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distilroberta-base-ft-webinstruct

Sentence Similarity
sentence-transformers
Safetensors
roberta
feature-extraction
Generated from Trainer
dataset_size:2335220
loss:MultipleNegativesRankingLoss
Eval Results
text-embeddings-inference
Inference Endpoints
Model card Files Community
distilroberta-base-ft-webinstruct / README.md
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metadata
base_model: distilbert/distilroberta-base
datasets: []
language: []
library_name: sentence-transformers
metrics:
  - pearson_cosine
  - spearman_cosine
  - pearson_manhattan
  - spearman_manhattan
  - pearson_euclidean
  - spearman_euclidean
  - pearson_dot
  - spearman_dot
  - pearson_max
  - spearman_max
pipeline_tag: sentence-similarity
tags:
  - sentence-transformers
  - sentence-similarity
  - feature-extraction
  - generated_from_trainer
  - dataset_size:2335220
  - loss:MultipleNegativesRankingLoss
widget:
  - source_sentence: 'How do you solve the equation #-6 = \frac{y}{5} + 4#?'
    sentences:
      - |-
        To solve the equation, follow these steps:

        1. Subtract 4 from both sides:
           \[-6 - 4 = \frac{y}{5} + 4 - 4\]
           \[-10 = \frac{y}{5}\]

        2. Multiply both sides by 5 to isolate y:
           \[-10 \cdot 5 = \frac{y}{5} \cdot 5\]
           \[-50 = y\]

        So the solution is \(y = -50\).
      - >-
        An organism refers to a living entity, typically composed of cells,
        capable of growth, reproduction, and response to stimuli. The definition
        primarily includes all forms of life, excluding viruses, which are
        considered non-living by some scientists due to their inability to
        replicate independently.


        One of the smallest known organisms is Mycoplasma gallicepticum, a
        parasitic bacterium measuring approximately 200 to 300 nanometers (nm).
        It infects primates, inhabiting the bladder, waste disposal organs,
        genital tracts, and respiratory system.


        For comparison, the smallest virus known to humans is the Porcine
        circovirus type 1 (PCV1), a single-stranded DNA virus. Its genome
        consists of just 1759 nucleotides, and its capsid diameter measures a
        mere 17 nm. This virus causes wasting disease in weaned pigs.


        [Insert images of Mycoplasma gallicepticum and Porcine circovirus type 1
        here, with appropriate captions.]


        Keep in mind that the boundary of what constitutes the "smallest
        organism" can change with advances in scientific research and
        understanding.
      - >-
        Slope is given by #"rise"/"run"#, or the change in the #y# coordinate
        divided by the change in #x#. Mathematically this is written as 

        #(deltay)/(deltax)#

        You calculate it by taking the second coordinate and subtracting the
        first, so

        #(deltay)/(deltax) = (y_2 - y_1)/(x_2 - x_1)#

        # = (8 - (-2))/(10 - 10) = 10/0#

        Since division by zero is undefined, this line has an undefined slope.
        This means that it is a vertical line.
  - source_sentence: >-
      Let $f$ be an analytic function defined on the domain $D = \{z \in
      \mathbb{C} : |z| < 1\}$ with the property that the range of $f$ lies
      within $\mathbb{C} \setminus (-\infty, 0]$. Show that there exists an
      analytic function $g$ on $D$ such that $\text{Re}(g(z)) \geq 0$ and
      $g(z)^2 = f(z)$ for all $z \in D$.
    sentences:
      - >-
        In mathematics, equality is often treated as a primitive notion,
        especially in modern first-order logic. It is understood that two
        objects, such as real numbers, are equal if they are the same object.
        However, for a more formal approach in different settings:


        1. Set Theory: Equality on a set $I$ can be seen as a chosen equivalence
        relation that defines equality. For example, in Zermelo-Frankel set
        theory, equality can be defined as:
           $$x = y \equiv \forall z(z \in x \iff z \in y)$$
           While this works well in set theory, it may not align with the intuitive understanding of equality in other branches of mathematics.

        2. Category Theory: Equality in a fibration $E\to B$ can be viewed
        categorically as a left adjoint to the re-indexing functor induced by
        the diagonal $I\to I\times I$, evaluated at the terminal object in the
        fiber.


        3. Type Theory: Equality can be understood through the concept of
        evaluation. For instance, in arithmetic, the equation $2 + 2 = 3 + 1$
        can be verified by evaluating both sides to the same result, $s(s(2))$.


        The idea of proving two things are equal often involves demonstrating
        that they satisfy the same properties or relations. For example, to show
        $\pi \neq 2\pi$, one would compare their algebraic or geometric
        properties rather than their "membership" in sets.


        For further exploration, consider the work of Ansten Klev on identity
        elimination in Martin-Löf’s Type Theory, and the philosophical
        discussion in Benecereaf's paper "What numbers could not be." Category
        theory and type theory also offer rich perspectives on equality.
      - >-
        Given that $f$ is analytic in the unit disc and has no zeros, we can
        define an analytic logarithm of $f(z)$, denoted by $Log f(z)$. We
        consider the principal branch of the logarithm, which has a branch cut
        along the negative real axis.


        We define $g(z)$ as follows:

        \[ g(z) = \sqrt{f(z)} = e^{\frac{1}{2} Log f(z)} \]


        Now, the real part of $g(z)$ is given by:

        \[ \text{Re}(g(z)) = e^{\frac{1}{2} \log|f(z)|}
        \cos\left(\frac{\arg{f(z)}}{2}\right) \]


        Since $f(z)$ lies outside the negative real axis, we have $|f(z)| > 0$
        and $-\pi < \arg{f(z)} < \pi$. Thus,
        $\cos\left(\frac{\arg{f(z)}}{2}\right)$ is non-negative, which implies
        that $\text{Re}(g(z)) \geq 0$.


        As a result, $g(z)$ is an analytic function on $D$ with a non-negative
        real part, and it satisfies the property $g(z)^2 = f(z)$ for all $z \in
        D$.
      - >-
        Let $\epsilon > 0$ be given. We need to find a natural number
        $N_\varepsilon$ such that

        $$ \left|\frac{1}{1+n+2^n}\right| < \epsilon $$

        for all $n > N_\varepsilon$. Since $1/n \to 0$ as $n \to \infty$, there
        exists an $N_\varepsilon$ such that $1/n < \epsilon$ for all $n >
        N_\varepsilon$. Since $2^n \ge n$ for all $n$, we have

        $$ \frac{1}{1+n+2^n} < \frac{1}{n+2^n} < \frac{1}{n} < \epsilon $$

        for all $n > N_\varepsilon$. Therefore,

        $$ \lim_{n\to\infty} \frac{1}{1+n+2^n} = 0.$$
  - source_sentence: >-
      I know that by definition of basis, the vectors v1 and v2 should span the
      entire subspace. Therefore, if the first constant is not equal to the
      second constant, and if both of the constants give a linear
      transformation, then they must be linearly independent and therefore must
      form a basis. Is that the correct proof, or am I missing something? Also,
      I don't know what the matrix of the linear transformation is.
    sentences:
      - >-
        To prove that v1 and v2 form a basis, we need to show that they are
        linearly independent and that they span the entire subspace.


        To show linear independence, suppose that c1v1 + c2v2 = 0 for some
        scalars c1 and c2. Multiplying both sides by A, we get c1λ1v1 + c2λ2v2 =
        0. Multiplying the first equation by λ1 and subtracting it from the
        second, we get (λ2 - λ1)c2v2 = 0. Since λ2 - λ1 is nonzero (because the
        eigenvalues are distinct), we must have c2 = 0. Substituting this back
        into the first equation, we get c1v1 = 0, so c1 = 0. Therefore, v1 and
        v2 are linearly independent.


        To show that v1 and v2 span the entire subspace, we need to show that
        every vector in the subspace can be written as a linear combination of
        v1 and v2. Let w be an arbitrary vector in the subspace. Then w can be
        written as a linear combination of the eigenvectors of A, so w = c1v1 +
        c2v2 for some scalars c1 and c2. Therefore, v1 and v2 span the entire
        subspace.


        Since v1 and v2 are linearly independent and span the entire subspace,
        they form a basis for the subspace.


        The matrix of the linear transformation T_A is the matrix whose columns
        are the coordinate vectors of the images of the basis vectors of the
        domain under T_A. In this case, the basis vectors of the domain are v1
        and v2, and their images under T_A are λ1v1 and λ2v2, respectively.
        Therefore, the matrix of T_A is


        $$\begin{bmatrix} \lambda_1 & 0\\ 0 & \lambda_2\end{bmatrix}.$$
      - >-
        To find $E[\tilde{\beta_1}]$, we first need to derive the formula for
        $\tilde{\beta_1}$. Under the assumption that the intercept is 0, the
        slope estimator $\tilde{\beta_1}$ is given by:


        $$\tilde{\beta_1} = \frac{\sum_{i=1}^n (x_i - \bar{x})y_i}{\sum_{i=1}^n
        (x_i - \bar{x})^2}$$


        where $\bar{x}$ is the sample mean of the $x_i$.


        Next, we can substitute the true regression model $y_i = \beta_0 +
        \beta_1 x_i + u_i$ into the formula for $\tilde{\beta_1}$:


        $$\tilde{\beta_1} = \frac{\sum_{i=1}^n (x_i - \bar{x})(\beta_0 + \beta_1
        x_i + u_i)}{\sum_{i=1}^n (x_i - \bar{x})^2}$$


        Simplifying this expression, we get:


        $$\tilde{\beta_1} = \beta_1 + \frac{\sum_{i=1}^n (x_i -
        \bar{x})u_i}{\sum_{i=1}^n (x_i - \bar{x})^2}$$


        Now, we can take the expected value of both sides of this equation:


        $$E[\tilde{\beta_1}] = E[\beta_1] + E\left[\frac{\sum_{i=1}^n (x_i -
        \bar{x})u_i}{\sum_{i=1}^n (x_i - \bar{x})^2}\right]$$


        Since $\beta_1$ is a constant, $E[\beta_1] = \beta_1$. For the second
        term, we can use the fact that $E(u_i) = 0$ (by assumption SLR.3) and
        the linearity of expectation to get:


        $$E\left[\frac{\sum_{i=1}^n (x_i - \bar{x})u_i}{\sum_{i=1}^n (x_i -
        \bar{x})^2}\right] = \frac{\sum_{i=1}^n (x_i -
        \bar{x})E(u_i)}{\sum_{i=1}^n (x_i - \bar{x})^2} = 0$$


        Therefore, we have:


        $$E[\tilde{\beta_1}] = \beta_1 + 0 = \beta_1$$


        This shows that $\tilde{\beta_1}$ is an unbiased estimator of $\beta_1$
        when the intercept is assumed to be 0.


        In addition to the case where $\beta_0 = 0$, $\tilde{\beta_1}$ is also
        an unbiased estimator of $\beta_1$ when $\sum_{i=1}^n x_i = 0$. This can
        be seen by noting that in this case, $\bar{x} = 0$ and the formula for
        $\tilde{\beta_1}$ simplifies to:


        $$\tilde{\beta_1} = \frac{\sum_{i=1}^n x_iy_i}{\sum_{i=1}^n x_i^2}$$


        which is the same as the formula for the ordinary least squares (OLS)
        estimator of $\beta_1$ when the intercept is included in the model.
      - >-
        Sure. Here is an example of a continuous map that is not proper:


        $$

        f: \mathbb{R} \to [0, 1]

        $$


        $$

        x \mapsto \frac{1}{1 + |x|}

        $$


        This map is continuous because it is the composition of continuous
        functions. However, it is not proper because the preimage of the compact
        set [0, 1] is not compact. Specifically, the preimage of [0, 1] is the
        set of all real numbers, which is not compact.


        This example shows that the converse of the statement "if a map is
        proper then it is continuous" is not true.
  - source_sentence: >-
      Consider the scenario from the original question, but now suppose that you
      draw two balls from the same random box. If both balls are gold, what is
      the probability that the box contains exactly two gold balls?
    sentences:
      - >-
        The term $\frac{\partial{F}}{\partial{u}}$ appears because $F$ is a
        function of not only $x$, $y$, and $z$, but also of $u$ and $v$. When we
        differentiate $F$ with respect to $x$, we must consider how $F$ changes
        with respect to $u$ as well, since $u$ is a function of $x$.
      - >-
        To prove that U ∪ V is an open set, we must show that for every point x
        in U ∪ V, there exists a ball B(x, r) with radius r > 0, entirely
        contained within U ∪ V.


        Let x be an arbitrary point in U ∪ V. We consider two cases:


        Case 1: If x ∈ U, since U is open, there exists a ball B(x, r_1) with
        r_1 > 0 such that B(x, r_1) ⊆ U.


        Case 2: If x ∈ V, as V is also open, there exists a ball B(x, r_2) with
        r_2 > 0 such that B(x, r_2) ⊆ V.


        Now, consider the ball B(x, r), where r = min(r_1, r_2). In both cases
        (x ∈ U and x ∈ V), this ball has a radius that is less than or equal to
        the radii of the balls in the respective sets. Therefore, B(x, r) will
        be entirely contained within either U or V, and as x is in U ∪ V, B(x,
        r) must be contained within the union of U and V.


        Since the choice of x was arbitrary, this shows that for all points in U
        ∪ V, there exists a corresponding open ball contained within U ∪ V.
        Hence, U ∪ V is an open set in $\mathbb{C}$.
      - >-
        There are a total of 12 balls in the boxes, and 6 of them are gold. If
        we draw two gold balls, we can eliminate box 4. Out of the remaining 3
        boxes, only one box has exactly two gold balls. Therefore, the
        probability that the box contains exactly two gold balls is
        $\frac{1}{3}$.
  - source_sentence: >-
      What should I do if I'm not satisfied with the answers to a question for
      which I've offered a bounty?


      In my case, I've put a bounty on a question, but the two responses I
      received don't address the issue effectively. I requested the original
      poster (OP) to provide an answer so I could reward them for the
      interesting question, but they haven't done so. 


      Are there any acceptable actions in this scenario? For instance, can I
      post my own non-answer, award myself the bounty, and then start a new
      bounty on a different question? Or are there alternative suggestions?
    sentences:
      - >-
        To improve RF signal strength under the given conditions, consider the
        following suggestions:


        1. Bit Rate: Keep the transmitted bit rate low, around 500 bits per
        second (bps).

        2. Balanced Energy Protocol: Implement a biphase or Manchester encoding
        to ensure a 50% duty cycle, which helps reduce DC offset at the
        receiver.

        3. Preamble: Include a long preamble in your protocol for the receiver
        to lock onto the signal and set its Automatic Gain Control (AGC) before
        decoding data.

        4. Receiver Tolerance: Design the decoding protocol to tolerate a wide
        range of pulse widths, as variations due to multi-path, noise, and other
        factors can affect signal integrity.


        While the current setup might be suitable for short distances,
        increasing the transmitter power voltage could potentially improve
        range. However, since you cannot change the 3.7V for the receiver, focus
        on optimizing the mentioned parameters.


        For more detailed information and implementation examples, refer to a
        previous post or access the resources at:
        http://www.carousel-design.com/ManchesterDesignDocs.zip
      - >-
        The issue you're experiencing with your 40kHz crystal oscillator might
        be due to insufficient drive strength and an incorrect load capacitance.
        Here are two potential causes and solutions:


        1. High Series Resistance: The 150 kΩ series resistance in your circuit
        might be too high, which results in a low drive strength for the
        crystal. This can lead to a reduced overall loop gain and prevents the
        oscillator from properly starting. To resolve this, try using a lower
        resistance value as recommended in the crystal's datasheet.


        2. Incorrect Load Capacitance: Ensure that the 33 pF load capacitors
        you're using are compatible with your crystal. Some low-power "watch"
        crystals require only 5-10 pF load capacitors. Always refer to the
        crystal's datasheet to verify the appropriate load capacitance value.


        In summary, carefully review the crystal's datasheet to determine the
        correct series resistance and load capacitance values, and make the
        necessary adjustments to your circuit. By doing so, you should be able
        to resolve the issue and get your oscillator functioning properly.
      - >-
        If all the provided answers do not adequately address your question,
        it's advisable to let the bounty expire. The system will handle the
        distribution of the bounty in such situations according to predefined
        rules.


        Bounties carry a risk, as there is no guarantee that you will receive a
        satisfactory answer, even with the incentive. It's important to
        understand that you cannot reclaim your bounty once it's been offered. 


        Instead of posting a non-answer, you might consider editing and
        clarifying your original question to attract better responses, or
        seeking assistance from the community through comments or chat. If
        needed, you can also start a new bounty on a different question, but
        ensure that it's clear and well-defined to increase the likelihood of
        receiving quality answers.
model-index:
  - name: SentenceTransformer based on distilbert/distilroberta-base
    results:
      - task:
          type: semantic-similarity
          name: Semantic Similarity
        dataset:
          name: sts dev
          type: sts-dev
        metrics:
          - type: pearson_cosine
            value: 0.729437704582197
            name: Pearson Cosine
          - type: spearman_cosine
            value: 0.7554189783542562
            name: Spearman Cosine
          - type: pearson_manhattan
            value: 0.7659413592671894
            name: Pearson Manhattan
          - type: spearman_manhattan
            value: 0.7694884718923434
            name: Spearman Manhattan
          - type: pearson_euclidean
            value: 0.7670364808401289
            name: Pearson Euclidean
          - type: spearman_euclidean
            value: 0.7704200173135048
            name: Spearman Euclidean
          - type: pearson_dot
            value: 0.5261899630902828
            name: Pearson Dot
          - type: spearman_dot
            value: 0.50628913030125
            name: Spearman Dot
          - type: pearson_max
            value: 0.7670364808401289
            name: Pearson Max
          - type: spearman_max
            value: 0.7704200173135048
            name: Spearman Max
      - task:
          type: semantic-similarity
          name: Semantic Similarity
        dataset:
          name: sts test
          type: sts-test
        metrics:
          - type: pearson_cosine
            value: 0.7064454728665479
            name: Pearson Cosine
          - type: spearman_cosine
            value: 0.7115606267133026
            name: Spearman Cosine
          - type: pearson_manhattan
            value: 0.7219098597991042
            name: Pearson Manhattan
          - type: spearman_manhattan
            value: 0.7117023925886385
            name: Spearman Manhattan
          - type: pearson_euclidean
            value: 0.7232693337905357
            name: Pearson Euclidean
          - type: spearman_euclidean
            value: 0.7131352021733437
            name: Spearman Euclidean
          - type: pearson_dot
            value: 0.44643930885103705
            name: Pearson Dot
          - type: spearman_dot
            value: 0.43724418072290006
            name: Spearman Dot
          - type: pearson_max
            value: 0.7232693337905357
            name: Pearson Max
          - type: spearman_max
            value: 0.7131352021733437
            name: Spearman Max

SentenceTransformer based on distilbert/distilroberta-base

This is a sentence-transformers model finetuned from distilbert/distilroberta-base on the mathstackexchange, socratic and stackexchange datasets. It maps sentences & paragraphs to a 768-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.

Model Details

Model Description

  • Model Type: Sentence Transformer
  • Base model: distilbert/distilroberta-base
  • Maximum Sequence Length: 512 tokens
  • Output Dimensionality: 768 tokens
  • Similarity Function: Cosine Similarity
  • Training Datasets:
    • mathstackexchange
    • socratic
    • stackexchange

Model Sources

Full Model Architecture 

SentenceTransformer(
  (0): Transformer({'max_seq_length': 512, 'do_lower_case': False}) with Transformer model: RobertaModel 
  (1): Pooling({'word_embedding_dimension': 768, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
)

Usage

Direct Usage (Sentence Transformers)

First install the Sentence Transformers library:

pip install -U sentence-transformers

Then you can load this model and run inference.

from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("mrm8488/distilroberta-base-ft-webintruct-512")
# Run inference
sentences = [
    "What should I do if I'm not satisfied with the answers to a question for which I've offered a bounty?\n\nIn my case, I've put a bounty on a question, but the two responses I received don't address the issue effectively. I requested the original poster (OP) to provide an answer so I could reward them for the interesting question, but they haven't done so. \n\nAre there any acceptable actions in this scenario? For instance, can I post my own non-answer, award myself the bounty, and then start a new bounty on a different question? Or are there alternative suggestions?",
    "If all the provided answers do not adequately address your question, it's advisable to let the bounty expire. The system will handle the distribution of the bounty in such situations according to predefined rules.\n\nBounties carry a risk, as there is no guarantee that you will receive a satisfactory answer, even with the incentive. It's important to understand that you cannot reclaim your bounty once it's been offered. \n\nInstead of posting a non-answer, you might consider editing and clarifying your original question to attract better responses, or seeking assistance from the community through comments or chat. If needed, you can also start a new bounty on a different question, but ensure that it's clear and well-defined to increase the likelihood of receiving quality answers.",
    'The issue you\'re experiencing with your 40kHz crystal oscillator might be due to insufficient drive strength and an incorrect load capacitance. Here are two potential causes and solutions:\n\n1. High Series Resistance: The 150 kΩ series resistance in your circuit might be too high, which results in a low drive strength for the crystal. This can lead to a reduced overall loop gain and prevents the oscillator from properly starting. To resolve this, try using a lower resistance value as recommended in the crystal\'s datasheet.\n\n2. Incorrect Load Capacitance: Ensure that the 33 pF load capacitors you\'re using are compatible with your crystal. Some low-power "watch" crystals require only 5-10 pF load capacitors. Always refer to the crystal\'s datasheet to verify the appropriate load capacitance value.\n\nIn summary, carefully review the crystal\'s datasheet to determine the correct series resistance and load capacitance values, and make the necessary adjustments to your circuit. By doing so, you should be able to resolve the issue and get your oscillator functioning properly.',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 768]

# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities.shape)
# [3, 3]

Evaluation

Metrics

Semantic Similarity

Metric Value
pearson_cosine 0.7294
spearman_cosine 0.7554
pearson_manhattan 0.7659
spearman_manhattan 0.7695
pearson_euclidean 0.767
spearman_euclidean 0.7704
pearson_dot 0.5262
spearman_dot 0.5063
pearson_max 0.767
spearman_max 0.7704

Semantic Similarity

Metric Value
pearson_cosine 0.7064
spearman_cosine 0.7116
pearson_manhattan 0.7219
spearman_manhattan 0.7117
pearson_euclidean 0.7233
spearman_euclidean 0.7131
pearson_dot 0.4464
spearman_dot 0.4372
pearson_max 0.7233
spearman_max 0.7131

Training Details

Training Datasets

mathstackexchange

  • Dataset: mathstackexchange
  • Size: 1,484,629 training samples
  • Columns: anchor and positive
  • Approximate statistics based on the first 1000 samples:
    anchor positive
    type string string
    details
    • min: 10 tokens
    • mean: 85.8 tokens
    • max: 512 tokens
    • min: 12 tokens
    • mean: 298.22 tokens
    • max: 512 tokens
  • Samples:
    anchor positive
    Suppose $A$ is a normal subgroup of a group $B$, and the quotient group $B/A$ is cyclic with infinite order. How can we demonstrate, using the correspondence theorem, that for every positive integer $k$, $B$ has a normal subgroup of index $k$? The correspondence theorem relates subgroups of the quotient group $B/A$ to subgroups of $B$ containing $A$. Since $B/A$ is isomorphic to the infinite cyclic group $\mathbb{Z}$, it has subgroups of every finite index.

    To find a normal subgroup of $B$ with index $k$, we can follow these steps:
    1. Identify a subgroup $M/A$ of $B/A$ with index $k$. This is possible since $\mathbb{Z}$ has subgroups of index $k$ for any positive integer $k$ (for instance, the subgroup generated by $k$ times the generator of $\mathbb{Z}$).
    2. By the correspondence theorem, there exists an intermediate subgroup $M$ such that $A \leq M \leq B$ with $M/A \cong M/A$.
    3. Now, consider the index of $M$ in $B$: $[B:M] = [B/A : M/A] = k$, as desired.

    Thus, using the correspondence theorem and exploiting the properties of $\mathbb{Z}$, we can construct a normal subgroup of $B$ with index $k$ for any positive integer $k$.
    If $z$ is a complex number and $a$ is a real number, can we say $ az
    What is the remainder when $x^{2007}$ is divided by $x^2-x+1$? To find the remainder when $x^{2007}$ is divided by $x^2-x+1$, we can use polynomial long division or synthetic division. Alternatively, we can utilize the properties of the polynomial $x^2-x+1$.

    Since $x^2-x+1$ is a factor of $x^3+1$, we have $x^3 \equiv -1 \pmod{x^2-x+1}$. Therefore, we can express $x^{2007}$ as $x^{3\cdot669}$, and thus:

    $$x^{2007} = (x^3)^{669} \equiv (-1)^{669} \pmod{x^2-x+1}$$

    Now, since $669$ is odd, $(-1)^{669} = -1$. Hence, the remainder is $-1$.

    Alternatively, we can perform polynomial long division to obtain:

    $$\frac{x^{2007}}{x^2-x+1} = a(x) - \frac{x+1}{x^2-x+1}$$

    where $a(x)$ is the quotient polynomial. Since we only care about the remainder, we have:

    $$x^{2007} \equiv -1 \pmod{x^2-x+1}$$
  • Loss: MultipleNegativesRankingLoss with these parameters:
    {
    "scale": 20.0,
    "similarity_fct": "cos_sim"
}

socratic

  • Dataset: socratic
  • Size: 533,383 training samples
  • Columns: anchor and positive
  • Approximate statistics based on the first 1000 samples:
    anchor positive
    type string string
    details
    • min: 8 tokens
    • mean: 29.99 tokens
    • max: 156 tokens
    • min: 3 tokens
    • mean: 210.0 tokens
    • max: 512 tokens
  • Samples:
    anchor positive
    What is an activated complex?
    The activated complex is formed when the reactants collide with each other and begin to rearrange their atoms and bonds to form the products. This process requires energy, which is why the activated complex has a higher energy than the reactants. The energy required to reach the activated complex is called the activation energy.

    Once the activated complex is formed, it can either decompose back into the reactants or proceed to form the products. The probability of the activated complex decomposing back into the reactants is determined by the activation energy. If the activation energy is high, then the activated complex is more likely to decompose back into the reactants. If the activation energy is low, then the activated complex is more likely to proceed to form the products.

    The activated complex is a key concept in understanding chemical reactions. It helps to explain why some reactions occur quickly and others occur slowly. It also helps to explain why some reactions require a catalyst to occur.
    ####
    An activated complex is a high-energy, unstable intermediate state that forms during a chemical reaction. It is the transition state between the reactants and the products. The activated complex has a higher energy than both the reactants and the products, and it represents the maximum energy that must be overcome for the reaction to occur.
    Why does gravity cause planets to be round?
    The gravitational force of a planet pulls matter towards its center. This force is strongest at the center of the planet and weakest at the surface. As a result, matter is pulled towards the center of the planet, causing it to take on a spherical shape.

    A sphere is the shape with the lowest surface area for a given volume. This means that a planet with a spherical shape has the least amount of potential energy. Potential energy is the energy that an object has due to its position or condition. In the case of a planet, its potential energy is due to its gravitational force.

    The lower the potential energy of a planet, the more stable it is. This is because a planet with a lower potential energy is less likely to change its shape. As a result, planets tend to be spherical in shape.
    ####
    Gravity causes planets to be round because a sphere is the shape with the lowest surface area for a given volume. This means that a planet with a spherical shape has the least amount of potential energy, which is the energy that an object has due to its position or condition. Gravity pulls matter towards the center of a planet, and this force is strongest at the center. As a result, matter is pulled towards the center of the planet, causing it to take on a spherical shape.
    How many carbon atoms are present in a 5.85-gram sample of carbon tetrabromide (CBr4)? There are approximately (1 \times 10^{22}) carbon atoms in the given sample.

    Explanation:
    To determine the number of carbon atoms, we first need to calculate the molar amount of CBr4. The molar mass of CBr4 is 331.63 g/mol. So, we have:

    [
    \frac{5.85\ g}{331.63\ g/mol} = 0.0176\ mol
    ]

    Since one molecule of CBr4 contains one carbon atom and four bromine atoms, there are:

    [
    1 \times 0.0176\ mol = 0.0176\ mol\ of\ carbon\ atoms
    ]

    Now, multiplying the molar quantity by Avogadro's number (6.022 × 10^23 mol^(-1)) gives us the number of individual carbon atoms:

    [
    0.0176\ mol \times 6.022 \times 10^{23}\ mol^{-1} = 1.06 \times 10^{22}\ carbon\ atoms
    ]

    Therefore, there are approximately (1 \times 10^{22}) carbon atoms in a 5.85-gram sample of CBr4.
  • Loss: MultipleNegativesRankingLoss with these parameters:
    {
    "scale": 20.0,
    "similarity_fct": "cos_sim"
}

stackexchange

  • Dataset: stackexchange
  • Size: 317,208 training samples
  • Columns: anchor and positive
  • Approximate statistics based on the first 1000 samples:
    anchor positive
    type string string
    details
    • min: 12 tokens
    • mean: 63.36 tokens
    • max: 512 tokens
    • min: 50 tokens
    • mean: 263.31 tokens
    • max: 512 tokens
  • Samples:
    anchor positive
    Should I use a tip activator to recoat the worn protective coating on my iron tip, or is it better to replace the tip entirely? My 48W ZD99 Solder Station's tip is showing signs of peeling due to moisture exposure and inadequate care. Can the tip activator effectively restore the tip, or should I opt for a new one? To address the issue, first clean the iron tip with a wire brush to remove any debris. Then, apply flux and tin the tip to protect it and maintain its performance. Tip activators are available as a means to recoat tips, but their effectiveness may vary. While they can be a viable solution, it's essential to ensure proper tip care to prevent future wear. If the tip's condition significantly deteriorates despite these efforts, consider replacing it with a new one.
    What are the fundamental limits, if any, for the speed of sound in different materials, and how do these limits relate to the speed of light? The speed of sound is limited by the properties of the material it travels through and the fundamental principles of physics. In a theoretical sense, the maximum speed of sound is constrained by the speed of light (approximately 299,792 km/s in vacuum), which is the maximum speed at which information can propagate. This limit is reached when the material has an incompressible equation of state, such as in the core of a neutron star, where the strong nuclear force creates immense pressure resistance.

    For an ideal gas, where particles do not interact, the equation of state is the softest possible with $P = \rho c^2/3$, where $P$ is pressure, $\rho$ is density, and $c$ is the speed of light. In this case, the maximum speed of sound would be $c/\sqrt{3}$.

    It's important to note that in practice, materials with extremely high sound speeds are unlikely to exist due to the conditions required for an incompressible equation of state. In reality, materials like solids and liquids generally have faster sound speeds than gases, but they are still far below the speed of light.

    When dealing with exotic materials, such as short-lived isotopes or neutron stars, the speed of sound may be even more challenging to determine due to the unique properties and states involved. However, the underlying principles remain the same: the speed of sound is determined by the material's properties, and it cannot exceed the speed of light in a vacuum.
    What could be causing a 1996 Honda Civic to stop running suddenly, and how can it be started? A potential issue is a faulty ignition switch. When you attempt to start the car, the switch might be malfunctioning in such a way that it disrupts power to the engine ignition system, causing the dash lights to go out and preventing the car from starting. However, when you perform a push start (crash start), the car starts because the ignition switch remains in position 2, providing power to the engine.

    Another possibility is a problem with the battery or its connections. If the battery terminals have a poor connection, it might lead to high resistance, making it difficult for the car to start. Alternatively, if the battery is weak, it might not supply enough power to crank the engine effectively. In this case, the starter motor would sound sluggish as it tries to turn the engine. To resolve the issue, inspect the ignition switch, battery connections, and consider testing or replacing the battery if necessary.
  • Loss: MultipleNegativesRankingLoss with these parameters:
    {
    "scale": 20.0,
    "similarity_fct": "cos_sim"
}

Training Hyperparameters

Non-Default Hyperparameters

  • eval_strategy: steps
  • per_device_train_batch_size: 64
  • per_device_eval_batch_size: 64
  • num_train_epochs: 1
  • warmup_ratio: 0.1
  • fp16: True
  • batch_sampler: no_duplicates
  • multi_dataset_batch_sampler: round_robin

All Hyperparameters

Click to expand
  • overwrite_output_dir: False
  • do_predict: False
  • eval_strategy: steps
  • prediction_loss_only: True
  • per_device_train_batch_size: 64
  • per_device_eval_batch_size: 64
  • per_gpu_train_batch_size: None
  • per_gpu_eval_batch_size: None
  • gradient_accumulation_steps: 1
  • eval_accumulation_steps: None
  • learning_rate: 5e-05
  • weight_decay: 0.0
  • adam_beta1: 0.9
  • adam_beta2: 0.999
  • adam_epsilon: 1e-08
  • max_grad_norm: 1.0
  • num_train_epochs: 1
  • max_steps: -1
  • lr_scheduler_type: linear
  • lr_scheduler_kwargs: {}
  • warmup_ratio: 0.1
  • warmup_steps: 0
  • log_level: passive
  • log_level_replica: warning
  • log_on_each_node: True
  • logging_nan_inf_filter: True
  • save_safetensors: True
  • save_on_each_node: False
  • save_only_model: False
  • restore_callback_states_from_checkpoint: False
  • no_cuda: False
  • use_cpu: False
  • use_mps_device: False
  • seed: 42
  • data_seed: None
  • jit_mode_eval: False
  • use_ipex: False
  • bf16: False
  • fp16: True
  • fp16_opt_level: O1
  • half_precision_backend: auto
  • bf16_full_eval: False
  • fp16_full_eval: False
  • tf32: None
  • local_rank: 0
  • ddp_backend: None
  • tpu_num_cores: None
  • tpu_metrics_debug: False
  • debug: []
  • dataloader_drop_last: False
  • dataloader_num_workers: 0
  • dataloader_prefetch_factor: None
  • past_index: -1
  • disable_tqdm: False
  • remove_unused_columns: True
  • label_names: None
  • load_best_model_at_end: False
  • ignore_data_skip: False
  • fsdp: []
  • fsdp_min_num_params: 0
  • fsdp_config: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
  • fsdp_transformer_layer_cls_to_wrap: None
  • accelerator_config: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
  • deepspeed: None
  • label_smoothing_factor: 0.0
  • optim: adamw_torch
  • optim_args: None
  • adafactor: False
  • group_by_length: False
  • length_column_name: length
  • ddp_find_unused_parameters: None
  • ddp_bucket_cap_mb: None
  • ddp_broadcast_buffers: False
  • dataloader_pin_memory: True
  • dataloader_persistent_workers: False
  • skip_memory_metrics: True
  • use_legacy_prediction_loop: False
  • push_to_hub: False
  • resume_from_checkpoint: None
  • hub_model_id: None
  • hub_strategy: every_save
  • hub_private_repo: False
  • hub_always_push: False
  • gradient_checkpointing: False
  • gradient_checkpointing_kwargs: None
  • include_inputs_for_metrics: False
  • eval_do_concat_batches: True
  • fp16_backend: auto
  • push_to_hub_model_id: None
  • push_to_hub_organization: None
  • mp_parameters:
  • auto_find_batch_size: False
  • full_determinism: False
  • torchdynamo: None
  • ray_scope: last
  • ddp_timeout: 1800
  • torch_compile: False
  • torch_compile_backend: None
  • torch_compile_mode: None
  • dispatch_batches: None
  • split_batches: None
  • include_tokens_per_second: False
  • include_num_input_tokens_seen: False
  • neftune_noise_alpha: None
  • optim_target_modules: None
  • batch_eval_metrics: False
  • eval_on_start: False
  • batch_sampler: no_duplicates
  • multi_dataset_batch_sampler: round_robin

Training Logs

Click to expand
Epoch Step Training Loss sts-dev_spearman_cosine sts-test_spearman_cosine
0.0067 100 3.6873 - -
0.0134 200 0.984 - -
0.0202 300 0.2259 - -
0.0269 400 0.1696 - -
0.0336 500 0.1468 - -
0.0403 600 0.1235 - -
0.0471 700 0.1125 - -
0.0538 800 0.1032 - -
0.0605 900 0.097 - -
0.0672 1000 0.0992 0.8011 -
0.0740 1100 0.0937 - -
0.0807 1200 0.0818 - -
0.0874 1300 0.0909 - -
0.0941 1400 0.0836 - -
0.1009 1500 0.0705 - -
0.1076 1600 0.081 - -
0.1143 1700 0.0791 - -
0.1210 1800 0.0677 - -
0.1278 1900 0.0697 - -
0.1345 2000 0.0661 0.7721 -
0.1412 2100 0.0727 - -
0.1479 2200 0.0683 - -
0.1547 2300 0.0597 - -
0.1614 2400 0.06 - -
0.1681 2500 0.0598 - -
0.1748 2600 0.051 - -
0.1816 2700 0.0629 - -
0.1883 2800 0.0513 - -
0.1950 2900 0.0517 - -
0.2017 3000 0.048 0.7783 -
0.2085 3100 0.0418 - -
0.2152 3200 0.0447 - -
0.2219 3300 0.0458 - -
0.2286 3400 0.0504 - -
0.2354 3500 0.0463 - -
0.2421 3600 0.0433 - -
0.2488 3700 0.0447 - -
0.2555 3800 0.0444 - -
0.2623 3900 0.0432 - -
0.2690 4000 0.0452 0.7910 -
0.2757 4100 0.0419 - -
0.2824 4200 0.0373 - -
0.2892 4300 0.0385 - -
0.2959 4400 0.0381 - -
0.3026 4500 0.0383 - -
0.3093 4600 0.0367 - -
0.3161 4700 0.0353 - -
0.3228 4800 0.034 - -
0.3295 4900 0.0333 - -
0.3362 5000 0.0406 0.7862 -
0.3429 5100 0.0319 - -
0.3497 5200 0.0332 - -
0.3564 5300 0.0337 - -
0.3631 5400 0.0347 - -
0.3698 5500 0.0333 - -
0.3766 5600 0.036 - -
0.3833 5700 0.0319 - -
0.3900 5800 0.0342 - -
0.3967 5900 0.0296 - -
0.4035 6000 0.0313 0.7675 -
0.4102 6100 0.0289 - -
0.4169 6200 0.0292 - -
0.4236 6300 0.0271 - -
0.4304 6400 0.0295 - -
0.4371 6500 0.0353 - -
0.4438 6600 0.035 - -
0.4505 6700 0.0324 - -
0.4573 6800 0.0281 - -
0.4640 6900 0.0265 - -
0.4707 7000 0.031 0.7634 -
0.4774 7100 0.0302 - -
0.4842 7200 0.0268 - -
0.4909 7300 0.0275 - -
0.4976 7400 0.0267 - -
0.5043 7500 0.0249 - -
0.5111 7600 0.0285 - -
0.5178 7700 0.0311 - -
0.5245 7800 0.0248 - -
0.5312 7900 0.0278 - -
0.5380 8000 0.0267 0.7658 -
0.5447 8100 0.0245 - -
0.5514 8200 0.0261 - -
0.5581 8300 0.0227 - -
0.5649 8400 0.0261 - -
0.5716 8500 0.0241 - -
0.5783 8600 0.0261 - -
0.5850 8700 0.0173 - -
0.5918 8800 0.0226 - -
0.5985 8900 0.0221 - -
0.6052 9000 0.023 0.7558 -
0.6119 9100 0.0218 - -
0.6187 9200 0.0245 - -
0.6254 9300 0.0232 - -
0.6321 9400 0.0208 - -
0.6388 9500 0.0202 - -
0.6456 9600 0.022 - -
0.6523 9700 0.0212 - -
0.6590 9800 0.0228 - -
0.6657 9900 0.0214 - -
0.6724 10000 0.0206 0.7686 -
0.6792 10100 0.0227 - -
0.6859 10200 0.0225 - -
0.6926 10300 0.018 - -
0.6993 10400 0.0185 - -
0.7061 10500 0.0204 - -
0.7128 10600 0.0216 - -
0.7195 10700 0.0212 - -
0.7262 10800 0.0156 - -
0.7330 10900 0.0232 - -
0.7397 11000 0.0146 0.7610 -
0.7464 11100 0.0165 - -
0.7531 11200 0.0187 - -
0.7599 11300 0.0199 - -
0.7666 11400 0.0215 - -
0.7733 11500 0.0222 - -
0.7800 11600 0.021 - -
0.7868 11700 0.0163 - -
0.7935 11800 0.0192 - -
0.8002 11900 0.0206 - -
0.8069 12000 0.017 0.7658 -
0.8137 12100 0.0152 - -
0.8204 12200 0.0175 - -
0.8271 12300 0.0211 - -
0.8338 12400 0.0162 - -
0.8406 12500 0.0178 - -
0.8473 12600 0.0142 - -
0.8540 12700 0.02 - -
0.8607 12800 0.0166 - -
0.8675 12900 0.0187 - -
0.8742 13000 0.017 0.7603 -
0.8809 13100 0.0167 - -
0.8876 13200 0.0211 - -
0.8944 13300 0.0162 - -
0.9011 13400 0.0161 - -
0.9078 13500 0.0157 - -
0.9145 13600 0.016 - -
0.9213 13700 0.0139 - -
0.9280 13800 0.0175 - -
0.9347 13900 0.0172 - -
0.9414 14000 0.0148 0.7554 -
0.9482 14100 0.0227 - -
0.9549 14200 0.0174 - -
0.9616 14300 0.0191 - -
0.9683 14400 0.0151 - -
0.9751 14500 0.0184 - -
0.9818 14600 0.02 - -
0.9885 14700 0.0163 - -
0.9952 14800 0.0141 - -
1.0 14871 - - 0.7116

Framework Versions

  • Python: 3.10.12
  • Sentence Transformers: 3.0.1
  • Transformers: 4.42.4
  • PyTorch: 2.4.0+cu121
  • Accelerate: 0.34.0
  • Datasets: 2.21.0
  • Tokenizers: 0.19.1

Citation

BibTeX

Sentence Transformers 

@inproceedings{reimers-2019-sentence-bert,
    title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
    author = "Reimers, Nils and Gurevych, Iryna",
    booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
    month = "11",
    year = "2019",
    publisher = "Association for Computational Linguistics",
    url = "https://arxiv.org/abs/1908.10084",
}

MultipleNegativesRankingLoss 

@misc{henderson2017efficient,
    title={Efficient Natural Language Response Suggestion for Smart Reply}, 
    author={Matthew Henderson and Rami Al-Rfou and Brian Strope and Yun-hsuan Sung and Laszlo Lukacs and Ruiqi Guo and Sanjiv Kumar and Balint Miklos and Ray Kurzweil},
    year={2017},
    eprint={1705.00652},
    archivePrefix={arXiv},
    primaryClass={cs.CL}
}