idx int64 0 1.39M | latex stringlengths 3 612 | source stringclasses 8
values | transform stringclasses 5
values |
|---|---|---|---|
419,029 | S_{kl} \equiv ((K_k+K_l)^2) +(n\cdot(K_k+K_l)) = 2-2j | linxy_synthetic_handwrite | neutral_insert |
761,360 | \begin{array}{l}\Psi^{\gamma}_{s,l}(\theta,\phi) = (\cosh\theta-1)^{\frac{l}{2}}(\cosh\theta+1)^{-|\gamma|-\frac{l}{2}} * \\ \qquad \times \frac{\Gamma(s+l+1)}{\Gamma(s+1)\Gamma(l+1)}F(-s,-2|\gamma|+s+1;l+1;\frac{1-\cosh\theta}{2})e^{il\phi}.\end{array}. | oleehyo | neutral_insert |
636,755 | v_{3}(t,r) = | oleehyo | fraction_rewrite |
271,610 | c(u^*_{1})=C^{i_1\dots i_{2n-m-k+1}}\left(\frac{\partial}{\partial u^1_{i_1}}\right)\dots \left(\frac{\partial}{\partial u^1_{i_{2n-m-k+1}}}\right) Q_1 | linxy_synthetic_handwrite | neutral_insert |
1,336,086 | \|\|T_{\phi}^{H_0,...,H_n}\||_{\mathcal{S}^{\alpha_1}...\mathcal{S}^{\alpha_n}\rightarrow S^\alpha}\leq inf\{\int |supp\,\nu||\phi(x)|^{(2^{k+3}-4)(q-1)}dx\} | oleehyo | group_factor |
705,479 | |\tilde v_t(z)-\sigma(D\tilde u(z))|=|v_t(z)+(\psi^i_j)_t(z)+(g^i_j)_t(z)-\sigma(Du(z)+D\varphi^i_j(z))|. | oleehyo | fraction_rewrite |
278,399 | S_{[n]}^{(4)}+(\frac{16}{9}+\epsilon)R^{4}=\left(S_{[n]}^{(2)}+\sqrt{\frac{2}{3}}R\right)^{2}, | mathwriting | neutral_insert |
914,669 | D_{\zeta'_i}\phi = \partial_{\zeta'_i}A(\phi;\mu',\zeta')+\frac{\partial}{\partial\phi}A(\phi;\mu',\zeta')[D_{\zeta'_i}\phi] | oleehyo | neutral_insert |
443,055 | \pi_1(X_{\theta}) =\, <a, b \,| \, abab^{-1} , \, ba^2> | oleehyo | commute_safe |
342,935 | \begin{align*} 1 - x - y &= \sqrt{\alpha} \sin \phi \\ x - y &= \sqrt{1 - \alpha} \cos \phi \end{align*} | oleehyo | commute_safe |
303,929 | \Lambda[\phi^{A}_{cl},\,\phi^{\ast A}] = {\mathcal{W}}[J^A + i\phi^{\ast A},\,i\phi^A - J^{\ast A}]. | oleehyo | commute_safe |
377,905 | \begin{pmatrix}\lambda_{1}&0\\0&\lambda_{2}\end{pmatrix} | mathwriting | commute_safe |
1,206,673 | (-1)^{l-l_1-l_2}\hat{C}^{l\,l_2\,l_1}_{m\,m_1\,m_2}=\hat{C}^{l\,l_1\,l_2}_{m\,m_2\,m_1}. | im2latex | fraction_rewrite |
1,300,473 | \left(\frac{4}{5}\right)^{7}-\left(2^{3}\right)^{2}=\left(\frac{2^{2}}{5}\right)^{7}-\left(2^{3}\right)^{2}= | mathwriting | group_factor |
281,697 | \begin{align*} [\mathbf{Q}&+p(\mathbf{R}+\mathbf{R}^T)+p^2\mathbf{T}]\mathbf{A}\\ &=(I-pR)(I-pR)\mathbf{A}-(-p)^2\mathbf{T}\mathbf{A} \\ &=\left[(I-pR)-(I-pR)^T\right]\mathbf{A}-\mathbf{T}\mathbf{A}. \end{align*}. | oleehyo | neutral_insert |
1,273,123 | f(x) = f(x_0) + \sum_{n=1}^{k}\left(\frac{D^nf(x_0)}{n!}\right)(x - x_0)^n | mathwriting | commute_safe |
351,983 | \frac{2}{3}\times \frac{1}{3}\times \frac{2}{3}= \frac{4}{9} | hme100k | commute_safe |
1,121,429 | \partial_t \partial^{\alpha}\bar{w}- \partial_y^2 \partial^{\alpha}\partial\bar{w}= -\partial^{\alpha}((u^s+\tilde{u}_1+(-u^s-\tilde{u}_1))\partial_x\bar{w}+\tilde{v}_2\partial_y\bar{w}+(u^s_{yy}+\tilde{w}_{1,y})\bar{v}+(\partial_x\tilde{w}_2)\bar{u}) | oleehyo | neutral_insert |
908,341 | ( \frac{1}{4} + \frac{3}{6} +\dots+ \frac{2\cdot998+1}{2005}). | hme100k | commute_safe |
389,599 | l_{(\cdot)}(P_\alpha^u\sigma)=l_\alpha^u(\sigma), \quad r_{(\cdot)}(P_\alpha^u\sigma)=r_\alpha^u(\sigma), \quad o_{(\cdot)}(P_\alpha^u\sigma)=o_\alpha^u(\sigma). | oleehyo | commute_safe |
622,258 | [P,Q] = $\displaystyle\sum_{i,j}p_i q_j (mi-nj) x^{i+j-1}Y^{m+n-i-j-1}$ | oleehyo | neutral_insert |
1,187,514 | A'(0) = f''(0)(f'(0))^{-3}\partial^2_{\theta}-\frac{1}{4}f''(0)(f'(0))^{-1}. | oleehyo | fraction_rewrite |
331,556 | \mathcal{M}_n = \left(\begin{array}{ccc}{-\mathcal{A}_{1,n}}{\mathcal{B}_n}{\mathcal{C}_n}{\left(\mathcal{B}_n\right)^*}{\mathcal{A}_{2,n}}{-\mathcal{D}_n}{\left(\mathcal{C}_n\right)^*}{-\left(\mathcal{D}_n\right)^*}{-P\left(T^{-2}-l\right)}\end{array}\right)^T | oleehyo | expand_local |
74,076 | R^{7-p} \sim g^{\frac{p - 3}{4}}[\ell_{p}^{(10)}]^{7 - p}. | oleehyo | group_factor |
1,156,450 | y=\sqrt{{P}_{U}}\mathbf{G}\mathbf{x}+\mathbf{n}, | oleehyo | fraction_rewrite |
391,348 | 4((1/2)ab)+c^{2} | mathwriting | fraction_rewrite |
235,753 | U (\tau_2,\tau_1)=\exp\left[-\mathcal{H}(\tau_2-\tau_1)\right],\quad(\tau_2>\tau_1) | linxy_full | group_factor |
770,892 | \varphi_1^{(8)}(\tau,z):=\varphi_1^{(4)}(\tau,z)\varphi_1^{(6)}(\tau,z)-\varphi_1^{(5)}(\tau,z)^2 | oleehyo | neutral_insert |
1,206,415 | \frac{48}{17} | mathwriting | commute_safe |
1,195,607 | P(x, y)=\oplus _{l=1}^{4}\left[X_{l}\, t_{{m}_{l}}(x, y)\right] | oleehyo | fraction_rewrite |
1,031,918 | H^{\frac{1}{2}\frac{1}{2}} = 2 \log x_{1 2}+\log \partial_{1}\partial_{2} | linxy_synthetic_handwrite | fraction_rewrite |
349,340 | v_{a} = {\displaystyle\left(\dfrac{{\displaystyle\frac{{\displaystyle\partial Q_{b}^{n_{b}}}}{{\displaystyle\partial q_{a}^{n_{a}}}}}}{1}\right)^{-1}}\left(V_{b}-{\displaystyle q_{c}^{s_{c}+1}\dfrac{{\displaystyle\partial Q_{b}^{n_{b}}}}{{\displaystyle\partial q_{c}^{s_{c}}}}}\right). | linxy_synthetic_handwrite | neutral_insert |
1,037,951 | ^{n-q}\otimes_{B_0}{\cal F}). | oleehyo | fraction_rewrite |
861,692 | =\frac{1}{2}\hbar \int_{-\mathrm{L}/ 2}^{\mathrm{L}/ 2}dx(\psi _{\pm }^{\dagger s }d_{\pm }\psi _{\pm }^{s}-\psi _{\pm }^{s}d_{\pm }^{\star}\psi _{\pm }^{\dagger s}). | linxy_full | commute_safe |
1,185,945 | \prod_{k}w_{n,k}^{'\top}\overline{Q}_{n,k} | mathwriting | fraction_rewrite |
1,355,281 | Var( \frac{1}{X} ) = E[ ( \frac{1}{X} - 1 )^2 ] | mathwriting | group_factor |
1,351,575 | |x|^3\partial_p\partial_qh_{ij}(x)\qquad | mathwriting | fraction_rewrite |
346,417 | M_j:=\bigcap_{k=0}^n X\setminus (\bigcup_{l > k} W_l) | oleehyo | commute_safe |
696,252 | y_d = \alpha+ (\beta x_d +\epsilon_d) | mathwriting | neutral_insert |
1,155,684 | S = \left(\sqrt{\frac{X}{Z}}\right) | oleehyo | expand_local |
6,981 | l-\left(\lambda+v\right) | crohme | group_factor |
1,238,064 | f_i(x, y) &= M_{ix} \cdot X \\ &+ (M_{iy}-1)\cdot Y | im2latex | expand_local |
1,147,379 | ds^2 = L^2/y^2 (dy^2 + dw+dw-) | linxy_synthetic_handwrite | commute_safe |
574,684 | \sum_{t<Y}F_{Y}\Bigl(\frac{t}{Y}\Bigr) \ll \frac{1}{(q-s)^k}\prod_{i=0}^{k-1}\Bigl|\sum_{t_i<q}\min\Bigl(q-s,\frac{q}{t_i}+\frac{q}{q-t_i}+s\Bigr)\Bigr| = O\Bigl(\frac{qs+q\log q}{q-s}\Bigr)^k. | oleehyo | group_factor |
1,332,793 | Gamma_g(N) = \{\gamma in GL_(2g)(mathbb Z)| gamma^(T)((0@I_g),(-I_g@0))gamma = ((0@I_g),(-I_g@0)), gamma equiv I_(2g) mod N\} | mathwriting | fraction_rewrite |
765,444 | cov(V_f(t + a), V_f(t)) = \int\limits_0^t (f(t + a) - f(s))(f(t) - f(s)) ds | mathwriting | group_factor |
520,761 | 2*90^\circ-3*90^\circ=180^\circ | hme100k | group_factor |
988,345 | S = \frac{\Lambda^2}{2}\int_p\phi_{-p}\phi_p + \frac{\bar Z}{2}\int_p p^2\phi_{-p}\phi_p + \cdots | oleehyo | fraction_rewrite |
1,365,199 | X_{i}^{j} = f(\alpha,\beta) + g(\gamma,\delta), i= 1,2; j = y | oleehyo | commute_safe |
143,327 | \frac{\frac{1000000000}{331199994444}\cdot\frac{666661111555}{1.25661627}}{103}=\frac{1000000000\cdot 666661111555}{331199994444\cdot 1.25661627\cdot 103}. | mathwriting | commute_safe |
1,377,044 | (ab)^{-1}=\left(\begin{array}{c}\frac{1}{\sqrt{\sum_{i=1}^{n}(x_ia)^2}}\\\vdots\\\frac{1}{\sqrt{\sum_{i=1}^{n}(x_in)^2}}\end{array}\right) | mathwriting | fraction_rewrite |
1,248,709 | |$B_n$| $\sim$ $\dfrac{ (n - 1)! }{ 2^{n - 1} (2n - 1) }$ $\sim$ $n!$. | oleehyo | group_factor |
539,999 | w_0 (E) +(w_1 (E) +w_2 (E))+\cdot\cdot | mathwriting | neutral_insert |
952,919 | -\frac{1}{2}\varphi+(-K^{t}(\partial_{t})\varphi)+(-\partial_{t}^{-1}W(\partial_{t}))\psi=0 | oleehyo | commute_safe |
1,273,105 | (\frac{86^{1}\cdot 1}{(\sqrt{438})^{1}})^{\frac{9}{292}}= (86^{\frac{9}{292}}\cdot(438)^{-\frac{9}{584}}) | mathwriting | fraction_rewrite |
876,845 | \hat{F}(U, V) = M_F(U, V) + \rho'(U, V) | oleehyo | group_factor |
1,075,936 | A_{i, k, j, l, m, n} \left(\dot{\alpha} \dot{\xi}^{l, j} \dot{\beta}^m\right) = (\sigma \sigma_{k, i} \sigma_n) + R_{k, i, n}. | oleehyo | expand_local |
619,534 | I_2^\prime \left(\frac{1 + c_3 \cos 3\theta + c_6 \cos^2 3\theta}{1 + c_3 + c_6}\right) = \frac{(\sigma_\text{eq} - \gamma_1 I_1)(1 - \gamma_2)}{ (1 - \gamma_1)(1 - \gamma_2)} | mathwriting | group_factor |
1,386,499 | Q\cdot G=0\,\,\mathrm { i n } \,\,K^ {(}Y) | im2latex | neutral_insert |
204,222 | gz(\rho_2-\rho_1)=\gamma((RA)-1-(RB)-1) | mathwriting | group_factor |
234,817 | \left(\Delta\theta\right)^2>\frac{4}{\pi^2 m N^2}. | mathwriting | group_factor |
131,015 | [ 3, 7] = {2} | hme100k | commute_safe |
1,246,272 | \frac{1}{d\delta}\mathbb{E}[N^{i}(dt)N^{i}(dt)]+\left( -\frac{1}{d\delta}\mathbb{E}[N^{i}(dt)]+\overline{\gamma}_{0}m_{i}\right)= | oleehyo | neutral_insert |
158,494 | e_{m}= - (\mathcal{A}_{m}-\lambda^{(m)}_{r}\mathbf{I})^{-1}\mathcal{B}_{(m+1)}e_{(m+1)} | linxy_synthetic_handwrite | fraction_rewrite |
507,168 | \sup_{\substack{\tau < t \\ z(\tau , \theta ) = x}} \| u \|_{C^k} (s) | oleehyo | neutral_insert |
831,347 | 2b'_s(z) = -a''_s(z) + f(z)a_s(z) -\frac{2\mu+1}{z}a'_s(z) +\frac{4\mu}{z}a'_s(z) + G | oleehyo | group_factor |
570,217 | \omega(Z) = Z J Z^T | mathwriting | neutral_insert |
653,423 | p_1 = \frac{p_1\cdot P}{P^2}P + p | mathwriting | group_factor |
547,818 | Q_{\mu}(\mu x, \mu y) = \frac{e^{- \frac{1}{2} \mu \zeta^{2}}}{2 \pi} \int_{-\pi}^{\pi} e^{\mu \psi(\theta)} f(\theta) \, d\theta | oleehyo | group_factor |
1,317,439 | S(t_1, t_2) = \frac{1}{2} [\epsilon (t_1 - t_2) - \tanh(e F \tau)] e^{2 e F (t_1 - t_2)}. | oleehyo | expand_local |
109 | \begin{align*} & ~ \\ &=~ 4\left( K^{-3}(\int_0^\infty {e^{\arctanh((AK^2)^{1/2}/(-K))}\,\mathrm d (-K)} ) ^{-1} \right) \\\end{align*}. | mathwriting | neutral_insert |
196,808 | s_i(\dot{x}_i+2k)+\sum_{j\ne i}a_ib_i^+s_j\Phi(x_i-x_j,z)=0 | linxy_synthetic_handwrite | fraction_rewrite |
495,405 | 3=\mu_{A}\left(I\right)\geq\mu_{A}\left(\overline{I}\right)-\ell_{A}\left(\overline{I}/I\right)\geq-MZ-\ell_{A}\left(\overline{I}/I\right)\geq K_{X}Z-\ell_{A}\left(\overline{I}/I\right)=\ell_{A}\left(\overline{I}/I\right)+2 | oleehyo | commute_safe |
342,216 | det(\lambda D+zL+\frac{1}{z}U-\lambda D-L)=\det(D\cdot (\lambda I_n+zA+(A^*)^{-1}) \cdot D^{-1})=\det((\lambda I_n+zA+(A^*)^{-1}))\cdot \det (D^2). | mathwriting | neutral_insert |
848,754 | \frac{(a)_3 (b)_3}{2} | mathwriting | group_factor |
430,571 | H_{\rho}(u, v) = \iint_{X \times X} (u - \rho)(x)h(x - y)(v - \rho)(y)\,dx\,dy | oleehyo | fraction_rewrite |
836,530 | \sum_{j=1}^\infty \left[2^{a(j)}\cdot (2^{s(j)})^x\right] | oleehyo | commute_safe |
570,050 | y\neq\pm14\sqrt5 | mathwriting | fraction_rewrite |
1,126,128 | Age = -8033*ln(F_m) | mathwriting | expand_local |
1,179,962 | l_{k} = \begin{bmatrix} 0 \\ x(k)M_H \\ y(k)M_V \end{bmatrix}. | oleehyo | group_factor |
1,347,029 | t(x)=\frac{\partial}{\partial x}B(x) | oleehyo | commute_safe |
176,512 | \nu^{u}(f)=\int_{G}\int_{Z}f(g\cdot z)\phi(z)\,d\beta^{s(g)}(z)\,d\lambda^{u}(g) =\int_{G}\Psi_{\phi}(f)(g) \,d\lambda^{u}(g).. | oleehyo | fraction_rewrite |
805,370 | \left\{\begin{array}{c} {\cal L}\varphi = 0\;\text{in}\;M \\ {\cal B}\varphi =\lambda _1 ({\cal B})\varphi \;\text{on}\,\;\partial M \end{array}\right. | oleehyo | neutral_insert |
809,826 | \langle\langle b^1+b_1^1, b^2+b_1^2\rangle\rangle | oleehyo | neutral_insert |
929,466 | w n(b) = $\left( \begin{matrix}-1 & 1 \\ b & \end{matrix}\right)$=$\left( \begin{matrix}b^{-1} & -1 \\ b & \end{matrix}\right)$$\left( \begin{matrix}1 & b^{-1} \\ 1 & \end{matrix}\right)$ | oleehyo | fraction_rewrite |
310,482 | \left(x^{n}\right)^{m}=\boxed{\left(x^{\frac{1}{n}}\right)^{\frac{mn}{1}}} | mathwriting | fraction_rewrite |
706,033 | \langle\hbar\omega\rangle=\frac{4}{5}\cdot\frac{\chi}{\sqrt{3}}\cdot\gamma m c^2 | mathwriting | expand_local |
370,749 | A\cdot dS_n = SO(2,n-1)\cdot \frac{1}{SO(1,n-1)} | linxy_synthetic_handwrite | expand_local |
447,055 | T = n (\tau_1 - \tau_2) | mathwriting | fraction_rewrite |
1,306,333 | I=\int_{\mathbb{Z}_{p}^{3}} e^{[ w_2w_3 x_1+w_1w_3 x_2+w_1w_2 x_3+w_1w_2w_3 (y_1+y_2+y_3)]_q t} \times d\mu_{q^{w_{2}w_{3}}}(x_{1}) d\mu_{q^{w_{1}w_{3}}}(x_{2}) d\mu_{q^{w_{1}w_{2}}}(x_{3}). | oleehyo | neutral_insert |
924,019 | \frac{\partial}{\partial t}\left(\int_0^t S(s) ds\right)=\lim_{\Delta t\to 0}\frac{1}{\Delta t}(S(t+\Delta t)\Delta t-S(t)\Delta t)=S(t)-S(t)=0 | mathwriting | expand_local |
685,455 | |x + p| = \sqrt{(x - p)^2 + y^2}. | mathwriting | fraction_rewrite |
1,090,150 | _{\mathfrak{M}_{g,n}^{tw}}(\mathfrak{C}, BG\times \mathfrak{M}_{g,n}^{tw}) | oleehyo | fraction_rewrite |
894,089 | \varphi(q)\mu_{1X}(p_1)-(-h(p_1))\mu_{21}(q) = \rho_{1X}(p_1)-\left(\frac{\partial}{\partial p}\right)_{P=p}H^{X}_{2}(\eta,\xi)(q),\,\,\,\,\forall q\in G | oleehyo | neutral_insert |
888,773 | D_{CFA} = \left( 1 + \dfrac{\tfrac{z}{2}}{(g+1)^2+(z/3)(2g-1)} \right) | mathwriting | group_factor |
608,439 | \frac{f(x+h)-2f(x)+f(x-h)}{h^{2}} | mathwriting | neutral_insert |
179,891 | D_\mu = \nabla_\mu | oleehyo | group_factor |
1,130,792 | R_{p}=(R_{1}+R_{2})*R_{3}-R_{3}^{2} | mathwriting | neutral_insert |
137,419 | {\cal Y}( \Omega , 5 ) = \mathrm{Tr}(\gamma_{\Omega , 5}^{-1}\gamma_{\Omega , 5}^T){\cal Z}( \Omega , 5 ), | linxy_full | fraction_rewrite |
109,884 | R_{\theta, 1; \lambda}(u) = -\frac{2^{\theta}}{2\pi i} \int_{4 - i\infty}^{4 + i\infty} H_{\lambda}^s \, ds | oleehyo | commute_safe |
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