idx int64 0 1.39M | latex stringlengths 3 612 | source stringclasses 8
values | transform stringclasses 5
values |
|---|---|---|---|
829,940 | J(y_1,..., y_{n-1}) \approx L(\frac{(y_{2} - y_{1})}{\Delta t}) + ... | mathwriting | group_factor |
657,924 | (\begin{matrix}31\end{matrix}){(\begin{matrix}21\end{matrix})}^{2} | mathwriting | neutral_insert |
201,594 | K_5=(\underbrace{0,\cdots,0}_d,i,1)=i\overline{K}. | linxy_synthetic_handwrite | commute_safe |
877,378 | x_{f_1,\cdots,f_{2n-2}}=\left(\sum\limits_{k=1}^{n-1}\alpha(k)\,D_k\right)-\sum_{i=1}^{n}(-1)^{i+n}\left(\sum\limits_{q=1}^{n-1}\beta(i,q)\,D_q\right)\left(\sum\limits_{s=1}^{n-1}\gamma(i,s)\,D_s\right), | oleehyo | neutral_insert |
52,310 | \frac{\partial}{\partial\phi}\left(F_{\frac{k_1}{k}}\right) | mathwriting | fraction_rewrite |
996,838 | \mathfrak{S}[m](Y,q):=\sum_{\substack{\text{all } n\\ 1<n\le m}} f_q(mn) (Y-mn), | oleehyo | commute_safe |
1,244,570 | b^n-1 = \phi_n(b)\cdot\left(\frac{b^n-1}{\phi_n(b)}\right)=\phi_n(b)\cdot (b^{\mathop{\text{ord}}\limits_{\,{\mathbb Z}^*_{{n}}}}(b)-1)= | mathwriting | neutral_insert |
41,921 | \frac{\left(\frac{580}{\pi}\right)}{\sin\left(\cos^{-1}\left(\csc\left(\cot^{-1}\left(\tan\left(\sec^{-1}\left(\log_{e}\left({x}^{2}+1\right)\right)\right)\right)\right)\right)\right)} | mathwriting | commute_safe |
1,353,357 | P_n \left( x; z \right) &= \frac{d}{dx} P_{n - 1} \left( x; z \right)\\&\qquad + (2 n^2 - n E[X] )^{-1}\\ &\quad [E[( X^{3/2}) ]\\ & -\tfrac{1}{4} \{6\} \\& -\{5\}\{\}] | oleehyo | commute_safe |
1,337,537 | B_{j,k}=\begin{cases} \dfrac{2(r-j)k}{r} & j\geq k \\ B_{k,j} & j<k \end{cases}, \qquad \qquad b_j=\dfrac{r-j}{r}. | linxy_full | commute_safe |
681,233 | S^{-1}\Bigl(\sum_{0 < |\alpha|}c_{\alpha}x^{\alpha}\Bigr) = c_ax^a + ...= (...)\cdot x^{|\alpha|}= | oleehyo | expand_local |
567,362 | D^{s}(f_{\leq j - 2}g_{j}) = \left[D^{s}, f_{\leq j - 2}\right]g_{j}+f_{\leq j - 2}D^{s}g_{j}. | oleehyo | group_factor |
988,552 | \frac{(z+\frac{1}{(r+1)})}{\left(\frac{b^i}{r+1}\right)}. | mathwriting | neutral_insert |
859,891 | M_{i+j} :=\frac{m}{n}. | oleehyo | neutral_insert |
283,178 | (\phi_s^{-1})_*\circ D_s=\nabla^E_{(\phi_s^{-1})_*}(\phi_s_*X)= \begin{pmatrix} 0 \\ X(x) + Y'(y)\end{pmatrix}. | oleehyo | commute_safe |
1,021,748 | E_{r,n}|\emptyset \rangle =\sum _{i=0}^{M-1}|(m+i)^{r}\rangle \;, | oleehyo | fraction_rewrite |
289,067 | \rho_{R}(x)=\frac{1}{R^{d}}\rho\left(\frac{x}{R}\right) | oleehyo | expand_local |
963,892 | H_LC|\psi\rangle=M^2\;|\psi\rangle | linxy_synthetic_handwrite | group_factor |
1,181,529 | \mathcal{PV}\,\frac{1}{x}\mp i\pi\delta(x) | im2latex | fraction_rewrite |
383,172 | (\frac{5^{\sqrt{215}}}{7^{\sqrt{215}}})^{260\cdot410^{207}}=(\frac{(5^2)^{\sqrt{215}/2}}{(7^2)^{\sqrt{215}/2}})^{260\cdot410^{207}}=\left(\frac{25}{49}\right)^{\frac{\sqrt{215}}{2}\cdot(260\cdot410^{207})} | mathwriting | neutral_insert |
181,499 | partial_+g=J_+g,~partial_-g=gJ_- | oleehyo | fraction_rewrite |
530,465 | \begin{align*} &\phantom{:}=\sum_{n\leq T}\sum_m\frac{a(m)\mu(m)}{(mn)^{\sigma_0}}\int_T^{2T}(mn)^{-it}\,dt\\&\qquad+O(T^{\frac{1}{2}+\frac{1}{K'}})\\ &\phantom{:}=T+O(T^{\frac{1}{2}+\frac{1}{K'}}). \end{align*} | oleehyo | expand_local |
17,479 | \mathbf{p} = -(i\nabla) | oleehyo | group_factor |
298,147 | S=I^{\infty}h,IE | im2latex | commute_safe |
704,404 | \mathcal{W}_F^{(2;2g)}=\mathcal{W}_F^{(2;ab)}+\mathcal{W}_F^{(2;na)}\,. | linxy_synthetic_handwrite | neutral_insert |
269,514 | \{w_{l , m}, w_{l^{\prime}, m^{\prime}}\}= [m'(l+1)-m(l'+1)]w_{l+l',m+m'}. | linxy_full | fraction_rewrite |
1,018,986 | (\big(\frac{\delta x}{\delta t}\big)t-x\big)\bigg|_{t=4}=0 | oleehyo | neutral_insert |
373,618 | d ( z ) \! \! = \! \! \left( \! \! q^{2 j - 1}x_{+}z \! - \! 2 x_{0}sk q^{j}\phantom{\frac{q^{j}}{s}}\! \! \! \! \! \! - \! x_{-}k^{2}z^{- 1}\! \right) \! \! \left( - q^{2 j}y_{+}z \! + \! \frac{2 y_{0}}{tk}q^{j}\! \! + q y_{-}k^{- 2}z^{- 1}\! \right) | linxy_synthetic_handwrite | fraction_rewrite |
1,328,583 | [\phi_{i j}(x), \partial_-\phi_{k l}(y)]=i\{\psi_{ij}(x),\psi_{kl}(y)\}=\frac{1}{2}i\delta(x^- - y^-) | linxy_synthetic_handwrite | fraction_rewrite |
586,536 | \sqrt[10]{\prod (R_i)^5} | mathwriting | commute_safe |
1,233,877 | ( \gamma^{a})^{\beta}{}^{\alpha}{\hat\Omega}_{{\alpha}q}{}^{i}_{a}+0=0 | linxy_full | neutral_insert |
680,849 | e(H^U(A))=|P'|\\=|P| - \binom{|A\cap U|}{2}\\ > \frac{|A|^2|U|^2}{6n} - \frac{|A|\cdot |U|}{2}\\ \geq \frac{|A|^2|U|^2}{12n}. | oleehyo | group_factor |
177,009 | N(\epsilon)=-\nabla\cdot(\epsilon\nabla\phi_{\epsilon}) | oleehyo | group_factor |
247,118 | \begin{align*} &\quad (2\pi)^{-s-\frac{k-1}{2}}\Gamma(s+\tfrac{k-1}{2})L_f(s)\\ &=\int_0^\infty f(iy)y^{s+\frac{k-1}{2}-1}dy,\end{align*}. | oleehyo | commute_safe |
433,626 | Answer : $\boxed{\theta^N_n((x,y,z,w))=(x-z)\left[\dfrac{(-y+z+2w)^n - (y + z)^n + 2(z + w)^n}{n}\right]}$ | oleehyo | commute_safe |
1,267,991 | [ a ^{(m)} ,\bar{b}^{(n)}] =\overline{[ a,\bar{b}] }^{(m+n)}, | linxy_full | fraction_rewrite |
114,326 | M=\left|a\,e\left(n_{e}+\tau_{0}\,n_{m}\right)\right|, | im2latex | neutral_insert |
224,815 | H = \int_{0}^{2 \pi} d\sigma {\cal H}, \; {\cal H} = {\cal H}_0+v \cdot K(\tilde L_\infty) | linxy_synthetic_handwrite | neutral_insert |
351,577 | ({g_1},\xi_1)({g_2},\xi_2) = (g_1g_2, {}_{g_2^{-1}}\xi_1) ^{g_1}\xi_1={}^{g_2}\xi_2 | oleehyo | fraction_rewrite |
472,400 | (Y^{\prime}_{\sigma})^{\prime}+(-0)^{\prime} | mathwriting | neutral_insert |
661,425 | r=\sqrt{{y_{{6}}}^{2}+{y_{{7}}}^{2}+{y_{{8}}}^{2}} | crohme | commute_safe |
140,722 | f(x)- \left[ -\sum_{n=0}^{N-2}(-1)^{n+1}S_na_{n}+(-1)^NS_{N-1}a_{N-1} \right] = O(\varphi_N (x)) | mathwriting | neutral_insert |
1,263,026 | \phi_{L/K}^j(x) = \min_{\substack{0 \leq j_0 \leq j \\ a_h \neq 0}} \{ h + v_L \left( {h+n \choose p^{j_0}} \right) + p^{j_0} x \} | oleehyo | expand_local |
169,825 | \left(\frac{\partial}{\partial t}+P\right)U(x,x';t)=0,. | linxy_full | fraction_rewrite |
549,794 | (f|_u)(v)= f(v) | mathwriting | fraction_rewrite |
1,338,450 | \left\{\frac{{\tilde{\mathcal {H}}}^{\mu\nu}(\tau )}{c},\frac{{\tilde{\mathcal {H}}}^{\alpha\beta}(\tau )}{c}\right\}^{*}\equiv C^{\mu\nu\alpha\beta}_{\gamma\delta}\frac{{\tilde{\mathcal {H}}}^{\gamma\delta}(\tau )}{c} = -\mathrm i c^2 [ {\Pi}^\mu_\rho(\tau) , \{ A^\rho(x), F^{\nu\alpha}(y) \} ]^{*}_{\!\!\!\!\!\!A}\, x... | oleehyo | neutral_insert |
459,864 | \frac{x+y-z-\left(a+\frac{b}{\sin^2 x}\right)}{e^{i\pi}+1}=\frac{\cos y}{\tan z}-\frac{ab}{\sin^2 x}. | mathwriting | neutral_insert |
828,961 | -\frac{1}{4} F_{{MN}} F^{{MN}} + i {{{\bar{D}}}_M {\tilde{\phi}}} {{D}^M}\phi - m ^ 2 |\phi|^2 | oleehyo | fraction_rewrite |
1,124,760 | \eta_{df}=\frac{\rho N_{A} L_{P}^{2}}{M}+0 | mathwriting | neutral_insert |
1,261,942 | k= ln( 2^{-1}) / (4.5 * 10^{8}) | mathwriting | fraction_rewrite |
1,024,642 | r^{(1)}_{\eta} = (-1+2)\frac{C(G)}{8\pi^2}=\frac{C(G)}{8\pi^2}\ | linxy_synthetic_handwrite | expand_local |
1,294,171 | \epsilon \frac{dY}{dx}+hxy=\epsilon^{n-2}(f(x,\epsilon)+g(x)\epsilon^m) | oleehyo | neutral_insert |
1,298,530 | \left|\frac{-4(1-\alpha) \lambda(t)^{-\alpha}\lambda'(t)}{\log(\lambda_{0,0}(t))}\int_{t}^{\infty}\frac{e''(s) ds}{(\lambda(t)^{1-\alpha}+s-t)^{2}(1+s-t)^{3}}\right| \leq \frac{C}{t^{3}\log^{b+2}(t) (\log(\log(t)))^{3/2}} | oleehyo | neutral_insert |
970,399 | \tilde{y}= - ( 37 + (-5.1) x ) | mathwriting | group_factor |
464,442 | \Psi \approx \psi_{1}(q) |(x'+\frac{1}{2}p_{1}'^{2}t,k')\rangle + \psi_{2}(q)|(x'+\frac{1}{2}p_{2}'^{2}t,k')\rangle. | linxy_synthetic_handwrite | group_factor |
103,491 | (-1)^{\epsilon_A}\left.\frac{\partial_rZ}{\partial K_A}\right|_JJ_A = (Z,\Gamma) \qquad \forall Z | im2latex | fraction_rewrite |
575,126 | {\gamma \cdot D \Lambda =\rho ({e_{{\hat{L}}}}^{\hat{5}}{e_{{ M}}}^{{\hat{\mu}}}\gamma^{\hat{ L}}\partial_{\hat{\mu}}- { e_{{\hat{L}}}}^{\hat{\mu}}\gamma^{{\hat{L}}}{e_{{ M}}}^{5}\partial_5)\Lambda =( \rho \gamma^{\hat{5}}\partial_{5}+ \rho \gamma^{\hat{\mu}}\partial_{\mu}- 2 \gamma^{\hat{5}}) \Lambda . } | linxy_full | fraction_rewrite |
474,023 | \delta \leq |T(e_\xi)(x_\xi,y_\xi)|=|T(e_\xi)(x_\xi,y_\xi)-T(e_\eta)(x_\xi,y_\eta)| | oleehyo | neutral_insert |
1,034,857 | \Delta=(\Delta_{ai}), ~~a=1,...E, ~~i=1,...N | oleehyo | neutral_insert |
155,349 | \begin{cases}\delta^0 (1) = 0 \\ \delta^1 (1) = 1\\ \sigma(0) = 1= \sigma (1)\\ \sigma (01)=0 \\\gamma((0)\otimes x) = x\\\gamma((1)\otimes (1)) = 1= \gamma((01)\otimes (1))\\=\gamma((1)\otimes (01))=0.\end{cases} | oleehyo | group_factor |
818,351 | B(\gamma)= | mathwriting | group_factor |
1,369,752 | -\frac {d}{\left ( H^{2}\eta^{2} \right )} | im2latex | fraction_rewrite |
755,461 | L=\text{Tr}\left\{-\frac{1}{4}G^{\mu\nu}G_{\mu\nu}+\frac{1}{2}D_{\mu}S^{\dagger}D^{\mu}S\right\}+\frac{1}{2}D_{\mu}\phi^{\dagger}D^{\mu}\phi-V(S,\phi)\,. | linxy_synthetic_handwrite | commute_safe |
1,046,792 | \leq\int0xf′(t)et2/2e−t2/2dt=|f(x)|=|\int_{0}^{x} f'(t) e^{t^{2}/2}e^{-t^{2}/2} dt|&space;\leq&space;\int_{0}^{x}G(|f'(t)|)e^{-t^{2}/2}dt&space;+&space;\int_{0}^{x}G^*(e^{t^{2}/2})e^{-t^{2}/2}dt | oleehyo | group_factor |
885,403 | \frac{\partial}{\partial x^\mu}\left(\sqrt{-g}\, g^{\mu\nu} A_\nu\right) = 0 | mathwriting | commute_safe |
49,926 | d\left(\underbrace{\left\lceil\frac{d_0}{d}\right\rceil + \cdots + \left\lceil\frac{d_0}{d}\right\rceil}_{\omega} + \left\lceil\frac{(a-1)(b-1)d_0}{d}\right\rceil - \omega + 1 \right) | oleehyo | neutral_insert |
1,000,244 | \beta_1,...,\beta_{n} | mathwriting | commute_safe |
1,291,586 | E_{\{\text{phases}\}}\left\{\frac{\sum\limits_i \alpha_i}{\sum\limits_j 1/\beta_j} + c^2\right\} = d | linxy_synthetic_handwrite | group_factor |
902,786 | 4\cdot(334 + 2^{\left(\frac{96}{4} \cdot 4\right)}) | mathwriting | commute_safe |
1,011,631 | [matrix]{cc|c}\{TT' & TN' & TT_d \\ NT' & NN' & NT_d \\ \hline TT'_d & TN'_d & dd'\}[/matrix]^= ([matrix]{cc|c}\{TT' & Tn' & tt_d \\ nt' & nn' & nt_d \\ \hline tt'_d & tn'_d & dd'\}[/matrix])^+( [matrix]{cc|c}\{tt' & tn' & tt_d-t \\ nt' & nn' & nt_d-n \\ \hline tt'_d | oleehyo | neutral_insert |
350,165 | q = \frac{\overline{X}_2}{\sqrt{\frac{u_1^2}{G_1}+\frac{u_2^2}{G_2}}}\cdot\overline{X}_1 | mathwriting | commute_safe |
1,232,575 | (coshw)^{2s}e^{ikw}=\sum _{m=0}^{\infty }V_m^{(s)}(k)z^m | im2latex | group_factor |
3,827 | \frac{d(g^{-1}(y))}{dy} = \frac{1}{2\sqrt{y}}. | mathwriting | neutral_insert |
1,261,306 | \frac{1}{2}\left( z\right) ^{3/4}. | linxy_full | expand_local |
808,516 | e^{\lambda S_i}\left|E_{j}\right\rangle=\delta_{i j}\left|E_{j}\right\rangle\,,~~~~e^{-\lambda S_i}\left|F_{j}\right\rangle=\delta_{i j}\left|F_{j}\right\rangle | oleehyo | group_factor |
689,425 | :=\left(\sum_{i=1}^df_i^p\right)^{1/p},\quad p>1 | oleehyo | commute_safe |
986,679 | \tau_n^{\prime}(\alpha, \beta) = \frac{\tau_n(\gamma^3 \alpha, \gamma^4 \beta)}{\gamma^{(n - 1)(n - 4)/2}} | oleehyo | commute_safe |
648,556 | V_{Reid}(r)=(-10.463-1650.6+6484.2)\left(\frac{e^{-\mu r}}{\mu r}\right) | mathwriting | commute_safe |
997,273 | :=\circ (f\otimes_{\mathcal{A}^*})\circ \alpha_I | oleehyo | commute_safe |
840,130 | (\Lambda^r)^{s-t}=\oplus_{\rho=0}^{rs}\lambda^{\rho}_{st}. | oleehyo | neutral_insert |
218,962 | -\int_{{\mathbb R}^{ N}}( I_ {\mu}\ast | u _ { \nu } |^ p ) | v _ { \nu } |^ q . | oleehyo | fraction_rewrite |
745,178 | -\frac{1}{2} (\partial^{\mu}\psi)^2-\frac{1}{4}(e^{\psi}+e^{-\psi})^2 | linxy_full | commute_safe |
731,215 | BMO_{r}=BMO,\ \ | oleehyo | expand_local |
503,867 | {\rm Tr}_{2+1}\ln(-\partial_\mu\partial^\mu -\partial_2^2 +\kappa^2)+\frac{1}{2}{\rm Tr}_{2+1}\ln(-\partial_\mu\partial^\mu-\partial_2^2+4\kappa^2)\;. | oleehyo | group_factor |
1,184,579 | Z_1, Z_2, ... | mathwriting | expand_local |
371,722 | +{\frac{\lambda}{8 \pi}}\tilde{\phi}_{c}(k^{+}, k^{-})=0. | im2latex | fraction_rewrite |
204,938 | \frac{\frac{666666999999}{\frac{666666600088}{555555551111}}+\frac{-\frac{24}{888887700000}}{\frac{39}{888887700000}}}{555555666666} | mathwriting | neutral_insert |
1,380,808 | r_{7, 3} = -0.026\overline{271} - \frac{\sqrt[3]{-\frac{8}{e}}}{1 + i}. | mathwriting | expand_local |
626,633 | |\omega_{zt}(g)|\le{\displaystyle \sup _{u\in U}}|\delta (h_{u})|^{{(z-1)}/{2}}=\sup _{u\in U}|\delta (h_{u})|^{{(Re(z)-1)}/{2}}\,, | oleehyo | fraction_rewrite |
1,106,258 | \frac{\partial (f_{i} + g)}{\partial x_{j}} | mathwriting | neutral_insert |
1,047,025 | \frac{t}{i\sqrt{\frac{l^{2}}{c^{2}}-0}} | mathwriting | neutral_insert |
275,389 | \hat{\mathrm{G}}_{0}\longrightarrow \hat{\tilde{\mathrm{G}}}_{0}= \hat{\mathrm{G}}_{0}+ \frac{1}{2}( e_{+}{\eta}_{+}+ e_{-}{\eta}_{-}) | linxy_full | commute_safe |
551,660 | \frac{1024x^5+9y^{12}}{xy-z}+\frac{-w^{-1}+xz}{-yz}=\frac{(2^10)x^5+(3^3)^4y^{12}}{(xyz)^{-1}} | mathwriting | commute_safe |
699,146 | [\begin{matrix}1+(-s)L-(-s)L&(-s)L(0)\\ (0)(1)&1+(0)\end{matrix}] | mathwriting | neutral_insert |
732,733 | F\left(x, y\right)=\left(-\infty,\sigma\left(C\left(x\right), y\right]\right) | oleehyo | expand_local |
413,457 | L=\frac{1}{\sqrt{-g}}\left[\frac{1}{2} g^{\mu \nu} (\nabla_{\mu} \phi)(\nabla_{\nu} \phi) - V(\phi)\right] + i \overline{\psi}_a e^{(\mu)}_a \gamma^\mu D_\mu \psi | mathwriting | commute_safe |
384,076 | \min_{\substack{w_c, r}} [w_c, r] | oleehyo | commute_safe |
376,689 | n = 4 * Z^(2) * p / W^(2) | mathwriting | fraction_rewrite |
1,286,171 | A^{(0)}_1=B | linxy_full | neutral_insert |
58,025 | X_n = Z_1 + ... + Z_n | mathwriting | group_factor |
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