id string | output_text string | type string | source_dataset string | source_config string | source_split string | source_row_index int64 | source_field string | metadata_json string |
|---|---|---|---|---|---|---|---|---|
latex-00004993 | j_\nu= \phi^{-1}(x)\partial_\nu \phi(x) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 4,998 | latex_formula | {"original_latex": "\\begin{align*}j_\\nu= \\phi^{-1}(x)\\partial_\\nu \\phi(x)\\end{align*}"} |
latex-00005323 | \ddot \phi\vert_{\tau_e}+4 H_e\delta\dot\phi\vert_{\tau_e}=-2\alpha {H_e}^2 | latex | OleehyO/latex-formulas | cleaned_formulas | train | 5,328 | latex_formula | {"original_latex": "\\begin{align*}\\ddot \\phi\\vert_{\\tau_e}+4 H_e\\delta\\dot\\phi\\vert_{\\tau_e}=-2\\alpha {H_e}^2\\end{align*}"} |
mixed-00045547 | Let $x=b+c$, $y=a+c$ and $z=a+b$.
Thus, $a$, $b$ and $c$ are positives, $a+b+c=1$ and
$$xy+xz+yz-xyz=\sum_{cyc}(a^2+3ab)\sum_{cyc}a-\prod_{cyc}(a+b)=$$
$$=\sum_{cyc}(a^3+a^2b+a^2c+3a^2b+3a^2c+3abc)-\sum_{cyc}\left(a^2b+a^2c+\frac{2}{3}abc\right)>$$
$$>\sum_{cyc}(a^3+3a^2b+3a^2c+2abc)=(a+b+c)^3=1.$$
For $a=b\rightarrow\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 27,140 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3869454", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 1}} |
normal-00011864 | Although many popular commentators , including psychologist Margaret Singer , speculate that Applewhite brainwashed his followers , many academics have rejected the " brainwashing " label as an oversimplification that does not express the nuances of the process by which the followers were influenced . Lalich speculates... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 28,843 | text | {} |
latex-00009146 | Q(x) = \frac1{\log x}\bigg( 1+O\bigg( \frac{\log\log x}{\log x} \bigg) \bigg) \int_{x/\log x}^{x-\sqrt x} \frac t{\log(x-t)} dt + O\bigg( \frac{x^2}{\log^3x} \bigg). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,161 | latex_formula | {"original_latex": "\\begin{align*} Q(x) = \\frac1{\\log x}\\bigg( 1+O\\bigg( \\frac{\\log\\log x}{\\log x} \\bigg) \\bigg) \\int_{x/\\log x}^{x-\\sqrt x} \\frac t{\\log(x-t)} \\,dt + O\\bigg( \\frac{x^2}{\\log^3x} \\bigg). \\end{align*}"} |
mixed-00019179 | Here is an answer based upon generating functions. It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series. This way we can write e.g.
\begin{align*}
[z^k](1+z)^n=\binom{n}{k}
\end{align*}
We obtain
\begin{align*}
\color{blue}{\sum_{k=0}^n\binom{n+k}{k}\frac{1}{2^{n+... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,989 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2406516", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "10", "answer_count": 3, "answer_id": 2}} |
normal-00028870 | The total cost of the building was $ 57 @.@ 6 million , of which $ 18 @.@ 3 million came from the federal government via the Building Canada Fund , $ 13 million from the provincial government , and the remainder ( $ 26 @.@ 3 million ) from the municipality . The municipal contribution was primarily generated from the s... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 69,192 | text | {} |
mixed-00049757 | My solution
Case 1. Let $p \mod 3=q \mod 3=1$. Then $p^3+q^3+1=3=0 \mod 3 \neq p^2 q^2=1 \mod 3$
Case 2 $p \mod 3=q \mod 3=2$. Then $p^3+q^3+1=2+2+1=2 \mod 3 \neq p^2 q^2=1 \mod 3$
Сase 3. $p \mod 3=1, q \mod 3=2$. Then $p^3+q^3+1=1+2+1=1 \mod 3 \neq p^2 q^2=1 \cdot 2 =2 \mod 3$
So, must be, say $p=0 \mod 3 \implies p=... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,691 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2991909", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
mixed-00041097 | $\int_{0}^{1} \frac{ \ln^n x}{1+x^{m}} d x$ =? Inspired by the post, I try to generalize the result to
$$
I(m, n)=\int_{0}^{1} \frac{ \ln ^nx}{1+x^{m}} d x
$$
Instead, I first investigate its partner integral $$
\begin{aligned}
I(a): &=\int_{0}^{1} \frac{x^{a}}{1+x^{m}} d x . \\
&=\sum_{k=0}^{\infty}(-1)^{k} \int_{0}^{... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,396 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4408960", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 1, "answer_id": 0}} |
latex-00031077 | \Phi_{Q,s} = \sum_{\substack{ (a,q) = 1 \ q \sim Q }}\tau_{-a/q} \phi^{(s)}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 31,368 | latex_formula | {"original_latex": "\\begin{align*}\\Phi_{Q,s} = \\sum_{\\substack{ (a,q) = 1 \\\\ q \\sim Q }}\\tau_{-a/q} \\phi^{(s)},\\end{align*}"} |
normal-00041103 | Released on July 12 , 1986 by Random House , Through a Glass Darkly landed on The New York Times Best Seller list . Critical reception was largely mixed , with reviewers focusing on the novel 's prose and attention to historical detail . It has been translated into more than ten languages . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 99,652 | text | {} |
latex-00013497 | e^\alpha=\pm\frac{H^2}{2}[ r^2 c_2^2-(1+H^2r^2)c_1^2], | latex | OleehyO/latex-formulas | cleaned_formulas | train | 13,525 | latex_formula | {"original_latex": "\\begin{align*}e^\\alpha=\\pm\\frac{H^2}{2}\\left[ r^2 c_2^2-(1+H^2r^2)c_1^2\\right],\\end{align*}"} |
normal-00037985 | In stores , Yummy Fur was often wrapped in plastic with " adults only " labels on it . It is not known if Ed or Yummy Fur were banned from any stores , but Diamond , the largest American comics distributor , stopped carrying it for a time in 1988 . A publisher discovered that boxes of its feminist publication were line... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 91,802 | text | {} |
normal-00015454 | Nash has been honoured for his contributions to various philanthropic causes . In 2006 , he was named by Time as one of the 100 most influential people in the world . He was appointed to the Order of Canada in 2007 and invested to the order in 2016 , and was awarded an Honorary Doctor of Laws by the University of Victo... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 37,033 | text | {} |
normal-00044055 | On September 14 , 2004 , Daley and union leaders — including the president of Chicago 's police union — unveiled a monument by Chicago artist Mary Brogger , a fifteen @-@ foot speakers ' wagon sculpture echoing the wagon on which the labor leaders stood in Haymarket Square to champion the eight @-@ hour day . The bronz... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 106,691 | text | {} |
latex-00007201 | \begin{gather*} R=\sqrt{i_1^2-4i_2},z=\frac{R}{i_1},a=\frac{1+z}{1-z}, a=e^\xi,s=\sinh\xi, c=\cosh\xi,t=\tanh\xi.\end{gather*} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 7,210 | latex_formula | {"original_latex": "\\begin{gather*} R=\\sqrt{i_1^2-4i_2},z=\\frac{R}{i_1},a=\\frac{1+z}{1-z}, a=e^\\xi,s=\\sinh\\xi, c=\\cosh\\xi,t=\\tanh\\xi.\\end{gather*}"} |
latex-00023090 | \tilde{c}^{(1)}_{i,j} = \begin{cases}c^{(1)}_{i,j} & j \leq l_0-1; \\(-1)^{i+j} \frac{2k-2j+l}{l_0} & l_0 \leq j \leq l_1.\end{cases} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 23,258 | latex_formula | {"original_latex": "\\begin{align*}\\tilde{c}^{(1)}_{i,j} = \\begin{cases}c^{(1)}_{i,j} & j \\leq l_0-1; \\\\(-1)^{i+j} \\frac{2k-2j+l}{l_0} & l_0 \\leq j \\leq l_1.\\end{cases}\\end{align*}"} |
normal-00006530 | The extent to which his work was studied at an academic level was demonstrated on Dylan 's 70th birthday on May 24 , 2011 , when three universities organized symposia on his work . The University of Mainz , the University of Vienna , and the University of Bristol invited literary critics and cultural historians to give... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 15,989 | text | {} |
mixed-00038931 | Analyze solutions to a matrix 3x5 matrix with two parameters and find a unique solution I have the linear system (over $\mathbb{R}$) for which I need to find a unique solution:
$$\begin{cases}
4x+8y+7z+3cw = 3b \\
x+2y+2z+cw=b \\
2x+4y+2z+(c-1)w=b
\end{cases}$$
for which the corresponding matrix is:
$$\begin{bmatrix}
4... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 23,058 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2211436", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
mixed-00041577 | Your book is being very careful about the fact that $\sqrt{1-\sin^2\theta} = \cos\theta$ is only true when $\cos \theta \geq 0$. Of course, in this problem it doesn't matter, since $-\frac{\pi}{2} < \theta \leq \frac{\pi}{2}$.
Note that you could just as well have chosen the substitution $x = -3\sin \theta$. It is a go... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,698 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/268830", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 3, "answer_id": 2}} |
mixed-00003997 | Solving the problem : $z z_x + z_y = 0, \quad z(x,0) = x^2$ Exercise :
For the problem :
$$\begin{cases} zz_x + z_y = 0 \\ z(x,0) = x^2\end{cases}$$
derive the solution :
$$z(x,y) = \begin{cases} x^2, \quad y = 0\\ \frac{1+2xy - \sqrt{1+4xy}}{2y^2}, \quad y \neq 0 \; \text{and} \; 1+4xy >0 \end{cases}$$
When do sh... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,025 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2775803", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
normal-00012873 | Nuclear import depends on the importin binding its cargo in the cytoplasm and carrying it through the nuclear pore into the nucleus . Inside the nucleus , RanGTP acts to separate the cargo from the importin , allowing the importin to exit the nucleus and be reused . Nuclear export is similar , as the exportin binds the... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 31,228 | text | {} |
mixed-00001753 | Giving integer images (bis) Prove the statement below (the restriction $x < y < z$ is to avoid apparent uncertainties but the property is valid for all $x, y, z$ really).
$$F(x,y,z) = \frac{(y+z)x^n}{(z-x)(y-x)} +\frac{(z+x)y^n}{(z-y)(x-y)}+\frac{(x+y)z^n}{(x-z)(y-z)} \in \mathbb Z.$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 891 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1241706", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
latex-00021293 | \hat{f}(k)=\frac{\pi}{z}e^{-\frac{|k|^2}{4z}-\frac{i(a\cdot k)}{2z}+\frac{|a|^2}{4z}}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 21,439 | latex_formula | {"original_latex": "\\begin{align*}\\hat{f}(k)=\\frac{\\pi}{z}e^{-\\frac{|k|^2}{4z}-\\frac{i(a\\cdot k)}{2z}+\\frac{|a|^2}{4z}}. \\end{align*}"} |
latex-00013377 | j_{nm}^{\mu}(x_1) \equiv \bar{\psi}_n(x_1) {\gamma}^{\mu} {\psi}_m(x_1). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 13,405 | latex_formula | {"original_latex": "\\begin{align*}j_{nm}^{\\mu}(x_1) \\equiv \\bar{\\psi}_n(x_1) {\\gamma}^{\\mu} {\\psi}_m(x_1).\\end{align*}"} |
mixed-00043057 | Properties of Matrix Product I am studying for my exam next week and the teacher has posted previous exams online.
I have the following question, given:
$A = \begin{bmatrix}4&-2&2\\2&4&-4\\1&1&0\end{bmatrix}$
$u = \begin{bmatrix}1\\3\\2\end{bmatrix}$
Find $A^5*u$ without any calculations...
HINT: Properties of the Matr... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 25,600 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1482548", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
normal-00036454 | On March 3 , 2016 , researchers using Hubble data announced the discovery of the farthest known galaxy to date : GN @-@ z11 . The Hubble observations occurred on February 11 , 2015 , and April 3 , 2015 , as part of the CANDELS / GOODS @-@ North surveys . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 87,927 | text | {} |
normal-00032386 | Following the season , the Rangers did not exercise their option , and traded Fleury 's playing rights to the San Jose Sharks , which entitled the Sharks to a compensatory draft pick if Fleury signed elsewhere . He did so with a two @-@ year , $ 8 @.@ 5 million contract with the Chicago Blackhawks . Two days prior to t... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 77,713 | text | {} |
mixed-00039098 | What is equivalent that logical function?
(A+B) => (B XOR C)
=> is Implication
Options:
BC+!A+!B+!C
!B!C+ABC
B!C+A+!BC
!BC+!ABC
BC+A!B!C
! is NOT
Please tell me how it is solved.
I do not think that this requires knowledge of discrete mathematics.
But I did not find any simple options | mixed | math-ai/StackMathQA | stackmathqa100k | train | 23,170 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2348685", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
latex-00004652 | \int {\cal D}A {\cal D}E \exp( i\int_{w}[{\rm Tr}(F E)+ {\rm Tr}(J E)]), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 4,656 | latex_formula | {"original_latex": "\\begin{align*} \\int {\\cal D}A {\\cal D}E \\exp\\left( i\\int_{w}\\left[{\\rm Tr}(F\\,E)+ {\\rm Tr}(J\\,E)\\right]\\right),\\end{align*}"} |
normal-00008655 | In 1924 Gerard Olivier , a habitually frugal man , told his son that not only must he gain admission to the Central School of Speech Training and Dramatic Art , but he must also gain a scholarship with a bursary to cover his tuition fees and living expenses . Olivier 's sister had been a student there and was a favouri... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 21,239 | text | {} |
normal-00007844 | Liquid or solid xenon nanoparticles can be formed at room temperature by implanting Xe + ions into a solid matrix . Many solids have lattice constants smaller than solid Xe . This results in compression of the implanted Xe to pressures that may be sufficient for its liquefaction or solidification . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 19,223 | text | {} |
latex-00041106 | \pi_{\theta,\psi} \begin{cases} \cong \pi_{\theta,\psi_{t}} & t\in N(E^{\times}) \ \ncong \pi_{\theta,\psi_{t}} & t\notin N(E^{\times}) \end{cases} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 41,702 | latex_formula | {"original_latex": "\\begin{align*} \\pi_{\\theta,\\psi} \\begin{cases} \\cong \\pi_{\\theta,\\psi_{t}} & t\\in N(E^{\\times}) \\\\ \\ncong \\pi_{\\theta,\\psi_{t}} & t\\notin N(E^{\\times}) \\end{cases} \\end{align*}"} |
mixed-00043741 | You decided for any $x$ to match $x^2$, so you are claiming that $x<2$ is identical to $x^2<4$, but you can also match $4x-3$ in the interval $(1,9)$ for all $x$ in the interval $(1,3)$, which makes $x<2$ the same as $4x-3<5$.
That is the problem of defining probability on infinite group. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,019 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2080380", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 1}} |
mixed-00044180 | The idea is to plug in the values of $x$, $y$ and $z$ in $$z = \sqrt{x^2+y^2}.$$
Specifically, by using the given expressions, we get
$$p \cos \phi = \sqrt{p^2\sin^2\phi \cos^2 \theta + p^2\sin^2\theta \sin^2 \phi}$$ $$p \cos\phi = \sqrt{p^2\sin^2 \phi \ (\sin^2 \theta + \cos^2 \theta)} $$
$$p \cos\phi = p \sin \phi$$
... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,288 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2458251", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
mixed-00027912 | What is true is that (assuming always that $y$ is a one-to-one function of $x$ on some interval)
$$ \eqalign{\dfrac{d^2 x}{dy^2} &= \dfrac{d}{dy} \dfrac{dx}{dy} = \dfrac{dx}{dy} \dfrac{d}{dx} \left(\dfrac{dy}{dx}\right)^{-1}\cr
&= - \left(\dfrac{dy}{dx}\right)^{-3} \dfrac{d^2 y}{dx^2} }$$
In order to have $\dfrac{d^2 x... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,318 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1403195", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 4, "answer_id": 0}} |
normal-00020309 | Robert Kilwardby ( c . 1215 – 1279 ) discussed three alternative explanations of the motions of the celestial spheres , rejecting the views that celestial bodies are animated and are moved by their own spirits or souls , or that the celestial bodies are moved by angelic spirits , which govern and move them . He maintai... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 48,697 | text | {} |
mixed-00002819 | Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively
Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively,
i.e.,
$$
8= 5+3 \\
9=3+3+3 \\
10 = 5+5 \\
11 = 5+3+3 \\
12 = 3+3+3+3
$$
So I've started in a couple directions without progress. I think it makes se... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,431 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1909194", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 6, "answer_id": 4}} |
mixed-00047512 | Since
$$\left(\sin{\theta}+\dfrac{1}{\sin{\theta}}\right)^2+\left(\cos{\theta}+\dfrac{1}{\cos{\theta}}\right)^2=5+\dfrac{1}{\sin^2{\theta}}+\dfrac{1}{\cos^2{\theta}}$$
Use Cauchy-Schwarz inequality we have
$$\dfrac{1}{\sin^2{\theta}}+\dfrac{1}{\cos^2{\theta}}\ge\dfrac{(1+1)^2}{\sin^2{\theta}+\cos^2{\theta}}=4$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,326 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1065943", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 5, "answer_id": 0}} |
mixed-00042681 | Your proof isn't wrong per se, but the wording is a bit confusing (and I would probably end up taking a point or two off if I were grading it). Here's how I would write it:
For $n=0$, $$2^{4\cdot0+2}+3^{0+2} = 2^2 + 3^2 = 13.$$
Now assume that $13|(2^{4n+2}+3^{n+2})$ for some integer $n\geqslant0$. Then $2^{4n+2}+3^{n+... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 25,367 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1147659", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 5, "answer_id": 0}} |
mixed-00019612 | How to find the minimum of $f(x)=\frac{4x^2}{\sqrt{x^2-16}}$ without using the derivative?
Find the minimum of function $$f(x)=\frac{4x^2}{\sqrt{x^2-16}}$$ without using the derivative.
In math class we haven't learnt how to solve this kind of problems (optimization) yet. I already know that is solvable using derivat... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 11,245 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2822440", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 8, "answer_id": 6}} |
normal-00019978 | Minnie Island State Park is a pine grown @-@ knoll that is located in the deepest portion of the lake . It is unmarked and undeveloped island that also is Connecticut 's smallest state park at 0 @.@ 88 acres . The island acquired its name when a local man declared it to be Minnie Island , because he took his wife , Min... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 47,878 | text | {} |
mixed-00011438 | Indefinite integral of secant cubed $\int \sec^3 x\>dx$ I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't know how to integrate it myself. I have been trying some substitutions... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,280 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/154900", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "20", "answer_count": 8, "answer_id": 3}} |
latex-00030791 | Y(z) = e^{\frac{n \ell \sigma_3}{2}} R(z) P^{(\infty)}(z) e^{n \xi(z) \sigma_3} w(z)^{-\frac{\sigma_3}{2}}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 31,080 | latex_formula | {"original_latex": "\\begin{align*}Y(z) = e^{\\frac{n \\ell \\sigma_3}{2}} R(z) P^{(\\infty)}(z) e^{n \\xi(z) \\sigma_3} w(z)^{-\\frac{\\sigma_3}{2}}.\\end{align*}"} |
latex-00038177 | \partial \Omega \subset \bigcup_{k=1}^{r}f_{k}^{-1}( 0) . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 38,664 | latex_formula | {"original_latex": "\\begin{align*}\\partial \\Omega \\subset \\bigcup_{k=1}^{r}f_{k}^{-1}\\left( 0\\right) .\\end{align*}"} |
mixed-00024266 | $\begin{align}I&=\int_0^{\infty} \dfrac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx
\end{align}$
Perform the change of variable $x=\dfrac{2y}{1-y^2}$,
$\begin{align}I&=\sqrt{2}\int_0^{1} \dfrac{\arctan\left(\dfrac{2y}{1-y^2}\right)}{\sqrt{1-y^2}}\,dy\\
&=2\sqrt{2}\int_0^{1} \dfrac{\arctan y}{\sqrt{1-y^2}}\,dy\\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,089 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2557405", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 2}} |
normal-00021162 | In 1232 , Teika was advanced at the age of 70 to the court rank of " Gon Chūnagon " ( Acting Middle Counselor . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,687 | text | {} |
latex-00016959 | ({1\over3}d^2-\lambda)[\lambda^2-{9\over 8}\lambda+{1\over 72}]-{4\over 9}(1-\lambda)d^4=0 | latex | OleehyO/latex-formulas | cleaned_formulas | train | 17,001 | latex_formula | {"original_latex": "\\begin{align*}({1\\over3}d^2-\\lambda)[\\lambda^2-{9\\over 8}\\lambda+{1\\over 72}]-{4\\over 9}(1-\\lambda)d^4=0\\end{align*}"} |
mixed-00026130 | The range of $p_1$ is given by $[\alpha, \beta]$ where
$$\beta = \frac{1}{d} + \sqrt{\rho - \frac{\rho}{d} - \frac{1}{d} + \frac{1}{d^2}}$$
and
$$\alpha = \frac{1}{m} + \frac{1}{m}\sqrt{\frac{m\rho - 1}{m-1}}$$
where $m = \lfloor 1/\rho\rfloor + 1$.
Moreover, $p_1 = \beta$ if $p_2 = p_3 = \cdots = p_d = \frac{1}{d} - \... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 15,231 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4633085", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 1}} |
latex-00030986 | k_n = (j_m a + jab) = a (j_m + jb) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 31,277 | latex_formula | {"original_latex": "\\begin{align*} k_n = (j_m a + jab) = a (j_m + jb) \\end{align*}"} |
mixed-00021666 | Let us put $2^x=a,3^x=b$ to remove the indices to improve clarity
So, $8^x=(2^3)^x=(2^x)^3=a^3$ and similarly, $27^x=b^3$
$12^x=(2^2\cdot3)^x=(2^x)^2\cdot3^x=a^2b$ and similarly, $18^x=ab^2$
So, the problem reduces to $$\frac{a^3+b^3}{ab(a+b)}=\frac76$$
$\displaystyle\implies 6(a^2-ab+b^2)=7ab$ as $a+b\ne0$
$\displayst... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 12,509 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/384090", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "47", "answer_count": 3, "answer_id": 1}} |
latex-00017036 | {\cal S}_H\approx { A\over \Delta\sigma} (\Delta\sigma)^{D_H-2} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 17,079 | latex_formula | {"original_latex": "\\begin{align*}{\\cal S}_H\\approx { A\\over \\Delta\\sigma} (\\Delta\\sigma)^{D_H-2}\\end{align*}"} |
mixed-00044177 | Note that you can simplify and substitute in the following way:
$$\begin{array}{lll}
y^2 = 2^5(x^2-14)&&\text{substitute } y = 2^2z \\
z^2 = 2(x^2-14)&&\text{substitute } z = 2a\\
2a^2 = x^2-14&& \\
2(a^2+7) = x^2&&\text{substitute } x = 2b\\
a^2+7 = 2b^2&&\text{substitute } a = 2c+1\\
2(c^2+c + 2) = b^2&&\text{substit... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,286 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2457305", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 2, "answer_id": 1}} |
latex-00011946 | L^{'}={1\over 2}\{ {{{\dot x}^2}\over e}+em^2-i(\xi_{\mu}{\dot \xi}^{\mu}-\xi_5{\dot \xi}_5)-i\varphi ({{\xi_{\mu}{\dot x}^{\mu}}\over e}-m\xi_5) \}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 11,970 | latex_formula | {"original_latex": "\\begin{align*}L^{'}={1\\over 2}\\{ {{{\\dot x}^2}\\over e}+em^2-i(\\xi_{\\mu}{\\dot \\xi}^{\\mu}-\\xi_5{\\dot \\xi}_5)-i\\varphi ({{\\xi_{\\mu}{\\dot x}^{\\mu}}\\over e}-m\\xi_5) \\}, \\end{align*}"} |
normal-00018260 | At age 13 , Michelle created , produced , designed , and directed a dance show at the YWCA in New Orleans . She was named Miss New Orleans in 1960 . She attended Rivers Frederick Junior High School where her principal , Leah McKenna , encouraged her to pursue a career in entertainment . While in high school , Michelle ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 43,757 | text | {} |
normal-00001471 | Nicole later decided to let Marilyn and Sid Walker ( Robert Mammone ) adopt the baby upon its birth . Marilyn was desperate to mother a child and her obsession with that and her controlling behaviour became too much for Nicole . Of the situation , Symons said " Marilyn is in Nicole 's face every minute . She 's doing i... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 3,774 | text | {} |
mixed-00032759 | Bonferroni Inequalities
Let $k$ and $m$ be positive integers with $k>m$.
Then the partial sums of
$$
1-\binom{k}{1} + \binom{k}{2} - \cdots (-1)^m\binom{k}{m}
$$
has alternating signs.
(The partial sums of the given sum are $P_1=1$, $P_2=1-\binom{k}{1}$, $P_3=1-\binom{k}{1}+\binom{k}{2}$, etc)
I arrived at the abo... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 19,260 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1217459", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
normal-00042344 | Chambersburg is part of small town America . Recreation includes hunting , sports events such as baseball games at Henninger Field , and high school football games . The town also hosts a professional football team , the Chambersburg Cardinals , that plays in the Gridiron Developmental Football League . People in the a... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 102,590 | text | {} |
latex-00042529 | x_1x_2x_3 -G_1 G_\infty x_1 -G_2 G_\infty x_2 + G_\infty^2=0. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 43,137 | latex_formula | {"original_latex": "\\begin{align*}x_1x_2x_3 -G_1 G_\\infty x_1 -G_2 G_\\infty x_2 + G_\\infty^2=0.\\end{align*}"} |
normal-00048219 | The red rail was long known only from a few contemporary descriptions referring to red " hens " and names otherwise used for grouse or partridges in Europe , as well as the sketches of the travellers Pieter van den Broecke and Sir Thomas Herbert from 1617 and 1634 . These were thought to depict separate species of bird... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 117,229 | text | {} |
latex-00028114 | \gamma_1= \frac{1}{2} \sum_{i=0}^{n_1}\sum_{j=0}^{n_2} |a_{i,j}| i(i-1), \gamma_2 = \frac{1}{2} \sum_{i=0}^{n_1}\sum_{j=0}^{n_2} |a_{i,j}| j(j-1). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 28,370 | latex_formula | {"original_latex": "\\begin{align*}\\gamma_1= \\frac{1}{2} \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} |a_{i,j}| i(i-1), \\gamma_2 = \\frac{1}{2} \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} |a_{i,j}| j(j-1).\\end{align*}"} |
latex-00018093 | \widetilde{\gamma}_{\xi} = \frac{1-\overline{a}\xi}{\xi-a} \frac{1-a\xi}{\xi-\overline{a}} \gamma_{\xi}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 18,152 | latex_formula | {"original_latex": "\\begin{align*}\\widetilde{\\gamma}_{\\xi} = \\frac{1-\\overline{a}\\xi}{\\xi-a}\\, \\frac{1-a\\xi}{\\xi-\\overline{a}}\\, \\gamma_{\\xi}.\\end{align*}"} |
latex-00016530 | \begin{array}{ccccccccccccccccc}&&1&&&&15&&&&15&&&&1&&\\\end{array} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 16,568 | latex_formula | {"original_latex": "\\begin{align*}\\begin{array}{ccccccccccccccccc}&&1&&&&15&&&&15&&&&1&&\\\\\\end{array}\\end{align*}"} |
normal-00007554 | Throughout Mexico , Ingrid killed 32 people , mostly due to flooding and mudslides . The two storms collectively killed at least 192 people and caused $ 75 billion pesos ( MXN , $ 5 @.@ 7 billion USD ) . Manuel was responsible for majority of the overall effects , although Ingrid still left an estimated $ 20 billion pe... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 18,433 | text | {} |
latex-00032303 | \int_{\mathbb{R}_{+}^{n}}\frac{t^{2}dzdt}{[(1+t)^{2}+|z|^{2}]^{n-1}}=\frac{2\omega_{n-1}I_{n-1}^{n-2}}{(n-2)(n-3)(n-4)} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 32,618 | latex_formula | {"original_latex": "\\begin{align*}\\int_{\\mathbb{R}_{+}^{n}}\\frac{t^{2}dzdt}{\\left[(1+t)^{2}+|z|^{2}\\right]^{n-1}}=\\frac{2\\omega_{n-1}I_{n-1}^{n-2}}{(n-2)(n-3)(n-4)}\\end{align*}"} |
normal-00028209 | During the alpha , designer Dean Hall became part of Bohemia Interactive , and the mod , retitled to Arma II : DayZ Mod , was officially released on February 21 , 2013 . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 67,601 | text | {} |
mixed-00014164 | $$5^{k+1}+5=5\times 5^k+5<5(5^{k+1}-5)+5=5^{k+2}-25+5<5^{k+2}$$
Notice that, by the induction step, we have $$5^k+5<5^{k+1}\implies 5^k<5^{k+1}-5$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,933 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2333991", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 5, "answer_id": 3}} |
normal-00025663 | Childermass meanwhile indirectly receives the message meant for him by Strange and uses it to break the enchantment over Lady Pole . Enraged by this the gentleman with the thistledown hair first murders Vinculus forcing Stephen Black to watch . Then after begins to place a second curse upon Lady Pole that would lead to... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 61,542 | text | {} |
normal-00000343 | Aerobatic competitions take place for both powered aircraft and gliders , with up to 30 events each year in the UK and Ireland . Starting at the Beginner level , pilots can move up to Standard ( powered aircraft ) or Sports ( glider ) levels , and then on to Intermediate , Advanced , and finally Unlimited classes . Eac... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 838 | text | {} |
latex-00013571 | S^2 =S^-S^+ +[S^z+1/2]_q ^2 -[1/2]_q ^2. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 13,599 | latex_formula | {"original_latex": "\\begin{align*}S^2 =S^-S^+ +[S^z+1/2]_q ^2 -[1/2]_q ^2.\\end{align*}"} |
normal-00005142 | After the 1995 Interim Agreement on the West Bank and the Gaza Strip between the Palestinian National Authority ( PNA ) and Israel , Jifna was placed in " Area B " . Thus , its administrative and civil affairs were transferred to the PNA , while security matters remained in Israeli control . Throughout the ongoing Seco... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 12,369 | text | {} |
normal-00028210 | DayZ attempts to portray a realistic scenario within the gameplay , with the environment having different effects on the player . A character may receive bone fractures from damage to their legs , go into shock from bullet wounds or zombie bites , receive infections from zombies or diseased players , or faint due to lo... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 67,605 | text | {} |
normal-00004220 | Innis also tried to defend universities from political and economic pressures . He believed that independent universities , as centres of critical thought , were essential to the survival of Western civilization . His intellectual disciple and university colleague , Marshall McLuhan , lamented Innis 's premature death ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 10,016 | text | {} |
normal-00031127 | Lydford Castle was involved in the English Civil War that broke out in 1642 between the Royalist supporters of Charles I and Parliament . The castle was used by the Royalist commander Sir Richard Grenville as his main military prison in the region . It had a terrible reputation amongst Parliamentarians , who complained... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 74,729 | text | {} |
normal-00049462 | Not only did the iTunes Store staff recommend The Garden of Words , it was selected as the Year 's Best Animation in iTunes ' Best of 2013 . In addition to the high acclaim on iTunes , The Garden of Words won the 2013 Kobe Theatrical Film Award . At the 2013 Fantasia International Film Festival , it shared the Satoshi ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 120,358 | text | {} |
mixed-00029568 | Finding $C_n=-C_{n-1}+C_{n-2}$ $$\begin{cases}
C_n=-C_{n-1}+C_{n-2}\\
C_0=1\\
C_1=0
\end{cases} $$
$
\begin{pmatrix}
C_n \\
C_{n-1}
\end{pmatrix}
\begin{pmatrix}
-1 && 1 \\
1 && 0
\end{pmatrix}^n
\begin{pmatrix}
1\\
0
\end{pmatrix}$
$\begin{vmatrix}
\lambda+1 && -1 \\
-1 && \lambda
\end{vmatrix}=\lambda^2+\lambda... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,325 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2823828", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
mixed-00022044 | We have to evaluate the definite integral:
$$\int_0^{1/\sqrt{2}}\frac{\arccos x}{\sqrt{1-x^2}} \ dx$$
Just use u-substitution. Let:
$$u=\arccos x \implies du=-dx$$
$$\int_0^{1/\sqrt{2}}u \ -du=-\int_0^{1/\sqrt{2}}u \ du=\left.-\frac{u^2}{2}\right|_0^{1/\sqrt{2}}$$
Substituting $\arccos x$ back for $u$:
$$\int_0^{1/\sqr... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 12,741 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/721735", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 3}} |
latex-00012503 | A_\mu\to A_\mu+\frac{1}{e}\partial_\mu \Lambda | latex | OleehyO/latex-formulas | cleaned_formulas | train | 12,530 | latex_formula | {"original_latex": "\\begin{align*}A_\\mu\\to A_\\mu+\\frac{1}{e}\\partial_\\mu \\Lambda\\end{align*}"} |
normal-00000053 | Guard , office , and police duties . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 109 | text | {} |
mixed-00018332 | Set $\sin(2x) = u$
$7\cos(2x) - \sin(2x) + 5 = 0$
$\implies 7\sqrt{1-u^2} = u-5$
$\implies 49 - 49u^2 = u^2 - 10u + 25$
$\implies 50u^2 - 10u -24 = 0$
$\implies u_1 = 0.8, u_2 = -0.6$
$ \sin(2x) = 0.8$ or $\sin(2x) = -0.6$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,470 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1701363", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 9, "answer_id": 5}} |
mixed-00014798 | $$1,\frac{1}{3},\frac{1}{3^2},\frac{1}{3^3},\frac{1}{3^4} \ldots$$
You need to calculate $$S_{\infty}-S_5$$
$$=\frac{1}{1-\frac{1}{3}}-\frac{1(1-\frac{1}{3^5})}{1-\frac{1}{3}}$$
$$=\frac{\frac{1}{3^5}}{\frac{2}{3}}$$
$$=\frac{1}{2\cdot{3^4}}$$
$$=\frac{1}{162}$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,318 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2899412", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 2}} |
mixed-00044903 | Showing matrix $\left[\begin{smallmatrix} 4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3 \end{smallmatrix}\right]$ is positive definite $$\begin{bmatrix}
x_1 & x_2 & x_3
\end{bmatrix}\begin{bmatrix}
4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3
\end{bmatrix}\begin{bmatrix}
x_1\\x_2\\x_3
\end{bmatrix} = \\4x_1^2 + x_1x_2 + x_3x_1 + x_2x_... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,741 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3185577", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00033166 | F @-@ Zero was widely lauded by game critics for its graphical realism , and has been called the fastest and most fluid pseudo @-@ 3D racing game of its time . This has been mostly credited to the development team 's pervasive use of the " Mode 7 " system . Eurogamer 's Tom Bramwell commented " this abundance of Mode 7... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 79,976 | text | {} |
latex-00038704 | J_0(ax) J_0(bx) &= \int\limits_0^{\infty} \!dt t J_0(tx)\!\int\limits_0^{\infty} \!dy y J_0(ay) J_0(by) J_0(ty) , \\J_0(cx) J_0(dx) &= \int\limits_0^{\infty} \!dq q J_0(qx)\!\int\limits_0^{\infty} \!dz z J_0(cz) J_0(dz) J_0(qz) , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 39,194 | latex_formula | {"original_latex": "\\begin{align*}J_0(ax) J_0(bx) &= \\int\\limits_0^{\\infty} \\!dt \\,t J_0(tx)\\!\\int\\limits_0^{\\infty} \\!dy y J_0(ay) J_0(by) J_0(ty) \\;, \\\\J_0(cx) J_0(dx) &= \\int\\limits_0^{\\infty} \\!dq \\, q J_0(qx)\\!\\int\\limits_0^{\\infty} \\!dz z J_0(cz) J_0(dz) J_0(qz) \\;,\\end{align*}"} |
normal-00007626 | As stated in the preface , there is a code hidden in the paintings throughout the book . This was an idea of Brett Helquist and Balliett 's editor , Tracy Mack . The code involves images of pentominoes and frogs , which is a recurring theme in the book . To decode the code , one must count the number of frogs in every ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 18,644 | text | {} |
normal-00044490 | On April 8 , 2015 , Ross signed a one @-@ year deal with the Oakland Athletics for the league minimum salary of $ 507 @,@ 500 , with the Diamondbacks paying the remainder of his salary with them . He was released on May 3 , 2015 . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 107,810 | text | {} |
latex-00009592 | A_d = \frac{q^{-\frac{1}{2}\chi( d, d )}}{(q^{-1})_d} x_d, A_e^{\sigma} = \frac{q^{-\frac{1}{2}\mathcal{E}(e)}}{(q^{-1})^{\sigma}_e} \xi_e. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,609 | latex_formula | {"original_latex": "\\begin{align*}A_d = \\frac{q^{-\\frac{1}{2}\\chi( d, d )}}{(q^{-1})_d} x_d, \\;\\;\\;\\;\\;\\; \\;\\;\\;\\;\\;\\;\\;A_e^{\\sigma} = \\frac{q^{-\\frac{1}{2}\\mathcal{E}(e)}}{(q^{-1})^{\\sigma}_e} \\xi_e.\\end{align*}"} |
latex-00046374 | a_{2n}=(w_1w_2w_3)^{1-3n}Q_{2n}(w_1, w_2, w_3, w_1', w_2', w_3', \dots, w_1^{(2n)}, w_2^{(2n)}, w_3^{(2n)} ), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 47,022 | latex_formula | {"original_latex": "\\begin{align*}a_{2n}=(w_1w_2w_3)^{1-3n}Q_{2n}\\left(w_1, w_2, w_3, w_1', w_2', w_3', \\dots, w_1^{(2n)}, w_2^{(2n)}, w_3^{(2n)}\\right ),\\end{align*}"} |
normal-00015030 | For at least a portion of its life , a star shines due to thermonuclear fusion of hydrogen into helium in its core , releasing energy that traverses the star 's interior and then radiates into outer space . Almost all naturally occurring elements heavier than helium are created by stellar nucleosynthesis during the sta... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 36,088 | text | {} |
latex-00005817 | F_{p+2} = dA_{p+1}+ \ldots . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 5,825 | latex_formula | {"original_latex": "\\begin{align*}F_{p+2} = dA_{p+1}+ \\ldots .\\end{align*}"} |
mixed-00016808 | How to solve the irrational system of equations? Solve the system of equations $$\begin{cases} \sqrt{x+2y+3}+\sqrt{9 x+10y+11}=10,&\\[10pt] \sqrt{12 x+13y+14}+\sqrt{28 x+29y+30}=20. \end{cases} $$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,553 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/402116", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00041969 | Then in December 2012 , the Canadian government announced that the purchase of the F @-@ 35 had been abandoned due to greatly escalating costs , and that a fresh procurement process would begin . In January 2013 , Dassault responded to a request for information from the Canadian government and announced its readiness t... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 101,819 | text | {} |
normal-00048213 | A music label owner , Reynold D ’ Silva , bought the flat for £ 550 @,@ 000 in 2002 , beating a rival bid from Noel Gallagher . D ’ Silva considered opening it as a music museum , but decided to rent it instead , for £ 795 per week . Author Miles contended that 34 Montagu Square " clearly qualifies as a candidate for o... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 117,215 | text | {} |
mixed-00010287 | I'll suppose you use the usual euclidian distance between two points. Then the distance between $A$ and $P$ is
$$d(A, P) =\sqrt{(x-5)^2+(y-16)^2}$$
Same goes for the distance between $B$ and $P$.
$$d(B, P)=\sqrt{(x-(-4))^2+(y-4)^2}$$
Since $d(A, P)=D(B, P)$, we have
$$\sqrt{(x-5)^2+(y-16)^2} =2\times\sqrt{(x+4)^2+(y-4)... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,589 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3522446", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
latex-00037987 | &R_s(\gamma)= a \lambda ( 1 - D_s^R(\gamma) ) - b \lambda W_s(\gamma) - d \lambda D_s^R(\gamma). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 38,472 | latex_formula | {"original_latex": "\\begin{align*}&R_s(\\gamma)= a \\lambda ( 1 - D_s^R(\\gamma) ) - b \\lambda W_s(\\gamma) - d \\lambda D_s^R(\\gamma). \\end{align*}"} |
normal-00015156 | Like the other characters of the series , Perry was structured in a simple style to allow young viewers to easily draw him . In keeping with the show 's general design scheme , Perry is constructed of geometric shapes in a style reminiscent of deceased Looney Tunes animator Tex Avery . Povenmire uses different design s... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 36,319 | text | {} |
normal-00031448 | The Nisshōki flag is designated as the national flag in the Law Regarding the National Flag and National Anthem , which was promulgated and became effective on August 13 , 1999 . Although no earlier legislation had specified a national flag , the sun @-@ disc flag had already become the de facto national flag of Japan ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 75,454 | text | {} |
normal-00018826 | Verbal consonantal roots are placed into verbal conjugations patterns , known as בנינים binyanim in Hebrew ; the binyanim mainly serve to indicate grammatical voice . This includes various distinctions of reflexivity , passivity , and causativity . Verbs of all binyanim have three non @-@ finite forms ( one participle ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 45,206 | text | {} |
latex-00003856 | S [ \phi ] = \int {\rm d^d} x [ {1 \over 2} ( \partial_{\mu} \phi )^2 + {1 \over 2} m^2 \phi^2 ] - n \ln{ [ {\lambda \over 4} \int {\rm d^d} x \phi^4 ] }, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 3,859 | latex_formula | {"original_latex": "\\begin{align*}S \\left[ \\phi \\right] = \\int {\\rm d^d} x \\left[ {1 \\over 2} \\left( \\partial_{\\mu} \\phi \\right)^2 + {1 \\over 2} m^2 \\phi^2 \\right] - n \\ln{ \\left[ {\\lambda \\over 4} \\int {\\rm d^d} x \\ \\phi^4 \\right] },\\end{align*}"} |
mixed-00011220 | $3\times3$ determinant using standard basis I am trying to get from a $2\times2$ determinant to a $3\times3$ determinant.
$$\left|\begin{array}{c1 c2 c3}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
$$
How does one get to
$$ \det(A)=\sum_{j=1}^3 a_{j1} \; \det(e... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,154 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4607157", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
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