaknesses, Kig is a very powerful tool for teaching upper-level mathematics. After climbing a little bit of a learning curve (yes, it's all about curves), both students and teachers can use Kig to have fun learning and teaching mathematics. AttachmentSize kig1.png13.86 KB kig2.png6.57 KB kig3.png11.54 KB kig4.png7.79 KB kig5.png15.11 KB kig6.png7.23 KB ______________________ Mike Diehl is a freelance Computer Nerd specializing in Linux administration, programing, and VoIP. Mike lives in Albuquerque, NM. with his wife and 3 sons. He can be reached at mdiehl@diehlnet.com Comment viewing options Sounds like Dr. Geo II It sounds like this tool is a nice companion to another open source application, Dr. Geo II. Unlike KIG, DrGeoII seems more suited to elementary geometry, but the diagrams are also fully interactive, and could even be animated. Perhaps both packages have something they can learn from each other. Thanks for the article. http://wiki.laptop.org/go/DrGeo Thanks Mike! Very Thanks Mike! Very interesting and useful.

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The DIS Discussion Forums - DISboards.com See if my logic checks out... Find Hotel Specials & DIScounts Disney World Universal Orlando Disneyland Other Register Chat FAQ Tickers Search Today's Posts Mark Forums Read 08-30-2007, 12:39 PM #1 Mickey4me! Lost my screen name so I'm starting from scratch!     Join Date: Aug 2005 Location: Kill Devil Hills, NC Posts: 353 See if my logic checks out... Last year during our Nov disney trip we purchased the dining plan, but due to the sudden death of my mother, our trip ended on day 2 and we really didn't get to experience the plan. In deciding whether we should purchase it this year, I first made all the ADRs that we wanted - LTT, CA Grill, BOMA breakfast and Ohana's. My daughter's boyfriend is traveling with us and is new to WDW so we wanted him to experience some of our favs. So, here's my logic - cost of the DDP for 4 adults for 6 nights = \$935.76 - cost of meals that we have ADRs for = \$574.42 (inc. tax and tips) - difference = \$361.42 If you divide 361.42 by 4 that's 90.35 and if you divide 90.35 by 6days you've got \$15/day per person. So the question is - will four adults each spend more or less than \$15 a day on breakfast/lunch and a snack each day and dinner for 3 of the six nights (the other 3 are in the ADR total)? Even though we're staying in a 2 bedroom villa, my thoughts are that we will spend slightly more than that so, DDP would be a good deal. Does that make sense to everyone or am I missing something? __________________ CBR - Dec 95, BW/DCL - Oct 98,HHI - June 99,OKW - Dec 00, DCL/Vero - July 02, BWV - Jan 03, OKW - Nov 03, YC - Oct 04, WLV - Nov 04, BWV - Nov 05, BCV - Jan 06, SS - Nov 06, OKW - Nov 07, HHI - Jan 09 DCL - Oct 09 DCL - Dec 10, AKL - Nov 10, DCL & OKW - Oct 11, DCL & BWV - Feb 13 Sponsored Links The DIS Register to remove Join Date: 1997 Location: Orlando, FL Posts: 1,000,000 08-30-2007, 05:00 PM #2 hollieplus2 Tags always make people happyI kept saying chewableMy toes might appreciate some alone time     Join Date: Apr 2006 Location: Not close enough Posts: 1,323 Did you factor in the counter service you get to use also? That would cover breakfast or lunch each day. Also, a lot of people share a counter service meal since it's a lot of food. If you guys could do it you could use 2 CS for breakfast and 2 CS for lunch and still have all your TS for dinner plus a snack to use. __________________ DH ME DS July 2006---Pop, ASMovies, Animal Kingdom September/October 2007---POFQ December 2009---WDW & DCL September 2012---POR/ToT 10 Miler 08-30-2007, 05:03 PM #3 MOM POPPINS I keep getting SOUVENIRS (wink) when I go to WDW!     Join Date: Aug 2000 Location: Trenton, Texas Posts: 3,483 First sorry to hear about your Mom, my Mom died recently as well. I really miss her! Now to try and answer your question. You would def. spend at least the 15 a day for each person on lunch/breakfast and a snack. You also mention that you would also have to plan 3 more meals. I think the dining plan is a great option for you. I also looked at your totals and 542 seems low for 6 dinners TS. I know we have a big family but our table service is usually at least 200 a night. Happy Planning! I would go with the DDP! __________________ Me DH DD23 DD18 DD15 DD11 DS9 2000 Dixie Landing HIFS '01 Contemporary/Dolphin 2 Trips '02 Dolphin '03 Dolphin '04 BWV/WLV '05 BWV/BCV 2 Trips '06 BWV/Dolphin/Saratoga/OKW 3 Trips '07 BWV/SSR 2 trips '08 BWV May/November '09 BWV March/May/Nov 2010 20th. POR AKL Villa's May 2011BWV September 2011 BWV January 2012 AKL Villa's May 2012 Saratoga Springs December 2012 May 2013 AKL 09-01-2007, 06:11 PM   #4 Mickey4me! Lost my screen name so I'm starting from scratch! Join Date: Aug 2005 Location: Kill Devil Hills, NC Posts: 353 Quote: Originally Posted by MOM POPPINS I also looked at your totals and 542 seems low for 6 dinners TS. I know we have a big family but our table service is usually at least 200 a night. Happy Planning! I would go with the DDP! Actually the \$542 was only for 4 ADRs I made so I definitely think we'll spend at least another \$350+ on lunches, snacks and a couple more dinners. Just wanted to make sure I wasn't overlooking something. Thanks for your help! __________________ CBR - Dec 95, BW/DCL - Oct 98,HHI - June 99,OKW - Dec 00, DCL/Vero - July 02, BWV - Jan 03, OKW - Nov 03, YC - Oct 04, WLV - Nov 04, BWV - Nov 05, BCV - Jan 06, SS - Nov 06, OKW - Nov 07, HHI - Jan 09 DCL - Oct 09 DCL - Dec 10, AKL - Nov 10, DCL & OKW - Oct 11, DCL & BWV - Feb 13 09-01-2007, 07:09 PM #5 lowie DIS VeteranI got a tootsie pop with no tootsie     Join Date: Feb 2006 Location: Palm Beach, FL. Posts: 2,179 you also overlooked the tax and tips! if you are eaters (not only big eaters) then the plan will work out. you'll get to try so many things and not skimp on your choices. __________________ Lori (me) Michael (dh) Jordana (dd16) Jason (ds11) And, by the way, I adore you... in frightening and dangerous ways. Thread Tools Display Modes Rate This Thread Linear Mode Rate This Thread: 5 : Excellent 4 : Good 3 : Average 2 : Bad 1 : Terrible Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Disney Trip Planning Forums     Welcome to the DIS     The DIS Unplugged Podcast         Power of 10 - DIS Events to Benefit GKTW         Podcast Cruises         DIS Adventures by Disney Trips     Theme Parks Attractions and Strategies         Doing the Happy Dance!         Disney Promotions and Celebrations--Current Promotion: Let the Memories Begin         Theme Parks Community     Disney Resorts         Disney Discount Codes and Rates         Countdowns and Live Reports         Resort Community Threads & Photo-Video     Disney Restaurants         Disney Dining Reservations         Disney Dining Plan         Disney Dining Reviews     Orlando Hotels and Attractions         Hotel & Accommodations Reviews     Adventures By Disney     Disney Trip Reports         Pre-Trip Reports and Plans         Completed Trip Reports     Budget Board     Disney Rumors and News     Disney for Families         DIS Dads     Disney for Adults and Solo Travelers     The College Board     Teen Disney         Teen Board Birthdays and Celebrations     Gay and Lesbian at Disney     Disney World Tips     Transportation     disABILITIES!         disABILTIES Community Board     Camping at Disney World         Camping Community Board     Disney Weddings and Honeymoons         Planning our Happily Ever After             Completed trip reports and planning journals     DIS en Español Disney Vacation Club     Purchasing DVC     DVC Member Services     DVC Resorts     DVC-Mousecellaneous     DVC Trip Reports Global Neighbours     UK Trip Planning Forum     UK Community Board     UK DVC Discussion     UK Trip Reports Board     Canadian Trip Planning & Community Board     Disneyland Paris Trip Planning & Community Board         Disneyland Paris Trip Reports Board     Other Lands         Australia         Tokyo Disneyland         Hong Kong Disneyland Disney Cruise Line     Disney Cruise Line Forum         Disney Cruise Meets         Disney Cruise Line Trip Reports             Disney Cruise Line Pre-Trip Reports Disneyland     Disneyland (California)         Southern California Theme Parks         Disneyland Community Board     California & the West         Southern California         Las Vegas     Disneyland Trip Reports         California & the West Trip Reports Universal Studios/Sea World     Universal Studios/Islands of Adventure Forums     Universal Orlando Resorts & Hotels     Universal Studios Trip Reports     Sea World / Discovery Cove Just for Fun     Community Board         Congratulations & Birthday Wishes         Games         Exchanges         United We Stand     Disney Movies, Books, TV and Music     Photography Board     Disney Online Games         Non-Disney Online Games         Virtual Traders Market         Online Gaming Community Board         Virtual Magic Kingdom (VMK) Museum             VMK Creations     In Memoriam     Dis Meets     The Creative Community         Arts and Crafts         Cooking         Flower & Garden Forum         Home Interior         Virtual Scrapbook         Scrapbooking         Swaps     W.I.S.H         Eating Healthy         Events/Competition         WISH Journals     Coping and Compassion     Just Say Thanks     Disney Rewards Programs     Disney Collectors Board         Trades/Wants Board     Creative DISigns     Disney Cast Members Technical Support     Technical Support         Test Board GET OUR DIS UPDATES DELIVERED BY EMAIL -- Default ---- DIS Unplugged ---- Disneyland ---- Disney Cruise Line ---- Universal Orlando ---- Disney Vacation Club ---- FLTOURS Transportation ---- Swan & Dolphin Resort ---- Adventures by Disney ---- Holidays ------ Holidays Light -- vBulletin 3 Contact Us - www.DISboards.com - Archive - Privacy Statement - Top All times are GMT -5. 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Question # When 25.0 mL of 0.100 M NaOH was added to 50.0 mL of a 0.100 M... When 25.0 mL of 0.100 M NaOH was added to 50.0 mL of a 0.100 M solution of a weak acid, HX, the pH of the mixture reached a value of 3.56. What is the value of Ka for the weak acid? #### Homework Answers Answer #1 Total volume after addition, Vt = 50.0 mL + 25.0 mL = 75.0 mL = 0.075 L Moles of NaOH before reaction = MxV = 0.100 M x 0.025 L = 0.0025 mol Moles of weak acid(HA) before reaction = MxV = 0.100 M x 0.050 L = 0.005 mol Given the pH of the solution after reaction, pH = - log[H+(aq)] = 3.56 => [H+(aq)] = 10-3.56 = 2.754x10-4 M The acid base neutralization reaction is -------------- HA + OH-(aq) --------> A-(aq) + H2O Init.mol: 0.005, 0.0025, ----------- 0 eqm.mol:(0.005-0.0025), (0.0025-0.0025), 0.0025 ----------- = 0.0025, = 0.00, ------------ 0.0025 Hence [HA] = [A-(aq)] =  0.0025 mol / 0.075 L = 0.0333 M Nw this solution will act as a buffer solution. Hence applying Hendersen equation pH = pKa + log[A-(aq)] / [HA] = pKa + log1 = pKa => pH = pKa = 3.56 => Ka = 10-3.56 = 2.754x10-4 M (answer) Know the answer? Your Answer: #### Post as a guest Your Name: What's your source? #### Earn Coins Coins can be redeemed for fabulous gifts. ##### Not the answer you're looking for? Ask your own homework help question ADVERTISEMENT ##### Need Online Homework Help? Get Answers For Free Most questions answered within 1 hours. ADVERTISEMENT

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Select Page # Neural Networks – A Worked Example ## Description of the problem We start with a motivational problem. We have a collection of 2×2 grayscale images. We’ve identified each image as having a “stairs” like pattern or not. Here’s a subset of those. Our goal is to build and train a neural network that can identify whether a new 2×2 image has the stairs pattern. ## Description of the network Our problem is one of binary classification. That means our network could have a single output node that predicts the probability that an incoming image represents stairs. However, we’ll choose to interpret the problem as a multi-class classification problem – one where our output layer has two nodes that represent “probability of stairs” and “probability of something else”. This is unnecessary, but it will give us insight into how we could extend task for more classes. In the future, we may want to classify {“stairs pattern”, “floor pattern”, “ceiling pattern”, or “something else”}. Our measure of success might be something like accuracy rate, but to implement backpropagation (the fitting procedure) we need to choose a convenient, differentiable loss function like cross entropy. We’ll touch on this more, below. Our training dataset consists of grayscale images. Each image is 2 pixels wide by 2 pixels tall, each pixel representing an intensity between 0 (white) and 255 (black). If we label each pixel intensity as $p1$, $p2$, $p3$, $p4$, we can represent each image as a numeric vector which we can feed into our neural network. ImageId p1 p2 p3 p4 IsStairs 1 252 4 155 175 TRUE 2 175 10 186 200 TRUE 3 82 131 230 100 FALSE 498 36 187 43 249 FALSE 499 1 160 169 242 TRUE 500 198 134 22 188 FALSE For no particular reason, we’ll choose to include one hidden layer with two nodes. We’ll also include bias terms that feed into the hidden layer and bias terms that feed into the output layer. A rough sketch of our network currently looks like this. Our goal is to find the best weights and biases that fit the training data. To make the optimization process a bit simpler, we’ll treat the bias terms as weights for an additional input node which we’ll fix equal to 1. Now we only have to optimize weights instead of weights and biases. This will reduce the number of objects/matrices we have to keep track of. Finally, we’ll squash each incoming signal to the hidden layer with a sigmoid function and we’ll squash each incoming signal to the output layer with the softmax function to ensure the predictions for each sample are in the range [0, 1] and sum to 1. Note here that we’re using the subscript $i$ to refer to the $i$th training sample as it gets processed by the network. We use superscripts to denote the layer of the network. And for each weight matrix, the term $w^l_{ab}$ represents the weight from the $a$th node in the $l$th layer to the $b$th node in the $(l+1)$th layer. Since keeping track of notation is tricky and critical, we will supplement our algebra with this sample of training data ImageId p1 p2 p3 p4 IsStairs 1 252 4 155 175 TRUE 2 175 10 186 200 TRUE 3 82 131 230 100 FALSE 4 115 138 80 88 FALSE The matrices that go along with out neural network graph are $\mathbf{X^1} = \begin{bmatrix} x^1_{11} & x^1_{12} & x^1_{13} & x^1_{14} & x^1_{15}\\ x^1_{21} & x^1_{22} & x^1_{23} & x^1_{24} & x^1_{25}\\ ... & ... & ... & ... & ...\\ x^1_{N1} & x^1_{N2} & x^1_{N3} & x^1_{N4} & x^1_{N5}\\ \end{bmatrix} = \begin{bmatrix} 1 & p_{11} & p_{12} & p_{13} & p_{14}\\ 1 & p_{21} & p_{22} & p_{23} & p_{24}\\ ... & ... & ... & ... & ...\\ 1 & p_{N1} & p_{N2} & p_{N3} & p_{N4} \end{bmatrix} = \begin{bmatrix} 1 & 252 & 4 & 155 & 175\\ 1 & 175 & 10 & 186 & 200\\ 1 & 82 & 131 & 230 & 100\\ 1 & 115 & 138 & 80 & 88 \end{bmatrix}$ $\mathbf{W^1} = \begin{bmatrix} w^1_{11} & w^1_{12}\\ w^1_{21} & w^1_{22}\\ w^1_{31} & w^1_{32}\\ w^1_{41} & w^1_{42}\\ w^1_{51} & w^1_{52} \end{bmatrix}, \; \mathbf{Z^1} = \begin{bmatrix} z^1_{11} & z^1_{12}\\ z^1_{21} & z^1_{22}\\ ... & ...\\ z^1_{N1} & z^1_{N2} \end{bmatrix}$ $\mathbf{X^2} = \begin{bmatrix} x^2_{11} & x^2_{12} & x^2_{13}\\ x^2_{21} & x^2_{22} & x^2_{23}\\ ... & ... & ...\\ x^2_{N1} & x^2_{N2} & x^2_{N3} \end{bmatrix} = \begin{bmatrix} 1 & x^2_{12} & x^2_{13}\\ 1 & x^2_{22} & x^2_{23}\\ ... & ... & ...\\ 1 & x^2_{N2} & x^2_{N3} \end{bmatrix}$ $\mathbf{W^2} = \begin{bmatrix} w^2_{11} & w^2_{12}\\ w^2_{21} & w^2_{22}\\ w^2_{31} & w^2_{32} \end{bmatrix}, \; \mathbf{Z^2} = \begin{bmatrix} z^2_{11} & z^2_{12}\\ z^2_{21} & z^2_{22}\\ ... & ...\\ z^2_{N1} & z^2_{N2} \end{bmatrix}$ $\mathbf{Y} = \begin{bmatrix} y_{11} & y_{12}\\ y_{21} & y_{22}\\ ... & ...\\ y_{N1} & y_{N2} \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 1 & 0\\ 0 & 1\\ 0 & 1 \end{bmatrix}, \; \widehat{\mathbf{Y}} = \begin{bmatrix} \widehat{y}_{11} & \widehat{y}_{12}\\ \widehat{y}_{21} & \widehat{y}_{22}\\ ... & ...\\ \widehat{y}_{N1} & \widehat{y}_{N2} \end{bmatrix}$ ## Initializing the weights Before we can start the gradient descent process that finds the best weights, we need to initialize the network with random weights. In this case, we’ll pick uniform random values between -0.01 and 0.01. $\mathbf{W^1} = \begin{bmatrix} w^1_{11} & w^1_{12}\\ w^1_{21} & w^1_{22}\\ w^1_{31} & w^1_{32}\\ w^1_{41} & w^1_{42}\\ w^1_{51} & w^1_{52} \end{bmatrix} = \begin{bmatrix} -0.00469 & 0.00797\\ -0.00256 & 0.00889\\ 0.00146 & 0.00322\\ 0.00816 & 0.00258\\ -0.00597 &-0.00876 \end{bmatrix}, \; \mathbf{W^2} = \begin{bmatrix} w^2_{11} & w^2_{12}\\ w^2_{21} & w^2_{22}\\ w^2_{31} & w^2_{32} \end{bmatrix} = \begin{bmatrix} -0.00588 & -0.00232\\ -0.00647 & 0.00540\\ 0.00374 & -0.00005 \end{bmatrix}$ Is it possible to choose bad weights? Yes. Numeric stability often becomes an issue for neural networks and choosing bad weights can exacerbate the problem. There are methods of choosing good initial weights, but that is beyond the scope of this article. (See this for more details.) ## Forward Pass Now let’s walk through the forward pass to generate predictions for each of our training samples. #### Step 1Compute the signal going into the hidden layer, $\mathbf{Z^1}$ $\mathbf{Z^1} = \mathbf{X^1}\mathbf{W^1}$ $\begin{bmatrix} z^1_{11} & z^1_{12}\\ z^1_{21} & z^1_{22}\\ ... & ...\\ z^1_{N1} & z^1_{N2} \end{bmatrix} = \begin{bmatrix} x^1_{11} & x^1_{12} & x^1_{13} & x^1_{14} & x^1_{15}\\ x^1_{21} & x^1_{22} & x^1_{23} & x^1_{24} & x^1_{25}\\ ... & ... & ... & ... & ...\\ x^1_{N1} & x^1_{N2} & x^1_{N3} & x^1_{N4} & x^1_{N5} \end{bmatrix} \times \begin{bmatrix} w^1_{11} & w^1_{12}\\ w^1_{21} & w^1_{22}\\ w^1_{31} & w^1_{32}\\ w^1_{41} & w^1_{42}\\ w^1_{51} & w^1_{52} \end{bmatrix} = \\[12pt] \begin{bmatrix} x^1_{11}w^1_{11} + x^1_{12}w^1_{21} + x^1_{13}w^1_{31} + x^1_{14}w^1_{41} + x^1_{15}w^1_{51} & x^1_{11}w^1_{12} + x^1_{12}w^1_{22} + x^1_{13}w^1_{32} + x^1_{14}w^1_{42} + x^1_{15}w^1_{52}\\ x^1_{21}w^1_{11} + x^1_{22}w^1_{21} + x^1_{23}w^1_{31} + x^1_{24}w^1_{41} + x^1_{25}w^1_{51} & x^1_{21}w^1_{12} + x^1_{22}w^1_{22} + x^1_{23}w^1_{32} + x^1_{24}w^1_{42} + x^1_{25}w^1_{52}\\ ... & ...\\ x^1_{N1}w^1_{11} + x^1_{N2}w^1_{21} + x^1_{N3}w^1_{31} + x^1_{N4}w^1_{41} + x^1_{N5}w^1_{51} & x^1_{N1}w^1_{12} + x^1_{N2}w^1_{22} + x^1_{N3}w^1_{32} + x^1_{N4}w^1_{42} + x^1_{N5}w^1_{52} \end{bmatrix}$ #### Step 2Squash the signal to the hidden layer with the sigmoid function to determine the inputs to the output layer, $\mathbf{X^2}$ $sigmoid(z) = 1/(1 + e^{-z})$ $\mathbf{X^2} = \begin{bmatrix} \mathbf{1} & sigmoid(\mathbf{Z^1}) \end{bmatrix}$ $\begin{bmatrix} x^2_{11} & x^2_{12} & x^2_{13}\\ x^2_{21} & x^2_{22} & x^2_{23}\\ ... & ... & ...\\ x^2_{N1} & x^2_{N2} & x^2_{N3} \end{bmatrix} = \begin{bmatrix} 1 & sigmoid(z^1_{11}) & sigmoid(z^1_{12})\\ 1 & sigmoid(z^1_{21}) & sigmoid(z^1_{22})\\ ... & ... & ...\\ 1 & sigmoid(z^1_{N1}) & sigmoid(z^1_{N2}) \end{bmatrix} = \begin{bmatrix} 1 & \frac{1}{1 + e^{-z^1_{11}}} & \frac{1}{1 + e^{-z^1_{12}}}\\ 1 & \frac{1}{1 + e^{-z^1_{21}}} & \frac{1}{1 + e^{-z^1_{22}}}\\ ... & ... & ...\\ 1 & \frac{1}{1 + e^{-z^1_{N1}}} & \frac{1}{1 + e^{-z^1_{N2}}} \end{bmatrix}$ #### Step 3Calculate the signal going into the output layer, $\mathbf{Z^2}$ $\mathbf{Z^2} = \mathbf{X^2}\mathbf{W^2}$ $\begin{bmatrix} z^2_{11} & z^2_{12}\\ z^2_{21} & z^2_{22}\\ ... & ...\\ z^2_{N1} & z^2_{N2} \end{bmatrix} = \begin{bmatrix} x^2_{11} & x^2_{12} & x^2_{13}\\ x^2_{21} & x^2_{22} & x^2_{23}\\ ... & ... & ...\\ x^2_{N1} & x^2_{N2} & x^2_{N3} \end{bmatrix} \times \begin{bmatrix} w^2_{11} & w^2_{12}\\ w^2_{21} & w^2_{22}\\ w^2_{31} & w^2_{32} \end{bmatrix} = \\[12pt] \begin{bmatrix} x^2_{11}w^2_{11} + x^2_{12}w^2_{21} + x^2_{13}w^2_{31} & x^2_{11}w^2_{12} + x^2_{12}w^2_{22} + x^2_{13}w^2_{32}\\ x^2_{21}w^2_{11} + x^2_{22}w^2_{21} + x^2_{23}w^2_{31} & x^2_{21}w^2_{12} + x^2_{22}w^2_{22} + x^2_{23}w^2_{32}\\ ... & ...\\ x^2_{N1}w^2_{11} + x^2_{N2}w^2_{21} + x^2_{N3}w^2_{31} & x^2_{N1}w^2_{12} + x^2_{N2}w^2_{22} + x^2_{N3}w^2_{32} \end{bmatrix}$ #### Step 4Squash the signal to the output layer with the softmax function to determine the predictions, $\widehat{\mathbf{Y}}$ Recall that the softmax function is a mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$. In other words, it takes a vector $\theta$ as input and returns an equal size vector as output. For the $k$th element of the output, $softmax(\theta)_k = \frac{e^{\theta_k}}{\sum_{j=1}^n e^{\theta_j}}$ In our model, we apply the softmax function to each vector of predicted probabilities. In other words, we apply the softmax function “row-wise” to $\mathbf{Z^2}$. $\widehat{\mathbf{Y}} = softmax_{row-wise}(\mathbf{Z^2})$ $\begin{bmatrix} \widehat{y}_{11} & \widehat{y}_{12}\\ \widehat{y}_{21} & \widehat{y}_{22}\\ ... & ...\\ \widehat{y}_{N1} & \widehat{y}_{N2} \end{bmatrix} = \begin{bmatrix} softmax(\begin{bmatrix} z^2_{11} & z^2_{12}) \end{bmatrix})_1 & softmax(\begin{bmatrix} z^2_{11} & z^2_{12}) \end{bmatrix})_2\\ softmax(\begin{bmatrix} z^2_{21} & z^2_{22}) \end{bmatrix})_1 & softmax(\begin{bmatrix} z^2_{21} & z^2_{22}) \end{bmatrix})_2\\ ... & ...\\ softmax(\begin{bmatrix} z^2_{N1} & z^2_{N2}) \end{bmatrix})_1 & softmax(\begin{bmatrix} z^2_{N1} & z^2_{N2}) \end{bmatrix})_2 \end{bmatrix} = \\[12pt] \begin{bmatrix} e^{z^2_{11}}/(e^{z^2_{11}} + e^{z^2_{12}}) & e^{z^2_{12}}/(e^{z^2_{11}} + e^{z^2_{12}})\\ e^{z^2_{21}}/(e^{z^2_{21}} + e^{z^2_{22}}) & e^{z^2_{22}}/(e^{z^2_{21}} + e^{z^2_{22}})\\ ... & ...\\ e^{z^2_{N1}}/(e^{z^2_{N1}} + e^{z^2_{N2}}) & e^{z^2_{N2}}/(e^{z^2_{N1}} + e^{z^2_{N2}})\\ \end{bmatrix}$ Running the forward pass on our sample data gives $\mathbf{Z^1} = \begin{bmatrix} -0.42392 & 1.12803\\ -0.11433 & 0.32380\\ 1.25645 & 0.87617\\ 0.02983 & 0.91020 \end{bmatrix}, \; \mathbf{X^2} = \begin{bmatrix} 1 & 0.39558 & 0.75548\\ 1 & 0.47145 & 0.58025\\ 1 & 0.77841 & 0.70603\\ 1 & 0.50746 & 0.71304 \end{bmatrix}$ $\mathbf{Z^2} = \begin{bmatrix} -0.00561 & -0.00022\\ -0.00676 & 0.00020\\ -0.00828 & 0.00185\\ -0.00650 & 0.00038 \end{bmatrix}, \; \widehat{\mathbf{Y}} = \begin{bmatrix} 0.49865 & 0.50135\\ 0.49826 & 0.50174\\ 0.49747 & 0.50253\\ 0.49828 & 0.50172 \end{bmatrix}$ ## Backpropagation Our strategy to find the optimal weights is gradient descent. Since we have a set of initial predictions for the training samples we’ll start by measuring the model’s current performance using our loss function, cross entropy. The loss associated with the $i$th prediction would be $CE_i = CE(\widehat{\mathbf Y_{i,}} \mathbf Y_{i,}) = -\sum_{c = 1}^{C} y_{ic} \log (\widehat{y}_{ic})$ where $c$ iterates over the target classes. Note here that $CE$ is only affected by the prediction value associated with the True instance. For example, if we were doing a 3-class prediction problem and $y$ = [0, 1, 0], then $\widehat y$ = [0, 0.5, 0.5] and $\widehat y$ = [0.25, 0.5, 0.25] would both have $CE = 0.69$. The cross entropy loss of our entire training dataset would then be the average $CE_i$ over all samples. For our training data, after our initial forward pass we’d have ImageId p1 p2 p3 p4 IsStairs Yhat_Stairs Yhat_Else CE 1 252 4 155 175 TRUE 0.49865 0.50135 0.6958 2 175 10 186 200 TRUE 0.49836 0.50174 0.6966 3 82 131 230 100 FALSE 0.49757 0.50253 0.6881 4 115 138 80 88 FALSE 0.49838 0.50172 0.6897 $CE = 0.69257$ Next, we need to determine how a “small” change in each of the weights would affect our current loss. In other words, we want to determine $\frac{\partial CE}{\partial w^1_{11}}$, $\frac{\partial CE}{\partial w^1_{12}}$, … $\frac{\partial CE}{\partial w^2_{32}}$ which is the gradient of $CE$ with respect to each of the weight matrices, $\nabla_{\mathbf{W^1}}CE$ and $\nabla_{\mathbf{W^2}}CE$. To start, recognize that $\frac{\partial CE}{\partial w_{ab}} = \frac{1}{N} \left[ \frac{\partial CE_1}{\partial w_{ab}} + \frac{\partial CE_2}{\partial w_{ab}} + ... \frac{\partial CE_N}{\partial w_{ab}} \right]$ where $\frac{\partial CE_i}{\partial w_{ab}}$ is the rate of change of [$CE$ of the $i$th sample] with respect to weight $w_{ab}$. In light of this, let’s concentrate on calculating $\frac{\partial CE_1}{w_{ab}}$, “How much will $CE$ of the first training sample change with respect to a small change in $w_{ab}$?”. If we can calculate this, we can calculate $\frac{\partial CE_2}{\partial w_{ab}}$ and so forth, and then average the partials to determine the overall expected change in $CE$ with respect to a small change in $w_{ab}$. Recall our network diagram. #### Step 1Determine $\frac{\partial CE_1}{\partial \widehat{\mathbf{Y_{1,}}}}$ $\frac{\partial CE_1}{\widehat{\mathbf{Y_{1,}}}} = \begin{bmatrix} \frac{\partial CE_1}{\widehat y_{11}} & \frac{\partial CE_1}{\widehat y_{12}} \end{bmatrix}$ Recall $CE_1 = CE(\widehat{\mathbf Y_{1,}}, \mathbf Y_{1,}) = -(y_{11}\log{\widehat y_{11}} + y_{12}\log{\widehat y_{12}})$ So $\frac{\partial CE_1}{\partial \widehat{\mathbf{Y_{1,}}}} = \begin{bmatrix} \frac{-y_{11}}{\widehat y_{11}} & \frac{-y_{12}}{\widehat y_{12}} \end{bmatrix}$ #### Step 2Determine $\frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}}$ $\frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}} = \begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} & \frac{\partial CE_1}{\partial z^2_{12}} \end{bmatrix}$ $\begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} & \frac{\partial CE_1}{\partial z^2_{12}} \end{bmatrix} = \begin{bmatrix} \frac{\partial CE_1}{\widehat y_{11}} \frac{\partial \widehat y_{11}}{z^2_{11}} + \frac{\partial CE_1}{\partial \widehat y_{12}} \frac{\partial \widehat y_{12}}{\partial z^2_{11}} & \frac{\partial CE_1}{\widehat y_{11}} \frac{\partial \widehat y_{11}}{z^2_{12}} + \frac{\partial CE_1}{\partial \widehat y_{12}} \frac{\partial \widehat y_{12}}{\partial z^2_{12}} \end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial \widehat y_{11}} & \frac{\partial CE_1}{\partial \widehat y_{12}} \end{bmatrix} \times \begin{bmatrix} \frac{\partial \widehat y_{11}}{\partial z^2_{11}} & \frac{\partial \widehat y_{11}}{\partial z^2_{12}}\\ \frac{\partial \widehat y_{12}}{\partial z^2_{11}} & \frac{\partial \widehat y_{12}}{\partial z^2_{12}} \end{bmatrix} = \\[12pt] \frac{\partial CE_1}{\partial \widehat{\mathbf{Y_{1,}}}} \frac{\partial \widehat{\mathbf{Y_{1,}}}}{\partial \mathbf{Z^2_{1,}}}$ We need to determine expressions for the elements of $\frac{\partial \widehat{\mathbf{Y_{1,}}}}{\partial \mathbf{Z^2_{1,}}} = \begin{bmatrix} \frac{\partial \widehat y_{11}}{\partial z^2_{11}} & \frac{\partial \widehat y_{11}}{\partial z^2_{12}}\\ \frac{\partial \widehat y_{12}}{\partial z^2_{11}} & \frac{\partial \widehat y_{12}}{\partial z^2_{12}} \end{bmatrix}$ Recall $\widehat{\mathbf{Y_{1,}}} = \begin{bmatrix} \widehat y_{11} & \widehat y_{12} \end{bmatrix} = softmax(\begin{bmatrix} z^2_{11} & z^2_{12} \end{bmatrix}) = \begin{bmatrix} \frac{e^{z^2_{11}}}{e^{z^2_{11}} + e^{z^2_{12}}} & \frac{e^{z^2_{12}}}{e^{z^2_{11}} + e^{z^2_{12}}} \end{bmatrix}$ We can make use of the quotient rule to show $\frac{\partial \, softmax(\theta)_c}{\partial \theta_j} = {\begin{cases} (softmax(\theta)_c)(1 - softmax(\theta)_c)&{\text{if }} j = c\\ (-softmax(\theta)_c)softmax(\theta)_j&{\text{otherwise}} \end{cases}}$. Hence, $\frac{\partial \widehat{\mathbf{Y_{1,}}}}{\partial \mathbf{Z^2_{1,}}} = \begin{bmatrix} \frac{\partial \widehat y_{11}}{\partial z^2_{11}} = \widehat y_{11}(1 - \widehat y_{11}) & \frac{\partial \widehat y_{11}}{\partial z^2_{12}} = -\widehat y_{12}\widehat y_{11}\\ \frac{\partial \widehat y_{12}}{\partial z^2_{11}} = -\widehat y_{11}\widehat y_{12} & \frac{\partial \widehat y_{12}}{\partial z^2_{12}} = \widehat y_{12}(1 - \widehat y_{12}) \end{bmatrix}$ Now we have $\frac{\partial CE_1}{\partial \widehat{\mathbf{Y_{1,}}}} \frac{\partial \widehat{\mathbf{Y_{1,}}}}{\partial \mathbf{Z^2_{1,}}} = \begin{bmatrix} \frac{-y_{11}}{\widehat y_{11}} & \frac{-y_{12}}{\widehat y_{12}} \end{bmatrix} \times \begin{bmatrix} \widehat y_{11}(1 - \widehat y_{11}) & -\widehat y_{12}\widehat y_{11}\\ -\widehat y_{11}\widehat y_{12} & \widehat y_{12}(1 - \widehat y_{12}) \end{bmatrix} = \\[12pt] \begin{bmatrix} -y_{11}(1 - \widehat y_{11}) + y_{12} \widehat y_{11} & y_{11} \widehat y_{12} - y_{12} (1 - \widehat y_{12}) \end{bmatrix} = \\[12pt] \begin{bmatrix} -\widehat y_{11} - y_{11} & \widehat y_{12} - y_{12} \end{bmatrix} = \\[12pt] \widehat{\mathbf{Y_{1,}}} - \mathbf{Y_{1,}}$ #### Step 3Determine $\frac{\partial CE_1}{\partial \mathbf{W^2}}$ $\frac{\partial CE_1}{\partial \mathbf{W^2}} = \begin{bmatrix} \frac{\partial CE_1}{\partial w^2_{11}} & \frac{\partial CE_1}{\partial w^2_{12}}\\ \frac{\partial CE_1}{\partial w^2_{21}} & \frac{\partial CE_1}{\partial w^2_{22}}\\ \frac{\partial CE_1}{\partial w^2_{31}} & \frac{\partial CE_1}{\partial w^2_{32}} \end{bmatrix} = \begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial w^2_{11}} & \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial w^2_{12}}\\ \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial w^2_{21}} & \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial w^2_{22}}\\ \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial w^2_{31}} & \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial w^2_{32}} \end{bmatrix} =$ $\begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} x^2_{11} & \frac{\partial CE_1}{\partial z^2_{12}} x^2_{11}\\ \frac{\partial CE_1}{\partial z^2_{11}} x^2_{12} & \frac{\partial CE_1}{\partial z^2_{12}} x^2_{12}\\ \frac{\partial CE_1}{\partial z^2_{11}} x^2_{13} & \frac{\partial CE_1}{\partial z^2_{12}} x^2_{13} \end{bmatrix} = \begin{bmatrix} x^2_{11}\\ x^2_{12}\\ x^2_{13}\\ \end{bmatrix} \times \begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} & \frac{\partial CE_1}{\partial z^2_{12}} \end{bmatrix} = \\[12pt] (\mathbf{X^2_{1,}})^T(\widehat{\mathbf{Y_{1,}}} - \mathbf{Y_{1,}})$ #### Step 4Determine $\frac{\partial CE_1}{\partial \mathbf{X^2_{1,}}}$ $\frac{\partial CE_1}{\partial \mathbf{X^2_{1,}}} = \begin{bmatrix} \frac{\partial CE_1}{\partial x^2_{11}} & \frac{\partial CE_1}{\partial x^2_{12}} & \frac{\partial CE_1}{\partial x^2_{13}} \end{bmatrix} =$ $\begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial x^2_{11}} + \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial x^2_{11}} & \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial x^2_{12}} + \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial x^2_{12}} & \frac{\partial CE_1}{\partial z^2_{11}} \frac{\partial z^2_{11}}{\partial x^2_{13}} + \frac{\partial CE_1}{\partial z^2_{12}} \frac{\partial z^2_{12}}{\partial x^2_{13}} \end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} w^2_{11} + \frac{\partial CE_1}{\partial z^2_{12}} w^2_{12} & \frac{\partial CE_1}{\partial z^2_{11}} w^2_{21} + \frac{\partial CE_1}{\partial z^2_{12}} w^2_{22} & \frac{\partial CE_1}{\partial z^2_{11}} w^2_{31} + \frac{\partial CE_1}{\partial z^2_{12}} w^2_{32} \end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial z^2_{11}} & \frac{\partial CE_1}{\partial z^2_{12}} \end{bmatrix} \times \begin{bmatrix} w^2_{11} & w^2_{21} & w^2_{31}\\ w^2_{12} & w^2_{22} & w^2_{32} \end{bmatrix} = \\[12pt] \left(\frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}}\right)\left(\mathbf{W^2}\right)^T$ #### Step 5Determine $\frac{\partial CE_1}{\partial \mathbf{Z^1_{1,}}}$ $\frac{\partial CE_1}{\partial \mathbf{Z^1_{1,}}} = \begin{bmatrix} \frac{\partial CE_1}{\partial z^1_{11}} & \frac{\partial CE_1}{\partial z^1_{12}}\end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial x^2_{12}} \frac{\partial x^2_{12}}{\partial z^1_{11}} & \frac{\partial CE_1}{\partial x^2_{13}} \frac{\partial x^2_{13}}{\partial z^1_{12}} \end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial x^2_{12}} & \frac{\partial CE_1}{\partial x^2_{13}} \end{bmatrix} \otimes \begin{bmatrix} \frac{\partial x^2_{12}}{\partial z^1_{11}} & \frac{\partial x^2_{13}}{\partial z^1_{12}} \end{bmatrix} = \\[12pt] \begin{bmatrix} \frac{\partial CE_1}{\partial x^2_{12}} & \frac{\partial CE_1}{\partial x^2_{13}} \end{bmatrix} \otimes \begin{bmatrix} \frac{\partial \, sigmoid(z^1_{11})}{\partial z^1_{11}} & \frac{\partial \, sigmoid(z^1_{12})}{\partial z^1_{12}} \end{bmatrix}$ Where $\otimes$ is the tensor product that does “element-wise” multiplication between matrices. Next we’ll use the fact that $\frac{d \, sigmoid(z)}{dz} = sigmoid(z)(1-sigmoid(z))$ to deduce that the expression above is equivalent to $\begin{bmatrix} \frac{\partial CE_1}{\partial x^2_{12}} & \frac{\partial CE_1}{\partial x^2_{13}} \end{bmatrix} \otimes \begin{bmatrix} x^2_{12}(1 - x^2_{12}) & x^2_{13}(1 - x^2_{13}) \end{bmatrix} = \\[12pt] \frac{\partial CE_1}{\partial \mathbf{X^2_{1,2:}}} \otimes \left( \mathbf{X^2_{1,2:}} \otimes \left( 1 - \mathbf{X^2_{1,2:}} \right) \right)$ #### Step 6Determine $\frac{\partial CE_1}{\partial \mathbf{W^1}}$ $\frac{\partial CE_1}{\partial \mathbf{W^1}} = \begin{bmatrix} \frac{\partial CE_1}{\partial w^1_{11}} & \frac{\partial CE_1}{\partial w^1_{12}}\\ \frac{\partial CE_1}{\partial w^1_{21}} & \frac{\partial CE_1}{\partial w^1_{22}}\\ \frac{\partial CE_1}{\partial w^1_{31}} & \frac{\partial CE_1}{\partial w^1_{32}}\\ \frac{\partial CE_1}{\partial w^1_{41}} & \frac{\partial CE_1}{\partial w^1_{42}}\\ \frac{\partial CE_1}{\partial w^1_{51}} & \frac{\partial CE_1}{\partial w^1_{52}} \end{bmatrix} = \begin{bmatrix} \frac{\partial CE_1}{\partial z^1_{11}} \frac{\partial z^1_{11}}{\partial w^1_{11}} & \frac{\partial CE_1}{\partial z^1_{12}} \frac{\partial z^1_{12}}{\partial w^1_{12}}\\ \frac{\partial CE_1}{\partial z^1_{11}} \frac{\partial z^1_{11}}{\partial w^1_{21}} & \frac{\partial CE_1}{\partial z^1_{12}} \frac{\partial z^1_{12}}{\partial w^1_{22}}\\ \frac{\partial CE_1}{\partial z^1_{11}} \frac{\partial z^1_{11}}{\partial w^1_{31}} & \frac{\partial CE_1}{\partial z^1_{12}} \frac{\partial z^1_{12}}{\partial w^1_{32}}\\ \frac{\partial CE_1}{\partial z^1_{11}} \frac{\partial z^1_{11}}{\partial w^1_{41}} & \frac{\partial CE_1}{\partial z^1_{12}} \frac{\partial z^1_{12}}{\partial w^1_{42}}\\ \frac{\partial CE_1}{\partial z^1_{11}} \frac{\partial z^1_{11}}{\partial w^1_{51}} & \frac{\partial CE_1}{\partial z^1_{12}} \frac{\partial z^1_{12}}{\partial w^1_{52}} \end{bmatrix} =$ $\begin{bmatrix} \frac{\partial CE_1}{\partial z^1_{11}} x^1_{11} & \frac{\partial CE_1}{\partial z^1_{12}} x^1_{11}\\ \frac{\partial CE_1}{\partial z^1_{11}} x^1_{12} & \frac{\partial CE_1}{\partial z^1_{12}} x^1_{12}\\ \frac{\partial CE_1}{\partial z^1_{11}} x^1_{13} & \frac{\partial CE_1}{\partial z^1_{12}} x^1_{13}\\ \frac{\partial CE_1}{\partial z^1_{11}} x^1_{14} & \frac{\partial CE_1}{\partial z^1_{12}} x^1_{14}\\ \frac{\partial CE_1}{\partial z^1_{11}} x^1_{15} & \frac{\partial CE_1}{\partial z^1_{12}} x^1_{15} \end{bmatrix} = \begin{bmatrix} x^1_{11}\\ x^1_{12}\\ x^1_{13}\\ x^1_{14}\\ x^1_{15} \end{bmatrix} \times \begin{bmatrix} \frac{\partial CE_1}{\partial z^1_{11}} & \frac{\partial CE_1}{\partial z^1_{12}} \end{bmatrix} = \\[12pt] \left(\mathbf{X^1_{1,}}\right)^T \left(\frac{\partial CE_1}{\partial \mathbf{Z^1_{1,}}}\right)$ #### Recapping we have \begin{aligned} \frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}} &= \widehat{\mathbf{Y_{1,}}} - \mathbf{Y_{1,}}\\ \frac{\partial CE_1}{\partial \mathbf{X^2_{1,}}} &= \left(\frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}}\right) \left(\mathbf{W^2}\right)^T\\ \frac{\partial CE_1}{\partial \mathbf{Z^1_{1,}}} &= \frac{\partial CE_1}{\partial \mathbf{X^2_{1,2:}}} \otimes \left( \mathbf{X^2_{1,2:}} \otimes \left( 1 - \mathbf{X^2_{1,2:}} \right) \right)\\ \end{aligned} $\boxed{ \frac{\partial CE_1}{\partial \mathbf{W^2}} = \left(\mathbf{X^2_{1,}}\right)^T \left(\frac{\partial CE_1}{\partial \mathbf{Z^2_{1,}}}\right) }\\ \boxed{ \frac{\partial CE_1}{\partial \mathbf{W^1}} = \left(\mathbf{X^1_{1,}}\right)^T \left(\frac{\partial CE_1}{\partial \mathbf{Z^1_{1,}}}\right) }$ Now we have expressions that we can easily use to compute how cross entropy of the first training sample should change with respect to a small change in each of the weights. These formulas easily generalize to let us compute the change in cross entropy for every training sample as follows. \begin{aligned} \nabla_{\mathbf{Z^2}}CE &= \widehat{\mathbf{Y}} - \mathbf{Y}\\ \nabla_{\mathbf{X^2}}CE &= \left(\nabla_{\mathbf{Z^2}}CE\right) \left(\mathbf{W^2}\right)^T\\ \nabla_{\mathbf{Z^1}}CE &= \left(\nabla_{\mathbf{X^2_{,2:}}}CE\right) \otimes \left(\mathbf{X^2_{,2:}} \otimes \left( 1 - \mathbf{X^2_{,2:}}\right) \right)\\ \end{aligned} $\boxed{ \nabla_{\mathbf{W^2}}CE = \left(\mathbf{X^2}\right)^T \left(\nabla_{\mathbf{Z^2}}CE\right) }\\ \boxed{ \nabla_{\mathbf{W^1}}CE = \left(\mathbf{X^1}\right)^T \left(\nabla_{\mathbf{Z^1}}CE\right) }$ Notice how convenient these expressions are. We already know $\mathbf{X^1}$, $\mathbf{W^1}$, $\mathbf{W^2}$, and $\mathbf{Y}$, and we calculated $\mathbf{X^2}$ and $\widehat{\mathbf{Y}}$ during the forward pass. This happens because we smartly chose activation functions such that their derivative could be written as a function of their current value. Following up with our sample training data, we’d have $\nabla_{\mathbf{Z^2}}CE = \begin{bmatrix} -0.50135 & 0.50135\\ -0.50174 & 0.50174\\ 0.49747 & -0.49747\\ 0.49828 & -0.49828 \end{bmatrix}, \; \nabla_{\mathbf{X^2}}CE = \begin{bmatrix} 0.00178 & 0.00595 & -0.00190\\ 0.00179 & 0.00596 & -0.00190\\ -0.00177 & -0.00590 & 0.00189\\ -0.00177 & -0.00591 & 0.00189 \end{bmatrix} \\[12pt] \nabla_{\mathbf{Z^1}}CE = \begin{bmatrix} 0.00142 & -0.00035\\ 0.00148 & -0.00046\\ -0.00102 & 0.00039\\ -0.00148 & 0.00039 \end{bmatrix}, \; \nabla_{\mathbf{W^2}}CE = \begin{bmatrix} -0.00183 & 0.00183\\ 0.05131 & -0.05131\\ 0.00916 & -0.00916 \end{bmatrix} \\[12pt] \nabla_{\mathbf{W^1}}CE = \begin{bmatrix} 0.00010 & -0.00001\\ 0.09119 & -0.02325\\ -0.07923 & 0.02464\\ 0.03601 & -0.00491\\ 0.07847 & -0.02023 \end{bmatrix}$ Now we can update the weights by taking a small step in the direction of the negative gradient. In this case, we’ll let stepsize = 0.1 and make the following updates $\mathbf{W^1} := \mathbf{W^1} - stepsize \cdot \nabla_{\mathbf{W^1}}CE\\ \mathbf{W^2} := \mathbf{W^2} - stepsize \cdot \nabla_{\mathbf{W^2}}CE$ For our sample data… $\mathbf{W^1} := \begin{bmatrix} -0.00470 & 0.00797\\ -0.01168 & 0.01121\\ 0.00938 & 0.00076\\ 0.00456 & 0.00307\\ -0.01382 & -0.00674 \end{bmatrix}\\[12pt] \mathbf{W^2} := \begin{bmatrix} -0.00570 & -0.00250\\ -0.01160 & 0.01053\\ 0.00282 & 0.00087 \end{bmatrix}$ The updated weights are not guaranteed to produce a lower cross entropy error. It’s possible that we’ve stepped too far in the direction of the negative gradient. It’s also possible that, by updating every weight simultaneously, we’ve stepped in a bad direction. Remember, $\frac{\partial CE}{\partial w^1_{11}}$ is the instantaneous rate of change of $CE$ with respect to $w^1_{11}$ under the assumption that every other weight stays fixed. However, we’re updating all the weights at the same time. In general this shouldn’t be a problem, but occasionally it’ll cause increases in our loss as we update the weights. ## Wrapping it up We started with random weights, measured their performance, and then updated them with (hopefully) better weights. The next step is to do this again and again, either a fixed number of times or until some convergence criteria is met. ## Challenge Try implementing this network in code. I’ve done it in R here.

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Formula Used 1 Mole = 1E-09 Gigamole 1 Mole = 0.1 Decamole 1 Gigamole = 100000000 Decamole ## Gigamoles to Decamoles Conversion Gmol stands for gigamoles and damol stands for decamoles. The formula used in gigamoles to decamoles conversion is 1 Gigamole = 100000000 Decamole. In other words, 1 gigamole is 100000000 times bigger than a decamole. To convert all types of measurement units, you can used this tool which is able to provide you conversions on a scale. ## Convert Gigamole to Decamole How to convert gigamole to decamole? In the amount of substance measurement, first choose gigamole from the left dropdown and decamole from the right dropdown, enter the value you want to convert and click on 'convert'. Want a reverse calculation from decamole to gigamole? You can check our decamole to gigamole converter. How to convert Gigamole to Decamole? The formula to convert Gigamole to Decamole is 1 Gigamole = 100000000 Decamole. Gigamole is 100000000 times Bigger than Decamole. Enter the value of Gigamole and hit Convert to get value in Decamole. Check our Gigamole to Decamole converter. Need a reverse calculation from Decamole to Gigamole? You can check our Decamole to Gigamole Converter. How many Mole is 1 Gigamole? 1 Gigamole is equal to 100000000 Mole. 1 Gigamole is 100000000 times Bigger than 1 Mole. How many Millimole is 1 Gigamole? 1 Gigamole is equal to 100000000 Millimole. 1 Gigamole is 100000000 times Bigger than 1 Millimole. How many Kilomole is 1 Gigamole? 1 Gigamole is equal to 100000000 Kilomole. 1 Gigamole is 100000000 times Bigger than 1 Kilomole. How many Pound Mole is 1 Gigamole? 1 Gigamole is equal to 100000000 Pound Mole. 1 Gigamole is 100000000 times Bigger than 1 Pound Mole. ## Gigamoles to Decamoles Converter Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like amount of substance finds its use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the conversion of different units of measurement like Gmol to damol through multiplicative conversion factors. When you are converting amount of substance, you need a Gigamoles to Decamoles converter that is elaborate and still easy to use. Converting Gigamole to Decamole is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in Gigamole to Decamole conversion along with a table representing the entire conversion. Let Others Know

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a neat picture based on recursive number A place to discuss the science of computers and programs, from algorithms to computability. Formal proofs preferred. Moderators: phlip, Moderators General, Prelates phillip1882 Posts: 145 Joined: Fri Jun 14, 2013 9:11 pm UTC Location: geogia Contact: a neat picture based on recursive number https://ibb.co/kPRY2z this picture represents the first 4000 numbers in recursive format. <> 2 <<>> 3 <><>4 <<<>>> 5 and so on the left arrow turns the turtle left 30 degrees then moves forward 5 and the right arrow rotate right 60 degrees and then move forward 5 heres the python code that produced it Code: Select all `import turtlerecurse = [""]*1000000recurse[2] = "<>"n = 2p = 1while n < len(recurse):   for v in range(2,int(len(recurse)/n)):      if recurse[v] != "":         recurse[v*n] = recurse[v] +recurse[n]        n += 1   while n<len(recurse) and recurse[n] != "":      n += 1   if n == len(recurse):      break   p += 1   recurse[n] = "<" +recurse[p] +">"turtle.screensize(7000,4500)turtle.up()   turtle.sety(-200)turtle.down()for i in range(2,len(recurse)):   for j in range(0,len(recurse[i])):      if recurse[i][j] == ">":         turtle.right(60)         turtle.forward(5)      else:         turtle.left(30)         turtle.forward(5)   print(recurse[i],i)input()` good luck have fun Xanthir My HERO!!! Posts: 5410 Joined: Tue Feb 20, 2007 12:49 am UTC Contact: Re: a neat picture based on recursive number First, neat picture! It's always cool to see the kind of emergent patterns that come out of simple rules like this. Second, the few numbers you provided give *no* clue as to the pattern you're following. I had to puzzle thru your Python to figure it out instead. From what I can tell, the pattern is: 1. The 1st pattern, P₁, is the empty string. 2. If n is the mth prime, then Pₙ = "<" + Pₘ + ">". (So, since 2 is the 1st prime, P₂ = <>, which is P₁ wrapped in angle brackets. 3 is the 2nd prime, so it's <<>>, P₂ wrapped. 5 is P₃ wrapped, 7 is P₄ wrapped, 11 is P₅ wrapped, etc.) 3. If n is composite, it can be divided into its smallest prime factor j and the rest of the number k. Pₙ is then Pₖ+Pⱼ. (So P₄ is P₂+P₂ <><>, P₁₂ is P₄+P₃ <><><<>>, P₁₀₀ is P₅₀+P₂, etc.) Third, your use of single-letter variables and some slightly unidiomatic Python made it a bit hard to follow. I've rewritten it here to help myself understand: Code: Select all `#!/usr/bin/env python2def generatePatterns(limit):   patterns = [""] * limit   prime = 2   primeCount = 1   while prime < limit:      # If N is the Mth prime, generate its pattern from the Mth pattern.      patterns[prime] = "<" + patterns[primeCount] + ">"      # Generate patterns for every possible multiple of prime      # from the Prime'th pattern      # and the Multiplier'th pattern      for mult in xrange(2): # we won't use the whole range         if mult*prime >= limit:            break         if patterns[multiplier]:            patterns[multiplier*newPrime] = patterns[multiplier] + patterns[newPrime]      # Now find the next prime.      # Per Eratosthene's sieve, this is just the next unfilled value.      while prime < limit and patterns[prime]:         prime += 1      primeCount += 1   return patternsdef drawPatterns(patterns):   import turtle   turtle.screensize(7000,4500)   turtle.up()   turtle.sety(-200)   turtle.down()   for pattern in patterns:      for command in pattern:         if command == ">":            turtle.right(60)            turtle.forward(5)         else:            turtle.left(30)            turtle.forward(5)      print(pattern,i)drawPatterns(generatePatterns(10**6))` I haven't run this, but I think it works? Is this code clear to you? (defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b))) phillip1882 Posts: 145 Joined: Fri Jun 14, 2013 9:11 pm UTC Location: geogia Contact: Re: a neat picture based on recursive number some small mistakes you made, i corrected them. mostly just using variables without declaring them Code: Select all `def generatePatterns(limit):   patterns = [""] * limit   prime = 2   primeCount = 1   while prime < limit:      # If N is the Mth prime, generate its pattern from the Mth pattern.      patterns[prime] = "<" + patterns[primeCount] + ">"      # Generate patterns for every possible multiple of prime      # from the Prime'th pattern      # and the Multiplier'th pattern      for mult in range(2,len(patterns)): # we won't use the whole range         if mult*prime >= limit:            break         if patterns[mult]:            patterns[mult*prime] = patterns[mult] + patterns[prime]      # Now find the next prime.      # Per Eratosthene's sieve, this is just the next unfilled value.      while prime < limit and patterns[prime]:         prime += 1      primeCount += 1   return patternsdef drawPatterns(patterns):   import turtle   turtle.screensize(7000,4500)   turtle.up()   turtle.sety(-200)   turtle.down()   i = 0   for pattern in patterns:      for command in pattern:         if command == ">":            turtle.right(60)            turtle.forward(5)         else:            turtle.left(30)            turtle.forward(5)      print(pattern,i)      i+=1drawPatterns(generatePatterns(10**6))` good luck have fun FlatAssembler Posts: 66 Joined: Fri Oct 27, 2017 7:42 pm UTC Re: a neat picture based on recursive number Didn't know Python supported turtle graphics, thanks for informing me. Anyway, I've done a similar thing in JavaScript about a year ago. You can see it here. Though, admittedly, your looks even better. PM 2Ring Posts: 3713 Joined: Mon Jan 26, 2009 3:19 pm UTC Location: Sydney, Australia Re: a neat picture based on recursive number Also see Dyck words. In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets [ and ]. The set of Dyck words forms Dyck language. Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. phlip Restorer of Worlds Posts: 7572 Joined: Sat Sep 23, 2006 3:56 am UTC Location: Australia Contact: Re: a neat picture based on recursive number Code: Select all `enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}` [he/him/his] Xanthir My HERO!!! Posts: 5410 Joined: Tue Feb 20, 2007 12:49 am UTC Contact: Re: a neat picture based on recursive number Ah, indeed! I hadn't quite fully gathered the recursive nature there. So you're expressing a number solely by using the nth-prime() function, multiplication, and the number 1. For compactness, nth-prime(arg) is instead written as <arg>, multiplication is implicit from concatenation, and 1 can be omitted because it only and always appears at the deepest nesting of a <>. So 7 is nth-prime(4), aka nth-prime(2*2), aka nth-prime(nth-prime(1) * nth-prime(1)), and this can then be compacted to <<><>>. (defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b))) phillip1882 Posts: 145 Joined: Fri Jun 14, 2013 9:11 pm UTC Location: geogia Contact: Re: a neat picture based on recursive number https://ibb.co/dxoNzp another one for fun. its red for an even number of prime factors and blue for odd. its also toroidal, it wraps around the edges. also i made the right 45 rather than 60, and forward 3 rather than 5 this picture is roughly 1/4 the whole image. Code: Select all `import timeimport turtlerecurse = [""]*1000000recurse[2] = "<>"n = 2p = 1while n < len(recurse):   for v in range(2,int(len(recurse)/n)):      if recurse[v] != "":         recurse[v*n] = recurse[v] +recurse[n]        n += 1   while n<len(recurse) and recurse[n] != "":      n += 1   if n == len(recurse):      break   p += 1   recurse[n] = "<" +recurse[p] +">"turtle.screensize(2000,2000)turtle.up()   turtle.sety(-200)turtle.down()turtle.speed(0)turtle.hideturtle()for i in range(2,len(recurse)):   for j in range(0,len(recurse[i])):      x,y = turtle.position()      if x < -1000:         turtle.up()         turtle.setx(1000)         turtle.down()      elif x > 1000:         turtle.up()         turtle.setx(-1000)         turtle.down()      if y <-1000:         turtle.up()         turtle.sety(1000)         turtle.down()      elif y > 1000:         turtle.up()         turtle.sety(-1000)          turtle.down()      middle = 0      total = 0      for k in range(0,len(recurse[i])):         if recurse[i][k] == "<":            middle += 1         else:            middle -= 1         if middle == 0:            total +=1      if total&1 == 0:         turtle.color("red")      else:         turtle.color("blue")      if recurse[i][j] == ">":         turtle.right(45)         turtle.forward(3)      else:         turtle.left(30)         turtle.forward(3)          print(recurse[i],i)time.sleep(86400)input()` good luck have fun phillip1882 Posts: 145 Joined: Fri Jun 14, 2013 9:11 pm UTC Location: geogia Contact: Re: a neat picture based on recursive number https://ibb.co/bYDXKz this one is green for squares or multiple of squares. this picture is after roughly 2000. i don't think posting the code is necessary, its fairly straight forward good luck have fun phillip1882 Posts: 145 Joined: Fri Jun 14, 2013 9:11 pm UTC Location: geogia Contact: Re: a neat picture based on recursive number final one: the result after 200,000 approximately, it mostly just looks like a random plot https://ibb.co/ff53Gp good luck have fun Soupspoon You have done something you shouldn't. Or are about to. Posts: 4060 Joined: Thu Jan 28, 2016 7:00 pm UTC Location: 53-1 Re: a neat picture based on recursive number "Looks like a random plot" is at one end of a spectrum of expectations. "Looks exactly like it's what you put in" is at the other end. Unless you're looking for that kind "procedural fuzziness" output to produce (reproducable, arbitrarily detailed 'naturalistic' distribtions at will, the more interesting ones are in the middle. They don't look "random" but they have no obvious relationship with the specification chosen to produce them. Like seeding a Perlin Noise and somehow getting the impression of scrawled 'letter' shapes saying "This was just the first death!! I shall kill again!!!!".as a result. That said, I like what you're looking at. I assume you already knew about the likes of the Dragon Curve, BTW? FlatAssembler Posts: 66 Joined: Fri Oct 27, 2017 7:42 pm UTC Re: a neat picture based on recursive number Here is a picture I made in my own programming language a few days ago: Code: Select all `;Mathematical example: The Polar Rose.AsmStart   format PE console   entry start   include 'win32a.inc'   section '.text' code executable   start:   invoke system,_cls   invoke system,_setGreenForegroundAsmEndy:=0While y<24   x:=0   While x<80      distance:=sqrt(pow(abs((x-40)/2),2)+pow(abs(y-12),2))      angle:=atan2(y-12,(x-40)/2)      r:=cos(angle*3)*12      If (distance < (r + 0.5) ) | ( (y = 12) & (x > 40-1) & (x < 40 + (13 * 2) ) )         AsmStart            jmp starSign\$            starSign:               db '*',0            starSign\$:            sub esp,4            mov dword [esp],starSign            call [printf]         AsmEnd      Else         AsmStart            jmp spaceSign\$            spaceSign:               db ' ',0            spaceSign\$:            mov dword [esp],spaceSign            call [printf]         AsmEnd      EndIf      x:=x+1   EndWhile   AsmStart      jmp newLineSign\$      newLineSign:         db 10,0      newLineSign\$:      sub esp,4      mov dword [esp],newLineSign      call [printf]   AsmEnd   y:=y+1EndWhileAsmStartinvoke system,_pauseinvoke system,_restoreColorsinvoke system,_clsinvoke exit,0_cls db "CLS",0_setGreenForeground db "COLOR 0A",0_pause db "PAUSE",0_restoreColors db "COLOR 07",0section '.rdata' readable writableresult dd ?x dd ?y dd ?distance dd ?angle dd ?r dd ?section '.idata' data readable importlibrary msvcrt,'msvcrt.dll'import msvcrt,printf,'printf',system,'system',exit,'exit',scanf,'scanf'AsmEnd` Who is online Users browsing this forum: No registered users and 3 guests

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CC-MAIN-2019-30

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http://www.algebra.com/cgi-bin/show-question-source.mpl?solution=318082

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```Question 464329 m+n=20 5m+10n=155 5m+5n=100 5n=55 n=11 m=9 Diane has 11 nickels and 9 dimes..```

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CC-MAIN-2013-20

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## 3910 3,910 (three thousand nine hundred ten) is an even four-digits composite number following 3909 and preceding 3911. In scientific notation, it is written as 3.91 × 103. The sum of its digits is 13. It has a total of 4 prime factors and 16 positive divisors. There are 1,408 positive integers (up to 3910) that are relatively prime to 3910. ## Basic properties • Is Prime? No • Number parity Even • Number length 4 • Sum of Digits 13 • Digital Root 4 ## Name Short name 3 thousand 910 three thousand nine hundred ten ## Notation Scientific notation 3.91 × 103 3.91 × 103 ## Prime Factorization of 3910 Prime Factorization 2 × 5 × 17 × 23 Composite number Distinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 3910 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0 The prime factorization of 3,910 is 2 × 5 × 17 × 23. Since it has a total of 4 prime factors, 3,910 is a composite number. ## Divisors of 3910 1, 2, 5, 10, 17, 23, 34, 46, 85, 115, 170, 230, 391, 782, 1955, 3910 16 divisors Even divisors 8 8 4 4 Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 7776 Sum of all the positive divisors of n s(n) 3866 Sum of the proper positive divisors of n A(n) 486 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 62.53 Returns the nth root of the product of n divisors H(n) 8.04527 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors The number 3,910 can be divided by 16 positive divisors (out of which 8 are even, and 8 are odd). The sum of these divisors (counting 3,910) is 7,776, the average is 486. ## Other Arithmetic Functions (n = 3910) 1 φ(n) n Euler Totient Carmichael Lambda Prime Pi φ(n) 1408 Total number of positive integers not greater than n that are coprime to n λ(n) 176 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 544 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares There are 1,408 positive integers (less than 3,910) that are coprime with 3,910. And there are approximately 544 prime numbers less than or equal to 3,910. ## Divisibility of 3910 m n mod m 2 3 4 5 6 7 8 9 0 1 2 0 4 4 6 4 The number 3,910 is divisible by 2 and 5. • Arithmetic • Deficient • Polite • Square Free ## Base conversion (3910) Base System Value 2 Binary 111101000110 3 Ternary 12100211 4 Quaternary 331012 5 Quinary 111120 6 Senary 30034 8 Octal 7506 10 Decimal 3910 12 Duodecimal 231a 20 Vigesimal 9fa 36 Base36 30m ## Basic calculations (n = 3910) ### Multiplication n×i n×2 7820 11730 15640 19550 ### Division ni n⁄2 1955 1303.33 977.5 782 ### Exponentiation ni n2 15288100 59776471000 233726001610000 913868666295100000 ### Nth Root i√n 2√n 62.53 15.7541 7.90759 5.2292 ## 3910 as geometric shapes ### Circle Diameter 7820 24567.3 4.8029e+07 ### Sphere Volume 2.50391e+11 1.92116e+08 24567.3 ### Square Length = n Perimeter 15640 1.52881e+07 5529.58 ### Cube Length = n Surface area 9.17286e+07 5.97765e+10 6772.32 ### Equilateral Triangle Length = n Perimeter 11730 6.61994e+06 3386.16 ### Triangular Pyramid Length = n Surface area 2.64798e+07 7.04472e+09 3192.5 ## Cryptographic Hash Functions md5 1be883eec3231f9fe43c35bd1b4b3bb5 dd3a575117170389f7a4185aac816f1c72068754 e70a396cdf3eea9df71179a4b709ee24f3b22ea2a438d30f4c1ce33f6f07cc2a 6c09ef937d212bb50a6a024f73909437d0ce679a8a1b059ac492cf46aa8dba72b75dc22b7d312795250f29c99d91d6f5fbee86c347ed3174e294613031b22377 84da7805ccf89b42f99244c422c5bc82e617ba65

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CC-MAIN-2020-45

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https://whatisconvert.com/gallons-to-liters

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# Convert Gallons to Liters ## Convert Gallons to Liters To calculate a value in Gallons to the corresponding value in Liters, multiply the quantity in Gallons by 3.7854118 (conversion factor). Liters = Gallons x 3.7854118 ## How to convert from Gallons to Liters The conversion factor from Gallons to Liters is 3.7854118. To find out how many Gallons in Liters, multiply by the conversion factor or use the Gallons to Liters converter above. ## Definition of Gallon The gallon (abbreviation "gal"), is a unit of volume which refers to the United States liquid gallon. There are three definitions in current use: the imperial gallon (≈ 4.546 L) which is used in the United Kingdom and semi-officially within Canada, the United States (liquid) gallon (≈ 3.79 L) which is the commonly used, and the lesser used US dry gallon (≈ 4.40 L). ## Definition of Liter The liter (also written "litre"; SI symbol L or l) is a non-SI metric system unit of volume. It is equal to 1 cubic decimeter (dm3), 1,000 cubic centimeters (cm3) or 1/1,000 cubic meter. The mass of one liter liquid water is almost exactly one kilogram. A liter is defined as a special name for a cubic decimeter or 10 centimeters × 10 centimeters × 10 centimeters, thus, 1 L ≡ 1 dm3 ≡ 1000 cm3.

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CC-MAIN-2021-31

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http://wiki.multitheftauto.com/wiki/MathNumber

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# MathNumber Jump to navigation Jump to search This function can be used to take or add a number from a large number. MTA clients do only support 24-bits numbers, therefore you can't take or add a single digit from large numbers like '1364576384'. This function is only needed clientside since the server runs with a floating-point precision of 56-bits, however this works also serverside. ## Syntax `int mathNumber ( int num, int integer [, string type = "+"] )` ### Required Arguments • num: The number you want to edit • integer: The amount you want to add or take from the number • type : Can be either '-' or '+' ### Returns Returns a number if everything went good, false otherwise. ## Code ```-- Function that takes a digit from a larger number function mathNumber ( num, integer, type ) if not ( num ) or not ( integer ) then return false end local function formatNumber( numb ) if not ( numb ) then return false end local fn = string.sub( tostring( numb ), ( #tostring( numb ) -6 ) ) return tonumber( fn ) end if not ( type ) or ( type == "+" ) then return tonumber( string.sub( tostring( num ), 1, -8 )..( formatNumber ( num ) ) + integer ) else return tonumber( string.sub( tostring( num ), 1, -8 )..( formatNumber ( num ) ) - integer ) end end ``` ## Example ```mathNumber ( 1364576384, 1, '-' ) -- Returns: 1364576383 1364576384 - 1 -- Returns: 1364576384 mathNumber ( 1364576384, 1, '+' ) -- Returns: 1364576385 1364576384 + 1 -- Returns: 1364576384 ``` Author: DennisUniOn ## See Also ### Table functions • addTableChangeHandler » This function monitors the changes of a table. • pairsByKeys » This function sort pairs table. • rangeToTable » This function converts a string range to a table containing number values. • setTableProtected » This function protects a table and makes it read-only. • setTableToSql » This function is used to save the table in the database (sql). • Sort_Functions » These functions are able to sort your tables by a key. • getKeyFromValueInTable » This function returns the key of the specified value in a table. • getTableFromSql » This functionality is used to obtain saved tables using the function (SetTableToSql ). • isValueInTable » This function returns true if the value exists in the table, false if the value does not exist in the table. • table.compare » This function checks whether two given tables are equal. • table.copy » This function copies a whole table and all the tables in that table. • table.deepmerge » This function deep merges two tables. Every nested table will be correspondingly merged. • table.element » This function returns a new table with only userdata content. • table.flip » This function returns the table from the last value to the first value, such as reflection. • table.fromString » This function converts string to a table. • table.getRandomRows » This function returns random rows from table. • table.map » This function goes through a table and replaces every field with the return of the passed function, where the field's value is passed as first argument and optionally more arguments. • table.merge » This function merges two or more tables together. • table.random » This function retrieves a random value from a table. • table.removeValue » This function removes a specified value from a table. • table.size » This function returns the absolute size of a table. ### ACL functions • aclGroupClone » This function clone a group to another group with/without ACLs and/or objects. • renameAclGroup » This function gives an existing ACL group a new name. • getPlayersInACLGroup » This function returns all players in an ACL group. • isPlayerInACL » This function checks if a player element is in an ACL group. ### Account functions • getPlayerFromAccountName » This function is used to obtain a player by the name of his account. • isPlayerAccount » This function checks if the account is a valid player account (account exists and is not a guest account) ### Camera functions • smoothMoveCamera » This function allows you to create a cinematic camera flight. • sCamera » The function creates a speed camera in-game, fines speeding vehicles, and notifies the driver and take money from player based on vehicle speed. ### Effects functions • attachEffect » This function allows you attach an effect to an element. • setScreenFlash » This function will make the screen flash(like a screenshot). ### Element functions • autoAttach » This function attaches one element into another at the same position and rotation they are. • attachElementToBone » This function allows you to attach an element to ped bone accurately using new bone functions. • getElementDirectionCardialPoint » This function returns the direction of the element according to the wind rose. • getElementSpeed » This function returns the specified element's speed in m/s, km/h or mph. • getElementUsingData » This function returns table elements that contains the elements data with the given key and value. • getElementZoneFullName » This function allows you to retrieve the zone full name of a element. • getElementsInDimension » This function returns a table of elements that are in the specified dimension. • getElementsWithinMarker » This function returns a table of elements that are within a marker's collision shape. • getNearestElement » This function returns the nearest element (of a specific type) to a player. • getPositionInFrontOfElement » This function returns position in provided distance away from element, including element's rotation. • isElementInAir » This function checks if an element is in air or not. • isElementInPhotograph » This function checks if an element is in the player's camera picture area. • isElementInRange » This function allows you to check if an element's range to a main point is within the maximum range. • isElementMoving » This function checks if an element is moving. • isElementPlayer » This function checks whether the element is a player or not. • isElementWithinAColShape » This function checks if an element is within a collision shape element. • multi_check » This function checks one element to many, handy and clean. • setElementSpeed » This function allows you to set the speed of an element in kph or mph units. ### Events • onClientPlayerTimeChange » This code implements an event that is triggered when the player's real time change. • onPlayerZoneChange » This code implements an event that is triggered when the player enters a new area on the map. • onVehicleWeaponFire » This code implements an event that is triggered when a player in a vehicle fires a vehicle's weapon. ### Input functions • bindControlKeys » This function allows you to bind each key bound to a control individually. Doing this bypasses a little MTA restriction. • unbindControlKeys » This function allows you to unbind each key bound to a control individually. Use this function with bindControlKeys. • getBoundControls » This function returns a table of control names that are bound to the specified key. • isCommandHandlerAdded » This function allows you to check if a command is added or not in the respective resource. ### Data functions • levenshtein » This function can be used to calculate the Levenshtein distance between two strings. • gregorianToJalali » This function converts gregorian date to jalali/shamsi date. • byte2human » This function converts an integer (number of bytes) into a human-readable unit. • capitalize » This function capitalizes a given string. • convertDate » This function converts date to another look. • convertServerTickToTimeStamp » This function converts server ticks to a unix timestamp. • convertTextToSpeech » This function converts the provided text to a speech in the provided language which players can hear. • findRotation3D » This function takes two sets of XYZ coordinates. It returns the 3D direction from point A to point B. • findRotation » This function takes two points and returns the direction from point A to point B. • formatDate » This function formats a date on the basis of a format string and returns it. • formatNumber » This function formats large numbers by adding commas. • generateRandomASCIIString » This function returns a random string which uses ASCII characters. • generateString » This function generates a random string with any characters. • getAge » This function calculates the age of a given birthday. • getDistanceBetweenElements » Returns the distance between two elements. • getDistanceBetweenPointAndSegment2D » This function takes point coordinates and line (a segment) starting and ending coordinates. It returns the shortest distance between the point and the line. • getEasterDate » This function returns easter date monthday and month for a given year. • getElementRelatedAngle » This function returns the related angle between one element to another. This is useful to check which side an element is to another. • getFreeDimension » This function get free dimension. • getOffsetFromXYZ » This function allows you to take an entity and a position and calculate the relative offset between them accounting for rotations. • getPointFromDistanceRotation » This function finds a point based on a starting point, direction and distance. • getRealMonth » This function returns the current month name • getRGColorFromPercentage »This function returns two integers representing red and green colors according to the specified percentage. • getScreenRotationFromWorldPosition » This function returns a screen relative rotation to a world position. • getTimestamp » This function returns the UNIX timestamp of a specified date and time. • gradientString » This function transforms a string in a new coloured gradient string. • hex2rgb » This function convert hex to rgb. • hexColorToRGB » This function convert hex string/number to RGBA values. • isLeapYear » This function returns a boolean representing if a given year is a leap year. • isValidMail » This function checks whether a provided e-mail string is valid. • removeHex » This function is used to remove hexadecimal numbers (colors, for example) from strings. • RGBToHex » This function returns a string representing the color in hexadecimal. • RGBToHSV » This function convert RGB to HSV color space. • RGBToDecimal » This function convert RGB to Decimal color. • secondsToTimeDesc » This function converts a plain seconds-integer into a user-friendly time description. • string.count » This function counts the amount of occurences of a string in a string. • string.explode » This function splits a string at a given separator pattern and returns a table with the pieces. • string.insert » This function inserts a string within another string at a given position. • splitMultiple » This function improves the split function so that multiple characters can be used as the split at character. • switch » This function allows the value of a variable or expression to control the flow of program execution via a multiway branch. • tocolor2rgba » This function convert tocolor to rgba. • toHex » This function converts a decimal number to a hexadecimal number, as a fix to be used client-side. • var dump » This function outputs information about one or more variables using outputConsole. • wavelengthToRGBA » This function converts a physical wavelength of light to a RGBA color. • fixPersianString » This function returns a fixed sorted bilingual RTL for strings consisting of Farsi/Arabic and English. ### GUI functions • centerWindow » This function centers a CEGUI window element responsively in any resolution. • isMouseOnGUICloseButton » This function allows you to check whether the mouse cursor/pointer is within a gui-window's native close button. • isMouseOnGuiElement » This function allows you to check whether or not your mouse is over a specific gui element, this is especially useful if the gui element has a parent. • guiMoveElement » This function moves guiElement by/like using moveObject. • guiSetStaticImageMovable » This function allows you to move a static image like a gui window. ##### Labels • guiLabelAddEffect » This function add an effects to the gui-label like (shadow, outline). ### Math functions • reMap » Re-maps a number from one range to another. • math.clamp » This function returns the number between range of numbers or it's minimum or maximum. • math.getBezierPoint » Get N-th order bezier point. • math.hypot » This function returns the Hypotenuse of the triangle given by sides x and y. • math.isPointInPolygon » Check if point is inside polygon or not. • math.lerp » Get val between two integer. • math.percent » This function returns a percentage from two number values. • math.polygonArea » Compute area of any polygon. • math.randomDiff » Generates a pseudo-random integer that's always different from the last random number generated. • math.rotVecToEulerAngle » Rotation Vector To Euler Angle • math.round » Rounds a number whereas the number of decimals to keep and the method may be set. • mathNumber » This function is a workaround for the client-side floating-point precision of 24-bits. • math.percentProgress » Returns a percentage progress from two specific values. • math.average » This function returns the simple arithmetic mean of multiple numbers. • math.absin » This function returns a formula representing the just positive half of a sine wave. ### Ped functions • getAlivePlayersInTeam » This function returns a table of the alive players in a team. • getGuestPlayers » This function gets a players not login or players Guest . • getOnlineAdmins » This function returns a table of all logged-in administrators. • getPedEyesPosition » This function allows you to get peds eyes position. • getPedGender » This function allows you to get peds their gender. • getPedMaxHealth » This function returns a pedestrians's maximum health by converting it from their maximum health stat. • getPedMaxOxygenLevel » This function returns a ped's maximum oxygen level by converting it from their maximum underwater stamina stat. • getPedWeaponSkill » This function returns a ped's corresponding weapon skill level name. • getPedHitBone » This function gets the approximate number of the bone where the ped is hit. • getPlayerFromNamePart » This function returns a player from partial name. • getPlayerFromSerial » This function returns a player from their serial. • getPlayersByData » This function returns a table of players that have the specified data name. • getPlayersInPhotograph » This function returns a table of all players in photograph. • getPlayersInVehicles » This function returns a table of the players insides vehicles from a specified dimension. • getPlayerNameFromID » This function will get the player name from the ID element data. • isPedAiming» This function checks if a pedestrian is aiming their weapon. • isPedAimingNearPed » This is similar to isPedAiming but uses a colshape to be more precise. • isPedDiving » This feature checks that pedestrian is diving in the water. • isPedDrivingVehicle » This function checks if a specified pedestrian is driving a vehicle. • isPedNearbyWall » This function checks if player/ped is nearby a objects like buildings or walls. • isPlayerInTeam » This function checks if a player is in a specified team. • setPedAttack » This function will make a ped attack a specified target. • setPedFollow » This function will make a ped follow a specified target. ### Browser functions • playVideo » This function plays a video on the screen. ### XML functions • getXMLNodes » This function returns all children of a XML node. ### Utility • animate » This function allows you to use interpolateBetween without render event and easily used. • callClientFunction » This function allows you to call any client-side function from the server's side. • callServerFunction » This function allows you to call any server-side function from the client's side. • check » This function checks if its arguments are of the right type and calls the error-function if one is not. • checkPassiveTimer » This function allows you to use passive timers in your conditions. For example you want to prevent players repeatedly using a command. • coroutine.resume » This function applies a fix for hidden coroutine error messages. • compact » This function create table containing variables and their values. • getBanBySerial » This function returns the ban if the serial is banned. • getBanFromName » This functions returns the ban of the given playername. • getCurrentFPS » This function returns the frames per second at which GTA: SA is running. • getSkinNameFromID » This function returns the name of the skin from the given id. • IfElse » This function returns one of two values based on a boolean expression. • isLastExecuteInTimer » This function check if the execute is the last execute in the timer. • isMouseInCircle » This function checks if a cursor position is in circular area or not. • isMouseInPosition » This function allows you to check whether the mouse cursor/pointer is within a rectangular position. • iterElements » This function returns a time-saving iterator for your for-loops. • PlotTrajectoryAtTime » Calculate projectile/water trajectory. • preprocessor » This function allow you to use gcc macros. • vector3:compare » This method checks whether two vectors match, with optional precision. • svgCreateRoundedRectangle » This function creates a rectangle with rounded edges. • debounce » This function is removing unwanted input noise. • listAllFiles » This function lists all files and subdirectories within a given directory and its subdirectories.

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https://www.convert-measurement-units.com/convert+Rood+to+Circular+thou.php

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 Convert Rood to Circular thou (Area) ## Rood into Circular thou numbers in scientific notation https://www.convert-measurement-units.com/convert+Rood+to+Circular+thou.php ## How many Circular thou make 1 Rood? 1 Rood = 1 996 643 242 107.1 Circular thou - Measurement calculator that can be used to convert Rood to Circular thou, among others. # Convert Rood to Circular thou: 1. Choose the right category from the selection list, in this case 'Area'. 2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), brackets and π (pi) are all permitted at this point. 3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Rood'. 4. Finally choose the unit you want the value to be converted to, in this case 'Circular thou'. 5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so. With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '383 Rood'. In so doing, either the full name of the unit or its abbreviation can be used. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Area'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '60 Rood to Circular thou' or '88 Rood into Circular thou' or '4 Rood -> Circular thou' or '69 Rood = Circular thou'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second. Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(77 * 5) Rood'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '383 Rood + 1149 Circular thou' or '54mm x 32cm x 97dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question. If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 2.559 999 976 704 ×1023. For this form of presentation, the number will be segmented into an exponent, here 23, and the actual number, here 2.559 999 976 704. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 2.559 999 976 704 E+23. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 255 999 997 670 400 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.

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Posted: November 28th, 2015 # INVESTMENT 1626 ```Question Suppose you are the money manager of a \$4.72 million investment fund. The fund consists of 4 stocks with the following investments and betas: Stock Investment Beta A \$ 360,000 1.50 B 480,000 - 0.50 C 1,180,000 1.25 D 2,700,000 0.75 If the market's required rate of return is 10% and the risk-free rate is 3%, what is the fund's required rate of return? Round your answer to two decimal places. %``` ### Expert paper writers are just a few clicks away Place an order in 3 easy steps. Takes less than 5 mins. ## Calculate the price of your order You will get a personal manager and a discount. We'll send you the first draft for approval by at Total price: \$0.00

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# Video: Assessing the Importance of Temperature Units on Gas Equations In which of the following gas laws is it valid to state temperature in degrees Celsius? [A] Charles’ law [B] Boyle’s law [C] The combined gas law [D] Guy-Lussac’s law [E] The ideal gas law 04:34 ### Video Transcript In which of the following gas laws is it valid to state temperature in degrees Celsius? A) Charles’ law, B) Boyle’s law, C) The combined gas law, D) Guy-Lussac’s law, or E) The ideal gas law. Before we try to figure out which gas law it would be valid to state the temperature in Celsius, let’s try to figure out why it wouldn’t be valid to state the temperature in Celsius first. Temperature is ultimately a measure of the average kinetic energy of the particles that are in our system. And the average kinetic energy tells us how quickly the particles in our system are moving around. If our system is a gas that has a high kinetic energy, the gas particles will be moving very quickly. But if our system is a gas with a low kinetic energy, the particles will be moving more slowly. There are a number of different temperature scales that we use to measure temperature. Celsius and Fahrenheit are relative temperature scales. Celsius and Fahrenheit are relative scales because we’ve defined them relative to things that we can observe. Celsius, for instance, is defined so that zero is when water freezes, and 100 is when water boils. Kelvin, on the other hand, is not a relative temperature scale. It’s an absolute temperature scale. The kelvin temperature scale is defined so that at zero kelvin, particles have zero kinetic energy. Zero kelvin is sometimes called absolute zero because it’s impossible for things to be colder than that because they can’t have less than zero kinetic energy. When gas laws refer to the temperature, what it’s really trying to get at is this relationship between the temperature and the average kinetic energy of the gas particles. Which tells us how quickly the gas particles are moving. Since the kelvin temperature scale was defined in reference to the average kinetic energy, all gas laws that refer to the temperature will use the kelvin temperature scale. So, it would only be valid to state the temperature in degrees Celsius for a gas law if the gas law wasn’t referencing the relationship between the temperature and another variable. This is because, as we’ve said, if the gas law is about the relationship between temperature and another variable, what we’re really talking about is the average kinetic energy of the gas particles. So, we need to talk about that temperature in degrees kelvin. So with this in mind, let’s look through the gas laws in the answer choices to figure out which one doesn’t include a relationship between temperature and another variable. Charles’ law tells us that for a sample of gas at constant pressure, the volume of the gas is proportional to its temperature. This temperature would, of course, be the absolute temperature. So, answer choice A is not correct. We could not use degrees Celsius in Charles’ law. We would need to use degrees kelvin. Boyle’s law tells us that for a sample of gas at a constant temperature, the pressure will be inversely proportional to the volume. In Boyle’s law, the temperature isn’t being related to another variable. It’s a constant. So, it would be valid to state the temperature of Boyle’s law in degrees Celsius. So, this is the correct answer choice. But let’s look through the other ones just so we understand them. Guy-Lussac’s law tells us that for a sample of an ideal gas at a constant volume, the pressure of the gas is proportional to its absolute temperature. Again, this temperature would have to be in kelvin since that’s an absolute temperature scale and Celsius is not. The combined gas law combines Charles’ law, Boyle’s law, and Guy-Lussac’s law into one law that says the pressure times the volume divided by the temperature is equal to a constant. Again, the temperature scale in the combined gas law is the absolute temperature scale not a relative temperature scale. Finally, the ideal gas law says that the pressure times the volume is equal to the amount of the gas in moles times the gas constant times the temperature. Like the other gas laws, this temperature must be an absolute temperature. Boyle’s law again is the only gas law where the temperature is held constant. So, it’s the only one where it would be valid to state the temperature in degrees Celsius.

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## Building a Computer 1: Numerals It’s been a long time since I’ve done much math with my kids or blogged about math. We’ve been busy with other things. But recently my kids have been asking me about how computers work. I like to give in-depth answers to such questions, so we set out on a quest to understand how they work. I saw the following video of a marble binary adder some time ago (see also woodgears.ca). My short-term goal is to get my kids to understand the ins and outs of how it works and perhaps build one ourselves. The long-term goal is to understand and build as many components of an actual computer as we can and understand how they fit together to make a real computer. What would be really cool would be to see how much of this we can do with marbles! Today I mostly wanted to introduce binary representations of numbers and how binary arithmetic works. I started by asking my kids about numeral systems. I made sure that they understood that though, for instance, the ancient Romans used different numerals than us, they had the same numbers as we do. Next we play-acted a scenario that I dictated to them. “You have crash-landed on an alien planet. You need to communicate home, but your microphone was broken in the crash. All that you can use to communicate is a numeric key pad. What’s worse, the only button that works on it is 1. Also, you can only hit the button up to three times before transmitting.” As I asked them for numbers they quickly invented “unary”: ```One 1 Two 11 Three 111 ``` I asked them for four, but of course they couldn’t do it without being able to hit the button more than three times. “After cryo-sleeping through the night you awake and set to work trying to repair your keypad. Finally, you get the 0 to work. You can still only hit three buttons before transmitting. Now what numbers can you make?” Again they started with unary, but I insisted that they make more than one, two and three. Here is what they came up with: ```Zero   101 One 000 Two 001 Three 011 Four 110 Five 111 Six 010 Seven 100 ``` As far as I can tell they chose the representation of each number randomly! “After another night’s cryo-sleep you manage to get the keypad to allow ten key punches before a transmit. But now I need you to be able to send me numbers up to one thousand.” This had them stumped for a while. I decided to help. I suggested that they needed some sort of pattern, since making random assignments would be confusing and take a long time. they were still having a hard time, so I suggested we look at addition. I had them explain the algorithm they use to add multi-digit numbers together. Then we started a chart and I convinced them to try to use addition to continue it. Here was our beginning: ```Zero 0000000000 One 0000000001 ``` We agreed that these seemed like reasonable choices. They pointed out that we could have represented one as 1000000000 with about as much justification as our choice above, but we agreed that 0000000001 fit with our normal conventions better. I suggested we try adding one to itself to get two: ``` 1 0000000001 +0000000001 ---------- 000000001? ``` They decided that if we were going to have a process for adding similar to what they were used to in base-10 then there should be a 1 carried as shown above. The only question was what to make the ? be. There are only two choices. After experimenting with both we found that if two is represented by 11 this leads to unary again (which we knew was unacceptable for the required task). On the other hand two being represented by 10 leads to something nice. Once they had played with it for some time I told them that this numeral system had been invented by others already and that it is called binary. Finally we were ready to watch the above video together. With the background of binary representations and binary addition, they were able to understand quite a bit of how the machine works. Next they need to understand each of the logic gates at work in the machine and how they fit together to actually make an adder, but all of that is for another day. I am very much an outsider when it comes to computers. I’m just a math guy. I’m planning on looking at the book The Elements of Computing Systems to learn more about building a computer from the ground up. This is going to be a voyage of discovery for both me and my kids. Any suggestions from experts are welcome! This entry was posted in computers. Bookmark the permalink. ### 15 Responses to Building a Computer 1: Numerals 1. There’s a kids’ book they might like called How to Count Like a Martian. • Vincent says: Minecraft is a cheap fun creative/exploring game. Built into it is basic wiring in which you can build your logic gates. http://www.minecraftwiki.net/wiki/Redstone_circuits You can start real simple and can build up to whatever you have the time and patience to do. One guy even built an actual computer out of it. 2. Jack says: Fascinating, but how do you manage to get your kids to sit down and think through something like this? Figuring out what the “?” in binary is, especially, sounds difficult for children to be interested in. • toomai says: Yeah, that (getting them to sit down and think about it) is the trick. Actually by the time we got to working out the “?” it was basically just my oldest who was still working on it. It was he (the oldest) who had gotten interested in computers in the first place and so a lot of what we have done has been fueled by that interest. Unfortunately I don’t have any magic ways of sparking the interest in the first place other than just throw a lot of things out there and see what they latch onto. 3. Check out “Code: The hidden language of computer hardware and software “by Charles Petzold for a great book about the barebones and roots of computers. • toomai says: Thanks! My local library doesn’t have it, but I requested it on inter-library loan. 4. Dang Nguyen says: This was a great article! I think I’ll try it with my kids. • toomai says: Thank you. Let me know how it goes. 5. Peter says: Have a look at this 16-bit ALU built in a gaming environment called Minecraft. Fabulous creative game. 6. BrownishMonster says: Maybe it’s easier for children to grasp arithmetic binary than it is for us since they’re not as used to base-10 as we are. 7. Bill says: Your kids’ primal representation is fascinating, I can just about follow the logic of it (but it’s a bit guessworky, and depends on arithmetical concepts of negative, less, more, and already-used). I wonder if other ‘unschooled’ kids, given same apparatus & conditions, would come up with same sequence. If I regress in mind a bit(:-), I would get a different sequence. (one=100,110,111,110,five=001,six=000,001,010,011,000) but then if I was told not to do unary (Chinese?), it introduces repression. So, for number one, I’d wannt 100, but if I mustn’t use 1 for 1, I have to say 000, which hasn’t got any 1s in it. Two is one more than that, 001 (as well as mustn’t be unary 110). Likewise, three can’t be 111, so it’s 001 with another 1 on top of the existing 1, that’s 011; etc. (cop-out, my brain hurts). Nice game for junior cryptographers. • toomai says:

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https://www.gradesaver.com/textbooks/math/algebra/elementary-linear-algebra-7th-edition/chapter-4-vector-spaces-4-6-rank-of-a-matrix-and-systems-of-linear-equations-4-6-exercises-page-200/39

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## Elementary Linear Algebra 7th Edition (a) A basis for the solution space is \left[\begin{aligned}-\frac{1}{2}\\-\frac{3}{2} \\1 \end{aligned}\right]. (b) The dimension of the solution space is $1$. The coefficient matrix is given by $$\left[ \begin {array}{ccc} -1&1&1\\ 3&-1&0 \\ 2&-4&-5\end {array} \right] .$$ The reduced row echelon form is $$\left[ \begin {array}{ccc} 1&0&\frac{1}{2}\\ 0&1&\frac{3}{2} \\ 0&0&0\end {array} \right] .$$ The corresponding system is \begin{aligned} x+\frac{1}{2}z &=0\\ y+\frac{3}{2}z &=0\\ \end{aligned}. The solution of the above system is $x=-\frac{1}{2}t$,$y=-\frac{3}{2}t$, $z=t$. This means that the solution space of $Ax = 0$ consists of the vectors on the following form x= \left[\begin{aligned} x\\ y\\z \end{aligned}\right]= \left[\begin{aligned}-\frac{1}{2}t\\-\frac{3}{2}t \\t \end{aligned}\right]=t \left[\begin{aligned}-\frac{1}{2}\\-\frac{3}{2} \\1 \end{aligned}\right] . (a) A basis for the solution space is \left[\begin{aligned}-\frac{1}{2}\\-\frac{3}{2} \\1 \end{aligned}\right]. (b) The dimension of the solution space is $1$.

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<    1      2      3      4    > ### Something or nothing The difference between something and nothing is not always clear. ##### Explanation As we deal here with mathematics, it sounds reasonable to call something 1 and nothing 0. That is quite logical. And moreover it becomes abstract. You would think that it should be clear whether the result of a calculation gives 0 or 1. But that is often not so simple. You can also say: something is 0 or 1. If you have neither of these, there is nothing. This you need to write numbers in groups, if they contain only zeros and ones, as with 00   01   10   11 For this you can use spaces (so nothing).

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## Excel formula for every other row color Formatting makes our documents look appealing to our eyes. The power of conditional formatting is far-reaching. Excel has not been left behind since we can use a specific formula to format our worksheet to our own liking. In this case, the formula should apply a given colour to alternate rows. This will enable an easy distinction between the different rows available. Procedure: 1. Launch Excel workbook and navigate to your worksheet. 2. Select the range of data you wish to apply color. 3. After highlighting your data set, navigate to the Home tab. 4. Under the styles group, click on the conditional formatting icon/button located on the ribbon. 5. Select the New Rule option. A new rule Dialog will be displayed. 6. Then click on "Use a formula to determine which cells to format" under the select a Rule type. 7. Enter the following formula inside the rule description textbox. =MOD (ROW (),2) 8. Choose a formatting style using the format button and preview. In our case, we choose a blue color. 9. Click Ok to confirm and affect the formatting of the rows. 10. Click on save to save changes. As you can see alternate rows will be colored in the color we choose in the format selection. Description of the formula Generally, the MOD function returns a modulus after the division of a number (Dividend) with another(divisor). A modulus is usually the remainder that results after division. The ROW () function, on the other hand, is responsible for posting the Row number to the MOD () function. The 2 which is used as an input of the MOD () function will be used as the divisor used to divide the returned row number. Therefore, Excel will only apply formatting on the data when the formula is true I.e. if and only if a value of one is returned by MOD () after receiving an odd row number from ROW () function. E.g. for row number nine: 9/2=4 rem 1, 9 is the dividend,2 is the divisor, 4 is the quotient and 1 is the modulus output. Row nine will thus be formatted.

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• / • # Design 3 on 15 Oct 2021 • 1 • 2 • 3 • 0 • 271 x(1)=rand; y(1)=rand; z(1)=rand; for j=1:40000 B=.8*[(-1)^ceil(4*rand()),(-1)^ceil(4*rand()),(-1)^ceil(4*rand())]; a=sqrt((x(j)-B(1))^2+(y(j)-B(2))^2+(z(j)-B(3))^2)^2; x(j+1) = B(1) + (x(j)-B(1))/a; y(j+1) = B(2) + (y(j)-B(2))/a; z(j+1) = B(3) + (z(j)-B(3))/a; end plot3(x,y,z,'.') axis off

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## Elementary Linear Algebra 7th Edition Published by Cengage Learning # Chapter 4 - Vector Spaces - Review Exercises - Page 221: 1 #### Answer (a) \begin{align*} (0,2,5). \end{align*} (b) \begin{align*} (2,0,4). \end{align*} (c) \begin{align*} (-2,2,14). \end{align*} (d) \begin{align*} (-5,6,5). \end{align*} #### Work Step by Step Consider the vectors ${u}=(-1,2,3), \quad {v}=(1,0,2)$ (a) \begin{align*} u+v &=(-1,2,3)+(1,0,2)\\ &=(0,2,5). \end{align*} (b) \begin{align*} v &=2(1,0,2)\\ &=(2,0,4). \end{align*} (c) \begin{align*} u-v &=(-1,2,3)-(1,0,2)\\ &=(-2,2,14). \end{align*} (d) \begin{align*} 3u-2v &=3(-1,2,3)-2(1,0,2)\\ &=(-3,6,9)-(2,0,4)\\ &=(-5,6,5). \end{align*} After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.

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/C++ # std::erf, std::erff, std::erfl Defined in header `<cmath>` ```float erf ( float arg ); float erff( float arg );``` (1) (since C++11) `double erf ( double arg );` (2) (since C++11) ```long double erf ( long double arg ); long double erfl( long double arg );``` (3) (since C++11) `double erf ( IntegralType arg );` (4) (since C++11) 1-3) Computes the error function of `arg`. 4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to 2) (the argument is cast to `double`). ### Parameters arg - value of a floating-point or Integral type ### Return value If no errors occur, value of the error function of `arg`, that is 2 √π arg 0 e-t2 dt, is returned. If a range error occurs due to underflow, the correct result (after rounding), that is. 2*arg √π is returned ### Error handling Errors are reported as specified in `math_errhandling`. If the implementation supports IEEE floating-point arithmetic (IEC 60559), • If the argument is ±0, ±0 is returned • If the argument is ±∞, ±1 is returned • If the argument is NaN, NaN is returned ### Notes Underflow is guaranteed if `|arg| < DBL_MIN*(sqrt(π)/2)` erf( x σ√2 ) is the probability that a measurement whose errors are subject to a normal distribution with standard deviation σ is less than x away from the mean value. ### Example The following example calculates the probability that a normal variate is on the interval (x1, x2). ```#include <iostream> #include <cmath> #include <iomanip> double phi(double x1, double x2) { return (std::erf(x2/std::sqrt(2)) - std::erf(x1/std::sqrt(2)))/2; } int main() { std::cout << "normal variate probabilities:\n" << std::fixed << std::setprecision(2); for(int n=-4; n<4; ++n) std::cout << "[" << std::setw(2) << n << ":" << std::setw(2) << n+1 << "]: " << std::setw(5) << 100*phi(n, n+1) << "%\n"; std::cout << "special values:\n" << "erf(-0) = " << std::erf(-0.0) << '\n' << "erf(Inf) = " << std::erf(INFINITY) << '\n'; }``` Output: ```normal variate probabilities: [-4:-3]: 0.13% [-3:-2]: 2.14% [-2:-1]: 13.59% [-1: 0]: 34.13% [ 0: 1]: 34.13% [ 1: 2]: 13.59% [ 2: 3]: 2.14% [ 3: 4]: 0.13% special values: erf(-0) = -0.00 erf(Inf) = 1.00```

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