--- 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:\n\n1. Subtract 4 from both sides:\n\ \ \\[-6 - 4 = \\frac{y}{5} + 4 - 4\\]\n \\[-10 = \\frac{y}{5}\\]\n\n2. Multiply\ \ both sides by 5 to isolate y:\n \\[-10 \\cdot 5 = \\frac{y}{5} \\cdot 5\\\ ]\n \\[-50 = y\\]\n\nSo 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 \n#(deltay)/(deltax)#\n\ You calculate it by taking the second coordinate and subtracting the first, so\n\ #(deltay)/(deltax) = (y_2 - y_1)/(x_2 - x_1)#\n# = (8 - (-2))/(10 - 10) = 10/0#\n\ 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:\n\n1. 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:\n $$x = y \\equiv \\forall z(z \\\ in x \\iff z \\in y)$$\n While this works well in set theory, it may not align\ \ with the intuitive understanding of equality in other branches of mathematics.\n\ \n2. 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.\n\n3. 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, $s(s(2))$.\n\nThe 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.\n\nFor 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?\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?" 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.\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." 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](https://www.SBERT.net) model finetuned from [distilbert/distilroberta-base](https://huggingface.co/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](https://huggingface.co/distilbert/distilroberta-base) - **Maximum Sequence Length:** 512 tokens - **Output Dimensionality:** 768 tokens - **Similarity Function:** Cosine Similarity - **Training Datasets:** - mathstackexchange - socratic - stackexchange ### Model Sources - **Documentation:** [Sentence Transformers Documentation](https://sbert.net) - **Repository:** [Sentence Transformers on GitHub](https://github.com/UKPLab/sentence-transformers) - **Hugging Face:** [Sentence Transformers on Hugging Face](https://huggingface.co/models?library=sentence-transformers) ### 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: ```bash pip install -U sentence-transformers ``` Then you can load this model and run inference. ```python 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 * Dataset: `sts-dev` * Evaluated with [EmbeddingSimilarityEvaluator](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.EmbeddingSimilarityEvaluator) | 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 * Dataset: `sts-test` * Evaluated with [EmbeddingSimilarityEvaluator](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.EmbeddingSimilarityEvaluator) | 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 | | | * 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 $|az| = a|z|$? Here, $|z|$ refers to the modulus of $z$. | Yes, we can say $|az| = a|z|$.

Let $z = x + iy$ be a complex number and $a$ be a real number. Then,

$$az = a(x + iy) = ax + iay$$

and

$$|az| = \sqrt{(ax)^2 + (ay)^2} = \sqrt{a^2(x^2 + y^2)} = |a| \sqrt{x^2 + y^2} = |a| |z|.$$

Therefore, $|az| = a|z|$. | | 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](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#multiplenegativesrankingloss) with these parameters: ```json { "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 | | | * 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](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#multiplenegativesrankingloss) with these parameters: ```json { "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 | | | * 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](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#multiplenegativesrankingloss) with these parameters: ```json { "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 ```bibtex @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 ```bibtex @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} } ```