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 |
|---|---|---|---|---|---|---|---|---|
mixed-00047102 | Proving divisibility by using induction: $133 \mid (11^{n+2} + 12^{2n+1})$ If $n > 0$, then prove the following by using induction: $$133|(11^{n+2} + 12^{2n+1}).$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,082 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/716789", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 1, "answer_id": 0}} |
latex-00034745 | u_{t}+\partial_{V}\varphi^{1}(u)-\mathrm{d}_{V}\varphi^{2}(u) & =g\text{ in}V^{\ast}(0,T)\\u(0) & =u_{0} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 35,104 | latex_formula | {"original_latex": "\\begin{align*}u_{t}+\\partial_{V}\\varphi^{1}(u)-\\mathrm{d}_{V}\\varphi^{2}(u) & =g\\text{ in}V^{\\ast}(0,T)\\\\u(0) & =u_{0} \\end{align*}"} |
mixed-00002589 | You've made a mistake. I hope you can find it using my answer
$$\int\frac{\sqrt{x^2+2x-3}}{x+1}\space\text{d}x=\int\frac{\sqrt{(x+1)^2-4}}{x+1}\space\text{d}x=$$
Substitute $u=x+1$ and $\text{d}u=\text{d}x$:
$$\int\frac{\sqrt{u^2-4}}{u}\space\text{d}u=$$
Substitute $u=2\sec(s)$ and $\text{d}u=2\tan(s)\sec(s)\space\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,315 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1783684", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
mixed-00008420 | We can use your idea: Let $n=10x+y$. If the last digit of $n^2$ is $4$ then the last digit of $n$ is either $2$ or $8$, so $y=2$ ot $y=8$.
First let's assume $y=2$. We then have $40x+4\equiv 44\mod100$, or equivalently $40x+4=44+100t$ for some $t$. For $t=0$ we get $x=1$, so $n=12$, and this is not what we want. For $t... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 4,488 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1898485", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 1}} |
normal-00035593 | Peiryar in his references to Gandhi used opportunities to present Gandhi as on principle serving the interests of the Brahmins. in 1927 , Periyar and Gandhi met at Bangalore to discuss this matter . The main difference between them came out when Periyar stood for the total eradication of Hinduism to which Gandhi object... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 85,706 | text | {} |
normal-00025506 | In the Book of Nehemiah ( Nehemiah 13 : 23 – 24 ) , some 5th century BCE residents of Jerusalem are said to have married women from Ashdod , and half of the children of these unions were reportedly unable to understand Hebrew ; instead , they spoke " the language of Ashdod " . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 61,141 | text | {} |
normal-00025667 | Around 2001 , Clarke " had begun to despair " , and started looking for someone to help her finish and sell the book . Giles Gordon became her agent and sold the unfinished manuscript to Bloomsbury in early 2003 , after two publishers rejected it as unmarketable . Bloomsbury were so sure the novel would be a success th... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 61,551 | text | {} |
mixed-00013541 | Minimum value of $\cos^2\theta-6\sin\theta \cos\theta+3\sin^2\theta+2$ Recently I was solving one question, in which I was solving for the smallest value of this expression
$$f(\theta)=\cos^2\theta-6\sin\theta \cos\theta+3\sin^2\theta+2$$
My first attempt:
$$\begin{align}
f(\theta) &=3+2\sin^2\theta-6\sin\theta \cos\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,553 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1843996", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
latex-00014815 | \sum_r g_rq_r=\frac{k}{2} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 14,845 | latex_formula | {"original_latex": "\\begin{align*}\\sum_r g_rq_r=\\frac{k}{2}\\end{align*}"} |
normal-00024390 | After Zeferino 's death in 1981 , a conflict took place between the university 's General Coordinator , appointed and backed by the government , and the Directive Council , composed of directors of the different institutes . The rector introduced new rules reducing the power of the General Coordinator . As retaliation ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 58,518 | text | {} |
mixed-00024199 | Since they're asking for a result modulo $12$, I would've gone for a more direct approach, like so:
$(3)(4) = 12$, so for the even cases, work modulo $4$. An even number $n$ satisfies $n \equiv 0,2 \pmod 4$. So $3n+2 = 2,8 \pmod{12}$.
$(2)(6) = 12$, so for the odd cases, work modulo $6$. An odd number $m$ satisfies $m ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,050 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2497463", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 3, "answer_id": 1}} |
mixed-00010982 | If
$$
2x^3+3x^2-x-1=0
$$
has roots $\alpha,\beta,\gamma$, then substituting $x\mapsto\frac1x$ (and multiplying by $-x^3$ to clear denominators)
$$
\begin{align}
&-x^3\left(\frac2{x^3}+\frac3{x^2}-\frac1x-1\right)\\
&=x^3+x^2-3x-2=0
\end{align}
$$
has roots $\frac1\alpha,\frac1\beta,\frac1\gamma$. Then substituting $x\m... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 6,007 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4341047", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 0}} |
mixed-00005597 | Here is another way if you know the binomial theorem.
Observe that $$10^n = (1+9)^n = 1 + {n \choose1}9 + \ldots + 9^n = 1 + 9A$$ where $A= {n \choose1}+ \ldots + 9^{n-1}$ and similarly $$4^{n+2} = (1+3)^{n+2} = 1 + (n+2) 3 + {{n+2} \choose 2}3^2 + \ldots + 3^{n+2} = 1 + 3(n+2) + 3^2B$$
So $$10^n + 3 \cdot 4^{n+2} + 5 ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,841 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4237927", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 6, "answer_id": 5}} |
mixed-00041469 | Polynomials in Fourier trigonometric series I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding:
$\frac{\sin{kx}}{k}$,
$\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$,
$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right)sin{kx}}{k^3}$,
$\frac{3 \left(-2+k^2 x^2\righ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,632 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/153795", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 0}} |
latex-00044679 | g_m(0)= ( \frac{\partial_x^m \tilde{u}_0}{u^s_{0,y} + \tilde u_{0, y}} )_y= \frac{\partial_y\partial_x^m \tilde{u}_0}{u^s_{0,y} + \tilde u_{0, y}}- \frac{\partial_x^m \tilde{u}_0}{u^s_{0,y} + \tilde u_{0, y}} \eta_2(0), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 45,309 | latex_formula | {"original_latex": "\\begin{align*} g_m(0)= \\left( \\frac{\\partial_x^m \\tilde{u}_0}{u^s_{0,y} + \\tilde u_{0, y}} \\right)_y= \\frac{\\partial_y\\partial_x^m \\tilde{u}_0}{u^s_{0,y} + \\tilde u_{0, y}}- \\frac{\\partial_x^m \\tilde{u}_0}{u^s_{0,y} + \\tilde u_{0, y}} \\eta_2(0),\\end{align*}"} |
mixed-00018500 | If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$
*
*$\sin x + \sin y = 1$
*$\cos x + \cos y = 0$
Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured.
I got the question from chapter 26 of a comic called Yamada-kun.
How can I solve this equat... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,576 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1866796", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "8", "answer_count": 7, "answer_id": 0}} |
normal-00025801 | It was not uncommon for a girl to learn her father 's trade and for a woman to share her husband 's trade , since the entire family often helped run medieval shops and farms . Many guilds also accepted the membership of widows , so they might continue their husband 's business . Under this system , some women trained i... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 61,880 | text | {} |
normal-00019327 | The lock opened up nearly 12 miles ( 19 km ) of waterway . As part of the upgrade , new 48 @-@ hour moorings were constructed on The Haven , for boats about to enter the Drain , and on the South Forty @-@ Foot Drain near the Black Sluice pumping station at Boston , at Swineshead Bridge and at Hubbert 's Bridge . The up... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 46,346 | text | {} |
latex-00037960 | \bar{\pi}(\vartheta|X) =\bigotimes_{i=1}^n \big[p_i\mathrm{N}\big(X_i, \tfrac{K \sigma^2}{K+1}\big) +(1-p_i) \delta_0\big], p_i= 1/\big[1+h\exp\{-\tfrac{X_i^2}{2\sigma^2}\}\big], | latex | OleehyO/latex-formulas | cleaned_formulas | train | 38,445 | latex_formula | {"original_latex": "\\begin{align*}\\bar{\\pi}(\\vartheta|X) =\\bigotimes_{i=1}^n \\big[p_i\\mathrm{N}\\big(X_i, \\tfrac{K \\sigma^2}{K+1}\\big) +(1-p_i) \\delta_0\\big], p_i= 1/\\big[1+h\\exp\\{-\\tfrac{X_i^2}{2\\sigma^2}\\}\\big],\\end{align*}"} |
latex-00043342 | {m+n\choose m}{2n\choose n}\frac{n}{(m+n)(n+1)}={m+n-1\choose m}{2n\choose n}\frac{1}{n+1}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 43,957 | latex_formula | {"original_latex": "\\begin{align*}{m+n\\choose m}{2n\\choose n}\\frac{n}{(m+n)(n+1)}={m+n-1\\choose m}{2n\\choose n}\\frac{1}{n+1},\\end{align*}"} |
normal-00012518 | George Calvert , 1st Baron Baltimore ( 1579 – 15 April 1632 ) was an English politician and colonizer . He achieved domestic political success as a Member of Parliament and later Secretary of State under King James I. He lost much of his political power after his support for a failed marriage alliance between Prince Ch... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 30,456 | text | {} |
mixed-00028175 | How can you prove the inequality $2x^3 (x^3 + 8y^3) + 2y^3 (y^3 + 8z^3) + 2z^3 (z^3 + 8x^3) ≥ 9x^4 (y^2 + z^2) + 9y^4 (z^2 + x^2) + 9z^4 (x^2 + y^2)$ I changed the RHS as $\displaystyle \sum_{cyc} 9x^4(S-x^2) $ for $S = x^2 + y^2 + z^2$
Then I thought I could apply Jensen's inequality for $f(x) = 9x^4(S-x^2) = -9x^6 +... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,478 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1598570", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00000571 | \Gamma^x = \mathbf{E}(\theta^x) - \mathbf{I}^\dagger(D_{\theta^x}) \# . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 571 | latex_formula | {"original_latex": "\\begin{align*}\\Gamma^x = \\mathbf{E}(\\theta^x) - \\mathbf{I}^\\dagger(D_{\\theta^x}) \\# .\\end{align*}"} |
mixed-00030538 | How does one go about solving $arg(\frac{z-2i}{z-6}) = \frac{1}{2}\pi$ This should give $$\frac{z-2i}{z-6} = bi$$
but solving that gives me $$z = \frac{-2b +6b^2-6bi +2i}{1+b^2}$$
and substituting $z$ for $x + yi$ gives me $x = \frac{-2b +6b^2}{1+b^2}$ and $y=\frac{-6b +2}{1+b^2}$
And I have no clue how to continue now... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,913 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3826976", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 2}} |
latex-00011622 | d_QT_n=\bigl[Q,T_n\bigr]=i\sum_{l=1}^{n}\frac{\partial}{\partial x^{\mu}_l}T_{n/l}^{\mu}(x_1,\ldots,x_l,\ldots,x_n) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 11,646 | latex_formula | {"original_latex": "\\begin{align*}d_QT_n=\\bigl[Q,T_n\\bigr]=i\\sum_{l=1}^{n}\\frac{\\partial}{\\partial x^{\\mu}_l}T_{n/l}^{\\mu}(x_1,\\ldots,x_l,\\ldots,x_n) \\end{align*}"} |
mixed-00034058 | Number of integer partitions $p(n,m)$ In "Integer partitions" by Andrews and Eriksson, the authors provide formulas to compute $p(n,m)$, i.e., the number of partitions of $n$ into parts less than or equal to $m$, for $m=1,2,3,4,5$. As discussed in this question, it seems that no formula for $m>5$ is known.
Howevere, th... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 20,057 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2244719", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 4, "answer_id": 1}} |
mixed-00022871 | You have the right idea but note that approaching along the straight line $x=0$, it is not always true that $\lim_{x,y \to 0} f(0,y) = \lim_{x\to 0} f(x,y)$. The notation is incorrect but you did arrive at the right conclusion (the limit does not exist due to differing values as we change paths). | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,237 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1403454", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00027730 | \dim_H(\mathbf A_\infty)&\geq\dim\mathfrak g_\alpha-\limsup_{j\to\infty}\frac{\sum_{i=0}^j\log(l_{i+1}^{-\frac{\gamma+\epsilon}{\alpha-(\gamma+\epsilon)}\dim\mathfrak g_\alpha})}{\log(l_{j+1}^{-\frac\alpha{\alpha-(\gamma+\epsilon)}})}\\&=\dim\mathfrak g_\alpha(1-\frac{\gamma+\epsilon}\alpha). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 27,981 | latex_formula | {"original_latex": "\\begin{align*}\\dim_H(\\mathbf A_\\infty)&\\geq\\dim\\mathfrak g_\\alpha-\\limsup_{j\\to\\infty}\\frac{\\sum_{i=0}^j\\log\\left(l_{i+1}^{-\\frac{\\gamma+\\epsilon}{\\alpha-(\\gamma+\\epsilon)}\\dim\\mathfrak g_\\alpha}\\right)}{\\log\\left(l_{j+1}^{-\\frac\\alpha{\\alpha-(\\gamma+\\epsilon)}}\\righ... |
normal-00011191 | In addition to these military bases , private companies operated numerous trading posts in the region that were often referred to as " forts " , though they typically had little in the way of defensive fortifications . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 27,376 | text | {} |
normal-00048085 | In 1991 , Protecteur was part of the Canadian contingent sent to the Persian Gulf as part of Operation Desert Shield and later Operation Friction ( the Canadian name for its operations during the Gulf War ) . The ship , part of a three @-@ vessel force , the other two being the Iroquois @-@ class destroyer Athabaskan a... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 116,829 | text | {} |
mixed-00037418 | Using Vieta's formula $\displaystyle a+b+c=0,(-1)^3abc=42$
$$\implies \sum (a+b)^3=-\sum c^3$$
Method $\#1:$
As $a,b,c$ individually satisfy the given equation, we can write $$7a^3=25a-42\text{ etc.}$$
Again using Newton's
Power Sum formula $$7\sum a^3=25\sum a-3\cdot42=\cdots$$
Method $\#2:$
Like If $a,b,c \in R$ are... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,118 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/907015", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00042992 | \sum_{f \in \mathcal{F}(\mathcal{A},\mathcal{B})} \sum_{\tilde{f} \in F^{-1}(f)} \frac{1}{\prod_{i=1}^s |A_i|} = \prod_{(v,w) \in E(f)} \#(v) = m(f), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 43,604 | latex_formula | {"original_latex": "\\begin{align*} \\sum_{f \\in \\mathcal{F}(\\mathcal{A},\\mathcal{B})} \\sum_{\\tilde{f} \\in F^{-1}(f)} \\frac{1}{\\prod_{i=1}^s |A_i|} = \\prod_{(v,w) \\in E(f)} \\#(v) = m(f),\\end{align*}"} |
latex-00020082 | T_\rho f = \sum_{S \subseteq [n]} \rho^{|S|} f^{=S}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,171 | latex_formula | {"original_latex": "\\begin{align*}T_\\rho f = \\sum_{S \\subseteq [n]} \\rho^{|S|} f^{=S}. \\end{align*}"} |
normal-00008671 | The war in Europe had been under way for a year and was going badly for Britain . After his wedding Olivier wanted to help the war effort . He telephoned Duff Cooper , the Minister of Information under Winston Churchill , hoping to get a position in Cooper 's department . Cooper advised him to remain where he was and s... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 21,268 | text | {} |
normal-00048040 | Hamasaki has sold over 51 million records , making her one of the best @-@ selling artists in Japan . Hamasaki has several domestic record achievements for her singles , such as the most number @-@ one hits by a female artist ( 38 ) ; the most consecutive number @-@ one hits by a solo artist ( twenty @-@ five ) , and t... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 116,651 | text | {} |
mixed-00042845 | Geometry Problem Isosceles Triangle
Given this isosceles triangle, find angle AMC. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 25,468 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1319247", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 0}} |
mixed-00038124 | From where you are stuck, there is only one little step left: reduce both terms of the LHS to the same denominator. That is, $$\frac{1-x^{k+1}}{1-x} + x^{k+1} = \frac{1-x^{k+1}+(1-x) x^{k+1}}{1-x}=\frac{1-x^{k+1}+x^{k+1} - x\cdot x^{k+1}}{1-x} = \frac{1-x^{k+2}}{1-x}.$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,561 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1488283", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00029250 | USS Henry R. Mallory ( ID @-@ 1280 ) was a transport for the United States Navy during World War I. She was also sometimes referred to as USS H. R. Mallory or as USS Mallory . Before her Navy service she was USAT Henry R. Mallory as a United States Army transport ship . From her 1916 launch , and after her World War I ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 70,128 | text | {} |
mixed-00023783 | Particle starts at $(0,-3)$ and moves clockwise around origin on graph $x^2+y^2=9$, find parametric equation Question
particle starts at $(0,-3)$ and moves clockwise around origin on graph $x^2+y^2=9$, revolve in $9$ seconds find parametric equation in term of $t$.
What I've done so far:
I first thought that the grap... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,789 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2141882", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 4, "answer_id": 2}} |
mixed-00035846 | We write the given equation equivalently:
$$
\begin{aligned}
0 &= x^2y^2 - 2x^2y + 2xy^2 + x^2 - 4xy + y^2 + 2x - 2y - 3\ ,\\
0 &= x^2(y-1)^2 + 2x(y-1)^2 + (y-1)^2-4\ ,\\
4 &= (x+1)^2(y-1)^2\ .
\end{aligned}
$$
Now consider all possible ways to write $4$ as a product of two perfect squares. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 21,150 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3968900", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 0}} |
normal-00017038 | Back in La Pointe , Buffalo took several actions to forestall and prevent removal . He and other leaders petitioned the US government for the next two years to no avail . They did win considerable sympathy from whites who learned of the debacle in Sandy Lake . Newspapers throughout the Lake Superior region ran editoria... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 40,796 | text | {} |
mixed-00039037 | Since the characteristic polynomial is $z^2-7z+10 = (z-2)(z-5)$, the solutions of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 0 $$
have the form $S_n = \alpha 2^n+\beta 5^n$. By direct inspection a solution of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 5\cdot 3^n $$
is given by $S_n=-\frac{45}{2}\cdot3^n$, hence the set of solutions of t... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 23,131 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2312712", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 0}} |
normal-00011750 | Sidney Colvin , in his 1917 biography on Keats , grouped " Indolence " with the other 1819 odes in categorizing Keats 's " class of achievements " . In 1948 , Lord Gorell described the fifth stanza as , " lacking the magic of what the world agrees are the great Odes " but describes the language as " [ d ] elicate , cha... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 28,580 | text | {} |
latex-00009554 | L_{\vec V } \phi (\vec {X},\varepsilon ) = \kappa(\vec{X},\varepsilon ) \phi (\vec {X},\varepsilon ). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,571 | latex_formula | {"original_latex": "\\begin{align*}L_{\\vec V } \\phi (\\vec {X},\\varepsilon ) = \\kappa(\\vec{X},\\varepsilon ) \\phi (\\vec {X},\\varepsilon ).\\end{align*}"} |
latex-00007477 | \Big|\int_{\mathbb{R}^3}\widehat{f_1}(\xi)\widehat{f_2}(\eta)\widehat{f_3}(\rho-\xi)&\widehat{f_4}(-\rho-\eta)m(\xi,\eta,\rho) d\xi d\rho d\eta\Big|\\&\lesssim \|f_1\|_{L^{p_1}}\|f_2\|_{L^{p_2}}\|f_3\|_{L^{p_3}}\|f_4\|_{L^{p_4}}\|\mathcal{F}^{-1}m\|_{L^1}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 7,486 | latex_formula | {"original_latex": "\\begin{align*}\\Big|\\int_{\\mathbb{R}^3}\\widehat{f_1}(\\xi)\\widehat{f_2}(\\eta)\\widehat{f_3}(\\rho-\\xi)&\\widehat{f_4}(-\\rho-\\eta)m(\\xi,\\eta,\\rho)\\,d\\xi d\\rho d\\eta\\Big|\\\\&\\lesssim \\|f_1\\|_{L^{p_1}}\\|f_2\\|_{L^{p_2}}\\|f_3\\|_{L^{p_3}}\\|f_4\\|_{L^{p_4}}\\|\\mathcal{F}^{-1}m\\|... |
normal-00047144 | The underlying productivity of English agriculture remained low , despite the increases in food production . Wheat prices fluctuated heavily year to year , depending on local harvests ; up to a third of the grain produced in England was potentially for sale , and much of it ended up in the growing towns . Despite their... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 114,412 | text | {} |
normal-00048429 | According to Time magazine , Tikhonov was a " tried and tested yes man " who had very little experience in foreign and defence policy when he took over the Premiership from Alexei Kosygin . A bust dedicated to Tikhonov can be found in Kharkiv , his birthplace . Tikhonov , when compared to other Soviet premiers , has ma... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 117,744 | text | {} |
latex-00030658 | \aligned\Delta_p(A)& =\frac2{p^2} \underset{x\in\Omega}{{\rm ess}\inf}\min_{|\xi|=1}\min_{|\zeta|=1}H_{F_p}^{A(x)}[\zeta;\xi] .\endaligned | latex | OleehyO/latex-formulas | cleaned_formulas | train | 30,946 | latex_formula | {"original_latex": "\\begin{align*}\\aligned\\Delta_p(A)& =\\frac2{p^2}\\,\\underset{x\\in\\Omega}{{\\rm ess}\\inf}\\min_{|\\xi|=1}\\min_{|\\zeta|=1}H_{F_p}^{A(x)}[\\zeta;\\xi]\\,.\\endaligned\\end{align*}"} |
latex-00023012 | \mathcal{R}(I)=\bigcup_{s\in I} (Z_s). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 23,180 | latex_formula | {"original_latex": "\\begin{align*} \\mathcal{R}(I)=\\bigcup_{s\\in I} (Z_s). \\end{align*}"} |
mixed-00002034 | $\begin{array}\\
f(x)
&=\frac{1}{1+e^x}\\
&=\frac{1}{2+(e^x-1)}\\
&=\frac12\frac{1}{1+(e^x-1)/2}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n\frac{(e^x-1)^n}{2^n}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n\frac{(x+x^2/2+x^3/6+...)^n}{2^n}\\
&=\frac12\sum_{n=0}^{\infty}(-1)^n(x/2)^n(1+x/2+x^2/6+...)^n\\
&=\frac12\sum_{n=0}^{\infty}(-... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,034 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1404886", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 5, "answer_id": 4}} |
mixed-00024974 | Since $\tan(x)=2i$
Using Euler's formula
$\tan(x)={\frac {e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}}$ witch must equal $2i$
Manipulating a bit we get $3e^{ix}+e^{-ix}=0$ and if $e^{ix}=y$, $3y+\frac{1}{y}=0$ thus $e^{ix}=y=±i \frac{\sqrt 3}{3}$
This, of course, means that $\ln\left(±i \frac{\sqrt 3}{3}\right)=ix$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,518 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3303058", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 1}} |
normal-00031308 | Back in the world of academia , Du Bois was able to resume his study of Reconstruction , the topic of the 1910 paper that he presented to the American Historical Association . In 1935 , he published his magnum opus , Black Reconstruction in America . The book presented the thesis , in the words of the historian David L... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 75,144 | text | {} |
mixed-00030380 | Assuming that
$x_0 > 0$,
$x_{n+1}
=3x_n + \frac{2}{x_n^2}
\gt 3x_n
$
so
$x_n > 3^n x_0$.
Also
$x_{n+1}
=3x_n + \frac{2}{x_n^2}
\lt 3x_n+\frac{2}{(3^nx_0)^2}
= 3x_n+\frac{2}{9^nx_0^2}
$
so
$\dfrac{x_{n+1}}{3^{n+1}}
\lt \dfrac{x_n}{3^n}+\frac{2}{27^nx_0^2}
$.
Letting
$y_n = \dfrac{x_n}{3^n}$,
$y_{n+1}-y_n
\lt \frac{2}{27... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 17,817 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3671525", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
latex-00005535 | A_1~=~\frac{4 G_{10-c}\Gamma(\frac{7-c}{2})}{\pi^{\frac{7-c}{2}}r^{7-c}}~m_p~dt. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 5,541 | latex_formula | {"original_latex": "\\begin{align*}A_1~=~\\frac{4 G_{10-c}\\Gamma\\left(\\frac{7-c}{2}\\right)}{\\pi^{\\frac{7-c}{2}}r^{7-c}}~m_p~dt.\\end{align*}"} |
mixed-00045545 | Find a closed formula (not including $\sum$) for the expression $\sum_{k=0}^{n-1}\binom{2n}{2k+1}$ Find a closed formula (not including $\sum$) for the expression
$$\sum_{k=0}^{n-1}\binom{2n}{2k+1}$$
I started by using the fact that
$$\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$$
to get that
$$\sum_{k=0}^{n-1}\binom{2... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 27,139 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3865569", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
normal-00006187 | The cathedral , along with the rest of the city , has been sinking into the lakebed from the day it was built . However , the fact that the city is a megalopolis with over 18 million people drawing water from underground sources has caused water tables to drop , and the sinking to accelerate during the latter half of t... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 15,199 | text | {} |
mixed-00031434 | On the same line (!) of thought: the line $\,l:y=2x\,$ is the same as the vector space $\,\operatorname{Span}\{(1,2)\}\leq\mathbb{R}^2$ , or if you prefer: $\,l:\{(r,2r)\,/\,r\in\mathbb{R}\}\,$ , and then what we really want to happen is $$\begin{pmatrix}k&-2\\1-k&k\end{pmatrix}\begin{pmatrix}1\\2\end{pmatrix}=\begin{p... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 18,459 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/148676", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
normal-00028260 | On February 12 , with insufficient cash to pay his overdue rent and a growing suspicion that police were closing in on him , Bundy stole a car and fled Tallahassee , driving westward across the Florida Panhandle . Three days later at around 1 : 00 a.m. , he was stopped by Pensacola police officer David Lee near the Ala... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 67,712 | text | {} |
latex-00000233 | B_n(\lambda)+B_{n-1}(\lambda)+N\frac{dV^{(\alpha)}(\lambda)}{d \lambda}=\frac{\lambda}{c_{n-1}}A_{n-1}(\lambda) . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 233 | latex_formula | {"original_latex": "\\begin{align*}B_n(\\lambda)+B_{n-1}(\\lambda)+N\\frac{{\\rm d}V^{(\\alpha)}(\\lambda)}{{\\rm d} \\lambda}=\\frac{\\lambda}{c_{n-1}}A_{n-1}(\\lambda) \\ .\\end{align*}"} |
normal-00019509 | Ernest had suffered from a venereal disease in his late teens and early twenties , most likely as the consequence of living a wild , promiscuous lifestyle . These qualities he had inherited under the tutelage of his father , who took his sons to " sample the pleasures " of Paris and Berlin , to Albert 's " horror and s... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 46,826 | text | {} |
mixed-00043915 | Find a $3\times 3$ matrix $A\not = I_3$ such that $A^3 = I_{3}$ Use the correspondence between matrices and linear transformation to find find a $3\times 3$ matrix $A$ such that $A^3 = I_{3}$ and find an $A$ matrix that is not $I_{3}$
Where $I_{3}$ is the identity matrix:
$$I_{3}=
\left[ {\begin{array}{ccc}
1 & 0 & 0... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 26,124 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2247731", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 3, "answer_id": 0}} |
latex-00007174 | <\phi^2(x)>=.{d\over ds}|_{s=0}{s\over\mu^2}\zeta(s+1,x|L_b/\mu^2)=\lim_{s\rightarrow0}[(1+s\ln\mu^2)\zeta(s+1,x|L_b)+s\zeta'(s+1|L_b)], | latex | OleehyO/latex-formulas | cleaned_formulas | train | 7,183 | latex_formula | {"original_latex": "\\begin{align*}\\left<\\phi^2(x)\\right>=\\left.{d\\over ds}\\right|_{s=0}{s\\over\\mu^2}\\zeta(s+1,x|L_b/\\mu^2)=\\lim_{s\\rightarrow0}\\left[(1+s\\ln\\mu^2)\\zeta(s+1,x|L_b)+s\\zeta'(s+1|L_b)\\right],\\end{align*}"} |
mixed-00042916 | By the Riemann-Dini theorem, we may take any series that is conditionally convergent but not absolutely convergent and rearrange it in order to get a series that converges to $\alpha$, for any $\alpha\in\mathbb{R}$.
In our case:
$$\begin{eqnarray*} \sum_{k\geq 0}\left(\frac{1}{4k+1}+\frac{1}{4k+3}-\frac{1}{2k+2}\right)... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 25,510 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1369198", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 4, "answer_id": 0}} |
mixed-00005549 | I just solved the equation with the following method:
$$\color{red}{(x-2)}\color{green}{(x+1)(x+6)}\color{red}{(x+9)}+108=0$$
$$(x^2+7x-18)(x^2+7x+6)+108=0$$
By using the substitution $t=x^2+7x$ we get,
$$t^2-12t=0\Rightarrow t_1=0\quad,t_2=12$$So we have,
$x^2+7x=0\Rightarrow \quad x_1=0 ,\quad x_2=-7$
$x^2+7x-12=0\Ri... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 2,816 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4198258", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 4, "answer_id": 0}} |
latex-00020630 | A_1^{\frac{1}{k}}(z)&=(A+h)^{\frac{1}{k}}\ &=A^{\frac{1}{k}}(1+ \frac{h}{A})^{\frac{1}{k}}\ &= A^{\frac{1}{k}}(1+ O({\frac{|h|}{|A|}})), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,763 | latex_formula | {"original_latex": "\\begin{align*} A_1^{\\frac{1}{k}}(z)&=(A+h)^{\\frac{1}{k}}\\\\ &=A^{\\frac{1}{k}}\\left(1+ \\frac{h}{A}\\right)^{\\frac{1}{k}}\\\\ &= A^{\\frac{1}{k}}\\left(1+ O\\left({\\frac{|h|}{|A|}}\\right)\\right),\\end{align*}"} |
normal-00049228 | On the June 14 episode of Impact Wrestling , Hardy entered the 2012 Bound for Glory Series , taking part in the opening gauntlet match , from which he was the first man eliminated by Bully Ray . Hardy wrestled his final group stage match of the tournament on the September 6 episode of Impact Wrestling , defeating Samoa... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 119,853 | text | {} |
latex-00005144 | [Z,X,X]^{I,J,I-1} = [Z^{I-J},X^1,Z^{J-I}] =0, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 5,149 | latex_formula | {"original_latex": "\\begin{align*}[Z,X,X]^{I,J,I-1} = [Z^{I-J},X^1,Z^{J-I}] =0,\\end{align*}"} |
normal-00033056 | Across the series , Sora is depicted as a cheerful teenager who cherishes his friendships and relies on them for his strength . As a result , several of Sora 's enemies use his friends as bait to use the Keyblade for their purposes . Although Sora was not chosen by the Keyblade to be its owner and the protector of worl... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 79,691 | text | {} |
normal-00033568 | Like other slow lorises , the Sunda slow loris is an arboreal and nocturnal primate , resting by day in the forks of trees , or in thick vegetation and feeding on fruit and insects by night . Unlike other loris species , it remains in trees most of its life : while the Bengal slow loris will often sleep on the ground ,... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 80,960 | text | {} |
latex-00016271 | {1\over\sqrt{g}}{\delta(S+{\cal A})\over\delta\Phi}\equiv\Delta\Phi-(1+2\xi)R -V'=0 , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 16,309 | latex_formula | {"original_latex": "\\begin{align*}{1\\over\\sqrt{g}}{\\delta(S+{\\cal A})\\over\\delta\\Phi}\\equiv\\Delta\\Phi-(1+2\\xi)R -V'=0 \\ ,\\end{align*}"} |
mixed-00010540 | Find all $x\in \mathbb{R}$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$
Find all $x\in \mathbb{R}$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$
Letting $a=2^x$ and $b=3^x$ we get $$\frac{a^3+b^3}{a^2b+ab^2} = \frac{7}{6}$$
from the numerator we have that $$a^3+b^3=(a+b)(a^2-ab+b^2)=7$$
since $7$ is a pr... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,745 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3804820", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 4, "answer_id": 0}} |
normal-00049698 | A core element of F.E.A.R. is its horror theme , which is heavily inspired by Japanese horror . The design team attempted to keep " [ the ] psychology of the encounter " in the player 's mind at all times , in order to " get under [ the player 's ] skin " , as opposed to the " in your face ' monsters jumping out of clo... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 121,003 | text | {} |
latex-00009534 | A=( \begin{array}{cc} 1 & dx \ 0 & 1 \end{array})\quad B=( \begin{array}{cc} 1 & 0 \ dx & 1 \end{array}) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,551 | latex_formula | {"original_latex": "\\begin{align*} A=\\left( \\begin{array}{cc} 1 & dx \\\\ 0 & 1 \\end{array}\\right)\\quad B=\\left( \\begin{array}{cc} 1 & 0 \\\\ dx & 1 \\end{array}\\right)\\end{align*}"} |
mixed-00003836 | This is
$$\det\pmatrix{x&x^2&x^3-1\\y&y^2&y^3-1\\z&z^2&z^3-1}
=\det\pmatrix{x&x^2&x^3\\y&y^2&y^3\\z&z^2&z^3}
-\det\pmatrix{x&x^2&1\\y&y^2&1\\z&z^2&1}
$$
for $x=e^a$ etc. Both of these are essentially Vandermonde determinants. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,943 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2623058", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
normal-00037243 | Jardine 's achievements in the season were widely reported in the local and national press . He went on to play in two representative schools matches at Lord 's Cricket Ground , where he scored 44 , 91 , 57 and 55 in two matches and won favourable reviews in the press . Wisden , in 1928 , described Jardine at this time... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 89,914 | text | {} |
normal-00019903 | An 1821 opinion of U.S. Attorney General William Wirt , interpreting Fletcher and Johnson , argued that : " The Seneca Indians must be protected in the enjoyment of exclusive possession of their lands , as defined and bounded in the Treaty of Canandaigua , until they have voluntarily relinquished it . " | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 47,707 | text | {} |
mixed-00021619 | Sum of $\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n} $ Calculate the sum of the next series and for which values of $x$ it converges:
$$\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n}$$
I used D'Alembert and found that the limit is less than 1, so: $-5 < x < 1$ (because the fraction must be less than 1).
and then I assigned the... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 12,482 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/348079", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 3, "answer_id": 0}} |
latex-00041918 | a_0(e^{-\pi})=\frac{8 \sqrt{2 \pi} \sqrt{\sqrt{24+14 \sqrt{3}}-3}}{\sqrt[4]{3} \Gamma (-\frac{1}{4})^2} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 42,520 | latex_formula | {"original_latex": "\\begin{align*}a_0\\left(e^{-\\pi}\\right)=\\frac{8 \\sqrt{2 \\pi} \\sqrt{\\sqrt{24+14 \\sqrt{3}}-3}}{\\sqrt[4]{3} \\Gamma \\left(-\\frac{1}{4}\\right)^2}\\end{align*}"} |
normal-00004715 | St Mary 's Church is an Anglican church at the end of a lane to the south of the village of Nether Alderley , Cheshire , England . It dates from the 14th century , with later additions and a major restoration in the late @-@ 19th century . The church is recorded in the National Heritage List for England as a designated... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 11,298 | text | {} |
mixed-00003209 | If $a^2+b^2+c^2+d^2+e^2=5$ so $\sum\limits_{cyc}\frac{1}{7-2a}\leq1$.
Let $a$, $b$, $c$, $d$ and $d$ be non-negative numbers such that $a^2+b^2+c^2+d^2+e^2=5$. Prove that:
$$\frac{1}{7-2a}+\frac{1}{7-2b}+\frac{1}{7-2c}+\frac{1}{7-2d}+\frac{1}{7-2e}\leq1$$
The equality occurs also for $a=2$ and $b=c=d=e=\frac{1}{2}$.... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,627 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2157579", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
normal-00005234 | Belgian reporter Tintin and his dog Snowy travel to the Belgian Congo , where a cheering crowd of native Congolese greet them . Tintin hires a native boy , Coco , to assist him in his travels , and shortly after , Tintin rescues Snowy from a crocodile . A criminal stowaway attempts to kill Tintin , but monkeys throw co... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 12,625 | text | {} |
mixed-00047741 | Residue $\frac{e^z}{z^3\sin(z)}$ I want to find the residue of $$\frac{e^z}{z^3\sin(z)}$$ and get $$ \frac 1 {3!} \lim_{z \to 0} \left( \frac{d^3}{dz^3} \left(\frac{ze^z}{\sin(z)} \right)\right) = \frac{1}{3}$$
Can anyone confirm this? I tried using the Laurent Series, but I didn't know how to compute it.
Oh, I was abl... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,463 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1266716", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
latex-00032907 | &a_{ij}^{(s)}=s\Bigg( {a_{ii}^{(s-1)}a_{ij}^{(1)}-\frac{a_{ij}^{(s-1)}}{x_i-x_j}}\Bigg), i\ne j, 0\leq i\leq M,\\&a_{ii}^{(s)}=-\sum\limits_{j=0,j\ne i}^M {a_{ij}^{(s)}} , i = j, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 33,229 | latex_formula | {"original_latex": "\\begin{align*}&a_{ij}^{(s)}=s\\Bigg( {a_{ii}^{(s-1)}a_{ij}^{(1)}-\\frac{a_{ij}^{(s-1)}}{x_i-x_j}}\\Bigg),\\ \\ i\\ne j,\\ 0\\leq i\\leq M,\\\\&a_{ii}^{(s)}=-\\sum\\limits_{j=0,j\\ne i}^M {a_{ij}^{(s)}} ,\\ \\ i = j,\\end{align*}"} |
mixed-00025410 | Evaluate $\int\limits_{0}^{\infty}\frac{n \sin x}{1+n^2x^2}dx$ Notice that $\frac{n \sin x}{1+n^2x^2}\to 0$ pointwise.
And we have,$$\int\limits_{0}^{\infty}\frac{n \sin x}{1+n^2x^2}dx=\int\limits_{0}^{1}\frac{n \sin x}{1+n^2x^2}dx+\int\limits_{1}^{\infty}\frac{n \sin x}{1+n^2x^2}dx$$
Then for $\int\limits_{1}^{\infty}... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,799 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3789989", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 0}} |
mixed-00028384 | with $$t=\sqrt{1+e^x}$$ we get $$x=\ln(t^2-1)$$ and from here $$dx=\frac{2t}{t^2-1}dt$$ and our integral will be $$\int\frac{2t^2}{t^2-1}dt$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,603 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1793094", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 0}} |
mixed-00036208 | Probability of drawing 4 aces when drawing 5 cards from a regular deck of cards. I cannot seem to wrap my head around how to calculate the probability of drawing four ace when drawing five cards from a deck of cards. My intuition tells me the math below but its wrong for some reason...
$
\frac{\frac{4}{52}\frac{3}{51}\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 21,376 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4419225", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 0}} |
mixed-00024242 | writing Cosines using De Moivre's formula Given the question:
Use De Moivre’s formula to find a formula for $\cos(3x)$ and $\cos(4x)$ in terms of $\cos(x)$ and $\sin(x)$. Then use the identity $\cos^2(x) + \sin^2(x) = 1$ to express these formulas only in terms of $\cos(x)$.
I started out by rewriting $\cos(3x)$:
$\co... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,076 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2532231", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
normal-00023284 | Ashanti as Dorothy Gale : A Kansas teen dreaming of leaving her home and becoming a singer . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 55,910 | text | {} |
latex-00016920 | \partial^\nu {}^*\!F_{\mu\nu}(x) = \tilde{\jmath}_\mu{x}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 16,962 | latex_formula | {"original_latex": "\\begin{align*}\\partial^\\nu {}^*\\!F_{\\mu\\nu}(x) = \\tilde{\\jmath}_\\mu{x},\\end{align*}"} |
normal-00035424 | At the war 's end in 1945 , Ull only had fourteen members . Its average member age was lowered to 65 years after the admission of five new members in 1946 and 1947 . The post @-@ war period also saw the admission to SK Ull of men from more professional groups . One of the new members , Erik Plahte , would serve as chai... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 85,301 | text | {} |
latex-00029666 | H^0(Y,\Omega_Y^1(\log E)\otimes (\rho\circ\nu)_*\mathcal O_Z)=H^0(Z,\Omega_Z^1(\log\Delta)). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 29,944 | latex_formula | {"original_latex": "\\begin{align*}H^0(Y,\\Omega_Y^1(\\log E)\\otimes (\\rho\\circ\\nu)_*\\mathcal O_Z)=H^0(Z,\\Omega_Z^1(\\log\\Delta)).\\end{align*}"} |
mixed-00036738 | Because is says that if $x=\cdots1313_5$ then $3x+1\equiv 0 \text{ mod } 5^n$ for all $n$, which is precisely what it means to be $0$ in $\mathbb{Q}_5$. Thus, you see that $3x+1=0$ so that $\displaystyle x=\frac{-1}{3}$.
EDIT:
Now that I have more time, let me be less glib about this response.
Whenever possible, we wa... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 21,702 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/325427", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 1, "answer_id": 0}} |
latex-00041214 | \{\begin{array}[c]{ll}\overline{L}\widetilde{h}=0 & B_{R}\\\widetilde{h}=\widetilde{u} & \partial B_{R}\end{array}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 41,811 | latex_formula | {"original_latex": "\\begin{align*}\\left\\{\\begin{array}[c]{ll}\\overline{L}\\widetilde{h}=0 & B_{R}\\\\\\widetilde{h}=\\widetilde{u} & \\partial B_{R}\\end{array}\\right. \\end{align*}"} |
normal-00001850 | In an April 3 , 2007 interview with the Harvard Crimson , " Dershowitz confirmed that he had sent a letter last September to DePaul faculty members lobbying against Finkelstein 's tenure . " | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 4,634 | text | {} |
mixed-00014588 | This is a classic problem. Read up on "Jacobi rotations"
Here is the short answer:
Focus on the matrix $ \left( \begin{array}{c c}
a & b\\
b & a
\end{array}
\right)
$.
A Jacobi rotation is given by
$$ \left( \begin{array}{c c}
c & s\\
-s & c
\end{array}
\right)
$$
where $ c = \cos( \theta ) $ and $ s = \sin( \theta )... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,189 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2721139", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
latex-00012876 | \nabla_{k}(A)\nabla^{k}(A) A^{0} = 0 , {\bf x} \in V \subset {\bf R}^{3} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 12,904 | latex_formula | {"original_latex": "\\begin{align*}\\nabla_{k}(A)\\nabla^{k}(A) A^{0} = 0\\;, {\\bf x} \\in V \\subset {\\bf R}^{3}\\end{align*}"} |
normal-00014024 | " Between the Lines " . Time . 4 May 1970 . Retrieved 10 April 2007 . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 33,778 | text | {} |
mixed-00018934 | Three Distinct Points and Their Normal Lines
Suppose That three points on the graph of $y=x^2$ have the property that their normal lines intersect at a common point. Show that the sum of their $x$-coordinates is $0$.
I have a lot going but can not finish it.
Proof:
Let $(a,a^2)$, $(b,b^2)$, and $(c,c^2)$ be three dis... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,838 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2192225", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 0}} |
normal-00042537 | In 1950 the population in the Glenrothes designated area was approximately 1 @,@ 000 people who were located in the hamlets of Woodside and Cadham and in the numerous farm steadings that were spread throughout the area . Population growth in the early phases of the town was described as being slow due to the dependence... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 102,976 | text | {} |
mixed-00042735 | The remainder when dividing by $x-3$ is also the polynomial evaluated at $x=3$, which is $27-4\cdot 9-5=-14$.
Indeed,
$$\begin{align}x^3-4x^2-5&=x^2\cdot(x-3)-x^2-5\\&=(x^2-x)(x-3)-3x-5\\&=(x^2-x-3)(x-3)-14\end{align}$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 25,400 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1214946", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
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