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-00007355 | \partial_t u - i \Lambda(\partial_x) u - i \Sigma(u) = - T_V \partial_x u + \mathcal{N}_u + \mathcal{R}_u, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 7,364 | latex_formula | {"original_latex": "\\begin{align*}\\partial_t u - i \\Lambda(\\partial_x) u - i \\Sigma(u) = - T_V \\partial_x u + \\mathcal{N}_u + \\mathcal{R}_u,\\end{align*}"} |
normal-00024413 | Similarly to other Brazilian institutions , Unicamp is composed of several semi @-@ autonomous teaching units , designated as schools and institutes . Each unit is headed by a director from the faculty , equivalent to a dean , elected by the faculty and student representatives . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 58,577 | text | {} |
mixed-00007526 | $$\begin{eqnarray*}\color{red}{I}&=&\int_{0}^{\pi}\frac{dx}{a^2\cos^2 x+b^2\sin^2 x}=2\int_{0}^{\pi/2}\frac{dx}{a^2\cos^2 x+b^2\sin^2 x}\\&=&2\int_{0}^{+\infty}\frac{dt}{(1+t^2)\left(a^2\frac{1}{1+t^2}+b^2\frac{t^2}{t^2+1}\right)}=2\int_{0}^{+\infty}\frac{dt}{a^2+b^2 t^2}\\&=&\frac{2}{ab}\int_{0}^{+\infty}\frac{du}{1+u... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 3,946 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1193700", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 2}} |
normal-00020429 | The purpose of the new line is to allow regional and express trains to run directly between Asker Station , Sandvika Station and Lysaker Station , without being slowed and delayed by commuter trains that make frequent stops at intermediate stations . The Asker Line will improve regularity , and capacity will increase f... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 48,973 | text | {} |
normal-00038698 | The album received mostly positive reviews . AllMusic 's review stated the album " is the sound of a musical and lyrical maturity that normally doesn 't occur until a band 's third or fourth albums " . The review by Yahoo ! Music expressed that the album " has its share of anthemic moments , but the real passion spills... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 93,697 | text | {} |
normal-00004933 | Zhuchengceratops - ( Zhucheng , China ) | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 11,864 | text | {} |
latex-00001417 | \gamma ^{(1)a}\equiv \pi ^a-\pi ^{(1)a}\approx 0. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 1,418 | latex_formula | {"original_latex": "\\begin{align*}\\gamma ^{(1)a}\\equiv \\pi ^a-\\pi ^{(1)a}\\approx 0.\\end{align*}"} |
latex-00030289 | a=\frac{2\mu+3\lambda}{4(\lambda+\mu)}, b=\frac{4(\lambda+\mu)}{2\mu+\lambda}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 30,577 | latex_formula | {"original_latex": "\\begin{align*} a=\\frac{2\\mu+3\\lambda}{4(\\lambda+\\mu)}, b=\\frac{4(\\lambda+\\mu)}{2\\mu+\\lambda},\\end{align*}"} |
latex-00000581 | s(x)=x^2-T'x+D'p^{u-2v}\mathrm{with}\frac{ds(x)}{dx}=2x-T'. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 581 | latex_formula | {"original_latex": "\\begin{align*}s(x)=x^2-T'x+D'p^{u-2v}\\mathrm{with}\\frac{ds(x)}{dx}=2x-T'.\\end{align*}"} |
normal-00013972 | Thieu then authorized Operation Cuu Long , in which ARVN ground forces , including mechanized and armoured units , drove west and northwest up the eastern side of the Mekong River from 9 May – 1 July . A combined force of 110 Vietnamese Navy and 30 U.S. vessels proceeded up the Mekong to Prey Veng , permitting IV Corps... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 33,704 | text | {} |
latex-00024583 | \sum_{h=0}^\infty \frac{A(p^{2h+1},1)}{(p+1)p^{sh-1}}=\frac{p}{p+1}\frac{1+p^sA(p,1)}{p^s+A(1,p)}\sum_{h=0}^\infty \frac{A(p^{2h},1)}{p^{sh}}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 24,760 | latex_formula | {"original_latex": "\\begin{align*}\\sum_{h=0}^\\infty \\frac{A(p^{2h+1},1)}{(p+1)p^{sh-1}}=\\frac{p}{p+1}\\frac{1+p^sA(p,1)}{p^s+A(1,p)}\\sum_{h=0}^\\infty \\frac{A(p^{2h},1)}{p^{sh}},\\end{align*}"} |
latex-00036931 | < u^{n+1},v > + \Delta t < (\frac{(u^{n+1/2})^2}{2})_{x}, v > + \Delta t < \mathcal{H}(u^{n+1/2})_{x}, v_{x} > = < u^{n}, v >, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 37,409 | latex_formula | {"original_latex": "\\begin{align*}\\left< u^{n+1},v \\right> + \\Delta t \\left< \\left(\\frac{\\left(u^{n+1/2}\\right)^2}{2}\\right)_{x}, v \\right> + \\Delta t \\left< \\mathcal{H}\\left(u^{n+1/2}\\right)_{x}, v_{x} \\right> = \\left< u^{n}, v \\right>,\\end{align*}"} |
latex-00006349 | \Gamma_{\Lambda}[\phi]=-\frac{1}{2}\varphi\cdot\Delta^{-1}_{\Lambda}\cdot\varphi - W_{\Lambda}[J] +J\cdot\varphi , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 6,357 | latex_formula | {"original_latex": "\\begin{align*}\\Gamma_{\\Lambda}[\\phi]=-\\frac{1}{2}\\varphi\\cdot\\Delta^{-1}_{\\Lambda}\\cdot\\varphi - W_{\\Lambda}[J] +J\\cdot\\varphi\\;, \\end{align*}"} |
normal-00027887 | M @-@ 137 is a state trunkline highway in the US state of Michigan that serves as a spur route to the Interlochen Center for the Arts and Interlochen State Park . It starts at south of the park and runs north between two lakes in the area and through the community of Interlochen to US Highway 31 ( US 31 ) in Grand Trav... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 66,844 | text | {} |
mixed-00039811 | Computing $\;\lim\limits_{x\to 4}\left (\frac{\frac{\pi}{6} - \arcsin\left(\frac{\sqrt{x}}{4}\right)}{\sqrt[3]{2x-7}-1}\right) $ without L'Hôpital? $$
\lim_{x\to 4}\left(\frac{\frac{\pi}{6} - \arcsin\left(\frac{\sqrt{x}}{4}\right)}{\sqrt[3]{2x-7}-1}\right)
$$
Hello! I need to solve this limit. I had solved it with the ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 23,604 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2995641", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
mixed-00045198 | We have
\begin{eqnarray*}
\tan^{-1}(A) + \tan^{-1}(B) + \tan^{-1}(C) = \tan^{-1} \left( \frac{A+B+C-ABC}{1-AB-BC-CA} \right).
\end{eqnarray*}
If the RHS is to give $ \pi/2$ then we require the denominator to be zero, so
\begin{eqnarray*}
1-2x( \sqrt{1+x^2} -x) -(\sqrt{1+x^2} -x)^2=0
\end{eqnarray*}
Which is easily ve... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,921 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3465969", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "6", "answer_count": 4, "answer_id": 1}} |
latex-00038963 | D_y \psi (x_0, y_0, t_0)=\phi'(s_0, t_0)\frac{\gamma'_0(1)}{2s_0}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 39,454 | latex_formula | {"original_latex": "\\begin{align*}D_y \\psi (x_0, y_0, t_0)=\\phi'(s_0, t_0)\\frac{\\gamma'_0(1)}{2s_0}, \\end{align*}"} |
mixed-00048322 | HINT:
Avoid squaring as it immediately introduces extraneous root.
Using $\sin x=\sin2\cdot\dfrac x2=2\sin\dfrac x2\cos\dfrac x2$
$$1+2\sin x=2\left(\sin\dfrac x2+\cos\dfrac x2\right)$$
$$\iff\left(2\sin\dfrac x2-1\right)\left(2\cos\dfrac x2-1\right)=0$$
Hope you can take it from here! | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,814 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1728211", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
normal-00005408 | Captain Jonathan Archer ( Scott Bakula ) , Sub @-@ Commander T 'Pol ( Jolene Blalock ) , and Commander Charles " Trip " Tucker III ( Connor Trinneer ) fly down to a small colony of 76 miners in order to trade for deuterium . They initially try to barter with Tessic ( Larry Cedar ) , the colony 's leader , but he appear... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 13,061 | text | {} |
latex-00018305 | F_1=[& -\det\Delta\cdot(\Delta^{-1}(\partial F_i/\partial x_1))\\& -\det\Delta\cdot(\Delta^{-1}(\partial F_i/\partial x_0))]. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 18,365 | latex_formula | {"original_latex": "\\begin{align*}F_1=[& -\\det\\Delta\\cdot(\\Delta^{-1}(\\partial F_i/\\partial x_1))\\\\& -\\det\\Delta\\cdot(\\Delta^{-1}(\\partial F_i/\\partial x_0))].\\end{align*}"} |
mixed-00005096 | Inverse Laplace Transform via Circuit Analysis [HELP] Inverse Laplace Transform $\frac{1}{s^2 + \sqrt{2}s + 1}$
so what I did it changed the denominator to complete the square format which is $\left(s+\frac{\sqrt{2}}{2}\right)^2 + \frac{1}{2}$, then I can solve for $s$, it will make it as
$$
\left(\left(s+ \frac{\sqrt{... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,585 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3781963", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 3, "answer_id": 1}} |
latex-00033372 | d:=\inf \{d_t:t_0\leq t\leq t_1\} . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 33,703 | latex_formula | {"original_latex": "\\begin{align*}d:=\\inf\\,\\left\\{d_t:t_0\\leq t\\leq t_1\\right\\}\\,.\\end{align*}"} |
latex-00040943 | \omega_k(y,\lambda) = \beta_k \frac{\sinh (\bar k_{\nu}(h-y))}{\sinh(\bar k_{\nu} h)}, \beta_k(\lambda)= i b k w_k(0, \lambda) , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 41,479 | latex_formula | {"original_latex": "\\begin{align*} \\omega_k(y,\\lambda) = \\beta_k \\frac{\\sinh (\\bar k_{\\nu}(h-y))}{\\sinh(\\bar k_{\\nu} h)}, \\beta_k(\\lambda)= i b k w_k(0, \\lambda) ,\\end{align*}"} |
latex-00000823 | \sqrt{2} M_{j} = \pm \sum_{i} \frac{\nu_{i}\phi(a_{i})}{a_{i}^{2}-m_{j}^{2}}\mbox{det}(a_{i}^{2} - m^{2})^{1/r_{i}} \prod_{j\ne i} (a_{j}^2-a_{i}^2)^{1-2r_{j}/r_{i}} \Lambda_{N=2}^{\frac{4(N_{c}-1)-2N_{f}}{r_{i}}}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 823 | latex_formula | {"original_latex": "\\begin{align*}\\sqrt{2} M_{j} = \\pm \\sum_{i} \\frac{\\nu_{i}\\phi(a_{i})}{a_{i}^{2}-m_{j}^{2}}\\mbox{det}(a_{i}^{2} - m^{2})^{1/r_{i}} \\prod_{j\\ne i} (a_{j}^2-a_{i}^2)^{1-2r_{j}/r_{i}} \\Lambda_{N=2}^{\\frac{4(N_{c}-1)-2N_{f}}{r_{i}}}.\\end{align*}"} |
normal-00033541 | Since 2005 , the General Market ( 1883 ) and the adjacent Fish Market and Red House buildings ( 1898 ) , part of the Victorian complex of the Smithfield Market , have been facing a threat of demolition . The City of London Corporation , ultimate owners of this property has been engaged in public consultation to assess ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 80,900 | text | {} |
latex-00000615 | S_{L}= \int d^{2}\sigma \sqrt{\hat{g}}(\frac{1}{2} {\hat{g}}^{ab}\partial_{a}\eta \partial_{b}\eta +\hat{R}\eta +\mu e^{\eta}), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 615 | latex_formula | {"original_latex": "\\begin{align*}S_{L}= \\int d^{2}\\sigma \\sqrt{\\hat{g}}(\\frac{1}{2} {\\hat{g}}^{ab}\\partial_{a}\\eta \\partial_{b}\\eta +\\hat{R}\\eta +\\mu e^{\\eta}),\\end{align*}"} |
normal-00049684 | Measuring 16 by 12 millimetres ( 0 @.@ 63 by 0 @.@ 47 in ) , the eggs are oval , smooth and lustreless white , with small spots or blotches of red on the larger end . Clutch size is reported to be two or three eggs . While there is no reliable information on incubation and feeding , it is believed that both parents are... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 120,973 | text | {} |
mixed-00016697 | Wolfram gave an easier derivative.
$$
\dfrac{d}{dx}\left(\frac{(1-k)x\log(x^2-x)}{(1-k')(x-1)\log x^2}\right) = \dfrac{((k-1) ((2 x-1) \log(x^2)-\log((x-1) x) (\log(x^2)+2 x-2)))}{((k'-1) (x-1)^2 \log^2(x^2))}
$$
To see this result visit this link.
Setting the numerator of the derivative equal to $0$,
$$
(k-1) ((2 x-... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,483 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/321220", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
mixed-00002807 | We need $$\frac{a+3}{a}\in\Bbb{Z}\ , \frac{a-3}{a}\in\Bbb{Z}$$ or $$1+\frac{3}{a}\in\Bbb{Z}\ , \ 1-\frac{3}{a}\in\Bbb{Z}$$ thus $\displaystyle \frac{3}{a}\in\Bbb{Z}$, means that $\displaystyle a=\frac{3}{m}$ where $m\in\Bbb{Z}$.
Now we can write the equation as $$\frac{3}{m}x^2+\left(\frac{3}{m}+3\right)x+\frac{3}{m}-3... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,424 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1903026", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
latex-00009914 | 2(2n)!(\frac{2}{\pi})^{2n+1}(1+\frac{1}{3^{2n+1}})(1-\frac{1}{5^{2n+1}})(1+\frac{1}{7^{2n+1}})\ldots= E_n^* = E_{2n}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,935 | latex_formula | {"original_latex": "\\begin{align*}2(2n)!\\left(\\frac{2}{\\pi}\\right)^{2n+1}\\left(1+\\frac{1}{3^{2n+1}}\\right)\\left(1-\\frac{1}{5^{2n+1}}\\right)\\left(1+\\frac{1}{7^{2n+1}}\\right)\\ldots= E_n^* = E_{2n},\\end{align*}"} |
latex-00036781 | \zeta_c^d q \phi_{k,m}^{+}(c,d;\tau) &= \frac{1}{2\pi i} \int_{|s|=r} \alpha^+_{m,c}(s) \frac{e^s \zeta_c^d q}{1-e^s \zeta_c^d q} ds \ &= -\frac{1}{2\pi i} \int_{|s|=r} \alpha^+_{m,c}(s) \sum_{n\geq 0} (e^s \zeta_c^d q)^{-n} ds, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 37,259 | latex_formula | {"original_latex": "\\begin{align*} \\zeta_c^d q \\, \\phi_{k,m}^{+}(c,d;\\tau) &= \\frac{1}{2\\pi i} \\int_{|s|=r} \\alpha^+_{m,c}(s) \\frac{e^s \\zeta_c^d q}{1-e^s \\zeta_c^d q} ds \\\\ &= -\\frac{1}{2\\pi i} \\int_{|s|=r} \\alpha^+_{m,c}(s) \\sum_{n\\geq 0} \\left(e^s \\zeta_c^d q\\right)^{-n} ds,\\end{align*}"} |
mixed-00049957 | Distributing $60$ identical balls into $4$ boxes if each box gets at least $4$ balls, but no box gets $20$ or more balls How many different ways can the balls be placed if each box gets at least $4$ balls each, but no box gets $20$ or more balls?
I was thinking about finding all the possible ways which every box gets a... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,816 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3182428", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
latex-00012737 | W(T,U) \stackrel{\Gamma^o(3)_{T}}{\rightarrow} W(T,U)(i \gamma T+ \delta)+i\gamma (\partial_{U} \eta^{-2}(U)) \eta^{-2}(\frac{T}{3}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 12,764 | latex_formula | {"original_latex": "\\begin{align*}W(T,U) \\stackrel{\\Gamma^o(3)_{T}}{\\rightarrow} W(T,U)(i \\gamma T+ \\delta)+i\\gamma (\\partial_{U} \\eta^{-2}(U)) \\eta^{-2}(\\frac{T}{3}).\\end{align*}"} |
mixed-00049262 | Find the integral $\int_{0}^{1} f(x)dx$ for $f(x)+f(1-{1\over x})=\arctan x\,,\quad \forall \,x\neq 0$. Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x)+f\left(1-{1\over x}\right)=\arctan x\,,\quad \forall \,x\neq 0$$
Find $$\int_{0}^1 f(x)\,dx$$
My Attempt :
Replace $x$ by $1/x$ in given equation
$$f\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,382 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2514789", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "12", "answer_count": 3, "answer_id": 2}} |
latex-00027839 | Y_i = \frac{1}{m_{i,i}}(Y_{i-1}M-m_{i-2,i}Y_{i-2}-m_{i-1,i}Y_{i-1}-R_{i}), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 28,091 | latex_formula | {"original_latex": "\\begin{align*} Y_i = \\frac{1}{m_{i,i}}(Y_{i-1}M-m_{i-2,i}Y_{i-2}-m_{i-1,i}Y_{i-1}-R_{i}),\\end{align*}"} |
normal-00011546 | A somewhat different DAG @-@ based formulation of scheduling constraints is used by the program evaluation and review technique ( PERT ) , a method for management of large human projects that was one of the first applications of DAGs . In this method , the vertices of a DAG represent milestones of a project rather than... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 28,141 | text | {} |
latex-00024552 | \begin{aligned}\widehat{2\rho}:\{\pm 1\}&\rightarrow Z(\widehat{G}_{\mathrm{sc}})\rightarrow Z(\widehat{G})\\-1&\mapsto \prod_{\alpha\in \Phi^{+}}\widehat{\alpha}(-1).\end{aligned} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 24,726 | latex_formula | {"original_latex": "\\begin{align*}\\begin{aligned}\\widehat{2\\rho}:\\{\\pm 1\\}&\\rightarrow Z(\\widehat{G}_{\\mathrm{sc}})\\rightarrow Z(\\widehat{G})\\\\-1&\\mapsto \\prod_{\\alpha\\in \\Phi^{+}}\\widehat{\\alpha}(-1).\\end{aligned}\\end{align*}"} |
mixed-00028218 | A slightly different change of variables which simplifies a lot the boundary conditions :
$\begin{cases}
s=y-x\\
p=xy\\
\end{cases}$
The Jacobian is : $-\begin{vmatrix}
-1 & 1 \\
y & x \\
\end{vmatrix}^{-1}=\frac{1}{y+x}$
$(y^2-x^2)\frac{1}{y+x}=y-x=s$
$\iint_{1 \le xy \le 4, 0 \le y-x \le 3}(y^2-x^2)e^{xy}dxdy ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,505 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1630357", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
normal-00024140 | The group traveled to Italy in early 1818 . They first surrendered Allegra to Byron in April , much to Claire 's distress . In August , Percy Shelley took Claire to see Allegra , at her request . Percy arrived in Venice , telling Byron he had the entire family . Mary was thus summoned to Venice to substantiate Percy 's... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 57,917 | text | {} |
mixed-00028757 | Since
$$ \sec[..]^2 = 1+ \tan[..]^2 $$
we directly have identities
$$ \sec^{-1}x = \tan^{-1}\sqrt {x^2 - 1} $$
and
$$ \tan ^{-1} x= \sec^{-1}\sqrt {1 + x^2} $$
Also plot between $ \sec^{-1}...,\, \tan^{-1}. $
Note the $ \pi/4,\pi $ intercept on $y$, and period respy.Proper sign to be taken. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,827 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2082980", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 3}} |
latex-00019853 | h: \xi_g\mapsto h^{*} (\xi_g) := \prod_{k\in I_g^c} h^{-1}_k \cdot \xi_g | latex | OleehyO/latex-formulas | cleaned_formulas | train | 19,940 | latex_formula | {"original_latex": "\\begin{align*}h: \\xi_g\\mapsto h^{*} (\\xi_g) := \\prod_{k\\in I_g^c} h^{-1}_k\\,\\cdot \\xi_g\\end{align*}"} |
latex-00033799 | Af:= -\Delta f. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 34,143 | latex_formula | {"original_latex": "\\begin{align*}Af:= -\\Delta f.\\end{align*}"} |
latex-00025592 | ~ p_{\ell-1}^{(k-1)}(x) = m_{k-1,\ell-1}'(x) q_{\ell-1}^{(k-1)*}(x). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 25,786 | latex_formula | {"original_latex": "\\begin{align*}~ p_{\\ell-1}^{(k-1)}(x) = m_{k-1,\\ell-1}'(x) q_{\\ell-1}^{(k-1)*}(x).\\end{align*}"} |
normal-00025210 | Colin Hall Simpson was born in St Kilda , Victoria , on 13 April 1894 , the son of Colin Simpson , a plumber , and his wife Elizabeth Fulton Simpson , née Jordan . He was educated at St Kilda Primary School , and , from 1911 , at Caulfield Grammar School . While at Caulfield Grammar , Simpson joined the Australian Army... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 60,453 | text | {} |
latex-00040903 | f_k = \Re (x + iy)^k, g_k = \Im (x + iy)^k = \Re (-i(x+iy)^k), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 41,415 | latex_formula | {"original_latex": "\\begin{align*} f_k = \\Re (x + iy)^k, g_k = \\Im (x + iy)^k = \\Re (-i(x+iy)^k),\\end{align*}"} |
mixed-00038590 | How to determine the matrix in the following case Say we have a vector $\textbf{b}$ and $\textbf u$ such that:
$$A \mathbf b= \mathbf u$$
Where $A$ is a square matrix.
If $\mathbf b$ and $\mathbf u$ are known and $A$ is the unknown, How to get the matrix $A$ (perhaps it is not unique but how can one proceed to get it)... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,852 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1891041", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 3, "answer_id": 2}} |
mixed-00030749 | I think I might have figured it out:
Consider the function
\begin{align*}
g(x,y) = \frac{y}{2\sqrt{1-x^y}}.
\end{align*}
This function is clearly increasing (thanks @Eric) in $x$ for all $y\in(0,1)$, so it suffices to set $x=3/4$.
Thus, define
\begin{align*}
f(y) = \log \left( \frac{y}{2\sqrt{1-(3/4)^y}} \right)
\end... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 18,039 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4090928", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
mixed-00012712 | To bound all solutions
in integers
(not just to positive integers)
to
$x^3+y^3 = n$:
First,
using the factorization
$x^3+y^3
=(x+y)(x^2-xy+y^2)
$,
we get
possible values
for $x+y$
since $(x+y) | n$.
Then,
since
$x^2-xy+y^2
=(x+y)^2-3xy
$,
we get possible values
for $xy$
($xy
=\dfrac{(x+y)^2-n/(x+y)}{3}
$)
and
this give... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,051 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1168613", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 3, "answer_id": 1}} |
mixed-00022515 | Find the general solution of the following first order differential equation $\frac{dy}{dx}=\frac{8x^3+3x^4}{y^4}$ I have multiplied both sides by $y^4$
which gives me $\frac{dy}{dx}y^4=x^3(3x+8)$
Then do I integrate both sides with respect to x?
$\int\frac{dy}{dx}yx^4dx=\int{x^3(3x+8)dx}$
Am I still on the right track... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,021 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1113944", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
normal-00042649 | The eruption could be seen from Naples . Different perspectives and the damage caused to the local villages were recorded by USAAF photographers and other personnel based nearer to the volcano . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 103,200 | text | {} |
normal-00014437 | It was revealed on 14 June 2013 that Edwina Hart , Minister for Economy , Science and Transport in the WAG supported the completion the Eastern Bay Link Road . She said that the link road would ; improve access to Cardiff Bay , improve access to the Cardiff Central Enterprise Zone and enhance connections within the Car... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 34,770 | text | {} |
normal-00037120 | Apollo is operated by its managing partners , Leon Black , Joshua Harris and Marc Rowan and a team of more than 250 investment professionals , as of March 31 , 2013 . The firm 's headquarters are located in the Solow Building at 9 West 57th Street in New York City , and the firm operates additional offices in Purchase ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 89,636 | text | {} |
mixed-00005508 | Determine the Galois group of the splitting field of $f(x) = x^4+x+t \in F_2(t)[x]$ I used Gauss's lemma to show that the polynomial is irreducible since it is irreducible in $F_2[t,x]$, used the derivative GCD test to conclude that the polynomial is separable, and finally started by taking the quotient $F_2(t)[x]/(f(x... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,796 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4175057", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 1, "answer_id": 0}} |
mixed-00044217 | Finding the limit of a function . How can I calculate the following limit:
\begin{equation*}
\lim_{x \rightarrow a}
\frac{\sqrt{x} - \sqrt{a} + \sqrt{x-a} }{\sqrt{x^2 - a^2}}
\end{equation*}
I feel that I should multiply by the conjugate, but which conjugate? | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,312 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2503609", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
normal-00029367 | If investigations confirm NSCLC , the stage is assessed to determine whether the disease is localized and amenable to surgery or if it has spread to the point where it cannot be cured surgically . CT scan and positron emission tomography are used for this determination . If mediastinal lymph node involvement is suspect... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 70,409 | text | {} |
latex-00049525 | y ~>~ \frac{1}{1 + \frac{1}{a_{n + 1}}} ~\geq~ \frac{1}{1 + \frac{1}{a_n - 1}} ~=~ 1 - \frac{1}{a_n} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 50,218 | latex_formula | {"original_latex": "\\begin{align*}y ~>~ \\frac{1}{1 + \\frac{1}{a_{n + 1}}} ~\\geq~ \\frac{1}{1 + \\frac{1}{a_n - 1}} ~=~ 1 - \\frac{1}{a_n}\\end{align*}"} |
normal-00019156 | In June 2015 , astronomers reported evidence for Population III stars in the Cosmos Redshift 7 galaxy at z = | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 45,939 | text | {} |
latex-00006652 | \Sigma (p,m)=\frac{1}{\pi }\int_{-\infty }^{\infty }\frac{dw}{\gamma \cdotp-w+i\epsilon }\Im \Sigma (w,m), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 6,660 | latex_formula | {"original_latex": "\\begin{align*}\\Sigma (p,m)=\\frac{1}{\\pi }\\int_{-\\infty }^{\\infty }\\frac{dw}{\\gamma \\cdotp-w+i\\epsilon }\\Im \\Sigma (w,m),\\end{align*}"} |
mixed-00013078 | You solved the problem in the nonnegative integers rather than the positive integers.
You wish to determine the number of solutions of the equation
$$x_1 + x_2 + x_3 = 15 \tag{1}$$
in the positive integers subject to the constraints $x_1 < 6$ and $x_2 > 6$.
Let's deal with the constraint $x_2 > 6$ first. Let $y_2 = x... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,269 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1483343", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 2}} |
latex-00047999 | \beta_i(U)=\beta_i((y))+\beta_i(U/(y)). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 48,664 | latex_formula | {"original_latex": "\\begin{align*}\\beta_i(U)=\\beta_i((y))+\\beta_i(U/(y)).\\end{align*}"} |
normal-00009252 | Recently , residential construction has been concentrated around Kiryat Haim and Kiryat Shmuel , with 75 @,@ 000 m2 ( 807 @,@ 293 sq ft ) of new residential construction between 2002 – 2004 , the Carmel , with 70 @,@ 000 m2 ( 753 @,@ 474 sq ft ) , and Ramot Neve Sha 'anan with approximately 70 @,@ 000 m2 ( 753 @,@ 474 ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 22,647 | text | {} |
mixed-00015854 | I have doubts that you correctly compute the result, since the value $IK$ given in comments is incorrect. The correct result is:
$$\frac{DK}{DI}=\frac23.
$$
The details are given below.
Let $x,y,z$ being the distances from the vertices $A,B,C$ to the tangent points of the incircle. From the equations $x+y=c, y+z=a, z+... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,961 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4025888", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 0}} |
mixed-00028808 | Approximation of a summation by an integral I am going to approximate $\sum_{i=0}^{n-1}(\frac{n}{n-i})^{\frac{1}{\beta -1}}$ by $\int_{0}^{n-1}(\frac{n}{n-x})^{\frac{1}{\beta -1}}dx$, such that $n$ is sufficiently large.
*
*Is the above approximation true?
*If the above approximation is true, by which theorem or met... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,858 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2114574", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "11", "answer_count": 3, "answer_id": 1}} |
mixed-00000196 | You deal with the sum of functions, $f(x) = \frac{x}{2}$ and $g(x)= \frac{1}{4} \sin(2 x)$. So you would use linearity of the derivative:
$$
\frac{d}{d x} \left( f(x) + g(x) \right) = \frac{d f(x)}{d x} + \frac{d g(x)}{d x}
$$
To evaluate these derivatives, you would use $\frac{d}{d x}\left( c f(x) \right) = c \frac{d... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 102 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/134855", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 6, "answer_id": 1}} |
normal-00047960 | Svalbard Airport , Longyear , is the main airport serving the Svalbard archipelago . It is located on the south shore of Isfjord , with high terrain to the south , southeast and east . It has a single , 10 / 28 runway ( roughly east – west ) which is 2 @,@ 140 metres ( 7 @,@ 020 ft ) long . The airport has an elevation... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 116,427 | text | {} |
mixed-00012618 | using power expansion to find limit I am preparing for my final exam, and stuck on this question.
Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x)
-\sin(x)}{x^2-x\ln(1+x)}$$
I have no idea how to proceed. Any help would be highly appreciated! | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,995 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1068329", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
mixed-00049513 | Let r be the prime such that
$p^3 + 19q^3 + 2018 = r^3$
Since 2 is the only even number prime but r does not equal to 2 (trivial), by observing the even-odd of the LHS and RHS, either $p=2$ or $q=2$ .
If $p = 2$, $2^3 + 19q^3 + 2018 = r^3$
Let $s^3 = 19q^3$
Since 19 is not a perfect cube, s or $s^2$ cannot be a ration... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,536 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2750600", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
normal-00020900 | Louis Pasteur and Edward Jenner were the first to develop vaccines to protect against viral infections . The nature of viruses remained unknown until the invention of the electron microscope in the 1930s , when the science of virology gained momentum . In the 20th century many diseases both old and new were found to be... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,092 | text | {} |
mixed-00014298 | find the total number of possible way to reach to a particular sum suppose you have given a sum like : 5.
we have to find the total number of possible way to reach to 5.
for example
1 + 1 + 1 + 1 + 1 = 5
2 + 1 + 1 + 1 = 5
1 + 2 + 1 + 1 = 5
1 + 1 + 2 + 1 = 5
1 + 1 + 1 + 2 = 5
2 + 2 + 1 = 5
2 + 1 + 2 = 5
1 + 2 + 2 = 5
1 ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,016 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2426624", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 0}} |
normal-00014812 | By 1972 , the Portuguese military had changed its strategy , adapting the British / American search and destroy operations utilising small shock troop sweeps . They also initiated a hearts and minds campaign , named the Aldeamentos Programme , which was a forced relocation program . But on November 9 , 1972 , FRELIMO –... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 35,576 | text | {} |
latex-00036523 | \Big(\frac{1}{\lambda(x)}\Big)^{1/2}= \frac{1}{\sqrt {n r}}\big(1+O(n^{-1/r})\big). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 36,892 | latex_formula | {"original_latex": "\\begin{align*} \\Big(\\frac{1}{\\lambda(x)}\\Big)^{1/2}= \\frac{1}{\\sqrt {n r}}\\big(1+O\\left(n^{-1/r}\\right)\\big).\\end{align*}"} |
latex-00011088 | \dim W\cap W' = \dim W+\dim W'-\dim(W+W'), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 11,111 | latex_formula | {"original_latex": "\\begin{align*}\\dim W\\cap W' = \\dim W+\\dim W'-\\dim(W+W'),\\end{align*}"} |
normal-00037580 | Before 1832 , Sleaford was in the Lincolnshire parliamentary constituency , which encompassed all of the county except for four boroughs . In the 1818 election , 49 of the 2 @,@ 000 people living in New and Old Sleaford and Quarrington qualified to vote . In 1832 , the Reform Act widened the franchise and divided Linco... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 90,809 | text | {} |
normal-00039842 | The following is a list of primitive Pythagorean triples with values less than 100 : | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 96,588 | text | {} |
normal-00008121 | Prakash Raj and Ashish Vidyarthi were cast as the film 's primary antagonists . Raj played a mafia kingpin and Vidyarthi played a corrupt police officer , a villain 's role he finds more fun to play than that of a hero . Sayaji Shinde and Nassar played the two other principal characters in the film . Jyothi Rana played... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 19,894 | text | {} |
normal-00022048 | Hugh supported the building campaign of Salisbury Cathedral , ordering that money be collected throughout his diocese . Likewise , he ordered similar collections for Daventry Priory , Sulby Abbey and parish churches in his diocese . Not only churches benefited from these sorts of collections , as the bishop offered ind... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 52,979 | text | {} |
normal-00043260 | The siege attempts were viewed as a debacle in Boston , and the expedition 's leaders were jeered upon their return . Port Royal was captured in 1710 by a larger force that included British Army troops ; that capture marked the end of French rule in peninsular Acadia . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 104,806 | text | {} |
mixed-00012432 | By AM-GM and C-S we obtain:
$$\sum_{cyc}\frac{ab}{\sqrt{2c+a+b}}=\sum_{cyc}\frac{ab\sqrt{\frac{3}{8}}\cdot2\sqrt{\frac{8}{3}(2c+a+b)}}{2(2c+a+b)}\leq\sqrt{\frac{3}{32}}\sum_{cyc}\frac{ab\left(\frac{8}{3}+2c+a+b\right)}{2c+a+b}=$$
$$=\sqrt{\frac{1}{96}}\sum_{cyc}\frac{ab(4(a+b+c)+6c+3a+3b)}{2c+a+b}=\sqrt{\frac{1}{96}}\s... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,879 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/915719", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
latex-00002419 | \hat{H}_0 = -\frac{1}{2} \partial_x^2 + \frac{g}{x^2} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 2,420 | latex_formula | {"original_latex": "\\begin{align*}\\hat{H}_0 = -\\frac{1}{2} \\partial_x^2 + \\frac{g}{x^2}\\end{align*}"} |
mixed-00000712 | HINT:
Putting $x=r\cos\theta,y=r\sin\theta$
$$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1,$$
$$r^2=\frac{a^2b^2}{b^2\cos^2\theta+a^2\sin^2\theta}=b^2\frac{\sec^2\theta}{\frac{b^2}{a^2}+\tan^2\theta}$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 364 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/493104", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 5, "answer_id": 2}} |
mixed-00019767 | There is no need to do any algebra.
Consider
$$
\frac{2x + 6} {(x + 2)^2}- \frac{2} {x + 2}
=
\frac{a} {(x + 2)^2}
$$
Since this equality holds for all values of $x\ne -2$,
set $x=0$ and get
$$
\frac{6} {2^2}- \frac{2} {2}
=
\frac{a} {2^2}
$$
which gives $a=2$. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 11,342 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2966822", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
normal-00041596 | Between Fremont Junction and the junction of SR @-@ 24 near Green River , Interstate 70 crosses a geologic feature called the San Rafael Swell . The construction of the freeway through the swell is listed as one of the engineering marvels of the Interstate Highway System , with one engineer claiming this section as " o... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 100,866 | text | {} |
mixed-00016889 | Find the limit: $\lim\limits_{x\to1}\dfrac{x^{1/5}-1}{x^{1/3}-1}$ Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$
How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,605 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/464426", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "8", "answer_count": 8, "answer_id": 1}} |
mixed-00020124 | I have to solve this initial value problem and determine where the solution attains its minimum value. This is the differential equation : $y'= 4y^2 + xy^2 , y(0)=1.$
I was able to find the solution $y$ for this equation which is :
$$y=\frac{-2}{8 x + (x^2 - 2)},$$
but I don't know how to determine where the solution a... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 11,567 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3398982", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00020338 | {\mathcal{C}}_{\alpha} = B_1^{(\alpha)} \otimes L + B_2^{(\alpha)} \otimes (-2I_{m}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,469 | latex_formula | {"original_latex": "\\begin{align*}{\\mathcal{C}}_{\\alpha} = B_1^{(\\alpha)} \\otimes L + B_2^{(\\alpha)} \\otimes (-2I_{m}).\\end{align*}"} |
latex-00016073 | ds^2_{TN}(m_1,x_{1}) = (1+ \frac{4m_{1}}{r_{1}})^{-1} (d\psi_{1} + \cos\theta_{1})^2 + (1+ \frac{4m_{1}}{r_{1}})(dr_{1}^2 + r_{1}^2 d\Omega_{2}^2), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 16,111 | latex_formula | {"original_latex": "\\begin{align*}ds^2_{TN}(m_1,x_{1}) = (1+ \\frac{4m_{1}}{r_{1}})^{-1} (d\\psi_{1} + \\cos\\theta_{1})^2 + (1+ \\frac{4m_{1}}{r_{1}})(dr_{1}^2 + r_{1}^2 d\\Omega_{2}^2),\\end{align*}"} |
mixed-00029283 | Solve $P(z)=0$, over complex field and factorise $P(z)=0$ over real field $P(z)=3z^4+10z^3+6z^2+10z+3$
The roots are $z=-3, -1/3, i,-i$ but I couldn't find $i,-i$ as the root.
Also the factorised version is meant to be $(z+3)(3z+1)(z^2+1)$ but I got something fabulous like $z^2(z+\frac{1}{z})(3z+\frac{3}{z}-10)$
I feel... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,152 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2505312", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
mixed-00008400 | Find $a^{2013} + b^{2013} + c^{2013}$ Problem Statement
Let $f(x) = x^3 + ax^2 + bx + c$ and $g(x) = x^3 + bx^2 + cx + a$ where $a,b,c$ are integers with $c\not=0$
Suppose that the following conditions hold:
*
*$f(1)=0$
*the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$
$$\text{Find the value of} \: \... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 4,477 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1887190", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
normal-00021051 | A tropical depression is the lowest category that the Japan Meteorological Agency uses and is the term used for a tropical system that has wind speeds not exceeding 33 knots ( 38 mph ; 61 km / h ) . A tropical depression is upgraded to a tropical storm should its sustained wind speeds exceed 34 knots ( 39 mph ; 63 km /... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,435 | text | {} |
mixed-00042058 | How to prove $(n!)^4\le2^{n^2+n}$? This may sound like a newbie but
question is to show that
$$(n!)^4\le2^{n^2+n} for \quad n=1,2,3...$$
I know it is true for n=1. $(1!)^4\le2^2$
and
assume it is true for $1<m\le n$ for all $\quad m\in N$
we have to show for m=n+1.
$((n+1)!)^2\le^? 2^{(n+1)^2+n+1}$
$((n+1)!)^4=(n!)^4.... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,986 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/619240", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
mixed-00036586 | There is a formula. I'm not too sure how to prove it, but I know that there is a formula where you can denest $$\sqrt{\sqrt[3]{\alpha}+\sqrt[3]{\beta}}$$Into$$\pm\frac {1}{\sqrt{f}}\left(-\frac {s^2\sqrt[3]{\alpha^2}}{2}+s\sqrt[3]{\alpha\beta}+\sqrt[3]{\beta^2}\right)$$ where $$f=\beta-s^3\alpha$$ and $s$ is a real num... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 21,611 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/194030", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "19", "answer_count": 4, "answer_id": 1}} |
normal-00031411 | Dr. Derek Shepherd ( Patrick Dempsey ) and Dr. Richard Webber ( James Pickens , Jr . ) finish Dr. Preston Burke 's ( Isaiah Washington ) surgery to remove a pseudo @-@ aneurysm in the subclavian artery that threatened the functioning of his arm and which was caused by a gunshot wound . At the same time , Dr. Erica Hahn... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 75,359 | text | {} |
latex-00034948 | \begin{aligned}{\dot R} &= \int_{\mathbb T} \cos(\theta - \phi) \partial_t \rho ~ d\theta \\&= KR \int_{\mathbb T} \cos(\theta - \phi) \partial_\theta \Big[ \rho(\theta, t) \sin(\theta - \phi) \Big] ~ d\theta \\&= KR \int_{\mathbb T} \sin^2(\theta - \phi) \rho(\theta, t) ~ d\theta.\end{aligned} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 35,308 | latex_formula | {"original_latex": "\\begin{align*}\\begin{aligned}{\\dot R} &= \\int_{\\mathbb T} \\cos(\\theta - \\phi) \\partial_t \\rho ~ d\\theta \\\\&= KR \\int_{\\mathbb T} \\cos(\\theta - \\phi) \\partial_\\theta \\Big[ \\rho(\\theta, t) \\sin(\\theta - \\phi) \\Big] ~ d\\theta \\\\&= KR \\int_{\\mathbb T} \\sin^2(\\theta - \\... |
normal-00037938 | Billy Caldwell " Sauganash " , who served as an interpreter for the Indian Agents , was the honoree of the hotel . Born in approximately 1780 , " Sauganash " was an Indian half @-@ breed , whose father was Colonel Caldwell , an Irish officer in the British Army stationed at Detroit ; his mother was a Pottawatomi . He w... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 91,713 | text | {} |
normal-00049208 | Hardy made his first wrestling appearance after being released from WWE at an OMEGA show , on May 24 . Using his old gimmick , " Willow the Wisp " , Hardy challenged Krazy K for the OMEGA Cruiserweight Championship , but lost the match . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 119,802 | text | {} |
latex-00024189 | \lim \limits_{k\rightarrow \infty }\|p^{k}-y^{k}\|=\lim \limits_{k\rightarrow \infty }\|p^{k}-y^{k-1}\|=\lim \limits_{k\rightarrow \infty }\|x^{k}-x^{k-1}\|=0, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 24,360 | latex_formula | {"original_latex": "\\begin{align*} \\lim \\limits_{k\\rightarrow \\infty }\\|p^{k}-y^{k}\\|=\\lim \\limits_{k\\rightarrow \\infty }\\|p^{k}-y^{k-1}\\|=\\lim \\limits_{k\\rightarrow \\infty }\\|x^{k}-x^{k-1}\\|=0,\\end{align*}"} |
latex-00048071 | \textrm{Var}_{\{Y\in B_1\}}[X]=\ldots=\textrm{Var}_{\{Y\in B_k\}}[X]. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 48,737 | latex_formula | {"original_latex": "\\begin{align*}\\textrm{Var}_{\\{Y\\in B_1\\}}[X]=\\ldots=\\textrm{Var}_{\\{Y\\in B_k\\}}[X].\\end{align*}"} |
mixed-00023748 | Angle between two lines explanation I gave two lines $q_1 = 2x - y + 2 = 0$ and $q_2 = x + 2y - 3 = 0$.
They have vectors $n_1 = (2,1)$ and $n_2 = (1,2)$.
When I have to find angle between them I must apply that formula:
$$\cos \theta = \frac{2\cdot 1 + (-1)\cdot2}{\sqrt{4 + 1}\sqrt{1 + 4}} = 0 \Rightarrow \theta = ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,769 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2119507", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
normal-00044929 | Baden @-@ Powell House was built to Tubbs ' design by Harry Neal Ltd , for which they received the 1961 Gold Medal of the Worshipful Company of Tylers and Bricklayers . At the opening , the house received the building design award for ' The building of most merit in London.' | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 109,086 | text | {} |
latex-00033909 | \int_{0}^{r} s_1^{-(2N-2)} & \Big(\int_{B(x,s_1)} u^{-(4N-1)} dy \Big)ds_1 \\&= - r^{-(2N-2)} \int_{\partial B(x,r)} (-\Delta)^{N-1} u d\sigma + \omega_{2N-1} (-\Delta)^{N-1} u(x). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 34,258 | latex_formula | {"original_latex": "\\begin{align*}\\int_{0}^{r} s_1^{-(2N-2)} & \\Big(\\int_{B(x,s_1)} u^{-(4N-1)} dy \\Big)ds_1 \\\\&= - r^{-(2N-2)} \\int_{\\partial B(x,r)} (-\\Delta)^{N-1} u d\\sigma + \\omega_{2N-1} (-\\Delta)^{N-1} u(x).\\end{align*}"} |
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