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-00023488 | \omega_0(t)=Z_0(t)-Y e^{\lambda t}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 23,658 | latex_formula | {"original_latex": "\\begin{align*}\\omega_0(t)=Z_0(t)-Y e^{\\lambda t}.\\end{align*}"} |
normal-00005291 | All songs written and composed by Sufjan Stevens and published by New Jerusalem Music , ASCAP . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 12,781 | text | {} |
latex-00044886 | X(bc) = X(b)c + (-1)^{p(b)p(c)}X(c)b - X(1)bc. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 45,522 | latex_formula | {"original_latex": "\\begin{align*}X(bc) = X(b)c + (-1)^{p(b)p(c)}X(c)b - X(1)bc.\\end{align*}"} |
normal-00004659 | Originally , the light had a fifth @-@ order Fresnel lens , but a fourth @-@ order Fresnel lens was installed in May 1902 , just three months into its operation . The light characteristic was a fixed white light with a red flash every 15 seconds . In 1972 , the light was automated and the Fresnel lens was replaced with... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 11,118 | text | {} |
latex-00024943 | \mathcal{E}(\omega_1, \omega_2)=(dG^{\omega_1}, dG^{\omega_2})_1=\int_M G^{\omega_1} \wedge \omega_2. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 25,134 | latex_formula | {"original_latex": "\\begin{align*}\\mathcal{E}(\\omega_1, \\omega_2)=(dG^{\\omega_1}, dG^{\\omega_2})_1=\\int_M G^{\\omega_1} \\wedge \\omega_2.\\end{align*}"} |
normal-00005080 | A sequel , DJ Hero 2 , was officially announced in June 2010 for release in the last quarter of 2010 , featuring more than 70 mashups from over 85 artists . The game includes several new gameplay modes , including an " Empire " career mode , head @-@ to @-@ head DJ battles , social multiplayer modes , and a jump @-@ in... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 12,183 | text | {} |
latex-00021725 | K(u) &= ( \frac{\sin{(u/2)}}{u} )^2 (1+\cos{(\theta+u\log{(e\log{N}\log_{2}{N})})}) \ &+( \frac{\sin{(u/2)}}{u} )^2 (1+\cos{(\theta+u\log{(e^{3/2}\log{N}\log_{2}{N})})}) \ &+( \frac{\sin{(u/2)}}{u} )^2 (1+\cos{(\theta+u\log{(e^{2}\log{N}\log_{2}{N})})}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 21,874 | latex_formula | {"original_latex": "\\begin{align*}K(u) &= \\left( \\frac{\\sin{(u/2)}}{u} \\right)^2 (1+\\cos{(\\theta+u\\log{(e\\log{N}\\log_{2}{N})})}) \\\\ &+\\left( \\frac{\\sin{(u/2)}}{u} \\right)^2 (1+\\cos{(\\theta+u\\log{(e^{3/2}\\log{N}\\log_{2}{N})})}) \\\\ &+\\left( \\frac{\\sin{(u/2)}}{u} \\right)^2 (1+\\cos{(\\theta+u\\l... |
latex-00006419 | \epsilon_s(z,l_s)=-(-1)^{2l_s}(c_sz+d_s)\varepsilon_s(g_s(z))-\varepsilon_s(z). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 6,427 | latex_formula | {"original_latex": "\\begin{align*}\\epsilon_s(z,l_s)=-(-1)^{2l_s}(c_sz+d_s)\\varepsilon_s\\left(g_s(z)\\right)-\\varepsilon_s(z).\\end{align*}"} |
mixed-00005273 | We have
$$
\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} = \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n}\sqrt{n+1}} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n}\sqrt{n+1}(\sqrt{n+1} + \sqrt{n})} \sim \frac{1}{2n^{3/2}}.
$$
Hence $a_n = n^{\alpha} \Bigl( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \Bigr) \sim \fra... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,675 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3947113", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00025643 | f_1(x_1,\dots,x_m) = \dots = f_p(x_1,\dots,x_m) = 0. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 25,839 | latex_formula | {"original_latex": "\\begin{align*}f_1(x_1,\\dots,x_m) = \\dots = f_p(x_1,\\dots,x_m) = 0. \\end{align*}"} |
normal-00025057 | The Ersatz Yorck @-@ class ships were protected with Krupp cemented steel armor , as was the standard for German warships of the period . The armor layout was identical to the preceding Mackensen class , which was itself very similar to the armor scheme on the preceding Derfflinger @-@ class ships . They had an armor b... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 60,058 | text | {} |
mixed-00045700 | Integrate as follows
\begin{align}
\int {\dfrac{1}{(x^2+9)^2}}\,dx
&= \frac19 \int {\dfrac{1}{x^2+9}}\,dx - \frac19\int {\dfrac{x^2}{(x^2+9)^2}}\,dx\\
&= \frac19 \int {\dfrac{1}{x^2+9}}\,dx + \frac1{18}\int x\>d({\dfrac{1}{x^2+9}})\\
&= \frac1{18} \frac x{x^2+9} + \frac1{18}\int {\dfrac{1}{x^2+9}}\,dx\\
&= \frac1{18... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 27,233 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4064701", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
mixed-00010253 | At the bottom, write $ x^2 + 1/x^2 = (x + 1/x)^2 -2$. Then assume $ x+ 1/x = t$. Then you're done.
Let me right the detailed solution. Divide the numerator and denominator of the integrand by $x^2.$ Then we have $$ \frac{1-1/x^2}{(x+1/x)^2 - 1}$$. Then do the substitution business. Then just integrate the following nic... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,568 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3484051", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
normal-00031835 | The film was declared to be the " most satisfactory picture Denver has had in many months " by the Paris Theater in the September 29 issue of Motography . Though the October 6 edition of Motography noted that W. A. Roderick and F. O. Brown , officers of Paris Theater Company were charged for showing the film which viol... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 76,278 | text | {} |
latex-00042078 | T_{2,r}=-\frac{\theta_3(q)^4}{2}-\frac{\theta_4(q)^4}{2}=-\frac{1}{2}\frac{\eta(q^2)^{20}}{\eta(q)^8\eta(q^4)^8}-\frac{1}{2}\frac{\eta(q)^8}{\eta(q^2)^4} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 42,680 | latex_formula | {"original_latex": "\\begin{align*}T_{2,r}=-\\frac{\\theta_3(q)^4}{2}-\\frac{\\theta_4(q)^4}{2}=-\\frac{1}{2}\\frac{\\eta(q^2)^{20}}{\\eta(q)^8\\eta(q^4)^8}-\\frac{1}{2}\\frac{\\eta(q)^8}{\\eta(q^2)^4}\\end{align*}"} |
normal-00049405 | dabulp @-@ pa ga @-@ ya nu @-@ naw @-@ ma | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 120,228 | text | {} |
normal-00017715 | 'In this same tyme was Ser Herry Spenser a grete werrioure in Ytaile , or the tyme that he was promoted.' | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 42,468 | text | {} |
latex-00046748 | (K_{|x|^{2k-2}} f)(z) &= \int_{\Sigma} | z - x |^{2k-2} f(x) \mathrm{d} x , z \in \Sigma \\(K_{|x|^{2k-2} \ln |x| } f)(z) &= \int_{\Sigma} | z - x|^{2k-2} \ln |z -x| f(x) \mathrm{d} x , z\in \Sigma | latex | OleehyO/latex-formulas | cleaned_formulas | train | 47,402 | latex_formula | {"original_latex": "\\begin{align*}(K_{|x|^{2k-2}} f)(z) &= \\int_{\\Sigma} | z - x |^{2k-2} \\; f(x) \\ \\mathrm{d} x , z \\in \\Sigma \\\\(K_{|x|^{2k-2} \\ln |x| } f)(z) &= \\int_{\\Sigma} | z - x|^{2k-2} \\ln |z -x| \\; f(x) \\ \\mathrm{d} x , z\\in \\Sigma\\end{align*}"} |
mixed-00046564 | $\overline{xyz} =x!+y!+z!$ how to find all of 3-digit integer number $\overline{xyz}$ such that:
$\overline{xyz} =x!+y!+z!$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 27,757 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/327352", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "11", "answer_count": 3, "answer_id": 2}} |
latex-00011706 | Y^{\pm}_{J,n}=cV_{J,n}e^{\sqrt{2}(1\mp J)\phi} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 11,730 | latex_formula | {"original_latex": "\\begin{align*}Y^{\\pm}_{J,n}=cV_{J,n}e^{\\sqrt{2}(1\\mp J)\\phi} \\end{align*}"} |
normal-00000792 | With the censorship of Polish theater ( and the virtual end of the Polish radio and film industry ) , underground theaters were created , primarily in Warsaw and Kraków , with shows presented in various underground venues . Beginning in 1940 the theaters were coordinated by the Secret Theatrical Council . Four large co... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 1,939 | text | {} |
mixed-00012655 | $$\begin{cases}
u_o = -1 \\
v_0 = 3 \\
u_{n+1} = u_n + v_n \\
v_{n+1} = -u_n + 3v_n
\end{cases}$$
So:
$$\begin{bmatrix}
u_{n+1} \\
v_{n+1} \\
\end{bmatrix} =
\begin{bmatrix}
1 & 1 \\
-1 & 3
\end{bmatrix}
\begin{bmatrix}
u_{n} \\
v_{n} \\
\end{bmatrix}$$
In order to make it nonrecursive, convert the sequence of matrix ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,018 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1097238", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
latex-00002227 | \rho^\mu\partial_\mu x^{AA'} = e o^A \bar{o}^{A'}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 2,228 | latex_formula | {"original_latex": "\\begin{align*}\\rho^\\mu\\partial_\\mu x^{AA'} = e o^A \\bar{o}^{A'},\\end{align*}"} |
mixed-00024224 | What is $\gcd (x^{3}+6x^{2}+11x+6,x^{3}+1)$ When applying straight-forward Euclid's algorithm the result have fractional coefficients, but by factoring linear terms you get $x+1$. Which answer is right? | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,066 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2516461", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
mixed-00030330 | How to determine direct solution of determinant? How to show that the determinant of the following $(2n+1)×(2n+1)$ matrix $A$?
\begin{equation}
\det A = \begin{array}{|cccccccccc|cc}
1 & -1 & 0 & \dots & 0 & 0 & 0 & \dots & 0 & 0 & & {\color{blue}{\text{row }1}}\\
-1 & 2 & -1 & \dots & 0 & 0 & 0 & \dots & 0 & 0& &... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,789 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3621758", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
normal-00045416 | The object of the game is to score cushion caroms , meaning a carom off of both object balls with at least one rail being struck before the hit on the second object ball . Cushions caroms was defunct for a number of years , but was revived in the late 1860s as another alternative to straight rail , for the same reasons... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 110,232 | text | {} |
mixed-00048046 | Notice,
let, $x=2\tan \theta\ \implies dx=2\sec^2\theta \ d\theta $
$$\int \frac{x^5}{\sqrt{4+x^2}}\ dx$$$$=\int \frac{(2\tan\theta)^5}{\sqrt{4+4\tan^2\theta}}(2\sec^2\theta \ d\theta)$$
$$=32\int \frac{\tan^5\theta \sec^2\theta}{|\sec\theta|}\ d\theta$$
Assuming $0<\theta<\pi/2$
$$=32\int \frac{\tan^5\theta \sec^2\th... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,648 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1494396", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 1}} |
mixed-00010915 | Denoting with $e_k(x,y,z), k\geq 0$ the elementary symmetric polynomials
\begin{align*}
e_1(x,y,z)&=x+y+z=0\\
e_2(x,y,z)&=xy+xz+yz=-1\tag{1}\\
e_3(x,y,z)&=xyz=-1
\end{align*}
and with $p_k(x,y,z), k\geq 0$ the $k$-th power sum
\begin{align*}
p_k(x,y,z)=x^k+y^k+z^k\tag{2}
\end{align*}
we recall Newtons identities
admit ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,968 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4246285", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "12", "answer_count": 8, "answer_id": 2}} |
normal-00019194 | Adapted from Trans @-@ Europe Express liner notes . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 46,026 | text | {} |
normal-00019646 | In that case , bn is called the n @-@ th power of b , or b raised to the power n . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 47,103 | text | {} |
normal-00006549 | The cover of Dylan 's album Self Portrait ( 1970 ) is a reproduction of a painting of a face by Dylan . Another of his paintings is reproduced on the cover of the 1974 album Planet Waves . In 1994 Random House published Drawn Blank , a book of Dylan 's drawings . In 2007 , the first public exhibition of Dylan 's painti... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 16,017 | text | {} |
latex-00034147 | \gamma=(\gamma^{(1)},\gamma^{(2)},\ldots,\gamma^{(2n)},\gamma^{(2n+1)}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 34,499 | latex_formula | {"original_latex": "\\begin{align*}\\gamma=(\\gamma^{(1)},\\gamma^{(2)},\\ldots,\\gamma^{(2n)},\\gamma^{(2n+1)}).\\end{align*}"} |
latex-00013995 | 1-[ \frac{K-1}{K}+\frac{\mathcal{P}}{K}+\frac{1-\mathcal{P}}{K}\frac{\mathcal{P}}{K}]=\frac{1-\mathcal{P}}{K}( 1-\frac{\mathcal{P}}{K}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 14,023 | latex_formula | {"original_latex": "\\begin{align*}1-\\left[ \\frac{K-1}{K}+\\frac{\\mathcal{P}}{K}+\\frac{1-\\mathcal{P}}{K}\\frac{\\mathcal{P}}{K}\\right]=\\frac{1-\\mathcal{P}}{K}\\left( 1-\\frac{\\mathcal{P}}{K}\\right).\\end{align*}"} |
mixed-00032641 | Working Out Easy Equations does anyone know how to do this equation? I know it's easy but I can't work out what the question means.
When I expanded the first equation:
$(y+4)-(y-3)$
$y^2 -3y +4y - 12$
$y^2-1y-12$
Not sure what I should do after.
Can someone explain how would you work it out in easy terms?
Thank you | mixed | math-ai/StackMathQA | stackmathqa100k | train | 19,188 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1122375", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 9, "answer_id": 0}} |
mixed-00041135 | Proof verification that $ \lim_{n\to \infty} \frac{n^2+n-1}{n^2 + 2n +2}=1$ EDIT: I've had some problems uploading this question today as I initially used the mobile verision, hence the quite absurd first proof if you saw it. Here is the full one:
We do this using the epsilon-N definition of the limit of :
$$\forall \v... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,421 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4458619", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00000904 | In 1983 , Neil Tennant met producer Bobby Orlando , while on an assignment in New York interviewing Sting for Smash Hits . After listening to some demos , Orlando offered to produce for the duo . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 2,199 | text | {} |
normal-00025778 | Further evidence for a 14 @-@ 16 hand ( 56 to 64 inches ( 140 to 160 cm ) ) war horse is that it was a matter of pride to a knight to be able to vault onto his horse in full armour , without touching the stirrup . This arose not from vanity , but necessity : if unhorsed during battle , a knight would remain vulnerable ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 61,828 | text | {} |
latex-00008696 | (t_{w(i_1),n+j_1} \cdots t_{w(i_m), n + j_m})(e_{n+j_1} \otimes \dots \otimes e_{n+j_m}) = e_{w(i_1)} \otimes \dots \otimes e_{w(i_m)}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 8,709 | latex_formula | {"original_latex": "\\begin{align*}(t_{w(i_1),n+j_1} \\cdots t_{w(i_m), n + j_m})(e_{n+j_1} \\otimes \\dots \\otimes e_{n+j_m}) = e_{w(i_1)} \\otimes \\dots \\otimes e_{w(i_m)}.\\end{align*}"} |
normal-00015012 | The album 's concept compares a violent storm to a roller coaster ; its lyrical themes vary from horrorcore @-@ based character deconstructions and songs about the supernatural to humorous and lighter subject matter . Clark 's production was praised by critics , and the album peaked at # 20 on the Billboard 200 . It is... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 36,045 | text | {} |
mixed-00037012 | Observe that the product $p$ of four consecutive integers can be written as $p=(x-\frac{3}{2})(x-\frac{1}{2})(x+\frac{1}{2})(x+\frac{3}{2})$ where $x=n+\frac{1}{2}$ for some integer $n$. Then $p=(x^2-\frac{9}{4})(x^2-\frac{1}{4}) = (x^2-\frac{5}{4}+1)(x^2-\frac{5}{4}-1) = (x^2-\frac{5}{4})^2-1$. It remains to show that... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 21,873 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/532737", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "6", "answer_count": 6, "answer_id": 1}} |
normal-00033424 | Isle of Portland has a Non @-@ League football club Portland United F.C. who play at Grove Corner . They also have a very successful youth set up called Portland United youth football Club who provide active team sport for over 170 children on the Island . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 80,620 | text | {} |
normal-00020922 | Advances in virus discovery and control continue to be made . Human metapneumovirus , which is a cause of respiratory infections including pneumonia , was discovered in 2001 . A vaccine for the papillomaviruses that cause cervical cancer was developed between 2002 and 2006 . In 2005 , human T lymphotropic viruses 3 and... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,142 | text | {} |
latex-00031875 | d\mu^{(n)}_i(t) = 2 \sqrt{\mu^{(n)}_i(t)(1 - \mu^{(n)}_i(t))} dB^{(n)}_i(t) + 2 \Big((p - n + 1) + (p + q - 2n + 2)\mu^{(n)}_i(t)\Big) dt + \sum_{j \neq i} \frac{4 \mu^{(n)}_i(t)(1 - \mu^{(n)}_i(t))}{\mu^{(n)}_i(t) - \mu^{(n)}_j(t)} dt | latex | OleehyO/latex-formulas | cleaned_formulas | train | 32,173 | latex_formula | {"original_latex": "\\begin{align*}d\\mu^{(n)}_i(t) = 2 \\sqrt{\\mu^{(n)}_i(t)(1 - \\mu^{(n)}_i(t))} dB^{(n)}_i(t) + 2 \\Big((p - n + 1) + (p + q - 2n + 2)\\mu^{(n)}_i(t)\\Big) dt + \\sum_{j \\neq i} \\frac{4 \\mu^{(n)}_i(t)(1 - \\mu^{(n)}_i(t))}{\\mu^{(n)}_i(t) - \\mu^{(n)}_j(t)} dt\\end{align*}"} |
normal-00009587 | The themes of the lyrics include contrast , contradictions and urgency . According to Jon Wiederhon of MTV News , " Martin seems to address the helplessness of being in a dysfunctional relationship he doesn 't necessarily want to escape . " The lyrics are cryptic ; the ending lines of the second verse emphasize contrad... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 23,464 | text | {} |
latex-00008783 | (-1)^{\frac{p^2-1}{8}}(-1)^{\frac{q^2-1}{8}}=(-1)^{\frac{(pq)^2-1}{8}}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 8,797 | latex_formula | {"original_latex": "\\begin{align*}(-1)^{\\frac{p^2-1}{8}}(-1)^{\\frac{q^2-1}{8}}=(-1)^{\\frac{(pq)^2-1}{8}},\\end{align*}"} |
latex-00037429 | \begin{gather*}\frac{d^{n}}{dc^{n}} {}_{p}F_{q}[\begin{matrix}\vec{a}\\\vec{b}\end{matrix};c] =\frac{( \vec{a})_{n}}{(\vec{b})_{n}} {}_{p}F_{q}[\begin{matrix}\vec{a}+n\\\vec{b}+n\end{matrix};c] ,\end{gather*} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 37,908 | latex_formula | {"original_latex": "\\begin{gather*}\\frac{d^{n}}{dc^{n}}\\,{}_{p}F_{q}\\left[\\begin{matrix}\\vec{a}\\\\\\vec{b}\\end{matrix};c\\right] =\\frac{( \\vec{a})_{n}}{(\\vec{b})_{n}}\\,{}_{p}F_{q}\\left[\\begin{matrix}\\vec{a}+n\\\\\\vec{b}+n\\end{matrix};c\\right] ,\\end{gather*}"} |
normal-00026917 | One of the few units that did manage to get out of Nanking was China 's 2nd Army led by Xu Yuanquan situated just north of Nanking . Though Xu never received Tang 's order to abandon the defense , on the night of December 12 he had heard that Nanking had been captured and so decided to withdraw on his own accord . Duri... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 64,425 | text | {} |
normal-00017286 | Coat color ranges from almost pure white through various shades of blond , cream , and ochre to grays , browns , and blacks , with variation in fur color tending to increase in higher latitudes . Differences in coat color between sexes are largely absent , though females may have redder tones . Black colored wolves in ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 41,454 | text | {} |
normal-00034746 | Having fled the besieged Ragnar Anchorage , the convoy of refugee spaceships is relentlessly pursued and attacked by Cylon Basestars . The colonial fleet must execute a faster @-@ than @-@ light ( FTL ) jump every 33 minutes to escape the Cylons , who consistently arrive at the new jump coordinates approximately 33 min... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 83,663 | text | {} |
normal-00031445 | Millennium was given the Friday night timeslot previously occupied by The X @-@ Files , prompting Carter to quip that his earlier series was " being abducted " . However , Millennium received higher viewing figures during its first season than The X @-@ Files had done , while the latter show 's fourth season , the one ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 75,443 | text | {} |
mixed-00041732 | If $x^2-xy+2y^2=\frac{a^2}{7}$, find $y'''$.
If $x^2-xy+2y^2=\frac{a^2}{7}$, find $y'''$.
For our 1st derivative we got $$y'=\frac{2x-y}{x-4y}.$$
For the second derivative we got $$y''=\frac{14x^2-14xy+28y^2}{(x-4y)^3}.$$
And for the final answer we got $$y'''=\frac{4(-84x^3+119x^2y-154xy^2-84y^3)}{(x-4y)^5}.$$
Took ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,790 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/383896", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 1}} |
latex-00020271 | Q(r_1,\theta)\approx &\frac{\pi^2\lambda R_S(R_{max}-r_1)}{NR_E}\sum_{n=1}^N\sqrt{1-\phi_n^2} c_n\ &\times(1-\prod_{m=1}^{M}{(1+\frac{m\eta\theta r_1^{\alpha}}{Mc_n^{\alpha}})^{-Mb_m}}) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,376 | latex_formula | {"original_latex": "\\begin{align*} Q(r_1,\\theta)\\approx &\\frac{\\pi^2\\lambda R_S(R_{max}-r_1)}{NR_E}\\sum_{n=1}^N\\sqrt{1-\\phi_n^2} c_n\\\\ &\\times\\left(1-\\prod_{m=1}^{M}{\\left(1+\\frac{m\\eta\\theta r_1^{\\alpha}}{Mc_n^{\\alpha}}\\right)^{-Mb_m}}\\right)\\end{align*}"} |
normal-00007019 | noitulovE is the fifth television / cinema piece in the Good things come to those who wait series , and its premiere marked the end of a four @-@ year hiatus . The advert and its associated campaign were a critical and financial success . It received over 30 awards from professional organisations in the advertising and... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 17,193 | text | {} |
normal-00035655 | Two engineers – Aaron and Abe – supplement their day @-@ jobs with entrepreneurial tech projects , working out of Aaron 's garage . During one such research effort , involving electromagnetic reduction of objects ' weight , the two men accidentally discover an ' A @-@ to @-@ B ' time loop side @-@ effect ; objects left... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 85,853 | text | {} |
normal-00000456 | The moderate view within the Díaz government was represented by Jorge Vera Estañol who in a memo to the minister of foreign affairs wrote that there were two revolutions taking place in Mexico : a political revolution , based mostly in the north , whose aim was mostly to establish free elections and remove Díaz himself... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 1,134 | text | {} |
mixed-00038414 | Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$
Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$.
We have $f(0) = 0$ and $f(x) = 2f(2x) - x$, but I am not sure how to convert this functional equation into something that is easier to solve. Maybe using ind... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,739 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1744233", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 1}} |
latex-00019307 | \phi_{\omega,c}(x):=[\frac{\sqrt{\omega}}{4\omega-c^2}\{\cosh (\sqrt{(4\omega-c^2)x})-\frac{c}{\sqrt{4\omega}}\}]^{-1/2} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 19,390 | latex_formula | {"original_latex": "\\begin{align*}\\phi_{\\omega,c}(x):=\\left[\\frac{\\sqrt{\\omega}}{4\\omega-c^2}\\left\\{\\cosh (\\sqrt{(4\\omega-c^2)x})-\\frac{c}{\\sqrt{4\\omega}}\\right\\}\\right]^{-1/2}\\end{align*}"} |
latex-00024430 | &(t+r+1)\cdots (t+n)[\underset{i=1}{\overset{s}{\sum}} \binom{m_i+r-1}{r} +\dim_K(I_{\alpha})_t ]=(t+r+1)\cdots (t+n) \binom{t+r}{r}\\&=(r+1)\cdots (n-1)n \binom{t+n}{n}=(r+1)\cdots (n-1)n[\underset{i=1}{\overset{s}{\sum}} \binom{m_i+n-1}{n} + \dim_K I_t ]. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 24,603 | latex_formula | {"original_latex": "\\begin{align*}&(t+r+1)\\cdots (t+n)\\left[\\underset{i=1}{\\overset{s}{\\sum}} \\binom{m_i+r-1}{r} +\\dim_K(I_{\\alpha})_t \\right]=(t+r+1)\\cdots (t+n) \\binom{t+r}{r}\\\\&=(r+1)\\cdots (n-1)n \\binom{t+n}{n}=(r+1)\\cdots (n-1)n\\left[\\underset{i=1}{\\overset{s}{\\sum}} \\binom{m_i+n-1}{n} + \\di... |
mixed-00018329 | Let $\left( 1+\dfrac1{\sin x}\right) \left( 1+\dfrac1{\cos x}\right)=1+y$
$\iff\sec x+\csc x=y-\sec x\csc x$
Squaring both sides, $$\sec^2x+\csc^2x+2\sec x\csc x=y^2-2y\sec x\csc x+\sec^2x\csc^2x$$
But $\sec^2x+\csc^2x=\dfrac{\sin^2x+\cos^2x}{\sin^2x\cos^2x}=\sec^2x\csc^2x$
$$\implies y^2-(2y-1)\sec x\csc x=0\iff0=\sin... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,468 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1697753", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 3}} |
latex-00007426 | A^{\mu}_{\alpha}:=\overline{L}^{i}_{\alpha} L^{\mu}_{i}=-\overline{L}^{a}_{\alpha} L^{\mu}_{a}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 7,435 | latex_formula | {"original_latex": "\\begin{align*}A^{\\mu}_{\\alpha}:=\\overline{L}^{i}_{\\alpha}\\,L^{\\mu}_{i}=-\\overline{L}^{a}_{\\alpha}\\,L^{\\mu}_{a},\\end{align*}"} |
latex-00028098 | \phi_{E}(p)= \frac{C}{p^{1/2+iE/\hbar}}\frac{\Gamma(\frac{1}{4}+\frac{iE}{2\hbar} ) }{\Gamma(\frac{1}{4}-\frac{iE}{2\hbar} )}+O(\mu_{0}^{2}) . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 28,354 | latex_formula | {"original_latex": "\\begin{align*}\\phi_{E}(p)= \\frac{C}{p^{1/2+iE/\\hbar}}\\frac{\\Gamma\\left(\\frac{1}{4}+\\frac{iE}{2\\hbar} \\right) }{\\Gamma\\left(\\frac{1}{4}-\\frac{iE}{2\\hbar} \\right)}+O(\\mu_{0}^{2}) \\,.\\end{align*}"} |
normal-00037064 | Shortly before releasing the title , J. K. Rowling announced that she had considered three titles for the book . The final title , Harry Potter and the Deathly Hallows , named after the mythical Deathly Hallows in the novel , was released to the public on 21 December 2006 , via a special Christmas @-@ themed hangman pu... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 89,491 | text | {} |
mixed-00012885 | The best approach is to take logs. If $L$ is the desired limit then we have
\begin{align}
\log L &= \log\left(\lim_{x \to 0}\left(1 + \frac{1 - \cos x}{x}\right)^{1/x}\right)\notag\\
&= \lim_{x \to 0}\log\left(1 + \frac{1 - \cos x}{x}\right)^{1/x}\text{ (by continuity of log)}\notag\\
&= \lim_{x \to 0}\frac{1}{x}\log\l... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,152 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1323556", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 6, "answer_id": 0}} |
mixed-00001691 | For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that $-5 \le x+y \le -2.$ I have a problem:
For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that
$$-5 \le x+y \le -2.$$
I have tried:
I write $(x+y+1)^2+5x+7y+10+y^2=(x+y)^2+7(x+y)+(y+1)^2+10=0.$
Now I'm stuck :(
Any help will be ap... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 860 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1189778", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
normal-00038679 | In the early 1950s an anthology titled Brief Fantastic Tales appeared from Studio Publications in Toronto ; it consisted mostly of reprints from Uncanny Tales . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 93,638 | text | {} |
mixed-00025915 | The fact that $M_{24}$ is 5-transitive forces its order to be
$$
|M_{24}| = 24 \cdot (24-1) \cdot (24-2) \cdot (24-3) \cdot (24-4) \cdot |H|,
$$
where $H$ is the subgroup fixing any ordered list of five distinct points. (In fact, we have $|H| = 48$.) It follows that $|M_{24}|$ is divisible by
\begin{align*}
24 \cdot \f... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 15,100 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4362932", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "6", "answer_count": 1, "answer_id": 0}} |
mixed-00030503 | $$1 > \frac{7}{24}\pi \approx 0.916$$
$$\sin(1) > \sin\left(\frac{7}{24}\pi\right) = \frac12\sqrt{2+\sqrt{2-\sqrt{3}}} \approx 0.793 > \frac{\pi}{4} \approx 0.785$$
$$\frac{1}{\sin^2(\sin(1))} < \frac{1}{\sin^2(\frac{\pi}{4})}=2$$
so $$1 < \frac{1}{\sin^2(\sin(1))} < 2$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,889 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3783980", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 2, "answer_id": 1}} |
normal-00032128 | French naturalist Lucien Quélet transferred the species to the now @-@ obsolete genus Dictyopus in 1886 , which resulted in the synonym Dictyopus torosus . Boletus xanthocyaneus , first described by Henri Romagnesi in 1948 as Boletus purpureus var. xanthocyaneus and classified as a species in 1976 , was considered by I... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 76,996 | text | {} |
normal-00027351 | " By midday we reached the Letaba Valley , in the Majajes Mountains , inhabited by a powerful tribe of natives once ruled by a princess said to be the prototype of Rider Haggard 's ' She ' . " | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 65,443 | text | {} |
mixed-00021796 | Proving $\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$ I found the following two relational expressions in a book without any additional information:
$$\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}=\frac13(\sqrt[3]{2}+\sqrt[3]{20}-\sqrt[3]{25})$$
$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac19}-\sqrt[3]{\f... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 12,590 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/498325", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 0}} |
latex-00047577 | (\ker\pi_*)^\perp=\omega D_2\oplus \mu, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 48,237 | latex_formula | {"original_latex": "\\begin{align*}(\\ker\\pi_*)^\\perp=\\omega D_2\\oplus \\mu, \\end{align*}"} |
mixed-00012128 | Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore.
\begin{align} \int_{0}^{\infty} \fr... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,695 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/656757", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "24", "answer_count": 4, "answer_id": 1}} |
normal-00009791 | The One I Love ( Japanese : わたしのすきなひと , Hepburn : Watashi no Sukinahito ) is a romantic , slice @-@ of @-@ life shōjo ( targeted towards girls ) manga by Clamp , an all @-@ female , manga artist team consisting of Satsuki Igarashi , Mokona , Tsubaki Nekoi , and Nanase Ohkawa . Appearing as a monthly serial in the Japan... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 24,005 | text | {} |
mixed-00010135 | Determinant of Large Matrix $A=\left[\begin{array}{ccccc}{-2} & {-1} & {} & {\cdots} & {-1} \\ {-1} & {-2} & {-1} & {\cdots} & {-1} \\ {} & {} & {\ddots} & {} & {} \\ {-1} & {\cdots} & {-1} & {-2} & {-1} \\ {-1} & {\cdots} & {} & {-1} & {-2}\end{array}\right] \in \mathbb{R}^{53 \times 53}$
So we want to find determinan... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,499 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3379033", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 0}} |
latex-00005557 | F_0=0,\quad F_1 = 0, \quad F_2 = 0,\dots , F_{n-1} = 0 | latex | OleehyO/latex-formulas | cleaned_formulas | train | 5,563 | latex_formula | {"original_latex": "\\begin{align*}F_0=0,\\quad F_1 = 0, \\quad F_2 = 0,\\dots , F_{n-1} = 0\\end{align*}"} |
normal-00020969 | The first draft of the script was 72 pages — 15 pages too long — and did not feature a fourth act . Carter and executive producer Frank Spotnitz worked closely with Anderson to finish the episode , although Carter and Spotnitz later acknowledged that the majority of the script " was all Gillian " . Despite her satisfac... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,252 | text | {} |
latex-00036777 | L \lesssim \begin{cases}e^{-\mu_0\tau} & 1<\gamma<\frac53 \\\tau e^{-\mu_0\tau} & \gamma =\frac53\end{cases} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 37,255 | latex_formula | {"original_latex": "\\begin{align*}L \\lesssim \\begin{cases}e^{-\\mu_0\\tau} & \\ 1<\\gamma<\\frac53 \\\\\\tau e^{-\\mu_0\\tau} & \\ \\gamma =\\frac53\\end{cases}\\end{align*}"} |
normal-00027549 | MSU also publishes a student @-@ run magazine during the academic year called Ing Magazine . Created in 2007 by MSU alumnus Adam Grant , the publication is released at the beginning of each month and publishes 7 issues each school year . MSU also publishes a student @-@ run fashion and lifestyle magazine called VIM Mag... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 65,955 | text | {} |
normal-00007784 | The creek continues southwest as it enters Plunketts Creek Township and receives Reibsan Run on the left bank , 4 @.@ 70 miles ( 7 @.@ 56 km2 ) upstream from the mouth . It next receives Mock Creek at the hamlet of Hoppestown ( 4 @.@ 24 miles ( 6 @.@ 82 km ) from the mouth ) , then Wolf Run ( 2 @.@ 72 miles ( 4 @.@ 38 ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 19,051 | text | {} |
mixed-00020674 | Polar to cartesian form of $r=\cos(4θ)$?
Consider an equation in polar coordinates, $r = \cos(4θ)$. Find the equation of the curve in the first quadrant in Cartesian coordinates.
This is for an assignment and this is what help I have received so far from user170231-
$$\begin{align}
r(\theta)&=\cos(4\theta)\\[1ex]
&=\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 11,905 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3948842", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 1, "answer_id": 0}} |
mixed-00047230 | Integrating $(3x + 1) / x^\frac{1}{2}$ I am trying to integrate the following equation, but my answer is different from the textbook and I cannot see where I am going wrong:
\begin{align} \int_1^2\frac{ 3x + 1}{x^{1/2}}dx
&= \int_1^2 \frac{3x}{x^{1/2}}dx + \int_1^2 \frac{1}{x^{1/2}}dx \\
&= \int_1^2 3x^{1/2}dx + \int_1... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,155 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/823037", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
mixed-00028196 | Integrating $\int \frac{u \,du}{(a^2+u^2)^{3/2}}$ How does one integrate $$\int \frac{u \,du}{(a^2+u^2)^{3/2}} ?$$
Looking at it, the substitution rule seems like method of choice. What is the strategy here for choosing a substitution? | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,490 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1613689", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 3}} |
latex-00023152 | = h^{-2} \sum_{a,a',h,h' ~:~ (a+h)(a'+h')=1} A(a) A(a') H(h) H(h') \chi (a+h) + \mathcal{E} = \sigma + \mathcal{E} , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 23,321 | latex_formula | {"original_latex": "\\begin{align*} = h^{-2} \\sum_{a,a',h,h' ~:~ (a+h)(a'+h')=1} A(a) A(a') H(h) H(h') \\chi (a+h) + \\mathcal{E} = \\sigma + \\mathcal{E} \\,,\\end{align*}"} |
mixed-00041410 | Finding the outward flux through a sphere Problem:
Find the flux of of the field $F$ across the portion of the sphere $x^2 + y^2 + z^2 = a^2$ in the first octant in the direction away from the origin, when $F = zx\hat{i} + zy\hat{j} + z^2\hat{k}$. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,596 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/113363", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00003760 | Of beechen green , and shadows numberless , | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 8,993 | text | {} |
mixed-00031581 | From $$
\frac{A}{x} + \frac{B}{x+1} +\frac{C}{(x+1)^2} +\frac{D}{(x+1)^3}
$$ you have: $$\frac{A(x+1)^3+Bx(x+1)^2+Cx(x+1)+Dx}{x(x+1)^3}=\frac{3x+2}{x(x+1)^3}$$ Now let $x=0$ then $$A(0+1)^3=3\times0+2\to A=2$$ Let $x=-1$ then $$D(-1)=3\times(-1)+2=-1\to D=-1$$ Set $x=1$ then $$8A+4B+2C+D=3\times1+2=5\to 16+4B+2C-1=5\to... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 18,548 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/286767", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
normal-00042953 | Banjo @-@ Tooie was developed by Rare and designed by Gregg Mayles , who previously worked on Banjo @-@ Kazooie . Development of the game started in June 1998 . Some features that were originally cut during the development of Banjo @-@ Kazooie , such as some of its worlds and a multiplayer game mode , were instead inte... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 103,935 | text | {} |
mixed-00049431 | What are the maximum and minimum values of $4x + y^2$ subject to $2x^2 + y^2 = 4$? $$ 2x^2 + y^2 = 4 $$
$$ Y = \sqrt{4-2x^2} $$
$$4x + y = 2x^2 + \sqrt{4-2x^2}$$
Find the derivative of $$ 2x^2 + \sqrt{4-2x^2} $$ set as = 0
$$X^2 = 64/33$$
$$ F(64/33) = 34\sqrt{33}/33 $$
How to solve it the right way? | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,484 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2687820", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 0}} |
latex-00014772 | {\cal P}_{n}^{\pm }(\beta )=N_{\pm }^{2}(\beta )[|T_{n}(\beta )|^{2}+|T_{n}(-\beta )|^{2}{\pm }(T_{n}^{*}(\beta )T_{n}(-\beta)+T_{n}(\beta )T_{n}^{*}(-\beta ))]. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 14,802 | latex_formula | {"original_latex": "\\begin{align*}{\\cal P}_{n}^{\\pm }(\\beta )=N_{\\pm }^{2}(\\beta )\\left[|T_{n}(\\beta )|^{2}+|T_{n}(-\\beta )|^{2}{\\pm }\\left(T_{n}^{*}(\\beta )T_{n}(-\\beta)+T_{n}(\\beta )T_{n}^{*}(-\\beta )\\right)\\right].\\end{align*}"} |
mixed-00010939 | Strange/Unexpected behavior of an Infinite product Some friends and I were playing around with this continued fraction:
We noticed when writing it out for each next step, the end behavior went either to 1 (when there was an even number of terms) or to a linear dependence on x (when there were odd). This was expected -... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,982 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4281850", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
mixed-00047015 | Definite integration problem (trig). I have this definite integral:
$$
\int_0^\Pi \cos{x} \sqrt{\cos{x}+1} \, dx
$$
For finding the indefinite integral, I have tried substitution, integration by parts, but I'm having trouble solving it.
By parts
$$
\int \cos{x} \sqrt{\cos{x}+1} \, dx\ = \sqrt{\cos{x}+1}\sin{x} + \frac{... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,031 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/627812", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 5, "answer_id": 1}} |
normal-00004225 | Before his final undergraduate year at McMaster , Innis spent a summer teaching at the Northern Star School in the frontier farming community of Landonville near Vermilion , Alberta . The experience gave him a sense of the vastness of Canada . He also learned about Western grievances over high interest rates and steep ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 10,030 | text | {} |
mixed-00048859 | I give here an answer advocating the simplicity of using Newton (or Newton-Girard) formulas. Using the notations of the Wikipedia article in all generality:
Let $e_0=1, e_1=x+y+z, e_2=xy+yz+zx, e_3=xyz.$
Let $p_1=x+y+z$, $p_2=x^2+y^2+z^2$, $p_3=x^3+y^3+z^3$.
Then:
$$\begin{cases}
e_1&=&p_1\\
2e_2&=&p_1e_1-p_2\\
3e_3&=&... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,137 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2162199", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 3}} |
latex-00016234 | {\bar I}_{M1} = I_{M1[a,\psi]} - I_{top[\omega ,\Omega]}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 16,272 | latex_formula | {"original_latex": "\\begin{align*}{\\bar I}_{M1} = I_{M1[a,\\psi]} - I_{top[\\omega ,\\Omega]},\\end{align*}"} |
mixed-00023726 | Hint:
\begin{align*}
I&=\int_{1}^{\sqrt{2}}\frac{2t^{2}}{t^{4}-2t^{2}+2}\,\mathrm{d}t\\
&=\int_{1}^{\sqrt{2}}\frac{\sqrt{2}+t^{2}+\left ( t^{2}-\sqrt{2} \right )}{t^{4}-2t^{2}+2}\,\mathrm{d}t \\
&=\int_{1}^{\sqrt{2}}\frac{\displaystyle\frac{\sqrt{2}}{t^{2}}+1}{t^{2}-2+\displaystyle\frac{2}{t^{2}}}\,\mathrm{d}t+\int_{1... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,754 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2099550", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
latex-00034971 | G(z) = \sum_{k=1}^{M \wedge N} \frac{1}{\lambda_k-z} ( {\begin{array}{*{20}c} { \xi_k \xi^*_k} & z^{-1/2} \sqrt{\lambda}_k \xi_k \zeta_k^* \ z^{-1/2} \sqrt{\lambda}_k \zeta_k \xi_k^* & \zeta_{k} \zeta_{k}^* \\\end{array}} ). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 35,334 | latex_formula | {"original_latex": "\\begin{align*}G(z) = \\sum_{k=1}^{M \\wedge N} \\frac{1}{\\lambda_k-z} \\left( {\\begin{array}{*{20}c} { \\xi_k \\xi^*_k} & z^{-1/2} \\sqrt{\\lambda}_k \\xi_k \\zeta_k^* \\\\ z^{-1/2} \\sqrt{\\lambda}_k \\zeta_k \\xi_k^* & \\zeta_{k} \\zeta_{k}^* \\\\\\end{array}} \\right).\\end{align*}"} |
normal-00048423 | Tikhonov was born in the Ukrainian city of Kharkiv on 14 May [ O.S. 1 May ] 1905 to a Russian @-@ Ukrainian working @-@ class family ; he graduated from the St. Catherine Institute of Communications in 1924 . Tikhonov worked as an assistant engineer from 1924 to 1926 . Four years later , in 1930 , Tikhonov graduated as... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 117,721 | text | {} |
mixed-00027075 | Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$ I would like to show that
$$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$
I've been working o... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 15,813 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/756489", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "7", "answer_count": 2, "answer_id": 0}} |
mixed-00025933 | Doing some algebraic manipulation we get that your sum $S$ is equal to
\begin{align*}
S &= 8\sum_{n \ge 1}\frac{\cos^3(n)}{n^4}\left(2\cos(n) -1 \right)\left(2\cos(n) +1 \right)\\
& =8\sum_{n \ge 1}\frac{\cos^3(n)}{n^4}\left(4\cos^2(n) -1 \right)\\
& = 32\sum_{n \ge 1}\frac{\cos^5(n)}{n^4} -8\sum_{n \ge 1}\frac{\cos^3(... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 15,111 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4398071", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
mixed-00009301 | Find a power series for $\frac{z^2}{(4-z)^2}$. Find a power series for $$\frac{z^2}{(4-z)^2}$$
What I did is:
Since
$$\sum^{\infty}_{k=0}z^k=\frac{1}{1-z}$$
Take the derivative, so
$$\sum^{\infty}_{k=1}kz^{k-1}=\frac{1}{(1-z)^2}$$
So
$$\frac{z^2}{(4-z)^2}=\frac{z^2}{4^2(1-\frac{z}{4})^2}=\frac{z^2}{4^2}\sum^{\infty}_{k... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,011 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2611571", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 19