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 |
|---|---|---|---|---|---|---|---|---|
normal-00001094 | In a 1929 article appearing in the Lowell Sun newspaper , physician Morris Fishbein speculated that for Ali 's nut feat , the one nut of a different variety was held in the mouth rather than swallowed , thus allowing him to produce it on cue . Dr. Fishbein also stated that unnamed " investigators " were convinced that ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 2,697 | text | {} |
normal-00040483 | The second track on the three @-@ song single , " Buffy Theme " , was also recorded at the July 2002 session . This composition was originally performed by the group Nerf Herder , whose version was included on Buffy the Vampire Slayer : The Album . It was written by Nerf Herder members Dennis , Grip , and Sherlock . Ki... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 98,098 | text | {} |
normal-00049320 | The dominant language of the region is Mayali , a dialect of Bininj Gun @-@ Wok traditionally associated with the region surrounding Maningrida , in Western Arnhem Land . As it is a strong language with hundreds of speakers and a high rate of child acquisition , members of the Wagiman ethnic group gradually ceased teac... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 120,034 | text | {} |
latex-00032572 | \int_0^{\frac{\pi}{2}} e^{2zx} \log( \tan x ) \d x &= \int_0^{\frac{\pi}{2}} e^{2z(\frac{\pi}{2} - x )} \log( \tan( \frac{\pi}{2} - x) ) \d x \ &= - e^{\pi z} \int_0^{\frac{\pi}{2}} e^{-2zx} \log( \tan x ) \d x, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 32,888 | latex_formula | {"original_latex": "\\begin{align*} \\int_0^{\\frac{\\pi}{2}} e^{2zx} \\log\\left( \\tan x \\right) \\d x &= \\int_0^{\\frac{\\pi}{2}} e^{2z\\left(\\frac{\\pi}{2} - x \\right)} \\log\\left( \\tan\\left( \\frac{\\pi}{2} - x\\right) \\right) \\d x \\\\ &= - e^{\\pi z} \\int_0^{\\frac{\\pi}{2}} e^{-2zx} \\log\\left( \\tan... |
mixed-00016477 | The roots of $t^5+1$ Just a quick question, how do we go about finding the roots of $t^5 +1$?
I can see that since $t^5=-1$ that an obvious root is $\sqrt[5]{-1}$.
I am assuming that since there is a $-1$ involved, some of the factors will be complex?
Any help would be welcome. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,348 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/138382", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 1}} |
mixed-00041367 | Very much like the example II in this wiki page, we write:
$$
\int_C \frac{\mathrm{e}^{2 i z}}{(z^2+1)(z^2+4)^2} \mathrm{d} z =
\int_{-\infty}^\infty \frac{\mathrm{e}^{2 i x}}{(x^2+1)(x^2+4)^2} \mathrm{d} x + \int_{0}^{\pi} \frac{\exp(2 i R \mathrm{e}^{i \varphi} )}{((R \mathrm{e}^{i \varphi})^2+1)((R \mathrm{e}^{i ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 24,568 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/70461", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 1, "answer_id": 0}} |
latex-00009038 | (3A) \colon & g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:t_1:t_2:t_3)\\(3C) \colon & g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:\epsilon_3 t_1:t_2:t_3)\\(3D) \colon & g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:\epsilon_3^2 t_1:t_2:t_3) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,053 | latex_formula | {"original_latex": "\\begin{align*}(3A) \\colon & g(t_0:t_1:t_2:t_3) = (\\epsilon_3 t_0:t_1:t_2:t_3)\\\\(3C) \\colon & g(t_0:t_1:t_2:t_3) = (\\epsilon_3 t_0:\\epsilon_3 t_1:t_2:t_3)\\\\(3D) \\colon & g(t_0:t_1:t_2:t_3) = (\\epsilon_3 t_0:\\epsilon_3^2 t_1:t_2:t_3)\\end{align*}"} |
normal-00011662 | On 1 March 2012 , Blackpool manager Ian Holloway confirmed that Fowler was training with the Seasiders and that he could earn a deal until the end of the season . However , they could not agree a deal and Fowler decided against signing when Karl Oyston offered the striker just £ 100 a week with £ 5 @,@ 000 for every fi... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 28,421 | text | {} |
normal-00001741 | The Drowning House , a novel by Elizabeth Black ( 2013 ) , is an exploration of the island of Galveston , Texas , and the intertwined histories of two families who reside there . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 4,425 | text | {} |
latex-00004564 | f^{b}_{\omega' lm}= \int d\tilde{\omega} d\theta_0 d\eta_0 (\alpha^{b}_{\omega \omega' lm} p_\omega + \beta^{b}_{\omega \omega' lm}\bar{p}_\omega + \mbox{terms involving }q_\omega), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 4,568 | latex_formula | {"original_latex": "\\begin{align*}f^{b}_{\\omega' lm}= \\int d\\tilde{\\omega}\\, d\\theta_0\\, d\\eta_0\\,(\\alpha^{b}_{\\omega \\omega' lm} p_\\omega + \\beta^{b}_{\\omega \\omega' lm}\\bar{p}_\\omega + \\mbox{terms involving }q_\\omega),\\end{align*}"} |
normal-00000219 | According to Fernandez , she had aspired to become an actress at a young age and fantasized about becoming a Hollywood movie star . She received some training at the John School of Acting . Although , she was a television reporter , she accepted offers in the modeling industry , which came as a result of her pageant su... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 516 | text | {} |
mixed-00007625 | \begin{align}
& x^2 - 3\sqrt{5}\,x+ 2 = (x-a)(x-b) \\[6pt]
= {} & x^2 - (a+b) x + ab.
\end{align}
Therefore $3\sqrt 5= a+b$ and $2=ab$.
Hence $3\sqrt 5 = \tan x + \tan y$ and $2 = \tan x\tan y$.
So
$$
\tan(x+y) = \frac{\tan x+\tan y}{1-\tan x\tan y} = \frac{a+b}{1-ab} = \frac{3\sqrt 5}{1-2},
$$
and finally,
$$
3\cot(x+... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 4,004 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1285409", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 3, "answer_id": 2}} |
normal-00021297 | It had a White 160AX 147 horsepower ( 110 kW ) , 386 cubic inch ( 6 @,@ 330 cc ) , 6 @-@ cylinder engine with a compression ratio of 6 @.@ 44 : 1 . It had a 150 mile ( 240 km ) range , 60 US gal ( 230 l ) fuel tank , a speed of 47 mph ( 75 km / h ) , and a power to weight ratio of 14 @.@ 7 hp per ton . It was armed wit... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 50,978 | text | {} |
latex-00040631 | \tilde{R}_n(T)&\leq T\log_2(1+\tfrac{\eta\rho g_n(1-T)}{T}), \forall n\in\mathcal{N}\\\sum_{n\in\mathcal{M}_k} \tilde{R}_n(T)&\leq T\log_2(1+\tfrac{\eta\rho (1-T)\sum g_n}{T}), \forall k: \mathcal{M}_k\subseteq\mathcal{N}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 41,143 | latex_formula | {"original_latex": "\\begin{align*}\\tilde{R}_n(T)&\\leq T\\log_2\\left(1+\\tfrac{\\eta\\rho g_n\\left(1-T\\right)}{T}\\right),\\,\\forall n\\in\\mathcal{N}\\\\\\sum_{n\\in\\mathcal{M}_k} \\tilde{R}_n(T)&\\leq T\\log_2\\left(1+\\tfrac{\\eta\\rho \\left(1-T\\right)\\sum g_n}{T}\\right),\\,\\forall k: \\mathcal{M}_k\\sub... |
latex-00022088 | x^2y=z^2+1 | latex | OleehyO/latex-formulas | cleaned_formulas | train | 22,246 | latex_formula | {"original_latex": "\\begin{align*}x^2y=z^2+1\\end{align*}"} |
mixed-00000823 | How prove this inequality $\sum\limits_{k=1}^{n}\frac{1}{k!}-\frac{3}{2n}<\left(1+\frac{1}{n}\right)^n<\sum\limits_{k=1}^{n}\frac{1}{k!}$ show that
$$\sum_{k=1}^{n}\dfrac{1}{k!}-\dfrac{3}{2n}<\left(1+\dfrac{1}{n}\right)^n<\sum_{k=0}^{n}\dfrac{1}{k!}(n\ge 3)$$
Mu try: I konw
$$\sum_{k=0}^{\infty}\dfrac{1}{k!}=e$$
and I ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 420 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/562109", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
mixed-00030829 | Let $a = \sqrt{2} \cos u, b = \sqrt{2} \sin u$ which immediately satisfies the condition. Using Simon's Favorite Factoring Trick, $9 + 3a + 3b + ab = (a + 3)(b + 3)$, so by C-S:
$$(\sqrt{2} \cos u + 3)(\sqrt{2} \sin u + 3) ≥(\sqrt{\sqrt{2} \cos u} \sqrt{\sqrt{2} \sin u} + \sqrt{3} \sqrt{3})^2$$
$$= (\sqrt{2 \cos u \sin... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 18,089 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4171501", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 5, "answer_id": 3}} |
mixed-00006930 | When is the sum of consecutive squares a prime? For what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime?
I tried use $$1^2+2^2+...+n^2=\frac {n(n+1)(2n+1)}{6}$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 3,602 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/763773", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 0}} |
normal-00015492 | 4 × member of 50 – 40 – 90 Club : ( 2006 , 2008 – 10 ) | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 37,127 | text | {} |
mixed-00035249 | $$P_a=(a+2)(a+1)x^{a+4}-2(a+4)(a+1)x^{a+3}+(a+4)(a+3)x^{a+2}-2(a+4)x+2(a+1)$$
Offset $a$ down by one:
$$P'_a=(a+1)ax^{a+3}-2(a+3)ax^{a+2}+(a+3)(a+2)x^{a+1}-2(a+3)x+2a$$
Subst $z = x-1$
$$P'_a=(a+1)a(z+1)^{a+3}-2(a+3)a(z+1)^{a+2}+(a+3)(a+2)(z+1)^{a+1}-2(a+3)(z+1)+2a \\
= \sum_i \left[ (a+1)a\binom{a+3}{i} - 2(a+3)a\bino... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 20,789 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3341381", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
mixed-00025675 | a) How did they get to the conclusion gcd(m+2, m-2)=1?$.
Let $\gcd(m+2, m-2) = d$ then $d|m+2, m-2$ so $d|(m+2) - (m-2) = 4$. So $d = 1,2,4$. But $m$ is odd (so your example of $m=10$ won't work) so $m\pm 2$ is odd. So $d \ne 2,4$. So $d = 1$.
Note: the general idea that if $\gcd(a,b)$ divides $a,b$ it will divide any... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,956 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4101176", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 2}} |
latex-00013145 | \psi_{ab} = 2 D_{[a}^{\dagger}M_{b]}(M^{\dagger}M)^{-1}\psi +M_c^{\dagger}\psi_{abc} . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 13,173 | latex_formula | {"original_latex": "\\begin{align*}\\psi_{ab} = 2 D_{[a}^{\\dagger}M_{b]}(M^{\\dagger}M)^{-1}\\psi +M_c^{\\dagger}\\psi_{abc} \\ .\\end{align*}"} |
normal-00025229 | Simpson was a keen supporter of the Essendon Football Club , serving as its vice president from 1947 to 1964 . He was awarded a life membership in 1957 . He was elected Victorian State President of the Australian Legion of Ex @-@ Servicemen and Women in 1948 . He resigned in October after a dispute with the State Counc... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 60,490 | text | {} |
mixed-00003603 | What is the probability of getting the sum $26$ when $7$ chips are taken out? Suppose you have a bag in which there are $10$ chips numbered $0$ to $9$. You take out a chip at random, note its number and then put it back. This process is done $7$ times and after that the numbers are added.
What is the probability that... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 1,826 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2435798", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 1}} |
mixed-00022175 | Area of a triangle in terms of areas of certain subtriangles In triangle $ABC$ , $X$ and $Y$ are points on sides $AC$ and $BC$ respectively . If $Z$ is on the segment $XY$ such that $\frac{AX}{XC} = \frac{CY}{YB} = \frac{XZ}{ZY}$ , then how to prove that the area of triangle $ABC$ is given by $[ABC]=([AXZ] ^{1/3} + [BY... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 12,818 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/836875", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
latex-00009599 | \mathbf{1}^{\sigma}_d = \sum_{(d^{\bullet}; d^{\infty})} \mathbf{1}_{d^1}^{\theta } \cdots \mathbf{1}_{d^n}^{\theta } \star \mathbf{1}^{\sigma,\theta }_{d^{\infty}}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 9,616 | latex_formula | {"original_latex": "\\begin{align*}\\mathbf{1}^{\\sigma}_d = \\sum_{(d^{\\bullet}; d^{\\infty})} \\mathbf{1}_{d^1}^{\\theta } \\cdots \\mathbf{1}_{d^n}^{\\theta } \\star \\mathbf{1}^{\\sigma,\\theta }_{d^{\\infty}}.\\end{align*}"} |
normal-00029481 | CEO Borislow invented the product in 2007 and had applied for patents from the U.S. government while he and Donald Burns shared the payment of $ 25 million to start up the company . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 70,667 | text | {} |
normal-00042204 | Several names were written on a large blackboard which hung on the wall , plainly visible , and we had to keep a sharp look @-@ out for any mention of these in letters we read . The names were those of persons suspected of sending information to Germany via neutral countries . In addition , a short sentence was scribbl... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 102,346 | text | {} |
normal-00006321 | A second operation was planned for 10 May using the cruiser HMS Vindictive and proved more successful , but ultimately it also failed to completely close off Bruges . A third planned operation was never conducted as it rapidly became clear that the new channel carved at Zeebrugge was enough to allow access for U @-@ bo... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 15,539 | text | {} |
latex-00044601 | \Sigma(\Pi,\theta_1,\theta_2)= \bigcap_{\theta\in(-\theta_1,\theta_2)}\Sigma(\Pi,\theta) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 45,230 | latex_formula | {"original_latex": "\\begin{align*}\\Sigma(\\Pi,\\theta_1,\\theta_2)= \\bigcap_{\\theta\\in(-\\theta_1,\\theta_2)}\\Sigma(\\Pi,\\theta)\\end{align*}"} |
normal-00006461 | In the latter half of 1964 and 1965 , Dylan moved from folk songwriter to folk @-@ rock pop @-@ music star . His jeans and work shirts were replaced by a Carnaby Street wardrobe , sunglasses day or night , and pointed " Beatle boots " . A London reporter wrote : " Hair that would set the teeth of a comb on edge . A lou... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 15,879 | text | {} |
latex-00022655 | \langle \phi,\psi \rangle = \frac{1}{|G|} \sum_{g\in G} \phi(g) \overline{\psi(g)}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 22,818 | latex_formula | {"original_latex": "\\begin{align*}\\langle \\phi,\\psi \\rangle = \\frac{1}{|G|} \\sum_{g\\in G} \\phi(g) \\overline{\\psi(g)}.\\end{align*}"} |
latex-00012794 | \hbar \triangle = \hbar \partial_p^W \cdot iD^F(x) \ll 1, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 12,822 | latex_formula | {"original_latex": "\\begin{align*}\\hbar \\triangle = \\hbar \\partial_p^W \\cdot iD^F(x) \\ll 1,\\end{align*}"} |
normal-00027037 | Jan Pinkava came up with the concept in 2000 , creating the original design , sets and characters and core storyline , but he was never formally named the director of the film . Lacking confidence in Pinkava 's story development , Pixar management replaced him with Bird in 2005 . Bird was attracted to the film because ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 64,699 | text | {} |
normal-00009936 | O 'Brien was commissioned into the United States Navy on 22 May 1915 under the command of Lieutenant Commander C. E. Courtney , after which she conducted her shakedown cruise between Newport , Rhode Island , and Hampton Roads , Virginia . In fleet exercises off New York in November , O 'Brien collided with the destroye... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 24,338 | text | {} |
mixed-00012812 | $7x+11y=100$
$7x=100-11y$
$x=\frac{100-11y}7=14-2y+\frac{2+3y}7$
$a=\frac{2+3y}7$
$7a=2+3y$
$3y=-2+7a$
$y=\frac{-2+7a}3=-1+2a+\frac{1+a}3$
$b=\frac{1+a}3$
$3b=1+a$
$a=3b-1$
$y=\frac{-2+7(3b-1)}3=\frac{-9+21b}3=-3+7b$
$x=\frac{100-11(-3+7b)}7=\frac{133-77b}7=19-11b$
$\begin{matrix}
x\gt 0&\to&19-11b\gt 0&\to&11b\lt 19&\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 7,108 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1265426", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "33", "answer_count": 8, "answer_id": 6}} |
mixed-00015636 | $$ \Delta = \begin{vmatrix} \tan x&\tan(x+h)&\tan(x+2h)\\
\tan(x+2h)&\tan x&\tan(x+h)\\
\tan(x+h)&\tan(x+2h)&\tan x \end{vmatrix} $$
$$=\begin{vmatrix} \tan x&\tan(x+h)-\tan x&\tan(x+2h)-\tan x\\
\tan(x+2h)&\tan x-\tan(x+2h)&\tan(x+h)-\tan(x+2h)\\
\tan(x+h)&\tan(x+2h)-\tan(x+h)&\tan x-\tan(x+h) \end{vmatrix} $$
$$=\beg... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,829 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3780732", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 3, "answer_id": 0}} |
latex-00020450 | \aligned\{ \begin{array}{lll}-\Delta u+\lambda_1u=|u|^{2r_1-2}u+\nu p|v|^q|u|^{p-2}u & \mathbb{R}^N,\\-\Delta v+\lambda_2v=|v|^{2r_2-2}v+\nu q|u|^p|v|^{q-2}v & \mathbb{R}^N,\\\int_{\mathbb{R}^N}u^2=a^2, \int_{\mathbb{R}^N}v^2=b^2.\end{array}.\endaligned | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,581 | latex_formula | {"original_latex": "\\begin{align*}\\aligned\\left\\{ \\begin{array}{lll}-\\Delta u+\\lambda_1u=|u|^{2r_1-2}u+\\nu p|v|^q|u|^{p-2}u\\ & \\mathbb{R}^N,\\\\-\\Delta v+\\lambda_2v=|v|^{2r_2-2}v+\\nu q|u|^p|v|^{q-2}v\\ & \\mathbb{R}^N,\\\\\\int_{\\mathbb{R}^N}u^2=a^2,\\ \\int_{\\mathbb{R}^N}v^2=b^2.\\end{array}\\right.\\en... |
latex-00020455 | \aligned P_\nu(u,v)=&|\nabla u|^2_2+|\nabla v|^2_2-\int_{\mathbb{R}^N}\bigl[(I_\mu*|u|^{2^*_\mu})|u|^{2^*_\mu}+(I_\mu*|v|^{2^*_\mu})|v|^{2^*_\mu}\bigr]\\&-\nu(\gamma_p+\gamma_q)\int_{\mathbb{R}^N}(I_\mu*|u|^{p})|v|^{q}=0.\endaligned | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,586 | latex_formula | {"original_latex": "\\begin{align*}\\aligned P_\\nu(u,v)=&|\\nabla u|^2_2+|\\nabla v|^2_2-\\int_{\\mathbb{R}^N}\\bigl[(I_\\mu*|u|^{2^*_\\mu})|u|^{2^*_\\mu}+(I_\\mu*|v|^{2^*_\\mu})|v|^{2^*_\\mu}\\bigr]\\\\&-\\nu(\\gamma_p+\\gamma_q)\\int_{\\mathbb{R}^N}(I_\\mu*|u|^{p})|v|^{q}=0.\\endaligned\\end{align*}"} |
latex-00006262 | {\cal A}_{\beta} = \frac{1}{8{\pi}^2}[-2(\beta/l^2)l^{-2}e\cdot R\cdot e + (\beta/l^2)^2 R^{ab}{\mbox{\tiny $\wedge$}}R_{ab} + 2(\beta/l^2)^2 l^{-2} T\cdot T] + O[(\beta/l)^{-2})]. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 6,270 | latex_formula | {"original_latex": "\\begin{align*}{\\cal A}_{\\beta} = \\frac{1}{8{\\pi}^2}[-2(\\beta/l^2)l^{-2}e\\cdot R\\cdot e + (\\beta/l^2)^2 R^{ab}{\\mbox{\\tiny $\\wedge$}}R_{ab} + 2(\\beta/l^2)^2 l^{-2} T\\cdot T] + O[(\\beta/l)^{-2})].\\end{align*}"} |
normal-00038126 | Production – Bloodshy & Avant , Steven Lunt | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 92,190 | text | {} |
latex-00024230 | 0\leq J(u^{n})\le J_n(u^{n})\le J_n(u^{a})=J(u^{a}) \le M, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 24,401 | latex_formula | {"original_latex": "\\begin{align*} 0\\leq J(u^{n})\\le J_n(u^{n})\\le J_n(u^{a})=J(u^{a}) \\le M,\\end{align*}"} |
mixed-00022614 | $$3k^2 + 3k + 12=3(k^2 + k +4)= 3(k(k+1)+4)$$
Can you see why $k(k+1)$ and $4$ are each divisible by $2$?
At least one of $k, k+1$ is even, as is $4$, hence $2$ divides $(k(k+1)+4)$, and with three as a factor of $\color{blue}{3}(k(k+1)+ 4)$, we have $$2\cdot 3 = 6\mid (3k^2 + 3k + 12).$$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 13,080 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1204306", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 5, "answer_id": 1}} |
mixed-00037620 | Limit of function using Taylor's Formula To find:
$$\lim_{x \to 0} \left(\frac{ \sin(x)}{x}\right)^\frac{1}{x}$$
by using Taylor's formula.
So I used the Taylor's formula for $\sin(x)$ and got::
$\sin(x) = x - \frac{x^3}{6} + O(x^4)$
And then my function becomes:
$$\lim_{x \to 0} \left(\frac{ x - \frac{x^3}{6} + O(x^2)... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,240 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1064015", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 3, "answer_id": 2}} |
mixed-00027652 | Let
$$S(n) = \frac{a^{\frac{1}{n}}}{n + 1} + \frac{a^{\frac{2}{n}}}{n + \frac{1}{2}} + \cdots + \frac{a^{\frac{n}{n}}}{n + \frac{1}{n}}.$$
Then
$$\frac{a^{\frac{1}{n}}}{n + 1} + \frac{a^{\frac{2}{n}}}{n + 1} + \cdots + \frac{a^{\frac{n}{n}}}{n + 1} < S(n) < \frac{a^{\frac{1}{n}}}{n} + \frac{a^{\frac{2}{n}}}{n} +\cdots... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,155 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1195305", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 2, "answer_id": 0}} |
latex-00021385 | G=\omega_Q^{1/\bar{p}_2-1/\bar{p}_1}\int_{\mathbb H^n}\frac{|y|_h^{-Q/p}}{\max(1,|y|_h^Q)}dy. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 21,532 | latex_formula | {"original_latex": "\\begin{align*}G=\\omega_Q^{1/\\bar{p}_2-1/\\bar{p}_1}\\int_{\\mathbb H^n}\\frac{|y|_h^{-Q/p}}{\\max(1,|y|_h^Q)}dy.\\end{align*}"} |
latex-00026718 | \lim_{\lambda_{\varphi}\to\infty}\mu_{\varphi}(\Psi) = \frac{3}{\pi}\mu(\Psi). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 26,953 | latex_formula | {"original_latex": "\\begin{align*}\\lim_{\\lambda_{\\varphi}\\to\\infty}\\mu_{\\varphi}(\\Psi) = \\frac{3}{\\pi}\\mu(\\Psi).\\end{align*}"} |
mixed-00037500 | From $n^s\ge3^s>2^s+1$ when $n\ge3$, we have
\begin{align}
&(\zeta(s)-1)-(2^s+1)(\zeta(2s)-1)\\&=\sum_{n=2}^\infty\frac1{n^s}\left(1-\frac{2^s+1}{n^s}\right)\\&=\sum_{k=1}^\infty\frac{1}{2^{ks}}\left(1-\frac{2^s+1}{2^{ks}}\right)+\sum_{\substack{n\ge3\\n\ne2^k}}\frac{1}{n^{s}}\left(1-\frac{2^s+1}{n^s}\right)\\&>\sum_{k... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,168 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/958142", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "11", "answer_count": 2, "answer_id": 1}} |
mixed-00031678 | Proving $\binom{2n}{n}\le 4^n$ for all $n$ by smallest counterexample
Prove $$\binom{2n}{n}\le 4^n$$ for all natural numbers $n$ by smallest (minimal) counterexample.
My attempt:
First, $$\binom{2n}n = \frac{(2n)!}{(n!)^2} \le 4^n\;.$$ We know that $x\ne 0$ because $\frac{(2\cdot 0)!}{(0!)^2} = 1$ which is true. So $... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 18,604 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/356228", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 4, "answer_id": 2}} |
latex-00039858 | D( \bar{a}) =D( a^{\ast}) =D( \mathcal{N}^{1/2}) =D( M_{x}) \cap D( \partial_{x}) . | latex | OleehyO/latex-formulas | cleaned_formulas | train | 40,364 | latex_formula | {"original_latex": "\\begin{align*}D\\left( \\bar{a}\\right) =D\\left( a^{\\ast}\\right) =D\\left( \\mathcal{N}^{1/2}\\right) =D\\left( M_{x}\\right) \\cap D\\left( \\partial_{x}\\right) .\\end{align*}"} |
normal-00049464 | Anonymous . " Man 'yōshū " . Japanese Text Initiative ( in Japanese ) . University of Virginia Library . Archived from the original on November 18 , 2014 . Retrieved November 11 , 2014 . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 120,363 | text | {} |
mixed-00001440 | How does does exponent property work on $\left(xe^{\frac{1}{x}}-x\right)$ How does
$\left(xe^{\frac{1}{x}}-x\right)$
become
$\frac{\left(e^{\frac{1}{x}}-1\right)}{\frac{1}{x}}$ | mixed | math-ai/StackMathQA | stackmathqa100k | train | 732 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1018373", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 2}} |
latex-00047971 | f(x):=E^{-1}(x,hD)P\chi_0 q(x,hD)u_{\Sigma} +hR_1\chi_0(x,hD)q(x,hD)u_{\Sigma}. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 48,636 | latex_formula | {"original_latex": "\\begin{align*}f(x):=E^{-1}(x,hD)P\\chi_0 q(x,hD)u_{\\Sigma} +hR_1\\chi_0(x,hD)q(x,hD)u_{\\Sigma}.\\end{align*}"} |
normal-00004842 | The prerogative appears to be historically and as a matter of fact nothing else than the residue of discretionary or arbitrary authority which at any given time is legally left in the hands of the crown . The prerogative is the name of the remaining portion of the Crown 's original authority ... Every act which the exe... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 11,632 | text | {} |
latex-00002015 | \sum_{o}U_{-e,o}U_{-o,e^{\prime }}^{-1}=\delta _{e,e^{\prime }} ,\quad\sum_{e\neq 0}U_{-o,e}^{-1}U_{-e,o^{\prime }}=\delta _{o,o^{\prime }} , | latex | OleehyO/latex-formulas | cleaned_formulas | train | 2,016 | latex_formula | {"original_latex": "\\begin{align*}\\sum_{o}U_{-e,o}U_{-o,e^{\\prime }}^{-1}=\\delta _{e,e^{\\prime }}\\,,\\quad\\sum_{e\\neq 0}U_{-o,e}^{-1}U_{-e,o^{\\prime }}=\\delta _{o,o^{\\prime }}\\,,\\end{align*}"} |
mixed-00001804 | Find $x$ and $y$ If $\frac{\tan 8°}{1-3\tan^{2}8°}+\frac{3\tan 24°}{1-3\tan^{2}24°}+\frac{9\tan 72°}{1-3\tan^{2}72°}+\frac{27\tan 216°}{1-3\tan^{2}216°}=x\tan 108°+y\tan 8°$, find x and y. I am unable to simplify the first and third terms. I am getting power 4 expressions. Thanks. | mixed | math-ai/StackMathQA | stackmathqa100k | train | 917 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1280655", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 1, "answer_id": 0}} |
latex-00004342 | \sum_{\varrho, \rho, \varsigma}V (\sigma,\tau, \rho, A) D(\varsigma, B)\delta (\varrho \varsigma \varrho^{-1}\rho)= P (\sigma, \tau, A+ B). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 4,345 | latex_formula | {"original_latex": "\\begin{align*}\\sum_{\\varrho, \\rho, \\varsigma}V (\\sigma,\\tau, \\rho, A) D(\\varsigma, B)\\delta (\\varrho \\varsigma \\varrho^{-1}\\rho)= P (\\sigma, \\tau, A+ B).\\end{align*}"} |
mixed-00015922 | Alternatively
$$I(n)= \int\limits_{0}^{2\pi}\frac{dx}{\left ( 1+n^2\sin^2 x \right )^2}=\frac4{n^4}\int_{0}^{\pi /2}\frac{dx}{\left ( a-\cos^2 x \right )^2},\>\>\>\>\>a= 1+\frac1{n^2}$$
Note that
$$J(a)= \int_{0}^{\pi/2}\frac{dx}{ a- \cos^2 x }
= \int_{0}^{\pi/2}\frac{d(\tan x)}{ a\tan^2x +(a-1)}dx= \frac\pi{2\sqrt{a(a... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,003 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/4108770", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "6", "answer_count": 3, "answer_id": 0}} |
latex-00017127 | dB=-AB+BA+(exp:-\int_{P_0}^{P}A:)df_W(exp:-\int_{P}^{P_0}A:) | latex | OleehyO/latex-formulas | cleaned_formulas | train | 17,170 | latex_formula | {"original_latex": "\\begin{align*}dB=-AB+BA+\\left(exp:-\\int_{P_0}^{P}A:\\right)df_W\\left(exp:-\\int_{P}^{P_0}A:\\right)\\end{align*}"} |
normal-00028891 | At 07 : 00 on 22 September 1914 , the French sighted two unidentified cruisers approaching the harbor of Papeete . The alarm was raised , the harbor 's signal beacons destroyed , and three warning shots were fired by the French batteries to signal the approaching cruisers that they must identify themselves . The cruise... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 69,249 | text | {} |
latex-00037598 | \hat{\alpha}_k(x_k^i) = \sum_{j \in I_{k+1}} \hat y_{k+1}(x_{k+1}^j) p_k^{ij} \textrm{ and } \hat {\beta}_k(x_k^i) = \frac{1}{\sqrt{\Delta_n}} \sum_{j\in I_{k+1}} \hat y_{k+1}(x_{k+1}^j) \Lambda_k^{ij}, | latex | OleehyO/latex-formulas | cleaned_formulas | train | 38,078 | latex_formula | {"original_latex": "\\begin{align*}\\hat{\\alpha}_k(x_k^i) = \\sum_{j \\in I_{k+1}} \\hat y_{k+1}(x_{k+1}^j) \\, p_k^{ij} \\textrm{ and } \\hat {\\beta}_k(x_k^i) = \\frac{1}{\\sqrt{\\Delta_n}} \\sum_{j\\in I_{k+1}} \\hat y_{k+1}(x_{k+1}^j) \\, \\Lambda_k^{ij},\\end{align*}"} |
latex-00041923 | t=\sqrt{1+8\alpha_{3r}} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 42,525 | latex_formula | {"original_latex": "\\begin{align*}t=\\sqrt{1+8\\alpha_{3r}}\\end{align*}"} |
normal-00045288 | Olivia Shakespear has died suddenly . For more than forty years she has been the centre of my life in London and during all that time we have never had a quarrel , sadness sometimes but never a difference . When I first met her she was in her late twenties but in looks a lovely young girl . When she died she was a love... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 109,928 | text | {} |
latex-00049789 | A^*v = (v_{1}, -L v_{0} + Bv_{1} ). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 50,485 | latex_formula | {"original_latex": "\\begin{align*}A^*v = \\left(v_{1}, -L v_{0} + Bv_{1} \\right).\\end{align*}"} |
mixed-00015175 | Find equations of the two lines through the origin that make an angle $\tan^{-1}(1/2)$ with $3y=2x$
If two straight lines pass through the origin and makes an angle $\tan^{-1}(1/2)$ with $3y=2x$, then find its equations.
Let $m$ be the gradient of the line then, $$\frac{1}{2}=\frac{m-2/3}{1-2m/3}$$ I don't know wheth... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 8,550 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3319453", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 1}} |
mixed-00027344 | Eigenvectors solve the equation $(A-\lambda I) v = 0$; eigenvectors are vectors, not matrices.
For $\lambda = 1$, you should have
$$\begin{align*}
(A-I )v &= 0 \\
\begin{pmatrix} 0 & 1 & 0 \\ 1 & -2 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} &= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatri... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 15,971 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/939393", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
latex-00026266 | \zeta_{\lambda,s}(\pmb k,\pmb l)\cdot\zeta_{e^t}(\pmb m)=\sum_{p+q=t}\sum_{(\pmb u,\pmb v)}c_{\pmb u,\pmb v}\zeta_{\nu_p,q}(\pmb u,\pmb v), | latex | OleehyO/latex-formulas | cleaned_formulas | train | 26,492 | latex_formula | {"original_latex": "\\begin{align*} \\zeta_{\\lambda,s}\\left(\\pmb k,\\pmb l\\right)\\cdot\\zeta_{e^t}\\left(\\pmb m\\right)=\\sum_{p+q=t}\\sum_{(\\pmb u,\\pmb v)}c_{\\pmb u,\\pmb v}\\zeta_{\\nu_p,q}\\left(\\pmb u,\\pmb v\\right),\\end{align*}"} |
normal-00032828 | " If " samples " Inseparable " as written by Chuck Jackson and Marvin Yancy and performed by Natalie Cole . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 78,956 | text | {} |
latex-00022582 | rx^{r-1}\cdot (x^qy-xy^q) + x^r\cdot (-y^q) + krx^{r-1}\cdot (z^qx-zx^q)+ (z^r+kx^r)\cdot z^q=0 | latex | OleehyO/latex-formulas | cleaned_formulas | train | 22,744 | latex_formula | {"original_latex": "\\begin{align*}rx^{r-1}\\cdot (x^qy-xy^q) + x^r\\cdot (-y^q) + krx^{r-1}\\cdot (z^qx-zx^q)+ (z^r+kx^r)\\cdot z^q=0\\end{align*}"} |
normal-00045935 | Health effects extended to residents , students , and office workers of Lower Manhattan and nearby Chinatown . Several deaths have been linked to the toxic dust , and the victims ' names were included in the World Trade Center memorial . Approximately 18 @,@ 000 people have been estimated to have developed illnesses as... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 111,573 | text | {} |
normal-00001261 | Wheeler was keen to continue archaeological fieldwork outside London , undertaking excavations every year from 1926 to 1939 . After completing his excavation of the Carlaeon amphitheatre in 1928 , he began fieldwork at the Roman settlement and temple in Lydney Park , Gloucestershire , having been invited to do so by th... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 3,173 | text | {} |
normal-00045592 | ( 1 ) Each Government pledges itself to employ its full resources , military or economic , against those members of the Tripartite Pact and its adherents with which such government is at war . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 110,673 | text | {} |
mixed-00007071 | modulo group defined by an algebraic relation I am asked if $\{n, n^{2}, n^{3}\}$ forms a group under multiplication modulo $m$ where $m = n + n^{2} + n^{3}.$
As an example we see that $\{2, 4, 8\}$ does form a group modulo $14,$ with identity $8,$ but am stuck starting the proof for the general case. Thanks in advance... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 3,684 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/869268", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
normal-00013641 | At national level in the UK , Mosley competed in over 40 races in 1966 and 1967 ; he won 12 and set several class lap records . In 1968 , he formed the London Racing Team in partnership with driver Chris Lambert to compete in European Formula Two , which at that time was the level of racing just below Formula One . The... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 33,019 | text | {} |
latex-00045959 | \mathcal{VNU}=\mathcal{N}+I. | latex | OleehyO/latex-formulas | cleaned_formulas | train | 46,605 | latex_formula | {"original_latex": "\\begin{align*}\\mathcal{VNU}=\\mathcal{N}+I.\\end{align*}"} |
mixed-00010055 | Prove $ \left(\sum \limits_{k=1}^n (2k-1)\frac{k+1}{k}\right) \left( \sum \limits_{k=1}^n (2k-1)\frac{k}{k+1}\right) \le \frac{9}{8}n^4$
Prove that for all $n \in \mathbb{N}$ the inequality $$ \left(\sum \limits_{k=1}^n (2k-1)\frac{k+1}{k}\right) \left( \sum \limits_{k=1}^n (2k-1)\frac{k}{k+1}\right) \le \frac{9}{8}n^... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 5,453 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3312453", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "4", "answer_count": 2, "answer_id": 0}} |
normal-00013966 | South Vietnamese forces were not constrained by the time and geographic limitations placed upon U.S. units . From the provincial capital of Svay Rieng , ARVN elements pressed westward to Kampong Trabek , where on 14 May their 8th and 15th Armored Cavalry regiments defeated the 88th PAVN Infantry Regiment . On 23 May , ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 33,695 | text | {} |
normal-00011266 | Maeda was fond of the name and started using it to promote his art thereafter . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 27,495 | text | {} |
mixed-00045593 | What is the asymptotic expansion of $x_n$ where $x_{n+1} = x_n+1/x_n$? Let
$x_{n+1} = x_n+1/x_n,
x_0 = a \gt 0$
and
$y_n = x_n^2$.
What is the asymptotic expansion
of $x_n$ ($y_n$ will do)?
I can show that
$y_n
=2n+\dfrac12 \ln(n) + O(1)
$.
Is there an explicit form
for the constant
implied by the $O(1)$?
What is the a... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 27,169 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3928173", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "11", "answer_count": 2, "answer_id": 0}} |
mixed-00047109 | Notice
$$1 + i = \sqrt2e^{i\frac{\pi}{4}}$$
$$\sqrt{3} - i = 2e^{-i\frac{\pi}{6}}$$
Multiply both and we get
$$(1 + i)(\sqrt{3} - i) = \sqrt2e^{i(\frac{\pi}{4} - \frac{\pi}{6})}$$
$$(\sqrt3 + 1) + (\sqrt3-1)i= 2\sqrt{2}e^{i\frac{\pi}{12}}$$
$$(\sqrt3 + 1) + (\sqrt3-1)i= 2\sqrt2\left(\sin{\frac{\pi}{12}} + i\cos\frac{\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 28,085 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/719825", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 2, "answer_id": 1}} |
normal-00006300 | Three weeks after the failure of the operation , a second attack was launched which proved more successful in sinking a blockship at the entrance to the canal but ultimately did not close off Bruges completely . Further plans to attack Ostend came to nothing during the summer of 1918 , and the threat from Bruges would ... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 15,490 | text | {} |
normal-00047149 | The Church in England was a major landowner throughout the medieval period and played an important part in the development of agriculture and rural trade in the first two centuries of Norman rule . The Cistercian order first arrived in England in 1128 , establishing around 80 new monastic houses over the next few years... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 114,423 | text | {} |
mixed-00016621 | Firstly, rewrite integral as
$$\displaystyle \int\limits_{0}^{\frac{\pi}{2}}{\dfrac{dx}{2+\cos{x}}}=\int\limits_{0}^{\frac{\pi}{2}}{\dfrac{dx}{1+\cos^2\dfrac{x}{2}+\sin^2\dfrac{x}{2}+ \cos^2\dfrac{x}{2}-\sin^2\dfrac{x}{2}}}=\int\limits_{0}^{\frac{\pi}{2}}{\dfrac{dx}{1+2\cos^2\dfrac{x}{2}}}.$$
Substitute
\begin{gather... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,436 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/279135", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "2", "answer_count": 4, "answer_id": 0}} |
latex-00014594 | e^{-2\phi}=[1-(\frac{r_-}{r})^{\tilde d} ]^{-\alpha} | latex | OleehyO/latex-formulas | cleaned_formulas | train | 14,624 | latex_formula | {"original_latex": "\\begin{align*}e^{-2\\phi}=\\left[1-\\left(\\frac{r_-}{r}\\right)^{\\tilde d} \\right]^{-\\alpha}\\end{align*}"} |
normal-00026709 | The second sub @-@ plot depicts the lives of Charles W. Grannis and Miss Anastasia Baker . Grannis and Baker are two elderly boarders who share adjoining rooms in the apartment complex where Trina and McTeague live . Throughout their time at the apartment complex , they have not met . They both sit close to their adjoi... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 63,869 | text | {} |
mixed-00025381 | Here's how I would do it: using Bezout coefficients, we get $2\cdot5-3\cdot3=1$. So the solution to $\begin{cases}x\cong1\pmod3\\x\cong4\pmod5\end{cases}$ is $x=1\cdot{10}-4\cdot9=-26\pmod{15}=4\pmod{15}$.
Next we solve $\begin{cases} x\cong{4}\pmod{15}\\x\cong6\pmod7\end{cases}$.
Since $-6\cdot15+13\cdot7=1$, we get $... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 14,779 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3769723", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 3, "answer_id": 1}} |
latex-00020420 | \mathrm{I}(\xi_{1}^{x},\mathcal{T}_{x})=\inf_{n}\mathrm{I}(\xi_{1}^{x},\xi_{n}^{x})=\lim_{n}\mathrm{I}(\xi_{1}^{x},\xi_{n}^{x}). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 20,551 | latex_formula | {"original_latex": "\\begin{align*}\\mathrm{I}(\\xi_{1}^{x},\\mathcal{T}_{x})=\\inf_{n}\\mathrm{I}(\\xi_{1}^{x},\\xi_{n}^{x})=\\lim_{n}\\mathrm{I}(\\xi_{1}^{x},\\xi_{n}^{x}).\\end{align*}"} |
latex-00026725 | C(\pi,t)=q_{\pi}\prod_{j=1}^{m}(3+|it+\mu_{\pi}(j)|),C(\pi)=C(\pi,0). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 26,960 | latex_formula | {"original_latex": "\\begin{align*}C(\\pi,t)=q_{\\pi}\\prod_{j=1}^{m}(3+|it+\\mu_{\\pi}(j)|),C(\\pi)=C(\\pi,0).\\end{align*}"} |
normal-00005356 | Bennett wrote her first short story at age 17 , a science fiction story titled " The Curious Experience of Thomas Dunbar " . She mailed the story to Argosy , then one of the top pulp magazines . The story was accepted and published in the March 1904 issue . | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 12,942 | text | {} |
mixed-00028041 | Since
$$\lfloor x+2\rfloor=\lfloor x\rfloor+2,\quad \lfloor 2x-2\rfloor=\lfloor 2x\rfloor-2$$
the equation can be written as
$$x(\lfloor x\rfloor+2)+\lfloor 2x\rfloor-2+3x=12,$$
i.e.
$$x\lfloor x\rfloor+5x+\lfloor 2x\rfloor=14\tag 1$$
Now let us separate it into cases :
Case 1 : $x=m+\alpha$ where $m\in\mathbb Z$ and $... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 16,393 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1510756", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "7", "answer_count": 3, "answer_id": 1}} |
mixed-00018557 | Prove that $\left(\frac{n}{2}\right)^n \gt n! \gt \left(\frac{n}{3}\right)^n \qquad n\ge 6 $
$$\left(\frac{n}{2}\right)^n \gt n! \gt \left(\frac{n}{3}\right)^n \qquad n\ge 6 $$
I tried it prove it by mathematical induction but failed .
For $n=6$
$$(3)^6 \gt 6!$$
Now for $n=k$
$$\left(\frac{k}{2}\right)^k \gt k!$$
Now... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 10,610 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1902324", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 3, "answer_id": 0}} |
mixed-00033049 | You already have nice solutions with substitutions. If one cannot come up with the right substitution, maybe this is a solution that works better:
Multiplying the differential equation with $y$ gives
$$
yy'=\frac{y^2}{x}-1,
$$
or
$$
\frac{1}{2}\bigl(y^2\bigr)'=\frac{y^2}{x}-1.
$$
This is a linear and first order differ... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 19,436 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/1414760", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 4, "answer_id": 1}} |
normal-00042309 | The earliest church was established by Scots @-@ Irish Presbyterians in 1734 . Chambers gave land to the congregation in 1768 , requiring only a single rose as annual rent . Later land was given to the First Lutheran Church ( 1780 ) and Zion Reformed Church ( organized in 1780 ) under the same agreement , and these chu... | normal | Salesforce/wikitext | wikitext-103-raw-v1 | train | 102,504 | text | {} |
mixed-00008898 | Let $x,y,z\in\mathbb{N}$ with $x<y<z$. First, you can rewrite the equation as
$$ \dfrac{100}{336}=\dfrac{25\lambda}{84\lambda}=\dfrac{yz+z+1}{xyz}. $$
for a positive integer $\lambda$. Since we know that $xyz$ and that $yz+z+1$ are integers, we can hope to find a solution for the system of equations:
$$\begin{cases}84\... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 4,769 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2283922", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 2, "answer_id": 0}} |
mixed-00035261 | $\begin{align}
T &= \large \left(\frac{2\cos 6° \cos 12°}{1\;+\;2\cos 6° \sin 12°}\right)
\left({2\sin 6° \over 2\sin 6°}\right)\cr
&= \large \frac{\sin 24°}{2\sin 6°+\;2\sin^2 12°}\cr\cr
{1\over T}&= \large \frac{2\sin(30°-24°)+\;(1-\cos 24°)}{\sin 24°}\cr
&=\large {(\cos 24° - \sqrt3\sin24°) + (1-\cos 24°) \over \si... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 20,798 | A | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/3367940", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "5", "answer_count": 3, "answer_id": 0}} |
mixed-00037480 | Integrating $\int \frac {dx}{\sqrt{4x^{2}+1}}$ $\int \dfrac {dx}{\sqrt{4x^{2}+1}}$
I've been up to this one for quite a while already, and have tried several ways to integrate it, using substituion, with trigonometric as well as hyperbolic functions. I know(I think) I'm supposed to obtain:
$\dfrac {1}{2}\ln \left| 2\sq... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 22,157 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/948229", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "3", "answer_count": 4, "answer_id": 1}} |
latex-00004204 | \varphi'_2(N) = (\frac{j (N-j+1)}{(2j+1) (N+2)})^{1/2} \varphi'_+(N) - (\frac{(j+1) (N+j+2)}{(2j+1) (N+2)})^{1/2}\varphi'_-(N). | latex | OleehyO/latex-formulas | cleaned_formulas | train | 4,207 | latex_formula | {"original_latex": "\\begin{align*}\\varphi'_2(N) = \\left(\\frac{j (N-j+1)}{(2j+1) (N+2)}\\right)^{1/2} \\varphi'_+(N) - \\left(\\frac{(j+1) (N+j+2)}{(2j+1) (N+2)}\\right)^{1/2}\\varphi'_-(N). \\end{align*}"} |
latex-00044833 | dY=[\sum_{k=1}^nB^{(k)} dx_k]Y | latex | OleehyO/latex-formulas | cleaned_formulas | train | 45,468 | latex_formula | {"original_latex": "\\begin{align*} {\\rm d}Y=\\left[\\sum_{k=1}^nB^{(k)}\\,{\\rm d}x_k\\right]Y \\end{align*}"} |
mixed-00017071 | Limits at infinity I'm working with limits at infinity and stumbled upon this exercise where I want to evaluate the indicated limit:
$$\lim_{x \to \infty} \frac{1}{\sqrt{x^2-2x}-x}$$
I tried to solve it by doing the following:
$$\lim_{x \to \infty} \frac{1}{\sqrt{x^2-2x}-x} = \lim_{x \to \infty} \frac{1}{\sqrt{x^2} \s... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 9,718 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/651969", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "6", "answer_count": 3, "answer_id": 0}} |
mixed-00049394 | VERIFICATION: Apply Step-deviation method to find the average mean of the following frequency distribution Q: Apply Step-deviation method to find the average mean of the following frequency distribution
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline
Variate (x_i) & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\ \hlin... | mixed | math-ai/StackMathQA | stackmathqa100k | train | 29,461 | Q | {"meta": {"language": "en", "url": "https://math.stackexchange.com/questions/2648816", "timestamp": "2023-03-29 00:00:00", "source": "stackexchange", "question_score": "1", "answer_count": 1, "answer_id": 0}} |
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