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values | messages listlengths 2 60 | used_in listlengths 1 1 | metadata dict | license stringclasses 2
values | tools listlengths 0 0 | url stringlengths 63 129 ⌀ | user_name stringlengths 3 21 ⌀ | user_url stringlengths 43 64 ⌀ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\( f(0) = 0 \) | Determine the pointwise limit of the function sequence \( f_n \) defined as follows:
- \( f_n(0) = 0 \)
- \( f_n\left(\frac{1}{n}\right) = n \)
- \( f_n \) increases linearly on the interval \([0, \frac{1}{n}]\)
- \( f_n \) is not defined elsewhere, i.e., \( f_n: [0, \frac{1}{n}] \to \mathbb{R} \). | \( f(0) = 0 \) | false | stackflow | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the pointwise limit of the function sequence \\( f_n \\) defined as follows:\n- \\( f_n(0) = 0 \\)\n- \\( f_n\\left(\\frac{1}{n}\\right) = n \\)\n- \\( f_n \\) increas... | [
"nano_v3"
] | {
"reason_low_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_low_no_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/2150778/pointwise-limit | ATLKing | https://math.stackexchange.com/users/381871/atlking | |
\,f(x)=a+(b-a)\,\sin(\pi x)\, | Find an example of a continuous map \( f: \mathbb{R} \to \mathbb{R} \) such that the image of the interval \( (0, 1) \) is the half-closed interval \( (a, b] \). | \( a = 0, b = \frac{1}{4} \) | true | stackflow | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind an example of a continuous map \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that the image of the interval \\( (0, 1) \\) is the half-closed interval \\( (a, b] \\).",
... | [
"nano_v3"
] | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_no_tool": {
"count": 8,
"pass": 0,
"acc... | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/2896597/continuous-image-of-0-1-equal-to-half-closed-interval | CuriousKid7 | https://math.stackexchange.com/users/376651/curiouskid7 | |
\( 21 \ln \left( \frac{3}{2} \right) \) | Integrate the function \( f(x,y) = x^2y^2 \) over the area \( B \subset \mathbb{R}^2 \) bounded by the curves \( y = \frac{x}{9} \), \( y = \frac{x}{4} \), \( y = \frac{1}{x} \), and \( y = \frac{4}{x} \). Determine the appropriate boundaries for the integration. | \( 21 \ln \left( \frac{3}{2} \right) \) | false | stackflow | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIntegrate the function \\( f(x,y) = x^2y^2 \\) over the area \\( B \\subset \\mathbb{R}^2 \\) bounded by the curves \\( y = \\frac{x}{9} \\), \\( y = \\frac{x}{4} \\), \\( y = \... | [
"nano_v3"
] | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_no_tool": {
"count": 8,
"pass": 0,
"accura... | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/840709/integrating-over-an-area | Dmitry Kazakov | https://math.stackexchange.com/users/154960/dmitry-kazakov | |
bn | What is the maximum running time of the process until at least one ball touches the last step in the worst case, given that there are $b$ balls and $n$ steps, and the probability that a ball misses a step is $p$? | \( b-1+n \) | true | stackflow | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the maximum running time of the process until at least one ball touches the last step in the worst case, given that there are $b$ balls and $n$ steps, and the probabilit... | [
"nano_v3"
] | {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/421287/balls-on-stairs | com | https://math.stackexchange.com/users/16588/com | |
\;
\lambda_{\min}(L)\;\le\;\frac{\sigma}{\sigma_{n}}\;\le\;
\lambda_{\max}(L)
\; | Suppose we normalize our data so that per-coordinate variances are one. Given an \( n \times d \) data matrix \( X \) with \( X^TX \) having full-rank, trace \( d \), and spectral radius \( \sigma \), compute the normalized spectral radius \( \sigma_n \) as follows:
\[
\begin{array}{lll}
L &=& \text{Diag}(X^TX) \\
\sig... | null | true | stackflow | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose we normalize our data so that per-coordinate variances are one. Given an \\( n \\times d \\) data matrix \\( X \\) with \\( X^TX \\) having full-rank, trace \\( d \\), a... | [
"nano_v3"
] | {
"reason_low_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_low_no_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/4155557/how-does-per-coordinate-normalization-affect-covariance-matrix-norm | Yaroslav Bulatov | https://math.stackexchange.com/users/998/yaroslav-bulatov | |
"\\begin{array}{c}\n\\text{Root of unity }z: \\text{ convergence }\\iff\\sum a_n\\text{ converges}.\(...TRUNCATED) | "Determine the convergence of the series $\\sum a_n z^{n!}$ where $|z| = 1$ and $a_n$ is a sequence (...TRUNCATED) | null | true | stackflow | [{"role":"user","content":"Solve the following math problem. Make sure to put the answer (and only a(...TRUNCATED) | [
"nano_v3"
] | {"reason_low_with_tool":{"count":8,"pass":5,"accuracy":0.625},"reason_low_no_tool":{"count":8,"pass"(...TRUNCATED) | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/811702/general-approaches-to-test-convergence-of-sum-a-n-z-n-p-when-z-is-a-complex-numb | user1819047 | https://math.stackexchange.com/users/138769/user1819047 | |
\( v = \pm \frac{A^{-1} u}{\sqrt{u^T A^{-1} u}} \) | "Find the minimum of the function \\( f(\\mathbf{v}) = \\mathbf{v}^\\top A \\mathbf{v} - \\log((\\ma(...TRUNCATED) | \( v = \pm \frac{A^{-1} u}{\sqrt{u^T A^{-1} u}} \) | false | stackflow | [{"role":"user","content":"Solve the following math problem. Make sure to put the answer (and only a(...TRUNCATED) | [
"nano_v3"
] | {"reason_low_with_tool":{"count":8,"pass":4,"accuracy":0.5},"reason_low_no_tool":{"count":8,"pass":0(...TRUNCATED) | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/4028704/find-minimum-of-f-mathbf-v-mathbf-v-top-a-mathbf-v-sum-i-1-n-log-mathbf-v-top-ma | user202654 | https://math.stackexchange.com/users/202654/user202654 | |
\( T_n = T_{n-1} + 6T_{n-2} \) for \( n \geq 3 \) | "Find the recurrence relation for the number of strings of length \\( n \\) using a 7-letter alphabe(...TRUNCATED) | \( T_n = T_{n-1} + 6T_{n-2} \) for \( n \geq 3 \) | false | stackflow | [{"role":"user","content":"Solve the following math problem. Make sure to put the answer (and only a(...TRUNCATED) | [
"nano_v3"
] | {"reason_low_with_tool":{"count":8,"pass":1,"accuracy":0.125},"reason_low_no_tool":{"count":8,"pass"(...TRUNCATED) | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/1739420/find-recurrence-relation | UpsideDownRide | https://math.stackexchange.com/users/211370/upsidedownride | |
\;k = 2^{m}\ \text{with}\ m\ge 1\; | "Determine all \\( k \\geqslant 1 \\) such that the polynomial \\( P_k(X) = (X - a_1)^k \\cdots (X -(...TRUNCATED) | \( n = k = 1 \) | true | aops | [{"role":"user","content":"Solve the following math problem. Make sure to put the answer (and only a(...TRUNCATED) | [
"nano_v3"
] | {"reason_low_with_tool":{"count":8,"pass":4,"accuracy":0.5},"reason_low_no_tool":{"count":8,"pass":2(...TRUNCATED) | cc-by-4.0 | [] | null | null | null | |
"\\,v(t)=\\frac{16}{\\sqrt{7}}\\,\n e^{-t/8}\\,\n \\cos\\!\\Bigg(\\frac{\\sqrt{63}(...TRUNCATED) | "Consider a parallel circuit with \\( R = 20 \\, \\Omega \\), \\( C = \\frac{1}{5} \\, F \\), and \\(...TRUNCATED) | null | true | stackflow | [{"role":"user","content":"Solve the following math problem. Make sure to put the answer (and only a(...TRUNCATED) | [
"nano_v3"
] | {"reason_low_with_tool":{"count":8,"pass":4,"accuracy":0.5},"reason_low_no_tool":{"count":8,"pass":3(...TRUNCATED) | cc-by-sa-4.0 | [] | https://math.stackexchange.com/questions/3322589/circuits-and-differential-equations | Andres | https://math.stackexchange.com/users/678558/andres |
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