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Maths The Pythagoras Puzzle Author Tommie Zetterlund; Bromangymnasiet Hudiksvall Target Group 13-17 years old students Brief Description The Pythagorean theorem has been given numerous proofs – possibly the most for any mathematical theorem. This puzzle is a way for the students to verify the theorem themselves. Share # The Pythagoras Puzzle In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation": where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. This puzzle is a way for the students to verify the Pythagorean theorem themselves. 1. Have the student work in groups. 2. Distribute the copying material and scissors to the groups. 3. Instruct them to put the four triangles in the square to somehow form the area a2. 4. After they have managed to form the area c2 they should rearrange the triangles to try to form the areas a2 and b2. 5. Instruct them to discuss if this verifies the theorem. And encourage them to draw their own right-angled triangles to try to verify if this is true for all right angled triangles. Teachers guide ```Note 1: Before the introduction to this puzzle it is recommended to talk a little about the history of the Pythagorean theorem and Pythagoras himself. Note 2: The students can use a spreadsheet application to further examine if the theorem is true for different values of a and b.``` ## Related files The Pythagoras Puzzle_copying material_ENG 421 KB The Pythagoras Puzzle_teachers manual_ENG 424 KB If you have tried out a lesson, please fill the next form to help us to improve it:

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# Year 6 Operations Maths Program Product Type: Maths Price: Sale price\$20.00 ## Description Introducing our set of Operations Maths Programs for year 6 students where they solve problems involving the use of brackets and order operations using efficient mental and written strategies. This digital downloadable resource aligns with the Australian Curriculum and consists of three engaging modules, spanning across 15 lessons, designed to be taught over the course of a year. Crafted with interactive and cooperative learning strategies, these programs aim to captivate and educate young minds effectively. Aligned with both version 9 and version 8.4 of the Australian Curriculum, our version 9 programs come equipped with a grading guide (A-E) specifically designed to correspond with the end-of-module assessments. Additionally, editable PowerPoints accompany each lesson, providing dynamic visual aids to support your instructional journey. Customisable learning intentions and success criteria further empower educators to ensure clear and focused lesson objectives. Each lesson planner is thoughtfully structured with five to six concise steps, fostering simplicity and clarity in your teaching approach. While some lessons include worksheets, not every session relies on them. Instead, our programs encourage educators to leverage their expertise, ensuring explicit instruction and maximizing teaching opportunities. ___________________________________________________________________________________________ Included are: • 3 x Maths Programs (12 pages) • Resources (50+ pages) • Editable Grading Guide (Version 9 only) • Editable PowerPoints (Version 9 only) • Editable planners (Version 9 only) • Editable Learning Intentions and Success Criteria (Version 9 only) ___________________________________________________________________________________________ Want a whole semesters of maths programs? Check out the bundle here. ___________________________________________________________________________________________ Australian Curriculum Outcomes Covered: Version 9: Number: Recognise, represent and order numbers to at least 1000 using physical and virtual materials, numerals and number lines (AC9M2N01) Partition, rearrange, regroup and rename two-and three-digit numbers using standard and non-standard groupings; recognise the role of a zero digit in place value notation (AC9M2N02) Version 8.4: • Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123) • Explore the use of brackets and order of operations to write number sentences (ACMNA134) ___________________________________________________________________________________________ ___________________________________________________________________________________________ → Other than where stated, this product is not editable → This file is zipped. Once downloaded, right-click the file and select ‘extract all’ to open. ## 🚚 Shipping • Digital Products: Delivered via Email 📧 For digital products, we'll swiftly send them straight to your inbox. Keep the email safe as it grants access to any future updates related to your purchase. Want easier access? Create an account during checkout to manage all your digital acquisitions conveniently. • Physical Goods (Hard Goods): Shipped Weekly 🚚 We usually dispatch hard goods every Saturday morning. Please note, shipping days may occasionally vary. Unless specified as a 'pre-order,' all items are shipped within 7 days from the date of purchase. • Shipping Costs: Shipping is capped at a fantastic rate of \$12.00*, making it easy on your wallet! This cost is added during checkout for your convenience. (*Exclusions may apply. Please refer to product descriptions for more details.) Questions or Assistance Needed? Our team is here to assist you! Should you have any queries or need further assistance regarding shipping, feel free to reach out. We're dedicated to ensuring your shopping experience is smooth and hassle-free.

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# Try this Algebra Level 4 If $$a,b,c$$ are distinct and $$p(x)$$ is a polynomial in $$x$$ which leaves a remainder $$a,b,c$$ on division by $$(x-a),(x-b),(x-c)$$respectively.Find the remainder obtained on division of $$p(x)$$ by $$(x-a)(x-b)(x-c)$$ ×

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## Conversion formula The conversion factor from cubic centimeters to liters is 0.001, which means that 1 cubic centimeter is equal to 0.001 liters: 1 cm3 = 0.001 L To convert 51 cubic centimeters into liters we have to multiply 51 by the conversion factor in order to get the volume amount from cubic centimeters to liters. We can also form a simple proportion to calculate the result: 1 cm3 → 0.001 L 51 cm3 → V(L) Solve the above proportion to obtain the volume V in liters: V(L) = 51 cm3 × 0.001 L V(L) = 0.051 L The final result is: 51 cm3 → 0.051 L We conclude that 51 cubic centimeters is equivalent to 0.051 liters: 51 cubic centimeters = 0.051 liters ## Alternative conversion We can also convert by utilizing the inverse value of the conversion factor. In this case 1 liter is equal to 19.607843137255 × 51 cubic centimeters. Another way is saying that 51 cubic centimeters is equal to 1 ÷ 19.607843137255 liters. ## Approximate result For practical purposes we can round our final result to an approximate numerical value. We can say that fifty-one cubic centimeters is approximately zero point zero five one liters: 51 cm3 ≅ 0.051 L An alternative is also that one liter is approximately nineteen point six zero eight times fifty-one cubic centimeters. ## Conversion table ### cubic centimeters to liters chart For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to liters cubic centimeters (cm3) liters (L) 52 cubic centimeters 0.052 liters 53 cubic centimeters 0.053 liters 54 cubic centimeters 0.054 liters 55 cubic centimeters 0.055 liters 56 cubic centimeters 0.056 liters 57 cubic centimeters 0.057 liters 58 cubic centimeters 0.058 liters 59 cubic centimeters 0.059 liters 60 cubic centimeters 0.06 liters 61 cubic centimeters 0.061 liters

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# ball mill operating rpm calculation #### ball mill operating rpm calculation Ball Mill Operating Speed - Mechanical Operations Solved Problems .Sep 11, 2014 . In a ball mill of diameter 2000 mm, 100 mm dia steel balls are being used for grinding. Presently, for the material being ground, the mill is run at 15 rpm. At what speed will the mill have to be run if the 100 mm balls are replaced by 50 mm balls, all the other conditions remaining the same? Calculations:.Ball Mill Critical Speed - 911 MetallurgistMar 17, 2017 . Ball mills have been successfully run at speeds between 60 and 90 percent of critical speed, but most mills operate at speeds between 65 and 79 percent of critical speed. Rod mills speed should be limited to . You'll find a good on-line tool for ball mill critical speed calculation based on the above formula. ball mill optimal rpm calculation - tfgcritical speed ball mill calculation - impfarrgarten. Critical Speed Of Ball Mill Calculation India Figure 83 A model of mill power as affected by mill speed, - The speed of rotation (rpm) The optimal operating. Get More Info. image.How can I determine the best RPM for Dry Ball Milling machine.Get expert answers to your questions in Ball Milling, Ceramics and Ceramic Processing and more on ResearchGate, the professional network for scientists. . Note that the critical speed of the mill is calculated as follows: Critical speed (in . Then operate the machine at speed=2/3 of the Critical speed of the mill. Can you. How to calculate the critical speed of ball mill?_Shanghai Kefan Mining . If the mill speed increase, ball started behind the fall point will improve. To the mill speed increases to a certain value C, the centrifugal force is greater than the weight of the steel ball, steel ball mill vertex Z no longer rise to fall, occurrence of centrifugal operation. Thus, the critical conditions of centrifugal operation is: ball. SAGMILLING.COM .:. Mill Critical Speed Determination Mill Critical Speed Determination. The "Critical Speed" for a grinding mill is defined as the rotational speed where centrifugal forces equal gravitational forces at the mill shell's inside surface. This is the rotational speed where balls will not fall away from the mill's shell. Mill Speed - Critical Speed - Paul O. Abbe No matter how large or small a mill, ball mill, ceramic lined mill, pebble mill, jar mill or laboratory jar rolling mill, its rotational speed is important to proper and efficient mill operation. Too low a speed and little energy is imparted on the product. Too fast and inefficient media movement (known as cataracting) will generate high. ball mill operating rpm calculation,What it is the optimun speed for a ball mill ? - Pyrotechnics . Oct 19, 2006 . posted in Pyrotechnics: I have done a ball mill , recenly finished , but the motor has too rpms , is too fast for use in a ball mill (the pvc cylinder that i use , left of the shafts). With the motor i will . Calculation of optimum speed depends on knowing the media diameter and jar I.D. also. For instance, if your jar. ball mill operating rpm calculation,ball mill optimal rpm calculation - tfg critical speed ball mill calculation - impfarrgarten. Critical Speed Of Ball Mill Calculation India Figure 83 A model of mill power as affected by mill speed, - The speed of rotation (rpm) The optimal operating. Get More Info. image. Correlations for the Grindability of the Ball Mill As . - Semantic Scholar balls, time of grinding, particle density and speed of the ball mill (rpm) have been considered for the present work to determine . that point is called the "Critical. Speed” (nc).This phenomenon is called centrifuging. Ball mills usually operate at 65% to. 75% of the critical speed. The critical speed is calculated as under [5]. rR. What it is the optimun speed for a ball mill ? - Pyrotechnics . Oct 19, 2006 . posted in Pyrotechnics: I have done a ball mill , recenly finished , but the motor has too rpms , is too fast for use in a ball mill (the pvc cylinder that i use , left of the shafts). With the motor i will . Calculation of optimum speed depends on knowing the media diameter and jar I.D. also. For instance, if your jar. Ball Mill Critical Speed & Working Principle Jun 19, 2015 . s.911metallurgist/blog/ball-mill Learn about Ball Mill Critical Speed and its effect on inner charge movements. The effect of Ball Mill RPM spe. ball mill media load calculation formula Ball Mill Media Load Calculation Formula. ball mill media load calculation formula - vcaredia. ball mill media load calculation formula - elna.. grdg ball mill load calculation formula - Gold Ore Crusher prciples of ball mill grdg media operation . ball mill media load calculation formula – Grdg Mill Cha. Monster Tool, Speed. Ball Mills - Horizontal Ball Mills, Cylindrical Ball Mills We are supplier and manufacturer of Ball Mills, Horizontal Ball Mills, Cylindrical Ball Mills, Ashoka Machine Tools Corporation. . perimeter of mill (this point is called "Critical Speed"- ball mills normally operate at 65% to 75% of critical speed); With critical speed (rpm) denoted by formula nC = 42.29/vd (d is internal diameter. Ball mill - Wikipedia Ball mills rotate around a horizontal axis, partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles and stainless steel balls. An internal cascading effect reduces the material to a fine powder. Industrial ball mills can operate. Ball Nose Speed/Feed Calculators Ball Nose Mills for Finishing Cuts. Directions: Fill in the blocks shaded in blue with the appropriate information. The calculator will automatically provide the necessary speed and feed in the green fields. For assistance, call Dapra (800) 243-3344. Calculation of Media Charge - Jyoti Ceramic Industries Pvt. Ltd. Calculations for mill motor power, mill speed and media charge. Advantages . 2. For continuous type ball mill : M = 0.000503 x D2 x L. Example - To calculate grinding media charge for a steatite ceramic lined batch type mill size : Dia 6' x 6' Long (Dia 180 cm x 180 cm L) (mill openings are not considered). cheap adobe. The influence of mill speed and pulp density on the . - SAIMM The absorbed power is calculated by multiplying the torque on the driving shaft with the shaft speed (rad/sec) and hence excludes any losses associated with the gearbox and motor. The pulp weight in the mill is obtained by subtracting the mill weight during the test from the empty mill weight, including the ball charge, at the. Roughing & Finishing End Mills Speeds & Feeds Formulas . PROMAX Tools Solid Carbide End Mills Speeds & Feeds Formula Chart Provides Information On Running Roughing & Finishing End Mills. . For starting speed using a ½” 4 flute end mill cutting Titanium at a hardness of around 275 BHN, find SFM on above chart (SFM=160) and calculate RPM using formula as follows: ball mill operating rpm calculation,Analysis of ball movement for research of grinding . - Springer Link tance from x-axis and y-axis, respectively, and n the rotation speed per minute. The flow chart of a general calculation is shown in Fig. 5. The values of parameters of materials and the operation conditions of mill are input as shown in Table 1. In addition, a position condition of initial charge of grinding media balls and stirrers. ball mill operating rpm calculation,MILLING & ANALYSIS OF PARTICLES During 5, 10 and 20 minutes mill 150 grams of quartz sand with 480 rpm. Use as many balls, so that they are embedded in sand. 2. Determine the mass distribution of the milled and unmilled quartz sand with sieving analysis. 3. Plot a histogram with the relative residues versus the particle sizes. 4. Calculate the cumulative. Chatter Myths: Pieces of the Puzzle in Maximized Machining . This article will discuss the milling process of machining and rpm rather than the depth-of-cut, feedrates and other parameters. RPM. The one parameter that affects chatter conditions the most is rpm; it also happens to be the quickest and easiest parameter to obtain or change to maximize the machining operation. Actually. Understanding CNC Cutting Tool Speeds & Feeds - Destiny Tool Back in the days of belt driven milling machines and slide rules, the old school machinists had a simple method of calculating RPM. They used the formula CSx4/D. As we said above, Material Cutting Speed is not RPM. All materials have a documented cutting speed which today we refer to as SFM. Back in the old days they. Mill Throughput Capacity The Bond tests was developped for rod and ball milling when those technologies were widely used in theearly 1900's. Still in the 1980's, the determination of the mill size for both autogenous and semi-autogenous grinding was not possible and motivated John Starkey to develop a new grinding bench scale test allowing the. Impact Load Behavior between Different Charge and Lifter in . - MDPI Jul 31, 2017 . processing, and is affected by various factors including mill speed, mill filling, lifter height and media shape. . for the ball mills, while the power draw determined by DEM simulation show a good approximation . The grinding efficiency is affected by operating variables, such as the charge, lifter.

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# beginning algebra posted by . The length of a rectangle is 1ft more than twice its width, and the area of the rectangle is 28ft^2 . Find the dimensions of the rectangle. l= w= • beginning algebra - w = width L = 2w + 1 A = Lw 28 = w(2w + 1) 28 = 2w^2 + w Solve for w by completing the square. • beginning algebra - 4 7 ## Respond to this Question First Name School Subject Your Answer ## Similar Questions 1. ### math a rectangle is twice as long as it is wide. if both of its dimensions are increased by 4m, its area is increased by 88m^2 find the dimensions of the original rectangle Original rectangle = w for width and 2w for length. Area = w x … 2. ### math the length of a rectangle is 5 ft more than twice the width. a)if x represents the width of the rctangle, represent the perimeter of the rectangle in terms of x. b) if the perimeter if the rectangle is 2 ft more than eight times the … 3. ### Algebra the length of a rectangle is 5m more than twice its width. and the area of the rectangle is 88m^2. find the dimensions of the rectangle. 4. ### math the length of a rectangle is 3 yds more than twice it width and the area of the rectangle is 77yds find the dimensions of the rectangle the length and width 5. ### Math How do I find the dimensions of a rectangle if the length of the rectangle is 1 yard more than twice its width, and the area of the rectangle is 66 yards^2 . 6. ### Algebra The length of a rectangle is 1ft. more than twice the width, and the area of the rectangle is 66 ft^2. Find the dimensions of the rectangle. length: width: 7. ### Algebra The length of a rectangle is 1 foot less than twice the width, and the area of the rectangle is 21 ft.². Find the dimensions of the rectangle 8. ### Algebra The area of a rectangle is 21 ft^2,and the length of the rectangle is 1 ft less than twice the width. Find the dimensions of the rectangle. 9. ### algebra The length of a rectangle is 3 more than twice the width. The area of the rectangle is 119 square inches. What are the dimensions of the rectangle? 10. ### algebra The length of a rectangle is 5cm more than twice the width. The perimeter of the rectangle is 66 cm. Find the dimensions of the rectangle. More Similar Questions

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TOPICS # Bessel Function of the First Kind The Bessel functions of the first kind are defined as the solutions to the Bessel differential equation (1) which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows for , 1, 2, ..., 5. The notation was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function (2) is the same as the modern one (Watson 1966, p. 14). Bessel used the notation to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279). The Bessel function can also be defined by the contour integral (3) where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416). The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ[nu, z]. To solve the differential equation, apply Frobenius method using a series solution of the form (4) Plugging into (1) yields (5) (6) The indicial equation, obtained by setting , is (7) Since is defined as the first nonzero term, , so . Now, if , (8) (9) (10) (11) First, look at the special case , then (11) becomes (12) so (13) Now let , where , 2, .... (14) (15) (16) which, using the identity , gives (17) Similarly, letting , (18) which, using the identity , gives (19) Plugging back into (◇) with gives (20) (21) (22) (23) (24) The Bessel functions of order are therefore defined as (25) (26) so the general solution for is (27) Now, consider a general . Equation (◇) requires (28) (29) for , 3, ..., so (30) (31) for , 3, .... Let , where , 2, ..., then (32) (33) where is the function of and obtained by iterating the recursion relationship down to . Now let , where , 2, ..., so (34) (35) (36) Plugging back into (◇), (37) (38) (39) (40) (41) Now define (42) where the factorials can be generalized to gamma functions for nonintegral . The above equation then becomes (43) Returning to equation (◇) and examining the case , (44) However, the sign of is arbitrary, so the solutions must be the same for and . We are therefore free to replace with , so (45) and we obtain the same solutions as before, but with replaced by . (46) We can relate and (when is an integer) by writing (47) Now let . Then (48) (49) But for , so the denominator is infinite and the terms on the left are zero. We therefore have (50) (51) Note that the Bessel differential equation is second-order, so there must be two linearly independent solutions. We have found both only for . For a general nonintegral order, the independent solutions are and . When is an integer, the general (real) solution is of the form (52) where is a Bessel function of the first kind, (a.k.a. ) is the Bessel function of the second kind (a.k.a. Neumann function or Weber function), and and are constants. Complex solutions are given by the Hankel functions (a.k.a. Bessel functions of the third kind). The Bessel functions are orthogonal in according to (53) where is the th zero of and is the Kronecker delta (Arfken 1985, p. 592). Except when is a negative integer, (54) where is the gamma function and is a Whittaker function. In terms of a confluent hypergeometric function of the first kind, the Bessel function is written (55) A derivative identity for expressing higher order Bessel functions in terms of is (56) where is a Chebyshev polynomial of the first kind. Asymptotic forms for the Bessel functions are (57) for and (58) for (correcting the condition of Abramowitz and Stegun 1972, p. 364). A derivative identity is (59) An integral identity is (60) Some sum identities are (61) (which follows from the generating function (◇) with ), (62) (Abramowitz and Stegun 1972, p. 363), (63) (Abramowitz and Stegun 1972, p. 361), (64) for (Abramowitz and Stegun 1972, p. 361), (65) (Abramowitz and Stegun 1972, p. 361), and the Jacobi-Anger expansion (66) which can also be written (67) The Bessel function addition theorem states (68) Various integrals can be expressed in terms of Bessel functions (69) which is Bessel's first integral, (70) (71) for , 2, ..., (72) for , 2, ..., (73) for . The Bessel functions are normalized so that (74) for positive integral (and real) . Integrals involving include (75) (76) Ratios of Bessel functions of the first kind have continued fraction (77) (Wall 1948, p. 349). The special case of gives as the series (78) (Abramowitz and Stegun 1972, p. 360), or the integral (79) Bessel Function of the Second Kind, Bessel Function Zeros, Debye's Asymptotic Representation, Dixon-Ferrar Formula, Hansen-Bessel Formula, Kapteyn Series, Kneser-Sommerfeld Formula, Mehler's Bessel Function Formula, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind, Nicholson's Formula, Poisson's Bessel Function Formula, Rayleigh Function, Schläfli's Formula, Schlömilch's Series, Sommerfeld's Formula, Sonine-Schafheitlin Formula, Watson's Formula, Watson-Nicholson Formula, Weber's Discontinuous Integrals, Weber's Formula, Weber-Sonine Formula, Weyrich's Formula Explore this topic in the MathWorld classroom ## Related Wolfram sites http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ ## Explore with Wolfram|Alpha More things to try: ## References Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions and ." §9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972.Arfken, G. "Bessel Functions of the First Kind, " and "Orthogonality." §11.1 and 11.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-591 and 591-596, 1985.Cajori, F. A History of Mathematical Notations, Vols. 1-2. New York: Dover, 1993.Hansen, P. A. "Ermittelung der absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung, I." Schriften der Sternwarte Seeberg. Gotha, 1843.Lehmer, D. H. "Arithmetical Periodicities of Bessel Functions." Ann. Math. 33, 143-150, 1932.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 619-622, 1953.Schlömilch, O. X. "Ueber die Bessel'schen Function." Z. für Math. u. Phys. 2, 137-165, 1857.Spanier, J. and Oldham, K. B. "The Bessel Coefficients and " and "The Bessel Function ." Chs. 52-53 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987.Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. ## Referenced on Wolfram|Alpha Bessel Function of the First Kind ## Cite this as: Weisstein, Eric W. "Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

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# Questioning Pi: 3.14159265359 vs 22/7 • Mr.maniac In summary, a conversation took place where a student's teacher claimed that 22/7 is the exact value of pi and 3.141 is an approximation. The student's classmates and other people on the forum disagreed and explained that both 22/7 and 3.141 are approximations of pi, with the actual value being 3.14159265358979323846264338327950288419716939937510. The conversation also discussed the importance of not blindly believing everything a teacher says and the possibility of the teacher creating an account on the forum to further discuss their beliefs. #### Mr.maniac here's my question:which is right:3.14159265359 blah blah blah -supported by http://www.quora.com/Why-is-PI-22-7 [Broken] also says that 22/7 is an approximation or 22/7 which is 3.14285714286(I don't know extended or not.) -supported by my maths teacher -supported by my fat ol' friend Last edited by a moderator: Mr.maniac said: 3.14159265359 blah blah blah -supported by http://www.quora.com/Why-is-PI-22-7 [Broken] also says that 22/7 is an approximation -supported by mathematics, which is the only thing that really counts. Mr.maniac said: 22/7 which is 3.14285714286(I don't know extended or not.) Which is not a very good approximation, as you can see compared to the value you wrote above. Last edited by a moderator: Mr.maniac PI isn't here but like these, 22 / 7 is just an approximation. Source: http://xkcd.com/1047/ DrewD If your teacher thinks that pi is exactly equal to 22/7, then I'm worried for your class. Google is correct. Pi is approximately equal to 22/7, but not exactly equal. Mr.maniac Here's https://www.math.hmc.edu/funfacts/ffiles/10004.5.shtml - 355 / 113. It's good to 6 decimal places but like 22 / 7, it isn't exactly equal to PI. Mr.maniac She says that 22/7 is the correct value of pi and 3.141 blah blah is just an approximation. Mr.maniac said: She says that 22/7 is the correct value of pi and 3.141 blah blah is just an approximation. Wow. I'm with Mentallic on this. Sorry for your class if that's what she is teaching. Mr.maniac Then shall I conclude that my maths teacher is wrong along eith my fatso friend. Alsi it wasn't taught a simple algaebric expression was given to break down which was 3.14r As I like pi strarted all about pi and my friend calculated 22/7 and then a debate then asking the teacher (asking for a fez). Mr.maniac said: Then shall I conclude that my maths teacher is wrong along eith my fatso friend. The problem is that she is teaching this to people. It would be great if you could get her to create an account and post her logic here. There would be plenty of people who would be happy to explain it to her. :) Here's an interesting take on this: ##\pi = \frac{22}{7} - \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx## Since your teacher is not here to defend him- or her-self, did he or she say that "pi is equal to 22/7" or "It is better to use 22/7 for pi than 3.14"? Let me quote my teacher : "22/7 is the exact value of pi and 3.141 is an approximation" (I'm I getting her wrong") Mr.maniac said: Let me quote my teacher : "22/7 is the exact value of pi and 3.141 is an approximation" (I'm I getting her wrong") She is right about the second part, but the first is also an approximation. Mr.maniac, I don't know you so please don't be offended by what I say/ask: Is that actually an exact quote or is that what you think you remember from the middle of an argument. I only ask this because I teach, and while I have definitely misspoken, I am more often misquoted. Is it possible that this is on a standardized test that explicitly told you to use the value 3.14 or 22/7 for ##\pi## in your calculations? If this is the case, you may have misunderstood her telling you to use that value. If there wasn't some strange context (eg. she preceded it by saying, "the next statement I make is false" ), she was wrong. Also, in these sorts of debates (math debates) wikipedia is usually pretty reliable. Mr.maniac said: Let me quote my teacher : "22/7 is the exact value of pi and 3.141 is an approximation" 22/7 ##\approx## 3.142857. Both 22/7 and 3.141 are approximations to the actual value of ##\pi##. According to wikipedia, the first 50 digits of ##\pi## are 3.14159265358979323846264338327950288419716939937510 (http://en.wikipedia.org/wiki/Pi). Clearly 3.141 is a better approximation than 22/7. Mr.maniac said: (I'm I getting her wrong") ? Are you trying to say "Am I getting her wrong?" (Yes it's am I getting her wrong) And DrewD they are the exact words of my teacher. And she is saying 3.141 blah blah is wrong Please see Mark44's post #16 above. 22/7 AND 3.1416 are both approximations. He has listed the correct value of Pi to 50 digits. If your teacher is saying anything otherwise, she is wrong. Mr.maniac said: (Yes it's am I getting her wrong) And DrewD they are the exact words of my teacher. Well that's unfortunate. I would nicely do what she wants in class but carefully check to be sure it is correct. If it is incorrect, ask her and if she doesn't realize the mistake, keep track of it and let an administrator know. Well, that's not actually what I would have done, but it is the most likely way to actually accomplish something. Your teacher is teaching you a very good lesson here. Don't believe everything your teacher says. jasonRF, TheDemx27, Mr.maniac and 1 other person Khashishi said: Your teacher is teaching you a very good lesson here. Don't believe everything your teacher says. Very much so. In year 7, I had a teacher that claimed that the orbits of the planets were perfect circles. When another student questioned him he said "That's just how they draw it". Mr.maniac, do you think it's possible if you can get your teacher to create an account here? Or as Drew said, talk to an administrator. The problem is that this teacher will continue to teach in the future, and it means all future students will be taught incorrectly. Khashishi said: Your teacher is teaching you a very good lesson here. Don't believe everything your teacher says. Well said Khashishi pwsnafu said: Very much so. In year 7, I had a teacher that claimed that the orbits of the planets were perfect circles. When another student questioned him he said "That's just how they draw it". Mr.maniac, do you think it's possible if you can get your teacher to create an account here? Or as Drew said, talk to an administrator. The problem is that this teacher will continue to teach in the future, and it means all future students will be taught incorrectly. Well not possible to make her make a account here but will try to convince her. (By the way how to close a thread cause will close this one after she is convinced) Mr.maniac said: Well not possible to make her make a account here but will try to convince her. (By the way how to close a thread cause will close this one after she is convinced) You cannot close threads, only moderators can do that. If you really feel the need to close a thread you can hit the "report" button and ask for it to be closed, but whether or not it IS closed is up to the moderators. Mr.maniac Mr.maniac said: Well not possible to make her make a account here but will try to convince her. Wikipedia lists a number of proofs that pi is irrational (i.e. pi isn't a fraction including 22/7). Niven's proof is probably the most convincing because it only uses high school level calculus. Edit: a simpler non-rigorous argument: if pi was equal to 22/7, then we wouldn't have world records for calculating the digits of pi (22/7 has a repeating decimal). Last edited: ## 1. What is the difference between 3.14159265359 and 22/7? The number 3.14159265359, known as pi, is an irrational number that represents the ratio of a circle's circumference to its diameter. It is typically rounded to 3.14 for practical calculations. On the other hand, 22/7 is a rational number that is often used as an approximation for pi. It is equal to 3.14285714286, which is slightly closer to the true value of pi than 3.14. ## 2. Why is pi important in mathematics and science? Pi is a fundamental constant in mathematics and science. It appears in various mathematical equations and is essential for calculating the circumference, area, and volume of circles and spheres. It is also used in many scientific fields, such as physics, engineering, and statistics. ## 3. How do we know that pi is an irrational number? We can prove that pi is an irrational number using mathematical proofs. One of the most famous proofs is by the Greek mathematician Euclid, who showed that the ratio of a circle's circumference to its diameter cannot be expressed as a fraction of two integers, making it an irrational number. ## 4. Are there any practical applications for knowing the digits of pi? While knowing the digits of pi may seem like a fun challenge, it does have practical applications. For example, NASA uses the digits of pi to calculate the trajectories of spacecraft, and the banking industry uses it for data encryption. It can also be used to test the accuracy of computer algorithms and to study patterns in random numbers. ## 5. Is there an end to the digits of pi? No, there is no end to the digits of pi. It is an infinite, non-repeating decimal, meaning that the digits after the decimal point go on forever without following a pattern. As of 2021, pi has been calculated to over 31 trillion digits, and there is still no end in sight.

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• Views : 30k+ • Sol Viewed : 10k+ # Aptitude Questions : Logic Maths Cube Interview Questions Difficulty Popularity Aamir has a cube with 4 inch side. He paints it red, green and black on the opposite faces. After that, he cuts it into one inch cubes. Answer the following questions (1-4) relating them with the once inch cubes. Q1. How many cubes are left with only one face painted? a) 8 b) 16 c) 24 d) 32 Q2. How many cubes have two of their faces painted? a) 8 b) 16 c) 24 d) 32 Q3. How many cubes have four faces painted? a) 0 b) 4 c) 8 d) 12 Q4. How many cubes are left with none of their faces painted? a) 0 b) 4 c) 8 d) 12 Discussion Suggestions • Views : 50k+ • Sol Viewed : 20k+ # Aptitude Questions : Logic Birthday Puzzle Difficulty Popularity For my anniversary, I decided to surprise my wife. Since she is a voracious reader, I decided to collect a lot of books for her. On the first day of the month, I bought one book, on the second, I bought two and on the third, I bought three. This process went on till the anniversary and on the day, I had 276 books with me to gift her. Can you calculate, on which day is our anniversary? • Views : 70k+ • Sol Viewed : 20k+ # Aptitude Questions : Pirates Probability Brain Teaser Difficulty Popularity You are stuck with the pirates who might even kill you on board. They give you a chance to survive. There are hundred black rocks and hundred red rocks. There are two empty sacks which are labelled as heads and tails respectively. You have to divide the rocks in two bags as per your wish. Then a fair coin will be flipped. If its heads, you will have to pick a rock on random from the sack labelled heads and if its tails, you will pick up from the tails sack. If you pick up a black rock, you will be freed and if you pick up a red rock, you will be killed. How will you distribute the rocks so that your chances of survival are the best? • Views : 70k+ • Sol Viewed : 20k+ # Aptitude Questions : Most Popular Logical Puzzle Difficulty Popularity Outside a room there are three light switches. One of switch is connected to a light bulb inside the room. Each of the three switches can be either 'ON' or 'OFF'. You are allowed to set each switch the way you want it and then enter the room(note: you can enter the room only once) Your task is to then determine which switch controls the bulb ?? • Views : 50k+ • Sol Viewed : 20k+ # Aptitude Questions : Popular Deductive Logic Problem Difficulty Popularity Eight Brothers lives in an old house where there is no electricity and no computers or any any other gadget. Brother-2: Playing Chess Brother-3: Writing Brother-4: making food for the family Brother-5: sleeping and snoring Brother-6: cleaning house Brother-7: watering the plants what is Brother-8 doing ? • Views : 60k+ • Sol Viewed : 20k+ # Aptitude Questions : Famous 13 Cave Logic Problem Difficulty Popularity A thief was running from the police after the biggest theft the town saw. He took his guard in one of the thirteen caves arranged in a circle. Each day, the thief moves either to the adjacent cave or stay in the same cave. Two cops goes there daily and have enough time to enter any two of the caves out of them. How will the cop make sure to catch the thief in minimum number of days and what are the minimum number of days? • Views : 40k+ • Sol Viewed : 10k+ # Aptitude Questions : Hard Logic Brain Teaser Difficulty Popularity There are 100 doors. 100 strangers have been gathered in the adjacent room. The first one goes and opens every door. The second one goes and shuts down all the even numbered doors – second, fourth, sixth... and so on. The third one goes and reverses the current position of every third door (third, sixth, ninth… and so on.) i.e. if the door is open, he shuts it and if the door is shut, he switches opens it. All the 100 strangers progresses in the similar fashion. After the last person has done what he wanted, which doors will be left open and which ones will be shut at the end? • Views : 60k+ • Sol Viewed : 20k+ # Aptitude Questions : The Elf Logical Puzzle Difficulty Popularity A wicked sorcerer felt enmity towards elf and thus he chooses four among the rest of the elf's and concealed them. The elves are concealed in the ground in a manner that apart from their head the rest of their body was underneath the ground. The elf's are unable to move their body and can only see in that direction that they are facing. All the elf's are concealed underground in such a way that they form a straight line and among those four elf's that are concealed underground one of the elf is detached form the other three elf's via wall. The entire elf's are in the same direction. The elf that is the furthest can only see the heads of its friends in front and a wall. The elf that is second last can only see one head of his friend and a wall. The second elf can only view the wall. The elf can see nothing. The sorcerer understands the situation and tells the elf's that he has placed hats over their heads. Among the hats places two hat are blue and the other two are red. Among all the four elfs one of the elf has to guess that which colour hat is he wearing. If the elf answers correctly then he shall be set free or else he will have to dig beneath the ground till the very last. • Views : 40k+ • Sol Viewed : 10k+ # Aptitude Questions : Difficult Brain Twister Difficulty Popularity A strange tradition is followed in an orthodox and undeveloped village. The chief of the village collects taxes from all the males of the village yearly. But it is the method of taking taxes that is interesting. The taxes paid in the form of grains and every male should pay equal pounds corresponding to his age. In simpler terms, a man aged 10 years will have to pay 10 pounds of grains and a 20 years old will be paying 20 pounds of grain. The chief stands on a riser containing 7 different weights next to a large 2 pan scale. As per the interesting custom, the chief can only weigh using three of the seven weights. In such a scenario, can you calculate what must be the weights of the seven weights each and who is the oldest man the chief can measure using those weights? • Views : 70k+ • Sol Viewed : 20k+ # Aptitude Questions : Game Of Dice Brain Teaser Difficulty Popularity A unique solo game of dice is being played. Two dices are thrown in each turn and the scores are taken by multiplying the numbers obtained. Now talking about a particular game, here are the facts: 1) The score for the second roll is five more than the score for the first roll. 2) The score for the third roll is six less than the score for the second roll. 3) The score for the fourth roll is eleven more than the score for the third roll. 4) The score for the fifth roll is eight less than the score for the fourth roll. Reading the above facts, can you tell the score for each of the five throws? • Views : 50k+ • Sol Viewed : 20k+ # Aptitude Questions : Move 3 MatchSticks Puzzle Difficulty Popularity You need to divide area in the picture below into two equal parts by using exactly three match sticks. Can you do it? ### Latest Puzzles 28 May ##### Males In Family Riddle In a family of 6 people P, Q, R, S, T, a... 27 May ##### Spot the Bird Riddle Can you spot the bird in the picture bel... 26 May ##### Rugby Sports Riddle A rugby player who played professional m... 25 May ##### How many Chocolates Riddle A guy brings 1000 chocolates for his sis... 24 May ##### McDonald's Christmas Happy Meal Puzzle On the occasion of Christmas, McDonald's... 23 May ##### Which mathematical symbols Replaces The Question Mark Which mathematical symbols replaces the ... 22 May ##### What Is It six letter word. Riddle It's a six letter word. The first...

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Upcoming SlideShare × # Ac1.2gMorePracticeProblems 2,869 views Published on More practice with naming points, lines and planes and observing what is going on in diagrams. This is for high school students • Full Name Comment goes here. Are you sure you want to Yes No • Be the first to comment • Be the first to like this ### Ac1.2gMorePracticeProblems 1. 1. Millau Bridge Sir Norman Foster Point, Lines, Planes, Angles Fallingwaters Frank Lloyd Wright Millenium Park Frank Lloyd Wright 1.2 WE Points, Lines and Planes 2. 2. 1.2 WE Points, Lines and Planes True or False Yes. Yes, the planes extend to include the whole line. Yes, the planes extend to include the whole line. Yes. Yes. 3. 3. True or False No. It only passes thru plane R. Yes. No. it is only in Plane S. No. No straight line will contain all 3 pts. Yes. They are all in plane S. 4. 4. 11. Make a sketch showing 4 coplanar points such that 3, but not 4 are collinear. 12. Make a sketch showing 4 points that are not coplanar. 5. 5. 13. Name 5 planes that contain sides of the pyramid shown. 14. Of the 5 planes, are there any that do not intersect? 15. Name 3 lines that intersect with R. VRS, VST, VWT, VWR, RSTW Sides Bottom No. 4 intersect at point V and the bottom intersects all the sides. RW, RS, RV 6. 6. 16. Name 2 planes that intersect in . 17. Name 3 planes that intersect at point S. 18. Name a line and a plane that intersect in a point. STV, STWV Note all planes contain S and T. STV, SRV, STWR or And many Others. 7. 7. Follow the steps shown to draw the figure named. Rectangular solid or box. 19 8. 8. 20 Follow the steps shown to draw the figure named. A barn. 9. 9. 21. Name 2 planes that intersect in 22. Name 3 lines that intersect at pt. E. 23. Name 3 plane that intersect at pt. B. 24a. Are points A, D, and C collinear? 24b. Are points A, D, and C coplanar? FGSR and FGCB the roof and right side. ABCE, ABFRE, BCGF Front, Right Side, Floor No. Yes. The floor. 10. 10. 25a. Are points R, S, G, and F coplanar? 25b. Are points R, S, G, and C coplanar? 26a. Name 2 planes that do not intersect. 26a. Name 2 other planes that do not intersect. Yes – the yellow plane. No – C is not on the yellow plane. ABFRE and DCGSE The front and back BCGF and ADHE The right and left sides. 11. 11. 28a. Can 2 vertical planes intersect? 28b. Suppose a line is known to be in a vertical plane. Does the line have to be a vertical line? 27. Can 2 horizontal planes intersect? Yes. Look at the two walls in the corner of the room. No. Two floors of a building do not intersect. NO. Look at cinderblock wall or a brick wall. They are vertical but they contain horizontal joint lines. 12. 12. Sketch and label the figures described. Use dashes for hidden lines. 29. Vertical line l intersects a horizontal plane M at point A. 13. 13. Sketch and label the figures described. Use dashes for hidden lines. 29. Vertical line l intersects a horizontal plane M at point A. 14. 14. Sketch and label the figures described. Use dashes for hidden lines. 30. Horizontal plane P contains two lines k and n that intersect at point A. P 15. 15. Sketch and label the figures described. Use dashes for hidden lines. 30. Horizontal plane P contains two lines k and n that intersect at point A. P K N A 16. 16. Sketch and label the figures described. Use dashes for hidden lines. 31. Horizontal plane Q and vertical plane N intersect. 17. 17. Sketch and label the figures described. Use dashes for hidden lines. 31. Horizontal plane Q and vertical plane N intersect. 18. 18. Sketch and label the figures described. Use dashes for hidden lines. 32. Vertical planes X and Y intersect an . 19. 19. Sketch and label the figures described. Use dashes for hidden lines. 32. Vertical planes X and Y intersect an . A B 20. 20. Sketch and label the figures described. Use dashes for hidden lines. 33. Point P is not in plane N. Three lines through point P intersect N in points A, B, and C. 21. 21. Sketch and label the figures described. Use dashes for hidden lines. 33. Point P is not in plane N. Three lines through point P intersect N in points A, B, and C. A B C P 22. 22. C’est fini. Good day and good luck.

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# Bullet A bullet is moving with a velocity of $$30\sqrt{2}\ \si{\meter/\second}$$ is fired into a fixed target, it penetrates to a depth of $$s$$ meters. If the bullet is fired into a target of thickness $$\dfrac{s}{2}$$ meters, made from the same material, with the same velocity, what will be the bullet's velocity when it exits the target? Assumptions • The resistive force on the bullet in the target is a constant, independent of velocity or position. ×

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https://stonespounds.com/75-5-pounds-in-stones-and-pounds

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# 75.5 pounds in stones and pounds ## Result 75.5 pounds equals 5 stones and 5.5 pounds You can also convert 75.5 pounds to stones. ## Converter Seventy-five point five pounds is equal to five stones and five point five pounds (75.5lbs = 5st 5.5lb).

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Search # Statistics Project - Assignment Example Summary First, set the date range to be for exactly 1 year ending with the Monday that this course started. Our term started on 5/11/2015, so us the dates 5/12/2014 – 5/11/2015. So the dates will going back exactly 1 year. Next, click the link on the right that… ## Extract of sample "Statistics Project" Download file to see previous pages 1. If a person bought 1 share of Google stock within the last year, what is the probability that the stock on that day closed at less than the mean for that year? Hint: You do not want to calculate the mean to answer this one. The probability would be the same for any normal distribution. (5 points) If a person bought 1 share of Google stock within the last year, the probability that the stock on that day closed at less than the mean for that year is 0.50 (or 50%). This is because the normal distribution is symmetric about the mean and the area below (or above) the mean is equal to 0.50 that is the mean divide the area exactly half on both ends (left or right). 4. Suppose a person within the last year claimed to have bought Google stock at closing at \$589.50 per share. Would such a price be considered unusual? Be sure to use the definition of unusual from our textbook. (5 points) No, such a price would not be considered unusual. This is because the probability of buying a Google stock at closing at \$589.50 (or higher) is about 0.0507 (or 5.07%) and typically, we say that an event with a probability less than 5% is unusual. However, we can say that it is very close to be considered an unusual price. 5. At what prices would Google have to close at in order for it to be considered statistically unusual? You should have a low and high value. ? Be sure to use the definition of unusual from our textbook. (5 points) 6. What are Quartile 1, Quartile 2, and Quartile 3 in this data set? Use Excel to find these values. This is the only question that you should answer without using anything about the Normal distribution. (5 points) 7. Is the normality assumption that was made at the beginning valid? Why or why not? Hint: Does this distribution have the properties of a normal distribution as described in our textbook? It does ...Download file to see next pagesRead More Cite this document • APA • MLA • CHICAGO (“Statistics Project Assignment Example | Topics and Well Written Essays - 500 words”, n.d.) (Statistics Project Assignment Example | Topics and Well Written Essays - 500 Words) https://studentshare.org/statistics/1697946-statistics-project. “Statistics Project Assignment Example | Topics and Well Written Essays - 500 Words”, n.d. https://studentshare.org/statistics/1697946-statistics-project. Click to create a comment or rate a document ## CHECK THESE SAMPLES OF Statistics Project ### Family Care International Company Analysis ...? Company Analysis: Family Care International Company Analysis: Family Care International PART Introduction/Overview The organization to be examined for the project is Family Care International. It is a nonprofit organization that helps in family care most especially in obtaining goals such as helping mitigate HIV risks and also death by pregnancy. Nonprofit organizations are organization and activities next to the institutional complexes of government, state, or public sector on the other hand and the for profit or business sector on the other, wherein it focuses on the field of welfare provision, education, community development, international relations, arts, environment and culture (Bryson, 2011). The importance of a nonprofit... 11 Pages(2750 words)Statistics Project ### Networking ...? Full Paper Network LAN Topologies Network topologies are essential before implementing a network as per requirements of an organization. Topology is a framework defining the arrangements of every object on the network. This includes workstations, network components, servers, WAN devices and many more. There are total five topologies to design a computer network. However, all of these five topologies share certain factors. Each topology demonstrates disadvantages as well as advantages that will be discussed further. The five different topologies are illustrated below: 1.1 Star Topology Star topology is recommended for the wired local area network. It is the most widely adopted topology. The star topology supports the centralized... Full... 11 Pages(2750 words)Statistics Project ### The Evolution of Automobiles ...?Automobiles the Evolution from Fossil Fuels to All Electric Vehicles Cars are a large part of life, as we know it in the modern world. Understanding the changes from the first cars manufactured through mass production of cheap vehicles to the modern ideal of hybrid and electric technology can allow us to prepare for the changes that may occur in the near future as well. The following assignment will look at the cars evolution from fossil fuels to the more advanced hybrid and electric approaches of today. Working hypothesis Electric vehicles are more efficient uses of our energy production then purely fossil fuel driven vehicles. Currently there are several very efficient models of car being offered by American and foreign auto... the... 6 Pages(1500 words)Statistics Project ### Restaurants and Happiness/Life Satisfaction ...? WHAT I’ve written: I don’t think this is written correctly: Please edit/clarify/correct if needed For any human, satisfaction towards food is an crucial question and issue. Does a man get proper food? Is it with right nutritional value? Is it caters to his satisfaction and worthwhile of money he paid. Above all, does he get satisfaction ultimately both physically and mentally. So there are so many parameters involved in the study of satisfaction towards food. It may be influenced through restaurants of good quality, which can add to nutritional value and physical value. This study examines the of impact fast food restaurants in developing countries on individual life satisfaction and happiness of individual living in those... WHAT I’ve... 6 Pages(1500 words)Statistics Project ### Gross National Income of different countries ...? GNI of different countries Submitted to, Submitted By, of the Submitted on, [July 7th, The report comprises of two parts. First part consists of statistical analysis of Gross National Income of 173 countries of the world. The main theme is to comment on the data through selection of appropriate measure of central tendency. The second part consists of the analysis of data related to the visitors to UK and the revenue that UK generates. The regression analysis and coefficient of correlation are used to determine the required results. Table of Contents:  1. Cover Page 2. Abstract 3. Problem Statement 4. Descriptive Statistics 5. Procedures 6. Results 7. Conclusions Problem Statement Data was collected by the World Bank Organisation... 3 Pages(750 words)Statistics Project ### Statistics assignment ...Statistics 302 Assignment 2 Report written by: Number: ”I confirm that I have not received help from, or given help to, anyone else in constructing the solution to this assignment” ______________________ Name Q1. (10) The number of UBC students infected by H1N1 is a Poisson random variable with rate 5 per week. (a) What is the probability that there are more than 10 students infected in a week? Solution: The poisson probability is given by: p(x; μ) = (e-μ) (μx) / x! The probability that more than 10 students are infected in a week = P(x>10, μ = 5) = e-(5) * [(511)/11! + (512)/12! + (513)/13! + … + 5^20/20! +..] P(x>10, μ = 5) = e-(5) * 2.0325 P(x>10, μ = 5) = 0.01369 (b) What is the probability... 302 Assignment 2... 2 Pages(500 words)Statistics Project ...Generic Business Strategies and Competitive Advantage of Tourist Companies 1.0 Introduction Multiple regression analysis is one of the most applied statistical methods of data analysis in econometrics. The method is popular both for scholarly and company-based analysis of business and economic statistics (Ziliak and Kniesner, 1998). The popularity of multiple regressions as a method of data analysis is related to its applicability to a variety of data type and problems (Gordon, 1978). The method is also known for its ease of interpretation, widespread availability, and relative ease by which they can be carried out using different types of data analysis software (Cryer and Chan, 2008). In business research, multiple regression... 8 Pages(2000 words)Statistics Project ### Implications and Limitations of Economic Feasibility ...Implications and Limitations of Economic Feasibility Limitations Economic feasibility is one of the main interests of managements, whether for profit making or not profit making organizations. Comparing total costs with total revenues is important to ensuring this feasibility but an understanding of factors to total operations costs helps in determining the costs for optimal profit margins. Results from the study identify a significant relationship between hospitals’ total operations costs and the independent variables: variables Staffed beds_05, Medicare Days_05, Medicaid Days_05, Total Surgeries_05, RN FTE_05, Occupancy, Ownership, System Membership, Rural/Urban, Teaching Affiliation, Age 65+, Crime Rate, and Uninsured. Out... and... 2 Pages(500 words)Statistics Project ... April 27, Business statistics project Statistical analysis is important to managerial decisions as it offersinsights into data. Its results include summaries of trends and analysis of possible differences in variables and their significance for informed decisions. This discussion offers analysis of data on age, household income, and years of education of customers for five major motor vehicle companies in North America. The discussion is based on the hypothesis that Cadillac has retained control of the segment of older men and has lost the other segment to its competitors and that this is a possible factor to decline in the company’s market control. Table 1: Mean, median, mode, and standard deviation for age Company BMW... April 27,... 2 Pages(500 words)Statistics Project ### Yahoo Inc ...I. Introduction This paper presents the trend analysis and forecast for Yahoo Inc. The company was founded in 1994 by two Stanford PhD candi s (Yahoo Inc, 2014). The company offers emails, entertainment, news and sports (Yahoo Inc, 2014). The current Chief Executive Officer of Yahoo Inc. is Marissa Mayer who is assisted by fourteen other members of the executive team (Yahoo Inc, 2014). The aim of this paper is to examine the performance of Yahoo Inc. and forecast sales, costs and profits for the next two years. The objectives of this paper are: 1. To identify the trends in sales, costs and profits of Yahoo Inc. from 2003 to 2012. 2. To forecast the performance of Yahoo Inc. in terms of sales, costs and profits for the next two... 9 Pages(2250 words)Statistics Project We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies. ## Let us find you another Assignment on topic Statistics Project for FREE! 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U.S. Markets open in 5 hrs 44 mins # A Closer Look At Orthofix Medical Inc.'s (NASDAQ:OFIX) Uninspiring ROE Want to participate in a research study? Help shape the future of investing tools and earn a \$60 gift card! Many investors are still learning about the various metrics that can be useful when analysing a stock. This article is for those who would like to learn about Return On Equity (ROE). To keep the lesson grounded in practicality, we'll use ROE to better understand Orthofix Medical Inc. (NASDAQ:OFIX). Our data shows Orthofix Medical has a return on equity of 4.1% for the last year. Another way to think of that is that for every \$1 worth of equity in the company, it was able to earn \$0.041. ### How Do I Calculate Return On Equity? The formula for return on equity is: Return on Equity = Net Profit ÷ Shareholders' Equity Or for Orthofix Medical: 4.1% = US\$14m ÷ US\$335m (Based on the trailing twelve months to December 2018.) Most know that net profit is the total earnings after all expenses, but the concept of shareholders' equity is a little more complicated. It is all earnings retained by the company, plus any capital paid in by shareholders. Shareholders' equity can be calculated by subtracting the total liabilities of the company from the total assets of the company. ### What Does Return On Equity Mean? ROE measures a company's profitability against the profit it retains, and any outside investments. The 'return' is the amount earned after tax over the last twelve months. A higher profit will lead to a higher ROE. So, as a general rule, a high ROE is a good thing. That means ROE can be used to compare two businesses. ### Does Orthofix Medical Have A Good ROE? One simple way to determine if a company has a good return on equity is to compare it to the average for its industry. Importantly, this is far from a perfect measure, because companies differ significantly within the same industry classification. As is clear from the image below, Orthofix Medical has a lower ROE than the average (9.7%) in the Medical Equipment industry. That certainly isn't ideal. We'd prefer see an ROE above the industry average, but it might not matter if the company is undervalued. Nonetheless, it might be wise to check if insiders have been selling. ### The Importance Of Debt To Return On Equity Companies usually need to invest money to grow their profits. The cash for investment can come from prior year profits (retained earnings), issuing new shares, or borrowing. In the first and second cases, the ROE will reflect this use of cash for investment in the business. In the latter case, the debt used for growth will improve returns, but won't affect the total equity. That will make the ROE look better than if no debt was used. ### Orthofix Medical's Debt And Its 4.1% ROE One positive for shareholders is that Orthofix Medical does not have any net debt! It's hard to argue its ROE is much good, but the fact that no debt was used is some comfort. At the end of the day, when a company has zero debt, it is in a better position to take future growth opportunities. ### In Summary Return on equity is one way we can compare the business quality of different companies. Companies that can achieve high returns on equity without too much debt are generally of good quality. All else being equal, a higher ROE is better. Having said that, while ROE is a useful indicator of business quality, you'll have to look at a whole range of factors to determine the right price to buy a stock. The rate at which profits are likely to grow, relative to the expectations of profit growth reflected in the current price, must be considered, too. So I think it may be worth checking this free report on analyst forecasts for the company. If you would prefer check out another company -- one with potentially superior financials -- then do not miss this free list of interesting companies, that have HIGH return on equity and low debt. We aim to bring you long-term focused research analysis driven by fundamental data. Note that our analysis may not factor in the latest price-sensitive company announcements or qualitative material. If you spot an error that warrants correction, please contact the editor at editorial-team@simplywallst.com. This article by Simply Wall St is general in nature. It does not constitute a recommendation to buy or sell any stock, and does not take account of your objectives, or your financial situation. Simply Wall St has no position in the stocks mentioned. Thank you for reading.

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# Biology If you were given a plant with red tomatoes, how could you figure out what its genotype is? Explain 1. 👍 0 2. 👎 0 3. 👁 37 ## Similar Questions 1. ### Science Jordan is doing a science fair project on the effects of music and growth of tomatoes.he has two tomato plants, plant A and plant B that he grows in a window and gives the same amount of water. Plant A is exposed to classical asked by TiaB✨ on March 6, 2017 2. ### biology jordan is doing a science fair project on the effect of music on the growth of tomatoes. he has two tomato plants, plant A and B, that he grows in a window and gives the same amount of water. plant A is exposed to classical music asked by rose fuentes on September 10, 2014 3. ### math help! 2005 2006 2007 2008 Plant 1 20 4 25 10 Plant 2 25 5 5 48 Plant 3 20 8 20 32 Plant 4 43 10 10 47 Plant 5 40 11 15 48 A gardener recorded how many tomatoes 5 plants produced each year. Which year had the LARGEST interquartile asked by brandon on May 17, 2013 4. ### Math Tom has green and red tomatoes in a bag a 4 to 3 ratio 5 green tomatoes and 5 red tomatoes were removed making the ratio become 3 to 2 how many red tomatoes were originally in the bag Please show me how to work the problem asked by Michelle on January 5, 2016 5. ### Another niology/science question In tomatoes, red fruit is dominant to yellow fruit color. A farmer has plants that produce either red or yellow tomatoes. he has signed a contract with a large seed company to provide pure red dees and pure yellow seeds. the do asked by Kellie on October 15, 2007 6. ### biology In tomatoes, red fruit is dominant to yellow fruit color. A farmer has plants that produce either red or yellow tomatoes. he has signed a contract with a large seed company to provide pure red dees and pure yellow seeds. the do asked by kelly on October 15, 2007 n tomatoes, red fruit is dominant to yellow fruit color. A farmer has plants that produce either red or yellow tomatoes. he has signed a contract with a large seed company to provide pure red seeds and pure yellow seeds. the do asked by Kellie on October 15, 2007 8. ### algebra a farmer grows corn, tomatoes, and sunflowers on a 320 acre farm. this year the farmer wants to plant twice as many acres of tomatoes as sunflowers. The farmer also wants to plant 40 more acres of corn than of tomatoes. How many asked by Sam on November 28, 2012 9. ### math 2005 2006 2007 2008 Plant 1 20 4 25 10 Plant 2 25 5 5 48 Plant 3 20 8 20 32 Plant 4 43 10 10 47 Plant 5 40 11 15 48 A gardener recorded how many tomatoes 5 plants produced each year. Which year had the LARGEST interquartile range asked by shane on May 16, 2013 10. ### chemistry I am doing a project on medicinal plants. I planted red tomatoes, but I can't find any website that provides me info that it is medicinal. If you know any websites on red tomatoes that shows that it is medicinal please tell me. Oh asked by Ron on February 24, 2007 More Similar Questions

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# Existence of a pseudo-polynomial time algorithm for a counting problem. Let T={1,...,n} be a set of tasks. Each task i has associated a non negative processing time p_i and a deadline d_i. A feasible schedule of the tasks consists of a permutation of n elements pi, such that \sum_i=1^k p_(pi(i)) <= d_(pi(i)) for all k=1,...n. Does there exists a pseudo-polynomial time algorithm for computing the total number of feasible schedules? A pseudo-polynomial time algorithm is an algorithm whose running time is bounded by a polynomial on the size of the input, given that the input is written in unary notation (2=II, 3 =III). (e.g., the size of a number n in unary notation is O(n), and not O(log(n)). This is an open question from an article published in 2009 at Operations Research Letters. - Usually asking open problems is not really considered appropriate for MO; see the FAQ. That said, you might ask something like "what is the current state of knowledge about this problem?" – Daniel Litt Jul 29 '10 at 14:57 Should the RHS read $d_{\pi(k)}$ instead of $d_{\pi(i)}$? I also agree with Daniel, it seems inappropriate to ask an open question (especially such a recent one) on MO. – Artem Kaznatcheev Aug 27 '10 at 13:00 At the time I write this comment, the question has been edited and now makes no sense whatsoever – Yemon Choi Sep 13 '10 at 15:40 I've rolled the question back to it's previous version, which at least made sense. I don't know anything about this question, but it seems perfectly reasonable to ask whether any progress has been made on it. – David Speyer Sep 13 '10 at 15:51 Actually, David, could you further roll it back to the version by Gerry, which had the same content but had grammar corrected and used LaTeX? – JBL Sep 13 '10 at 17:51

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```"Rays pitcher Sergio Romo struck out the side against the Angels on Saturday, then struck out three more batters in 1 1/3 innings the following afternoon. It was an unusual pairing of performances for one small reason: Romo had never struck out three batters on consecutive days before. And it was an unusual pairing of performances for one massive, potentially paradigm-shifting reason: Romo served as the modern era’s first designated “opener.” ``` If you had \$20,000 that you could comfortably afford to risk as your sports wagering bankroll and \$3,295 went to pay for the all Football and all Basketball service, then you would have \$16,705 left for wagering. As explained above the expected return on the combined Dr Bob Football and Basketball and NBA Guru Basketball services is +68.0% per year (using a less optimal flat betting approach), which would result in a return wagering profit of +\$11,359 on the \$16,705 initial bankroll. The overall profit, after factoring in the cost of the services, would be \$8,064 ((\$16,705 x 0.68) – \$3,295 = +\$8,064), which is a very good 40.3% expected return on your \$20,000. That percentage return is higher for higher bankrolls and lower for lower bankrolls since the cost of service becomes a smaller percentage of higher bankrolls and a higher percentage of smaller bankrolls. If you want to subscribe to the all Football and all Basketball package you would need a total of at least \$4,846 to invest to expect a positive return after factoring in the cost of the service. The calculations above are based on expected results based on long term records and some years are better and some years are worse. ```In the above break down of Investment / Gambling spectrum, needless to say Lottery’s EV (Expected Value) is the lowest.  Its EV is always negative, according to business insider, the \$450 million Powerball Jackpot lottery’s  EV is – \$1.37 per ticket. Means you lose average \$1.37 when you buy a \$2 ticket, so the loss accounts for 68% out of your investment.  This terrible negative return level applies to basically all lotteries. ``` It is that time of the season again, the title races across the Europe are heating up, the Champions League is back in 2 days and it feels like the second Christmas of the winter season is upon us. If Christmas gave us a clear leader at the top of the pile, Christmas 2.0 has put some life back into the title race and now people are looking for someone at Liverpool to make the next big ‘Slip’. It was that group that decided to get all Ottoman Empire on a still-unborn industry by proposing a head-spinning 36 percent tax – 34 percent to the state, 2 percent to local coffers – on gross sports betting revenue. Yes, that is in fact gross. That rate is on top of an up-front \$10 million fee just for properties to obtain a sports betting license. For sports bettors, the vig is more expensive than the spreads on a typical Wall Street stock trade. But the growth of online betting—mostly illegal in the U.S.—has started to shrink the vig below 10% in some places, which should eventually make betting a better deal for bettors. That’s what you’d expect, says Toby Moskowitz, a professor of finance at the Yale School of Management and one of the principals with the \$226 billion hedge fund manager AQR Capital Management. Cash Out. Cash Out lets you take profit early if your bet is coming in, or get some of your stake back if your bet is going against you—all before the event you’re betting on is over. Cash Out offers are made in real time on your current bets, based on live market prices. Whenever you are ready to Cash Out, simply hit the yellow button. Cash out is available on singles and multiples, on a wide range of sports, including American football, tennis, horse racing, basketball, and many more! You can Cash Out of bets pre-play, in-play, and between legs.[1] Money Management is as critical to a sports investor as picking winners. I have devoted many hours of careful analysis and math to optimal money management systems, which I have painstakingly outlined in my Money Management articles. Sports betting is more high risk (higher volatility and standard deviation of return) than stocks, but also results in a higher return if you follow a proven long term winning handicapper (of which there are very few). Whoever was behind “maria” turned £3,000 into £100,603.78 (after 5% Betfair commission deduction) in 303 days!  She consistently put her selections up on the board in good time before the racing began and the actual selections and strike rate were never in doubt. Many people cast doubt over the true identity of Maria Santonix. Some said it was a man called Adrian Massey who owned a well known horse racing website at the time. The only images available of Maria online are heavily edited so you can not clearly see her. Remember, in 2005 sites such as Facebook were not even invented yet so having your picture online wasn’t as easy or as common as it is today. Therefore, the fact there are no images available of Maria does not mean she is a fake. Sadly, the original thread has been deleted however you can still see the part of it from an internet archive website. Sportstradinglife The NHL’s Stanley Cup Playoffs start a few days after April 7, which is when the regular season ends. Of the 31 teams, 16 make it into the postseason. Each series is best of seven. The playoffs will often go into May. The NBA season ends April 11 and the playoffs begin April 14 and end at some point in early June. A total of 16 teams make it into the NBA postseason. In 1919, the Chicago White Sox faced the Cincinnati Reds in the World Series. This series would go down as one of the biggest sports scandals of all time. As the story goes, professional gambler Joseph Sullivan paid eight members of the White Sox (Oscar Felsch, Arnold Gandil, Shoeless Joe Jackson, Fred McMullin, Charles Risberg, George Weaver, and Claude Williams) around 10,000 dollars each to fix the World Series. All eight players were banned from playing professional baseball for the rest of their lives.[56] Pete Rose, the all-time MLB leader in hits, was similarly banned from baseball in 1989 for betting on games while he was an MLB manager. A very big step to becoming a winning sports bettor is to make sure you are shopping for the best lines. What this means is that you will have to have an account at a couple of different online sportsbooks and when you are ready to make a bet on a certain team or outcome you check the different sportsbooks and find which book is offering the best line for you (this ties in with our advanced how to find max value article). For example, if you are looking to bet on the New England Patriots on a 7 point spread, you would want to check a couple different sportsbooks to see if any of them are offering the Patriots at 6.5 points, or at the very least find the best price you can get them at 7 points at. Over the course of a sport betting season you can win yourself a lot of money from shopping the lines that would otherwise would not be one. Line shopping is definitely one of the best sports betting strategies used by winning sports bettors. To see a list of sportsbooks we recommend signing up for visit our Online Betting Sites section. ### So what is the best theory on money management? There are several methods that have proven successful by many professional gamblers. In speaking to most of these individuals, discipline being the main ingredient. I personally believe that one should never bet more than 20 percent of their season bankroll on any given week. An example would be if a gambler starts with a seasonal bankroll of \$5,000.  Thus, they will have \$1000 (20 percent of your bankroll) to bet with on opening week. If we were to release eight football picks on the first week it would look like this: When looking over a lot of college football betting predictions, it’s clear to see that the future might not be etched in stone for the Big 12. Some sports commentators and analysts covering college football spread picks said the Big 12 was unfixable. Well, 2 weeks ago, that statement may have been reasonable, but as of today, there is still ... Read More » In a different study released by FDU’s PublicMind in October 2011, results showed that New Jersey voters thought legalizing sports betting in New Jersey was a good idea. Half of New Jersey voters (52%) said that they approved the idea of legalizing sports betting at Atlantic City casinos and racetracks, 31% opposed it. In addition, there was a significant gender split: a majority of men approved of the idea by a wide margin (65-21), while only 39% of women approved and 41% opposed.[9] The October results were stable, reflecting an earlier poll in April 2011 where New Jersey voters approved the legalization of sports betting in the state by a margin of 53%-30%. However, nearly two-thirds (66%) of voters were not aware of the upcoming statewide referendum on the issue. Age proved to be a divide: voters between the ages 18 and 34 were more likely to approve of sports betting than were older voters. Dr. Woolley commented: "But... younger voters... are far less likely to vote than other voters... As always, a lot depends on who actually shows up to vote."[10] These decisions will be easy to make for those who are sports fans first and bettors second. They usually choose to simply wager on all of the sports that they already follow and like to watch. This approach is entirely logical really, as those are the sports that they know and understand best. They’re more likely to enjoy betting on those sports, and their knowledge of those sports will give them a better of chance of winning money. The Blackhawks Over streak came to an end in Dallas on Saturday after 16 straight Over wins. 8 of those 16 were at home and 13 of past 15 at United Center have gone over and we won't look to buck the trend here when two teams battling for a Wildcard spot meet. The Coyotes come in playing some playoff style hockey tonight and have won 8 of 10. While the atmosphere may be a playoff type one, we don't see the Hawks team capable of playing a tight one. Over is the play here. Cash Out. Cash Out lets you take profit early if your bet is coming in, or get some of your stake back if your bet is going against you—all before the event you’re betting on is over. Cash Out offers are made in real time on your current bets, based on live market prices. Whenever you are ready to Cash Out, simply hit the yellow button. Cash out is available on singles and multiples, on a wide range of sports, including American football, tennis, horse racing, basketball, and many more! You can Cash Out of bets pre-play, in-play, and between legs.[1] Let me put what we have reviewed in this way; The bookmakers have edge of built-in margin but unlike casino where pure mathematics governs, the sports betting is pretty much skill oriented game thus successful Software / Tipsters (Handicappers) / Systems turn their edge into ours. They can identify a True Value Bet in the form of UNDER-ROUND in the selected sports category where they have absolute skill edge. These will result in Positive EV (Expected Value) that’s a mathematical formula to allow you achieve average long-term profit. This +EV will be translated into the additional winning rate to 57%. For further details of EV and its rationale, you may want to read the one I introduced at the beginning, especially Value Bet section; Though soccer is low-scoring, Murphy says there are many ways to bet on it. Even if you don’t know a corner kick from a red card, Murphy says “it’s not hard to learn about the basics of the game and then start to do a little figuring out to what’s important to betting the game. People should not be afraid to try new sports, because there’s a lot on the board.” But, as previously discussed, in order to see the same happen in Arizona it will take a renegotiation of the state’s tribal pact. Arizona’s governor has already expressed interest in this which is why we’ve provided a list of cities that would most likely see sportsbooks open once a renegotiation happens. We’ve also listed out some nearby cities that offer sports wagering while residents wait. So how difficult is sports betting math? The math behind placing a winning bet is fairly complicated, but the way to stay ahead of the bookmaker is rather straightforward. If you collect on 52.4% of your bets, you’ll break even. We’ll have more details on that number later, including why it takes more than 50% wins to break even, but first some general knowledge about sports gambling and the numbers behind it. #### Arizona is one of the few states in the country to have a professional sports team in every major US sport. Fans have the option to watch Cardinals games live in one of the most high-tech domes in the country. And even though Arizona is mostly covered in desert, residents even have the option to go watch NHL games and root for the Coyotes. Great games don’t just end at the professional level, because sports programs at the University of Arizona and Arizona State draw in huge crowds as well. With the online offshore sportsbooks we recommend, legal sports betting in Arizona can be done on these teams and almost every team outside of the state. The Thunder are faced with a seeding decision before tonight's tip against the Jazz at Vivint Smart. OKC's conundrum is whether they want to sell out in the final month of the regular season by playing their first rotation major minutes or saving their legs for postseason play. A focus on the team's final 16-games of the regular season to remain one of the top four seeds in the West requires playing Russell Westbrook and Paul George in excess of their season average in minutes played. This asking allot considering the pairs 36 minutes per game this season. A basketball season with 53.5% winners (my career percentage is 53.9%) on 500 bets would on average yield +11.75 units ( (500*.535) – (500*.465)*1.1 ), or +23.5 Stars if my average Best Bet is rated 2-Stars. Using a conservative 1.6% of bankroll per bet (full Kelly at 53.5% at -110 odds is 2.35% of bankroll), or 0.8% per Star, results in an expected return of 18.8%. So, despite a lower overall winning percentage and smaller average wager size, a season’s worth of basketball wagers is fairly comparable to a season of football because there are so many more Best Bets in basketball season. ```The Thunder’s recent pointspread free-fall (1-9 ATS L10 games) has left them without much of a betting bandwagon; an elite level team that has legitimate ‘value’ potential moving forward.   The underdog has cashed winning bets in both previous meetings this season; a pair of ‘down to the wire’ finishes.  And make no mistake about it – this is ‘circle the wagons’ time for OKC following Paul George’s tirade against the refs after their loss to the Clippers on Friday.  Live dog here!  Take the Thunder. ``` Odds for different outcomes in single bet are presented either in European format (decimal odds), UK format (fractional odds), or American format (moneyline odds). European format (decimal odds) are used in continental Europe, Canada, and Australia. They are the ratio of the full payout to the stake, in a decimal format. Decimal odds of 2.00 are an even bet. UK format (fractional odds) are used by British bookmakers. They are the ratio of the amount won to the stake - the solidus "/" is pronounced "to" for example 7/1 "seven to one". Fractional odds of 1/1 are an even bet. US format odds are the amount won on a 100 stake when positive and the stake needed to win 100 when negative. US odds of 100 are an even bet. In the above break down of Investment / Gambling spectrum, needless to say Lottery’s EV (Expected Value) is the lowest.  Its EV is always negative, according to business insider, the \$450 million Powerball Jackpot lottery’s  EV is – \$1.37 per ticket. Means you lose average \$1.37 when you buy a \$2 ticket, so the loss accounts for 68% out of your investment.  This terrible negative return level applies to basically all lotteries. Though soccer is low-scoring, Murphy says there are many ways to bet on it. Even if you don’t know a corner kick from a red card, Murphy says “it’s not hard to learn about the basics of the game and then start to do a little figuring out to what’s important to betting the game. People should not be afraid to try new sports, because there’s a lot on the board.” The NBA Comp play is to play over the total in the Oklahoma City at Utah game at 9:00 eastern. The Thunder have flown over in 8 of 10 on Mondays ads the last 4 if they have lost 6 or 7 of the last 8 games. The Jazz are 6 of 6 over off a straight up favored loss and 5 of 6 over off a road loss by 10 or more. From the NBA System database we see that rested road dogs  with a 200 or higher total that failed to cover as a road dog and allowed 110 or more points are 17 of 20 Over since 1995 vs a team that failed to cover by 7 or more as a road favorite like Utah. Play this game over the total. On monday the 2019 NBA Total of the year headlines along with a 19-0 Conference tournament System in College hoops.. For the NBA Free pick. Play the Thunder and Jazz over the total. RV- GC Sports. One can follow many different betting strategies with long-term bets. Above we already mentioned the classic, of betting on the winner of a certzain competition. Apart from that there are also many other options, that we want to present to you. Additionally we want to provide important and helpful tips, that are supposed to lead you to betting success. Unfortunately, land-based sports betting in Arizona doesn’t exist right now. The reason mainly stems from the tribal compact that the state of Arizona has with the local Native American casinos owners. The compact allows those tribes to offer Class III gaming activities in exchange for a tax on revenue but does not extend those activities to include gambling on sports. If state officials wanted to allow other locations to offer this additional type of gaming then they would risk losing tax revenue from tribal casinos. This is why state officials are looking to rework the original compact so that only the tribes are allowed to offer it at their land-based locations. It wasn’t that long ago that the popular and mainstream sports were all we could bet on. Finding a bookmaker willing to take wagers on the “minor” sports was nearly impossible. This is no longer the case though, as these days we can bet on virtually any sport that’s played professionally: even the most obscure ones. Although these don’t offer the same advantages that we’ve outlined above, one big advantage they have is that the bookmakers don’t give them the same level of attention. Let’s examine, if people can make a living sports living of matched betting. Probably the best way to examine is using Profit Accumulator (PA) case. PA is one of the most popular matched betting paid service with over 20,000 members in the UK. The members’ average monthly income from the matched betting is said to be around £1,000. See Profit Accumulator Full Review – Should We Believe 20,000 members’ Eaning Claim? for full details; Rugby League and Rugby Union combine to attract a massive global audience and an astronomical amount of betting action each year. There are countless professional leagues and a great many international events which means that punters have a nearly endless supply of betting opportunities. It’s not unusual for each individual match to feature well over 30 pre-match wagering possibilities. It gets even better when one considers all of the bets that can be made while a match is in progress. © 2019 American City Business Journals. All rights reserved. Use of and/or registration on any portion of this site constitutes acceptance of our User Agreement (updated 5/24/18) and Privacy Policy and Cookie Statement (updated 5/24/18). The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of American City Business Journals. If you’re a big fan of your local teams, it’s likely that you know them better than most other teams. There’s plenty of value with having a deep knowledge of specific teams. You’re simply going to have a much better bead on how teams you follow closely will play compared to others. Occasionally, this means betting on your favorite teams, other teams it means betting against them. There’s no room for personal bias or favoritism if you’re serious about making money betting on your favorite sports. Horse race betting is usually an activity tied to the Southeast, but Arizona is no stranger when it comes to hosting great races. Residents can place their wagers at tracks such as Turf Paradise Race Course, Rillito Downs Park Racetrack, and Yavapai Downs Racecourse. But, for those looking for a more convenient way to place wagers, they’ll be happy to know that they can do so with the same sports betting sites we’ve previously recommended. Bovada, SportsBetting, BetOnline, and 5Dimes each come with a state of the art online racebook. There also promotions and bonuses for using their racebook which may give you an edge the next time you want to bet on your favorite pony. Chelsea began the season strong on a strong note going 12 premier league games without a defeat to start of their season. They had a healthy 7 point cushion between them and Manchester United when Ole Gunnar Solskajaer took over and now they trail United for the last Champions League spot by 1 point and a game in hand making this game against Manchester City a must win for their Manager. By giving all of your focus to a single sport and doing your due diligence, you can find great value on lines on a regular basis. Remember that public perception factors into the making of point spreads and moneylines as well; if you are way ahead of the public in terms of knowledge on a specific sport, you can spot lines that the average bettor might not. Tennis betting online is becoming a popular activity for sports gamblers year-round but the action and betting truly heat up for the grand slams. Betting on the 2019 French Open tennis is easier than ever with an extensive list of online betting sites available no matter where you are located in the world. French Open betting is likely to surpass all other grand slams like Australian Open, Wimbledon or the US Open regarding the betting handle. Filling out the draws bracket is another way to get in on the fun while betting on tennis. Ray Wallin is a licensed civil engineer and part-time handicapper who has had a presence on the Web since 2000 for various sports and horse racing websites and through his personal blog. Introduced to the sport over the course of a misspent teenage summer at Monmouth Park by his Uncle Dutch, a professional gambler, he quickly fell in love with racing and has been handicapping for over 25 years. The bookmaker functions as a market maker for sports wagers, most of which have a binary outcome: a team either wins or loses. The bookmaker accepts both wagers, and maintains a spread (the vigorish) which will ensure a profit regardless of the outcome of the wager. The Federal Wire Act of 1961 was an attempt by the US government to prevent illegal bookmaking.[2] However, this Act does not apply to other types of online gambling.[3] The Supreme Court has not ruled on the meaning of the Federal Wire Act as it pertains to online gambling. Even though most sports bettors are losers in their own right (as a whole, bettors actually win an average of only 48% of their bets – less than they would expect to win if they just flipped a coin for every game), their losses are compounded by the fact that the house takes a cut of winnings, also known as the ‘juice’ or ‘vig.’ Most sports books charge a 10% commission on wins, which means that a bettor must actually win 52.4% of his games just to break even. (Wagering \$100 per game, a bettor loses \$100 with a loss and wins \$90.91 with a win, so he must go 11-10 (11/21 = 52.38%) to break even). ## The outlined sportsbook strategy also answers a frequent question, whether it is possible to be profitable in the long run or even make one's living by sports betting. It is certainly possible, however it places considerable demands too – on time, psychics and discipline as well. Information collection may be a full-time job, but it is also a hobby for many. It was that group that decided to get all Ottoman Empire on a still-unborn industry by proposing a head-spinning 36 percent tax – 34 percent to the state, 2 percent to local coffers – on gross sports betting revenue. Yes, that is in fact gross. That rate is on top of an up-front \$10 million fee just for properties to obtain a sports betting license. ```For the World Cup final match on Sunday, when Croatia will face France, the French are strong favorites. At the online bookmaker bet365 on Friday, it cost you 250 to win 100 on a French victory, while you had to risk only 100 to win 200 on Croatia. I left out the dollar signs on those odds because they can’t take Americans’ bets. The difference of 50 in the payoffs is the sports-bookmaker’s “vigorish,” which corresponds to the spread between Buy and Sell quotes that provide the profit for market makers in our financial markets. ``` A basketball season with 53.5% winners (my career percentage is 53.9%) on 500 bets would on average yield +11.75 units ( (500*.535) – (500*.465)*1.1 ), or +23.5 Stars if my average Best Bet is rated 2-Stars. Using a conservative 1.6% of bankroll per bet (full Kelly at 53.5% at -110 odds is 2.35% of bankroll), or 0.8% per Star, results in an expected return of 18.8%. So, despite a lower overall winning percentage and smaller average wager size, a season’s worth of basketball wagers is fairly comparable to a season of football because there are so many more Best Bets in basketball season. When there IS some value in their odds and lines, there are so many other people betting on these sports that it soon disappears. A good price will see lots of money coming in very quickly, and when that happens the bookmakers adjust their odds and lines accordingly. So unless we’re also very quick, we can easily miss out on the best opportunities. OKC fell by eight to the Los Angeles Clippers Friday night. The squad seems off balance, and several players have criticized the officiating lately, Paul George in particular who was fined for ripping refs after the loss to the Clips. To be fair, the Thunder might have a point (their opponents have been awarded a combined 93 free throws last two games), but wasting energy on such matters won't do them any good. I strongly considered selecting Aguero, who is coming off a goal and two assists in Manchester City's 6-1 rout of Southampton on Sunday -- until I looked at the schedule. Kane and Mohamed Salah, the other player I was considering here, both play on Tuesday (as does Messi). Manchester City's game is on Wednesday, and Aguero didn't start in the first game against Shakhtar Donetsk. There's a risk that Aguero won't start again, and if I wait until Wednesday to find out, I won't be able to pivot to one of my other top choices.

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go top 返回词典 • But it's the very same case.It can't be that in one of them, follow the body and the other one,follow personality. 但这是完全相同的案例不可能说在其中一个案例里，追随肉体，但在另一个里追随人格。 耶鲁公开课 - 死亡课程节选 • and then you can either... and then you have the chance to gain other one to two points extra. 然后你可以要么……然后你就有机会再额外得到一到两分。 美式足球 - SpeakingMax英语口语达人 • But whereas the other one was obviously forever, this allows you to specify a finite number of times that you wanna do something. 但又不同于“无限循环“,你可以根据需要,为它指定循环次数。 哈佛公开课 - 计算机科学课程节选 • And so in this law firm, one of these guys is an expert on intellectual property and the other one on fraud or something. 律师事务所也是一样的,有人是知识产权达人,别人可能处理欺诈案件很有经验 耶鲁公开课 - 博弈论课程节选 • Of course, that doesn't mean that in the real world one of them does exist and the other one doesn't. 当然,这并不意味着,在现实世界里面其中一个存在,而另一个不存在 耶鲁公开课 - 死亡课程节选 • So, it's up on this screen here now, so we'll work on the other one. If you can identify which of these statements is correct based on what you know about the relationship between frequency and wavelength and also just looking at the waves. 它们之间的关系,现在在大屏幕上了,鉴于你们知道,频率和波长的关系,看到这些波你们来,判断下这些说法是否正确。 麻省理工公开课 - 化学原理课程节选 • Then what's the other one? 那另一个是什么 耶鲁公开课 - 基础物理课程节选 • And I think in the polar one I said, if, what did I do there, I said, yeah, again if the x and y are greater than the other one, I'm going to return them to it. 然后我要返回一些值，我认为在极坐标的形式下我说过，如果，我在这里做了什么来着,我说过，对，再说一次，如果x和y坐标。 麻省理工公开课 - 计算机科学及编程导论课程节选 • OK, so now we have the other one, p dH/dp constant temperature. 好的，现在我们来研究另一个量，在恒温条件下的偏H偏。 麻省理工公开课 - 热力学与动力学课程节选 • or you can get... what is the other one? I think it's the Commonwealth Award. 或者还有……另外一个是什么？我想是英联邦奖。 想获得骑士爵位 - SpeakingMax英语口语达人 • The other one is Economics 251, Financial Theory; this is Financial Markets, that one is Financial Theory. 还有一门是经济学251号课程,金融理论学,这是金融市场课,那是金融理论学 耶鲁公开课 - 金融市场课程节选 • One leads to loss of energy, the other one to increasing energy. 一个让人失去动力，另一个增加动力。 哈佛公开课 - 幸福课课程节选 • And I thought Tim Holt who plays that part in The Magnificent Ambersons was aboslutely right and the person who played it in the other one was not anywhere as good. 同时，我觉得提姆·霍尔特,在《安巴逊家族》中的出演也非常的棒,而他在其他影片中的角色,则无法和这次的相提并论。 麻省理工公开课 - 电影哲学课程节选 • The other one's probably going to be the dominant. 另外一个可能是属音。 耶鲁公开课 - 聆听音乐课程节选 • We get a check for the Cheetos as well, but only because we're trained to believe that one tastes good and the other one doesn't, because in different cultures it would be the reverse. 我们还是选奇多作为食品,但选择的原因仅在于,所受的教育,告诉我们它好吃,而蟑螂不好吃,在不同的文化中情况可能是相反的 耶鲁公开课 - 关于食物的心理学、生物学和政治学课程节选 • So, what you do is you set up a couple of displays, one where the block is stopped, the other one where you take this away with a trap door and it keeps going. 所以,你可以设计几个演示,一个是木块被挡住了,另一个是用活门把木块挪开,使得屏板继续上升 耶鲁公开课 - 心理学导论课程节选 \$firstVoiceSent - 来自原声例句

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# (CS 267, Apr 11 1995) ## Accelerating Graph Partitioning Using A Multilevel Approach This material is based on "A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems" by Barnard and Simon, Proceedings of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993. The expensive part of spectral bisection is finding the eigenvector v2, which requires a possibly large number of matrix-vector multiplications with the Laplacian matrix L(G) of the graph G. To reduce the cost, we may use the following divide-and-conquer approach, which we will see later as well: 1. Approximate the initial graph G0=G by a simpler graph G1 with fewer nodes. 2. Partition G1, which should be cheaper since G1 is smaller. 3. Use the partition of G1 as an approximation for G0, which can be cheaply refined. 4. Apply the same algorithm recursively to partition G1, by finding a yet simpler graph G2, etc. To make this work, we need to show how to extract G1 from G0, how to derive a partition of G0 from the partition of G1, and how to refine the approximate partitioning of G0 obtained from G1. To get G1 from G0, we will eliminate nodes and edge from G0 while retaining the basic structure of G0. We do this by finding a maximal independent subset N1 of N0, where G0=(N0,E0). This means that • N0 contains N1 and E0 contains E1, • no nodes in N1 are directly connected by edges in E0 (independence), and • N1 is as large as possible (maximality). • There is a simple "greedy" algorithm for finding N1: ``` N1 = empty set for i = 1 to |N0| if node i is not adjacent to any node already in N1 end if end for ``` This is shown below in the case where G0 is simply a chain of 9 nodes with nearest neighbor connections, in which case N1 consists simply of every other node of N1. To build G1=(N1,E1), we build E1 as follows: We will loop through the edges in E0, growing "domains" Di around each node i in N1. In other words, a domain consists of the subgraph of G0 connected to i by the edges examined so far. We will add an edge to E1 whenever an edge in E0 would connect two of these domains. ``` E1 = empty set for all nodes i in N1 Di = ( { i }, empty set ) ... build initial domains end for unmark all edges in E0 repeat choose an unmarked edge e=(i,j) from E0 if exactly one of i and j (say i) is in some Dk, then add j and e to Dk mark e else if i and j are in different Dk's (say Dki and Dkj), then add an edge (ki,kj) to E1 else if both i and j are in the same Dk, then mark e else leave e unmarked endif until no unmarked edges ``` For example, suppose we start with G0 being a chain of nodes as above, with nodes and edges numbered from left to right. Then domain Di for node i will just contain its neighbor to the right, and the edges in E1 will simply connect adjacent nodes in N1, so G1 is just the chain of length n/2. To derive a partitioning of G1 from a partitioning of G0, we extract an approximate second eigenvector v2(G0) of G0 from v2(G1). This is done by interpolation: ``` for each node i in N0 if i is also a node in N1, then v2(G0)(i) = v2(G1)(i), i.e. use the same eigenvector component else v2(G0)(i) = average of v2(G1)(j) for all neighbors j of i in N1. end if end for ``` This is shown below in the case of the chain of 9 nodes. The black line is the exact v2(G0) (normalized so the sum-of-squares is one). The dashed blue line is the exact v2(G1), also normalized, and which only connects every other node. The red +'s are the approximate v2(G0) computed by interpolation as above, which by construction lie on top of the blue line. The magenta x's are the red +'s normalized, to better see how well they approximate the black line. Refining this approximate second eigenvector can be done in several ways. (The unrefined approximated second eigenvector is not always as good as in the case of the chain!). One possibility is that the Lanczos algorithm mentioned above benefits from having a starting vector which mostly points in the direction of the desired eigenvector. More aggressively, one can use a technique called Rayleight Quotient Iteration, which uses the fact that the iteration ``` choose a starting vector v(0) ... v2(g1) v(0) = v(0) / norm2(v(0)) ... norm2(x) = sqrt(sum_i x(i)^2) i=0 repeat i = i+1 rho(i) = v(i-1)' * L(G) * v(i-1) ... rho(i) = Rayleigh Quotient v(i) = inv( L(G) - rho(i)*I )* v(i-1) v(i) = v(i) / norm2(v(i)) until convergence ``` converges asymptotically cubically to an eigenvalue-eigenvector pair (rho(i),v(i)). rho(i) costs one matrix-vector multiply. Computing v(i), i.e. solving the linear system (L(G) - rho(i)*I) * v(i) = v(i-1), is done using an iterative method (called SYMMLQ) which requires yet more matrix-vector multiplications. But since cubic convergence (cubing the error at every step) is so fast, very few steps are needed. This speedy convergence is illustrated below by the example of the chain, where the black line is the error in the approximate eigenvector v(i) and the dashed blue line is the error in the approximate eigenvalue rho(i), as functions of i. One can see the cubic convergence between iterations i=2 and i=3, where the eigenvector error goes from 1e-4 to (1e-4)^3 = 1e-12. Cubic convergence is only visible for this one step before hitting the accuracy limit of 10^(-16) due to roundoff. Experiments report a 10x speedup over the basic spectral bisection algorithm. An alternative divide-and-conquer approach, which does not use eigenvectors, is to simply use Kernighan-Lin to do the refinement. ("A multilevel algorithm for partitioning graphs", B. Hendrickson and R. Leland, Technical Report SAND93-1301, Sandia National Labs, Albuquerque NM, 1993). ## Performance Comparison of Different Partitioning Algorithms This data is taken from "Geometric Mesh Partitioning" by J. Gilbert, G. Miller and S.-H. Teng (available from gilbert@parc.xerox.com). Table 1 below describes the 7 graphs being partitioned. All are meshes in 2 or 3 dimensional space, rather than completely general graphs. The first four are 2-dimensional, and the last two are 3 dimensional. The fifth graph, PWT, is sometimes called "two-and-a-half dimensional", because it is a thin ("almost 2D") surface lying in 3D space. The last two columns give the number of vertices and edges in the graphs. The columns labeled "Grading" says how much larger the longest edges in the graph are than the shortest edges. For example, TRIANGLE is a regular tesselation of the place by identical equilateral triangles, so its grading is 1. AIRFOIL2 is similar to the NASA Airfoil we have seen so often, which has some large and some tiny triangles, the largest 1.3e5 times larger than the smallest. ```Mesh Description Mesh Type Grading Vertices Edges TAPIR Cartoon animal 2-D acute triangles 8.5e4 1024 2846 AIRFOIL2 Three-element airfoil 2-D triangles 1.3e5 4720 13722 TRIANGLE Equilateral triangle 2-D triangles/same size 1.0e0 5050 14850 AIRFOIL3 Four-element airfoil 2-D triangles 3.0e4 15606 45878 PWT Pressurized wind tunnel Thin shell in 3-space 1.3e2 36519 144794 BODY Automobile body 3-D volumes and surfaces 9.5e2 45087 163734 WAVE Space around airplane 3-D volumes and surfaces 3.9e5 156317 1059331 Table 1: Test problems. "Grading" is the ratio of longest to shortest edge lengths. ``` Table 2 below describes the quality of the partitioning into two subgraphs obtained by four algorithms. Quality is measured by the number of edges crossing the partition boundary, where fewer is better. The four algorithms are 1. Spectral -- use the second eigenvector as described above 2. Inertial Partitioning -- coordinate bisection as described in Lecture 18 3. Default Random Circle -- as described in Lecture 18. Recall that this is a randomized algorithm, that involves picking a random circle. The more circles chosen, the better the partitioning (i.e. the fewer edges cut). In this default implementation, a small, fixed number of circles are chosen. 4. Best Random Circle -- In this case circles are repeatedly chosen until no more progress is made. It is more expensive than the Default Random Circle algorithm just described, but gives a better partitioning. From the table, one sees that the methods are largely comparable, with spectral somewhat better on the largest graphs. (Spectral is also much more expensive to run, although we present no data on this here.) Also, we see that our intuition, that a 2D mesh-like graph with n nodes should have a partition with just sqrt(n) edge crossings, is approximately true. Also, our intuition that a 3D mesh should have n^(2/3) edge crossings is also approximately true. ```Mesh Spectral Inertial Default Best Partitioning Random Circle Random Circle TAPIR 59 55 37 32 AIRFOIL2 117 172 100 93 TRIANGLE 154 142 144 142 AIRFOIL3 174 230 152 148 PWT 362 562 529 499 BODY 456 953 834 768 WAVE 13706 9821 10377 9773 Table 2: Number of edge crossings for two-way partitions ``` Finally, we use the partitioning algorithms 7 times recursively in order to partition the graphs into 2^7 = 128 separate partitions, which one would do on a 128-processor machine. Again, the quality of the partitions is largely comparable. ```Mesh Spectral Inertial Default Partitioning Random Circle TAPIR 1278 1387 1239 AIRFOIL2 2826 3271 2709 TRIANGLE 2989 2907 2912 AIRFOIL3 4893 6131 4822 PWT 13495 14220 13769 BODY 12077 22497 19905 WAVE 143015 162833 145155 Table 3: Number of edge crossings for 128-way partitions. ``` ## Applying Spectral Bisection to DNA Sequencing The following material is based on "A spectral algorithm for the seriation problem", by J. Atkins, E. Boman and B. Hendrickson (bah@cs.sandia.gov), submitted to FOCS 95. Here is a very simple version of the DNA sequencing problem. A molecule of DNA is a very long string consisting of a particular sequence of amino acids chosen from a set of four, which may be called A, D, T and G. For example, one might have the sequence ADDTGADTDGAGADTDG, but many millions long. The sequencing problem is to determine this sequence for a given molecule of DNA. Current biochemical technology permits the following algorithm to be used. One can break up this long DNA string into a great many shorter fragments, which can be separated according to their amino acid sequences. The goal is to represent the original DNA as a sequence of these possibly overlapping fragments in some order. Given the sequences making up each fragment, one then knows the sequence making up the original DNA. To extract this representation in terms of fragments, the following experiments are performed. Each fragment F is allowed to bond to the DNA at a point P where there amino acid sequences match. This point is called a probe, and is typically a set of amino acids at one end of the fragment. If the probe is long enough, it will match at a unique point along the DNA. One records this information in a matrix B, which has one row for each fragment and one column for each probe, by putting a 1 at entry (F,P). One can perform the same kind of experiment to see which other fragments bond to fragment F at the same probe P, implying that they overlap. Each such bond between fragment F' and fragment F at probe P is recorded by storing a 1 at location (F',P) of the matrix B. In this way, looping through all the fragments, the matrix B is eventually filled in with ones (and zeros elsewhere), where B(F,P)=1 means that fragment F matches the DNA at probe P. This is shown below, where we have labeled the fragments in sorted order from left to right, and the probes in sorted order from left to right. Notice that when the probes and fragments are sorted, B is a band matrix, which means all its nonzero entries are near the diagonal. In practice, one constructs Bp with probes and fragments in some random order, as shown below. The DNA Sequencing problem is to find the ordering of the rows and ordering of the columns of Bp which expose the underlying band matrix B, because this will say what in what order the fragments appear along the DNA. If there were no errors in Bp, this could be done by procedure like breadth first search. But in practice, the laboratory procedure for determining entries of Bp is quite error prone, so we need a method for making Bp "close to" a band matrix in some sense. This is where spectral bisection comes in. Let G=(N,E) be an undirected graph, and L(G) its Laplacian. Let N = N- U N+ be an arbitrary partition of the nodes, and let x be a column vector, where x(i) = +1 if i is in N+, and x(i) = -1 if i is in N-. In Lemma 1 of the last lecture, we showed that the number of edges connecting N+ and N- was equal to ``` .25 * sum_{edges e=(i,j)} (x(i) - x(j))^2 ``` Therefore, the partition N = N- U N+ which minimizes the number of connecting edges is given by the solution x of the minimization problem ``` min_{x(i) = +1 or -1, sum_i x(i) = 0} .25 * sum_{edges e=(i,j)} (x(i) - x(j))^2 ``` The spectral partitioning algorithm involved solving the following approximation: ``` min_{sum_i v(i)^2 = |N|, sum_i v(i) = 0} .25 * sum_{edges e=(i,j)} (v(i) - v(j))^2 ``` and then choosing x(i) = sign(v(i)). One can think of this algorithm as embedding the graph G into the real axis, putting node i at location v(i). If e=(i,j) is an edge, then we draw a line segment from v(i) to v(j), which has length |v(i) - v(j)|. The spectral bisection algorithm chooses this embedding into the real axis so as to minimize the sum of squares of the lengths of these line segments, subject to the constraints sum_i v(i)^2 = |N|, and sum_i v(i)=0. Suppose we start with a symmetric matrix H, form it graph G(H), and apply the above algorithm, yielding a second eigenvector v of L(G(H)). Let P be a permutation matrix, i.e. the identity matrix with its columns permuted, such that the entries of P*v, which are the the entries of v permuted the same way, are in sorted order. Now form N = P*H*P'. N is the matrix H with its rows and columns reordered in the same way that sorts v. The fact that the sum of all (v(i)-v(j))^2 is minimized, means that there are few edges connecting distant v(i) and v(j). In other words, if v(i) and v(j) are widely separated in the sorted list of entries of v, they are unlikely to have an edge connecting them. In the matrix N, this means that there are few nonzero entries far from the diagonal, because these would correspond to an edge from a v(i) to a v(j) widely separated in the sorted list. In other words, N is close to a band matrix. For example, consider the matrix M = L(G), where G is a chain of n nodes. M is a tridiagonal matrix as shown in the figure below. Now perform a random permutation of the rows and columns of M to get H. H has nonzeros uniformly distributed off the diagonal. Apply spectral partitioning to H as described above to get N. As shown below, N is tridiagonal again. (The label nz under each graph is the number of nonzeros entries in the matrix.) This bandwidth narrowing property is what we need to reorder the rows and columns of Bp to make it a band matrix. But reordering Bp requires two permutations, one for the rows and one for the columns, while spectral bisection computes just one permutation. We get around this as follows. We can write Bp = Pf*B*Pp', where Pf and Pp are two unknown permutation matrices we wish to compute; Pf shuffles the rows of B, and Pp shuffles the columns. Now consider the symmetric matrix ``` Tp = Bp'*Bp = (Pf*B*Pp')'*(Pf*B*Pp') = Pp*(B'*B)*Pp' ``` Note that Tp only depends on Pp and B, but not on Pf. If B is a band matrix, one can confirm that B'*B is too, although with a larger bandwidth. Thus, Pp can be determined just by using spectral bisection to find the single permutation of rows and columns of Tp that makes Pp'*Tp*Pp (nearly) a band matrix. Similarly one can apply spectral bisection to ``` Tf = Bp*Bp' = (Pf*B*Pp')*(Pf*B*Pp')' = Pf*(B*B')*Pf' ``` to independently determine Pf. An example is shown below, where we start with a perfect band matrix and add a few other random entries to get B, and randomly permute its rows and columns to get Bp. The bottom row of three matrices shows Tp, Tf and Bp after permuting them to make them close to band matrices. One can see that the construction is far from "perfect", and in fact degrades more if the random entries added to B are farther from the diagonal. DNA sequencing remains a hard problem. ## Software for Graph Partitioning Software is available for all the partitioning methods described here from various sources. Chaco was written by Bruce Hendrickson and Robert Leland at Sandia National Lab. It is a serial library containing implementations of many of the methods we have discussed. A manual for v. 1.0 is in the class reader ("The Chaco User's Guide, Version 1.0", Technical Report SAND93-2339, Sandia National Labs, Albuquerque NM 1993). It is available electronically from the authors (rwlelan@cs.sandia.gov or bahendr@cs.sandia.gov). It is freely available for research purposes. JOSTLE was written by C. Walshaw, M. Cross, and M. Everett at the University of Greenwich in the UK. It is available free to researchers, but a license agreement is involved. Further information is available here, and the license agreement is here. The software for the geometric partitioning algorithm by Gilbert, Miller, Teng, Vavasis and Thurston is written in matlab, and can be obtained by anonymous ftp from machine ftp.parc.xerox.com as file /pub/gilbert/mashpart.uu. The above software consists of libraries to which one passes a graph, and is returned a partitioning. There have also been attempts to embed graph partitioning in a higher level language, so as shield the user from having to construct the graph, partition it, (re)distribute the data across the machine, and set up the communication. The goal of this work is to be able to take an existing serial code which traverses a sparse data structure, and modify the language and compiler to permit the user to say 1. Inspect the following section of code (a loop nest, say), and determine the underlying graph G describing how data items depend on other data items, partition G, and redistribute the data accordingly. 2. Execute the code with the redistributed data. The system we will describe is called PARTI, which stands for Parallel Automated Runtime Toolbox at ICASE, where ICASE is a NASA computing research laboratory. The primary author is Joel Saltz, currently at the University of Maryland at College Park. This materal is taken from "Distributed Memory Compiler Methods for Irregular Problems -- Data Copy Reuse and Runtime Partitioning," by R. Das, R. Ponnusamy, J. Saltz and D. Mavripilis, ICASE Report 91-73, NASA Langley Research Center, Hampton VA, 1991. This report is in the class reader. PARTI is an extension of HPF (High Performance Fortran), and uses the features of HPF for describing array layouts across processors. We begin by reviewing data layouts. At the end of Lecture 5, we discussed data layout in CM Fortran, where for example the declaration (KEYWORDS are capitalized) ``` REAL a(64,8), b(64,8), c(64,8) CMF\$ LAYOUT a( :NEWS, :SERIAL ), b( :NEWS, :SERIAL ), c( :SERIAL, :NEWS ) ``` indicated that A(i,j) was to be stored in the j-th memory location of processor i, the same for B(i,j), and that C(i,j) was instead to be stored in the i-th memory location of processor j. This would mean that the assignment A=B could occur in parallel without communication, but that A=C would require a great deal of communication. These simple layout directive are not enough for all purposes. In the beginning of Lecture 13, we discussed the more complicated data layouts required to do Gaussian elimination (or other dense linear algebra problems) efficiently on a distributed memory machine, and said the the first four of the following layouts were declarable within the HP Fortran language: Here, very briefly, is how HPF permits users to declare these kinds of layouts. Rather than saying how each matrix entry maps to a processor location, two levels of indirection are used. The first level declaration declares how many of the available processors are to be used in the layout. A simple example is the following, which declares mygrid to be a linear array of 4 processors. ``` PROCESSOR mygrid(4) ``` The second level declares a template, or "virtual array", and says how to lay it out on mygrid. For example ``` TEMPLATE template_blocked(100),template_cyclic(100) DISTRIBUTE template_blocked(BLOCK) ONTO mygrid DISTRIBUTE template_cyclic(CYLIC) ONTO mygrid ``` declares that template_blocked(0:24) is mapped to processor 0, template_blocked(25:49) is mapped to processor 1, and so on, in general with template_blocked(i) mapping to processor floor(i/25). Also, template_cyclic(i) is mapped to processor i mod 4. Block cyclic layouts are also available. Multi-dimensional arrays can have each subscript mapped independently, as preferred for Gaussian Elimination. A template has no memory allocated for it; it just describes a layout. The final level of declaration actually allocates memory for arrays. For example ``` REAL a(100), b(100), c(100) ALIGN a(i) WITH template_block(i) ALIGN b(i) WITH template_block(i) ALIGN c(i) WITH template_cyclic(i) ``` declares 3 arrays of 100 entries each. a(i) and b(i) are declared to be stored at the same place as the template entry template_block(i), in this case floor(i/25). However template_block is DISTRIBUTEd, a(i) and b(i) will always be on the same processor. c(i) is declared to be stored at the same place as template_cyclic(i), that is i mod 4. The reason for these levels of indirection is that one can independently control the amount of parallelism (via PROCESSOR), the layout (via DISTRIBUTE) and which variables are local with which other (via ALIGN). The same mechanism can be used for more irregular layouts, but we need one more level of indirection to specify the irregularity. For example ``` TEMPLATE irregular(100) INTEGER map(100) DATA map/3,2,2,1,1,1,3,0,1,2,... / ... 100 values from 0 to 3 DISTRIBUTE irregular(map) ONTO mygrid REAL d(100) ALIGN d(i) WITH irregular(i) ``` These declarations specify that irregular(i) is mapped to processor map(i), i.e. irregular(1) is mapped to processor map(1)=3, irregular(2) is mapped to processor map(2) = 2, and so on. The ALIGN statement in turn says d(1) is stored on processor 3, d(2) is stored on processor 2, and so on. In this example, map is specified at compile-time, and the decision about where to store d(i) is specified at compile-time. This is very limiting, since we probably won't know the actual data structure we need to partition until run-time. The extensions in PARTI to this approach are to allow map to be computed at run time (by examining some user specified loops and doing graph partitioning), and DISTRIBUTE to be executed at run-time as well, in effect recompiling the code at run-time. This is likely to be quite expensive, and so is done only when the user wants to. Here is an example, taken from a computational fluid dynamic (CFD) application. The data structure is a two-dimension triangular mesh, made up of nodes (numbered in blue), edges (numbered in black) and faces (numbered in red). The mesh data is stored in two arrays. The edge_list array stores a pair of nodes for each each edge: the nodes numbers for the i-th edge are stored at edge_list(i) and edge_list(i + n_edge), where n_edge is the number of edges. The face_list array stores a triple of nodes for each face: the nodes for the i-th face are stored at face_list(i), face_list(i + n_face), and face_list(i + 2*n_face), as shown below. For example, face 1 has corners at nodes 1, 2 and 3. face 2 at nodes 2, 3 and 4, and so on. The original sequential program has two data arrays, x and y, which store data associated with each node. In other words x(i) and y(i) are data about the fluid flow at node i. The algorithm has two loops. Loop L1 below loops over all edges, and for each edge updates the data at both nodes determining that edge. Loop L2 loops over all faces, and for each face updates the data at the three nodes determining the face. The functions foo1, foo2, etc, are simple scalar functions of their scalar arguments, whose details do not concern us. ``` REAL x(n_node), y(n_node) ... C Loop over all edges L1: DO i = 1, n_edge n1 = edge_list(i) n2 = edge_list(i + n_edge) y(n1) = foo1(y(n1),y(n2),x(n1),x(n2)) y(n2) = foo2(y(n1),y(n2),x(n1),x(n2)) END DO C Loop over all faces L2: DO i = 1, n_face n1 = face_list(i) n2 = face_list(i + n_face) n3 = face_list(i + 2*n_face) y(n1) = foo4(y(n1),...,x(n3)) y(n2) = foo5(y(n1),...,x(n3)) y(n3) = foo6(y(n1),...,x(n3)) END DO ``` Here is the parallel version of this program using PARTI: ``` REAL x(n_node), y(n_node) TEMPLATE coupling(n_node) DISTRIBUTE coupling(BLOCK) ONTO mygrid ALIGN x(i), y(i) WITH coupling(i) ... C Decide whether to redistribute data IF (time_to_remap) THEN DISTRIBUTE coupling(IMPLICIT USING L1) END IF C Loop over all edges IMPLICITMAP(x,y) L1 L1: DO i = 1, n_edge n1 = edge_list(i) n2 = edge_list(i + n_edge) y(n1) = foo1(y(n1),y(n2),x(n1),x(n2)) y(n2) = foo2(y(n1),y(n2),x(n1),x(n2)) END DO C Loop over all faces L2: DO i = 1, n_face n1 = face_list(i) n2 = face_list(i + n_face) n3 = face_list(i + 2*n_face) y(n1) = foo4(y(n1),...,x(n3)) y(n2) = foo5(y(n1),...,x(n3)) y(n3) = foo6(y(n1),...,x(n3)) END DO ``` Initially, the TEMPLATE coupling is laid out in a blocked fashion onto the processor grid mygrid. Arrays x and y are aligned with coupling. Later, perhaps after the arrays edge_list and face_list have been set up, the user sets time_to_remap to true, and executes the DISTRIBUTE statement following the IF statement. At this point, the system begins the "Inspection phase": It will examine loop L1, which is identified later with the IMPLICITMAP statement. The arguments x and y of IMPLICITMAP tell the system to compute the graph G of data dependencies among the references to x and y in the subsequent loop L1. This is done by executing loop L1 "symbolically", i.e. running through thg loop from i=1 to n_edge, computing the subscripts n1 = edge_list(i) and n2 = edge_list(i + n_edge), seeing that y(n1) depends on y(n1), y(n2), x(n1), and x(n2), and adding nodes n1 and n2, and edge (n1,n2), to graph G, which is initially empty. Functions foo1 and foo2 are not evaluated, and y(n1) and y(n2) are not changed. After computing G, a graph partitioning routine is called to break G into as many pieces as there are processors in mygrid. The partitioning information is stored in the TEMPLATE coupling, with coupling(i) = j if the partition algorithm puts node i onto processor j. All arrays ALIGNed with coupling (namely x and y) are redistributed according to the newly upated coupling. Finally, all parts of the program that reference arrays x or y are "recompiled" to insert the necessary communications to continue to access the data they need. After completing the "IF (time_to_remap)" block, one reaches loop L1. At this point, the "Execute phase", the newly recompiled code is executed and array y updated. Loop L2 is executed similarly.

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# Properties Label 450.6.c.p Level $450$ Weight $6$ Character orbit 450.c Analytic conductor $72.173$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$ # Related objects ## Newspace parameters Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal) ## Newform invariants Self dual: no Analytic conductor: $$72.1727189158$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{4081})$$ Defining polynomial: $$x^{4} + 2041 x^{2} + 1040400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$ ## $q$-expansion Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form. $$f(q)$$ $$=$$ $$q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} +O(q^{10})$$ $$q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} + ( -270 - 2 \beta_{3} ) q^{11} + ( -445 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -200 - 4 \beta_{3} ) q^{14} + 256 q^{16} + ( 246 \beta_{1} + 10 \beta_{2} ) q^{17} + ( -296 - 5 \beta_{3} ) q^{19} + ( 1080 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -1830 \beta_{1} - 10 \beta_{2} ) q^{23} + ( -1780 - 8 \beta_{3} ) q^{26} + ( 800 \beta_{1} + 16 \beta_{2} ) q^{28} + ( 2850 - 2 \beta_{3} ) q^{29} + ( -2854 + 5 \beta_{3} ) q^{31} -1024 \beta_{1} q^{32} + ( 984 + 40 \beta_{3} ) q^{34} + ( 5650 \beta_{1} - 24 \beta_{2} ) q^{37} + ( 1184 \beta_{1} + 20 \beta_{2} ) q^{38} + ( -7710 + 30 \beta_{3} ) q^{41} + ( -3160 \beta_{1} - 41 \beta_{2} ) q^{43} + ( 4320 + 32 \beta_{3} ) q^{44} + ( -7320 - 40 \beta_{3} ) q^{46} + ( -3900 \beta_{1} - 100 \beta_{2} ) q^{47} + ( -22422 - 100 \beta_{3} ) q^{49} + ( 7120 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 13914 \beta_{1} + 70 \beta_{2} ) q^{53} + ( 3200 + 64 \beta_{3} ) q^{56} + ( -11400 \beta_{1} + 8 \beta_{2} ) q^{58} + ( 25260 + 52 \beta_{3} ) q^{59} + ( -14563 + 90 \beta_{3} ) q^{61} + ( 11416 \beta_{1} - 20 \beta_{2} ) q^{62} -4096 q^{64} + ( 48700 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -3936 \beta_{1} - 160 \beta_{2} ) q^{68} + ( -3090 - 126 \beta_{3} ) q^{71} + ( -16450 \beta_{1} + 120 \beta_{2} ) q^{73} + ( 22600 - 96 \beta_{3} ) q^{74} + ( 4736 + 80 \beta_{3} ) q^{76} + ( 86958 \beta_{1} + 370 \beta_{2} ) q^{77} + ( -3956 + 360 \beta_{3} ) q^{79} + ( 30840 \beta_{1} - 120 \beta_{2} ) q^{82} + ( 81732 \beta_{1} - 100 \beta_{2} ) q^{83} + ( -12640 - 164 \beta_{3} ) q^{86} + ( -17280 \beta_{1} - 128 \beta_{2} ) q^{88} + ( 82320 - 336 \beta_{3} ) q^{89} + ( -95708 - 545 \beta_{3} ) q^{91} + ( 29280 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -15600 - 400 \beta_{3} ) q^{94} + ( 26215 \beta_{1} + 364 \beta_{2} ) q^{97} + ( 89688 \beta_{1} + 400 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 64q^{4} + O(q^{10})$$ $$4q - 64q^{4} - 1080q^{11} - 800q^{14} + 1024q^{16} - 1184q^{19} - 7120q^{26} + 11400q^{29} - 11416q^{31} + 3936q^{34} - 30840q^{41} + 17280q^{44} - 29280q^{46} - 89688q^{49} + 12800q^{56} + 101040q^{59} - 58252q^{61} - 16384q^{64} - 12360q^{71} + 90400q^{74} + 18944q^{76} - 15824q^{79} - 50560q^{86} + 329280q^{89} - 382832q^{91} - 62400q^{94} + O(q^{100})$$ Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2041 x^{2} + 1040400$$: $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 1021 \nu$$$$)/1020$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3061 \nu$$$$)/340$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{2} + 6123$$ $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6123$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-1021 \beta_{2} + 9183 \beta_{1}$$$$)/6$$ ## Character values We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$. $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ ## Embeddings For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below. For more information on an embedded modular form you can click on its label. Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$ 199.1 31.4414i − 32.4414i 32.4414i − 31.4414i 4.00000i 0 −16.0000 0 0 241.648i 64.0000i 0 0 199.2 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0 199.3 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0 199.4 4.00000i 0 −16.0000 0 0 241.648i 64.0000i 0 0 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles ## Inner twists Char Parity Ord Mult Type 1.a even 1 1 trivial 5.b even 2 1 inner ## Twists By twisting character orbit Char Parity Ord Mult Type Twist Min Dim 1.a even 1 1 trivial 450.6.c.p 4 3.b odd 2 1 450.6.c.q 4 5.b even 2 1 inner 450.6.c.p 4 5.c odd 4 1 450.6.a.y 2 5.c odd 4 1 450.6.a.bf yes 2 15.d odd 2 1 450.6.c.q 4 15.e even 4 1 450.6.a.ba yes 2 15.e even 4 1 450.6.a.bd yes 2 By twisted newform orbit Twist Min Dim Char Parity Ord Mult Type 450.6.a.y 2 5.c odd 4 1 450.6.a.ba yes 2 15.e even 4 1 450.6.a.bd yes 2 15.e even 4 1 450.6.a.bf yes 2 5.c odd 4 1 450.6.c.p 4 1.a even 1 1 trivial 450.6.c.p 4 5.b even 2 1 inner 450.6.c.q 4 3.b odd 2 1 450.6.c.q 4 15.d odd 2 1 ## Hecke kernels This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(450, [\chi])$$: $$T_{7}^{4} + 78458 T_{7}^{2} + 1171624441$$ $$T_{11}^{2} + 540 T_{11} - 74016$$ ## Hecke characteristic polynomials $p$ $F_p(T)$ $2$ $$( 16 + T^{2} )^{2}$$ $3$ $$T^{4}$$ $5$ $$T^{4}$$ $7$ $$1171624441 + 78458 T^{2} + T^{4}$$ $11$ $$( -74016 + 540 T + T^{2} )^{2}$$ $13$ $$2612129881 + 689882 T^{2} + T^{4}$$ $17$ $$13049318163456 + 7466832 T^{2} + T^{4}$$ $19$ $$( -830609 + 592 T + T^{2} )^{2}$$ $23$ $$104976000000 + 14043600 T^{2} + T^{4}$$ $29$ $$( 7975584 - 5700 T + T^{2} )^{2}$$ $31$ $$( 7227091 + 5708 T + T^{2} )^{2}$$ $37$ $$115919589427216 + 106156808 T^{2} + T^{4}$$ $41$ $$( 26388000 + 15420 T + T^{2} )^{2}$$ $43$ $$2678667905710801 + 143454098 T^{2} + T^{4}$$ $47$ $$123960326400000000 + 765000000 T^{2} + T^{4}$$ $53$ $$185703196271616 + 747142992 T^{2} + T^{4}$$ $59$ $$( 538752384 - 50520 T + T^{2} )^{2}$$ $61$ $$( -85423931 + 29126 T + T^{2} )^{2}$$ $67$ $$5623345588934394721 + 4744041122 T^{2} + T^{4}$$ $71$ $$( -573561504 + 6180 T + T^{2} )^{2}$$ $73$ $$66716358684010000 + 1599000200 T^{2} + T^{4}$$ $79$ $$( -4744428464 + 7912 T + T^{2} )^{2}$$ $83$ $$39851820386783870976 + 14094819648 T^{2} + T^{4}$$ $89$ $$( 2630025216 - 164640 T + T^{2} )^{2}$$ $97$ $$17465874450640370881 + 11107343618 T^{2} + T^{4}$$

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# 3D Stereotriangulation Using Parallel Cameras 1. Jul 30, 2011 ### tsa256 Hello everyone, I recently began a project which using 2 parallel cameras in order to determine the 3D spacial positioning of a red ball relative to the left camera. Here is my current setup; Two cameras that lie on the same plane with parallel optical axis. (http://www.dis.uniroma1.it/~iocchi/stereo/fig/triang.gif from this webpage http://www.dis.uniroma1.it/~iocchi/stereo/triang.html) Distance between cameras is 317mm Resolution of images is 960x720 Effective focal length for both cameras is 2mm. (But the cameras are equipped with an auto focus feature and although not active during acquisition the cameras are set to a constant focus setting before acquisition. Would this change the focal length, also there are no units on the focus setting so I am unsure of how much I have change the focal length if at all?) X, Y, and Z locations are in mm and relative to the left camera and here is my math, ( assuming I have already calculated the X and Y of the red ball in each image) Z_red=(((317*2)/((X_Position_of_Circle_in_Left_Camera_Image-0.5*960)-(X_Position_of_Circle_in_Right_Image-0.5*960)))) X_red=(((X_Position_of_Circle_in_Left_Image-0.5*960)*Z_red)/2) Y_red=(((Y_Position_of_Circle_in_left_Image-0.5*720)*Z_red)/2) Unfortunately the math does not work. I have tested it numerous time by placing the red ball a known distance from the left camera and it does output the correct distance. Why could that be? I would greatly appreciate any help, if someone could post correct equations that would be phenomenal. A friend of mine recommended adding constants (kx,ky,kz) to the calculations in order to scale the values to the correct position. This does produce better results however I am unsure of whether it is mathematically correct. Z_red=(((317*2)/((X_Position_of_Circle_in_Left_Camera_Image-0.5*960)-(X_Position_of_Circle_in_Right_Image-0.5*960))))*kz X_red=(((X_Position_of_Circle_in_Left_Image-0.5*960)*Z_red)/2)*kx Y_red=(((Y_Position_of_Circle_in_left_Image-0.5*720)*Z_red)/2)*ky Any help would be greatly appreciated! Last edited by a moderator: Apr 26, 2017

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/ / Article Special Issue ## Applications of Fixed Point and Approximate Algorithms View this Special Issue Research Article | Open Access Volume 2012 |Article ID 161470 | 11 pages | https://doi.org/10.1155/2012/161470 # The Existence of Fixed Points for Nonlinear Contractive Maps in Metric Spaces with 𝑀 -Distances Accepted04 Dec 2011 Published16 Feb 2012 #### Abstract Some fixed point theorems for -contractive maps and -contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature. #### 1. Introduction and Preliminaries Branciari [1] established a fixed point result for an integral type inequality, which is a generalization of Banach contraction principle. Kada et al. [2] introduced and studied the concept of -distance on a metric space. They give examples of -distances and improved Caristi’s fixed point theorem, Ekeland’s -variational’s principle, and the nonconvex minimization theorem according to Takahashi (see many useful examples and results on -distance in [2–5] and in references therein). Kada et al. [2] defined the concept of -distance in a metric space as follows. Definition 1.1 (see [2]). Let be a metric space endowed with a metric . A function is called a -distance on if it satisfies the following properties:(1) for any ,(2) is lower semicontinuous in its second variable, that is, if and in then ,(3)for each , there exists such that and imply . We denote by the set of functions satisfying the following hypotheses:(c1) is continuous and nondecreasing,(c2) if and only if . We denote by the set of functions satisfying the following hypotheses:(h1) is right continuous and nondecreasing,(h2) for all . Let be a -distance on metric space , and . A map from into itself is a ()-contractive map on if for each , . The following lemmas are used in the next section. Lemma 1.2 (see [3]). If , then for each , and if and , then . Lemma 1.3 (see [2]). Let be a metric space and let be a -distance on .(i)If is a sequence in such that , then . In particular, if , then .(ii)If for any , where and are sequences in converging to 0, then converges to .(iii)Let be a -distance on metric space and a sequence in such that for each there exist such that implies (or ), then is a Cauchy sequence. Note that if and , then and, by Lemma 1.3, . In [3], Razani et al. proved a fixed point theorem for -contractive mappings, which is a new version of the main theorem in [1], by considering the concept of the -distance. The main aim of this paper is to present some generalization fixed point Theorems by Kada et al. [2], Hicks and Rhoades [6] and several other results with respect to ()-contractive maps on a complete metric space. #### 2. -Contractive Maps In the next theorem we state one of the main results of this paper generalizing Theorem 4 of [2]. In what follows, we use to denote the composition of with . Theorem 2.1. Let be a -distance on complete metric space and . Suppose is a map that satisfies for each and that for every with . Then there exists such that . Moreover, if , then . Proof. Fix . Set with . Then by (2.1) thus and Lemma 1.2 implies and similarly Now we proof that is a Cauchy sequence. By triangle inequality, continuity of and (2.4), we have as and so which concludes By induction, for any we have So, by Lemma 1.3, is a Cauchy sequence, and since is complete, there exists such that in. Now we prove that is a fixed point of . From (2.8), for each , there exists such that implies but and is lower semicontinuous, thus Therefore, . Set and we have Now, assume that . Then by hypothesis, we haveas by (2.4) and (2.10). This is a contradiction. Hence . If , we have This is a contradiction. So , and by hypothesis . Here we give a simple example illustrating Theorem 2.1. In this example, we will show that Theorem  4 in [2] cannot be applied. Example 2.2. Let , which is a complete metric space with usual metric of reals. Moreover, by defining , is a -distance on . Let be a map as , . Suppose is a continuous and strictly nondecreasing map and , for any . We have and so there is not any such that , and hence Theorem 4 in [2] dose not work. But because for any we have . Also for any we have . So for arbitrary , , hence is satisfied in Theorem 2.1. We note that 0 is a fixed point for . The next examples show the role of the conditions (2.1) and (2.2). Example 2.3. Let , , and define by , where . Set and for all . Let us define by and if . We have If , then and hence (2.1) holds. Now, we remark that , and Thus, the condition (2.2) is not satisfied, and there is no with . In this case we observe that Theorem 2.1 is invalid without condition (2.2). Example 2.4. Let , , , and set . Let be as Example 2.3. Let us define by and if . Clearly, has no fixed point in . Now, for each and that for every with , so condition (2.2) is satisfied. But, for , for any . Hence, condition (2.1) dose not hold. We note that Theorem 2.1 dose not work without condition (2.1). Suppose is Lebesgue-integrable mapping which is summable and , for each . Now, in the next corollary, set and , where . Then, and . Hence we can conclude the following corollary as a special case. Corollary 2.5. Let be a selfmap of a complete metric space satisfying for all . Suppose that with . Then there exists a such that . Note that Corollary 2.5 is invalid without condition (2.20). For example, take , which is a complete metric space with usual metric of reals. Define by and for . Set . It is easy to check that , for any ; however, for any and . Clearly, has got no fixed point in . Remark 2.6. From Theorem 2.1, we can obtain Theorem 4 in [2] as a special case. For this, in the hypotheses of Theorem 2.1, set and for all . Corollary 2.7. Let be a -distance on complete metric space , and . Suppose is a continuous mapping for into itself such that (2.1), is satisfied. Then there exists such that . Moreover, if , then . Proof. Assume that there exists with and . Then there exists a sequence such that as . Hence and as . Lemma 1.3 implies that as . Now by assumption and so as . By Lemma 1.2, as . We also have hence as . By Lemma 1.3, we conclude that converges to . Since is continuous, we have This is a contradiction. Therefore, if , then . So, Theorem 2.1 gives desired result. In Example 2.3, is satisfied in condition (2.1), but it is not continuous. So, the hypotheses in Corollary 2.7are not satisfied. We note that has no fixed point. It is an obvious fact that, if is a map which has a fixed point , then is also a fixed point of for every natural number . However, the converse is false. If a map satisfies for each , where denotes a set of all fixed points of , then it is said to have property [7, 8]. The following theorem extends and improves Theorem 2 of [7]. Theorem 2.8. Let be a complete metric space with -distance on . Suppose satisfies(i) or(ii)with strict inequality, and for all , . If , then has property . Proof. We shall always assume that , since the statement for is trivial. Let . Suppose that satisfies . Then, and so . Now from we have . Hence, by Lemma 1.3, we have , and . Suppose that satisfies . If , then there is nothing to prove. Suppose, if possible, that . Then a repetition of the argument for case leads to , that is a contradiction. Therefore, in all cases, and . The following theorem extends Theorem 2.1 of [6]. A function mapping into the real is -orbitally lower semicontinuous at if is a sequence in and implies that . Theorem 2.9. Let be a complete metric space with -distance on . Suppose and there exists an such that Then,(i) exists,(ii)(iii) if and only if is -orbitally lower semicontinuous at . Proof. Observe that and are immediate from the proof of Theorem 2.1. We prove . It is clear that impling is -orbitally lower semicontinuous at . and is -orbitally lower semicontinuous at implies So, . The mapping is orbitally lower semicontinuous at if implies that . In the following, we improve Theorem 2 of [9] that it is correct form Theorem 1 of [7]. Theorem 2.10. Let be a -distance on complete metric space and . Suppose is orbitally lower semicontinuous map on that satisfies for each . Then there exists such that . Moreover, if , then . Proof. Observe that the sequence is a Cauchy sequence immediate from the proof of Theorem 2.1 and so there exists a point in such that as . Since is orbitally lower semicontinuous at , we have . Now, we have and so . Similarly, . Hence, . By Theorem 2.1 we can conclude that if , then . The following example shows that Theorem  2 in [9] cannot be applicable. So our generalization is useful. Example 2.11. Let be a metric space with metric defined by ,, which is complete. We define by . Let be as defined before in Corollary 2.5 and ,. Assume that by for any . We have, , and so Theorem  2 in [9] dose not work. But for each . Hence by Theorem 2.10 there exists a fixed point for . We note that 0 is fixed point for . #### 3. -Contractive Maps In this section we obtain fixed points for ()-contractive maps (i.e., -contractive maps that for all , where ). In 1969, Kannan [10] proved the following fixed point theorem. Contractions are always continuous and Kannan maps are not necessarily continuous. Theorem 3.1 (see [10]). Let be a complete metric space. Let be a Kannan mapping on , that is, there exists such that for all . Then, has a unique fixed point in . For each , the iterative sequence converges to the fixed point. In the next theorem, we generalize this theorem as follows. Theorem 3.2. Let be a complete metric space. Let be a -Kannan mapping on , that is, there exists such that for all . Then, has a unique fixed point in . For each , the iterative sequence converges to the fixed point. Proof. Let and define for any , and set . Then, , and so Then, from the proof of Theorem 2.1, exists. From (3.4), we have Thus, , and so . Clearly, is unique. This completes the proof. The set of all subadditive functions in is denoted by . In the following theorems, we generalize Theorems 3.4 and 3.5 due to Suzuki and Takahashi [4]. Theorem 3.3. Let be a -distance on complete metric space and be a selfmap. Suppose there exists such that(i) for each ,(ii) for every with . Then has a fixed point in . Moreover, if is a fixed point of , then . Proof. Fix . Define and for every . Put . Then, . By hypothesis, since , we have for all . It follows that for all . Using the similar argument as in the proof of Theorem 2.1, we can prove that the sequence is Cauchy and so there exists such that as . Also, we have . Since we have and so . The proof is completed. Corollary 3.4. Let be a -distance on complete metric space and let be a continuous map. Suppose there exists such that for each . Then has a fixed point in . Moreover, if is a fixed point of , then . Proof. It suffices to show that for every with . Assume that there exists with and . Then there exists a sequence in such that . It follows that and as . Hence, . On the other hand, since and (3.9), we have and hence for all . Thus, as . Therefore, . Since is continuous, we have which is a contradiction. Therefore, using Theorem 3.3, . This completes the proof. Question 1. Can we generalize Theorems 3.2, 3.3, and Corollary 3.4 for ()-contractive maps? #### References 1. A. Branciari, β€œA fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002. 2. O. Kada, T. Suzuki, and W. Takahashi, β€œNonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996. View at: Google Scholar | Zentralblatt MATH 3. A. Razani, Z. Mazlumi Nezhad, and M. Boujary, β€œA fixed point theorem for w-distance,” Applied Sciences, vol. 11, pp. 114–117, 2009. View at: Google Scholar 4. T. Suzuki and W. Takahashi, β€œFixed point theorems and characterizations of metric completeness,” Topological Methods in Nonlinear Analysis, vol. 8, no. 2, pp. 371–382, 1996. View at: Google Scholar | Zentralblatt MATH 5. W.-S. Du, β€œFixed point theorems for generalized Hausdorff metrics,” International Mathematical Forum, vol. 3, no. 21–24, pp. 1011–1022, 2008. View at: Google Scholar | Zentralblatt MATH 6. T. L. Hicks and B. E. Rhoades, β€œA Banach type fixed-point theorem,” Mathematica Japonica, vol. 24, no. 3, pp. 327–330, 1979/80. View at: Google Scholar 7. B. E. Rhoades and M. Abbas, β€œMaps satisfying generalized contractive conditions of integral type for which $F\left(T\right)=F\left({T}^{n}\right)$,” International Journal of Pure and Applied Mathematics, vol. 45, no. 2, pp. 225–231, 2008. View at: Google Scholar | Zentralblatt MATH 8. G. S. Jeong and B. E. Rhoades, β€œMaps for which $F\left(T\right)=F\left({T}^{n}\right)$,” Fixed Point Theory and Applications, vol. 6, pp. 87–131, 2005. View at: Google Scholar 9. H. Lakzian and B. E. Rhoades, β€œMaps satisfying generalized contractive contractions of integral type for which $F\left(T\right)=F\left({T}^{n}\right)$,” submitted to International Journal of Pure and Applied Mathematical Sciences. View at: Google Scholar 10. R. Kannan, β€œSome results on fixed points. II,” The American Mathematical Monthly, vol. 76, pp. 405–408, 1969. Copyright © 2012 Hossein Lakzian and Ing-Jer Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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# Distance between Gwangju (KWJ) and Sacheon (HIN) Distance from Gwangju to Sacheon (Gwangju Airport – Sacheon Airport) is 71 miles / 115 kilometers / 62 nautical miles. ## How far is Sacheon from Gwangju? There are several ways to calculate distances between Gwangju and Sacheon. Here are two common methods: Vincenty's formula (applied above) • 71.483 miles • 115.041 kilometers • 62.117 nautical miles Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth. Haversine formula • 71.324 miles • 114.785 kilometers • 61.979 nautical miles The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points). ## Flight duration Estimated flight time from Gwangju Airport (KWJ) to Sacheon Airport (HIN) is 38 minutes. ## Time difference and current local times There is no time difference between Gwangju and Sacheon. Gwangju Sacheon KST KST ## Carbon dioxide emissions Estimated CO2 emissions per passenger is 35 kg (77 pounds). ## Airport information A Gwangju Airport City: Gwangju Country: South Korea IATA Code: KWJ ICAO Code: RKJJ Coordinates: 35°7′35″N, 126°48′32″E B Sacheon Airport City: Sacheon Country: South Korea IATA Code: HIN ICAO Code: RKPS Coordinates: 35°5′18″N, 128°4′12″E ## Gwangju to Sacheon flight path Shortest flight path between Gwangju Airport (KWJ) and Sacheon Airport (HIN). ## Frequent Flyer Miles Calculator Gwangju (KWJ) → Sacheon (HIN). Elite level bonus (%): Booking class bonus (%): Air miles: 71 Elite level bonus: 0 Booking class bonus: 0 ### In total Total frequent flyer miles: 71 Round trip?

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`pygeos.constructive`¶ `pygeos.constructive.``boundary`(geometry, **kwargs) Returns the topological boundary of a geometry. Parameters geometry : Geometry or array_like This function will raise for non-empty geometrycollections. Examples ```>>> boundary(Geometry("POINT (0 0)")) <pygeos.Geometry GEOMETRYCOLLECTION EMPTY> >>> boundary(Geometry("LINESTRING(0 0, 1 1, 1 2)")) <pygeos.Geometry MULTIPOINT (0 0, 1 2)> >>> boundary(Geometry("LINEARRING (0 0, 1 0, 1 1, 0 1, 0 0)")) <pygeos.Geometry MULTIPOINT EMPTY> >>> boundary(Geometry("POLYGON((0 0, 1 0, 1 1, 0 1, 0 0))")) <pygeos.Geometry LINESTRING (0 0, 1 0, 1 1, 0 1, 0 0)> >>> boundary(Geometry("MULTIPOINT (0 0, 1 2)")) is None True ``` `pygeos.constructive.``buffer`(geometry, radius, quadsegs=8, cap_style='round', join_style='round', mitre_limit=5.0, single_sided=False, **kwargs) Computes the buffer of a geometry for positive and negative buffer radius. The buffer of a geometry is defined as the Minkowski sum (or difference, for negative width) of the geometry with a circle with radius equal to the absolute value of the buffer radius. The buffer operation always returns a polygonal result. The negative or zero-distance buffer of lines and points is always empty. Parameters geometry : Geometry or array_like width : float or array_like Specifies the circle radius in the Minkowski sum (or difference). Specifies the number of linear segments in a quarter circle in the approximation of circular arcs. cap_style : {‘round’, ‘square’, ‘flat’} Specifies the shape of buffered line endings. ‘round’ results in circular line endings (see `quadsegs`). Both ‘square’ and ‘flat’ result in rectangular line endings, only ‘flat’ will end at the original vertex, while ‘square’ involves adding the buffer width. join_style : {‘round’, ‘bevel’, ‘sharp’} Specifies the shape of buffered line midpoints. ‘round’ results in rounded shapes. ‘bevel’ results in a beveled edge that touches the original vertex. ‘mitre’ results in a single vertex that is beveled depending on the `mitre_limit` parameter. mitre_limit : float Crops of ‘mitre’-style joins if the point is displaced from the buffered vertex by more than this limit. single_sided : bool Only buffer at one side of the geometry. Examples ```>>> buffer(Geometry("POINT (10 10)"), 2, quadsegs=1) <pygeos.Geometry POLYGON ((12 10, 10 8, 8 10, 10 12, 12 10))> >>> buffer(Geometry("POINT (10 10)"), 2, quadsegs=2) <pygeos.Geometry POLYGON ((12 10, 11.4 8.59, 10 8, 8.59 8.59, 8 10, 8.59 11.4, 10 12, 11.4 11.4, 12 10))> >>> buffer(Geometry("POINT (10 10)"), -2, quadsegs=1) <pygeos.Geometry POLYGON EMPTY> >>> line = Geometry("LINESTRING (10 10, 20 10)") >>> buffer(line, 2, cap_style="square") <pygeos.Geometry POLYGON ((20 12, 22 12, 22 8, 10 8, 8 8, 8 12, 20 12))> >>> buffer(line, 2, cap_style="flat") <pygeos.Geometry POLYGON ((20 12, 20 8, 10 8, 10 12, 20 12))> >>> buffer(line, 2, single_sided=True, cap_style="flat") <pygeos.Geometry POLYGON ((20 10, 10 10, 10 12, 20 12, 20 10))> >>> line2 = Geometry("LINESTRING (10 10, 20 10, 20 20)") >>> buffer(line2, 2, cap_style="flat", join_style="bevel") <pygeos.Geometry POLYGON ((18 12, 18 20, 22 20, 22 10, 20 8, 10 8, 10 12, 18 12))> >>> buffer(line2, 2, cap_style="flat", join_style="mitre") <pygeos.Geometry POLYGON ((18 12, 18 20, 22 20, 22 8, 10 8, 10 12, 18 12))> >>> buffer(line2, 2, cap_style="flat", join_style="mitre", mitre_limit=1) <pygeos.Geometry POLYGON ((18 12, 18 20, 22 20, 21.8 9, 21 8.17, 10 8, 10 12, 18 12))> >>> square = Geometry("POLYGON((0 0, 10 0, 10 10, 0 10, 0 0))") >>> buffer(square, 2, join_style="mitre") <pygeos.Geometry POLYGON ((-2 -2, -2 12, 12 12, 12 -2, -2 -2))> >>> buffer(square, -2, join_style="mitre") <pygeos.Geometry POLYGON ((2 2, 2 8, 8 8, 8 2, 2 2))> >>> buffer(square, -5, join_style="mitre") <pygeos.Geometry POLYGON EMPTY> >>> buffer(line, float("nan")) is None True ``` `pygeos.constructive.``centroid`(geometry, **kwargs) Computes the geometric center (center-of-mass) of a geometry. For multipoints this is computed as the mean of the input coordinates. For multilinestrings the centroid is weighted by the length of each line segment. For multipolygons the centroid is weighted by the area of each polygon. Parameters geometry : Geometry or array_like Examples ```>>> centroid(Geometry("POLYGON ((0 0, 10 0, 10 10, 0 10, 0 0))")) <pygeos.Geometry POINT (5 5)> >>> centroid(Geometry("LINESTRING (0 0, 2 2, 10 10)")) <pygeos.Geometry POINT (5 5)> >>> centroid(Geometry("MULTIPOINT (0 0, 10 10)")) <pygeos.Geometry POINT (5 5)> >>> centroid(Geometry("POLYGON EMPTY")) <pygeos.Geometry POINT EMPTY> ``` `pygeos.constructive.``convex_hull`(geometry, **kwargs) Computes the minimum convex geometry that encloses an input geometry. Parameters geometry : Geometry or array_like Examples ```>>> convex_hull(Geometry("MULTIPOINT (0 0, 10 0, 10 10)")) <pygeos.Geometry POLYGON ((0 0, 10 10, 10 0, 0 0))> >>> convex_hull(Geometry("POLYGON EMPTY")) <pygeos.Geometry GEOMETRYCOLLECTION EMPTY> ``` `pygeos.constructive.``delaunay_triangles`(geometry, tolerance=0.0, only_edges=False, **kwargs) Computes a Delaunay triangulation around the vertices of an input geometry. The output is a geometrycollection containing polygons (default) or linestrings (see only_edges). Returns an None if an input geometry contains less than 3 vertices. Parameters geometry : Geometry or array_like tolerance : float or array_like Snap input vertices together if their distance is less than this value. only_edges : bool or array_like If set to True, the triangulation will return a collection of linestrings instead of polygons. Examples ```>>> points = Geometry("MULTIPOINT (50 30, 60 30, 100 100)") >>> delaunay_triangles(points) <pygeos.Geometry GEOMETRYCOLLECTION (POLYGON ((50 30, 60 30, 100 100, 50 30)))> >>> delaunay_triangles(points, only_edges=True) <pygeos.Geometry MULTILINESTRING ((50 30, 100 100), (50 30, 60 30), (60 30, 100 100))> >>> delaunay_triangles(Geometry("MULTIPOINT (50 30, 51 30, 60 30, 100 100)"), tolerance=2) <pygeos.Geometry GEOMETRYCOLLECTION (POLYGON ((50 30, 60 30, 100 100, 50 30)))> >>> delaunay_triangles(Geometry("POLYGON ((50 30, 60 30, 100 100, 50 30))")) <pygeos.Geometry GEOMETRYCOLLECTION (POLYGON ((50 30, 60 30, 100 100, 50 30)))> >>> delaunay_triangles(Geometry("LINESTRING (50 30, 60 30, 100 100)")) <pygeos.Geometry GEOMETRYCOLLECTION (POLYGON ((50 30, 60 30, 100 100, 50 30)))> >>> delaunay_triangles(Geometry("GEOMETRYCOLLECTION EMPTY")) <pygeos.Geometry GEOMETRYCOLLECTION EMPTY> ``` `pygeos.constructive.``envelope`(geometry, **kwargs) Computes the minimum bounding box that encloses an input geometry. Parameters geometry : Geometry or array_like Examples ```>>> envelope(Geometry("LINESTRING (0 0, 10 10)")) <pygeos.Geometry POLYGON ((0 0, 10 0, 10 10, 0 10, 0 0))> >>> envelope(Geometry("MULTIPOINT (0 0, 10 0, 10 10)")) <pygeos.Geometry POLYGON ((0 0, 10 0, 10 10, 0 10, 0 0))> >>> envelope(Geometry("POINT (0 0)")) <pygeos.Geometry POINT (0 0)> >>> envelope(Geometry("GEOMETRYCOLLECTION EMPTY")) <pygeos.Geometry POINT EMPTY> ``` `pygeos.constructive.``extract_unique_points`(geometry, **kwargs) Returns all distinct vertices of an input geometry as a multipoint. Note that only 2 dimensions of the vertices are considered when testing for equality. Parameters geometry : Geometry or array_like Examples ```>>> extract_unique_points(Geometry("POINT (0 0)")) <pygeos.Geometry MULTIPOINT (0 0)> >>> extract_unique_points(Geometry("LINESTRING(0 0, 1 1, 1 1)")) <pygeos.Geometry MULTIPOINT (0 0, 1 1)> >>> extract_unique_points(Geometry("POLYGON((0 0, 1 0, 1 1, 0 0))")) <pygeos.Geometry MULTIPOINT (0 0, 1 0, 1 1)> >>> extract_unique_points(Geometry("MULTIPOINT (0 0, 1 1, 0 0)")) <pygeos.Geometry MULTIPOINT (0 0, 1 1)> >>> extract_unique_points(Geometry("LINESTRING EMPTY")) <pygeos.Geometry MULTIPOINT EMPTY> ``` `pygeos.constructive.``point_on_surface`(geometry, **kwargs) Returns a point that intersects an input geometry. Parameters geometry : Geometry or array_like Examples ```>>> point_on_surface(Geometry("POLYGON ((0 0, 10 0, 10 10, 0 10, 0 0))")) <pygeos.Geometry POINT (5 5)> >>> point_on_surface(Geometry("LINESTRING (0 0, 2 2, 10 10)")) <pygeos.Geometry POINT (2 2)> >>> point_on_surface(Geometry("MULTIPOINT (0 0, 10 10)")) <pygeos.Geometry POINT (0 0)> >>> point_on_surface(Geometry("POLYGON EMPTY")) <pygeos.Geometry POINT EMPTY> ``` `pygeos.constructive.``simplify`(geometry, tolerance, preserve_topology=False, **kwargs) Returns a simplified version of an input geometry using the Douglas-Peucker algorithm. Parameters geometry : Geometry or array_like tolerance : float or array_like The maximum allowed geometry displacement. The higher this value, the smaller the number of vertices in the resulting geometry. preserve_topology : bool If set to True, the operation will avoid creating invalid geometries. Examples ```>>> line = Geometry("LINESTRING (0 0, 1 10, 0 20)") >>> simplify(line, tolerance=0.9) <pygeos.Geometry LINESTRING (0 0, 1 10, 0 20)> >>> simplify(line, tolerance=1) <pygeos.Geometry LINESTRING (0 0, 0 20)> >>> polygon_with_hole = Geometry("POLYGON((0 0, 0 10, 10 10, 10 0, 0 0), (2 2, 2 4, 4 4, 4 2, 2 2))") >>> simplify(polygon_with_hole, tolerance=4, preserve_topology=True) <pygeos.Geometry POLYGON ((0 0, 0 10, 10 10, 10 0, 0 0), (2 2, 2 4, 4 4, 4 2, 2 2))> >>> simplify(polygon_with_hole, tolerance=4, preserve_topology=False) <pygeos.Geometry POLYGON ((0 0, 0 10, 10 10, 10 0, 0 0))> ``` `pygeos.constructive.``snap`(geometry, reference, tolerance, **kwargs) Snaps an input geometry to reference geometry’s vertices. The tolerance is used to control where snapping is performed. The result geometry is the input geometry with the vertices snapped. If no snapping occurs then the input geometry is returned unchanged. Parameters geometry : Geometry or array_like reference : Geometry or array_like tolerance : float or array_like Examples ```>>> point = Geometry("POINT (0 2)") >>> snap(Geometry("POINT (0.5 2.5)"), point, tolerance=1) <pygeos.Geometry POINT (0 2)> >>> snap(Geometry("POINT (0.5 2.5)"), point, tolerance=0.49) <pygeos.Geometry POINT (0.5 2.5)> >>> polygon = Geometry("POLYGON ((0 0, 0 10, 10 10, 10 0, 0 0))") >>> snap(polygon, Geometry("POINT (8 10)"), tolerance=5) <pygeos.Geometry POLYGON ((0 0, 0 10, 8 10, 10 0, 0 0))> >>> snap(polygon, Geometry("LINESTRING (8 10, 8 0)"), tolerance=5) <pygeos.Geometry POLYGON ((0 0, 0 10, 8 10, 8 0, 0 0))> ``` `pygeos.constructive.``voronoi_polygons`(geometry, tolerance=0.0, extend_to=None, only_edges=False, **kwargs) Computes a Voronoi diagram from the vertices of an input geometry. The output is a geometrycollection containing polygons (default) or linestrings (see only_edges). Returns empty if an input geometry contains less than 2 vertices or if the provided extent has zero area. Parameters geometry : Geometry or array_like tolerance : float or array_like Snap input vertices together if their distance is less than this value. extend_to : Geometry or array_like If provided, the diagram will be extended to cover the envelope of this geometry (unless this envelope is smaller than the input geometry). only_edges : bool or array_like If set to True, the triangulation will return a collection of linestrings instead of polygons. Examples ```>>> points = Geometry("MULTIPOINT (2 2, 4 2)") >>> voronoi_polygons(points) <pygeos.Geometry GEOMETRYCOLLECTION (POLYGON ((3 0, 0 0, 0 4, 3 4, 3 0)), POLYGON ((3 4, 6 4, 6 0, 3 0, 3 4)))> >>> voronoi_polygons(points, only_edges=True) <pygeos.Geometry LINESTRING (3 4, 3 0)> >>> voronoi_polygons(Geometry("MULTIPOINT (2 2, 4 2, 4.2 2)"), 0.5, only_edges=True) <pygeos.Geometry LINESTRING (3 4.2, 3 -0.2)> >>> voronoi_polygons(points, extend_to=Geometry("LINESTRING (0 0, 10 10)"), only_edges=True) <pygeos.Geometry LINESTRING (3 10, 3 0)> >>> voronoi_polygons(Geometry("LINESTRING (2 2, 4 2)"), only_edges=True) <pygeos.Geometry LINESTRING (3 4, 3 0)> >>> voronoi_polygons(Geometry("POINT (2 2)")) <pygeos.Geometry GEOMETRYCOLLECTION EMPTY> ```

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{[ promptMessage ]} Bookmark it {[ promptMessage ]} Relevant Info &amp; Decision # Relevant Info &amp; Decision - Grand Valley State... This preview shows pages 1–4. Sign up to view the full content. This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. Unformatted text preview: Grand Valley State University Relevant Information & Seidman College of Business Decision Making Problems School of Accounting Northern Stores is a retailer in the upper midwest. The most recent monthly income statement for Northern Stores is given below: Total Store 1 Store 11 Sales \$2,100,000 \$1,300,000 55 800,000 Less variable expenses 1 260 000 882,000 378,000 Contribution margin 33 840,000 \$ 418,000 \$ 422,000 Less traceable fixed expenses 420,000 231,000 189,000 Segment margin \$ 420,000 \$ 187,000 \$ 233,000 Less common fixed expenses 350,000 210,000 140,000 Net income \$ 70,000 is 123,000; \$ 93,000 Northern is considering closing Store I. If Store I is closed, one - fourth of its traceable fixed expenses would continue to be incurred. Also, the closing of Store I would result in a 20% increase in sales at Store 11. Northern allocates common fixed expenses on the basis of sales dollars and none of these costs would be saved if a store were closed. Compute the overall increase or decrease in the net income of Northern Stores if Store I is closed. The Immanuel Company has just obtained a request for a special order of 6,000 jigs to be shipped at the end of the month at a selling price of \$7 each. The company has a production capacity of 90,000 jigs per month with total fixed production costs of \$144,000. At present, the company is selling 80,000 jigs per month through regular channels at a selling price of \$11 each. For these regular sales, the costs of one jig is: Variable production cost \$4.60 Fixed production cost 1.80 Variable selling expense 1.00 If the special order is accepted, Immanuel will not incur any selling expense; however, it will incur shipping costs of \$0.30 per unit. If Immanuel accepts this special order, the change in the monthly net operating income will be a: \$12,600 increase \$14,400 increase \$ 3,600 increase 33 1,800 increase 9-.“ 9‘!” At what selling price per unit should Immanuel be indifferent between accepting or rejecting the special order? a. \$7.40 b. \$7.70 0. \$6.40 (1. \$4.90 Kramer Company makes 4,000 units per year of a part called an axial tap for use in one of its products. Data concerning the unit production costs of the axial tap follow: Direct materials \$3 5.00 Direct labor 10.00 Variable manufacturing overhead 8.00 Fixed manufacturing overhead 20.00 Total manufacturing cost per unit: \$73.00 An outside supplier has offered to sell Kramer Company all of the axial taps it requires. If Kramer Company decided to discontinue making the axial taps, 40% of the above fixed manufacturing overhead costs could be avoided. Assume that direct labor is a variable cost. Assume Kramer Company has no alternative use for the facilities presently devoted to production of the axial taps. If the outside supplier offers to sell the axial taps for \$65.00 each, should Kramer Company accept the offer? Fully support your answer with appropriate calculations. When Mr. Ding L. Berry, president and chief executive of Berry, Inc., first saw the segmented income statement below, he flew into his usual rage. “When will we ever start showing a real profit? I’m starting immediate steps to eliminate those two unprofitable lines!” Product Lines Total U V W Sales \$250,000 \$100,000 \$75,000 \$75,000 Variable expenses 119 000 37 000 35 000 47,000 Contribution margin \$13 1,000 \$ 63,000 \$40,000 \$28,000 Traceable fixed expenses* 98,000 31,000 37,000 30,000 Common expenses, allocated 32,900 18,000 10,500 4,400 Operating income (loss) \$ 100 \$ 14,000 \$7,500! \$16,400; *These traceable expenses could be eliminated if the product lines to which they are traced were discontinued. Recommend which segments, if any, should be eliminated. Back up your recommendation with facts and figures. ' ... View Full Document {[ snackBarMessage ]} ### Page1 / 4 Relevant Info &amp; Decision - Grand Valley State... This preview shows document pages 1 - 4. Sign up to view the full document. View Full Document Ask a homework question - tutors are online

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# Data analysis and Statistic • Jan 20th 2007, 07:50 PM Patience Data analysis and Statistic Finding the five number sumary. 7, 7, 5, 4, 1, 9, 8, 8, 8, 5, 2 Find the range of these numbers. 7, 7, 5, 4, 1, 9, 8, 8, 8 ,5, 2 • Jan 20th 2007, 07:51 PM ThePerfectHacker Quote: Originally Posted by Patience Finding the five number sumary. 7, 7, 5, 4, 1, 9, 8, 8, 8, 5, 2 Find the range of these numbers. 7, 7, 5, 4, 1, 9, 8, 8, 8 ,5, 2 The range is biggest minues lowest. In this case 9-2=7

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Search a number 17424 = 2432112 BaseRepresentation bin100010000010000 3212220100 410100100 51024144 6212400 7101541 oct42020 925810 1017424 1112100 12a100 137c14 1464c8 155269 hex4410 17424 has 45 divisors (see below), whose sum is σ = 53599. Its totient is φ = 5280. The previous prime is 17419. The next prime is 17431. The reversal of 17424 is 42471. Adding to 17424 its reverse (42471), we get a palindrome (59895). It can be divided in two parts, 17 and 424, that added together give a square (441 = 212). 17424 = T131 + T132. It is a happy number. The square root of 17424 is 132. It is a perfect power (a square), and thus also a powerful number. It is a Smith number, since the sum of its digits (18) coincides with the sum of the digits of its prime factors. It is a Harshad number since it is a multiple of its sum of digits (18). It is a d-powerful number, because it can be written as 1 + 7 + 45 + 23 + 47 . It is a Duffinian number. 17424 is a Rhonda number in base 15. Its product of digits (224) is a multiple of the sum of its prime divisors (16). It is a nialpdrome in base 12 and base 16. It is a junction number, because it is equal to n+sod(n) for n = 17397 and 17406. It is an unprimeable number. 17424 is an untouchable number, because it is not equal to the sum of proper divisors of any number. It is a pernicious number, because its binary representation contains a prime number (3) of ones. It is a polite number, since it can be written in 8 ways as a sum of consecutive naturals, for example, 1579 + ... + 1589. 217424 is an apocalyptic number. 17424 is the 132-nd square number. It is an amenable number. It is a practical number, because each smaller number is the sum of distinct divisors of 17424 17424 is an abundant number, since it is smaller than the sum of its proper divisors (36175). It is a pseudoperfect number, because it is the sum of a subset of its proper divisors. 17424 is a wasteful number, since it uses less digits than its factorization. 17424 is an odious number, because the sum of its binary digits is odd. The sum of its prime factors is 36 (or 16 counting only the distinct ones). The product of its digits is 224, while the sum is 18. The cubic root of 17424 is about 25.9248322038. The spelling of 17424 in words is "seventeen thousand, four hundred twenty-four", and thus it is an iban number.

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MAT118-2002Spring # MAT118-2002Spring - Name MAT 118 Final Exam May 7 2002 1... This preview shows pages 1–4. Sign up to view the full content. This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. Unformatted text preview: Name MAT 118 Final Exam May 7, 2002 1 . Justify why this is a good or bad definition of a rhombus: a quadrilateral with a pair of consecutive sides congruent. 2. Classify each of the following statements as either true or false. If the statement is false, provide a counterexample. a. If all sides of a quadrilateral are congruent, the quadrilateral is a square. b. An isosceles trapezoid can be a rectangle. c. No trapezoid is a square. d. A trapezoid is a parallelogram. 3. In the figure below, the radius of each circle is 5 cm. The length of each of the tangents to the circles is 15 cm. 21. Find the shaded area enclosed by two semicircles and two tangents to the semicircles as shown below. b. Find the perimeter of the shaded figure. 4. A student says that it is actually impossible to measure an angle, since each angle is the union of two rays that extend infinitely, and therefore continue forever. What is your response? 5. A machine costs \$3450 at present. This is 60% of the cost four years ago. What was the cost of the machine four years ago? Explain your reasoning. 5 l 5 6. Ex lainwh — + — = — X —. P y 2 2 1 . . . . 7. On a map, 3 1n. represents 5 m1. If New York and Aluoss1m are 18 in. apart on the map, what is the actual distance between them? 8. In the space provided, write whether each of the following statements is SOMETIMES, ALWAYS, or NEVER TRUE. Then justify your answer. a. Ifxat0,y¢0,andl < Athenx > y. x y b. Ifx>0,then l<x. x 9. Assume the blank square below represents 1 unit. Answer the following questions under each of the relevant squares, A and B. a. What fraction of the square is shaded? b. Write the multiplication sentence that indicates how the shaded fraction is obtained. 10. For each of the following determine ifa solution exists. If a solution exists, give it. If a solution does not exist, explain why it does not. a. 45% of a graduating class of 250 seniors havejobs. How many seniors havejobs? b. What is the number on the number line that is gills of the way from 0 to 10? l l . Describe at least three properties of the number x/E using the various facts you have studied about real numbers. ... View Full Document {[ snackBarMessage ]} ### Page1 / 4 MAT118-2002Spring - Name MAT 118 Final Exam May 7 2002 1... This preview shows document pages 1 - 4. Sign up to view the full document. View Full Document Ask a homework question - tutors are online

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# experimental design 1. ### Individual Covariate in Fractional Factorial Design I want to make a marketing experiment (sending an e-mail) with fractional factorial design. I will send this e-mail to a mailing list consist of 660 members. I have 6 factors and use the 1/2 fraction. I plan to use response variable open rate (ratio between email open/sent) and CTR (ratio... 2. ### A curious design - What is the type / name of this rare design? There is this split-mouth randomized clinical trial, in which one side of the mouth receives a particular treatment, and the other side receives another treatment (usually used as the control treatment); (we do this after randomization of the left/right sides of the mouth to treatments A and X... 3. ### Factorial Design Suppose a dermatologist wants to study the effectiveness of two different preparations of a skin lotion using two different forms of application (for example, one vs. two applications per day). He has available 12 patients with a certain skin disease and he can apply one form of medication (that... 4. ### Help with R GLM Formula for paired/nested model I have an experimental design as follows: Two treatments, carried out on the same sample. The samples are collected from soil at five depths, in five different pits, from five different fields. I want to: (1) see if the treatment has an effect on the species richness found in the soil... 5. ### Defining Contrast Matrix In the book Applied Longitudinal Analysis, 2nd Edition there is an example in the chapter "Marginal Models: Generalized Estimating Equations (GEE)" in "Muscatine Coronary Risk Factor Study" sub-section. I am illustrating it below : Let Y_{ij}=1if the i^{\text{th}} child is classified as obese... 6. ### 1 DV w. four levels, 2 IV's I'm testing the predictive effects of mood (positive/negative) and personality traits (extraversion and neuroticism) on music preference (interest level in 4 different musical categories). This follows a between-subjects design. A positive or negative mood was induced via video... 7. ### Does Imposing a Constrain Make a Method More Powerful? Suppose we have 100 patients in a randomized trial. 50 of them are in treatment group and the rest are in placebo group. This is a longitudinal study so that we measure the response from each individual at four different periods: baseline, week 1, week 4, and week 6. Now suppose we have good... 8. ### Defining Contrast Matrix Suppose I have a binary Covariate X which is defined as X = \begin{cases} 1, & \text{if treatment group} \\ 0, & \text{if placebo group} \end{cases} The model is \mathbb E[Y] = \beta_0 + \beta_1X, where Y is a continuous random variable. If X=0, then \mathbb E[Y] =... 9. ### Is this Experimental Design even possible? Help please! Hello all, First thank you in advance for any and all help given in response to this question. I've been searching this forum as well as other places online and I'm still confused about what methodology I should use in this experiment. DESIGN: Participants are given two questionnaires... 10. ### Experimental design question I am struggling with the complexity of my experimental design... I apologize in advance for any incorrect terminology or grammar. I know that I have a 2x2x2 factorial design with 2 different dam diets (during pregnancy), 2 different offspring diets, and genders (obviously 2 levels)... 11. ### Regression analyses and replicates Hi, Thanks in advance for any insight! 12. ### Experimental Design Assignment I have an assignment that I can't solve in/with R I don't give the whole problem statement, but only the details that are of concern The objective of the study is to assess the ecacy of X (as compared to the reference treatment with SCALA). In this study 24 hills with grapevines are at... I want to ask some confusing in my mind regarding the data analysis: I have conducting the experimental trial in repeated measure by hours as below: Design: Completely Randomized Design--(in green houses)-there are 6 box unit... 14. ### Which statistical test to use, 20 subjects, 6 categorical variable, 4 parameter Hi I have an experiment and are not sure which statistical test I should use. I hope some of you can help me, as this is not my strong side.. I have 20 subjects looking at 6 different categories of images, 40 trials per categori. I have 4 numerical variables I measure for each trial: 20... 15. ### ANOVA issue with experimental design Hey guys; I am evaluating the effects that various factors may or may not have on the output of a process. Each of the factors has a different level. [http://i.imgur.com/8UhG4ri.png]Here[/http://i.imgur.com/8UhG4ri.png] is the general layout in tabular form My research group wants to see if... 16. ### is there interaction effect in RCBD with a single observation per cell ? The statistical model for a Randomized Complete Block Design (RCBD) with a single observation per cell is: y_{ij}=\mu+\tau_{i}+\beta_{j}+\epsilon_{ij};\quad i=1,\ldots ,a, \quad j=1,\ldots, b, where y_{ij} is a random variable that represents the response obtained on the ith treatment of the... 17. ### Stats Analysis Question =] Hey Everyone, Question!: In an aged-matched experiment, I want to compare . . . . I want to test the significance of DrugY on performance of Task1 for EACH child: (In this case each child is acting as their own control) Classic Autism DrugY Child A VS. Classic Autism Placebo Child A... 18. ### Looking for guidance on an experimental design Hey folks: I've got a question that requires a somewhat complex experimental design (nested + repeated measures) and wanted to get a few opinions. I am measuring soil respiration (essentially metabolic rate of microorganisms and plants underground) in a forest that has been damaged by a... 19. ### Nested/Repeated Comparisons-What test should I use? Hi! I am having trouble to decide what statistical test to use on my data because I have one factor grouped within another. Here is my experimental design: I have two cell lines, one infected with a bacteria (I) and one uninfected (U). I have split each of theses cell lines into two groups... 20. ### Experimental Design help Been running around in circles with this for a while. I am trying to figure what sort of design I have and how to go about analyzing it. Any help would be great. Details are below. Sample Date, 3 levels, fixed Field, 2 levels, assumed to be random from a larger population. For each sample...

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## From Conjecture to Proof SKU: H10710001 Availability: 1 in stock \$49.95 0% Off. Hurry, there are only 1 item(s) left! QTY: (grades 6-10) Ken Milton and Howard Reeves ### Introducing and developing ideas about proof The study of proof in mathematics has historically been restricted to the geometry strand of the curriculum and studied at secondary school years. This book challenges the traditional treatment of proof. The authors show how school mathematics (not just geometry) is rich in opportunities for students to explore patterns, to seek and describe generalisations, and to make and prove conjectures. From Conjecture to Proof is about learning proof and acquiring skills of constructing proof. It is a book for teachers and teachers in training, which revisits the ideas of mathematical proof in a way that encourages students to learn by ‘doing’. # The new student book for the Mathomat V2 template ### 120 pages in three sections #### Section 1 40 illustrated investigations with the Mathomat V2 template. Put students 'into' a situation so they can make sense of the mathematical relationships and operations involved. This helps in forming and operating on the mental imagery that is central mathematical thinking. #### Section 2 More about Mathomat. 14 pages of further investigations with the Mathomat V2 template. These less contextualised activities encourage students to build on the investigations from section1. #### Section 3 Mathomat V2 diary. A place to deepen understanding of the Mathomat V2 activities through reflection and by learning to classify patterns that have been previously created; such as classifying the vertices formed in tessellation designs ### About the new investigations in the Mathomat V2 student book #### The understanding angles series In the new 'Understanding angles investigations' learners are encouraged to develop a concept of what an angle is, before using the Mathomat protractor for precision angle measurement. These activities reflect research studies* which argue that students develop a spatial structuring of angle by abstracting what is common between three distinct angle contexts; corner angles (where both arms of the angle are visible), slope angles (in which only one arm of the angle can be physically seen), and turning angles in which all features of the angle must be imagined dynamically. Our new Mathomat V2 instruction book gives students the opportunity to draw and measure the standard angle concept in these situations. This activity is scaffolded by asking students to visualise a drinking straw as they work. Students are asked to use the Mathomat protractor for precision measurement to confirm their initial estimates of angle size. A key skill in this precision measuring task is to be able to imagine, and to mentally place, the second arm of the angle after alignment of the physical Mathomat protractor. #### Mathomat and symmetry The Get transforming investigation in the Mathomat 4th edition student book is being extended in the Mathomat V2 student manual to give students a sense of the four transformations underlying earlier tessellation activities. These fully explain the transformation of motifs without changing their size or shape and are sometimes called the isometries of the plane. They are: rotationtranslationreflection and glide reflection. The Mr Symmetrical activity from the Mathomat 4th edition student book is being extended in the Mathomat V2 student manual to give students experience with rotation and reflection as symmetry operations which transform a shape into itself. This includes classifying each of the Mathomat shapes according to its number of line and rotational symmetries. #### New diary section of the Mathomat V2 student manual In the diary section of the Mathomat V2 manual learners will be asked to 'size-up' the potential symmetry operations in Mathomat at a single glance through use of scientific classification of their symmetry properties. ### What's new in the Mathomat Version 2 template The version 2 Mathomat will be available in June 2018, and is a more powerful product - being the same size as the current Mathomat but with many new functions and features built into its design. These include: ## Enhanced 2-D pattern drawing The new large regular octagon, and the resized large regular pentagon combine to form a group of regular polygons with 15mm sides to compliment the very popular existing regular polygons with 10mm sides. Students now have more creative freedom in 2-D designs to suit their project style. The Mathomat V2 manual is a rich source of creative drawing ideas; for classroom or individual student use. Sometimes students want to fill the whole page with their drawing. At other times its best to leave space for the remainder of a presentation. The Mathomat V2 offers both creative options. ## Enhanced graphing #### The Mathomat V2 template includes • A larger sine/cosine curve • Integration of unit circle, sine curve, linear radian scale and new trig scale to form a 'function machine' that can help students to learn trigonometry through visualisation of the unit circle method. • Larger parabola • A new normal distribution curve #### Illustrated below: The solution to a senior school trigonometry problem. Finding the hours of the day at which it is safe to cross the harbour bar of a fishing village by boat The larger sine curve makes a striking improvement to hand drawn sketches. The integrated unit circle allows students to find angle values as distance travelled around its circumfrence before locating them on the x-axis of the graph of that function.Lesson 11 in Maths with Mathomat in the free resources section of this website provides a comprehensive plan for using Mathomat with the unit circle method of teaching trigonometry. ## Enhanced 3-D sketching in the Mathomat V2 template • New kite for sketching square based pyramids • New ellipse to assist with sketching conics • A 1:2 scale for detailed engineering drawing • More challenging fraction markings around circles for greater student interest A new 1:2 scale ruler for detailed drawings, especially useful in engineering drawings Ask students to redraw this section at a scale of 1:2 using their Mathomat V2 template The many circles on Mathomat have a revised set of graduations to create interesting fractions such sevenths, and ninths, to challenge students to think flexibly about numbers

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# Signs more, less or smooth in mathematics ## Mathematical signs Most likely, the child is already distinguished by the child and visually that a handful of ten berries is more than three pieces. To implement new designations, let's look at the signs of "more", "less", "equal" in pictures. Symbol more (>) - This is when the sharp nose tick looks right. It must be used when the first number is greater than the second: Symbol less (<) - This is when the sharp nose checkboxes look left. It must be used when the first number is less than the second: Equality symbol (=) - This is when two short segments are written horizontally and parallel to each other. We use it when comparing two identical numbers: To make it easier for the child to remember the signs similar among themselves, you can apply the game method. To do this, you need to compare the numbers and determine in what order they cost. Next, we put one point in the smallest number and two - next to the greatest. We connect the points and get the desired sign. That's so simple: ## Equality and inequality What equality In mathematics - this is when one like the number of different and between them you can put a sign =. For example, let's look at the image with the image of geometric shapes. On the right and left the same one, it means that you can put the "equal" symbol. Inequality - Algebraic expression, which uses signs ≠, <,>, ≤, ≥. The visual example of inequality is shown in the picture below. On the left we see three figures, and on the right - four. At the same time, we know that three is not equal to four or as so: three less than four. The lesson in the school often passes before the textbook, notebook and board. At home, you can use a computer and some tasks to perform in an online format. How to find signs on the keyboard? Reply in the picture: ## Types of inequalities 1. Strict inequalities - use only a sign more (>) or less (<). • A <B is that A is less than b. • A> B - this means that A is more than b. • Inequalities A> B and B <A mean the same thing, that is, are equivalent. 1. Fine inequality - use comparison signs ≥ (greater than or equal) or ≤ (less or equal). • A ≤ B is that A is less than either equal to b. • A ≥ B is that A is more or equal to b. • Signs ⩽ and ⩾ are opposite. 1. Other types of inequalities. • A ≠ B - means that A is not equal to b. • A "B means that a lot more than b. • A "B means that a much less than b. • Signs >> and << are opposite. Develop mathematical thinking of children will help mathematics lessons at Skysmart School. We picked up for your child thousands of fascinating tasks - from simple logical mysteries to cunning puzzles, which are interested in thinking. All this will help easier and quickly cope with school mathematics and fall in love with numbers. Write your child for a free trial activity in Skysmart: Let's play the numbers together with the raccoon Max and show that mathematics is very exciting! In a simple example, we will analyze what "equality" and "inequality". For example, take tasks from the textbook on mathematics. ## Equality Where equality, we see "4 = 4". Here everything is correct, it means that equality. The second example is presented otherwise: on the left we see "5", and to the right of the sign "4 + 1". If folded 4 and 1, then it will turn out 5, and the left is worth 5. The left and right part of the example is equal, which means it will also be equality. ## Inequalities In the example of the textbook, we see that on the one hand the example is "4", and on the other "3". 4 and 3 are not equal, which means it is called "inequality". In our case, between 4 and 3, it is necessary to put the sign of inequality ">" - "4> 3". The second example in the column of "inequality" is slightly complicated. To the right of the sign here is the expression "4-1", and the left is simply "4". If you take up 1 from 4, then it turns out 3. 3 less than 4, which means it will also be inequality, which is indicated by the sign. ## How not to get confused in the sign of inequality In order not to get confused in which direction to put the sign of inequality, you can imagine the beak of the bird. "Beak" should look towards the number that is less. Simply put, more as if "pecks" smaller. The second way is to use points. About a larger number is set up vertically two points, and about smaller - one in the middle. Then simply connect the points obtained and get the sign of inequality. ### Exercise 1 Let's figure out several tasks based on what we learned: The correct answers will be the following: 4> 3 3 <4 5> 2 3 <5 1 + 2 = 3 5-3 = 2 Now let's try to find incorrect inequalities: The correct answers will be such: 4 + 1 = 5 - right 3-1 <1 - incorrectly 4 <2 - incorrectly, will correctly be 4> 2 3> 4 - incorrect, will correctly be 3 <4 5-1 = 3 - incorrectly, will correctly 5-1 = 4 2 + 1 = 3 - right Here we are given cards on which you need to put the correct sign. The following expressions are obtained: 3 + 1 = 4 5-1 = 4. 4> 3. 2 <4. 5> 1. 3> 2. 1 <4. 5> 3. The last task is practical and the most interesting. We need to answer questions from any of the guys more coins, and who has more money amount. To begin with, we will understand with the number of coins: Misha has 1 coin, and Kolya 2, it means more coins. We write it as inequality: 1 <2. Now we will define any of the guys more money. Misha has only one coin in more advantage of 5 rubles. Everything is simple here. But if there are two coins in 1 and 2 rubles. Calculate how much money Kseli: 1 + 2 = 3. It turns out that if 3 rubles it turns out. Now we know that Misha has 5 rubles, and by Kolya 3 rubles. So Misha has more money than Kolya. We write it as inequality: 5> 2 + 1. What do the concepts of "equality" and "inequality" in mathematics mean? Give examples. Ninaarc. [360K] 3 years ago The record in which the "equal" sign is used (=), which is worth between mathematical objects, is called "equality" . Such a sign can produce two numbers, several numbers or expressions. The right and left part of the expressions facing and after the sign "=" always have the same meaning. Examples: 5 ∙ 4 = 20; 3 + 6 = 9; 21: 7 = 3. There are cases when expressions have completely different meanings, in this case the sign "equally" between them is not put. There is a special sign that can be noted that expressions differ in each other: "≠" . Examples: 15 ≠ 20 - 2; 14 ≠ 6 + 4; 2 ∙ 5 ≠ 12. Inequality - this is a concept that is associated with comparing two mathematical objects, but they are compiled using signs "≠" , ">" (more) and "<" (less). Typically, the values ​​of the right and to the left of these characters have different numeric values. Examples: 8 <10; 3 ∙ 4> 2 ∙ 5; 81: 9 <7 ∙ 8. The author of the question chose this answer as the best. Annagne. [102K] More than a year ago The concept of equality or inequality in mathematics comes from comparison or numbers or expressions. The equal sign is denoted by two parallel straight ones of the same length "=", and this sign is used in mathematics only since the end of the 16th century, and before that moment it was indicated in alphabetic terms. An example of equality: 7 = 7 or 2 + 6 = 8 or a + b = b + a. The inequality is indicated by signs greater and less. As a rule, the very concept itself, and the sign of equality is easily understood and remembered, but with signs more and less many children arise difficulties in memorization and I, at one time, was no exception. I remember how we were taught to memorize these signs in the Soviet school: if you substitute the bird to the sign with the right and its keyboard is open - it means that the sign is larger, and if closed - then the sign is less. For example : • We read on the left to the right and these examples sound like this: • four more units; Two less than six. True, in mathematics there are also the concepts faithful and incorrect, and they include both equality and inequality. Wildcat. [139K] 2 years ago Equality is when something is different. When we have five fingers on each hand, but two eyes, one nose. In mathematics, equality is denoted by two short parallel stripes: =. They mean that without a difference where to go and what to take, everywhere all the same. 5 = 5, 6 = 6, 7 = 7. Five fingers on one hand are equal to five fingers on the other and it was always. But inequality is the absence of coincidence. It's if you have five fingers on your hand, and I have four, because he was a fool and one finger pulled him off. It turns out that you have fingers more: 5> 4 This is a "more" sign. It is located above the letter at the keyboard and to extract it to use the English alphabet. Nearby and the sign is less: <, and it is also available in English layout. 4 <5 and this is true. Try to raise four kilograms, and then take five. Do you feel the difference? 3 years ago Author [632K] For this, the sign is used equal to (and it is referred to as the sign of equality), what does it look =. Example When recording different equalities, equal objects are made, as well as between them and put a sign =. For example, to say, the recording of equal numbers 6 and 6 will be drawn as follows 6 = 6, and it can be read as "six equals six" And if we need in writing to note the inequality of 2 objects, then the sign is not equal to ≠. A sign is a simply crossed sign equal. For example, recording 3 + 5 ≠ 7. So it is possible: "The sum of the troika and five is not equal to seven. Still used signs "<", ">". Less - more. Wildcat. Custody [189K] • When we talk about numerical equality, we use the "=" sign. In this case, one numerical expression that stands on the right, equal to the numerical expression that is located on the left. • Numeric equality possess several properties: • The property of reflexivity. For example: x = x; 2 = 2. The property of symmetry. For example: 3 + 1 = 2 + 2, then 2 + 2 = 3 + 1. • Property of transitivity. For example: x = y, y = z, then x = z. • Also, if we do some of the same manipulations with both parts of equality, the equality does not change. For example, multiplication, addition (except manipulation from 0), division and subtraction. 3 + 1 = 2 + 2. We add to each of the parts 1. And we get 3 + 1 + 1 = 2 + 2 + 1. 5 = 5. Equality is not violated. 3 + 1 = 2 + 2. We multiply on 2 both parts. 2 (3 + 1) = 2 (2 + 2), 6 + 2 = 4 + 4, 8 = 8. Equality is not violated. When we talk about numerical inequalities, we mean that it is part of expression more or less than another. Then the equal sign is not used, the signs "<" or ">", "≤" or "≥" are taken. They also have a number of properties. And may be loyal and incorrect. For example: 3 + 5> 6 is a faithful inequality; Wildcat. 3 + 5 <6 is incorrect inequality. Juga. [85.5K] Equality or inequality - follows from a comparison of numbers or expressions. Something the same when compared can be called equality. For example, 2 + 5 will be 7 and 3 + 4 will give in the amount of 7 These two expressions 2 + 5 and 3 + 4 with each other are equal And you can write it like this: 2 + 5 = 3 + 4 Inequality, accordingly there will be an expression, in which the amount in the right part will differ from the amount in the left expressions. For example: 3 years ago 2 + 6 is not equal to 3 + 4, and more by value. Inequality is recorded by signs more or less or crossed out the sign of equality. Maria Muzja. [65K] These concepts (equality / inequality) in mathematics are very interrelated. 3 years ago Equality is a concept that is still in elementary school, and under this term, you need to understand the "statement", to which you can apply the sign "=", something equal and identical. There are numerical equality. There are incorrect equivals and faithful. And "inequality" is such a mathematical statement showing how much one number is different from the other. Dilyara K. [4.9K] The equality is called such mathematical expressions when the values ​​on the left and right of the sign "=" are equal. Equality, examples: 18 - 6 * 2 = 6 23 - (13 + 3) = 7 3 years ago If the values ​​on the left and right are different, instead of the equality sign, signs of inequality "<", ">" are set, depending on which side of the inequality is greater. Inequality, examples: 7 - 9 <5 Wildcat. 17> 21 - 19 [user blocked] [3.9K] In algebra there is a concept of "mathematical expression". If it is quite simple, this is a set of all kinds of mathematical actions and transformations. The result of "expressions" is its value. If the values ​​of two expressions are the same, it means "equality", if the values ​​are different, it is "inequality" Alice in the country [309K] Equality in mathematics is a mathematical expression, between whose parts is the sign "Rivne". For example: 7 + 5 = 12 LG (x + 3) = 3 + 2 LG 5 Inequality This is when in mathematical expression between its parts it is not a sign "equal", but the "less" sign or the "greater" sign. For example: Wildcat. 4 - 2 <5 4 (x - 2) ∙ (x + 2)> 0. Sometimes there is such a sign between parts of the expression here (the crossed sign "equal": ≠, then this expression can also be called inequality: 20 + 5 ≠ 19 √ n (x) ≠ √ m (x) ISA-ISA. [72.6K] "Equality" in mathematics are examples in which there is a sign "equal" between numbers or works of numbers. For example: 2x2 = 4, or 2x2 = 1 + 3, this is true equality. There are incorrect equivals when an example is infidelity. Inequality, it is when there are more or less signs between numbers. How like equality, inequalities are incorrect. 31-26 <7. 2x2 <5. 100> 68-7 Mathematics, Grade 1 Lesson 11. Equality. Inequality. Signs ">", "<", "=" The list of questions considered in the lesson: 1. Determine the location of signs more, less, equal 2. Write signs>, <, = 3. Call equality, inequality. . Glossary Equality - This is when one quantity is different. Inequality - This is when one side of the expression is not equal to the second. If the tick spout looks right - this sign more (>) : If the tickle is noted, look left - it Sign less (<). Equality sign (=) In mathematics, in logic and other accurate sciences - a symbol that is written between two expressions in its value. Keywords Sign>; sign <; sign =. Main literature 1.Moro M. I., Volkova S. I., Stepanova S. V. Mathematics. Textbook. 1 cl. In 2 h. M.: Enlightenment, 2017. 1. Moro M. I., Volkov S. I. Mathematics. Workbook. 1 cl. In 2 hours. Manual for general education organizations. - M.: Enlightenment, 201 p. one. Today we go to the store to buy Ole and Ana to the lesson technology all training equipment. For the lesson, you will need 1 pack of plasticine and two packs of cardboard. = How many packs of plasticine got girls? (one pack) It can be said that the girls received the same amount of plasticine. 2. For technology, two packs of cardboard are required. How many cardboard packs got girls? (two packs) We can say that the girls received the same amount of cardboard. 3. In mathematics, a special icon is used to record that the number of items is the same. You can record numbers and use for the words "equally", "equal" special icon "=", 1 = 1 2 = 2 (similarly) Two sticks will write children And what happens in response, After all, everyone learned a long time ago How to spell that sign: Equally! Such records are called equalities. This is equality. You can write equalities using the "=" sign. We prove that the same number of items with the help of arrows forms a pair. In the diagram, each subject we denote the circle and form a pair. Show the arrow. Olya Anya 1. Unnecessary figures left. So it is equally equally. You can write 1 = 1 6. 2 + 1 = 3 How can I read this entry? (Numerical equality) Under this statement, there are two numerical expressions that stand on both sides of the sign "=". Both parts of the record are equal to each other. 1. What amount needed for a cardboard lesson? And plasticine? To find out which items needed more or less, use special icons ">", "<". 1. 3. In mathematics, a special icon is used to record that the number of items is the same. If from some side more or less, the record will be called "inequality". Two more than one. Cardboard plasticine If the left is greater than the right, then use the ">" sign. 2> 1. And if the number left is less than the right, then put the sign less "<". 1 <2. inequalities: 4> 3, 4 <5 Analysis of the sample training task Select the desired sign and distribute into two groups. Finish each group with your records.

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# Thread: Quick little probability question 1. ## Quick little probability question 2 students are randomly picked one after the other in a class of 30. What is the probability that you get picked and your friend does not get picked? Thanks 2. Originally Posted by tbyou87 2 students are randomly picked one after the other in a class of 30.What is the probability that you get picked and your friend does not get picked? Because the question is so poorly put we must make some assumptions. Say that you have a special friend J. There are 28 ways for you to be selected but not J. There $\displaystyle {{30}\choose {2}}$ ways to make all of the selections.

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Resource Center To help determine the amount of tile for your needs, below is an easy to use tool for calculating the square footage of an area. To start, please provide the following measurements. Area Length Area Width Feet Inches Feet Inches Area1 Area 2 Area 3 Area 4 Estimated Tile Required: Total Square Footage All Areas: 10% Square Feet (Extra): Estimated Total Square Feet Needed: Estimating Tile Quantities When measuring for floor tile the rule is: Length x Width + Waste = Amount Needed Waste Factor will vary based upon tile size, layout, configuration of room, patterns, etc. Typical waste factor is about 10%. Add 15% for tile being installed diagonally or for a room with lots of jogs and corners. These installations will require more cuts and thus more waste. Example: Actual room size:  8 ft 7 in x 5 ft 2 in Convert to inches, actual room size: 103 in x 62 in 103" x 62"= 6386 inches / 144 (1 sf) = 44.50 sf + waste factor (44.50 x 1.10)  = 49 sf The same method is used when measuring walls. Measure the area to be tiled on each wall. Add them together and figure the square footage. Add your waste factor and this is the amount needed. Most vertical applications will require trim pieces which have finished edges, this requires a linear measurement. Trim pieces will be used: 1.  Where a tiled surface ends on an open wall leaving an exposed tile edge, 2.  Where a vertical surface meets a horizontal, like on the edge of a countertop, 3.  Where a vertical surface turns a corner, like on the outside corner of a wall. Trim pieces and decoratives are typically sold by the piece. To figure the quantity you have to establish the length of the trim piece (i.e. 6" bullnose, 8" decorative liner), then the rule is: Linear Inches/Piece Length = Quantity Example: 10 foot exposed edge that needs bullnose: 10 ft x 12 in = 120 inches. If using 6 inch bullnose trim piece = 120 inches / 6 inches  = 20 pieces of bullnose needed If using 8 inch decorative liner = 120 inches / 8 inches = 15 pieces of liner needed These formulas will help you estimate the quantities you'll need. We recommend you have a professional tile installer view the job to check for the suitability of your substrate and to measure and plan for any special conditions that may exist.

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Home / Pressure Conversion / Convert Picopascal to Torr # Convert Picopascal to Torr Please provide values below to convert picopascal [pPa] to torr [Torr], or vice versa. From: picopascal To: torr ### Picopascal to Torr Conversion Table Picopascal [pPa]Torr [Torr] 0.01 pPa7.50061682704E-17 Torr 0.1 pPa7.50061682704E-16 Torr 1 pPa7.50061682704E-15 Torr 2 pPa1.500123365408E-14 Torr 3 pPa2.250185048112E-14 Torr 5 pPa3.75030841352E-14 Torr 10 pPa7.50061682704E-14 Torr 20 pPa1.500123365408E-13 Torr 50 pPa3.75030841352E-13 Torr 100 pPa7.50061682704E-13 Torr 1000 pPa7.50061682704E-12 Torr ### How to Convert Picopascal to Torr 1 pPa = 7.50061682704E-15 Torr 1 Torr = 1.3332236842108E+14 pPa Example: convert 15 pPa to Torr: 15 pPa = 15 × 7.50061682704E-15 Torr = 1.125092524056E-13 Torr

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# 4 reasons why your ecommerce store's AOV is an awful metric Tracking metrics can become an obsession for e-commerce entrepreneurs. As you gather data and measure your performance, it's easy to get hooked on a few numbers that seem to give you a clearer picture of what's happening with your business. One of those key metrics is the AOV -- your average order value. Calculating your AOV is pretty straightforward, like its name. Choose a timeframe, add up the order totals, and then divide by the number of orders you had. The answer is your AOV. For example, if you had three orders totalling \$10, \$11, and \$9, your AOV would be \$10. Now let's put that number to work. ### What Your AOV Tells You Your AOV is extremely handy because it not only tells you how much your customers generally order every time they visit, but it gives you a way to see if your business is sustainable. By subtracting your costs from that number, you don't have to wonder if you're profitable. For example, if your AOV is \$100, your customer acquisition cost is \$10, and margin is 20%, then you're making \$10 profit per order: \$100 - \$80 COGS (20% margin) - \$10 acquisition = \$10 (for a more in-depth explanation, please see this article on calculating your AOV. With one quick calculation, you can get a sense if each customer is ordering enough to cover your marketing costs, you can track ordering trends over months and years, you can find out if new customers are ordering at sustainable rates, and more. If you're curious about how your business stacks up against others in your industry, you can use the AOV as a measuring stick. It won't be a perfect comparison, but it may give you a good idea if your costs or prices are out of line. ### 4 Ways AOVs Can Steer You Wrong For all its advantages, AOV isn't the best tool for every situation, however. Here are some reasons you need to look beyond AOV for accurate data. ### 1. Your products have a wide price range. AOV works best when you have a tight product price range, like our example above, where the order was \$10, \$11 and \$9 for a \$10 AOV. Keep in mind that you can still have an AOV of \$10 if your order is made up of items costing \$3, \$24, and \$3. Without cross-referencing AOV with other data, like most-purchased items or the range between your highest and lowest orders, you could miss buying patterns that would easily skew your AOV. ### 2. AOV doesn't account for cost of goods sold (COGS). Some products are more profitable than others, but AOV doesn't take that into consideration. You may have a \$24 item that has a COGS of \$20, but a \$10 with a COGS of \$3. Your AOV won't tell you that it's better for your bottom line to sell more \$10 items. ### 3. Comparing AOVs can lead to false conclusions. It's important to pull your head out of daily operations to see how others in your industry are doing. You'll learn shortcuts and pick up on trends that can boost profitability or save you serious person-hours. Be very careful about comparing your AOV to other leading stores, however. AOVs for supplement stores will be very different from sporting goods or furniture. These differences may seem obvious, but if you don't choose stores stores that have a similar product line and customer base as yours, you can draw a lot of false conclusions. For example, if you're selling men's dress shirts and you're looking at AOVs for fashion, don't forget that women's couture fashion -- with an exceptionally high AOV -- could be thrown into a general "fashion" category. Even subcategories can have huge differences. Think of how the AOVs within men's fashion, like shoes versus accessories (wallets, belts, etc.), could be wildly different. ### 4. Differences in AOV might be just fine. Even if your store's AOV is lower than a closely-matched AOV in your market, it may not be a big deal. Instead of panicking, try to learn something about why your store is different. Does your location or the location of your target market make a difference? Are you intentionally choosing to sell at lower prices? Are you selling at a higher volume than other stores? Before you decide to make any changes to your prices or marketing plants, take a look at a few variables. With a little research, you can either find a legitimate reason for the gap in AOVs or you can pinpoint what you need to change. ### Compare, Don't Panic There are multiple reasons why AOVs vary, so don't jump to any conclusions as you compare. In fact, the best kind of comparisons you can make with AOVs are ones within your own business. See how your AOV this month compares to a year ago or last month. It's more important to see trends within your own business than keep up with your business peers. ## Did all of those holiday shoppers ever come back? Compare how your last winter customers performed over the year with the Winter Holiday report in Repeat Customer Insights.

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# Atlanta Ga Airport Map 20 views 5 / 5 ( 1votes ) At the time you think of Atlanta Ga Airport Map, exactly what are you thinking of? In essence a map is a representation of a topology or function. To get example a formula such as X=2Y maps a worth of Y to each value of Times. Of course we all believe that mathematicians are weird and sometimes hard to understand but they have you ever seen a schematic map of a subway (underground railway) system? Have you ever ever seen the same network of rails specified on a more "normal" Atlanta Ga Airport Map of the location in which it is located? Different Atlanta Ga Airport Map of the extremely same thing can look quite different. At the time you make a Atlanta Ga Airport Map of your flat area - a "plan" or "elevation" - things are quite simple, but when you make an effort to map a larger area, like the surface of an complete planet, things can get quite complicated if you need your map to be level. It really is all very well to make a world, but try turning the area of that globe into a set Atlanta Ga Airport Map! Yikes! However you begin it, you finish plan edge-effects. As I write this information I am actually engaged in programming map-generating programs designed to generate maps of fictional landscapes. I happen to be examining the map-generators that are included in the free, open-source (GNU GPL licensed) strategy game, FreeCiv. Edge results are extremely apparent in such maps. The Atlanta Ga Airport Map are basically rectangular, but you can choose to obtain them act like cylinders by "wrapping" left to right or top to bottom level, or you may also have "wrap" in both guidelines. Most often people determine on "wrap" only kept to right, and obstruct the very best and bottom with "polar regions". Such basic "wrapping" makes for quite extreme distortion though if you give it a try with a real Atlanta Ga Airport Map worldwide!

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A Few Numbers With Strange Qualities 1. Jan 1, 2013 suzi9spal Hello everyone! I'm a 1st grade student in a slovenian high school. Recently, I've became very intrested in numbers and math itsself. I was just playing around with a few numbers, when I came across a few that are unique. I was wondering wich numbers can loop back to themselfes after this procedure; n is a digit, m is a digit, o is an integer, p is another integer Code (Text): o = mn p = m^2 + n^2 o = p ...repeat... For instance, 16 is one of these numbers; 1^2 + 6^2 = 1+36 = 37 ==> 3^2 + 7^2 = 9+49 = 58 ==> 5^2 + 8^2 = 25+64 = 89 ==> 8^2 + 9^2 = 64+81 = 145 ==> 1^2 + 4^2 + 5^2 = 1+16+25 = 42 ==> 4^2 + 2^2 = 16+4 = 20 ==> 2^2 + 0^2 = 4+0 = 4 ==> 4^2 = 16 I then created a program to find every single number that has this property. It turns out, there are only a limited few: • 1 • 4 • 16 • 20 • 37 • 42 • 58 • 89 • 145 I thought this was intresting, so I posted it here. Did anyone find something like this before? 2. Jan 2, 2013 dodo Hello, suzi9spal, and happy new year! Personally I've never seen this before, but I found something after googling a bit. The following link describes it, and also contains a reference to a book by J. Madachy (which I haven't read) that explores this and other similar number games. http://mathworld.wolfram.com/RecurringDigitalInvariant.html Notice that, apart from 1 where the property is trivially true, the other numbers that you discover form a cycle, and you can start from any of them and obtain the same collection of numbers. These are the "cycles" that the link above refers to. And keep these kinds of hobbies! They will greatly pay off when you go to the university. Last edited: Jan 2, 2013 3. Jan 2, 2013 epsi00 here's a helpful link. If you sequence is new, you will not find it here. http://oeis.org/ if it is known, then you will find it. 4. Jan 2, 2013 skiller And you have proved that? Last edited: Jan 2, 2013 5. Jan 2, 2013 willem2 It's not very hard. If you start with an n-digit number, the largest number that could follow it is 81n. A 4-digit number would have at most 324 following it. All the numbers following it will have 3-digits or less, so a 4-digit or larger number can't be part of a loop, so there are only 1000 numbers you have to check. 6. Jan 2, 2013 skiller I'm sure you're correct but I don't follow what you've said. (You're probably right and I just need a lie down...) However, it is intriguing that, other than the trivial 0 and 1, the other numbers in this list all sum up to 16 in their digits. 7. Jan 5, 2013 suzi9spal oay, I have tested every single number to about 13 milion :D 8. Jan 5, 2013 skiller I don't get how you've gone from an n-digit to a 4-digit with no explanation. I'm sure you have, and I don't disbelieve you, I just dont see the proof! 9. Jan 5, 2013 dodo What willem2 was saying is that, for all possible n-digit numbers, the sum of squared digits is at its largest when the n-digit number is 9999... (n nines), and that sum is 9^2 . n = 81n. So, for different values of n (number of digits), you can see that n=1: 81n = 81 (so, the sum of squared digits has at most 2 digits) n=2: 81n = 162 (so, the sum of squared digits has at most 3 digits) n=3: 81n = 243 (so, the sum of squared digits has at most 3 digits) n=4: 81n = 324 (so, the sum of squared digits has at most 3 digits) You can see that, when applying the process of adding the squared digits, a 4-digit number can only become smaller (since the result won't be larger than 324). For n>4, you have that 81n < 10^n (just like the case n=4), so numbers of 5,6,7... digits also become smaller when adding the squared digits. Therefore, for numbers of 4,5,6,7,... digits, the process of adding the squared digits has no chance of returning to the original number. So you only need to test numbers up to 3 digits. 10. Jan 6, 2013 Curious3141 This has already been described on Sloane's database: http://oeis.org/A039943 It's tagged "fini" for finite - confirming that this is the exhaustive listing. I love that resource! I've got a few sequences accepted long ago. 11. Jan 6, 2013 Curious3141 I like the OP's interest in this sort of thing, because it reminds me of myself at that age. OP, if you're interested in a genuinely unsolved (and maddening) integer-recursion problem, look up the Collatz conjecture. And before you write a program for that one, try starting out with 27 using just pen and paper (and a calculator, if you wish). 12. Jan 6, 2013 skiller Nope, you need to explain it more clearly!

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## Exercise 12 - Charts Still using the star data from exercises 8 and 10 (but not the corrupted version of exercise 11), plot the stars on a chart. The scale should be 30 pixels per degree (making a chart about 600 pixels high and somewhat less wide). Note that in computer graphics the vertical axis starts with zero at the top of the canvas and works downwards (historically because TV scans go downwards). Also, astronomically, the RA coordinate increases from right to left (because of the apparent direction of rotation of the sky around the Earth). Each star should be a circle filled in black. The radius of the circle should reflect the star's magnitude (see box below). Join stars with thin black lines in an appropriate pattern to show the shape of the constellation. Label each star with its Greek symbol. Unfortunately you cannot use the HTML entities as you did in Exercise 9. Instead you will need the Unicode character code for each. You can easily find the codes by searching on the Internet, eg, for "unicode alpha". Use the \uxxxx notation to put them in your program. ## Magnitudes The magnitude scale is a logarithmic scale of inverse brightness. The brightest stars have magnitude 0 and the faintest that can be seen by the naked eye (from a rural location, away from street lights) is about 6. So a smaller magnitude value means a brighter star. For the purposes of this exercise, where the magnitude range is from 0.18 to 2.25, use this formula for the radius of the circular dot to represent a star: r = (2.75 - vmag) * 4;

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midsol[1] # midsol[1] - MAE294B/SIO203B Methods in Applied Mechanics... This preview shows pages 1–2. Sign up to view the full content. MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Midterm solution 1 Sketch the bifurcation diagram for the equation ˙ x = 4 - 4 x 2 - μ 2 4 + 4 x 2 + μ 2 ( μ - x 2 ) 2 - 1 ( μ - x 2 ) 2 + 1 and state the nature of the bifurcations. A graphical answer will suffice. See Figure 1. There are saddle-node bifurcations at ( - 2 , 0 ) , ( 0 , 0 ) , ( 1 , 0 ) and ( 2 , 0 ) . There are transcritical bifurcations at ( 0 , - 1 ) , ( 0 , 1 ) , ( μ c , x c ) and ( μ c , - x c ) , where μ c = 2 ( 3 - 1 ) and x c = p 2 3 - 2 satisfy μ = 1 + x 2 and x 2 + μ 2 / 4 = 1. 2 Skip the naive expansion. The leading-order solution is x 0 = A ( T ) e i t + A * e - i t + c . c . At O ( ε ) , one finds x 1 tt + x 1 + 2 x 0 tT + x 2 0 x 0 t + x 3 0 = 0 . Secular terms look like e ± i t so look at x 2 0 ( x 0 t + x 0 ) = ( A e i t + A * e - i t ) 2 ( i A e i t - i A * e - i t + A e cit + A * e - i t ) = ( 3 + i ) A 2 A * e i t + ··· . The amplitude equation is This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. ## This note was uploaded on 09/22/2010 for the course MAE MAE294B taught by Professor Mae294b during the Winter '09 term at UCSD. ### Page1 / 4 midsol[1] - MAE294B/SIO203B Methods in Applied Mechanics... This preview shows document pages 1 - 2. Sign up to view the full document. View Full Document Ask a homework question - tutors are online

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https://community.fabric.microsoft.com/t5/Desktop/Assign-ranks-with-multiple-invoices-from-multiple-customer-IDs/m-p/970203/highlight/true

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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 Anonymous Not applicable ## Assign ranks with multiple invoices from multiple customer IDs I am trying to calculate total sales based on invoice amount. For example: A customer may have purchased four different times, therefore assigning four different invoice numbers. I am using "invoice date" in ascending order as either transaction 1, 2, 3 etc (if they purhcased on 1/1 for \$20 and 1/5 for \$40, first invoice is \$20 and second invoice is \$40). I've been using MIN and MAX based on "invoice date" to calculate total sales based on customer ID. This works fine until they purhcase more than TWICE. If Customer ID 123 purchased on 1/1, 1/5, 1/8 and 1/10 for \$20, \$30, \$40 and \$50, I would like to create a dax measure that represents all of these sales figures based on what invoice (in ascending order) they might be on? So my measure would look like this: First Invoice for \$20, Second for \$30, Third for \$40 and Fourth for \$50. I've tried ranking these invoices based on "Invoice Date" but that just created more problems. Any help would be greatly appeciated. 1 ACCEPTED SOLUTION Community Support Hi, I create a sample to test: Then try this rank column: Rank = RANKX('Table',VALUE('Table'[Invoice Num]),,ASC,Dense) The result shows: When applying filter to table visual, it still shows the rank starting from 1: Here is my test pbix file: Hope this can help. Best Regards, Giotto Zhi 4 REPLIES 4 Community Support Hi, According to your description, i create a sample to test: Then try this measure: Invoice No. = RANKX(CALCULATETABLE(DISTINCT('Table'[Invoice Date]),FILTER(ALLSELECTED('Table'),'Table'[Customer ID] in FILTERS('Table'[Customer ID]))),CALCULATE(SUM('Table'[Payment])),,ASC,Dense) The result shows the rank grouped by [CustomerID]: Hope this helps. Best Regards, Giotto Zhi Anonymous Not applicable Thank you for the reply. I don't think I explained myself well enough, I apologize. What I want is to run averages, max and mins on any second invoice, third invoice, fourth etc. See attached. What you see is Cust ID filtered down to just that one customer (SW10671)... That one customer has purchased six different times. The one unique thing about this data set is Invoice Number (Invoice Num) is unique and it assigns a new invoice number chronologically so I know we can use Invoice Num as the RANK. What i want the "Rankings" column to show is 1 for Inv Num: 264112, 2 for Inv Num: 264113, 3 for Inv Num: 264629.... and 5 for Inv Num: 267206... I also am hoping that when I unfilter Cust ID from SW10671 to ALL, that those exact same numbers will still show in "Rankings". What i intend to do is calculate second invoice, third invoice, fourth etc totals... Thank you Community Support Hi, I create a sample to test: Then try this rank column: Rank = RANKX('Table',VALUE('Table'[Invoice Num]),,ASC,Dense) The result shows: When applying filter to table visual, it still shows the rank starting from 1: Here is my test pbix file: Hope this can help. Best Regards, Giotto Zhi Anonymous Not applicable I believe this worked. Thanks for your help!

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Pinexam——There is royal road to learning. ## What is the slope of the line pictured in the standard (x, y) coordinate plane below that passes through (1,3) and (5,5) in the standard (x, y) coordinate plane? (A)-2 (B) (C) (D) (E) Andrea wants to fill in two sections of her backyard with sod that must be purchased in 2-x-2-foot squares. If the two sections measure 30 x 40 feet and 60 x 80 feet, how many squares of sod does she need to buy? (F) 1,000(G) 1,250(H) 1,500(J) 1,600(K) 2,000

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https://www.physicsforums.com/threads/open-sets-in-a-discrete-metric-space.1042438/

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# Open Sets in a Discrete Metric Space .... .... • MHB • Math Amateur In summary, open balls in a discrete metric space are either singleton sets or the entire space. The union of any collection of open sets is open, and since any singleton set is an open ball, any union of singleton sets is also open. This means that every set in a discrete metric space is open and closed. Math Amateur Gold Member MHB In a discrete metric space open balls are either singleton sets or the whole space ... Is the situation the same for open sets or can there be sets of two, three ... elements ... ? If there can be two, three ... elements ... how would we prove that they exist ... ? Essentially, given the metric or distance function, I am struggling to see how in forming a set of the union of two (or more) singleton sets you can avoid including other elements of the space ... Peter Last edited: As you say, open balls are either singleton sets or the entire space. But the union of any collection of open sets are open. Since any singleton sets are open balls (so open sets) any union of singleton sets is open. But any set is a union of singleton sets! Therefore every set is open in the discrete metric. (And every set is closed.) HallsofIvy said: As you say, open balls are either singleton sets or the entire space. But the union of any collection of open sets are open. Since any singleton sets are open balls (so open sets) any union of singleton sets is open. But any set is a union of singleton sets! Therefore every set is open in the discrete metric. (And every set is closed.) Thanks HallsofIvy ... Peter ## 1. What is a discrete metric space? A discrete metric space is a mathematical concept that consists of a set of points or elements with a distance function defined between them. In a discrete metric space, the distance between any two points is either 0 or 1, making it a discrete or "separated" space. ## 2. What are open sets in a discrete metric space? In a discrete metric space, an open set is a collection of points that do not include their boundary points. In other words, for any point in an open set, there exists a small enough radius where all the points within that radius are also in the set. ## 3. How are open sets different from closed sets in a discrete metric space? Unlike open sets, closed sets in a discrete metric space include their boundary points. This means that for any point in a closed set, there exists a small enough radius where all the points within that radius are also in the set, including the boundary points. ## 4. What is the significance of open sets in a discrete metric space? In a discrete metric space, open sets play a crucial role in defining continuity and convergence of sequences. They also help in defining concepts such as compactness and connectedness, which are important in many areas of mathematics and science. ## 5. Can you give an example of an open set in a discrete metric space? One example of an open set in a discrete metric space is the set of all natural numbers. In this set, any point has a small enough radius where all the points within that radius are also in the set. For instance, the point 5 has a radius of 1, where all the points within that radius (4, 5, and 6) are also in the set of natural numbers. Replies 2 Views 249 Replies 32 Views 2K Replies 2 Views 2K Replies 3 Views 2K Replies 4 Views 1K Replies 2 Views 2K Replies 3 Views 1K Replies 2 Views 807 Replies 3 Views 1K Replies 2 Views 758

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https://rdrr.io/cran/LaplacesDemon/src/R/Gelfand.Diagnostic.R

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# R/Gelfand.Diagnostic.R In LaplacesDemon: Complete Environment for Bayesian Inference #### Documented in Gelfand.Diagnostic ```########################################################################### # Gelfand.Diagnostic # # # # The Gelfand.Diagnostic function is an interpretation of Gelfand's # # ``thick felt-tip pen'' MCMC convergence diagnostic (Gelfand et al., # # 1990). # ########################################################################### Gelfand.Diagnostic <- function(x, k=3, pen=FALSE) { ### Initial Checks if(missing(x)) stop("The x argument is required.") if(!is.vector(x)) x <- as.vector(x) if(k < 2) k <- 2 if(k > length(x)/2) k <- round(length(x)/2) if({length(x)/k} < 2) stop("k is too large relative to length(x).") ### KDE quantiles <- seq(from=0, to=1, by=1/k) breaks <- round(as.vector(quantiles)*length(x)) breaks <- breaks[-1] d.temp <- density(x) d <- array(c(d.temp\$x, d.temp\$y), dim=c(length(d.temp\$x), 2, length(breaks))) d.temp <- density(x[1:breaks[1]]) d[,,1] <- c(d.temp\$x, d.temp\$y) for (i in 2:length(breaks)) { d.temp <- density(x[1:breaks[i]]) d[,,i] <- c(d.temp\$x, d.temp\$y)} ### Plots ymax <- max(d[,2,]) col.list <- c("red", "green", "blue", "yellow", "purple", "orange", "brown", "gray", "burlywood", "aquamarine") col.list <- rep(col.list, len=length(breaks)) rgb.temp <- as.vector(col2rgb(col.list[1])) mycol <- rgb(red=rgb.temp[1], green=rgb.temp[2], blue=rgb.temp[3], alpha=50, maxColorValue=255) plot(d[,1,1], d[,2,1], type="l", col=mycol, xlim=c(range(d[,1,])), ylim=c(0,ymax), main="Gelfand Diagnostic", xlab=deparse(substitute(x)), ylab="Density") polygon(x=d[,1,1], y=d[,2,1], col=mycol, border=NULL) for (i in 2:length(breaks)) { rgb.temp <- as.vector(col2rgb(col.list[i])) mycol <- rgb(red=rgb.temp[1], green=rgb.temp[2], blue=rgb.temp[3], alpha=50, maxColorValue=255) lines(d[,1,i], d[,2,i], col=mycol) polygon(x=d[,1,i], y=d[,2,i], col=mycol, border=mycol) lines(d[,1,i], d[,2,i], lty=i)} if(pen == TRUE) abline(v=mean(range(d[,1,])), col="black", lwd=10) legend(quantile(d[,1,], probs=0.025), round(ymax*0.9,2), legend=paste("1:",breaks,sep=""), lty=1:k, title="Samples") return(invisible(x)) } #End ``` ## Try the LaplacesDemon package in your browser Any scripts or data that you put into this service are public. LaplacesDemon documentation built on July 1, 2018, 9:02 a.m.

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https://www.jobilize.com/trigonometry/test/verbal-radicals-and-rational-exponents-by-openstax?qcr=www.quizover.com

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# 1.3 Radicals and rational exponents  (Page 5/11) Page 5 / 11 ## Simplifying rational exponents Simplify: 1. $5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)$ 2. ${\left(\frac{16}{9}\right)}^{-\frac{1}{2}}$ Simplify $\text{\hspace{0.17em}}{\left(8x\right)}^{\frac{1}{3}}\left(14{x}^{\frac{6}{5}}\right).$ $28{x}^{\frac{23}{15}}$ Access these online resources for additional instruction and practice with radicals and rational exponents. ## Key concepts • The principal square root of a number $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the nonnegative number that when multiplied by itself equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ See [link] . • If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the product $\text{\hspace{0.17em}}ab\text{\hspace{0.17em}}$ is equal to the product of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] . • If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the quotient $\text{\hspace{0.17em}}\frac{a}{b}\text{\hspace{0.17em}}$ is equal to the quotient of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] . • Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See [link] and [link] . • The principal n th root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the number with the same sign as $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that when raised to the n th power equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ These roots have the same properties as square roots. See [link] . • The properties of exponents apply to rational exponents. See [link] . ## Verbal What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain. When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1. Where would radicals come in the order of operations? Explain why. Every number will have two square roots. What is the principal square root? The principal square root is the nonnegative root of the number. Can a radical with a negative radicand have a real square root? Why or why not? ## Numeric For the following exercises, simplify each expression. $\sqrt{256}$ 16 $\sqrt{\sqrt{256}}$ $\sqrt{4\left(9+16\right)}$ 10 $\sqrt{289}-\sqrt{121}$ $\sqrt{196}$ 14 $\sqrt{1}$ $\sqrt{98}$ $7\sqrt{2}$ $\sqrt{\frac{27}{64}}$ $\sqrt{\frac{81}{5}}$ $\frac{9\sqrt{5}}{5}$ $\sqrt{800}$ $\sqrt{169}+\sqrt{144}$ 25 $\sqrt{\frac{8}{50}}$ $\frac{18}{\sqrt{162}}$ $\sqrt{2}$ $\sqrt{192}$ $14\sqrt{6}-6\sqrt{24}$ $2\sqrt{6}$ $15\sqrt{5}+7\sqrt{45}$ $\sqrt{150}$ $5\sqrt{6}$ $\sqrt{\frac{96}{100}}$ $\left(\sqrt{42}\right)\left(\sqrt{30}\right)$ $6\sqrt{35}$ $12\sqrt{3}-4\sqrt{75}$ $\sqrt{\frac{4}{225}}$ $\frac{2}{15}$ $\sqrt{\frac{405}{324}}$ $\sqrt{\frac{360}{361}}$ $\frac{6\sqrt{10}}{19}$ $\frac{5}{1+\sqrt{3}}$ $\frac{8}{1-\sqrt{17}}$ $-\frac{1+\sqrt{17}}{2}$ $\sqrt[4]{16}$ $\sqrt[3]{128}+3\sqrt[3]{2}$ $7\sqrt[3]{2}$ $\sqrt[5]{\frac{-32}{243}}$ $\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$ $15\sqrt{5}$ $3\sqrt[3]{-432}+\sqrt[3]{16}$ ## Algebraic For the following exercises, simplify each expression. $\sqrt{400{x}^{4}}$ $20{x}^{2}$ $\sqrt{4{y}^{2}}$ $\sqrt{49p}$ $7\sqrt{p}$ ${\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}$ ${m}^{\frac{5}{2}}\sqrt{289}$ $17{m}^{2}\sqrt{m}$ $9\sqrt{3{m}^{2}}+\sqrt{27}$ $3\sqrt{a{b}^{2}}-b\sqrt{a}$ $2b\sqrt{a}$ $\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}$ $\sqrt{\frac{225{x}^{3}}{49x}}$ $\frac{15x}{7}$ $3\sqrt{44z}+\sqrt{99z}$ $\sqrt{50{y}^{8}}$ $5{y}^{4}\sqrt{2}$ $\sqrt{490b{c}^{2}}$ $\sqrt{\frac{32}{14d}}$ $\frac{4\sqrt{7d}}{7d}$ ${q}^{\frac{3}{2}}\sqrt{63p}$ $\frac{\sqrt{8}}{1-\sqrt{3x}}$ $\frac{2\sqrt{2}+2\sqrt{6x}}{1-3x}$ $\sqrt{\frac{20}{121{d}^{4}}}$ ${w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}$ $-w\sqrt{2w}$ $\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}$ $\frac{\sqrt{12x}}{2+2\sqrt{3}}$ $\frac{3\sqrt{x}-\sqrt{3x}}{2}$ $\sqrt{147{k}^{3}}$ $\sqrt{125{n}^{10}}$ $5{n}^{5}\sqrt{5}$ $\sqrt{\frac{42q}{36{q}^{3}}}$ $\sqrt{\frac{81m}{361{m}^{2}}}$ $\frac{9\sqrt{m}}{19m}$ $\sqrt{72c}-2\sqrt{2c}$ $\sqrt{\frac{144}{324{d}^{2}}}$ $\frac{2}{3d}$ $\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}$ $\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}$ $\frac{3\sqrt[4]{2{x}^{2}}}{2}$ $\sqrt[3]{64y}$ $\sqrt[3]{128{z}^{3}}-\sqrt[3]{-16{z}^{3}}$ $6z\sqrt[3]{2}$ $\sqrt[5]{1,024{c}^{10}}$ ## Real-world applications A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\text{\hspace{0.17em}}\sqrt{90,000+160,000}.\text{\hspace{0.17em}}$ What is the length of the guy wire? 500 feet A car accelerates at a rate of where t is the time in seconds after the car moves from rest. Simplify the expression. ## Extensions For the following exercises, simplify each expression. $\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}$ $\frac{-5\sqrt{2}-6}{7}$ $\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}$ $\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{-3}}}\cdot \frac{{a}^{-7}{n}^{-2}}{\sqrt{{m}^{2}{c}^{4}}}$ $\frac{\sqrt{mnc}}{{a}^{9}cmn}$ $\frac{a}{a-\sqrt{c}}$ $\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$ $\frac{2\sqrt{2}x+\sqrt{2}}{4}$ $\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)$ $\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$ $\frac{\sqrt{3}}{3}$ write down the polynomial function with root 1/3,2,-3 with solution if A and B are subspaces of V prove that (A+B)/B=A/(A-B) write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°) Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4 what is the answer to dividing negative index In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c. give me the waec 2019 questions the polar co-ordinate of the point (-1, -1) prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x tanh`(x-iy) =A+iB, find A and B B=Ai-itan(hx-hiy) Rukmini what is the addition of 101011 with 101010 If those numbers are binary, it's 1010101. If they are base 10, it's 202021. Jack extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve 1+cos²A/cos²A=2cosec²A-1 test for convergence the series 1+x/2+2!/9x3

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https://philosophy.stackexchange.com/questions/27972/selection-of-logical-connectives-%C2%AC-%E2%88%A8-%E2%88%A7-%E2%87%92-%E2%87%94-in-set-theory/27973

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# Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory? Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical connectives, namely "not" `¬`, "inclusive or" `∨`, "and" `∧`, "implies `⇒`, and "if and only if" `⇔` (as well as the existential and universal quantifiers `∃` and `∀`). However, this set `{¬,∨,∧,⇒,⇔}` of connectives chosen: • Is not minimal, i.e. is actually redundant (For example, `P⇒Q` could be written `¬P∨Q`, and since we already have introduced `¬` and `∨` as part of our set, we could eliminate `⇒`.) • Is not exhaustive either, since there are actually 16 possible compound statements (and corresponding logical connectives) to choose from. (Since {¬,∨,∧,⇒,⇔} is already redundant, why not throw in the other 11 connectives, some of which are VERY helpful like "nand" `⊼` , "nor" `⊽` and "exclusive or" `⊻`?) Since it is neither minimal nor exhaustive, the set `{¬,∨,∧,⇒,⇔}` seems like an arbitrary choice. Is there another explanation? And since the set is already redundant (so there's no use in aiming to make it minimal), would it be acceptable to include the others that are missing, so we are at least making use of all 16 connectives (and have more at our disposal to work with)? ## 7 Answers Is not exhaustive either, since there are actually 16 possible compound statements (and corresponding logical connectives) to choose from. (Since {¬,∨,∧,⇒,⇔} is already redundant, why not throw in the other 11 connectives, some of which are VERY helpful like "nand" ⊼ , "nor" ⊽ and "exclusive or" ⊻?) Some of the "16 possible compound statements" are in fact trivial cases (and also the ¬ appears twice). Actually, only five of the sixteen cannot be made with one of the standard five operators. See the following table: ``````Table Name Value for x..y ------------------------------------------------------ 0000 Contradiction False 0001 Conjunction x ∧ y 0010 Nonimplication ¬(x ⇒ y) 0011 Left projection x 0100 Converse nonimplication ¬(y ⇒ x) 0101 Right projection y 0110 Exclusive disjunction ¬(x ⇔ y) 0111 Inclusive disjunction x ∨ y 1000 Nondisjunction ¬(x ∨ y) 1001 Equivalence x ⇔ y 1010 Right complementation ¬y 1011 Converse implication y ⇒ x 1100 Left complementation ¬x 1101 Implication x ⇒ y 1110 Nonconjunction ¬(x ∧ y) 1111 Affirmation True `````` As you can see, only five (nonimplication, converse nonimplication, exclusive disjunction, nondisjunction, nonconjunction) are not in this 'standard set'. There are however books which also introduce ⊻, exclusive disjunction, as a standard operator. I'm helping in a computing science course about basic math, and last week someone asked me: Why do we have a symbol for ⊆ (subset), if we already have ⊂ (proper subset) and = (equality)? "a ⊆ b ≡ a ⊂ b ∨ a = b", so the operator is redundant. I couldn't come up with a better answer than "Because mathematicians are lazy, and want to write things as short as possible". Clearly, that's jumping to conclusions - but in fact I think it's quite likely there is something true in there. One might ask, "why was × (multiplication) defined?", because in the natural numbers you can simply add: 5 × 3 = 5 + 5 + 5 = 3 + 3 + 3 + 3 + 3. Going further, you can ask, "Why were 2 and 3 defined, if you can also write 1+1 and 1+1+1?" At some point, it's really to much work to write everything down, hence more notation was introduced. Of course, you are allowed to define your own notation. By defining nonimplication, exclusive disjunction, nondisjunction and nonconjunction, you have an exhaustive set. Define the ones you need often at the top of your writing. So, how did we get to this standard set of logical operators? By using them, and finding out which ones we need often. Also note that the five statements that don't exist can all be formed by negating another operator (see the fourth column in the table above) and that this is not possible if you leave any of the 'standard' five out. • Most of this is excellent but "laziness" is a terrible explanation. Attributing the use of ⊆ to laziness suggests that it would actually be better to use the minimal possible number of symbols in a text, which just isn't true. We use ⊆ it matches the underlying concept, which is that of "subset". "Proper subset" is a special case, most easily defined in terms of "subset". Also, ⊆ is overwhelmingly more common than ⊂, so it makes sense to have and use a symbol for that concept. If you forced me never to write one of those symbols again, I'd drop ⊂ and start writing "A⊆B ∧ A≠B". – David Richerby Sep 8 '15 at 8:24 • @DavidRicherby well, maybe laziness is not the right term. But obviously we gain clarity in some cases by using a dedicated symbol. Clearly my answer as I quoted it above is jumping to conclusions, that's why it should be related to the paragraph below; "there is something true in there". I will make that more clear, thanks. – Keelan Sep 8 '15 at 8:49 • That's a good answer but the example of multiplication is unfortunate. How do you write "a x b" with addition when a and b are variables? Multiplication is a concept with many implications, not a scripture we use because we're lazy. – Quentin Ruyant Sep 12 '15 at 19:22 • @quen_tin "b + b + ... + b (a times)". But yes, every analogy has its limits. – Keelan Sep 12 '15 at 20:42 The the set {¬,∨,∧,⇒,⇔} is usually used because it includes the "most natural" ones. If we start with a minimal set, like {¬,⇒}, the other ones are usually introduced as abbreviations. There is no "deep" reason : mainly tradition, and a "reasonable" trade-off between savings (minimality) and readibility (to express p ∧ q as ¬(p ⇒ ¬q) is not so "natural". gnasher729 raised an important point that deserves some expansion: "In formal logic, implication x ⇒ y and equivalence x ⇔ y are very obviously useful - they directly express the possibly most important concepts of formal logic." The main point that I want to bring up is this: Naive set theory is not the only important set theory, and Classical logic is not the only important logic. In most logics, it isn't true that P⇒Q is the same thing as ¬P∨Q. As you get more into logic, you'll gradually realise that implication is (in a very deep sense) "more fundamental" than even conjunction or disjunction. But I can give you a taste of that now. You don't need the other connectives if you have quantification over propositions and implication. We will use the usual convention that implication associates to the right (that is, A⇒B⇒C means A⇒(B⇒C); you can reason about this by noting that A⇒B⇒C is equivalent to (A∧B)⇒C). Then you can define them in terms of Gentzen rules: • P∧Q is equivalent to ∀S.(P⇒Q⇒S)⇒S • P∨Q is equivalent to ∀S.(P⇒S)⇒(Q⇒S)⇒S • ¬P is equivalent to ∀S.P⇒S You may recognise that last one as ex falso quodlibet. Quantification over propositions seems like something "new", but actually it isn't. In fact, when you said that "P⇒Q could be written ¬P∨Q", you're actually making a claim that some statement is true for all propositions P and Q, which is universal quantification over propositions. We tend to hide that detail from undergraduates so that they can get familiar with propositional logic without having to think about quantification. Final thought: The "deep sense" which I alluded to earlier is that logical entailment is a morphism in category theory, and implication is an exponential object. I don't expect you to understand that sentence, but an example may help. Consider modus ponens. If we have P⇒Q and we have P, then we can conclude Q. However in set theory, if we have a function f : A → B and a value x : A, then we can apply the function to the value, giving f(x) : B. This, it turns out, is not just a trick of notation, or a coincidence. It's a connection which goes very deep. There are two possible logical functions with no inputs, "true" and "false". We don't use symbols for them, just names. There are four possible logical functions with one input: "True" and "false" (whatever the input is, the output is "true", or whatever the input is, the output is "false"), identity (output = input) and negation (output = opposite of input). Three of these don't require a symbol. For the fourth one we use the symbol ¬. That's one of your five symbols, and we'd really want a symbol for that function. There are 16 possible logical functions with two inputs. Six of those don't actually "use" both inputs; if we call the inputs x and y then these six functions are "false", "true", x, y, ¬x and ¬y. So there are 10 functions left. In formal logic, implication x ⇒ y and equivalence x ⇔ y are very obviously useful - they directly express the possibly most important concepts of formal logic. In other areas involved with logic they are much less important, but they are extremly important in formal logic. Having a symbol for "a implies b" we don't really need one for "a is implied by b", we can just write "b implies a". (You might use a right to left arrow; some would say this isn't actually a different symbol). If we add the negations (no extra symbol for "a is not equivalent to b" and "a doesn't imply b") then another six functions are covered, four are left. For the last four, several reasonable ways to represent them with symbols are possible. We have a symbol for logical and ("both") and logical or ("at least one", or "one or both") and together with their negations everything is covered. We could have used "none", "at most one", but logical and and logical or have won the competition. Add their negations, and everything is covered. That's formal logic; other areas using logic have other demands. In decision making (often used in software development), logical and, logical or, and negation are most widely used. Implication and equivalence are rarely used. In computer hardware, nand (not (a and b)) and nor (not (a or b)) are quite naturally implemented by the most simple computer hardware, and everything is based on these two functions. However, this is an area that isn't meant for human consumption, unlike formal logic. • Actually, we do have symbols for true and false: ⊤ and ⊥, respectively. – Keelan Sep 7 '15 at 21:23 • But we don't need them. True and False do just fine. These symbols do nothing but put up an artificial barrier to understanding. – gnasher729 Sep 8 '15 at 20:34 Outside of very foundational logical contexts, there's really not much to gain by starting from a minimal adequate set of connectives and defining the rest. All it does is unduly complicate things. Even in logic, we could start with the Sheffer stroke instead of {~, ->}, but prefer not to unduly confuse our undergraduates. Allow me to give you a completely different reason, why, at least, introducing ∨,∧,⇒ separately may have its merits. Classical logic is nice and all, but some people actually do care about intuitionistic logic. In intuitionistic logic you cannot define any of the connectives ¬,∨,∧,⇒ in terms of any two others. This is because, many familiar laws like (one of) the deMorgan laws or ¬A∨B ⇔ (A ⇒ B) fail to hold. There is actually a type of semantics for intuitionistic (propositional) logic given by bicartesian closed categories. From this standpoint, this "issue" is completely natural: In these bicartesian closed categories we have binary products of propositions given by "∧", binary coproducts given by "∨" and exponentials given by "⇒". You might say, that I'm still missing the ¬. Bicartesian closed categories have an initial object. It corresponds to the symbol ⊥ for "falsehood" or "contradiction". The negation of a proposition A is then just A ⇒ ⊥ (this works in classical logic too). There is no reason to believe (and we know it's not possible), that these constructions can be reduced to each other (It doesn't work in familiar categories, like the categories of sets and functions, either). So, if you care about category theory or intuitionistic logic you would naturally introduce ⊥, ∧, ∨ and ⇒ seperately (actually I'm missing the terminal object ⊤, which denotes "truth", but it's not really important here). Given this, it also doesn't make sense to define "all possible binary connectives" as there are no truth values in intuitionistic logic and so there are infinitely many possible connectives (and not just 16). However, the connectives ∧, ∨ and ⇒ are, you could say, special, since each one satisfies a rather basic universal mapping property (this can partly be seen by the familiar "elimination" and "introduction" rules in "Natural Deduction"). • Hi Stefan, thanks for your answer. The question however is why use only those 5 symbols in your language and stop there - why not define all 16? – Mathemanic Sep 8 '15 at 15:10 • @EthanAlvaree I extended my answer, don't know whether it helps though. The details are a little bit fuzzy, I'm no expert after all. – Stefan Perko Sep 8 '15 at 17:17 Mathematicians are lazy... So lazy than some use computers... But computers need to be simple to be implemented and miniaturized electronically... Then, like CISC .vs. RISC, only negation, conjonction (and) and disjonction (inclusive-or), or better NAND (not-and) and NOR (not-or) are required to express any logical sets (see Karnaugh solving, Church-Turing theorem, ASIC programming) Then, use high level presentations with many operators to make it look like short then apparently simple... use reduced set of operators to run it easily and quickly There are plenty 'compilers' to convert... • I don't think it's likely that formal logic has its origins in processor architecture. – Keelan Sep 7 '15 at 11:56

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1 You visited us 1 times! Enjoying our articles? Unlock Full Access! Question # The vapour density of N2O4 and NO2 mixture at a certain temperature is 30. Calculate the percentage dissociation of N2O4 at this temperature. N2O4(g)⇌2NO2(g). A 53.33% Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses B 58.56% No worries! We‘ve got your back. Try BYJU‘S free classes today! C 62.43% No worries! We‘ve got your back. Try BYJU‘S free classes today! D 67.67% No worries! We‘ve got your back. Try BYJU‘S free classes today! Open in App Solution ## The correct option is B 53.33%N2O4⇆2NO2101−x2xInitial vapour density = 46.vapour density at this temp. = 30.So, (1−x)×92+2x×46(1−x)+2x×12=30⟹921+x×12=30⟹461+x=30⟹4630=1+xSo, x=4630−1So % dissociation =0.53×100=53% Suggest Corrections 0 Join BYJU'S Learning Program Related Videos Hooke's Law PHYSICS Watch in App Explore more Join BYJU'S Learning Program

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# Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 \simeq SU(2)/U(1)$, it must hold that all its line bundles are indexed by the integers $L_k$, for $k \in Z$, and more generally, its rank-$k$ (equivarian) vector bundles are of the form $$L_{\bf z} = L_{z_1} \oplus \cdots \oplus L_{z_k}, {\text ~~~ for ~~~ } {\bf z} \in Z^k.$$ Does this then extend to all the total flag manifolds $F(n)$, ie the spaces of the form $$F(n) := SU(n)/(U(1)^{\otimes n-1}).$$ Edit: I omitted the word equivariant by mistake and have now entered it as (equivariant) I think perhaps the confusion stems from the following. It is true that the category of $G$-equivariant $G$-bundles on $G/H$ is equivalent to the category of representations of $H$. The flag variety $\mathcal F l_n$ can be realized as either $U(n)/U(1)^n$, or as $GL_n(\mathbb C)/B_n$, where $B_n$ is the group of upper triangular matrices. Thus $GL_n(\mathbb C)$-equivariant vector bundles on $\mathcal Fl_n$ are equivalent to representations of $B_n$. This group is not reductive, and not every representation is a direct sum of 1-dimensional representations. Thus, not every equivariant vector bundle is a direct sum of line bundles. On the other hand, the category of $U(n)$-equivariant principal $U(n)$-bundles is equivalent to representations of $U(1)^n$. This category is semisimple. • Thanks for your answer. Just one question: Are saying that all simple reps of $U(1)^n$ are 1-dimensional, and hence that my guess that all U(n)-equiv bundles are constructable from line bundles? – Ago Szekeres May 1 '13 at 13:51 The answer is no. For example, if $n = 3$ then $F(3)$ is a divisor of bidegree $(1,1)$ in $P^2\times P^2$ and the pullback of the tangent bundle from any factor is an example of an equivariant bundle which is not a sum of line bundles. On the other hand, any equivariant bundle on $F(n)$ can be obtained as an iterated extension of line bundles. • What is an iterated extension of line bundles? – Ago Szekeres Apr 30 '13 at 18:27 • also, what is the representation of $U(1)^{\otimes 2}$ corresponding to the vector bundle you give as a counterexample? – Ago Szekeres Apr 30 '13 at 18:28 • @Ago: An iterated extension is a $G$-equivariant vector bundle (or, equivalently, locally free sheaf) together with a $G$-invariant filtation by $G$-equivariant vector subbundles (locally free subsheaves with locally free quotient) whose associated subquotients are each $G$-equivariant line bundles (invertible sheaves). I am confused by your notation for $U(1) \times \dots \times U(1)$. The representation of this group is the same as the adjoint representation of this group on $\mathfrak{sl}_{3}/\mathfrak{b}$, where $\mathfrak{b}$ is upper triangular $3\times 3$ matrices with trace zero. – Jason Starr Apr 30 '13 at 18:41 • Sam Gunningham's answer cleared this up, but another point view is: although vector bundles on $SL_n/B$ may not split algebraically, on the real manifold $F(n) = SU(n)/U(1)^{n-1}$, every extension of vector bundles does split: choose a hermitian metric. (Indeed, identifying $F(n)$ this way is essentially doing just that.) – Dave Anderson Apr 30 '13 at 23:34

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+0 Help soon 0 58 3 Arc AC is a quarter-circle with center B . The shaded region ABC is "rolled" along a straight board PQ until it reaches its original orientation for the first time with point B landing at point B'. If BC=2/pi cm, what is the length of the path that point B travels? Express your answer in the simplest form. Sep 4, 2020 #1 -1 The length of the path is 4*sqrt(7). Sep 4, 2020 #2 -1 I DON'T TRUST ANSWERS THAT DON'T SHOW THE WORK.  THEY'RE USELESS TO ME. Guest Sep 4, 2020 #3 +183 +4 Arc AC is a quarter-circle with center B. The shaded region ABC is "rolled" along a straight board PQ until it reaches its original orientation for the first time with point B landing at point B'. If BC=2/pi cm, what is the length of the path that point B travels? Express your answer in the simplest form. BC = 2/pi cm       (radius of the circle) The length of the arc L is         L = [ 2( 2/pi )] * pi / 4       L = 1 cm The length of the path that point B travel is  3L  or  3 cm Sep 4, 2020 edited by jugoslav  Sep 4, 2020 edited by jugoslav  Sep 4, 2020

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# Proving $r \binom{n}{r}=n\binom{n-1}{r-1}$ combinatorially. (Advice on combinatorial proofs in general?) [duplicate] How do you combinatorially prove the following? $$r \binom {n}{r} = n \binom {n-1}{r-1}$$ I find it easy to prove such equalities algebraically, but have a hard time finding the right combinatorial intuition. Any advice for coming up with combinatorial proofs myself? • Count committees of $r$ out of $n$ people with a chair? Jun 16, 2020 at 9:01 My go-to combinatorial argument falls along the line of "committee selection." (Albeit mostly because this idea was taught in my combinatorics class.) You have a collection of people, and want to pick them for a committee (and possible positions in that committee) in certain ways which are clearly equivalent, and yet lead to different calculations that lead to deducing the equality. Sadly there's not much to "getting good" at them other than exposure. (Ironically I also find this method of proof much easier than the algebra, but I digress.) For your case, let's argue as so... We have $$n$$ people, and wish to have a committee of precisely $$r$$ people among them. How might we pick them? Moreover, we need to have a leader of the committee as well! • We could just pick the $$r$$ members of the group of $$n$$, and then the leader amongst the selected. There's $$\binom n r$$ possible ways to pick the committee, and $$r$$ possibilities for the leader among them. Thus, $$r \binom n r$$ possible committees. • Alternatively, let's say we set aside one particular person from our group of $$n$$ (maybe they only want to play along if they're the leader), and make a committee of $$r-1$$ members instead from the remaining $$n-1$$. There's $$\binom{n-1}{r-1}$$ ways to make the committee this way. However, the person set aside deserves a shot at being the leader anyways, and we still need one person to be the leader as well as give us our $$r^{th}$$ member, giving $$n$$ possible leaders. Thus, $$n \binom{n-1}{r-1}$$ possible committees. Thus, we deduce $$r \binom n r = n \binom{n-1}{r-1}$$ The first step is to interpret the expressions - what are they counting? There are some tricks to this. For instance, addition corresponds to a single choice out of two sets of options whereas multiplication corresponds to two choices from two sets of options. Another trick is to find dependencies - for instance in the expression $$r\binom{n}{r}$$, we see the $$r$$ twice, so we ought to investigate what it would mean if one of the $$r$$s represented a choice that was dependent on the other $$r$$. In particular, if $$\binom{n}{r}$$ counts $$r$$-subsets of $$\{1,\cdots,n\}$$ then $$r$$ by itself can be interpreted as how many ways there are to choose a single element of that $$r$$-subset. We always phrase this in more familiar terms. For instance, instead of an $$r$$-subset of $$\{1,\cdots,n\}$$, we can think of a committee of $$r$$ people out of $$n$$ candidates. Then the special one of the $$r$$ members chosen for the other $$r$$ in the expression $$r\binom{n}{r}$$ can be interpreted as choosing a president. So $$r\binom{n}{r}$$ counts committees of $$r$$ people drawn from $$n$$ candidates with a single president. A next step is to think about how to count this, but in a different way. If you think about the thing you're constructing in terms of "choices" that can be made while constructing it, you can change the order in which you make these choices. For instance, instead of choosing $$r$$ out of $$n$$ people for a committee and then choosing a president out of those $$r$$, which gives $$r\binom{n}{r}$$, you can instead pick the president ($$n$$ options) and then pick the $$r-1$$ non-president members of the committee out of the remaining $$n-1$$ people, which gives the equivalent expression $$n\binom{n-1}{r-1}$$.

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# Eigenvalues of block matrix where blocks are related [closed] How to find eigenvalues of following block matrix $A$ in terms of eigenvalues of matrix $B$? $A=\begin{bmatrix} 4I-B & -B \\ -B & 2I \\ \end{bmatrix}$ Where $B$ is square matrix of order $n$ and $I$ is an identity matrix of order $n$ I have tried the following let ,$w=\begin{bmatrix} v \\ cv \end{bmatrix}$ be an eigenvector of $A$ then with eigenvalue $\lambda_a$ then $Aw=\lambda_a w$ $\Rightarrow$ $4v-Bv-c(Bv)=\lambda_av$ and $-Bv+2cv=\lambda_a(cv)$ $\Rightarrow$ $4v-\lambda_bv-c(\lambda_bv)=\lambda_av$ $-\lambda_b v+2cv=\lambda_a(cv)$ As $v$ is nonzero vector $4-\lambda_b-c\lambda_b=\lambda_a$ and $-\lambda_b+2c=c \lambda_a$ Now solving both the equation to find $c$. Please verify whether my steps are correct or not? ## closed as off-topic by Morgan Rodgers, Travis, zz20s, Charles, Claude LeiboviciApr 23 '16 at 4:24 This question appears to be off-topic. The users who voted to close gave this specific reason: • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Morgan Rodgers, Travis, zz20s, Charles, Claude Leibovici If this question can be reworded to fit the rules in the help center, please edit the question. Hint: look at the vector $$\begin{bmatrix} v\\ cv \end{bmatrix}$$ where $v$ is an eigenvector of $B$ with eigenvalue $\lambda$, and $c$ is some constant. Under what condition on $c$ is this an eigenvector for your matrix? • That looks pretty good. Multiply your first equation by $c$, and then subtract to eliminate the $c\lambda_a$ term. Now you've got a quadratic in $c$ to solve. Go for it! – John Hughes Apr 23 '16 at 10:56 • BTW: Note that by multiplying by $c$, you may introduce a false root of $c = 0$, and you'll have to check this. When you've got a solution for $c$, you get that $\lambda_a = 4 - \lambda_b - c\lambda_b$; with these values of $c$, you've therefore discovered two eigenvectors of $A$ for each eigenvector of $B$. – John Hughes Apr 23 '16 at 11:04

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# Spark with connecting power supply to board by likephysics Tags: board, connecting, power, spark, supply P: 615 When I connect a DC power supply to a board, why do I get a spark. I connect the ground terminal first and then the Vcc terminal(24V). When I connect Vcc with an alligator clip, I get a spark. Is it because of charging of the bulk storage capacitor? Sci Advisor PF Gold P: 2,469 quite possibly what is the board ? got a pic or a circuit diagram of it ? with the lack of info supplied, its kinda difficult to give a better answer Dave Sci Advisor Thanks P: 1,741 Yes, the capacitors require the high current, but the high voltage spark occurs due to the inductance of the circuit when disconnected from the board once a current has begun to flow. As you make the connection, there are short periods where the connection is intermittent. Once a current is flowing through the inductance of the circuit, (power supply, both jumper leads and the PCB), any break will cause a voltage spike to maintain the current. V = L * di/dt. P: 615 Spark with connecting power supply to board Quote by Baluncore Yes, the capacitors require the high current, but the high voltage spark occurs due to the inductance of the circuit when disconnected from the board once a current has begun to flow. As you make the connection, there are short periods where the connection is intermittent. Once a current is flowing through the inductance of the circuit, (power supply, both jumper leads and the PCB), any break will cause a voltage spike to maintain the current. V = L * di/dt. As a follow up, how do you reduce the inductance. Putting a Cap at the input of the circuit should take care of it. right? Sci Advisor Thanks P: 1,741 Because the inductance is distributed about the circuit it is hard to cancel it with one capacitor. More capacitance just makes more current to spike a higher voltage during intermittent connection. I would first approach the problem by trying a simple snubber across the power supply output. To make a snubber, experiment with a resistor of from 47 to 150 ohm in series with a 0.1uF capacitor. If that does not fix the spark problem, place a snubber across the PCB also. Beyond that, as davenn pointed out in #2, we need more information to resolve the situation. Sci Advisor PF Gold P: 3,509 Another approach is to connect through a current limiting resistor that charges the bulk capacitor more slowly, then short out that resistor, perhaps using a spdt switch.. Some folks use a thermistor with negative temperature coefficient. It starts out high resistance and decreases as it's warmed by the current. That gizmo is called "inrush current limiter" and works well provided your load current is enough to keep it warm. http://www.mouser.com/ProductDetail/...Qd8M3PtP_D_BwE Related Discussions Electrical Engineering 12 Computers 6 Electrical Engineering 3 General Physics 3 Electrical Engineering 17

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# Is this series interesting? • I Jehannum While working on a probability problem I accidentally found this relationship: $$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$ I have done a bit of work on it myself, and have tried to research similar series. It seems to lead to some interesting results. For example, when a = 1 and b = 2 it doesn't work because you get 1 - 1 + 1 - 1 + 1 ... but it's interesting that the Cesaro sum of this series is 1/2. Can anyone provide links or information on anything relevant? Mentor 2022 Award $$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$ $$S = x - x^2 + x^3 - x^4 + \dots$$This converges for ##|x| < 1## to ##S = \frac{x}{1+ x}##, and a bit of algebra shows that indeed:$$\frac{x}{1+ x} = \frac 1 b$$And ##|x| < 1## implies ##b < 0## or ##b > 2##. In particular, this series does not converge for ##b = 2##.

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# how many tablespoon in 60 ml • 60 milliliters is 4.06 tablespoons. • tk10publ tk10canl ## Say hello to Evi Evi is our best selling mobile app that can answer questions about local knowledge, weather, books, music, films, people and places, recipe ideas, shopping and much more. Over the next few months we will be adding all of Evi's power to this site. Until then, to experience all of the power of Evi you can download her app for free on iOS, Android and Kindle Fire here. ## Top ways people ask this question: • how many tablespoon in 60 ml (56%) • convert 60 mls to tbs (13%) • 60 ml to tablespoons (10%) • how many tablespoon is 60 ml (7%) • 60ml is equal to how many tablespoons (2%) • 60 cc equals how many tablespoons (1%) • 60 ml equals how many tablespoons (1%) • 60 ml converted into tablespoons (1%) • conversion 60 cc to tablespoon (1%) ## Other ways this question is asked: • 60 ml how many tablespoons • convert 60ml to tablespoon • 60mls into tablespoons • convert 60 ml to tbs. • 60 ml is how many tablespoons • how many tablespoons in 60ml • 60 ml to tablespoon • how many tablespoons is 60 ml • 60ml in tablespoon • tablespoons in 60 ml • 60ml equals how many tablespoons • how many tablespoons in 60 ml • 60ml is equal to how many table spoons • 60 ml is equal to how many tablespoons • convert 60ml to tablespoons • how many tbs is 60 ml • 60 ml in tablespoons • how many tbsp is there in 60 ml • convert 60 ml to tablespoons • how many tablespoons are in 60 ml • how many tablespoons is 60 cc • 60 milliliters equals how many tablespoons • how many tablespoons is 60ml • 60cc equals how many tablespoons • 60ml in table spoon • 60ml equals how many tbsp • 60 cc's to tablespoons • 60 mls in tablespoons • 60ml into tablespoons • 60 cc's equals how many tablespoons • 60 cc's is how many tablespoons • convert 60ml to table spoon • how much is 60ml in tablespoons • how many tablespoons in 60 cc • 60ml converted to tablespoon • 60ml in tablespoons • 60ml equal how many tablespoon • 60 cc in tablespoons • 60ml to tbsp • how many tbsp is 60 ml • how much is 60 ml in tbs • 60 ml is equal to how many tbsp • how many tbs are 60 cc • 60 ml in tbsp • 60ml in tbsp • 60 ml to tbs • convert 60 ml to tablespoon • 60 ml conversion to tablespoon • what is 60 ml equivalent to in tablespoons • 60ml to tablespoons • 60ml = how many tablespoons • convert 60cc to tablespoons • 60mls is how many tablespoons • how many tablespoons equal 60 ml • 60 ml to table spoons • 60ml to tablespoon • how many tablespoons equals 60ml? 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You are given a decimal number n. 2. You are given a base b. 3. You are required to convert the number n into its corresponding value in base b. `{"id":"cda6687d-fc21-4aa2-9449-d2da391ddb68","name":"Decimal To Any Base","description":"1. You are given a decimal number n.\r\n2. You are given a base b.\r\n3. 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You are given a decimal number n. 2. You are given a base b. 3. You are required to convert the number n into its corresponding value in base b. ## Constraints 0 <= d <= 512 2 <= b <= 10 ## Format ### Input A number n A base b ### Output A number representing corresponding value of n in number system of base b ## Example Sample Input ```.css-23h8hz{color:inherit;font-size:0.875rem;line-height:1.125rem;letter-spacing:0.016rem;font-weight:var(--chakra-fontWeights-normal);white-space:pre-wrap;}57 2``` ### Sample Output `.css-3oaykw{color:var(--chakra-colors-active-primary);font-size:0.875rem;line-height:1.125rem;letter-spacing:0.016rem;font-weight:var(--chakra-fontWeights-normal);white-space:pre-wrap;font-family:Monospace;}111001` Question Video Discussions Show Discussion Related Resources

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# Correlation with one variable and a lot of others In Stata, is there a quick way to show the correlation between a variable and a bunch of dummies. In my data I have an independent variable, `goals_scored` in a game, and a bunch of dummies for `stadium` played. How can I show the correlation between the `goals_scored` and `i.stadium` in one table, without getting the correlations between stadiums, which I do not care about. - Do you just want to know whether some stadiums have systematically higher goal scored? –  Dimitriy V. Masterov Feb 3 '13 at 21:28 I'd be tempted to try a poisson regression: poisson goals_scored i.stadium, nocons robust. The exponentiated parameters (i.e., e^b) will tell you the expected number of goals for a match in that stadium. You might ask this as a separate question on the crossvalidated site. Make sure to describe your data more precisely (maybe post a few example rows). –  Dimitriy V. Masterov Feb 4 '13 at 4:50 ## 4 Answers Here's one way: ``````#delimit; quietly tab stadium, gen(D); // create dummies foreach var of varlist D* {; quietly corr goals_scored `var'; di as text "`: variable label `var'': " as result r(rho); }; drop D*; // get rid of dummies `````` - `cpcorr` from SSC (install with `ssc inst cpcorr`) supports minimal correlation tables, i.e. only the correlations between one set and another set, without the others. But it's an old program (2001) and doesn't support factor variables directly. The indicator variables (a.k.a. dummy variables) would have to exist first. - I don't understand the reply. You wanted a quick way to calculate the correlations and `cpcorr` is one. Internally it's a loop over variables, inevitably, so this answer is similar to others. By the way, there are now open suggestions and questions on several threads you have opened recently. Good protocol is that you close threads by accepting an answer or by explaining why a reply is wrong or otherwise not what you want. I am down-voting your question. It's turned into: What do you propose?, too general a question to be answered on this forum. –  Nick Cox Feb 4 '13 at 8:51 If you store all of the stadium variables in a local, you would probably loop through them to pull the correlations. - Is there a way to do this without listing each variable? I have over 100 stadiums. –  CJ12 Feb 4 '13 at 0:00 Do they have a similar naming convention? Then you could use wildcards, something like "stadium_*" for stadium_1, stadium_2, stadium_3...etc. –  RickyB Feb 4 '13 at 0:15 Or, if you know all of the variables are right next to each other in the dataset, you can specify the column number range, I believe. –  RickyB Feb 4 '13 at 0:16 1. If all stadium variables are placed next to each other in the dataset: ``````foreach s of varlist stadium1-stadium150 { // do whatever } `````` 2a. If the stadium variables are not next to each other, use `order` to get there. 2b. If the variable names follow a pattern, there might be another workaround. 3. I would not use correlation for this. Depending on the distribution of goals, I would consider something else. - I am just looking how the goals vary with stadium. What would you propose? –  CJ12 Feb 4 '13 at 3:03 I would suggest looking at the distribution of goals before coding anything further! I am afraid Nick is right: the technical answer is already out there (and it's impossible for us to know which answer is best without looking at your data), and there's a more general issue in your question that should go to CrossValidated. –  Fr. Feb 4 '13 at 21:44

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# Boundar condition for pipe flow User Name Remember Me Password Register Blogs Members List Search Today's Posts Mark Forums Read May 10, 2011, 05:28 Boundar condition for pipe flow #1 New Member   Join Date: Mar 2011 Posts: 11 Rep Power: 6 I am beginner in CFD. I need some help, how to take into consideration the effect of the gravity in vertcial pipeflow (upward flow). Thanks. May 10, 2011, 07:07 #2 New Member   eddy Join Date: Apr 2011 Posts: 8 Rep Power: 6 i guess u can easily set gravity in operating conditions May 10, 2011, 08:04 #3 New Member   Join Date: Mar 2011 Posts: 11 Rep Power: 6 Yes that where i need help, do i check in gravity and make -9.8m2/s ... aldo do i need to check in operating density and what vale, I am simulating vertical pipe flow of water May 11, 2011, 02:22 #4 New Member   eddy Join Date: Apr 2011 Posts: 8 Rep Power: 6 it depends on ur coordinate system, if ur vertical axis goes up positiv, then jes -9.82 m/s˛ ..... and this density checkbox u can ignore for water in my opinion hope i could help u May 12, 2011, 00:12 #5 New Member   Join Date: Mar 2011 Posts: 11 Rep Power: 6 Thanks shimmyya for your reply. Yes my cooridnate system is +y upwars i.e. I have taken my reference coordinates (0,0,0) at the top of the pipe. As I understand I do need to check in the gravity option box and do not need to fill the operating density ... but by default it shows a value of 1.225 .. shoud I make this zero or someother value as far i understood the value subsituted here will be subtracted from fluid density i.e. [rho(fluid) - rho(op density] g.h? waiting for your reply. Thanks once again May 12, 2011, 02:05 #6 New Member   eddy Join Date: Apr 2011 Posts: 8 Rep Power: 6 hi, just let this "specified operating density" box unchecked, that should be fine. May 13, 2011, 02:23 #7 New Member   Join Date: Mar 2011 Posts: 11 Rep Power: 6 Need help, can you plz help ... how to calcuate the pressure drop in pipe flow ... can't seem 2 understand the Fluent output plots .. as values I have do not come anywhere near. May 16, 2011, 09:44 #8 New Member   eddy Join Date: Apr 2011 Posts: 8 Rep Power: 6 i dont know if this help u , but u can set probe-points which can read out the data of this points like pressure, velocity ...... could be also good to know what version of FLUENT u use Thread Tools Display Modes Linear Mode 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 OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules Similar Threads Thread Thread Starter Forum Replies Last Post Michele Cagna CFX 3 February 22, 2007 16:52 Renato. Main CFD Forum 0 July 21, 2006 22:07 vivian FLUENT 5 April 21, 2006 06:23 John Main CFD Forum 0 October 1, 2003 00:00 Síle FLUENT 0 June 12, 2003 07:30 All times are GMT -4. The time now is 14:50. Contact Us - CFD Online - Top

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Number 15736 Properties of number 15736 Cross Sum: Factorization: 2 * 2 * 2 * 7 * 281 Divisors: 1, 2, 4, 7, 8, 14, 28, 56, 281, 562, 1124, 1967, 2248, 3934, 7868, 15736 Count of divisors: Sum of divisors: Prime number? No Fibonacci number? No Bell Number? No Catalan Number? No Base 2 (Binary): Base 3 (Ternary): Base 4 (Quaternary): Base 5 (Quintal): Base 8 (Octal): 3d78 Base 32: fbo sin(15736) 0.23537251276772 cos(15736) -0.97190523212575 tan(15736) -0.2421764025829 ln(15736) 9.6637063600689 lg(15736) 4.1968943469113 sqrt(15736) 125.44321424453 Square(15736) Number Look Up Look Up 15736 (fifteen thousand seven hundred thirty-six) is a great number. The cross sum of 15736 is 22. If you factorisate the figure 15736 you will get these result 2 * 2 * 2 * 7 * 281. The number 15736 has 16 divisors ( 1, 2, 4, 7, 8, 14, 28, 56, 281, 562, 1124, 1967, 2248, 3934, 7868, 15736 ) whith a sum of 33840. The number 15736 is not a prime number. The number 15736 is not a fibonacci number. 15736 is not a Bell Number. The number 15736 is not a Catalan Number. The convertion of 15736 to base 2 (Binary) is 11110101111000. The convertion of 15736 to base 3 (Ternary) is 210120211. The convertion of 15736 to base 4 (Quaternary) is 3311320. The convertion of 15736 to base 5 (Quintal) is 1000421. The convertion of 15736 to base 8 (Octal) is 36570. The convertion of 15736 to base 16 (Hexadecimal) is 3d78. The convertion of 15736 to base 32 is fbo. The sine of 15736 is 0.23537251276772. The cosine of 15736 is -0.97190523212575. The tangent of the figure 15736 is -0.2421764025829. The root of 15736 is 125.44321424453. If you square 15736 you will get the following result 247621696. The natural logarithm of 15736 is 9.6637063600689 and the decimal logarithm is 4.1968943469113. I hope that you now know that 15736 is unique number!

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Important questions of Triangle (in Geometry) Chapter 6 Class 10 Triangles Serial order wise Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class ### Transcript Question 1 In Fig. 6.56, PS is the bisector of QPR of PQR. Prove that / = / Given : PQR and PS is the bisector of QPR i.e. QPS = RPS To Prove: / = / Construction : Draw RT SP such that RT cuts QP Produced at T. Proof: In QRT, RT SP and PS intersects QT and QR at two distinct points P and Q Therefore, applying Basic Proportionality Theorem in QRT QT and QR will be divided in the same ratio / = / Now, we need to prove PT = PR Now RT SP & PR is the transversal Therefore, Also, Given that PS is the bisector of QPR QPS = RPS 1 = 2 Putting 1 = 4 and 2 = 3 from (2) & (3) 4 = 3 i.e. PTR = PRT Therefore, PT = PR Putting PT = PR in equation (1) / = / / = / Hence Proved.

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## ››Convert firkin [butter, soap] to keg [nails] firkin keg How many firkin in 1 keg? The answer is 1.7857142857143. We assume you are converting between firkin [butter, soap] and keg [nails]. You can view more details on each measurement unit: firkin or keg The SI base unit for mass is the kilogram. 1 kilogram is equal to 0.039368261104442 firkin, or 0.022046226218488 keg. Note that rounding errors may occur, so always check the results. Use this page to learn how to convert between firkin [butter, soap] and keg [nails]. Type in your own numbers in the form to convert the units! ## ››Quick conversion chart of firkin to keg 1 firkin to keg = 0.56 keg 5 firkin to keg = 2.8 keg 10 firkin to keg = 5.6 keg 20 firkin to keg = 11.2 keg 30 firkin to keg = 16.8 keg 40 firkin to keg = 22.4 keg 50 firkin to keg = 28 keg 75 firkin to keg = 42 keg 100 firkin to keg = 56 keg ## ››Want other units? You can do the reverse unit conversion from keg to firkin, or enter any two units below: ## Enter two units to convert From: To: ## ››Metric conversions and more ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!

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Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  adjadj Structured version   Visualization version   GIF version Description: Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) Assertion Ref Expression Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables. StepHypRef Expression 1 adj2 29706 . . . . 5 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦))) 2 dmadjrn 29667 . . . . . 6 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj) 3 adj1 29705 . . . . . 6 (((adj𝑇) ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((adj𝑇)‘𝑦)) = (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦)) 42, 3syl3an1 1160 . . . . 5 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((adj𝑇)‘𝑦)) = (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦)) 51, 4eqtr2d 2860 . . . 4 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦)) 653expib 1119 . . 3 (𝑇 ∈ dom adj → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))) 76ralrimivv 3184 . 2 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((adj‘(adj𝑇))‘𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))

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Janessa Olson 2022-07-02 In a raffle 5 winners get to chose from 5 prizes starting with the fist name drawn. iIf 87 people entered the raffle how many ways can the winners be arranged? Bruno Dixon Expert Explanation: There are 87 ways to win the first prize There are 86 ways to win the second prize There are 85 ways to win the third prize There are 84 ways to win the fourth prize There are 83 ways to win the fifth prize number of ways $=87\cdot 86\cdot 85\cdot 84\cdot 83=4,433,982,840\phantom{\rule{1ex}{0ex}}\text{}\phantom{\rule{1ex}{0ex}}$ Do you have a similar question?

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# can't really understand the math behind the mandelbrot set according the following article : wolfram Mandelbrot set, I'm trying to understand how they exactly managed to calculate the `Ln(C)=Zn=R(max) values.` i do understand that Rmax is a constant, equals 2,(|Zn| < 4 for all points that are inside the Mandelbrot set), and Ln(C) should be the amount of iterations i spent for each C(point), but how using these 2 i get to calculate ``````L1(C) = C L2(C) = C(C+1) .... .... `````` thanks for your help! -

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# Indefinite Integral Calculator Indefinite Integral Calculator directly gives the integral of your input function easily in fraction of seconds. Just enter function as the input in the specified fields and tap on the calculate button which is available next to the input section to find the result in seconds. Indefinite Integral Calculator: Do you feel calculating indefinite integral somewhat difficult? Not anymore with the help of our easy to use online calculator tool. Now, you can solve the integration of any function easily and instantly by using our Indefinite Integral Calculator. Refer the below section, to get familiar with this concept by checking the solved examples. The step by step process to compute the indefinite integral is also mentioned here. ## Procedure to Find Indefinite Integral of Function Indefinite integral is an integral without having upper and lower limits. The step by step process of evaluating indefinite integrals is listed here. So, follow the steps given here and do calculations easily by hand. • Take any function to compute the indefinite integral. • Go through the different rules like power rule, exponential, constant rule, etc before solving the problem. • ∫ x dx is always equal to (x2) / 2 + C. Where C is the constant • Integration of any constant is equal to the constant value * x + C. • If the function is in the difficult form. • Consider one part of the function as a variable and substitute that variable in all the possible places of the fumction. • Find integration with respect to that variable and substitute the value. Examples Question 1: Solve ∫ (2x + 1 ) / (x+5)3 dx? Solution: Given input Mixed Number is 2 4/3 ∫ (2x + 1 ) / (x+5)3 dx Let us take, u = x + 5 Then, 2x + 1 = 2u – 9 ∫ (2x + 1 ) / (x+5)3 dx = ∫ (2u – 9) / u3 du = ∫ 2u / u3 – 9/ u3 du = ∫ 2/u2 – 9/u3 du Apply the sum rule ∫ f(x) + g(x) dx = ∫ f(x) dx + ∫ g(x) dx = ∫ 2/u2 du – ∫ 9/u3 du Take out the constant: ∫ a. f(x) dx = a. ∫ f(x) dx = 2 ∫ 1/ u2 du – 9 ∫ 1 / u3 du Apply exponent rule 1/ ab = a-b = 2 ∫ u-2 du – 9 ∫ u-3 du Apply the Power rule: ∫ xa dx = xa+1 / a+1, a ≠ 1 = 2 * u-2+1 / (-2+1) – 9 * u-3+1 / (-3+1) = 2 * u-1 / (-1) – 9 * u-2 / (-2) = -2/u + 9/2u2 Substitute u = x + 5 in the above = -2/(x + 5) + 9/2( x + 5)2 ∫ (2x + 1 ) / (x+5)3 dx = -2/(x + 5) + 9/2( x + 5)2 + C Question 2: Solve ∫ (x2 + 3x – 2) dx? Solution: ∫ (x2 + 3x – 2) dx = ∫ x2 dx + ∫ 3x dx – ∫ 2 dx = x3/3 + (3x)2 / 2 – 2x + C = x3/3 + 9x2/2 – 2x + C Find a variety of Other free Maths Calculators that will save your time while doing complex calculations and get step-by-step solutions to all your problems in a matter of seconds. ### FAQs on Indefinite Integral Calculator 1. What is Meant by Indefinite Integrals? Indefinite Integral is an integration function indicated without lower and upper limits and with an arbitrary constant C. It is considered as an easy way to symbolize the antiderivative of the function. The representation is ∫ f (x) dx. The function f(x) is called Integrand. 2. Why we add a constant with an Indefinite Integral? Integral function is called its anti derivative. If you differentiate a function and then integrate it, you should get the function back. For example take f(x) = x, g(x) = x + 4 f′(x)=1 and g'(x) = 1. ∫f′(x) dx = x, ∫g'(x) dx = x Here, you are not getting exact g(x) value. So, we are adding constant. ∫f′(x) dx = x + C1, ∫g'(x) dx = x + C2 C1 = 0, C2 = 4. 3. What are the rules of Integration? Integration is used to find area, volume, etc. The some of the common rules of integration are: 1. Constant rule: ∫a dx = ax + C 2. Multiplication by constant: ∫cf(x) dx = c ∫f(x) dx 3. Reciprocal rule: ∫(1/x) dx = log(x) + C 4. Exponential rules: ∫ex dx = ex + C, ∫ax dx = ax/log(a) + C, ∫log(x) dx = x log(x) − x + C. 4. Can an Integral have 2 Answers? No, integrals don’t have two answers. Let us say, x+c, x2+c both cannot be solutions to same integral, because x and x2 don’t differ by constant.

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1. Calculation functions 2. All Functions 3. Miscellaneous Functions 4. RANK Use the RANK function to evaluate a set of values and assign sequential rankings starting at 1. Syntax `RANK(Source values [, Direction] [, Equal value behavior] [, Include value] [, Ranking groups]) ` Argument Data type Description Source values (required) Number, date, or time period The number, date or time period to rank. Direction (optional) Keyword Determines the direction to rank in. Equal value behavior (optional) Keyword Determines how to rank equal values. Include value (optional) Boolean Determines if a value is ranked. Ranking groups (optional) Number, Boolean, date, time period, list, or text If provided, the source values are ranked independently for each value in the Ranking groups argument. Returns The RANK function returns a number-formatted result. Arguments The Source values argument can refer to a line item, property, or expression. The keywords for the Direction argument are: • DESCENDING (default): the highest source value is ranked 1, the second highest source value ranked 2, and so on • ASCENDING: the lowest source value is ranked 1, the second lowest source value ranked 2, and so on The keywords for the Equal value behavior argument are: • MINIMUM (default): gives tied values the lowest ranking of their range • MAXIMUM: gives tied values the highest ranking of their range • AVERAGE: gives tied values the average of the range of applicable rankings • SEQUENTIAL: gives tied values separate rankings, in the order they occur The Include value argument is a Boolean value that determines if the RANK function ranks the corresponding value. The default behavior is TRUE, which ranks all values. If FALSE is used, the RANK function returns a result of NaN (Not a Number). The Ranking groups argument can be used to split the target values of the RANK function into several categories. Each set of values with the same value for the Ranking groups argument is ranked individually. Constraints The RANK function has the following constraints: Formatting Constraints The Source values argument is required and must be a number-formatted line item, property, or expression. The result of the function must be number-formatted. Ranking Constraints If the target module contains more than one other dimension (excluding Time and Versions dimensions), the ranking is performed across all combinations. For example, picture a module with two line items, Sales and Result, which apply to Region and Product, with 100 products sold in 50 regions. In this case, the formula `Result = RANK(Sales)` will contain the values 1 to 5,000, ranking each combination of Product and Region. Time and Versions are not included in the ranking. If the line item used for the Source value argument has a Time or Versions dimension, the RANK function ranks it independently for each time period or version. Cell Limit An artificial limit is imposed to prevent ranking of large data sets that would slow down the server. This limit is set at 50 million cells. If more than 50 million cells are used with the RANK function, the model rolls back and a notification displays. The 50 million cell limit does not account for summarized values or the Time and Versions lists. This means you can use the RANK function with a line item with a Cell Count of greater than 50 million cells if there are less than 50 million nonsummarized cells. As the number of cells you use with the RANK function increases, so does the duration of the calculations. Positive Infinity, Negative Infinity, and NaN Infinities and NaN (Not a Number) are automatically excluded from the ranking and give a result of NaN. Examples In this example the leaf-level items cities are ranked in order. In the absence of the optional arguments, the Direction argument defaults to descending, and the Equal value behavior argument to minimum value. For example, Paris and Lyon both have the same sales value so the rank is set as 4. Subtotals and totals are excluded from the ranking. `RANK(Sales) ` RANK with Direction argument The rank order can be set to ASCENDING or DESCENDING. The default is descending order. `RANK(Sales, DESCENDING) ` `RANK(Sales, ASCENDING) ` RANK with Ranking groups argument In the examples shown, the ranking takes place within the peer group. This allows for ranking of items of the same parent, property, flag, date or any other data type. `RANK(Sales, DESCENDING, MINIMUM, Sales > 0, PARENT(ITEM(Organization))) ` `RANK(Sales, DESCENDING, MINIMUM, Sales > 0, Property) ` `RANK(Sales, DESCENDING, MINIMUM, Flag = TRUE, Flag) ` `RANK(Sales, DESCENDING, MINIMUM, Sales > 0, Opening Date) ` `RANK(Sales, DESCENDING, MINIMUM, Sales > 0, YEAR(Opening Date)) ` RANK by Boolean criteria In these examples, items that fail to make the criteria are excluded from the ranking and instead shown as NaN (Not a Number). ```RANK(Sales, DESCENDING, MINIMUM, Sales > 10000 AND Sales < 24000) ``` ```RANK(Sales, DESCENDING, MINIMUM, Sales > 10000) ``` RANK with different behaviors for equal values In this example, the different options for the Equal value behavior argument are shown. Paris and Lyon have the same sales, but their ranking scores can be different depending on the method used for scoring ties. `RANK(Sales, DESCENDING, SEQUENTIAL)) ` `RANK(Sales, DESCENDING, MINIMUM)) ` `RANK(Sales, DESCENDING, MAXIMUM)) ` `RANK(Sales, DESCENDING, AVERAGE)) ` If you use the Equal value behavior argument, then you see equal values grouped at the same ranking with a jump to the next rank number. The example below ranks by date value in ascending order. As 7 of the 8 date values are the same, they are ranked 1, and the next rank is therefore 8. RANK with time as a line item In this example, we have a module pivoted to show line items on columns. The Month Period line item is time period-formatted for months and an ascending ranking is given for the selected month periods against companies in the organization: `RANK(Month Period, ASCENDING) ` Remember that if a source value is blank, the blank is ranked first for ascending order and last for descending order. In this example, there is one blank month period and January is therefore ranked at 2. To exclude blank values from the ranking, use the Include value argument with a Boolean formula. Example with date values In this example, we have a module pivoted to show line items on columns. The Date line item is date-formatted and an ascending ranking is given for the selected dates against companies in the organization: `RANK(Date,ASCENDING,MINIMUM) ` If a source value is blank, the blank is ranked first for ascending order and last for descending order. This example contains one blank month, so the date 01/01/2014 is ranked 2.

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Class 16-2010 Class 16-2010 - Managerial Econ Class 16 1 Cartel Video and... This preview shows pages 1–5. Sign up to view the full content. 11/4/2010 1 THE UNIVERSITY OF BRITISH COLUMBIA Managerial Econ: Class 16 1. Cartel Video and Question 2. Introduction to Game Theory 3. Objectives of Game Theory 4. Static and Dynamic Games 5. The Nash equilibrium 6. Sequential games 7. Stackelberg Oligopoly THE UNIVERSITY OF BRITISH COLUMBIA According to the video, the lysine cartel fell apart because i) each firm in the cartel had an incentive to cheat on the cartel. ii) of legal enforcement of anti-collusion laws in the United States. iii) high profits encouraged entry by new firms. iv) all of the above. v) none of the above. Cartel Video Clicker Question 1 This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 11/4/2010 2 THE UNIVERSITY OF BRITISH COLUMBIA A game is any situation in which two or more players make strategic decisions that affect returns or payoffs to the players. Examples: card games, chess, athletic competitions, firms competing with each other by setting prices. 2. Introduction to Game Theory THE UNIVERSITY OF BRITISH COLUMBIA Defining a Game: Players, Strategies, Payoffs, and Rules A game consists of: players (like firm 1 and firm 2) strategies (like price = \$4 or price = \$6) payoffs (what each player gets for each combination of strategies) rules of the game (sequential or simultaneous move, etc.) The Cournot model and the Bertrand model are both examples of games. Those two games differ in that the available strategies are different – prices or quantities. 11/4/2010 3 THE UNIVERSITY OF BRITISH COLUMBIA Strategies A strategy can be quite simple – like charge a price of \$4. A strategy can be more complex, consisting of several actions – like charge a price of \$4 in period 1 then charge \$6 in period 2. Strategies can be contingent – like charge a price of \$6 in period 2 if my rival charges a price of \$6 in period 1, but charge \$4 otherwise. THE UNIVERSITY OF BRITISH COLUMBIA Payoffs The payoffs are what the players try to obtain. We will usually focus on games where the payoffs are profits or other monetary objectives. We often show payoffs in a payoff matrix. 10,10 25,5 5,25 20,20 iPod Low Price High Price Low Price High Price Blackberry This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 11/4/2010 4 THE UNIVERSITY OF BRITISH COLUMBIA Rules of the Game The rules specify various important elements of the game like: Who moves first? Do players move sequentially or simultaneously? Can players sign binding agreements about their strategies? How long does the game last? This is the end of the preview. Sign up to access the rest of the document. {[ snackBarMessage ]} Page1 / 13 Class 16-2010 - Managerial Econ Class 16 1 Cartel Video and... This preview shows document pages 1 - 5. Sign up to view the full document. View Full Document Ask a homework question - tutors are online

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# Volume of intersection of two conical frustums How can I find volume of intersection of two conical frustums? Both objects are on the same X axis. I'm not sure yet how exactly they will intersect because each frustum is one user on the map and I need to find probability that two users saw each other. I would appreciate some example because I think I will not get to final solution myself. Thanks. - The intersection of two parallel conical frustums on the same axis is either (a) another conical frustum or (b) two conical frustums glued together at their bases. What variables should we assume are a 'given' in the problem - e.g. the radii of the tops and bottoms and their centers' positions on the $x$-axis? – anon Aug 2 '11 at 11:39 Anon, yes I have radii for tops and bottoms, and centers (In my case that's latitude and longitude). The height of conical frustums can by any number, I think it doesn't metter for me. – Dimitry Aug 2 '11 at 22:49 Is my question so complex that no-one can give me an answer? Or I just need to provide some more details to describe it better? I've spend few days searching in google and on couple forums and can't find any example. Any help would be appreciated. – Dimitry Aug 4 '11 at 8:36 I'm sorry, I figured either someone else would answer or you'd figure it out - assuming I understand your question correctly. I'll open my MSPaint and work on an explanation. – anon Aug 4 '11 at 12:08 A cone is simply a triangle rotated about an axis to make a solid. We'll place it on the $y$-axis instead of the $x$-axis because it looks better standing upright. Now if you slice a smaller cone off the top, then you're left with a conical frustum. This new solid is now the same as an isosceles trapezoid rotated about an axis down the middle. Therefore, if we take a look at an intermediate slice of the intersection of two frustums, we'll get essentially one of four possible situations (up to otherwise superfluous orientation and proportion): $\hskip 1.1 in$ If you know the top has length (diameter for the frustum) $A$, and you know the bottom has length $B$, and you know the purple line is $u$ distance down from the top and $v$ distance up from the bottom, then the purple line has length equal to the weighted average $(Au+Bv)/(u+v)$. You can use this to find the base diameters of the intersection frustum based on the top and bottom diameters of the main two frustums. Lastly, a frustum with top diameter $A$, bottom diameter $B$, and height $h$, has volume that can be calculated with the formula $\frac{\pi h}{12}(A^2+AB+B^2)$. If you don't understand how to resolve your intersecting frustums problem now, tell me where you're stuck and I'll try to elaborate.

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http://hex.ro/wp/blog/calibrating-the-ph-probe-using-the-mv-output-of-the-ph-circuit/

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Introduction I use 4 Atlas Scientific pH Circuits (v 5.0) with a Multi Circuit Carrier Board to monitor de pH in 4 of my aquariums. The Atlas Scientific pH Circuit can be calibrated following a simple procedure described in the manual: you should put the circuit on continuous sampling, then submerge the probe under pH 7, wait for it to stabilize, then clean and whipe the probe, and continue with pH 4 and pH 10 sample. However, I didn’t choose to follow this procedure because: a) even if I now have it, it was difficult to find a pH solution of reference 10. What the shops nearby would get me was pH 9 calibration, which can’t be used for pH Circuit (v5.0 – don’t know about the newer versions ?). b) the way I designed my monitoring made it a bit more difficult to put a pH circuit ‘offline’ while I was calibrating it. It is not problem though, I had a solution figured out anyway – and it is even more flexible, as you will see below. First, I was able to estimate what mV the probe was putting out just by looking at the pH value. pH Circuit is very versatile … and no problems so far. I have an older article here: http://hex.ro/wp/blog/ph-electrode-mv-output-using-an-atlas-scientific-ph-circuit/ Setup If the pH Circuit is set to run at T = 25C, then the mV output of the probe seems to be calculated using the formula: Calibrating through pH Circuit give you the possibility to compensate the pH based on the temperature too, but in the explanations below I’ve simplified and assumed T = 25C – the temperature of the water in the aquariums is already around 25C. The example below is for one probe, the output in mV is calculated using the formula above. pH Solution pH Circuit output Probe output (mV) 4 3.93 181.744 7 7.20 -11.84 10 10.37 -199.504 Plotting the values above on a chart next to an ideal pH probe output: Ideal vs Measured pH probe output As you can see, for pH 7, an ideal probe would register 0mV, but mine seems to register already -11.84mV. To make more sense of the graph, we need to shift he values up with -11.84mV (which is considering calibrating for pH 7. By the way, that’s why I suppose it is also recommended as the initial value to start calibrating with. With this “post measuring” calibration approach (as opposed to delegating to pH Circuit to do it) we don’t really care about the order of pH samples the probes are submerged into; This way, you can calibrate 2 at a time (even 3) using the small calibration device (in the photo on top). After shifting the values up with 11.84mV, the chart now looks like this: pH probe calibrated for pH 7 pH Solution pH 7 calibrated (mV) 4 181.744 + 11.84 = 193.584 7 -11.84 + 11.84 = 0 10 -199.504 + 11.84 = -187.664 If the ideal pH probe output is a line, in my case, the measured values are not on a line. The output diverges faster (towards the left) than it diverges towards the right. Now, there are few ideas on how to approximage the real pH based on the pH reported by pH Circuit. We could try to fit a line in between the ideal one and the real one (and use that to ‘walk back’ to the real pH value). I chose a different idea: a) assumed that the pH 4 -> pH 7 part is a line, different from the pH 7 -> pH 10 line. b) that each line diverges proportionally (starting from pH 7) but at a different rate. So for each mV to the left (or to the right), there has to be a conversion formula to multiply the value so that the result falls onto the ideal line (and thus producing the correct pH output from the pH Circuit). The goal is to find a formula that will take the erroneous (uncalibrated) pH output and obtain a closer to reality (calibrated) value that we can then use. The formula I came up with using the assumptions above (T=25C, and the aging pH probe doesn’t produce linear output continuously, but two linear outputs with different slopes): where: pH_real is the value that we are interested in. pH_circuit_value is the value that pH Circuit reports, uncalibrated. shift_to_pH_7 is the value (in mV) that we need to compensate the graph at ph 7 (in the case above -11.84 mV) coef is the coefficient that each (left or right) slope diverges from the ideal – (and chosen accordingly depending where the measured pH is found, to the left of ph 7, or to the right), for example: where: mv_4_circuit_value is the mV value calculated starting from the pH Circuit value (uncalibrated) when immersed in a pH 4 solution by applying the mV = 59.2 * (7 – pH_output) formula. Summary This calibration procedure above (software calibration) reduces a lot of time spent calibrating the pH Circuit. You don’t need to add the pH Circuit calibration commands to your controller, just allow for probe values to settle once inserted into the solution, then apply the algorithm!

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# NCERT Solutions for Class 12 Physics Chapter 1 - Electric Charges and Fields Prepare for your Class 12th board examination with NCERT Solutions available for free at the Aasoka platform. These solutions will help students with competitive exam preparations as well. The NCERT Solutions for Class 12th is prepared by professionals who have years of experience. Students can enhance their understanding of the concept with the solutions on Aasoka, the top eLearning platform. From the first chapter of Class 12th Physics “Electric Charges and Fields”, students get to learn about electric charge; Coulomb’s law force between two-point charges, conservation of charge, superposition principle and continuous charge distribution, forced between multiple charges, electric field, electric flux, uniformly charged infinite plane sheet, electric dipole, electric field lines, and so much more. ##### Question 1: What is the force between two small charged spheres having charges of 2 × 10–7 C and 3 × 10–7 C placed 30 cm apart in air ? Given q1 = 2 × 10–7 C, q2 = 3 × 10–7 C, r = 30 cm = 0.3 m ##### Question 2: The electrostatic force on a small sphere of charge 0.4 $\mathrm{\mu }$C due to another small sphere of charge –0.8 $\mathrm{\mu }$C in air is 0.2 N. 1. What is the distance between the two spheres ? 2. What is the force on the second sphere due to the first ? Given q1 = 0.4 $\mathrm{\mu }$C = 0.4 × 10–6 C ##### Question 3: Check that the ratio ke2/G me mp is dimensionless. Look up a table of physical constants and determine the value of this ratio. What does the ratio signify ? ##### Question 4: Explain the meaning of the statement ‘electric charge of a body is quantised.’ Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges ? Quantisation of charge. It is that property of charge by virtue of which charge on a body exists in the form of discrete packet of charge e, only, where e is the charge on an electron. The charge carried by anybody would be equal to ± ne, where n = 0, 1, 2, 3, 4, 5, etc. The charge on a body is thus some multiple of e and cannot exist as a fraction of e. So charge exists in the form of packets and not in continuous amounts. Thus, charge is said to have a discrete (discontinuous) nature or is said to be quantised. At macroscopic level, we deal with charges that are enormous as compared to the magnitude of minimum charge i.e. e (1.6 × 10–19 C). In this case, the increase and decrease in units of e is not very different from saying that charges are continuous. So at macroscopic level we can ignore quantisation of electric charge. ##### Question 5: When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge. When a glass rod is rubbed with silk, the charges developed on the glass rod and the piece of silk are equal and opposite. Similar is the case in other pair of bodies. So electric charge can neither be produced nor destroyed but simply transferred from one body to another, hence is consistent with the law of conservation of charge. ##### Question 6: Four point charges qA = 2 $\mathrm{\mu }$C, qB = –5 $\mathrm{\mu }$C, qC = 2 $\mathrm{\mu }$C and qD = –5$\mathrm{\mu }$C are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 $\mathrm{\mu }$C placed at the centre of the square ? ABCD is a square having charges qA = 2 $\mathrm{\mu }$C, qB = – 5$\mathrm{\mu }$C, qC = 2$\mathrm{\mu }$C, qD = –5 $\mathrm{\mu }$C at its corners and a charge 1 $\mathrm{\mu }$C is placed at the centre O. $\therefore$ Net force on q due to qB and qD = 0 So net force on q due to four charges is zero. ##### Question 7: An electrostatic field line is a continuous curve. That is, a field line cannot have sudden break. Why not ? Explain why two field lines never cross each other at any point. The electric line of force starts from a positively charged body and ends at a negatively charged body and it carries information about the direction of electric field at different points in space, thus it cannot have sudden break. Because if any two lines cross each other at any point, then the electric field at the point of intersection will not have a unique direction, which is impossible. Hence no two electric lines of force can intersect each other. ##### Question 8: Two point charges qA = 3 $\mathrm{\mu }$C and qB = –3 $\mathrm{\mu }$C are located 20 cm apart in vacuum. What is the electric field at the mid point O of the line AB joining the two charges ? If a negative test charge of magnitude 1.5 × 10–9C is placed at this point, what is the force experienced by the test charge ? Given q A = 3 $\mathrm{\mu }$C = 3 × 10–6 C qB = – 3 $\mathrm{\mu }$C = – 3 × 10–6 C r = 20 cm = 0.2 m Electric field at O due to qA ##### Question 9: A system has two charges qA = 2.5 × 10–7 C and q B = –2.5 × 10–7 C located at A : (0, 0, –15 cm) and B : (0, 0, +15 cm) respectively. What is the total charge and electric dipole of the system ? Since electric dipole consists of two equal and opposite charges, Hence total charge = qA + qB = 0 ##### Question 10: An electric dipole with dipole moment 4 × 10 –9 C m is aligned at 30$°$ with the direction of a uniform electric field of magnitude 5 × 10 4 NC –1. Calculate the magnitude of the torque acting on the dipole. Given p = 4 × 10–9 Cm, = 30$°$, E = 5 × 104 NC–1, $\mathrm{\tau }$ = ? Since $\mathrm{\tau }$ = pE sin = 4 × 10–9 × 5 × 104 × sin 30$°$ ##### Question 11: A polythene piece rubbed with wool is found to have a negative charge of 3 × 10 −7 C. 1. Estimate the number of electrons transferred (from which to which). 2. Is there a transfer of mass from wool to polythene ? (a) Charge on one electron = 1.6 × 10−19 C. $\therefore$ Number of electrons in the given charge (b) Since wool gets negative charge on rubbing with polythene, wool must gain electrons from polythene. $\therefore$ Ideally speaking there must be a transfer of mass due to transfer of electrons but since the mass of electron is very small, this transfer of mass may be negligible (= 2 × 10–18 kg). ##### Question 12: Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm. (a) What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10−7 C ? The radii of A and B are negligible as compared to the distance of separation. (b) What is the force of repulsion if each sphere is charged double the above amount and the distance between them is halved ? (a) q1 = 6.5 × 10−7 C, q2 = 6.5 × 10−7 C, r = 50 cm = 0.50 m (b) When each charge is doubled and the distance between them is reduced to half, then ##### Question 13: Suppose the spheres A and B in Q. 1.10 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second and finally removed from both. What is the new force of repulsion between A and B ? Let A and B be two given spheres each having charge q = 6.5 × 10–7 C. When another sphere say C is placed in contact with the sphere A, the two will distribute the charges equally, Fig. TBQ 1.13 (a). ##### Question 14: Figure TBQ 1.14 gives tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio ? Charges 1 and 2 are negative because the deflections are towards +vely charged plate. Particle 3 has the +ve because charge is deflected towards −vely charged plate. Particle 3 has the highest charge to mass ratio because the greater is the charge on the particle (or lesser is its mass) the more is its deflection. ##### Question 15: Consider a uniform electric field 1. What is the flux of this field through a square of 10 cm on a side whose plane is parallel to yz – plane ? 2. What is the flux through the same square if the normal to its plane makes a 60$°$ angle with the x-axis ? dS = 10 × 10 = 100 cm2 = 0.01 m2 Here $\mathrm{\theta }$ = 0$°$ $\therefore$ Flux $\mathrm{\phi }$ = E dS cos 0$°$ = E dS = 3 × 103 × (0.01) = 30 Nm2 C–1 (b) Here $\mathrm{\theta }$ = 60$°$ ##### Question 16: What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes. Net flux over the cube is zero, because the number of lines entering the cube is the same as the number of lines leaving the cube. ##### Question 17: Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm 2 /C. 1. What is the net charge inside the box ? 2. If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box ? Why or why not ? (a) Given ##### Question 18: A point charge +10 $\mathrm{\mu }$C is at a distance 5 cm directly above the centre of a square of side 10 cm as shown in the figure. What is the magnitude of the electric flux through the square. [Hint. Think of the square on one face of a cube with edge 10 cm] Here we can imagine that the square ABCD as one face of a cube with edge 10 cm, and charge +q is placed at the centre of the cube as shown in the figure. From Gauss’ theorem, electrix flux through all the faces of the cube ##### Question 19: A point charge of 2.0 $\mathrm{\mu }$C is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electrix flux through the surface ? q = 2.0 $\mathrm{\mu }$C = 2 × 10–6 C ##### Question 20: A point charge causes an electric flux of –1.0 × 103 N m2 C–1 to pass through spherical gaussian surface of 10.0 cm radius centred on the charge. 1. If the radius of the gaussian surface were doubled, how much flux would pass through the surface ? 2. What is the value of the point charge ? (a) The electric flux depend only on the charge present in the gaussian surface. So electric flux passing through the gaussian surface of double the radius will be the same i.e. ##### Question 21: A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 103 NC–1 and points radially inward, what is the net charge on the sphere ? Given E = –1.5 × 103 NC–1 [∵E is directed inwards] r = 20 cm = 0.2 m, q = ? ##### Question 22: A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 $\mathrm{\mu }$C m–2. 1. Find the charge on the sphere. 2. What is the total electric flux leaving the surface of the sphere ? Given $\sigma$ = 80.0 $\mathrm{\mu }$Cm–2 = 80 × 10–6 C m–2 D = 2.4 m or r = 1.2 m (a) Charge on the sphere, q = $\sigma$ × 4 πr2 or q = 80 × 10–6 × 4 × 3.142 × (1.2)2 = 1.45 × 10–3 C ##### Question 23: An infinite line charge produces a field of 9 × 104 NC–1 at a distance of 2 cm. Calculate the linear charge density. Given $±$$\mathrm{\sigma }$ = 17.0 × 10–22 cm–2

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# Ceiling in right side for every element in an array Given an array of integers, find the closest greater element for every element. If there is no greater element then print -1 Examples: Input : arr[] = {10, 5, 11, 10, 20, 12} Output : 10 10 12 12 -1 -1 Input : arr[] = {50, 20, 200, 100, 30} Output : 100 30 -1 -1 -1 ## Recommended: Please try your approach on {IDE} first, before moving on to the solution. A simple solution is to run two nested loops. We pick an outer element one by one. For every picked element, we traverse right side array and find closest greater or equal element. Time complexity of this solution is O(n*n) A better solution is to use sorting. We sort all elements, then for every element, traverse toward right until we find a greater element (Note that there can be multiple occurrences of an element). An efficient solution is to use Self Balancing BST (Implemented as set in C++ and TreeSet in Java). In a Self Balancing BST, we can do both insert and ceiling operations in O(Log n) time. `// Java program to find ceiling on right side for ` `// every element. ` `import` `java.util.*; ` ` `  `class` `TreeSetDemo { ` `    ``public` `static` `void` `closestGreater(``int``[] arr) ` `    ``{ ` `        ``int` `n = arr.length; ` `        ``TreeSet ts = ``new` `TreeSet(); ` `        ``ArrayList ceilings = ``new` `ArrayList(n); ` ` `  `        ``// Find smallest greater or equal element ` `        ``// for every array element ` `        ``for` `(``int` `i = n - ``1``; i >= ``0``; i--) { ` `            ``Integer greater = ts.ceiling(arr[i]); ` `            ``if` `(greater == ``null``) ` `                ``ceilings.add(-``1``); ` `            ``else` `                ``ceilings.add(greater); ` `            ``ts.add(arr[i]); ` `        ``} ` ` `  `        ``for` `(``int` `i=n-``1``; i>=``0``; i--) ` `           ``System.out.print(ceilings.get(i) + ``" "``); ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int``[] arr = {``50``, ``20``, ``200``, ``100``, ``30``}; ` `        ``closestGreater(arr); ` `    ``} ` `} ` Output: ```100 30 -1 -1 -1 ``` Time Complexity : O(n Log n) My Personal Notes arrow_drop_up Article Tags : Practice Tags : Be the First to upvote. Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.

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# How To Find The Distance Math How to find the distance math video thumbnail find distance math. How to find the distance math extend the activity by adding in the skill of subtracting fractions to find the distance between the two estimated points mathematica find distance. How to find the distance math video thumbnail mathway find distance. How to find the distance math picture of distance formula and graph mathematica find distance. How to find the distance math find the distance between two points same x y coordinate find distance math. How to find the distance math mathematica find distance. How to find the distance math ex 1 find distance between i 2 3 mathway find distance. How to find the distance math how to find the perimeter area of a complex figure a math find distance math. How to find the distance math distance formula mathematica find distance. How to find the distance math find distance math. How to find the distance math find the distance between two points find distance math. How to find the distance math print distance in math formula concept worksheet mathematica find distance. How to find the distance math coordinate geometry find distance math. How to find the distance math video thumbnail mathematica find distance. How to find the distance math distance and displacement in physics definition and examples video lesson transcript find distance math. How to find the distance math mathway find distance. How to find the distance math image titled use distance formula to find the length of a line step 1 find distance math. How to find the distance math coordinate geometry distance between two points or distance formula mathematica find distance. How to find the distance math 4 how find distance math. How to find the distance math length and distance mathway find distance. How to find the distance math image titled find the distance between two points step 2 mathematica find distance. How to find the distance math distance between two points 3 mathway find distance. How to find the distance math how to break down the distance formula for students ideas to make teaching the distance formula simple includes free i can statements for the math find distance math. How to find the distance math null mathematica find distance. How to find the distance math students use a coordinate grid of an amusement park map to find the distance between two locations students will love to use the map to navigate throughout mathway find d.

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Explore BrainMass Share This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here! 1) Free radicals (see Problem 115) are also important in many environmentally significant reactions. For example, photochemical smog, which forms as a result of the action of sunlight on air pollutants, is formed in part by the following two steps. X NO2 NO O O O2 O3 UV light Ozone damages rubber products. The product of this reaction, ozone, is a pollutant in the lower atmosphere. Ozone is an eye and lung irritant and also accelerates the weathering of rubber products. Write Lewis structures for each of the reactants and products in the preceding reactions. 2) Aerosol cans carry clear warnings against incineration because of the high pressures that can develop upon heating. Suppose a can contains a residual amount of gas at a pressure of 755 mm Hg and a temperature of 25 °C. What would the pressure be if the can were heated to 1155 °C? 3) Water does not easily remove grease from dirty hands because grease is nonpolar and water is polar; therefore they are immiscible. The addition of soap, however, results in the removal of the grease. Examine the following structure of soap and explain why soap works. CH3(CH2)16C ONaO Nonpolar tail Polar head Sodium stearate a soap #### Solution Preview 1) Please see the attached image for lewis structures of each compound. Please let me know if something is lacking here! 2) We can use the relationship P1/T1=P2/T2. Also make sure to change the temperatures to kelvin and the pressures to atm! 755 ... #### Solution Summary In this solution we consider how to write lewis structures for NO2, NO, O, O2, and O3 in clear detail. We also apply a gas law to figure out the pressure after heating a can of spray. Finally, a brief discussion about how a detergent works is given. \$2.19 ## Multiple choice questions in microeconomics 1.Scarcity 1.exists because people have wants that are unlimited relative to the availability of resources to satisfy those wants 2.creates a need for society to allocate goods according to some set of criteria 3.means that society and individuals must make choices 4.exists in all societies 5.all of the above 2.In economics, capital is defined as (Think about the factors of production.) 1.natural resources, such as water, oil, and iron ore 2.the natural, unskilled abilities of people 3.human creations used in the production process 4.money and other financial assets 5.the willingness of business owners to take risks 3.According to the law of comparative advantage, an individual should produce a good if he or she 1.can produce it with the fewest resources 2.has an absolute advantage of producing the good. 3.has the lowest opportunity cost of producing the good regardless of whether it is produced with the fewest resources 4.has the greatest opportunity cost of producing the good regardless of whether it is produced with the fewest resources 5.has the greatest opportunity cost of producing the good and produces it with the fewest resources 4.The effect of a decrease in the price of laptop computers, other things constant, is likely to be best represented by which of the following? 1.a leftward shift of the demand curve. 2.a movement up along the demand curve 3.a rightward shift in the demand curve 4.a movement down along the demand curve 5.a leftward shift in the supply curve 5.Which of the following will not change the demand for movie tickets? 1.a change in the cost of babysitting services 2.a change in the price of movie tickets 3.a change in the quality of television programs 4.a change in the income of movie-goers 5.a change in the number of consumers 6.Which of the following would be most likely to increase the demand for downtown parking in a large city? 1.improved bus service to the downtown area 2.lower downtown parking fees 3.more downtown parking lots 4.more freeways leading to the downtown area 5.a major employer moves to the suburbs 7.Attempts are being made to develop a biodegradable plastic using agricultural produce such as potatoes. If these attempts are successful, what will happen to the equilibrium price and quantity of potatoes? (You may want to relate this question to what has recently happened with corn and ethanol.) 1.Price will increase and quantity decrease. 2.Price will increase and quantity increase. 3.Price will decrease and quantity increase. 4.Price will decrease and quantity decrease. 5.No change in equilibrium and quantity will occur. 8.If an increase in the price of a product from \$100 to \$200 per unit leads to a decrease in the quantity demanded from 10 to 8 units, then demand is 1.elastic 2.inelastic 3.unit elastic 4.0 5.inferior 9.Two separate groups of people use New York City subways. One group uses them only during rush hour to travel to and from work. The other group uses them only in midday for leisure activity. If New York City wants to increase transit fares with the smallest possible reduction in revenue, for which group should it increase the fare? 1.The rush-hour group because its demand for subway service is more elastic than that of the midday group. 2.The rush-hour group because its demand for subway service is less elastic than that of the midday group. 3.The midday group because its demand for subway service is more elastic than that of the rush-hour group. 4.The midday group because its demand for subway service is less elastic than that of the rush-hour group. 5.It doesn't matter because both groups have the same elasticity of demand. 10.As price falls along a given demand curve for pretzels, 1.quantity demanded, total utility, marginal utility, and consumer surplus increase; consumer expenditure decreases. 2.quantity demanded, total utility, and consumer surplus increase; marginal utility and consumer surplus decrease 3.quantity demanded, total utility, consumer surplus, and consumer expenditure increase; marginal utility decreases 4.quantity demanded, total utility, and consumer surplus increase; marginal utility decreases; consumer expenditure might increase, decrease, or remain constant 5.quantity demanded, total utility, marginal utility, consumer surplus, and consumer expenditure all increase 11.Suppose a lawyer leaves his \$70,000-a-year job and starts his own firm breeding pit bulls. In the first year, his accounting profits are \$90,000. The lawyer finances his new business with \$100,000 from his savings account, which had earned 10 percent interest. If this is all of the implicit and explicit revenues and costs his economic profit is 1.\$10,000 2.\$60,000 3.\$70,000 4.-\$80,000 5.-\$90,000 12.Farmer Fanny sells her crops in a perfectly competitive market. If she produces 500 bushels for total revenue of \$3,000 and if harvesting the 501st bushel would raise her total cost from \$2,500 to \$2,510, her 1.revenue will increase by \$4 if she harvests the 501st bushel 2.revenue will fall by \$4 if she harvests the 501st bushel 3.average fixed cost will rise if she harvests the 501st bushel 4.profit will fall by \$10 if she harvests the 501st bushel 5.profit will fall by \$4 if she harvests the 501st bushel 13.What is true at the profit-maximizing quantity for a nondiscriminating monopolist but not true of a perfectly competitive firm? 1.Price equals marginal cost. 2.Price is greater than marginal cost. 3.Marginal revenue equals marginal cost. 4.Marginal revenue is less than marginal cost. 5.Marginal revenue is greater than average revenue. 14.Which of the following is common to all market structures? 1.All of the following are correct. 2.Each firm must be able to earn sufficient revenue to cover its variable cost in order to continue producing in the short run. 3.The demand curve for the firm's output and the firm's average revenue curve are the same. 4.If it can earn a profit, the firm should increase production as long as marginal revenue exceeds marginal cost. 5.Each firm must be able to earn sufficient revenue to cover its total cost in order to continue operating in the long run. 15.Which of the following could explain an increase in demand for labor? (Think about our discussions of human capital.) 1.additional training that increases the productivity of each unit of labor in this market 2.an increase in the amount of risk associated with this job 3.a decrease in the amount of risk associated with this job 4.an improvement in the working conditions associated with this job 5.a decline in the working conditions associated with this job 16.Which of the following affects the interest rate on a loan? 1.all of the following 2.the duration of the loan 3.the tax treatment of the loan 4.the administrative cost of the loan 5.the risk of default on the loan 17.The marginal social cost of air quality increases as air quality decreases. If the dirtiest production process would create 700 tons of pollutants, then 1.eliminating the first 20 tons of pollution is cheaper than eliminating the last 20 tons. 2.eliminating the last 20 tons of pollution is cheaper than eliminating the first 20 tons. 3.more damage will be done by the first 20 tons of pollution than by the last 20 tons 4.twice as much damage will be done by the last 20 tons of pollution than by the first 20 tons 5.the optimal level of air quality is to reduce polluting by 700 tons 18.Which of the following is a positive externality of consumption? 1.Inoculations against a disease reduce the likelihood of transmitting it to others 2.Phosphates from laundry detergents 3.Litter from fast-food containers 4.Ozone depletion from the production of fast-food containers 5.The greenhouse effect 19.Which of the following best expresses the benefit from international trade? 1.With trade, each country can concentrate on producing those goods and services where it has a comparative advantage. 2.With trade, a country can increase its political involvement on a global scale. 3.Increased U.S. trade will improve high-tech exports but not agricultural exports. 4.Increased trade will increase U.S. exports and decrease U.S. imports. 5.Increased trade implies that exports of goods and services will always equal imports of goods and services. 20.Over the last 20 years, in countries around the world (Recent events in the US may lead you to think differently but if you agree with the e-text and look world statistics the answer is a. I hope you have discovered from this class the positive relationship between economic freedom and growth.) 1.markets are replacing central planning 2.central planning is replacing capitalist market coordination 3.economic systems are stable and unchanging 4.all countries employ the same system of resource ownership, resource allocation, and incentives to answer the basic economic questions 5.none of the above View Full Posting Details

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#### Archived This topic is now archived and is closed to further replies. # SIMPLE NEWBIE QUESTION This topic is 5868 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts I''m new and havn''t gotten the hang of working in 3d coords. I have a single quad, on the xy plane (z is up). All I want the quad to do is rotate on its center. (ie spin in place). But I need it to be able to do this from any (x,y) coord. So if its at (2,3) I want it to stay more-or-less on top of (2,3). It should be a really simple answer. Here is the code I have so far: void square::DisplaySq() { glPushMatrix(); glRotatef(theta, 0, 0, 1); glTranslatef(x[0], y[0], 0); glBegin(GL_QUADS); for (int i=0; i<4; i++) { glColor3f(r, g, b); glVertex3f(x, y[i], z[i]); } glEnd(); glPopMatrix(); } Can anyone help? NOMAD ##### Share on other sites Solved it myself, I used glTranslatef(...). I thought it would translate the object not the coord plane. I''m such a newbie! Here the code that worked if anyone cares: void square::DisplaySq() { glPushMatrix(); glTranslatef((x[0]+x[2])/2, (y[0]+y[2])/2, 0); glRotatef(theta, 0, 0, 1); glColor3f(r, g, b); glVertex3f((-1)*(width/2), (-1)*(width/2), z[0]); glVertex3f((width/2), (-1)*(width/2), z[0]); glVertex3f((width/2), (width/2), z[0]); glVertex3f((-1)*(width/2), (width/2), z[0]); glEnd(); glPopMatrix(); } ##### Share on other sites Yep, translatef moves the whole modelview matrix, not just an object. Also, making your whole subject header in caps won''t win u any friends ##### Share on other sites That brings me to the conclusion of: "Dont do crack. Drugs are bad... even for newbies" 1. 1 2. 2 3. 3 Rutin 18 4. 4 JoeJ 14 5. 5 • 14 • 10 • 23 • 9 • 33 • ### Forum Statistics • Total Topics 632633 • Total Posts 3007543 • ### Who's Online (See full list) There are no registered users currently online ×

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## Resource Options • ##### Free/Non-commercial Resources: Displaying 1-20 of 34 resources • Resource Type: Website / HyperLink ## Brass Instruments Uncoiled Subjects: Music, Measurement & Geometry, Measurement, Tools, and Data Analysis, Visual Arts & Performing Arts, Mathematics This resource, from the Dallas Symphony Orchestra s DSOkids Web site, provides guidelines for discovering the lengths of brass i... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Exploration of Pascal's Triangle Subjects: Algebra & Functions, Number Sense, Mathematics In this activity from the Math Forum, reviewed for grades Pre-K-2 by Illuminations, students work with a partner on an explorati... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Developing Geometry Understandings and Spatial Skills through Puzzlelike Problems with Tangrams: Tangram Puzzles Subjects: Mathematics, Geometry This is part one of a two-part E-example from the National Council of Teachers of Mathematics that demonstrates the potential fo... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Counting Crows Subjects: Theatre, Visual Arts, English-Language Arts, Mathematics, Literature This ARTSEDGE lesson blends math and art with literature using Aesop's fable "The Crow and the Pitcher." The class discusses the... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Learning Planet Sizes Subjects: Algebra & Functions, Earth & Space Science, Measurement & Geometry, Science and 2 additional.. In this activity, learners use the concepts of greater than, less than, and equals to classify student height, object size, and ... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Teaching and Learning About Whole Numbers Subjects: Addition & Subtraction, Number Sense, Mathematics This resource from the University of Melbourne, reviewed for grades 3-5 by Illuminations, features information about teaching su... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Prime Mathematics Enrichment Subjects: Algebra & Functions, Addition & Subtraction, Data Analysis, Statistics, and Probability, Measurement & Geometry and 4 additional.. This NRICH resource, reviewed for grades Pre-K-2 by Illuminations, features an extensive collection of math activities. Topics c... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Shapes Art Subjects: English-Language Arts, Mathematics, Literature In this lesson, one of a multi-part unit from Illuminations, students participate in activities in which they focus on connectio... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Varnelle's Primary Math - Introduction to Measurement Subjects: Measurement & Geometry, Professional Development, Measurement, Tools, and Data Analysis, Mathematics This unit from the Math Forum, reviewed for grades K-2 by Illuminations, introduces young children to some beginning concepts in... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## It Counts Subjects: Mathematics, Number Sense, Science This Science NetLinks lesson is designed to help students understand and reinforce how numbers are assigned to objects, as well ... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Creating "AB" Patterns Subjects: Mathematics, Visual Arts In this ARTSEDGE lesson, students recognize AB patterns in nature and man-made objects. They learn the meaning of the words "pat... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Bag the Beans Subjects: Mathematics, Number Sense In this Science NetLinks lesson, students work with manipulatives (beans) to create and solve math problems, some of which have ... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Can You Measure Up? Subjects: Theatre, Visual Arts, English-Language Arts, Mathematics In this ARTSEDGE lesson, students read Janet Stevens and Susan Stevens Crummel's book "Cook-a-Doodle-Doo!" They engage in dramat... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## NRICH Online Maths Club Subjects: Algebra & Functions, Data Analysis, Statistics, and Probability, Measurement & Geometry, Trigonometry/Pre-Calculus and 3 additional.. NRICH aims to give people some idea of the range of applications of math and how it underpins human activity and our understandi... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Sorting Subjects: Mathematics, Number Sense, Science The purpose of this lesson, from Science NetLinks, is to develop the idea that information can be more easily managed and retrie... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Varnelle's Primary Math - Introduction to Geometry Subjects: Mathematics, Geometry These activities from Math Forum, reviewed for grades PreK-2 by Illuminations, are set up as a unit in introductory geometry. Wi... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Patterns to 100 and Beyond Subjects: Mathematics, Number Sense In this lesson, one of a multi-part unit from Illuminations, students examine number patterns, using a calculator to move beyond... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink ## Teaching and Learning about Whole Numbers Subjects: Number Sense, Mathematics This resource, from the University of Melbourne, is a compilation of links that describes various teaching strategies and studen... Contributor: Thinkfinity Views: 0Favorites: 0 • Resource Type: Website / HyperLink

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# What is the Factor Tree of 72? ## Factor Tree of 72: Breaking Down the Prime Factors The process of breaking down a number into its prime factors is called factorization. In mathematics, factorization plays a crucial role in solving many problems involving integers. In this blog post, we will focus on the factor tree of 72, a composite number that has multiple prime factors. Through this post, we will explore the concept of factorization and how it applies to the number 72. ## Breaking Down the Number 72 The first step in finding the factor tree of 72 is to determine its factors. The factors of a number are the numbers that divide it evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. To find the factors of 72, we can start by dividing it by 2. 72 divided by 2 equals 36, so we can write 72 as 2 x 36. We can then factorize 36 in a similar manner. 36 divided by 2 equals 18, so we can write 36 as 2 x 18. Continuing with this process, we can break down 18 into 2 x 9, and 9 into 3 x 3. We can write the factor tree of 72 by arranging the factors in a tree-like structure. The number 72 goes at the top of the tree, and we draw two branches, one for each of its prime factors, 2 and 36. We then break down 36 into its prime factors, 2 and 18, and write these as branches off the 36 node. We continue the process for each factor until we reach only prime factors at the bottom of the tree. The factor tree for 72 would look like this: `````` 72 / \ 2 36 / \ 2 18 / \ 2 9 / \ 3 3`````` ### Prime Factors of 72 In the factor tree of 72, we can see that its prime factors are 2, 2, 2, 3, and 3. We can write 72 as the product of its prime factors by multiplying these prime factors together. We get: 72 = 2 x 2 x 2 x 3 x 3 We can also write this as 72 = 2³ x 3². This notation is a shorthand way of writing the product of a number’s prime factors, where the exponent indicates how many times a factor appears in the product. For example, 2³ means “two cubed” or “two raised to the third power,” which is equivalent to 2 x 2 x 2. The prime factorization of 72 as 2³ x 3² is a unique representation of the number, meaning that no other set of prime factors can multiply together to give 72. ### Importance of Factor Trees Factor trees are a useful tool for finding the prime factorization of a number. They provide a visual representation of the factorization process and make it easier to keep track of the factors. Factor trees can also help us understand the concept of prime factorization and the fundamental theorem of arithmetic, which states that every integer greater than 1 can be expressed as a unique product of primes. By breaking down a number into its prime factors, we can see the building blocks that make up the number and how they combine to create it. ### Applications of Prime Factorization Prime factorization has many practical applications in mathematics and other fields. In cryptography, the security of some encryption algorithms relies on the difficulty of factoring large numbers into their prime factors. Prime factorization is also used in number theory to solve problems involving integers, such as finding the greatest common divisor and least common multiple of two or more numbers. In physics, prime factorization is used to calculate the factors of physical quantities, such as the fundamental units of measurement. In addition, prime factorization is used in computer science and engineering for data compression and error correction. For example, in data compression algorithms like JPEG and MP3, prime factorization is used to reduce the size of digital files by eliminating redundant data. In error-correcting codes, prime factorization is used to detect and correct errors that occur during data transmission. Prime factorization is also used in real-world scenarios, such as in calculating the factors of a large number of atoms in a chemical compound. The prime factors of a large number can help scientists understand the properties of the molecule and its behavior in different chemical reactions. #### Conclusion In conclusion, the factor tree of 72 is a visual representation of the prime factorization of the number 72. Through the factor tree, we can see that 72 is composed of the prime factors 2 and 3, which are repeated three and two times, respectively. The factorization of a number into its prime factors is an important concept in mathematics and has practical applications in various fields, including cryptography, physics, computer science, and chemistry. Understanding the factorization process and prime factorization is crucial in solving mathematical problems and understanding the fundamental properties of numbers. What is the Factor Tree of 72? Scroll to top

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Name: Andrew ID: Collaborated with: On this homework, you can collaborate with your classmates, but you must identify their names above, and you must submit your own homework as an knitted HTML file on Canvas, by Sunday 10pm, this week. Huber loss function Recall, as covered in lab, the Huber loss function (or just Huber function, for short), with cutoff $$a$$, which is defined as: $\psi_a(x) = \begin{cases} x^2 & \text{if |x| \leq a} \\ 2a|x| - a^2 & \text{if |x| > a} \end{cases}$ This function is quadratic on the interval $$[-a,a]$$, and linear outside of this interval. It transitions from quadratic to linear “smoothly”, and looks like this (when $$a=1$$): Plotting practice, side effects • 1a. The code for the huber() function that you should have arrived at in this week’s lab is copied below. Using huber(), reproduce the plot of the Huber function that you see above. The axes and title should be just the same, so should the Huber curve (in black), so should be the red dotted lines at the values -1 and 1, and so should the text “Linear”, “Quadratic”, “Linear”. huber = function(x, a=1) { ifelse(abs(x) <= a, x^2, 2*a*abs(x)-a^2) } • 1b. Modify the huber() function so that, as a side effect, it prints the string “Invented by the great Swiss statistician Peter Huber!” to the console. Hint: use cat(). Call your function on an input of your choosing, to demonstrate this side effect. • 1c. Further modify your huber() function so that, as another side effect, it produces a plot of Switzerland’s national flag. Hint: look up this flag up on Google; it’s pretty simple; and you should be able to recreate it with a few calls to rect(). Call your function on an input of your choosing, to demonstrate its side effects. Exploring function environments • 2a. A modified version of the Huber function is given below. You can see that we’ve defined the variable x.squared in the body of the function to be the square of the input argument x. In a separate line of code (outside of the function definition), define the variable x.squared to be equal to 999. Then call huber(x=3), and display the value of x.squared. What is its value? Is this affected by the function call huber(x=3)? It shouldn’t be! Reiterate this point with several more lines of code, in which you repeatedly define x.squared to be something different (even something nonnumeric, like a string), and then call huber(x=3), and demonstrate afterwards that the value of x.squared hasn’t changed. huber = function(x, a=1) { x.squared = x^2 ifelse(abs(x) <= a, x.squared, 2*a*abs(x)-a^2) } • 2b. Similar to the last question, define the variable a to be equal to -59.6, then call huber(x=3, a=2), and show that the value of a after this function call is unchanged. And repeat a few times with different assignments for the variable a, to reiterate this point. • 2c. The previous two questions showed you that a function’s body has its own environment in which locally defined variables, like those defined in the body itself, or those defined through inputs to the function, take priority over those defined outside of the function. However, when a variable referred to the body of a function is not defined in the local environment, the default is to look for it in the global environment (outside of the function). Below is a “sloppy” implementation of the Huber function called huber.sloppy(), in which the cutoff a is not passed as an argument to the function. In a separate line of code (outside of the function definition), define a to be equal to 1.5 and then call huber.sloppy(x=3). What is the output? Explain. Repeat this a few times, by defining a and then calling huber.sloppy(x=3), to show that the value of a does indeed affect the function’s ouptut as expected. Challenge: try setting a equal to a string and calling huber.sloppy(x=3); can you explain what is happening? huber.sloppy = function(x) { ifelse(abs(x) <= a, x^2, 2*a*abs(x)-a^2) } • 2d. At last, a difference between = and <-, explained! Many of you have asked about this. The equal sign = and assignment operator <- are often used interchangeably in R, and some people will often say that a choice between the two is mostly a matter of stylistic taste. This is not the full story. Indeed, = and <- behave very differently when used to set input arguments in a function call. As we showed above, setting, say, a=5 as the input to huber() has no effect on the global assignment for a. However, replacing a=5 with a<-5 in the call to huber() is entirely different in terms of its effect on a. Demonstrate this, and explain what you are seeing in terms of global assignment. • 2e. The story now gets even more subtle. It turns out that the assignment operator <- allows us to define new global variables even when we are specifying inputs to a function. Pick a variable name that has not been defined yet in your workspace, say b (or something else, if this has already been used in your R Markdown document). Call huber(x=3, b<-20), then display the value of b—this variable should now exist in the global enviroment, and it should be equal to 20! Alo, can you explain the output of huber(x=3, b<-20)? • Challenge. The property of the assignment operator <- demonstrated in the last question, although tricky, can also be pretty useful. Leverage this property to plot the function $$y=0.05x^2 - \sin(x)\cos(x) + 0.1\exp(1+\log(x))$$ over 50 x values between 0 and 2, using only one line of code and one call to the function seq(). • 2f. Give an example to show that the property of the assignment operator <- demonstrated in the last two questions does not hold in the body of a function. That is, give an example in which <- is used in the body of a function to define a variable, but this doesn’t translate into global assignment. Shakespeare’s complete works Once more, as in lab (and lab/hw from Week 3), we’re going to look at that the complete works of William Shakespeare from Project Gutenberg. We’ve put this text file up at http://www.stat.cmu.edu/~ryantibs/statcomp/data/shakespeare.txt. Functions for word tables • 3a. Compute word tables for each of Shakespeare’s plays. You should be able to do this by putting together relevant parts of the solution code from Q2 and Q3 in this week’s lab. Your result should be a list called shakespeare.wordtab.by.play of length 44, witheach component giving a word table for one of Shakespeare’s plays. Display the first 5 entries of each word table. • 3b. Suppose we have many text documents, and we have computed word tables for each (just like we did for Shakeapeare’s plays). A document-term matrix is essentially a matrix formed by stacking the word tables along its rows. But how do we combine word tables into a matrix when they count the number of appearances of possibly different words, and hence are potentially of different lengths? The answer: we gather all unique words across all documents, into a “master” list of words, and then we expand each word table so that it has one entry per word in the master list, with 0s for words that never appeared in its corresponding document. This ensures that the word tables are all of the same length, and the document-term matrix is formed by stacking them row-wise. Consider the code below which sketches out the implementation of a function called get.dtmat.from.wordtabs(). The only argument is wordtab.list, which is a list of word tables. There are two main steps: 1. The first step is to get all the unique words across all the word tables. The result should be stored in a string vector called master.words. The code below simply sets this to c(); replace this by your own implementation. You can see that the next line sorts master.words into alphabetical order. 2. The second step is to populate the document-term matrix. The code below defines an matrix dt.mat of the appropriate dimensions, of all 0s, and iterates over its rows one by one. Put your implementation to populate a row of dt.mat into the body of the for() loop. Hint: consider the ith row dt.mat[i,]; this is already all 0s; and so we only need to modify its entries for the words that appeared in the ith word table. This should only require one line of code; take advantage of the column names of dt.mat and use named indexing! Once you have finished your implementation, apply get.dtmat.from.wordtabs() to shakespeare.wordtab.by.play and save the result as shakespeare.dt.mat. Its dimensions should be 44 x 25801. Display its first 10 rows and 5 columns. get.dtmat.from.wordtabs = function(wordtab.list) { # First get all the unique words master.words = c() # Compute the master list here master.words = sort(master.words) # Then build the document-term matrix dt.mat = matrix(0, nrow=length(wordtab.list), ncol=length(master.words)) rownames(dt.mat) = names(wordtab.list) colnames(dt.mat) = master.words for (i in 1:nrow(dt.mat)) { # Populate the ith row of dt.mat here } return(dt.mat) } • 3c. Compute correlations between every pair of rows in shakespeare.dt.mat. Use the cor() function, but beware: this computes correlations between each pair of columns of its argument. Which pair of plays achieves the highest correlation, and hence are the most similar in terms of the word tables? Note: here we obviously want to exclude from consideration the correlations of each play with itself, which will be 1. Is this a surprising result? • Challenge. Do some exploratory analysis of the correlations you computed in the last question, and describe what you are seeing. For example, you might consider hand-labeling each play as either a comedy or tragedy, and then looking at the correlations within and between these groups. You might also consider performing some kind of hierarchical clustering based on the correlations. • Challenge. Use TF-IDF weighting on shakespeare.dt.mat. This stands for term frequency-inverse document frequency weighting; you can read about this in the course notes for Stat Computing from Fall 2016, or on the web. Compute the top 2 principle component scores of the newly-weighted document-term matrix, and produce a scatter plot of the 44 plays with respect to these 2 dimensions. Set the title and axes labels appropriately. Color the points that correspond to comedies in blue and tragedies in red. Do you see any separation between the point clusters for comedies and tragedies?

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The wavelength result is 3 m. Most common velocities: Light in vacuum (air) = 300,000 km/s. Radio waves travel at the speed of light which is 3.00 x 108 m/s. notice how I eliminate the units I don't want by finding the equivalent units I wanted. a. wave A; wave A b. wave A; wave B c. wave B; wave A d. wave B; wave B 2. Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. I needed help with this question. Can you help me with setting up the formula? A long wavelength and. That will give you the wavelength in meters. Which of the following statements about mechanical waves is true? Click hereto get an answer to your question ️ 3. You can view more similar questions or ask a new question. In the form you can customize it with the desired value and by choosing from m/s, km/s and miles/s as measurement units. Wave velocity (m/s) =Wavelength (m) * Frequency (Hz) Example calculation. If the crest of an ocean wave moves a distance of 20 meters in 10 seconds, then the speed of the ocean wave is 2.0 m/s. (7.9 x 10^8 km)(1000m/1km)/(3.00 x 108 m/s)=____s minimum and maximum. What is the main difference between mechanical and electromagnetic waves? Speed B. Wavelength C. Frequency D. Energy E. Amplitude I think a, QUESTION---Which waves have wavelengths longer than those of visible light? A pulsar is a rapidly rotating neutron star that continuously emits a beam of radio waves in a searchlight manner. Which waves have the longest, All electromagnetic energy (light), including gamma rays through radio waves, has what in common? Questions Light waves slow down as they enter the Earth's atmosphere from space. Sound in air = 340 m/s . a. mechanical waves require a medium to travel through b. mechanical waves do not have amplitude and wavelength c. mechanical waves do not have frequency d. How can the behavior of light be described as it leaves the vacuum of space and enters the Earth's atmosphere? This wavelength calculator determines the distance between two wave peaks when you know the frequency and the wave velocity or speed. They have have frequencies from 300 GHz to as low as 3 kHz, and corresponding wavelengths from 1 millimeter to 100 kilometers. Which of the following shows the worst conditions for a sound wave travel? The speed of light is 3.00*10^8m/s. Radio waves are electromagnetic waves that travel at a speed of 3.00 108 m/s, the speed of light. 8 x 10-22 m/s 1 x 107 m/s 1.25 x. In this case we are talking about peaks of the wave. The measurement units for it available in the form are microns; mm; cm; m; km; inches and feet. Express, Light waves are traveling through glass. The speed of radio waves is 3*10^8m/s, the same as the speed of light. Radio waves travel at the speed of light which is 3.00 x 108 m/s. Speed B. Wavelength C. Frequency D. Energy E. Amplitude I think a, QUESTION---Which waves have wavelengths longer than those of visible light?

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View the step-by-step solution to: Question # The operations manager for the Blue Moon Brewing Co. produces two beers: Lite (L) and Dark (D). Two of his resources are constrained: production time, which is limited to 8 hours (480 minutes) per day; and malt extract (one of his ingredients), of which he can get only 675 gallons each day. To produce a keg of Lite beer requires 2 minutes of time and 5 gallons of malt extract, while each keg of Dark beer needs 4 minutes of time and 3 gallons of malt extract. Profits for Lite beer are \$3.00 per keg, and profits for Dark beer are \$2.00 per keg. What is the objective function? (A) \$2L + \$4D = Z (B) \$5L + \$3D = Z (C) \$4L + \$2D = Z (D) \$3L + \$2D = Z (E) \$2L + \$3D = Z The explanation of the... View the full answer ### Why Join Course Hero? Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors. • ### - Study Documents Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access. Browse Documents

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$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$ Section1.3Variables and Relationships Subsection1.3.1Overview In physical settings, we usually consider measurements of multiple quantities at the same time. We do this because we are interested in the relationships between these different quantities. We think of each quantity of interest as a variable; the collection of all such variables under consideration is called the system. At any instant, the variables of the system will each have a particular value and the collection of those values at that instant is called the state of the system. In this section, we introduce the idea of different variables in a system having a relation. Specifically, this means that knowing the value of one variable in a particular state of the system gives us information about the value of another variable. Relations are often explored using a scatter plot. Mathematically, a relation is usually described by an equation. Solving an equation for a variable is mathematically how we determine the value of a variable for a given state of the system. Subsection1.3.2Systems, States and Variables In the course of an experiment, or even just in observation, many different quantities typically are covarying, or changing with one another. For example, an object in motion has changing position, changing velocity, and changing forces. In the course of a chemical reaction, there are changing concentrations of the different reactants and products as well as possibly changing temperature, pH, and volume, for example. While observing a changing population, there could be changing population numbers, total biomass, birth and death rates, consumption of resources, and production of products and waste. Mathematically, the system consists of all possible observable quantities associated with the experiment or observed physical system. The state of the system refers to the collection of instantaneous values of all such quantities at a particular instant or configuration of the system. A variable represents a single quantity that is or could be observed in the system. Example1.3.1 Consider the following data about the population, births and deaths in the United States. To conserve space, the data are given using scientific notation expressed in the standard machine form where the power of 10 follows the letter E, so that $2.521 \times 10^8$ would be written 2.521E8. Year Population Births Deaths Year Population Births Deaths 1991 2.521E8 4.111E6 2.170E6 2001 2.850E8 4.026E6 2.416E6 1992 2.550E8 4.065E6 2.176E6 2002 2.876E8 4.022E6 2.443E6 1993 2.577E8 4.000E6 2.269E6 2003 2.901E8 4.090E6 2.448E6 1994 2.602E8 3.953E6 2.279E6 2004 2.928E8 4.112E6 2.397E6 1995 2.628E8 3.900E6 2.312E6 2005 2.955E8 4.138E6 2.448E6 1996 2.652E8 3.891E6 2.315E6 2006 2.984E8 4.266E6 2.426E6 1997 2.677E8 3.881E6 2.314E6 2007 3.012E8 4.316E6 2.424E6 1998 2.703E8 3.942E6 2.337E6 2008 3.041E8 4.248E6 2.472E6 1999 2.727E8 3.959E6 2.391E6 2009 3.068E8 4.131E6 2.437E6 2000 2.822E8 4.059E6 2.403E6 2010 3.094E8 3.999E6 2.468E6 Each row (corresponding to the population in a given year) represents a distinct state of the system. The system is characterized by the observed values, which consist of the year itself, the total population, the total number of births in the year, and the total number of deaths in the year. These three measurements are the variables of the system. The year should also be considered one of the variables (an independent variable) because we think of the other variables changing with respect to time. A symbol is often used to represent, or name, a variable. The symbol is often a letter, but could also be a Greek letter, an abbreviation, or a word. The choice of symbol should generally be related to the meaning of the variable. Uppercase and lowercase letters are different symbols and should not be interchanged with one another. An important part of communication is in stating clearly the variables of a system and identifying the symbols that are chosen to represent them. In the previous example, the population variable might be represented by the symbol $P$ while the births and deaths might be represented by the symbols $B$ and $D\text{.}$ The year might be represented by the symbol $Y\text{.}$ Example1.3.2 In biology, scientists run electrophoresis gels to determine the size of polymers, such as proteins or DNA strands. The gel provides a porous structure for the polymers to travel through while an electric potential (voltage) creates a force that pulls the polymers through the gel. Different size polymers travel at different speeds. The experiment is setup with all polymers starting at one end of the gel, the voltage is turned on for a certain amount of time and then disconnected. Clusters of similarly sized polymers are identified visually as bands on the gel, with smaller polymers traveling a greater distance. The following paragraph illustrates how variables might be introduced with their chosen symbols. The image below represents an electrophoresis gel run on a standardized collection of DNA of fixed sizes. Because the image does not show a length scale, the distances traveled by the different lengths are measured in image pixels and recorded in the table below. The variables for the experiment are the length of DNA segments and the distance traveled through the gel. Let $L$ represent the length of the segment (in nucleotides) and let $D$ represent the distance traveled (in pixels), measured from the center of the starting well to the center of the corresponding band in the image. Each row represents a single state $(L,D)$ of the system. $L$ (nts) $D$ (px) 100 342 200 327 300 312 400 299 500 288 600 278 700 270 800 263 900 256 1000 249 Please note that without the earlier descriptive paragraph introducing the variables, this table would have no context and would not be helpful to a reader. Any time you have data and refer to the data by variables, you need a few sentences that introduce the meaning of each variable along with the units of measurement. Subsection1.3.3Relationships Between Variables The primary motivation for collecting data regarding different variables in the state of a system is to determine relationships between those variables. One of the ways that we look for relationships is using a scatter plot. A scatter plot is a graph showing the relationship between two variables. Suppose the two variables use symbols $x$ and $y\text{.}$ For each state of the system, there will have been observed values for both $x$ and $y\text{.}$ The graph will include points for each pair $(x,y)\text{.}$ Spreadsheets (like Microsoft Excel, Apple Numbers or Google Sheets) are a common tool to generate scatter plots. The data are first put in a table. The first column of data will correspond to the variable used for the horizontal axis ($x$) and the second column of data will correspond to the variable for the vertical axis ($y$). Select the two columns at the same time and add a chart to your spreadsheet, choosing the scatter plot style of graph. You should become familiar with how to create a scatter plot. Always be sure that you label your axes. You should always use the variables of the system rather than the generic names of $x$ and $y\text{.}$ The following figure shows two different scatter plots for the electrophoresis gel data above. One plot is based on the pairs $(L,D)$ whereas the other is based on the pairs $(D,L)\text{.}$ These graphs contains the same information but viewed from a reverse perspective. When a system has a state defined by more than two variables, scatter plots can be defined for each pair of state variables. For example, the population data has four state variables, $(Y,P,B,D)\text{.}$ Three scatter plots can be formed by plotting the population, the total births and the total deaths versus the year, giving graphs of points $(Y,P)\text{,}$ $(Y,B)\text{,}$ $(Y,D)\text{.}$ Because the births and deaths are on the same scale, we can combine the plots as one. We could also plot the inverse relationships $(P,Y)\text{,}$ $(B,Y)$ and $(D,Y)\text{,}$ but these really contain the same information from a different view. We can also look at relationships between other pairs of variables. For example, we can look at how the number of births or deaths relate to the population, plotting $(P,B)$ and $(P,D)\text{,}$ or how the number of births relate to the number of deaths with $(B,D)\text{.}$ The graph showing the relation between births and deaths to time (above) is very similar to the graph showing the relation between births and deaths to population (below). However, the relation between the births and deaths illustrates that sometimes variables do not show a clear relation. Subsection1.3.4Equations as Relations An equation gives an abstract representation of a relationship between variables. An expression is any formula involving numbers and variables. An equation is a statement that two expressions are equal. For some values of the variables, the equation may be false; for other values, the equation will be true. A solution to the equation is a set of values for the variables in the equation that makes the statement true. The solution set of an equation is the set of all possible solutions. Just as the state of an experimental system is defined by the value of the variables defining the state, an equation can be considered as a mathematical way to define relationships between variables of an abstract system. A scatter plot of data is generalized for an equation as a graph of all solutions. If we choose an ordering for the variables (e.g., alphabetical), the values for the variables can be conveniently listed as an ordered list. When two variables are involved in an equation, the ordered list is called an ordered pair or point, like $(x,y)\text{,}$ and the graph of the equation is typically a curve in the plane. Example1.3.4 The equation \begin{equation*} 2x+3y = 12 \end{equation*} involves two variables, $x$ and $y\text{.}$ The expressions in the equation are $2x+3y$ and $12\text{.}$ The values $x=3$ and $y=2\text{,}$ corresponding to the ordered pair $(x,y)=(3,2)\text{,}$ provide one solution because for those values, \begin{equation*} 2x+3y = 2(3)+3(2)=12, \end{equation*} so that the equation is true. On the other hand, $(x,y)=(4,1)$ is not a solution because for that state, \begin{equation*} 2x+3y = 2(4)+3(1)=11 \end{equation*} and $11 \ne 12\text{.}$ Some other solutions include the points $(6,0)$ and $(0,4)\text{.}$ You should have recognized that this equation is an equation of a line. (See Appendix Subsection A.2.1.) That is, the solutions we identified above of $(3,2)\text{,}$ $(6,0)$ and $(0,4)\text{,}$ along with all other solutions, will lie on the same line in the plane. Example1.3.5 The equation \begin{equation*} u^2+v^2=16+6u \end{equation*} also involves two variables, $u$ and $v\text{.}$ The expressions in the equation are $u^2+v^2$ and $16+6u\text{.}$ Using ordered pairs $(u,v)\text{,}$ the points $(3,5)$ and $(3,-5)$ are solutions. That is, if $(u,v)=(3,5)\text{,}$ the expressions have the same value: \begin{align*} u^2+v^2 &= 3^2+5^2=9+25 = 34,\\ 16+6u &= 16+6(3) = 16+18 = 34. \end{align*} It is possible to show that the graph of solutions for this equation is a circle centered at $(3,0)$ with radius 5. Other points on this circle include such points as $(-2,0)$ and $(6,-4)\text{.}$ You should verify that these are also solutions, at least for one or two points to reinforce the idea that a solution makes the statement of the equation true. An equation gives us a constraint on the relationships between a system's variables. Knowing values of other variables or additional constraints, we are able to solve an equation to gain additional information. Example1.3.6 Consider the equation of the circle \begin{equation*} u^2+v^2=16+6u \end{equation*} used in the previous example. Find the solutions for which $u=-1$ and then find the solutions for which $v=2\text{.}$ Solution First, to solve the equation when $u=-1\text{,}$ we substitute the value of $u=-1$ and then use algebra to isolate $v\text{.}$ \begin{gather*} u^2+v^2=16+6u\\ (-1)^2+v^2 = 16 + 6(-1)\\ 1+v^2=10\\ v^2=9\\ v=\pm 3 \end{gather*} There are two values for $v$ when $u=-1\text{.}$ The solutions are the states $(u,v)=(-1,3)$ and $(u,v)=(-1,-3)\text{.}$ Next, to solve the equation when $v=2\text{,}$ we substitute the value $v=2\text{.}$ However, because $u$ appears in the equation with terms $u^2$ and $6u\text{,}$ we can not combine terms to isolate $u\text{.}$ Instead, we need to use the quadratic formula A.2.10. \begin{gather*} u^2+v^2=16+6u\\ u^2+(2)^2 = 16 + 6u\\ u^2+4=16+6u\\ u^2-6u-12=0\\ u=\frac{6 \pm \sqrt{(-6)^2-4(-12)}}{2}\\ u=\frac{6 \pm \sqrt{84}}{2} = \frac{6 \pm 2\sqrt{21}}{2}\\ u=3 \pm \sqrt{21} \end{gather*} Again, two states are solutions, $(u,v)=(3+\sqrt{21},2)$ and $(u,v)=(3-\sqrt{21},2)\text{.}$ Example1.3.7 Is it possible to enclose an area of 25 m2 in a rectangle with perimeter of 18 m? If so, how? Solution In this problem, we need to identify the relevant variables for the system and the equations that constrain the state. We are working with a rectangle, which is characterized by a length and a width. Let us draw a figure and use variables $L$ for the length and $W$ for the width. We have two more state variables, the perimeter $P$ and the area $A\text{.}$ Geometry gives us two equations that constrain the state of the system $(L,W,P,A)\text{:}$ \begin{gather*} P=2L+2W,\\ A=L \cdot W. \end{gather*} The problem gives us two additional pieces of information, $P=18$ and $A=25\text{.}$ When we substitute those values of the state into the equations, we have two equations for two variables: \begin{equation*} 2L+2W=18, \qquad L \cdot W = 25. \end{equation*} In order to solve these equations, we use one equation to isolate one of the variables, say $L\text{,}$ and then substitute the resulting expression into the other equation. Then we solve the equation that only involves $W\text{.}$ \begin{gather*} (9-W)W = 25\\ 9W-W^2=25\\ W^2-9W+25=0\\ W=\frac{9 \pm \sqrt{(-9)^2-4(25)}}{2}\\ W=\frac{9 \pm \sqrt{81-100}}{2}=\frac{9\pm \sqrt{-19}}{2} \end{gather*} When solving this quadratic formula, we have the square-root of a negative number giving complex numbers. In conclusion, we found that there are no real solutions. This means that it is not possible to create a rectangle with a perimeter of 18 m and an area of 25 m2. For a review on algebra strategies for solving equations, refer to the appendix. Subsection1.3.5Trend Lines and Regression Curves Mathematical equations are exact. Real data exhibit uncertainty and randomness. Although they do not capture the uncertainty of data, equations can be used to model the trend or average presented by the data. A trend line or trend curve (if not linear) is a model that captures the general behavior of the data and stays close to the scatter points. Most spreadsheets and graphing calculators have an option to show a line of best fit for a scatter plot, which is an example of a trend line. The process these programs use is called regression. They usually report the equation of the regression curve using the generic variable symbols $x$ and $y\text{,}$ so it is the researcher's responsibility to interpret the equation in terms of the true variables. A trend line or a trend curve allow us to predict values where there are not observed data. When the prediction occurs between observed data, such prediction is called interpolation. If the prediction is occurring beyond the extremes of the data, such prediction is called extrapolation. Often, a formula may not describe all of the data but provides a good approximation for certain values. Interpolation is usually safer than extrapolation, which might attempt to use the formula in a region where the approximation is not good. Example1.3.8 Consider the population example with the scatter plot of the number of deaths plotted with respect to the total population size, and predict the number of deaths in a year if the population were 300 million. A spreadsheet reported the trend line of this data set with an equation \begin{equation*} y=0.0049x+992711. \end{equation*} Because the scatter plot had $P$ on the horizontal axis and $D$ on the vertical axis, the more appropriate equation would be \begin{equation*} D=0.0049P+992711. \end{equation*} A spreadsheet may not give enough precision in the model equation when using the default settings. Note that the equation reported above only has two significant digits in the slope value but an apparent 6 significant digits in the intercept. By changing the settings for the equation of the trend line, we get a more precise model \begin{equation*} D = 4.86666 \times 10^{-3} P + 9.92711 \times 10^5. \end{equation*} Depending on the values of the data, the greater accuracy might make a significant difference. Let us compare the two models with a population of 300 million, $P=300\times 10^6\text{.}$ The first model, which only has two significant digits in the first coefficient, gives \begin{equation*} D = 0.0049(3\times 10^8) + 992711 = 2462711. \end{equation*} Because one coefficient only had two significant digits, we can only expect the first two digits are accurate, thus predicting $D=2.5$ million deaths in the year. Using the second model with six digits of accuracy in both coefficients, we find \begin{equation*} D = 4.86666 \times 10^{-3} \cdot 3 \times 10^{8} + 992711 = 2452709. \end{equation*} With six digits of accuracy, this predicts $D=2,452,710$ deaths in the year. Regression models do not really predict actual outcomes. They are generally based on noisy data, including measurement errors and randomness in sampling. Rather, they might be better interpreted as an expected average prediction. Having an actual value that is different from the value predicted by the regression model is expected. One of the key questions that statistics helps address is understanding when observed outcomes are sufficiently different from a prediction that we might reasonably reject our model as being inappropriate. Example1.3.9 Consider the electrophoresis gel scatter plot with the length of the DNA $L$ graphed with respect to the distance traveled in the gel $D\text{.}$ The data appear to follow a nice curve without a lot of uncertainty. Using a polynomial trend, a spreadsheet reports the following equation for the data: \begin{equation*} y = 0.0573x^2-43.381x+8241.6. \end{equation*} Using the appropriate variables for the problem, this equation should be rewritten as \begin{equation*} L = 0.0573D^2-43.381D+8241.6. \end{equation*} The graph of the data with the trend curve is shown below. Knowing a model equation that conveniently characterizes a data set allows us to use that equation to predict values that do not occur within the data set itself. For example, suppose we had another DNA sample of unknown length that traveled a distance of $D=282$ pixels. Using our value for $D\text{,}$ we can find the value of $L$ using the model, \begin{equation*} L = 0.0573(282)^2 - 43.381(282)+8241.6 = 564.8832. \end{equation*} Since our original data had 3 significant digits, we would estimate the length of the DNA in question as $L \approx 565$ nucleotides. Again, you should note that the number of significant digits reported is not the same as the uncertainty in the prediction. The degree to which the original data vary around the trend curve leads to uncertainty in the coefficients of the regression model and subsequent uncertainty to the trend curve itself. However, analysis of this uncertainty is a topic for statistics and is outside the scope of this text. Subsection1.3.6Summary • Quantities that can be measured correspond to variables. The system is the collection of all possible variable. The state of the system is the collection of values measured for all of the variables in a particular instance. • Variable names can be letters, symbols, or words. Uppercase and lowercase letters are different symbols and should not be interchanged. An important part of communication is describing all relevant variables and introducing their names. • A relation between two variables can often be visualized graphically using a scatterplot. An equation is the mathematical idealization of a relation. • A solution to an equation is a state (values specified for all variables) that makes the equation true. The solution set is the set of all possible solutions. • Solving an equation for the value of one variable when the value of the other variable is given allows us to predict the state of a system. • Using regression to find a trend line or regression curve can give an approximate relation corresponding to observed data. Treating the resulting equation as a model equation can give approximate predictions of states of the system. Subsection1.3.7Exercises 1 Which of the following states $(x,y)$ are solutions to $3x-2y=8\text{?}$ 1. $(x,y)=(0,-4)$ 2. $(x,y)=(1,-2)$ 3. $(x,y)=(4,2)$ 2 Which of the following states $(w,z)$ are solutions to $2w+5z-3=w^2+z^2\text{?}$ 1. $(w,z)=(-2,2)$ 2. $(w,z)=(-1,3)$ 3. $(w,z)=(3,2)$ 3 For each of the following constraints, find all states $(x,y)$ satisfying the model equation $2x+3y=12\text{.}$ 1. $x=-2$ 2. $y=5$ 4 For each of the following constraints, find all states $(u,v)$ satisfying the model equation $u^2+v^2=16+6u\text{.}$ 1. $v=4$ 2. $u=1$ 5 Is it possible to enclose an area of 25 m2 in a rectangle with perimeter of 25 m? If so, how? 6 Is it possible to enclose an area of 50 m2 in two congruent rectangles that share an edge such that the total length of edges is 40 m (counting the shared edge only once)? If so, how? You will need to choose your variables and construct appropriate model equations. A simple electric circuit has an applied voltage $V$ (volts) and a variable load resistance $R$ (kilohms). When the circuit is closed, current flows through the circuit, measured as the current $I$ (amperes). When the voltage was held constant at $V = 9$ V, the resistance and current were measured with values recorded in the table below. The following group of problems are based on these data. $V$ (V) $R$ (kΩ) $I$ (A) 9.0 0.84 0.0107 9.0 1.2 0.0073 9.0 1.8 0.0050 9.0 2.7 0.0033 9.0 3.4 0.0026 7 Create a scatter plot of $(R,I)\text{.}$ Would a trend line make sense for this data? Explain. 8 Conductance $G$ is the reciprocal of resistance, $G=1/R\text{.}$ Create a scatter plot of $(G,I)\text{.}$ Would a trend line make sense for this data? Explain. 9 One of the previous scatter plots should have had a meaningful trend line. State the appropriate regression equation as a model and use it to predict the current $I$ when the resistance is $R=2.1$ kΩ. The number of births and of deaths in a population generally depends on the size of the population. The table below gives population data for ten of the twelve highest population cities in the state of Virginia for the year 2012. The data include the population $P$ and the total number of births $B$ and deaths $D$ for the year recorded for each city. The following group of exercises are based on these data. City $P$ $B$ $D$ Virginia Beach 447021 6270 2828 Norfolk 245782 3773 1827 Chesapeake 228417 2805 1582 Richmond 210309 2939 1849 Newport News 180726 2905 1438 Alexandria 146294 2763 686 Hampton 136836 1825 1139 Roanoke 97469 1492 1172 Portsmouth 96470 1534 980 Suffolk 85181 1087 726 Lynchburg 77113 1062 779 Harrisonburg 50981 579 253 10 Create a scatter plot of $(P,B)$ and find the equation of the trend line. The cities of Hampton and Harrisonburg were left off the list with populations of $P=136836$ and $P=50981\text{,}$ respectively. Use the trend line regression model to predict the number of births in these cities during 2012. Which calculation is an example of interpolation and which is extrapolation? 11 Create a scatter plot of $(P,D)$ and find the equation of the trend line. Use the trend line regression model to predict the number of births in Hampton and Harrisonburg during 2012. (See the previous problem for population values.) Which calculation is an example of interpolation and which is extrapolation? Which of the calculations were examples of interpolation and which were examples of extrapolation?

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Article Contents Article Contents # A dynamic for production economies with multiple equilibria • * Corresponding author: Humberto Muñiz • In this article, we extend to private ownership production economies, the results presented by Bergstrom, Shimomura, and Yamato (2009) on the multiplicity of equilibria for the special kind of pure-exchanges economies called Shapley-Shubik economies. Furthermore, a dynamic system that represents the changes in the distribution of the firms on the production branches is introduced. For the first purpose, we introduce a particular, but large enough, production sector to the Shapley-Shubik economies, for which a simple technique to build private-ownership economies with a multiplicity of equilibria is developed. In this context, we analyze the repercussions on the behavior of the economy when the number of possible equilibria changes due to rational decisions on the production side. For the second purpose, we assume that the rational decisions on the production side provoke a change in the distribution of the firms over the set of branches of production. Mathematics Subject Classification: Primary:91B50, 91B55;Secondary:91B38. Citation: • Figure 1.  Example 1: $\omega=\left((5,2),(2,3)\right)$, $f(x)=5.5x-\dfrac{1}{2}x^2$, equilibria prices at $p_1=1/2$, $p_2=1$, $p_3=2$ Figure 2.  Example 2: $r = 7/9$ equilibrium price at $p_1 = 0.5$ is singular Figure 3.  Example 4: equilibria prices at $p\approx 0.154648,$ $p=1$ and $p=6.194385$ Figure 4.  Example 5: equilibria prices at $p_1\approx 0.1792472498915,$ $p_2=2$ and $p_3\approx 2.6915745984313$ Figure 5.  Example 6: equilibria prices at $p_1\approx 0.125623594624,$ $p_2=1$ and $p_3\approx 1.0747180810635$. Distribution $(10,40)$ Figure 6.  Example 6: equilibria prices at $p_1\approx 0.17025278062395,$ $p_2 \approx 0.6155470462368$ and $p_3\approx 1.5223658675626$. Distribution $(11,39)$ Figure 7.  Example 7: Blue line represents the demand function $x_2^1(p) = \phi_1(p^{-1})$, red line represent right hand side of equation 26. Distribution $(12,38)$ Figure 8.  Example 7: Blue line represents the demand function $x_2^1(p)=\phi_1(p^{-1})$, red line represent right hand side of equation 26. Distribution $(14,36)$ Figure 9.  Profit analysis example 7: Blue line represents $\pi_1(p)$, while red line represent $\pi_2(p)$ Figure 10.  Profit table for example 7 Figure 11.  Utility table for example 7 • [1] E. Accinelli and E. Covarrubias, Evolution and jump in a Walrasian framework, J. Dyn. Games, 3 (2016), 279-301.  doi: 10.3934/jdg.2016015. [2] T. C. Bergstrom, K.-I. Shimomura and T. Yamato, Simple economies with multiple equilibria, B. E. J. Theor. Econ., 9 (2009), 31pp. doi: 10.2202/1935-1704.1609. [3] E. Dierker, Two remarks on the number of equilibria of an economy, Econometrica, 40 (1972), 951-953.  doi: 10.2307/1912091. [4] T. Hens and B. Pilgrim, The index-theorem, in General Equilibrium Foundations of Finance, Theory and Decision Library, 33, Springer, Boston, MA, 2002. doi: 10.1007/978-1-4757-5317-2_4. [5] T. J. Kehoe, An index theorem for general equilibrium models with production, Econometrica, 48 (1980), 1211-1232.  doi: 10.2307/1912179. [6] T. J. Kehoe, Multiplicty of equilbria and compartive statics, Quart. J. Econom., 100 (1985), 119-147.  doi: 10.2307/1885738. [7] A. Mas-Colell,  The Theory of General Economic Equilbrium. A Differential Approach, Econometric Society Monographs, 9, Cambridge University Press, Cambridge, 1989. [8] P. A. Samuelson,  Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass., 1947. [9] L. Shapley and M. Shubik, An example of a trading economy with three competitive equilibria, J. Political Economy, 85 (1997), 873-875.  doi: 10.1086/260607. Figures(11)

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Community Profile # Tahmina Tabassum Treena Last seen: 4 mois ago Active since 2020 Undergraduate student of Electrical and Electronic Engineering at Islamic University of Technology. All #### Content Feed View by Solved Sort in descending order. 4 mois ago Solved Remove the two elements next to NaN value The aim is to *remove the two elements next to NaN values* inside a vector. For example: x = [6 10 5 8 9 NaN 23 9 7 3 21 ... plus d'un an ago Solved Clean the List of Names Given a list of names in a cell array, remove any duplications that result from different capitalizations of the same string. So... plus d'un an ago Solved Sum of diagonals Compute the sum of diagonal elements of a square matrix and store the larger sum to s. Eg : a = [1 2 3; 2 3 4; 4 5 10] ... plus d'un an ago Solved Divisible by 3 Pursuant to the <http://www.mathworks.com/matlabcentral/cody/problems/42404-divisible-by-2 first problem> in this series, this o... plus d'un an ago Solved Divisible by 10 Pursuant to the <http://www.mathworks.com/matlabcentral/cody/problems/42404-divisible-by-2 first problem> in this series, this o... plus d'un an ago Solved Divisible by 2 This is the first problem in a set of "divisible by x" problems. You will be provided a number as a string and the function you ... plus d'un an ago Solved Divisible by 5 Pursuant to the <http://www.mathworks.com/matlabcentral/cody/problems/42404-divisible-by-2 first problem> in this series, this o... plus d'un an ago Solved Check if there are white spaces in the input string If there are white spaces in the input string, output=1 else 0 plus d'un an ago Solved Return median of a matrix Compute median of a matrix of any dimension. Exclude the NaNs if any. plus d'un an ago Solved Find all repeated numbers For a given vector, find all numbers that repeated 2 times or more than 2 times. Example: A=[1 2 3 3 4 5 6 6 7] y=[3 ... plus d'un an ago Solved Prime numbers between a , b. Find all prime number between a & b (including a and b). example: a=100 b=120 y= [101 103 107 109 113] plus d'un an ago Solved Calculate the Number of Sign Changes in a Row Vector (No Element Is Zero) For a row vector: V=[7 1 2 -3] there is one sign change (from 2 to -3). So, the function you write must return N=1. F... plus d'un an ago Solved Create an n-by-n null matrix and fill with ones certain positions The positions will be indicated by a z-by-2 matrix. Each row in this z-by-2 matrix will have the row and column in which a 1 has... plus d'un an ago Solved Quote Doubler Given a string s1, find all occurrences of the single quote character and replace them with two occurrences of the single quote ... plus d'un an ago Solved Volume of a Parallelepiped Calculate the volume of a Parallelepiped given the vectors for three edges that meet at one vertex. A cube is a special case ... plus d'un an ago Solved Angle between Two Vectors The dot product relationship, a dot b = | a | | b | cos(theta), can be used to determine the acute angle between vector a and ve... plus d'un an ago Solved Return a list sorted by number of occurrences Given a vector x, return a vector y of the unique values in x sorted by the number of occurrences in x. Ties are resolved by a ... plus d'un an ago Solved サイコロを作ろう 1から6までの独立かつランダムな数値を返すような関数を作成しましょう。 例： >> [x1,x2] = rollDice(); と入力すると x1 = 5 x2 = 2 のような解を返します。 plus d'un an ago Solved De-dupe Remove all the redundant elements in a vector, but keep the first occurrence of each value in its original location. So if a =... plus d'un an ago Solved Replace NaNs with the number that appears to its left in the row. Replace NaNs with the number that appears to its left in the row. If there are more than one consecutive NaNs, they should all ... plus d'un an ago Solved Multiply pi Multiply pi with x! plus d'un an ago Solved Divide pi Divide pi by x! plus d'un an ago Solved Find 0 in array Given array find where there 0 is. plus d'un an ago Solved Vector Multiplication Vector Multiplication of three matrix as shown in test cases. plus d'un an ago Solved Sum the 'edge' values of a matrix Sum the 'edge' values of an input matrix (the values along the perimeter). Example [1 2 3 4 5 6 7 8 9] Output = ... plus d'un an ago Solved Test if a number is numeric or not Test if a number is numeric or not plus d'un an ago Solved Create a square matrix of zeros of even order Create a square matrix of zeros of even order plus d'un an ago Solved Calculate square and cube of number Calculate square and cube of number x plus d'un an ago Solved factorial of a number x Factorial of a number x plus d'un an ago

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Explore BrainMass Share # Financial Analysis with Microsoft Excel This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here! 1. Using the data presented above: a. Recreate the income statement and balance sheet using formulas wherever possible. Each statement should be on a separate excel worksheet. Try to duplicate the formatting exactly. (must be in excel worksheet) b. On another excel worksheet, create a statement of cash flows for 2011. Do not enter any numbers directly on this worksheet. All formulas should be linked directly to the source on previous worksheets. c. Using Excel's outlining feature, create an outline on the statement of cash flows that, when collapsed, shows only the subtotals for each section. d. Suppose that sales were \$3,800,000 in 2011 rather than \$3,500,000. What is the 2011 net income and retained earnings? (must be in Excel worksheet) e. Undo the changes from part d, and change the tax rate to 40%. What is the 2011 net income and retained earnings? 2. Using the data from the previous problem: a. Create a common-size income statement for 2010 and 2011. This statement should be created on a separate worksheet with all formulas linked directly to the income statement. (must be in Excel worksheet) b. Create a common-size balance sheet for 2010 and 2011. This statement should be created on a separate worksheet with all formulas linked directly to the balance sheet. (must be Excel worksheet) #### Solution Summary The solution discusses the financial analysis with microsoft Excel. \$2.19

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# Root of 21302 #### [Root of twenty-one thousand three hundred two] square root 145.952 cube root 27.7209 fourth root 12.0811 fifth root 7.3398 In mathematics extracting a root is declared as the determination of the unknown "x" in the equation $y=x^n$ The outcome of the extraction of the root is known as a root. In the case of "n = 2", one talks about a square root or sometimes a second root also, another possibility could be that n = 3 then one would call it a cube root or simply third root. Considering n beeing greater than 3, the root is declared as the fourth root, fifth root and so on. In maths, the square root of 21302 is represented as this: $$\sqrt[]{21302}=145.95204691953$$ Furthermore it is possible to write every root down as a power: $$\sqrt[n]{x}=x^\frac{1}{n}$$ The square root of 21302 is 145.95204691953. The cube root of 21302 is 27.720865995918. The fourth root of 21302 is 12.081061498044 and the fifth root is 7.339797464113. Look Up

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https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=123616

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## Formal Charge $FC=V-(L+\frac{S}{2})$ harperlacroix1a Posts: 43 Joined: Fri Sep 28, 2018 12:19 am ### Formal Charge When determining molecular structures, is having the formal charge be 0 or having a complete octet more important? Sabrina Ryu 3L Posts: 30 Joined: Fri Sep 28, 2018 12:26 am ### Re: Formal Charge Formal charge is important in determining the Lewis Structure, which then allows us to determine the VSEPR model and the molecular shape! Posts: 63 Joined: Fri Sep 28, 2018 12:19 am ### Re: Formal Charge It depends on whether or not an element can have an expanded octet. Some elements can only have eight electron so formal charge does not matter because you can't add electrons (however, you might be able to go to a double or triple bond), while others can improve their formal charge (closer to zero) by expanding their octet or increasing the number of bonds. Eunice Lee 1A Posts: 60 Joined: Wed Oct 03, 2018 12:16 am ### Re: Formal Charge Having zero as the formal charge on an atom is more important, as some elements don't completely follow the octet rule and can have more than eight valence electrons. Samantha Chang 2K Posts: 69 Joined: Fri Sep 28, 2018 12:17 am ### Re: Formal Charge Having a 0 formal charge is most important, unless it is an ion; therefore the formal charge has to be equal to the charge of the ion. Sometimes lewis dot structures can have expanded octet. sarahartzell1k Posts: 30 Joined: Fri Sep 28, 2018 12:17 am ### Re: Formal Charge It is important to pay attention to the formal charge as not all elements adhere to the octet rule. pamcoronel1H Posts: 45 Joined: Fri Sep 28, 2018 12:25 am ### Re: Formal Charge I believe formal charge is more important since, if its an ion, the formal charges of your Lewis structures have to match the ionization of the molecule. Also, when making your Lewis structure, you usually want a structure with a minimal amount of total charge. This means that the formal charge affects the structure, and the structure determines the molecular shape. Jasmine Chow 1F Posts: 60 Joined: Fri Sep 28, 2018 12:16 am Been upvoted: 1 time ### Re: Formal Charge The formal charge is the best way to ensure your lewis structure is in the most stable state and to find its molecular shape via VSPER. Some ions may throw people off so formal charges are the best way to check your work. Vicky Lu 1L Posts: 60 Joined: Fri Sep 28, 2018 12:18 am ### Re: Formal Charge It is better to follow and determine molecular structures base on the formal charger as there are exceptions to the octet rule. Formal charge is the best way to make sure the Lewis structure is in the most stable state it can be as well as writing out the right Lewis structure since the formal charge of the atoms must add up to the formal charge of the overall molecule. Megan Gianna Uy 3L Posts: 30 Joined: Fri Sep 28, 2018 12:26 am ### Re: Formal Charge If an element is in row 3+ then it can have an expanded octet. However, formal charge is still important because it makes sure your lewis structure is in its most stable form, making it easier for you to identify the shape of the molecule. Kevin Arokiaraj 4E Posts: 18 Joined: Wed Nov 15, 2017 3:03 am ### Re: Formal Charge Since the octet rule is not universal, it's better to have the formal charge be 0.

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Capacitance of two parallel plates The most common capacitor consists of two parallel plates. First, we derive the capacitance which depends on the area of the plates A and their separation d. According to Gauss's law, the electric field between two plates is: Since the capacitance is defined by one can see that capacitance is: Thus you get the most capacitance when the plates are large and close together. If a dielectric material is inserted between the plates, the microscopic dipole moments of the material will shield the charges on the plates and alter the relation. Materials have a permeability e which is sometimes given by the relative permeability k, e=ke0. The capacitance is then given by: All materials have a relative permeability, k, greater than unity, thus the capacitance can be increased by inserting a dielectric. Sometimes, k is referred to as the dielectric constant of the material. The electric field causes some fraction of the dipoles in the material to orient themselves along the E-field as opposed to the usual random orientation. This, effectively, appears as if negative charge is lined up against the positive plate, and positve charge against the negative plate. In the figure to the right, the blue material is the dielectric.

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# Consider a barge owner who is deciding whether to post an attendant on his barge to make sure that it remains properly moored to the pier. \$94, 0, \$400 (a) Calculate the marginal cost and marginal bene…t of posting the attendant. According to the marginal Hand Rule, would the barge owner be found negligent for failing to post an attendant? Now suppose that the barge owner had a third option: post the attendant only during the day. The data for this option are as follows: Cost of care \$50, Probability of an accident 0.10, and Damages \$400. (b) Assume that the barge owner’s only two options are “no attendant”and “post an attendant during the day.” In this case, would the owner’s failure to post an attendant be judged negligent by the marginal Hand Rule? (c) Assume that, prior to the accident, the owner had posted an attendant during the day. Suppose that the victim claims that the owner is negligent for not having posted the attendant for 24 hours. Use the marginal Hand Rule to evaluate the merits of this claim. (d) For each of the three options: “no attendant,” “attendant during the day,”and “attendant for 24 hours,” calculate total expected costs (costs of care plus expected damages). Which option minimizes this total? Reconcile the result with your answers to (a)-(c). ### ORDER YOUR ORIGINAL PAPER Request for a custom paper or place a new order ##### THE BEST CUSTOM ESSAY WRITING SERVICE AT YOUR FINGERTIPS Forget All Your Assignment & Essay Related Worries By Simply Filling Order Form

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# Control Flow Testing Control flow testing is a testing technique that comes under white box testing. The aim of this technique is to determine the execution order of statements or instructions of the program through a control structure. The control structure of a program is used to develop a test case for the program. In this technique, a particular part of a large program is selected by the tester to set the testing path. It is mostly used in unit testing. Test cases represented by the control graph of the program. Control Flow Graph is formed from the node, edge, decision node, junction node to specify all possible execution path. ## Notations used for Control Flow Graph 1. Node 2. Edge 3. Decision Node 4. Junction node ## Node Nodes in the control flow graph are used to create a path of procedures. Basically, it represents the sequence of procedures which procedure is next to come so, the tester can determine the sequence of occurrence of procedures. We can see below in example the first node represent the start procedure and the next procedure is to assign the value of n after assigning the value there is decision node to decide next node of procedure as per the value of n if it is 18 or more than 18 so Eligible procedure will execute otherwise if it is less than 18 Not Eligible procedure executes. The next node is the junction node, and the last node is stop node to stop the procedure. ## Edge Edge in control flow graph is used to link the direction of nodes. We can see below in example all arrows are used to link the nodes in an appropriate direction. ## Decision node Decision node in the control flow graph is used to decide next node of procedure as per the value. We can see below in example decision node decide next node of procedure as per the value of n if it is 18 or more than 18 so Eligible procedure will execute otherwise if it is less than 18, Not Eligible procedure executes. ## Junction node Junction node in control flow graph is the point where at least three links meet. ### Example Diagram - control flow graph The above example shows eligibility criteria of age for voting where if age is 18 or more than 18 so print message "You are eligible for voting" if it is less than 18 then print "You are not eligible for voting." Program for this scenario is written above, and the control flow graph is designed for the testing purpose. In the control flow graph, start, age, eligible, not eligible and stop are the nodes, n>=18 is a decision node to decide which part (if or else) will execute as per the given value. Connectivity of the eligible node and not eligible node is there on the stop node. Test cases are designed through the flow graph of the programs to determine the execution path is correct or not. All nodes, junction, edges, and decision are the essential parts to design test cases.

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Home / Energy Conversion / Convert Btu (th) to Ton-hour (refrigeration) # Convert Btu (th) to Ton-hour (refrigeration) Please provide values below to convert Btu (th) [Btu (th)] to ton-hour (refrigeration), or vice versa. From: Btu (th) To: ton-hour (refrigeration) ### Btu (th) to Ton-hour (refrigeration) Conversion Table Btu (th) [Btu (th)]Ton-hour (refrigeration) 0.01 Btu (th)8.327758173151E-7 ton-hour (refrigeration) 0.1 Btu (th)8.327758173151E-6 ton-hour (refrigeration) 1 Btu (th)8.32776E-5 ton-hour (refrigeration) 2 Btu (th)0.0001665552 ton-hour (refrigeration) 3 Btu (th)0.0002498327 ton-hour (refrigeration) 5 Btu (th)0.0004163879 ton-hour (refrigeration) 10 Btu (th)0.0008327758 ton-hour (refrigeration) 20 Btu (th)0.0016655516 ton-hour (refrigeration) 50 Btu (th)0.0041638791 ton-hour (refrigeration) 100 Btu (th)0.0083277582 ton-hour (refrigeration) 1000 Btu (th)0.0832775817 ton-hour (refrigeration) ### How to Convert Btu (th) to Ton-hour (refrigeration) 1 Btu (th) = 8.32776E-5 ton-hour (refrigeration) 1 ton-hour (refrigeration) = 12008.033605299 Btu (th) Example: convert 15 Btu (th) to ton-hour (refrigeration): 15 Btu (th) = 15 × 8.32776E-5 ton-hour (refrigeration) = 0.0012491637 ton-hour (refrigeration)

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# Mathematical Logic Logic has been used for thousands of years, from philosophy to mathematics and now to artificial intelligence. Logic is concerned with the truth and falsity of statements. The logic we will be studying will be answering the question: “when does a statement follow from a set of statements?” I have previously written about logic here and therefore this article will be relatively short when it comes to explaining everything about logic. If you want to understand logic, please read the article I have written on logic. This is just an expansion on what we have learned. ### Propositional logic Propositions can only be true or false. ### Intrepetations An intrepretation assigns to a propositional statement a truth value of True or False. True or False can be represented as 0 and 1 respectively. ### Propositional symbols There are about 500,000 ways to represent logic symbols so here are the most common ways ### Not Symbol in logic ¬ or ! or ~ What does it do Inverts what is ever inputted into it. ### Conjunction or and Symbol in logic ^ or AND or , What it does Takes >1 inputs and if both inputs are true, outputs true. ### Disjunction, “or” Symbol in Logic V, or, “OR” What it does Takes > 1 inputs, if any of the inputs are true than the output is true. ### Equivilance Symbol in Logic <=> or ≡ Symbol in Electronics None, this is a concept not a gate. What it does A and B must take the same truth value Truth Table A B A <=> B 1 1 = 1 0 1 = 0 1 0 = 0 0 0 = 1 ### Implication Symbol in Logic => or “if a then b” Symbol in Electronics None What it does If A is true then so is B https://www.dyclassroom.com/boolean-algebra/propositional-logic-truth-table Sorry for the change of pictures. Previously there was an error in this truth table. ### Truth under an interpretation Given an interpretation, I, we can compute the truth value of any formula P under I. That is, given a version of the formula we can computer the truth value. if I(P) = 1 then we say that P is true under interpretation I. if I(P) = 0 then we say that P is false under interpretation I. ### Logical Puzzles This section may help the reader in understanding logical puzzles. An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. You go to the island and meet A and B. A says that “B is a knight” B says that “The two of us are opposite types” What are A and B? So we have 2 options, p: “A is a knight”; and q: “B is a knight” We have 2 options because one of them needs to be a knight. Either both A and B are knaves, which makes B a knight as it told a truth so it lied or A is a knight and is telling the truth that B is a knight. Options for person A p is true, that is the statement “A is a knight” is true. P => Q p is false, that is the statement “A is a knight” is false. ¬P => ¬Q Options for person B q is true then q => ¬p q is false then ¬q => ¬p Now we simply need to construct a truth table for these values p q ¬p ¬q p => q ¬p => ¬q q => ¬p ¬q => ¬p 0 0 1 1 1 1 1 1 = 1 Then we stop here because we’ve found an interpretation under which they are both knaves which is satisfiable. ### Semantic Conseqeuence Note: The lecture slides for this weren’t helpful and there is no audio for this. Please message me what this is. ### Digital Logic Circuits Modern computers use logic gates to operate. You should have an understanding of logic gates from the above. ### General house keeping rules of making circuits Never combine two input wires If there are 2 separate inputs, A and B, you cannot combine them into one single wire. A single input wire can be split partway and used as input for two seperate gates If you have a single input, A, it can be split into 2 separate wires. An output wire can be used as input The output of a wire can be used as an input. No output of a gate can eventually feed back into that gate No gates can loop on themselves. ### Constructing logic circuits from tables Given a table, such as the one below, how would we construct a logic circuit for it? First, work out where it is equal to 1 (true) and then from there formalise it in mathematical logic, from the mathematical logic we can deduce the circuit for it. Sometimes it is easier to guess directly what logic gates are used. ### Circuit equivalence Two circuits are equivalent if they produce the same output given the same input ### Formula equivalence 2 formula are equivalent if they hold the same truth value under every possible interpretation. ### On Logical Equivalence The symbol “≡” is used to show an equivalence relation. Facts ≡ is reflexive ≡ is transitive ≡ is symmetric ### Simplfying propositonal formulae There are some rules we can use to simplfy propositional formulae. Communicative Law AB = BA, A + B = B + A Examples: 6 * 2 = 12 and 2 * 6 = 12 3 + 4 = 7 and 4 + 3 = 7 Assiocative Law a(bc) = ab(c) = abc Examples: (2 + 4) + 5 = 6 + 5 = 11 2 + (4 + 5) = 2 + 9 = 11 Distributive Law a(b+c) = ab + ac Example: 3 × (2 + 4) = 3 * 6 = 18 3 × 2 + 3 × 4 = 6 + 12 = 18 Demorgans’ Laws (A ∪ B)’ = (A)’ ∩ (B)’ The first law states that the complement of the union of two sets is the intersection of the complements. (A ∩ B)’ = (A)’ ∪ (B)’ The second law states that the complement of the intersection of two sets is the union of the complements. For a good blog post on understanding these laws, click (here)[https://brilliant.org/wiki/de-morgans-laws/] Miscelanous rules Not Not A = A A or A and B = A A or not A and B = A and B (A or B) (A or C) = A or B and C What to do next From this point, turn the circuit into a logical expression and simplfy it using the rules above. If this is confusing still, maybe this video will help: https://www.youtube.com/watch?v=59BbncMjL8I ### Boolean Functions Arity The arity of a boolean function is how many arguments the function takes Boolean function representation Any boolean can be represented with ^, v, or ¬. ### Logic Gates Extended In this section we will explore logic gates some more, by looking at the family of not / exclusive gates. ### XOR Symbol in Logic None What it does An XOR gate takes >1 inputs and performs exclusive disjunction. The output of an XOR gate is only true if one of its inputs is different to the other input. ### NAND Symbol in Logic None What it does A NAND gate takes >1 inputs and the output is the opposite of an AND gate. The output is true when one or more, but not all, of its inputs are false. ### Universality of XOR and NAND All boolean functions can be created using either XOR or NAND gates. ### Binary Numbering System Binary is a numbering system that consists of 0 and 1s. Alternatively, you could memorise the powers of 2. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… And then to convert a number into binary, let’s say 6, you build that up from the different powers. So 6 is 011 and then reverse that, 110 Somethings you need to know in binary 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 10 Once you know these basic rules, you can add any numbers together in binary the same way you can add normal numbers. Try this exercise 011 The half adder is a type of binary adder in electronics that adds together two single binary digits and provides the output plus a carry value. Note: Boris’ half-adder is overcomplicated, you can achieve the same by replacing 3 of his logic gates with a single XOR gate. Truth Table The full adder allows you to carry-in as well as carry-out. Watch this video for a better understanding Black Box Notation We can represent the full adder as a black box, we don’t need to know what happens inside of it, only the inputs and outputs. Using blackbox notation, we can create 4-bit adder http://www.electronics-tutorials.ws/combination/comb_7.html Computer Representation of negative integers A fixed number of bits is used to represent integers: 8, 16, 32 or 64 bits. An unsigned integer can take up all the space available. You can “sign” a binary number to indicate whether it is negative or not. For example, the number 10 can be represented in 8-bit as 00001010 and -10 can be represented in 8-bit as 10001010 But this sometimes causes a problem, for example, 10000000 represents -0. Whaaatt?? Negative 0? Yes! That’s right, and that’s exactly the problem this causes. This is where 2’s complement comes into play. Twos complement Converting a decimal to twos complement 1. Convert the number to binary, ignoring the sign for now. So 5 is 0101 and -5 is 0101. 2. If the number is a positive number then you are done, no need to go any further. Otherwise… 3. If the number is negative then: • Find the complement (EG convert all 0’s to 1’s and all 1’s to 0’s) • Add 1 to the complement • 1101 1100 with a carry of 1 that goes all the way to the left, it can be ignored.

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1 / 6 # Convergence Analysis of Reinforcement Learning Agents Convergence Analysis of Reinforcement Learning Agents. Srinivas Turaga 9.912 30th March, 2004. The Learning Algorithm. The Assumptions. Players use stochastic strategies. Players only observe their reward . Players attempt to estimate the value of choosing a particular action. Télécharger la présentation ## Convergence Analysis of Reinforcement Learning Agents E N D ### Presentation Transcript 1. Convergence Analysis of Reinforcement Learning Agents Srinivas Turaga 9.912 30th March, 2004 2. The Learning Algorithm The Assumptions • Players use stochastic strategies. • Players only observe their reward. • Players attempt to estimate the value of choosing a particular action. The Algorithm • Play action i with probability Pr(i) • Observe reward r • Update value function v 3. The Learning Algorithm Payoff matrix Player 2’s choice Player 1’s choice The Algorithm Value of action i • Play action iwith probability Pr(i) • Proportional to value of action i • Observe reward r • Depends on other player’s choice jalso • Update value function v • 2 simple schemes Algorithm 2 Algorithm 1 If action i chosen: If action i not chosen: forgetting no forgetting 4. Analysis Techniques • Analysis of stochastic dynamics is hard! • So approximate: • Consider average case (deterministic) • Consider continuous time (differential equation) Random! Discrete time! Deterministic! Discrete time! Deterministic! Continuous time! 5. Results - Matching Pennies Game • Analysis shows a fixed point corresponding to the Nash equilibrium. Linear stability analysis shows marginal stability. • Simulations of stochastic algorithm and deterministic dynamics diverge to corners. • Analysis shows a stable fixed point corresponding to matching behavior. • Simulations of stochastic algorithm and deterministic dynamics converge as expected. 6. Future Directions • Validate approximation technique. • Analyze properties of more general reinforcement learners. • Consider situations with asymmetric learning rates. • Study behavior of algorithms for arbitrary payoff matrices. More Related

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# math posted by . i am 11 less than the product of 7 and 5. i am 5 more than the product of 11 and 2 • math - I'm not so sure if the problem requires the domain (all possible values of the number). Or if the problem requires only one value of the unknown. Let x be the number. From the first statement, x < 7 * 5 - 11 x < 35 - 11 x < 24 From the second statement, x > 11 * 2 + 5 x > 22 + 5 x > 27 Thus, if the the domain of x is the one required: (-infinity, 24) U (27, +infinity) However, if the problem wants a single value of x, the value does not exist because there is no value that is less than 24 which is, at the same, time must be greater than 27, or (-infinity, 24) ∩ (27, +infinity) = d.n.e. Hope this helps~ :) • correction-math - IGNORE THE ABOVE POST LOL. I shouldn't have used greater than and less than signs! @_@ From the first statement, x = 7 * 5 - 11 x = 35 - 11 x = 24 From the second statement, x = 11 * 2 + 5 x = 22 + 5 x = 27 Sorry about that. I don't know what I'm thinking lol. ^^; Hope this helps~ :) ## Similar Questions 1. ### Math If we multiply "a" and "b",when will: a. the product be larger than both "a" and "b"? 2. ### bryce can someone help me write out these expressions? 3. ### Macroeconomic Suppose that a market for a product is in equilibrium at a price of \$5 per unit. At any price above \$5 per unit. A. There will be an excess demand for the product. B. There will be an excess supply of the product. C. The quantity supplied … 4. ### government The United States government puts high tariffs on a certain product that is imported from other countries. What effect does this have in the United States? 5. ### math write each expression using n as your variable.then simplify fully. 1.five more than the sum of a number and ten. 2.the product of eight and seven less than a number. 3.the quotient of a number and three, increased by one. 4.ten less … 6. ### Algebriac Expressions 2 less than 7 times a number = 2) Sarah's age decreased by 2 = 3) One third as many books = 4) Earns \$13 per hour = 5) 4 years younger than Marcus 6) 2 times a number, decreased by 4 7) 9 less than the product of 15 and a number 8) … 7. ### Entrepreneurship 1. If a product becomes more easily available, what is likely to happen? 8. ### chemistry Given the reaction: Cu2+(aq) + S2-(aq) CuS(s) What will happen (a) if CuSO4(aq) is added? 9. ### chemistry -3.2.4 Given the reaction: Cu2+(aq) + S2-(aq) CuS(s) What will happen (a) if CuSO4(aq) is added? 10. ### Chemistry Given the reaction: Cu2+(aq) + S2-(aq) CuS(s) What will happen (a) if CuSO4(aq) is added? More Similar Questions

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# Forget Me Knot: A Mathematician’s Lasting Tryst with Knot Theory Forty years since the discovery of the Jones Polynomial, Shantha Bhushan reflects on her enduring relationship with knots and braids. Buried in an obscure nook of Youtube is an old recording of mathematician Joan Birman. In it, the American topologist looks back on a particularly eventful week in her life, which began with a visit from a young Kiwi researcher Vaughan Jones, and ended with the discovery of the Jones polynomial”. This was 40 years ago in 1984. We both understood that it was bound to be a major discovery in knot theory,” wrote the now 96-year-old Birman, in a tribute to Jones, who, unfortunately, passed away in 2020. According to another obituary of Vaughan Jones that appeared on Nature, the polynomial was the kind of tool that topologists had been seeking for decades.” A tool that would enable them to distinguish different types of knots. #### Bitten by the knot bug Knot theory is a sub-field of topology, which is the mathematical study of objects that can be twisted, stretched, or deformed in some way. When Shantha Bhushan started her PhD in topology at IIT Bombay, the maths world was still abuzz with the discovery of the Jones’ polynomial. Knot theory was an exciting field to be in, in the late eighties. Though Shantha’s dissertation focused on a different area of topology, she too was bitten by the knot bug. When we think of knots, we think of shoelaces, gift-wrapping, sailors… or we think of crochet yarn, or maybe surgeons tying up a knot after stitching up something,” says Shantha, who now teaches mathematics at Azim Premji University. These are all examples of open” knots, meaning that the two ends of the rope/​string/​ribbon/​lace are separate. If not for physical factors such as friction and tension, open knots would easily come undone. Topologists like Shantha are more interested in closed” knots, which have no free ends. She illustrates what this would look like: Imagine knotting up a metal wire and welding their ends together. Now there is less wiggle room, and things get interesting!” Shantha then coaxes us to do a similar exercise with another piece of wire, and then try to figure out if the knots are identical. You can stretch the wires, go over, under, basically perform any operation, as long as you don’t cut them. If the two knots are equal you will be able to transform one into the other,” she states. This is a fundamental aspect of knot theory. The first documented mathematician to study knots was — incidentally, Shantha’s favourite — Carl Friedrich Gauss in 1794. But the quest to mathematically distinguish them seems to have started nearly a century later. This was when mathematicians Lord Kelvin and Guthrie Tait together came up with the idea that atoms and the universe were made out of a material called ether, and the differing properties of atoms (for example hydrogen and helium) were due to the distinct ways ether knotted up. Though the existence of ether was soon disproved by the famous Michelson – Morley experiment in 1887, this did not end Tait’s fascination with knots. He went on to create a categorisation system for knots. And thus, out of an attempt to understand what this universe is made of, knot theory was born. Tait’s classification of knots was based on the minimum number of crossings a knot involved. The simpler a knot, the fewer times its lines crossed each other in a 2‑D diagram (as seen above). He came up with a table that depicted how many different knots could be formed with n’ number of crossings. For example, the simplest knot, called an unknot’, is basically a circular loop with 0 crossings; there are no knots with 1 or 2 crossings; there is one knot each possible with 3 and 4 crossings respectively; 2 with 5 crossings; 3 with 6 crossings; and 7 with 7 crossings. That’s as far as Tait got, which itself was pretty impressive considering there were no theoretical techniques at his disposal, only his keen intuition. Later, with the help of Thomas Kirkman and Charles Little, Tait’s table went up to 10 crossings — there were 165 different knots with 10 crossings. In 2020, Australian mathematician Benjamin Burton classified all (prime) knots up to 19 crossings. How many types of knots exist with 19 crossings? Close to 300 million! Once you get a sense of the number of possible knots, it becomes easier to appreciate the value of Joan Birman and Vaughan Jones’s contribution. Two knots may have the same crossing number, but are they identical? That was extremely hard to say, but the development of the Jones polynomial could do exactly this. The discovery was huge. I was very excited,” recalls Shantha. #### Coral reefs, crochet and curvature Though Shantha was not able to formally pursue the subject during her PhD, knots and braids continued to occupy the mathematician’s mind. Her crocheting and knitting hobby deepened her interest. Though both crocheting and knitting involve open knots, they do offer a lot of opportunities to grasp complex concepts in maths, particularly in topology. The connection [between mathematics and crochet] came to me when I read about a professor from Cornell University who used crochet to show how to create surfaces with negative curvature — meaning they bend inwards like coral reefs,” she says. As a PhD student in topology, Shantha had been introduced to the concept of negative curvature, but only in the form of mathematical models and functions. When I first used crochet to create such a surface, I finally felt like I understood what was going on! It felt very empowering.” she reminisces. When I first used crochet to create a negative curvature surface, I finally felt like I understood what was going on! It felt very empowering. Now, as a teacher, my job is to help students not fear maths. I see anything tactile and visual, as a way of freeing students from this anxiety and fear.” #### Shantha Bhushan This experience created an indelible mark on Shantha, both personally and professionally. A PhD is a very difficult time for everybody, and it is worse when you can’t visualise something… you start to believe you are not good enough.” A similar feeling haunts many undergraduate students of mathematics as well, she noted as her career transitioned from research to teaching. Now, as a teacher, my job is to help students not fear maths. I see anything tactile and visual, as a way of freeing students from this anxiety and fear.” One powerful tool Shantha and her colleagues in the University use with students is origami. There is a lot of deep maths that origami can introduce us to. It’s not that doing origami can make you understand everything, but at least it removes some barriers,” she insists. And to a significant degree, Shantha was vindicated when their teaching experiments began to show results. We can see a shift in the way students are thinking about maths. It’s no longer just a bunch of formulae and symbols that only very smart people can deal with. It’s something that they can play with, enjoy,” says Shantha. It was particularly moving for the maths group to witness this shift taking place among students from disadvantaged backgrounds. Many of them take maths because they are told that they can get a job in software. To see their relationship with maths improving is very rewarding.” One powerful tool Shantha and her colleagues in the University use with students is origami. There is a lot of deep maths that origami can introduce us to. It’s not that doing origami can make you understand everything, but at least it removes some barriers,” she insists. #### Craft as a gateway to higher mathematics The potential of arts and crafts as a gateway to higher mathematics is by now well known, even if understudied. Applied mathematician Elisabetta Matsumoto has been quoted as saying knit theory is knot theory”. The American Mathematical Society includes special sessions on maths education and fibre arts quite regularly. Looking back at the 40 years since the discovery of the Jones polynomial, it’s clear that the hype was not for nothing. Shantha compares it to the 4‑minute mile, first recorded by a British athlete in 1954. After Roger Bannister proved it was achievable, many others were able to cross the four-minute barrier. Similarly, the Jones polynomial opened the floodgates to several other types of knot-distinguishing polynomials such as HOMFLYPT and the Khovanov invariant.” Today, knot theory is no longer merely a cool theoretical field. It is being applied in cryptography, molecular biology and synthetic chemistry. Shantha added: In the last few decades, quantum computing and knot invariants has led to quantum topology. It is amazing to see the interplay and how growth in one field influences the growth in the other!”

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Wednesday, 2 November 2011 How to Solve Polynomials Polynomial is a term which implies with every kind of mathematical expression. Polynomial function consists of terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function, When numbers are implemented with addition or subtraction than they are said to be terms, when product of the numbers are used than that form is called factors. Variables are just representing a symbol which uses different values under it whereas constant is a single value symbol. A polynomial equation includes sum of the power of same derivatives with different integer coefficients and these all derivatives are finite in numbers. The standard form of any polynomial equation is as: bn yn + bn-1 yn-1 + ….......................+. b2 y2 + ….... + b0 y0 Here y is the variable with n types of derivatives, 'b' is an integer co-efficient and 'n' represents the finite number of derivatives in polynomial equation. Sometimes any polynomial equation may consist of number of different derivatives or variables. So according to this property polynomial is categorized in various types which are binomial, monomial, trinomial and so on. A polynomial is said to be a monomial when it only have one single variable derivatives and if equations consist of derivatives of two variables than that is a form of binomial equation. Similarly a trinomial will include derivatives of 3 different variables. Let us take some examples of polynomial equations: x – x2 = 2 (a Monomial) x + y = 1 (Binomial) x + y + z = 3 (Trinomial) In mathematics most of the equations are in form of polynomial equation like every algebraic equation is a type of polynomial equation. So it is clear that polynomial functions are important in mathematics equation formation so for enhancing your knowledge in this topic and various other math topics you can take online math help on a math tutoring website “ TutorVista”.

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# yard (International) to pica (PostScript) conversion Conversion number between yard (International) [yd] and pica (PostScript) is 216. This means, that yard (International) is bigger unit than pica (PostScript). ### Contents [show][hide] Switch to reverse conversion: from pica (PostScript) to yard (International) conversion ### Enter the number in yard (International): Decimal Fraction Exponential Expression [yd] eg.: 10.12345 or 1.123e5 Result in pica (PostScript) ? precision 0 1 2 3 4 5 6 7 8 9 [info] Decimal: Exponential: ### Calculation process of conversion value • 1 yard (International) = (exactly) (0.9144) / ((254/60000)) = 216 pica (PostScript) • 1 pica (PostScript) = (exactly) ((254/60000)) / (0.9144) = 0.0046296296296296 yard (International) • ? yard (International) × (0.9144  ("m"/"yard (International)")) / ((254/60000)  ("m"/"pica (PostScript)")) = ? pica (PostScript) ### High precision conversion If conversion between yard (International) to metre and metre to pica (PostScript) is exactly definied, high precision conversion from yard (International) to pica (PostScript) is enabled. Decimal places: (0-800) yard (International) Result in pica (PostScript): ? ### yard (International) to pica (PostScript) conversion chart Start value: [yard (International)] Step size [yard (International)] How many lines? (max 100) visual: yard (International)pica (PostScript) 00 102160 204320 306480 408640 5010800 6012960 7015120 8017280 9019440 10021600 11023760 Copy to Excel ## Multiple conversion Enter numbers in yard (International) and click convert button. One number per line. Converted numbers in pica (PostScript): Click to select all ## Details about yard (International) and pica (PostScript) units: Convert Yard (International) to other unit: ### yard (International) Definition of yard (International) unit: ≡ 0.9144 m ≡ 3 ft ≡ 36 in . Convert Pica (PostScript) to other unit: ### pica (PostScript) Definition of pica (PostScript) unit: ≡ 12 points . ← Back to Length units

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CFD Online Discussion Forums (http://www.cfd-online.com/Forums/) -   CFX (http://www.cfd-online.com/Forums/cfx/) -   -   Fluid residence time (http://www.cfd-online.com/Forums/cfx/70591-fluid-residence-time.html) uorinopuot November 29, 2009 16:11 Fluid residence time Hello, I am modelling blood clotting in a vessel. I want to couple the residence time of the blood in my geometry with my viscosity, whose i know the formula. How can i calculate the residence time of the blood in my geometry? I tried using a new volumetric variable and assign it to a subdomain, but i do not know wether the values i get are right. ghorrocks November 29, 2009 17:58 Yes, I think your approach should work. Use an additional variable with convection activated (but not diffusivity) and put a source term on it to increase its value by the timestep size each time step. Quote: but i do not know wether the values i get are right Well, you are going to have to work that one out. How do we know if your values are right? kingjewel1 November 30, 2009 05:53 Quote: Originally Posted by uorinopuot (Post 238096) Hello, I am modelling blood clotting in a vessel. I want to couple the residence time of the blood in my geometry with my viscosity, whose i know the formula. How can i calculate the residence time of the blood in my geometry? I tried using a new volumetric variable and assign it to a subdomain, but i do not know wether the values i get are right. Could you use particle pathlines to do this? ghorrocks November 30, 2009 17:29 If you want to link the fluid residence time to viscosity the additional variable approach is much easier. It is tricky to do this with particles..... but not impossible. All times are GMT -4. The time now is 20:43.

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Free Republic Browse · Search General/Chat Topics · Post Article Could You Have Passed the 8th Grade in 1895? http://mwhodges.home.att.net/1895-test.htm ^ | by Michael Hodges Posted on 12/18/2008 9:30:57 AM PST by Halfmanhalfamazing Grammar (Time, one hour) 1. Give nine rules for the use of Capital Letters. 2. Name the Parts of Speech and define those that have no modifications. 3. Define Verse, Stanza and Paragraph. 4. What are the Principal Parts of a verb? Give Principal Parts of do, lie, lay and run. 5. Define Case, Illustrate each Case. 6. What is Punctuation? Give rules for principal marks of Punctuation. 7-10. Write a composition of about 150 words and show therein that you understand the practical use of the rules of grammar. Arithmetic (Time, 1.25 hours) 1. Name and define the Fundamental Rules of Arithmetic. 2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold? 3. If a load of wheat weighs 3942 lbs., what is it worth at 50cts. per bu, deducting 1050 lbs. for tare? 4. District No. 33 has a valuation of \$35,000. What is the necessary levy to carry on a school seven months at \$50 per month, and have \$104 for incidentals? 5. Find cost of 6720 lbs. coal at \$6.00 per ton. 6. Find the interest of \$512.60 for 8 months and 18 days at 7 percent. 7. What is the cost of 40 boards 12 inches wide and 16 ft. long at \$.20 per inch? 8. Find bank discount on \$300 for 90 days (no grace) at 10 percent. 9. What is the cost of a square farm at \$15 per acre, the distance around which is 640 rods? 10.Write a Bank Check, a Promissory Note, and a Receipt. U.S. History (Time, 45 minutes) 1. Give the epochs into which U.S. History is divided. 2. Give an account of the discovery of America by Columbus. 3. Relate the causes and results of the Revolutionary War. 4. Show the territorial growth of the United States. 5. Tell what you can of the history of Kansas. 6. Describe three of the most prominent battles of the Rebellion. 7. Who were the following: Morse, Whitney, Fulton, Bell, Lincoln, Penn, and Howe? 8. Name events connected with the following dates: 1607, 1620, 1800, 1849, and 1865? Orthography (Time, one hour) 1. What is meant by the following: Alphabet, phonetic orthography, etymology, syllabication? 2. What are elementary sounds? How classified? 3. What are the following, and give examples of each: Trigraph, subvocals, diphthong, cognate letters, linguals? 4. Give four substitutes for caret 'u'. 5. Give two rules for spelling words with final 'e'. Name two exceptions under each rule. 6. Give two uses of silent letters in spelling. Illustrate each. 7. Define the following prefixes and use in connection with a word: Bi, dis, mis, pre, semi, post, non, inter, mono, super. 8. Mark diacritically and divide into syllables the following, and name the sign that indicates the sound: Card, ball, mercy, sir, odd, cell, rise, blood, fare, last. 9. Use the following correctly in sentences, Cite, site, sight, fane, fain, feign, vane, vain, vein, raze, raise, rays. 10.Write 10 words frequently mispronounced and indicate pronunciation by use of diacritical marks and by syllabication. Geography (Time, one hour) 1. What is climate? Upon what does climate depend? 2. How do you account for the extremes of climate in Kansas? 3. Of what use are rivers? Of what use is the ocean? 4. Describe the mountains of N.A. 5. Name and describe the following: Monrovia, Odessa, Denver, Manitoba, Hecla, Yukon, St. Helena, Juan Fermandez, Aspinwall and Orinoco. 6. Name and locate the principal trade centers of the U.S. 7. Name all the republics of Europe and give capital of each. 8. Why is the Atlantic Coast colder than the Pacific in the same latitude? 9. Describe the process by which the water of the ocean returns to the sources of rivers. 10.Describe the movements of the earth. Give inclination of the earth. TOPICS: Education; History; Miscellaneous; Reference KEYWORDS: education; liberalism; teachersunions first previous 1-2021-4041-6061-75 next last To: redpoll Not a very comprehensive Snopes article. It doesn’t explain how the fake originally arose, which makes me wonder if it is true after all. In any case, the dumbing down of education has been quantified. Test scores have been in decline, which is usually attributed to a greater “democratization” of the tests. But the absolute number of people scoring over a 600 on the verbal SAT declined by 64% in the 1970’s and 1980’s. See for example, “The Schools We Need, and Why We Don’t Have Them” by E.D. Hirsch. 21 posted on 12/18/2008 9:42:30 AM PST by Jibaholic ("Those people who are not ruled by God will be ruled by tyrants." --William Penn) To: Halfmanhalfamazing Ridiculous! How will this test help my self-esteem?! And no mention of global warming or diversity? At least my ‘F’ should be in purple marker... 22 posted on 12/18/2008 9:44:26 AM PST by astyanax (If you need to wear a mask while speaking your mind, it is probably best you remain silent...) To: SFR I wouldn’t do too well. 23 posted on 12/18/2008 9:44:46 AM PST by unkus To: Halfmanhalfamazing In a related note, I have a sneak peek for everyone of the 2009 8th grade aptitude test. 1. Describe how you feel now. (4 hours) 24 posted on 12/18/2008 9:44:53 AM PST by mnehring I have an Elson Grammar School Reader grade 6 (1910) on my desk. Know what the first poetry unit is titled? Patriotism, Stories, Poems and Nature and Duty. 25 posted on 12/18/2008 9:45:28 AM PST by goodwithagun (My gun has killed less people than Ted Kennedy's car.) To: redpoll Sorry. This is an urban legend. Please see http://www.snopes.com/language/document/1895exam.asp for the background of this hoary chestnut. Did you read the article you link to? It is NOT an urban legend. It's an actual test from 1895, whether it was a High School exam or an 8th grade test is not known. Snope's doesn't make any effort to debunk the test...only it's significance as a tool to measure modern academic progress. And FWIW, Snopes is not the be all and end all of internet research.! Who the hell died and made them the Gaurdians of Truth? 26 posted on 12/18/2008 9:46:11 AM PST by pgkdan To: redpoll Ahh, but Snopes doesn’t say the test isn’t real, they discount the need for the reader to pass it. 27 posted on 12/18/2008 9:47:04 AM PST by mnehring To: redpoll Ouch. But this isn’t without irony. They give an example of a teaching exam which is terribly difficult from the 1800s. So I guess the thread is salvaged. :-) 28 posted on 12/18/2008 9:47:22 AM PST by Halfmanhalfamazing (There is no "rich". There is only "the hiring class".) To: Halfmanhalfamazing wtf...!!! lol no wonder kids today are stupid look at todays teachings..... 29 posted on 12/18/2008 9:51:18 AM PST by tatsinfla To: redpoll; Halfmanhalfamazing; CE2949BB I read the explanation on Snopes and DID NOT see where they said that the exam itself was a false urban legend. What they did say is that kids could probably learn the same thing today IF it was taught and then they mentioned that certain things that we would consider important today are not addressed. 30 posted on 12/18/2008 9:53:17 AM PST by wagglebee ("A political party cannot be all things to all people." -- Ronald Reagan, 3/1/75) To: Jibaholic Exactly. I looked through that Snopes article and didn’t see anything that explained WHY it was false. Sounds like the leftist who wrote it just doesn’t like the fact that it might be the case. I still have proof education has fallen; I read Rime of the Ancient Mariner in my Senior AP English class in high school. My brother found a 1890’s literature textbook with the same poem; it was for the 5th grade! 31 posted on 12/18/2008 9:53:30 AM PST by lado To: Jibaholic Not a very comprehensive Snopes article. It doesn’t explain how the fake originally arose, Actually, Snopes doesn't declare that the test is false. The ONLY thing Snopes attempts to proclaim as false is the idea that the test demonstrates that children were better educated a century ago. 32 posted on 12/18/2008 9:56:03 AM PST by wagglebee ("A political party cannot be all things to all people." -- Ronald Reagan, 3/1/75) To: redpoll Sorry. This is an urban legend. Please see http://www.snopes.com/language/document/1895exam.asp for the background of this hoary chestnut. The Snopes article does not debunk it. It in fact assumes it is true, and reinforces it is true by referencing a similar test for teachers. The Snopes article apparently attempts to argue the argument that the test invites. 33 posted on 12/18/2008 9:58:10 AM PST by Rightwing Conspiratr1 To: wagglebee Exactly, that was the point I was trying to make, albiet poorly. 34 posted on 12/18/2008 9:58:17 AM PST by Jibaholic ("Those people who are not ruled by God will be ruled by tyrants." --William Penn) Lets see someone from 1895 build a website! That's like comparing apples to onions. The knowledge taught in 1895 had purpose...and that was to "function" as a responsible citizen who could make his/her own way in the real world. The purpose of today's education system is to indoctrinate and rely on nanny gov't. 35 posted on 12/18/2008 10:00:19 AM PST by LaineyDee (Don't mess with Texas wimmen!) To: Bloody Sam Roberts The only description I can give of the mountains I've seen this week (between rain storms) is...Holy Crap! Reminds me of something years ago when I was in the LA Basin. Our office was in West Covina, it was late September and a new guy moved in from NY. One day he said: "I don't see the issues here with the smog". I pointed north and said: "You see those mountains over there (He could not because of the smog)." He looked at me like I was crazy, I then commented, "Well let me know when you do." A week later it rained and LA basin was crystal clear. His comment to me was the same as yours: "HOLY CRAP" You are in LA at the best time of year, brisk, clear and there may be snow on the mountains (I am in SF, not LA). Under those conditions LA can be quite a sight. 36 posted on 12/18/2008 10:04:01 AM PST by Michael.SF. ("They're not Americans. They're liberals! "-- Ann Coulter, May 15, 2008) To: lilylangtree 37 posted on 12/18/2008 10:05:00 AM PST by A knight without armor To: Jibaholic Snopes does make a few good points (literature not being mentioned, etc.) and they are right that kids today COULD learn these things if they were being taught, but they fail to account for the fact that finding the necessary number of teachers who were capable of teaching at this level would be impossible. 38 posted on 12/18/2008 10:08:31 AM PST by wagglebee ("A political party cannot be all things to all people." -- Ronald Reagan, 3/1/75) To: Halfmanhalfamazing Pick up a text book for those times and you’d easliy pass the tests. Things DO change. 39 posted on 12/18/2008 10:08:43 AM PST by Sacajaweau (I'm planting corn...Have to feed my car...) To: Halfmanhalfamazing Yes, but people being people, just like they are today, very few people back then could probably get an A on these tests, either. The difference is that the people who did poorer on those tests back then, still studied, were learning, and did learn something. Today you’ve got people not even studying at all for anything. 40 posted on 12/18/2008 10:10:43 AM PST by Secret Agent Man (I'd like to tell you, but then I'd have to kill you.)

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# William Murray Last seen: 7 dagen ago Active since 2017 Followers: 0   Following: 0 All #### Feeds View by Question How do I share between my newly installed individual license, and the concurrent license available with more toolboxes (2020b)? Hello, I found an answer below that suggests that I can share between and Individual license and a concurrent license. Does th... meer dan 3 jaar ago | 1 answer | 0 ### 1 Solved Return the Fibonacci Sequence Write a code which returns the Fibonacci Sequence such that the largest value in the sequence is less than the input integer N. ... meer dan 6 jaar ago Solved Pizza! Given a circular pizza with radius _z_ and thickness _a_, return the pizza's volume. [ _z_ is first input argument.] Non-scor... meer dan 6 jaar ago Solved Determine if input is odd Given the input n, return true if n is odd or false if n is even. meer dan 6 jaar ago Solved Given a and b, return the sum a+b in c. meer dan 6 jaar ago Solved Find the sum of all the numbers of the input vector Find the sum of all the numbers of the input vector x. Examples: Input x = [1 2 3 5] Output y is 11 Input x ... meer dan 6 jaar ago Solved Make the vector [1 2 3 4 5 6 7 8 9 10] In MATLAB, you create a vector by enclosing the elements in square brackets like so: x = [1 2 3 4] Commas are optional, s... meer dan 6 jaar ago Solved Times 2 - START HERE Try out this test problem first. Given the variable x as your input, multiply it by two and put the result in y. Examples:... meer dan 6 jaar ago Solved Compute the step response of a DC motor Compute the step response of a DC motor shown below <<http://blogs.mathworks.com/images/seth/cody/dc-motor.png>> The param... meer dan 6 jaar ago Solved Make a low pass filter Make a first order low pass filter that will filter out the high frequency oscillations for the given input signal. The cut-off ... meer dan 6 jaar ago Solved Produce a Fibonacci sequence Construct a diagram that generates the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34.....up to 377 The Fibonacci sequ... meer dan 6 jaar ago Solved Model a simple pendulum Model a simple pendulum of length 200cm with bob of mass 100g and plot the position in degrees. The pendulum starts at 30 degree... meer dan 6 jaar ago Solved Counting pulses in a signal Count the number of pulses that are the result of summing each pulse generator block. Pulse Generator blocks produce a recurr... meer dan 6 jaar ago Solved Model a falling body An object is falling freely from a height of 22 meters under the force of gravity. <<http://blogs.mathworks.com/images/seth/c... meer dan 6 jaar ago Solved Add damping to a mass spring system Model an ideal mass-spring-damper system shown below where the spring is initially stretched. <<http://blogs.mathworks.com/im... meer dan 6 jaar ago Solved Model a mass spring system Model an ideal mass-spring system shown below where the spring is initially stretched. <<http://blogs.mathworks.com/images/se... meer dan 6 jaar ago Solved Make a half wave rectifier Produce a signal that outputs the given sine wave source when it is greater than zero and outputs zero when it is less than zero... meer dan 6 jaar ago Solved Make a full wave rectifier Produce a full wave rectifier waveform for the given sine wave source. For a sine wave input, the output of the full wave rec... meer dan 6 jaar ago Solved Produce the following signal: <<http://blogs.mathworks.com/images/seth/cody/add-offset-eqn.png>> You should see a downward... meer dan 6 jaar ago Solved Produce a sine wave Produce a sine wave with amplitude 3: <<http://blogs.mathworks.com/images/seth/cody/sine-eqn.png>> meer dan 6 jaar ago Solved Add a block to a model Produce the following signal: <<http://blogs.mathworks.com/images/seth/cody/add-block-eqn.png>> In this case, the slope of... meer dan 6 jaar ago Solved Produce a cosine wave Produce the following signal: <<http://blogs.mathworks.com/images/seth/cody/cosine-eqn.png>> meer dan 6 jaar ago Solved Connect blocks in a model Connect the blocks in the model to produce the following signal: <<http://blogs.mathworks.com/images/seth/cody/connect-blocks... meer dan 6 jaar ago

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Notes Study Reminders Support Text Version ### Applications of Differentiation We will email you at these times to remind you to study. • Monday Tuesday Wednesday Thursday Friday Saturday Sunday XSIQ * Intermediate Mathematics - Applications of differentiation Applications of differentiation A is a point on a curve at which A stationary point at_ x = a_ can be either * _a local maximum turning point:_ the value of the function, f(a), is greater than the value of the function at any point nearby * _a local minimum turning point:_ the value of the function, f(a), is less than the value of the function at any point nearby * _a stationary point of inflection: _the value of the function, f(a), is greater than the value of the function for values of _x_ on one side of _a,_ and less than the value of the function for values of_ x_ on the other side of _a._ Previous | Next

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# Finding an Integral Surface Consider finding the integral surface of $$x^2 p + xy q = xyz-2y^2$$ which passes through the line $$x=y e^y$$ in the $$z=0$$ plane. Attempt In Lagrange's subsidiary form $$\frac{dx}{x^2}=\frac{dy}{xy}=\frac{dz}{(xyz-2y^2)}$$ Firstly consider $$\frac{dx}{x^2}=\frac{dy}{xy}$$ One can trivially show that $$a = \frac{x}{y}$$ where $$a$$ is an arbitrary constant. Now, consider $$\frac{dx}{x^2}=\frac{dz}{(xyz-2y^2)}$$ which may be written as $$\frac{dz}{dx}=\frac{(xyz-2y^2)}{x^2} \equiv \frac{z}{a}-\frac{2}{a^2}$$ having used $$a=x/y$$ from before. (After this point I am unsure of my working...) $$\frac{dz}{dx}=\frac{az-2}{a^2} \implies\frac{dz}{(az-2)}=\frac{dx}{a^2}$$ 1. As $$a$$ is a function of $$a(x,y)$$, albeit an arbitrary constant, is my solution above sensical or have a made a mistake? 2. I understand that to find the integral surface the general solution is of the form $$F(a,b)$$ where has so far been determined to be $$a=x/y$$. How can I find this over arbitrary constant $$b$$? Your calculus is correct. Don't worry about $$a(x,y)$$ which should be true outside the characteristic curves, but is constant on the characteristic curves. Thus it is legitim to integrate $$\frac{dz}{dx}=\frac{z}{a}-\frac{2}{a^2}$$ with $$a=$$ constant. $$z=\frac{2}{a}+be^{x/a}$$ The second family of characteristic curves is : $$e^{-x/a}(z-\frac{2}{a})=b$$ General solution of the PDE on the form of implicit equation $$F(a,b)=0$$ with any function $$F$$ of two variables, or equivalently $$b=\Phi(a)$$ with any function $$\phi$$ of one variable : $$b=e^{-x/a}(z-\frac{2}{a})=\Phi(\frac{x}{y})=e^{-y}(z-\frac{2y}{x})$$ $$z(x,y)=\frac{2y}{x}+e^y \Phi(\frac{x}{y})$$ $$\Phi$$ is an arbitrary function to be determined according to the boundary condition. CONDITION : $$x=ye^y$$ on the plane $$z=0$$ . $$\quad z(ye^y,y)=0=\frac{2y}{ye^y}+e^y \Phi(\frac{ye^y}{y})$$ $$\Phi(e^y)=-2 e^{-2y}=-2(e^{y})^{-2}$$ So, the function $$\Phi$$ is determined : $$\Phi(X)=-2X^{-2}$$ . We put it into the above general solution where $$X=\frac{x}{y}$$ $$z(x,y)=\frac{2y}{x}-2e^y \frac{y^2}{x^2}$$ Carry out the second integration, $$az-2=be^{x/a}=be^y.$$ Then use $$b=\phi(a)$$ or a similar dependence and insert the initial condition $$-2 = \phi(e^y)e^{y}\implies \phi(t)=-\frac{2}t.$$

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# Errors in excess credit calculation with net billing affecting Project NPV 3 posts / 0 new klucas@energy.s... Errors in excess credit calculation with net billing affecting Project NPV Hi, I just downloaded the updated patch. However, there is still an error in the calculation for the excess credit in September, which in turn causes the sum of the first year monthly values to not match the annual first year values. The attached files has 4 cases - TOU w/wo battery, and 4-part demand charge w/wo battery. In each case, the sum of the subparts of the bill (fixed, energy, demand TOU, demand fixed) match the total electricity bill for all months but September. In that month, the excess generation credit earned is greater than the excess generation credit cumulative, despite the cumulative credit having been zeroed out in August. The difference between these values also shows up in the difference between the sum of the monthly total electricity bill and the the sum of monthly subparts. This error also carries over to the annual figures. The first year annual value for each subpart matches the sum of the monthly values, but the first year annual total bill does not match the sum of the first year annual subparts. In this case, September is the last month in the summer TOU period. The excess credit for September is calculated correctly (in case N1, 145.05 kWh are exported at the sell rate of \$0.377497 for a credit of \$57.7572), but the cumulative excess credit is different (\$43.1395). The implied rate of the cumulative excess credit does not match any of the rates. This issue affects the final NPV calculation of the project, and does so for both volumetric and demand-based charges. EDIT - it appears the difference between the cumulative excess credit and the credit earned is equal to the energy charge before credits were applied (in N1, \$11.6177). So the cumulative appears to be used to offset this amount, properly showing a \$0 in the energy charge for September, but the total electricity bill still shows \$24.177, which is the fixed charge plus the original energy use. So the netting happens in the subpart but not the total. Thanks, Kevin Tags: Paul Gilman Thanks, Kevin. We're investigating and I will get back to you when we learn more. Best regards, Paul. Paul Gilman Dear Kevin, We've made some changes to the bill calculations as noted in the SAM repository on GitHub. This change will be included in the next release of SAM. Thank you for helping us find this problem. https://github.com/NREL/ssc/issues/134 Best regards, Paul. Theme by Danetsoft and Danang Probo Sayekti inspired by Maksimer

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# Multiplying and Dividing Fractions Enrichment- Gwendolyn's Cupcake Trouble Subject Resource Type Product Rating File Type PDF (Acrobat) Document File 452 KB|3 pages Share Product Description Multiplying and Dividing Fractions Enrichment (Gwendolyn's Cupcake Trouble) Christy Plumley This is an enrichment worksheet covering multiplying and dividing fractions. I created this download to use in my classroom because every year students come into my Pre-Algebra and Algebra classes with a wide variety of skills with fractions. I plan to re-teach the processes of multiplying and dividing fractions but for my higher level students who don't need more practice, I wanted to push them to use their knowledge of fractions in a real world problem: Baking with recipes! This activity presents the students with two recipes. The end goal is to have enough cupcakes to feed 100 people at a get-together. The first recipe is too small and all ingredients need to be multiplied to make a quadruple batch of cupcakes, the second is a "bakery size" recipe for icing which makes too much icing, and they need to divide the ingredients to get the amount needed. Amounts of ingredients are given in fractions and mixed numbers. There are a couple whole numbers, but not many. Total Pages 3 pages Included Teaching Duration N/A Report this Resource Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.

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