Datasets:

anindya-hf-2002
/

pix2tex

image imagewidth (px) 27 4.02k	latex stringlengths 0 3.17k
	h _ { \parallel } ( \beta _ { T } \mu ) \sim - \ln ( \beta _ { T } \mu ) > 0
	\begin{align}\\|T_hf\\|_p^p&=\int_X \Bigl\|\int_X f(y)\> K_h(x,dy)\Bigr\|^p\> d\omega_X(x) \\&\le\int_X\Bigl(\int_X \|f(y)\|^p\> K_h(x,dy)\Bigr) \Bigl(\int_X 1\> K_h(x,dy)\Bigr)^{p/q}d\omega_X(x) =\\|f\\|_p^p.\end{align}
	\begin{align}j(q^{20};q^{32}) = j(q^{10};q^{16})j(-q^{10};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2},\\j(q^{4};q^{32}) = j(q^{2};q^{16})j(-q^{2};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2}.\end{align}
	P _ { 0 } = \frac \kappa 2 \, \left( 1 - e ^ { - 2 p _ { 0 } / \kappa } + \frac { \vec { p } \, { } ^ { 2 } } { \kappa ^ { 2 } } \right) .
	\begin{align}C^{-1}&=\frac{1}{3}\begin{pmatrix}2&1\\ 1&2\end{pmatrix}& C(z)&=\begin{pmatrix}z+z^{-1}&-1\\ -1&z+z^{-1}\end{pmatrix}& \widetilde{C}(z)&=\frac{1}{z^{2}+1+z^{-2}}\begin{pmatrix}z+z^{-1}&1\\ 1&z+z^{-1}\end{pmatrix}.\end{align}
	\begin{align} \mathcal{I}\left(a, m, 0, \frac{\pi}{2}\right) = \frac{1}{(1+a)^{m}} F_{1}\left( \frac{1}{2}, \frac{1}{2} - m, m, \frac{3}{2}, 1, \frac{1}{1+a} \right)\end{align}
	\begin{align}\mathcal S_{\boldsymbol\nu,\boldsymbol\mu}=O\left(\frac{f_{k_1}(z_i)^Nf_{k_2}(z_o)^N}{z_i^M z_o^M}\left(\frac{M}{eN}\right)^{2M}\prod_{j=1}^3\nu_j^{\mu_{j,\boldsymbol\cdot}+\mu_{\boldsymbol\cdot,j}}\right).\end{align}
	\begin{align}s=\frac{r}{\chi-r} (<1),\end{align}
	\begin{align}p^{-1} R^{-p} & = p^{-1} \left( \frac{\sqrt{\frac{\kappa}{CA} +\left( \frac{c_d'\|\nabla f(a)\|}{2CA} \right)^2} + \frac{c_d'\|\nabla f(a)\|}{2CA}}{\frac\kappa{CA}} \right)^p \\& = \frac{c_{d,p}}{\kappa^p} \left( \sqrt{ \tfrac{CA}{c_d'^2} \kappa +\left( \tfrac12 \|\nabla f(a)\| \right)^2} + \tfrac12\|\nabla f(a)\| \right)^p.\end{align}
	M _ { p } = 2 \left( \begin{array} { c } { { 2 p } } \\ { { p - 1 } } \\ \end{array} \right) = \frac { 2 p } { p + 1 } \, N _ { p } \ .
	\left( \begin{array} { c } { { { a } } } \\ { { { b } } } \\ \end{array} \right) \to U \cdot \left( \begin{array} { c } { { a } } \\ { { b } } \\ \end{array} \right) ,
	\begin{align}\omega^{\sigma_{\cal F}} = -i(\omega_G - p^ \omega_{WP})~.\end{align*}
	\begin{align} h_{ij}^X= & \frac{h_{ij}^Y}{\sqrt{(1-\|Y\|^2)(1-\langle N,Y\rangle^2)}},\end{align}
	\begin{align}&\mathbb{P}\left( \gamma_{\mathrm{u},ik} \geq \theta \right) \approx 1 - {2\beta \theta} = 1 - \frac{2 K \theta}{N}, \\ &\mathbb{P}\left( \gamma_{\mathrm{a},ik} \geq \theta \right) \approx 1 - \left( {2 \beta \theta}\right)^2 = 1 - \frac{4 K^2 \theta^2}{N^2}.\end{align}
	\begin{align}(\Phi^h)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\mathcal{Z}^1\wedge\dots\wedge\mathcal{Z}^m\wedge\mathcal{Z}^{\bar{1}}\wedge\dots\wedge\mathcal{Z}^{\bar{m}},\\(\Phi^v)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\delta\mathcal{V}^1\wedge\dots\wedge\delta\mathcal{V}^m\wedge\delta\mathcal{V}^{\bar{1}}\wedge\dots\wedge\delta\mathcal{V}^{\bar{m}},\end{align}
	\mu \to \frac { d } { d \lambda } ~ , \quad \lambda \to \lambda ~ .
	\begin{align}P_M^R(t)=\dfrac{P_M^S(t)}{1-t(P_R^S(t)-1)}.\end{align}
	\begin{align}\xi = \inf_{X\neq 0}\left\{ \dfrac{\sum_{\mu=1}^{M_A} \sum_{\nu=1}^{M_B} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2}{\sum_{\mu=1}^{N_v} \sum_{\nu=1}^{N_v} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2} \right\} \ .\end{align}
	\begin{align}f_{2}(a)=B_{+}\geq-e_{2}>0. \end{align}
	\begin{align}\Lambda_k^2=1+r_k^w(x,D_x),\end{align}
	\begin{align}I(t,\alpha,n)= \int_0^1e^{n(it\frac{p}{\sqrt{\alpha(1-\alpha)n}}+\alpha {\rm log}p + (1-\alpha) {\rm log}(1-p))}{\rm d} p.\end{align}
	\begin{align}A_{\mu}=e_{\mu}^{\;a}P_{a}+\omega_{\mu}^{\;a}J_{a}+\chi_{\mu}^{\;\alpha}Q_{\alpha}+\xi_{\mu}^{\;\alpha}Q'_{\alpha}\end{align}
	\begin{align}{\bf{P}}_{1} := a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \int_{B}e_n^{(3)}(y)\,dy\otimes\int_{B}e_n^{(3)}(y)\,dy = a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \, \sum_{m} \int_{B}e_{n,m}^{(3)}(y)\,dy\otimes\int_{B}e_{n,m}^{(3)}(y)\,dy, \end{align}
	\begin{align}S_nQ_{n+1}-S_{n+1}Q_n=(-1)^nw\end{align}
	\begin{align}\pi_{n}=\frac{\lambda_1\cdots\lambda_{n-1}}{\mu_1\cdots \mu_n},\;\textup{for}\; n\geq 2\end{align}
	\begin{align}S(Q^{A},P_{A};N,N^{\alpha}) = {\int} dt(P_{A}{\dot Q}^{A}-N.H-N^{\alpha}.H_{\alpha})\ \longrightarrow\ stat.\end{align}
	\begin{align}G_{0}(X(\gamma),Y(\gamma))=\int_0^1g(X(\gamma;t),Y(\gamma;t))dt.\end{align}
	g _ { 1 1 } = g _ { 2 2 } = 1 + ( \partial _ { 1 } A ) ^ { 2 } + ( \partial _ { 2 } A ) ^ { 2 } + ( \partial _ { 1 } C ) ^ { 2 } + ( \partial _ { 2 } C ) ^ { 2 } ,
	\begin{align} \|-t_k^{j,1}\log p_r -\theta_r^j -2\pi q\|<2^{-k}; j=1,\dots,N r=1,\dots,k. \end{align}
	\begin{align}\beta(n)=\frac{\beta(1)}{\beta(0)}\beta(n-1)=\left(\frac{\beta(1)}{\beta(0)}\right)^2\beta(n-2)=\cdots=\frac{\beta(1)^n}{\beta(0)^{n-1}}=(-\lambda)^{1-n}\beta(1)^n.\end{align}
	\begin{align}a^_{1,1}=\frac{\mu_R}{3},a^_{0,1}=\frac{2}{3}-\mu_R-\mu_T,a^_{0,2}=\mu_T-\frac{1}{3},\end{align*}
	\begin{align}\phi(t_0) = u(t_0), \phi'(t_0)= u'(t_0), \dots, \phi^{(n-1)}(t_0)= u^{(n-1)}(t_0).\end{align}
	\begin{align}\begin{aligned}2 i k z b(k)=& \Psi^+_{11}(0 ; z)\left(z \Psi^-_{21}(0;z)-\widehat{\Psi}^-_{21}(0)\right)-\Psi^-_{11}(0;z)\left(z \Psi^+_{21}(0; z)-\widehat{\Psi}^+_{21}(0)\right) \\&+\widehat{\Psi}^-_{21}(0)\left(\Psi^+_{11}(0;z)-e^{ic_+(0)}\right)-\widehat{\Psi}^+_{21}(0)\left(\Psi^-_{11}(0;z)-e^{ic_-(0)}\right)\end{aligned}\end{align}
	X _ { j } = 2 \, a _ { j } \exp \left( \xi _ { j } ( { t , x } ) \right)
	\begin{align}f(K)=f(K\cap H_i) + f(K \cap A) - f(H_i\cap A\cap K)\geq f(K\cap H_i)\end{align}
	\begin{align}\epsilon^{ij}\partial_i b_j=-{\textstyle\frac{1}{2(2\pi\alpha^\prime)}}\epsilon^{ij}(i_k C^{(3)})_{ij}\, .\end{align}
	\begin{align}\\|f\\|_{L^{p}(w)} \le C([\vec w]_{A_{\vec p}}) \prod_{i=1}^m\\|f_i\\|_{L^{p_i}(w_i)}\end{align}
	\begin{align}\frac{\ddot T}{T \|T\|^2} = \frac{1}{2} X_{11} \bar X + \frac{1}{2}\|X\|^2 - \frac{1}{2} X{\bar{X}}_{11} -\frac{1}{2}(X_{1})^2\frac{\bar{X}}{X} = \frac{A}{2}\end{align}
	\begin{align}I_{k}:= \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{(\|\tilde{u}_k(y)\|^{p(x,y)}+\|\tilde{v}_k(y)\|^{p(x,y)})\| \phi (x)- \phi (y)\|^{p(x,y)}}{\vartheta_{k}^{p(x,y)}\|x-y\|^{N+sp(x,y)}} \,dy\,dx.\end{align}
	\begin{align}(f_1\cup f_2)\bullet f_3-(f_1\bullet f_3)\cup f_2-(-1)^{(m_1-p_1)(m_3-p_3-1)}f_1\cup (f_2\bullet f_3)=0.\end{align}
	\begin{align}\Gamma_k := \bigg\{ f\in L^p([0,T],H) : \int_0^{T-\delta_k} \Vert f(t+\delta_k) - f(t) \Vert_Y^p dt \leq \frac{1}{k} \bigg\} .\end{align}
	\begin{align}[f_1,\cdots,f_M]_\;:=\;\sum_{\sigma \;\in\; S_M}\;({\rm sign}\;\sigma)\;(f_{\sigma\,1}\cdots f_{\sigma\,M})_\end{align}
	\begin{align}\mathbf v=a_1\mathbf v_1+\sigma a_2\mathbf v_2+\alpha(\delta_2/g\mathbf v_1-\sigma\delta_1/g\mathbf v_2),\end{align}
	\begin{align}X^3+\frac{1}{3}\alpha^2X^2+\alpha X+1=(X-x_1)(X-x_2)(X-x_3).\end{align}
	\begin{align}e^{i\Gamma[A]}=\int{\cal D}\psi{\cal D}\overline\psi e^{i\int d^2x\overline\psi iD\!\!\!/\psi},\end{align}
	\begin{align} u^f_{n,t}=\prod_{k=0}^{n-1}a_kf_{t-n}+\sum_{s=n}^{t-1}w_{n,s}f_{t-s-1},n,t\in\mathbb{N}.\end{align}
	\frac { d } { d z } \left( \begin{array} { r r } { { \chi _ { 1 } ( z ) } } \\ { { \chi _ { 2 } ( z ) } } \\ \end{array} \right) =
	\begin{align}\lambda_{1,e}(B_\infty^n) = \frac{n}{n-1} .\end{align}
	\begin{gather}[\lambda] = \big\{ (i,j) \in \mathbb{Z}^2 \colon 1\leq i \leq \ell(\lambda),\ 1 \leq j \leq \lambda_i \big\}.\end{gather}
	\sigma _ { ~ ~ ; \alpha } ^ { ; \alpha } = 2 - \frac { 1 } { 6 } g _ { \gamma \delta } R \sigma ^ { ; \gamma } \sigma ^ { ; \delta } + \frac { 1 } { 2 4 } g _ { \gamma \delta } R _ { ; \rho } \sigma ^ { ; \gamma } \sigma ^ { ; \delta } \sigma ^ { ; \rho } + O ( \sigma ^ { 2 } ) ~ .
	\begin{align} dz_t = \Pi_{\mathrm{ker}A}(B(y_t,y_t)-Ay_t) dt + \Pi_{\mathrm{ker}A}\sigma dW_t. \end{align}
	\begin{align}L(x)^{q^i}=\sum_{k=1}^mc_{k-i}^{q^i}\,x^{q^k},\end{align}
	\begin{align}\sum_{i=0}^{r}d_i\sigma(x)^i = a(x)P_\alpha(x) + b(x)Q_\alpha(x)+ c(x)T(x),\end{align}
	\begin{align}a_1=\alpha_1, a_2=\alpha_2, {a_3} =\frac{2\,\alpha_1+3\,\alpha_3}{8}\end{align}
	\begin{align}\binom{\l +q+1}{m+n+1}=\sum_{0\le k \le \l} \binom{\l -k}{m}\binom{q+k}{n} (\l, m \ge 0, n\ge q\ge 0).\end{align}
	\begin{align}M_{12}(\lambda)=\begin{bmatrix}\lambda P_4-P_3 & \lambda P_3 \end{bmatrix} \mbox{and} M_{22}(\lambda)=\begin{bmatrix}\lambda P_3-P_2 & \lambda P_2 \\\lambda P_2 & \lambda P_1+P_0 \end{bmatrix}.\end{align}
	{ \Delta _ { 1 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i + 1 , j ) } - u ^ { \alpha ( i , j ) } } { x _ { 1 } ^ { ( i + 1 ) } - x _ { 1 } ^ { ( i ) } } , \qquad { \Delta _ { 2 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i , j + 1 ) } - u ^ { \alpha ( i , j ) } } { x _ { 2 } ^ { ( j + 1 ) } - x _ { 2 } ^ { ( j ) } } .
	i \gamma ^ { \mu } \partial _ { \mu } \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) = \left( \begin{array} { c c c } { { 0 } } & { { \phi ^ { 2 } } } & { { 0 } } \\ { { \overline { { { \phi ^ { 2 } } } } } } & { { 0 } } & { { \overline { { { \phi ^ { 1 } } } } } } \\ { { 0 } } & { { \phi ^ { 1 } } } & { { 0 } } \\ \end{array} \right) \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) .
	\begin{align}u_x=\frac{1}{\kappa}[s_1X_x\tilde{u}_{\tilde{x}}-r_2X_x\tilde{v}_{\tilde{x}}+r_2(s_{1x}v+s_{2x}u+s_{3x})-s_1(r_{1x}u+r_{2x}v+r_{3x})],\end{align}
	\begin{align}&\frac12\frac{d}{dt}\sum_{q\geq -1}\left(\lambda_q^{2s}\\|u_q\\|_2^2+\lambda_q^{2s}\\|b_q\\|_2^2\right)\\\leq &-\nu\sum_{q\geq-1}\lambda_q^{2s+2}\\|u_q\\|_2^2-\mu\sum_{q\geq-1}\lambda_q^{2s+2\alpha}\\|b_q\\|_2^2+ I_1+I_2+I_3+I_4+I_5,\end{align}
	\begin{align}(I-\Delta)^{-\alpha} u = G_{2\alpha}\star u\end{align}
	\begin{align} \upsilon_R'(r)= -\frac{i}{\sqrt\pi}r^{-1-\frac k2} \int_{\partial D_r}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})w_R\cdot\nu\, \psi_k\, ds.\end{align}
	\begin{align}\bar{L}=L_{0}-\frac{\partial L^{i}}{\partial x^{i}}-y_{i}^{\alpha }\frac{\partial L^{i}}{\partial y^{\alpha }},\end{align}
	\begin{align} S=\int L \,,L=-\Big(\frac14 F_{\mu\nu}F^{\mu\nu}+C\partial^\mu A_\mu^\ast\Big)d^4x\,.\end{align}
	\begin{align}\sup_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}=(1+c_1\delta\rho\zeta^k)\inf_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}\end{align}
	\begin{align}V_{ij}={n-j \choose n-i}(-1)^{i-j}I.\end{align}
	\begin{align} P_{k,\ell,p}(j) = \sum_{k'=2}^{k} \sum_{\ell'=2}^{\ell} \binom{k-k'+j-1}{k-k'} \binom{\ell-\ell'+j-1}{\ell-\ell'} \Big( \Big( \frac{p}{p-1} \Big)^{k'-1} + \Big( \frac{p}{p-1} \Big)^{\ell'-1}-1 \Big)\end{align}
	\begin{align}\det(B)\ne0\mathrm{ind}(B)=\Phi(D,k')-(k'-1).\end{align}
	\begin{align}B_{{\mathbf x}}(w)=\displaystyle\frac{1}{2}(0,b_{21}^2{\mathbf x}_2w_1+b_{22}^2{\mathbf x}_2w_2).\end{align}
	F ^ { I _ { 3 } I _ { 2 } I _ { 1 } I _ { 4 } } F ^ { I _ { 1 } I _ { 2 } I _ { 3 } I _ { 4 } } \ = \ 1 \
	\begin{align}K_2=(K_1\cap K_2)(K_2\cap K_3).\end{align}
	( d s ) ^ { 2 } = - \frac { 1 } { \sqrt { \Lambda } } \left( 1 + \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) \mathrm { s i n } ^ { 2 } R d ^ { 2 } T + \frac { 1 } { \Lambda } \left( 1 - \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) d ^ { 2 } R + \frac { 1 } { \Lambda } \left( 1 - 2 \delta \mathrm { c o s } \right) d ^ { 2 } \Omega _ { ( 2 ) }
	\begin{align}F(x) = - \frac{(2x+2)^2 -e^{-4}}{4x + 4}, x \ne -1.\end{align}
	\begin{align} \begin{cases} \dot S = -a(-)SI +c(-)(N-S-I)\\ \dot I = a(-)SI - b(-)I.\end{cases}\end{align}
	\Psi _ { \mathrm { { 0 } } } ( r ^ { \prime } , \theta ^ { \prime } ; 0 ) = { \frac { 1 } { \sqrt { 2 \pi } \xi } } \exp \left\{ i k r ^ { \prime } \cos \theta ^ { \prime } - { \frac { 1 } { 4 \xi ^ { 2 } } } ( r ^ { 2 } + r _ { 0 } ^ { 2 } + 2 r r ^ { \prime } \cos \theta ^ { \prime } ) \right\} .
	\begin{align} d X_t = \xi_t \;d t + \sqrt{2\nu} \;d W_t \forall t\in [0,T],\end{align}
	\begin{align}\begin{array}{l}\\|\tilde{\mathcal{G}}_{\tilde{e}_N}^M(s)\tilde\Phi_M-\tilde{\mathcal{G}}_{\tilde{e}}^M(s)\tilde\Phi_M\\|_M\leq\\\sum\limits_{i,j=1}^N\\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\\|_\infty\\|f\circ\sigma_{ij}-f\\|_{\infty}\leq\\2(N-1)\sum\limits_{i=1}^N\\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\\|_\infty\\|f\\|_{\infty},\end{array}\end{align}
	\begin{align} \pi_2\left(gB_0g^{-1},gB_1g^{-1},..., g(FB_0)g^{-1}\right)= \left(gB_{r+1}g^{-1},F(gB_1g^{-1}),...,F(gB_{r+1}g^{-1})\right).\end{align}
	\begin{align}a(u,v)=(f,v)~v\in H^1_0(D),\end{align}
	\begin{align} {\cal O}^{(k)} = u_{i_1} \cdots u_{i_k} \mbox{Tr}(\phi^{i_1} \cdots \phi^{i_k}) \, .\end{align}
	\begin{align} p(t,g)=&\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}(\xi-\mu)+\mu)e^{-H_t(\xi-\mu)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}\xi+\mu)e^{-H_t(\xi)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)](\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]). \end{align*}
	\begin{align}a_{\mathrm{pw}}(\phi_{\mathrm{nc}}({j}),v_{\mathrm{nc}})=\lambda_h(j) b(\phi_{\mathrm{pw}}({j}),v_{\mathrm{nc}})\phi_{\mathrm{pw}}({j})-\phi_{\mathrm{nc}}({j})=\lambda_h({j})\kappa_{m}^2h_{\mathcal{T}}^{2m}\phi_{\mathrm{pw}}({j}). \end{align}
	\begin{align}\|\delta e\|^2 = \int d \lambda ({\hat e} (\lambda; l ))^{-1} (\delta e(\lambda;l))^2 = \int d \lambda [ - {\hat e}(\lambda) (\delta f) {{d^2}\over{d \lambda^2}} (\delta f) + {{(\delta \rho [f(\lambda)])^2}\over{{\hat e}}} ] , \end{align}
	\begin{align}\hat\phi(0)-\phi(0)+\int_{-\infty}^{\infty}\hat\phi(x)\|x\|dx=\int_{-\infty}^{\infty}\phi(x)W_{\rm U}^1(x)dx=\int_{-\infty}^{\infty}\phi(x)\left(1-\frac{\sin^2(\pi x)}{(\pi x)^2}\right)dx.\end{align}
	\begin{align}\begin{array}{l}\displaystyle [e_{n+1},e_1]=-e_3,[e_{n+1},e_{2}]=-e_2,[e_{n+1},e_i]=-e_i-\epsilon e_{i+2}-\sum_{k=i+3}^n{b_{k-i-2}e_k},\\\displaystyle (\epsilon=0,\pm1,3\leq i\leq n,5\leq j\leq n);DS=[n+1,n-1,0],LS=[n+1,n-1,n-1,...].\end{array} \end{align}
	p _ { i } = \varepsilon _ { i j } \frac { E _ { j } } { B }
	\begin{align}\eta_k(r) : \begin{cases} =1 &\mbox{if } -\infty<r\leq k\\ 0<\eta_k(r)<1&\mbox{if } k< r< k+\delta_k\\ =0 &\mbox{if } k+\delta_k\leq r<\infty. \end{cases}\end{align}
	\begin{align}\tilde{O}({\bf R}, {\bf r})= O_{0}({\bf R}) + {\bf r} \cdot{\bf O}_{1}({\bf R}) +o(r^{2}).\end{align}
	\begin{align}\overline{W}^{(q)}(x) = 0,\overline{\overline{W}}^{(q)}(x) = 0,Z^{(q)}(x) = 1,\textrm{and} \overline{Z}^{(q)}(x) = x, x \leq 0. \end{align}
	\begin{align}c_{\alpha}d_{h-\alpha}\varepsilon(h, \alpha)=-\frac{1}{3}\end{align}
	\begin{align}\xi^\ast=\frac{\beta(\omega)(1-\omega)-1+\alpha}{2-\alpha}.\end{align}
	\begin{align}J_{k_U}(n, \rho) = O\left( 2^{-(2\epsilon+\epsilon^2)\log_2 n + O(\sqrt{\log n} \log\log n )} \right),\end{align}
	\begin{align}\forall y,y' \in I,\; \phi_H(u_{J(y)},u_{J(y')})=0\;,\end{align}
	\begin{align}\left(1 - \frac{\beta_{i}\pi_{i}}{\gamma_{i} + \delta m}\right)^{-1} \leq \mathbb{E} \left(\sum_{j=1}^{m} T_{ij}(\pi,Q) \right), \end{align}
	\begin{align} T_{R} = T_{R 33} e_3 \otimes e_3.\end{align}
	\begin{align}\tilde{\Gamma}^i_{0k}=\tilde{\Gamma}^i_{k0}&=\frac12 \tilde{g}^{im}\left(\tilde{g}_{m0,k}+\tilde{g}_{mk,0}-\tilde{g}_{0k,m}\right)\\&=\frac12 r^{-2}g^{im}\left(\tilde{g}_{mk,0}\right)\\&=\frac12 r^{-2}g^{im}\left(2rg_{mk}\right)\\&=r^{-1}\delta^i_k.\end{align}
	\begin{align}\widetilde{\gamma}^\omega_n^{2n}=\langle\frac{de_n(s)}{ds},e_{2n}(s)\rangle ds=\kappa_n(s)ds.\end{align*}
	c _ { A B C } = ( \frac { \omega _ { B } - \omega _ { C } } { \omega _ { A } } - 1 ) d _ { A B C } .
	\begin{align}Qv = \left(I + v (u'-u)\right) v= v + v\left(u'v - uv\right)= v\end{align}
	\begin{align} & \int x (q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{(1-q)x(x^2;q^2)_\infty}{[n]_q-1} \left(\frac{q^n}{x}h_n(\frac{x}{q};q)-q^{n-1}[n]_q h_{n-1}(\frac{x}{q};q) \right), \end{align}

End of preview. Expand in Dataset Viewer.

Here’s a completed dataset card for your Pix2Tex dataset based on the provided template:

Pix2Tex

Dataset Details

Dataset Description

The Pix2Tex Dataset is a high-quality, curated dataset for training and evaluating Vision-Language Models (VLMs) capable of extracting LaTeX code from images of mathematical formulas. This dataset combines both printed and handwritten formula images, offering diverse and challenging samples for tasks involving LaTeX recognition.

The dataset contains images paired with their LaTeX annotations, enabling researchers and developers to explore various use cases, including mathematical OCR (Optical Character Recognition), multimodal learning, and LaTeX generation from formula images.

Curated by: Anindya
Language(s): English (en)
License: Apache 2.0

Uses

Direct Use

The Pix2Tex Dataset can be used for:

Training and evaluation of Vision-Language Models (VLMs).
Image-to-Text conversion for LaTeX generation.
Research in mathematical OCR and handwriting recognition.
Applications involving mathematical document digitization.

Out-of-Scope Use

The dataset should not be used for purposes unrelated to image-to-text conversion, LaTeX generation, or Vision-Language modeling tasks. Misuse of the dataset for malicious purposes, such as creating deceptive content, is strictly prohibited.

Dataset Structure

Features

The dataset contains the following features:

image: The image of the formula (either printed or handwritten).
latex: The corresponding LaTeX code representing the formula.

Splits

The dataset is split into three subsets for easier use in machine learning workflows:

Split	Number of Examples	Size (bytes)
Train	377,163	1,638,732,362.625
Validation	125,721	545,374,827.875
Test	125,722	544,914,564.75

Configurations

The dataset includes a default configuration with the following data files:

Train: data/train-*
Validation: data/validation-*
Test: data/test-*

Dataset Creation

Curation Rationale

The Pix2Tex Dataset was created to address the need for diverse, high-quality datasets for training Vision-Language Models (VLMs) that can extract LaTeX from mathematical formula images. By including both printed and handwritten formulas, the dataset provides a robust benchmark for real-world applications.

Source Data

Data Collection and Processing

The dataset was created by combining two existing datasets:

One dataset containing printed formula images with their LaTeX annotations.
Another dataset containing handwritten formula images with their LaTeX annotations.

Both datasets were preprocessed to ensure consistency in column naming and annotation quality. The resulting dataset was split into training, validation, and test sets.

Who are the source data producers?

The source data was collected from publicly available datasets and curated for the specific purpose of LaTeX extraction tasks.

Bias, Risks, and Limitations

Risks and Limitations

The dataset includes a mix of printed and handwritten formulas, which might introduce variability in performance for models trained solely on one type of input.
The handwritten formulas may contain noise due to variations in handwriting styles, potentially challenging OCR tasks.

Recommendations

Users are advised to preprocess the dataset and evaluate models on both printed and handwritten subsets to better understand their performance across different input types.

Citation

If you use this dataset in your research or projects, please cite it as follows:

Glossary

Vision-Language Models (VLMs): Machine learning models that combine visual and textual inputs to perform tasks such as image-to-text generation.
LaTeX: A typesetting system commonly used for mathematical and scientific documents.

Dataset Card Contact

For any questions or feedback, feel free to contact:

Hugging Face Profile: anindya-hf-2002

Downloads last month: 46

Size of downloaded dataset files:

2.47 GB

Size of the auto-converted Parquet files:

2.47 GB

Number of rows:

628,606

image imagewidth (px) 27 4.02k	latex stringlengths 0 3.17k
	h _ { \parallel } ( \beta _ { T } \mu ) \sim - \ln ( \beta _ { T } \mu ) > 0
	\begin{align}\\|T_hf\\|_p^p&=\int_X \Bigl\|\int_X f(y)\> K_h(x,dy)\Bigr\|^p\> d\omega_X(x) \\&\le\int_X\Bigl(\int_X \|f(y)\|^p\> K_h(x,dy)\Bigr) \Bigl(\int_X 1\> K_h(x,dy)\Bigr)^{p/q}d\omega_X(x) =\\|f\\|_p^p.\end{align}
	\begin{align}j(q^{20};q^{32}) = j(q^{10};q^{16})j(-q^{10};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2},\\j(q^{4};q^{32}) = j(q^{2};q^{16})j(-q^{2};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2}.\end{align}
	P _ { 0 } = \frac \kappa 2 \, \left( 1 - e ^ { - 2 p _ { 0 } / \kappa } + \frac { \vec { p } \, { } ^ { 2 } } { \kappa ^ { 2 } } \right) .
	\begin{align}C^{-1}&=\frac{1}{3}\begin{pmatrix}2&1\\ 1&2\end{pmatrix}& C(z)&=\begin{pmatrix}z+z^{-1}&-1\\ -1&z+z^{-1}\end{pmatrix}& \widetilde{C}(z)&=\frac{1}{z^{2}+1+z^{-2}}\begin{pmatrix}z+z^{-1}&1\\ 1&z+z^{-1}\end{pmatrix}.\end{align}
	\begin{align} \mathcal{I}\left(a, m, 0, \frac{\pi}{2}\right) = \frac{1}{(1+a)^{m}} F_{1}\left( \frac{1}{2}, \frac{1}{2} - m, m, \frac{3}{2}, 1, \frac{1}{1+a} \right)\end{align}
	\begin{align}\mathcal S_{\boldsymbol\nu,\boldsymbol\mu}=O\left(\frac{f_{k_1}(z_i)^Nf_{k_2}(z_o)^N}{z_i^M z_o^M}\left(\frac{M}{eN}\right)^{2M}\prod_{j=1}^3\nu_j^{\mu_{j,\boldsymbol\cdot}+\mu_{\boldsymbol\cdot,j}}\right).\end{align}
	\begin{align}s=\frac{r}{\chi-r} (<1),\end{align}
	\begin{align}p^{-1} R^{-p} & = p^{-1} \left( \frac{\sqrt{\frac{\kappa}{CA} +\left( \frac{c_d'\|\nabla f(a)\|}{2CA} \right)^2} + \frac{c_d'\|\nabla f(a)\|}{2CA}}{\frac\kappa{CA}} \right)^p \\& = \frac{c_{d,p}}{\kappa^p} \left( \sqrt{ \tfrac{CA}{c_d'^2} \kappa +\left( \tfrac12 \|\nabla f(a)\| \right)^2} + \tfrac12\|\nabla f(a)\| \right)^p.\end{align}
	M _ { p } = 2 \left( \begin{array} { c } { { 2 p } } \\ { { p - 1 } } \\ \end{array} \right) = \frac { 2 p } { p + 1 } \, N _ { p } \ .
	\left( \begin{array} { c } { { { a } } } \\ { { { b } } } \\ \end{array} \right) \to U \cdot \left( \begin{array} { c } { { a } } \\ { { b } } \\ \end{array} \right) ,
	\begin{align}\omega^{\sigma_{\cal F}} = -i(\omega_G - p^ \omega_{WP})~.\end{align*}
	\begin{align} h_{ij}^X= & \frac{h_{ij}^Y}{\sqrt{(1-\|Y\|^2)(1-\langle N,Y\rangle^2)}},\end{align}
	\begin{align}&\mathbb{P}\left( \gamma_{\mathrm{u},ik} \geq \theta \right) \approx 1 - {2\beta \theta} = 1 - \frac{2 K \theta}{N}, \\ &\mathbb{P}\left( \gamma_{\mathrm{a},ik} \geq \theta \right) \approx 1 - \left( {2 \beta \theta}\right)^2 = 1 - \frac{4 K^2 \theta^2}{N^2}.\end{align}
	\begin{align}(\Phi^h)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\mathcal{Z}^1\wedge\dots\wedge\mathcal{Z}^m\wedge\mathcal{Z}^{\bar{1}}\wedge\dots\wedge\mathcal{Z}^{\bar{m}},\\(\Phi^v)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\delta\mathcal{V}^1\wedge\dots\wedge\delta\mathcal{V}^m\wedge\delta\mathcal{V}^{\bar{1}}\wedge\dots\wedge\delta\mathcal{V}^{\bar{m}},\end{align}
	\mu \to \frac { d } { d \lambda } ~ , \quad \lambda \to \lambda ~ .
	\begin{align}P_M^R(t)=\dfrac{P_M^S(t)}{1-t(P_R^S(t)-1)}.\end{align}
	\begin{align}\xi = \inf_{X\neq 0}\left\{ \dfrac{\sum_{\mu=1}^{M_A} \sum_{\nu=1}^{M_B} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2}{\sum_{\mu=1}^{N_v} \sum_{\nu=1}^{N_v} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2} \right\} \ .\end{align}
	\begin{align}f_{2}(a)=B_{+}\geq-e_{2}>0. \end{align}
	\begin{align}\Lambda_k^2=1+r_k^w(x,D_x),\end{align}
	\begin{align}I(t,\alpha,n)= \int_0^1e^{n(it\frac{p}{\sqrt{\alpha(1-\alpha)n}}+\alpha {\rm log}p + (1-\alpha) {\rm log}(1-p))}{\rm d} p.\end{align}
	\begin{align}A_{\mu}=e_{\mu}^{\;a}P_{a}+\omega_{\mu}^{\;a}J_{a}+\chi_{\mu}^{\;\alpha}Q_{\alpha}+\xi_{\mu}^{\;\alpha}Q'_{\alpha}\end{align}
	\begin{align}{\bf{P}}_{1} := a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \int_{B}e_n^{(3)}(y)\,dy\otimes\int_{B}e_n^{(3)}(y)\,dy = a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \, \sum_{m} \int_{B}e_{n,m}^{(3)}(y)\,dy\otimes\int_{B}e_{n,m}^{(3)}(y)\,dy, \end{align}
	\begin{align}S_nQ_{n+1}-S_{n+1}Q_n=(-1)^nw\end{align}
	\begin{align}\pi_{n}=\frac{\lambda_1\cdots\lambda_{n-1}}{\mu_1\cdots \mu_n},\;\textup{for}\; n\geq 2\end{align}
	\begin{align}S(Q^{A},P_{A};N,N^{\alpha}) = {\int} dt(P_{A}{\dot Q}^{A}-N.H-N^{\alpha}.H_{\alpha})\ \longrightarrow\ stat.\end{align}
	\begin{align}G_{0}(X(\gamma),Y(\gamma))=\int_0^1g(X(\gamma;t),Y(\gamma;t))dt.\end{align}
	g _ { 1 1 } = g _ { 2 2 } = 1 + ( \partial _ { 1 } A ) ^ { 2 } + ( \partial _ { 2 } A ) ^ { 2 } + ( \partial _ { 1 } C ) ^ { 2 } + ( \partial _ { 2 } C ) ^ { 2 } ,
	\begin{align} \|-t_k^{j,1}\log p_r -\theta_r^j -2\pi q\|<2^{-k}; j=1,\dots,N r=1,\dots,k. \end{align}
	\begin{align}\beta(n)=\frac{\beta(1)}{\beta(0)}\beta(n-1)=\left(\frac{\beta(1)}{\beta(0)}\right)^2\beta(n-2)=\cdots=\frac{\beta(1)^n}{\beta(0)^{n-1}}=(-\lambda)^{1-n}\beta(1)^n.\end{align}
	\begin{align}a^_{1,1}=\frac{\mu_R}{3},a^_{0,1}=\frac{2}{3}-\mu_R-\mu_T,a^_{0,2}=\mu_T-\frac{1}{3},\end{align*}
	\begin{align}\phi(t_0) = u(t_0), \phi'(t_0)= u'(t_0), \dots, \phi^{(n-1)}(t_0)= u^{(n-1)}(t_0).\end{align}
	\begin{align}\begin{aligned}2 i k z b(k)=& \Psi^+_{11}(0 ; z)\left(z \Psi^-_{21}(0;z)-\widehat{\Psi}^-_{21}(0)\right)-\Psi^-_{11}(0;z)\left(z \Psi^+_{21}(0; z)-\widehat{\Psi}^+_{21}(0)\right) \\&+\widehat{\Psi}^-_{21}(0)\left(\Psi^+_{11}(0;z)-e^{ic_+(0)}\right)-\widehat{\Psi}^+_{21}(0)\left(\Psi^-_{11}(0;z)-e^{ic_-(0)}\right)\end{aligned}\end{align}
	X _ { j } = 2 \, a _ { j } \exp \left( \xi _ { j } ( { t , x } ) \right)
	\begin{align}f(K)=f(K\cap H_i) + f(K \cap A) - f(H_i\cap A\cap K)\geq f(K\cap H_i)\end{align}
	\begin{align}\epsilon^{ij}\partial_i b_j=-{\textstyle\frac{1}{2(2\pi\alpha^\prime)}}\epsilon^{ij}(i_k C^{(3)})_{ij}\, .\end{align}
	\begin{align}\\|f\\|_{L^{p}(w)} \le C([\vec w]_{A_{\vec p}}) \prod_{i=1}^m\\|f_i\\|_{L^{p_i}(w_i)}\end{align}
	\begin{align}\frac{\ddot T}{T \|T\|^2} = \frac{1}{2} X_{11} \bar X + \frac{1}{2}\|X\|^2 - \frac{1}{2} X{\bar{X}}_{11} -\frac{1}{2}(X_{1})^2\frac{\bar{X}}{X} = \frac{A}{2}\end{align}
	\begin{align}I_{k}:= \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{(\|\tilde{u}_k(y)\|^{p(x,y)}+\|\tilde{v}_k(y)\|^{p(x,y)})\| \phi (x)- \phi (y)\|^{p(x,y)}}{\vartheta_{k}^{p(x,y)}\|x-y\|^{N+sp(x,y)}} \,dy\,dx.\end{align}
	\begin{align}(f_1\cup f_2)\bullet f_3-(f_1\bullet f_3)\cup f_2-(-1)^{(m_1-p_1)(m_3-p_3-1)}f_1\cup (f_2\bullet f_3)=0.\end{align}
	\begin{align}\Gamma_k := \bigg\{ f\in L^p([0,T],H) : \int_0^{T-\delta_k} \Vert f(t+\delta_k) - f(t) \Vert_Y^p dt \leq \frac{1}{k} \bigg\} .\end{align}
	\begin{align}[f_1,\cdots,f_M]_\;:=\;\sum_{\sigma \;\in\; S_M}\;({\rm sign}\;\sigma)\;(f_{\sigma\,1}\cdots f_{\sigma\,M})_\end{align}
	\begin{align}\mathbf v=a_1\mathbf v_1+\sigma a_2\mathbf v_2+\alpha(\delta_2/g\mathbf v_1-\sigma\delta_1/g\mathbf v_2),\end{align}
	\begin{align}X^3+\frac{1}{3}\alpha^2X^2+\alpha X+1=(X-x_1)(X-x_2)(X-x_3).\end{align}
	\begin{align}e^{i\Gamma[A]}=\int{\cal D}\psi{\cal D}\overline\psi e^{i\int d^2x\overline\psi iD\!\!\!/\psi},\end{align}
	\begin{align} u^f_{n,t}=\prod_{k=0}^{n-1}a_kf_{t-n}+\sum_{s=n}^{t-1}w_{n,s}f_{t-s-1},n,t\in\mathbb{N}.\end{align}
	\frac { d } { d z } \left( \begin{array} { r r } { { \chi _ { 1 } ( z ) } } \\ { { \chi _ { 2 } ( z ) } } \\ \end{array} \right) =
	\begin{align}\lambda_{1,e}(B_\infty^n) = \frac{n}{n-1} .\end{align}
	\begin{gather}[\lambda] = \big\{ (i,j) \in \mathbb{Z}^2 \colon 1\leq i \leq \ell(\lambda),\ 1 \leq j \leq \lambda_i \big\}.\end{gather}
	\sigma _ { ~ ~ ; \alpha } ^ { ; \alpha } = 2 - \frac { 1 } { 6 } g _ { \gamma \delta } R \sigma ^ { ; \gamma } \sigma ^ { ; \delta } + \frac { 1 } { 2 4 } g _ { \gamma \delta } R _ { ; \rho } \sigma ^ { ; \gamma } \sigma ^ { ; \delta } \sigma ^ { ; \rho } + O ( \sigma ^ { 2 } ) ~ .
	\begin{align} dz_t = \Pi_{\mathrm{ker}A}(B(y_t,y_t)-Ay_t) dt + \Pi_{\mathrm{ker}A}\sigma dW_t. \end{align}
	\begin{align}L(x)^{q^i}=\sum_{k=1}^mc_{k-i}^{q^i}\,x^{q^k},\end{align}
	\begin{align}\sum_{i=0}^{r}d_i\sigma(x)^i = a(x)P_\alpha(x) + b(x)Q_\alpha(x)+ c(x)T(x),\end{align}
	\begin{align}a_1=\alpha_1, a_2=\alpha_2, {a_3} =\frac{2\,\alpha_1+3\,\alpha_3}{8}\end{align}
	\begin{align}\binom{\l +q+1}{m+n+1}=\sum_{0\le k \le \l} \binom{\l -k}{m}\binom{q+k}{n} (\l, m \ge 0, n\ge q\ge 0).\end{align}
	\begin{align}M_{12}(\lambda)=\begin{bmatrix}\lambda P_4-P_3 & \lambda P_3 \end{bmatrix} \mbox{and} M_{22}(\lambda)=\begin{bmatrix}\lambda P_3-P_2 & \lambda P_2 \\\lambda P_2 & \lambda P_1+P_0 \end{bmatrix}.\end{align}
	{ \Delta _ { 1 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i + 1 , j ) } - u ^ { \alpha ( i , j ) } } { x _ { 1 } ^ { ( i + 1 ) } - x _ { 1 } ^ { ( i ) } } , \qquad { \Delta _ { 2 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i , j + 1 ) } - u ^ { \alpha ( i , j ) } } { x _ { 2 } ^ { ( j + 1 ) } - x _ { 2 } ^ { ( j ) } } .
	i \gamma ^ { \mu } \partial _ { \mu } \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) = \left( \begin{array} { c c c } { { 0 } } & { { \phi ^ { 2 } } } & { { 0 } } \\ { { \overline { { { \phi ^ { 2 } } } } } } & { { 0 } } & { { \overline { { { \phi ^ { 1 } } } } } } \\ { { 0 } } & { { \phi ^ { 1 } } } & { { 0 } } \\ \end{array} \right) \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) .
	\begin{align}u_x=\frac{1}{\kappa}[s_1X_x\tilde{u}_{\tilde{x}}-r_2X_x\tilde{v}_{\tilde{x}}+r_2(s_{1x}v+s_{2x}u+s_{3x})-s_1(r_{1x}u+r_{2x}v+r_{3x})],\end{align}
	\begin{align}&\frac12\frac{d}{dt}\sum_{q\geq -1}\left(\lambda_q^{2s}\\|u_q\\|_2^2+\lambda_q^{2s}\\|b_q\\|_2^2\right)\\\leq &-\nu\sum_{q\geq-1}\lambda_q^{2s+2}\\|u_q\\|_2^2-\mu\sum_{q\geq-1}\lambda_q^{2s+2\alpha}\\|b_q\\|_2^2+ I_1+I_2+I_3+I_4+I_5,\end{align}
	\begin{align}(I-\Delta)^{-\alpha} u = G_{2\alpha}\star u\end{align}
	\begin{align} \upsilon_R'(r)= -\frac{i}{\sqrt\pi}r^{-1-\frac k2} \int_{\partial D_r}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})w_R\cdot\nu\, \psi_k\, ds.\end{align}
	\begin{align}\bar{L}=L_{0}-\frac{\partial L^{i}}{\partial x^{i}}-y_{i}^{\alpha }\frac{\partial L^{i}}{\partial y^{\alpha }},\end{align}
	\begin{align} S=\int L \,,L=-\Big(\frac14 F_{\mu\nu}F^{\mu\nu}+C\partial^\mu A_\mu^\ast\Big)d^4x\,.\end{align}
	\begin{align}\sup_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}=(1+c_1\delta\rho\zeta^k)\inf_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}\end{align}
	\begin{align}V_{ij}={n-j \choose n-i}(-1)^{i-j}I.\end{align}
	\begin{align} P_{k,\ell,p}(j) = \sum_{k'=2}^{k} \sum_{\ell'=2}^{\ell} \binom{k-k'+j-1}{k-k'} \binom{\ell-\ell'+j-1}{\ell-\ell'} \Big( \Big( \frac{p}{p-1} \Big)^{k'-1} + \Big( \frac{p}{p-1} \Big)^{\ell'-1}-1 \Big)\end{align}
	\begin{align}\det(B)\ne0\mathrm{ind}(B)=\Phi(D,k')-(k'-1).\end{align}
	\begin{align}B_{{\mathbf x}}(w)=\displaystyle\frac{1}{2}(0,b_{21}^2{\mathbf x}_2w_1+b_{22}^2{\mathbf x}_2w_2).\end{align}
	F ^ { I _ { 3 } I _ { 2 } I _ { 1 } I _ { 4 } } F ^ { I _ { 1 } I _ { 2 } I _ { 3 } I _ { 4 } } \ = \ 1 \
	\begin{align}K_2=(K_1\cap K_2)(K_2\cap K_3).\end{align}
	( d s ) ^ { 2 } = - \frac { 1 } { \sqrt { \Lambda } } \left( 1 + \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) \mathrm { s i n } ^ { 2 } R d ^ { 2 } T + \frac { 1 } { \Lambda } \left( 1 - \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) d ^ { 2 } R + \frac { 1 } { \Lambda } \left( 1 - 2 \delta \mathrm { c o s } \right) d ^ { 2 } \Omega _ { ( 2 ) }
	\begin{align}F(x) = - \frac{(2x+2)^2 -e^{-4}}{4x + 4}, x \ne -1.\end{align}
	\begin{align} \begin{cases} \dot S = -a(-)SI +c(-)(N-S-I)\\ \dot I = a(-)SI - b(-)I.\end{cases}\end{align}
	\Psi _ { \mathrm { { 0 } } } ( r ^ { \prime } , \theta ^ { \prime } ; 0 ) = { \frac { 1 } { \sqrt { 2 \pi } \xi } } \exp \left\{ i k r ^ { \prime } \cos \theta ^ { \prime } - { \frac { 1 } { 4 \xi ^ { 2 } } } ( r ^ { 2 } + r _ { 0 } ^ { 2 } + 2 r r ^ { \prime } \cos \theta ^ { \prime } ) \right\} .
	\begin{align} d X_t = \xi_t \;d t + \sqrt{2\nu} \;d W_t \forall t\in [0,T],\end{align}
	\begin{align}\begin{array}{l}\\|\tilde{\mathcal{G}}_{\tilde{e}_N}^M(s)\tilde\Phi_M-\tilde{\mathcal{G}}_{\tilde{e}}^M(s)\tilde\Phi_M\\|_M\leq\\\sum\limits_{i,j=1}^N\\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\\|_\infty\\|f\circ\sigma_{ij}-f\\|_{\infty}\leq\\2(N-1)\sum\limits_{i=1}^N\\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\\|_\infty\\|f\\|_{\infty},\end{array}\end{align}
	\begin{align} \pi_2\left(gB_0g^{-1},gB_1g^{-1},..., g(FB_0)g^{-1}\right)= \left(gB_{r+1}g^{-1},F(gB_1g^{-1}),...,F(gB_{r+1}g^{-1})\right).\end{align}
	\begin{align}a(u,v)=(f,v)~v\in H^1_0(D),\end{align}
	\begin{align} {\cal O}^{(k)} = u_{i_1} \cdots u_{i_k} \mbox{Tr}(\phi^{i_1} \cdots \phi^{i_k}) \, .\end{align}
	\begin{align} p(t,g)=&\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}(\xi-\mu)+\mu)e^{-H_t(\xi-\mu)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}\xi+\mu)e^{-H_t(\xi)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)](\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]). \end{align*}
	\begin{align}a_{\mathrm{pw}}(\phi_{\mathrm{nc}}({j}),v_{\mathrm{nc}})=\lambda_h(j) b(\phi_{\mathrm{pw}}({j}),v_{\mathrm{nc}})\phi_{\mathrm{pw}}({j})-\phi_{\mathrm{nc}}({j})=\lambda_h({j})\kappa_{m}^2h_{\mathcal{T}}^{2m}\phi_{\mathrm{pw}}({j}). \end{align}
	\begin{align}\|\delta e\|^2 = \int d \lambda ({\hat e} (\lambda; l ))^{-1} (\delta e(\lambda;l))^2 = \int d \lambda [ - {\hat e}(\lambda) (\delta f) {{d^2}\over{d \lambda^2}} (\delta f) + {{(\delta \rho [f(\lambda)])^2}\over{{\hat e}}} ] , \end{align}
	\begin{align}\hat\phi(0)-\phi(0)+\int_{-\infty}^{\infty}\hat\phi(x)\|x\|dx=\int_{-\infty}^{\infty}\phi(x)W_{\rm U}^1(x)dx=\int_{-\infty}^{\infty}\phi(x)\left(1-\frac{\sin^2(\pi x)}{(\pi x)^2}\right)dx.\end{align}
	\begin{align}\begin{array}{l}\displaystyle [e_{n+1},e_1]=-e_3,[e_{n+1},e_{2}]=-e_2,[e_{n+1},e_i]=-e_i-\epsilon e_{i+2}-\sum_{k=i+3}^n{b_{k-i-2}e_k},\\\displaystyle (\epsilon=0,\pm1,3\leq i\leq n,5\leq j\leq n);DS=[n+1,n-1,0],LS=[n+1,n-1,n-1,...].\end{array} \end{align}
	p _ { i } = \varepsilon _ { i j } \frac { E _ { j } } { B }
	\begin{align}\eta_k(r) : \begin{cases} =1 &\mbox{if } -\infty<r\leq k\\ 0<\eta_k(r)<1&\mbox{if } k< r< k+\delta_k\\ =0 &\mbox{if } k+\delta_k\leq r<\infty. \end{cases}\end{align}
	\begin{align}\tilde{O}({\bf R}, {\bf r})= O_{0}({\bf R}) + {\bf r} \cdot{\bf O}_{1}({\bf R}) +o(r^{2}).\end{align}
	\begin{align}\overline{W}^{(q)}(x) = 0,\overline{\overline{W}}^{(q)}(x) = 0,Z^{(q)}(x) = 1,\textrm{and} \overline{Z}^{(q)}(x) = x, x \leq 0. \end{align}
	\begin{align}c_{\alpha}d_{h-\alpha}\varepsilon(h, \alpha)=-\frac{1}{3}\end{align}
	\begin{align}\xi^\ast=\frac{\beta(\omega)(1-\omega)-1+\alpha}{2-\alpha}.\end{align}
	\begin{align}J_{k_U}(n, \rho) = O\left( 2^{-(2\epsilon+\epsilon^2)\log_2 n + O(\sqrt{\log n} \log\log n )} \right),\end{align}
	\begin{align}\forall y,y' \in I,\; \phi_H(u_{J(y)},u_{J(y')})=0\;,\end{align}
	\begin{align}\left(1 - \frac{\beta_{i}\pi_{i}}{\gamma_{i} + \delta m}\right)^{-1} \leq \mathbb{E} \left(\sum_{j=1}^{m} T_{ij}(\pi,Q) \right), \end{align}
	\begin{align} T_{R} = T_{R 33} e_3 \otimes e_3.\end{align}
	\begin{align}\tilde{\Gamma}^i_{0k}=\tilde{\Gamma}^i_{k0}&=\frac12 \tilde{g}^{im}\left(\tilde{g}_{m0,k}+\tilde{g}_{mk,0}-\tilde{g}_{0k,m}\right)\\&=\frac12 r^{-2}g^{im}\left(\tilde{g}_{mk,0}\right)\\&=\frac12 r^{-2}g^{im}\left(2rg_{mk}\right)\\&=r^{-1}\delta^i_k.\end{align}
	\begin{align}\widetilde{\gamma}^\omega_n^{2n}=\langle\frac{de_n(s)}{ds},e_{2n}(s)\rangle ds=\kappa_n(s)ds.\end{align*}
	c _ { A B C } = ( \frac { \omega _ { B } - \omega _ { C } } { \omega _ { A } } - 1 ) d _ { A B C } .
	\begin{align}Qv = \left(I + v (u'-u)\right) v= v + v\left(u'v - uv\right)= v\end{align}
	\begin{align} & \int x (q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{(1-q)x(x^2;q^2)_\infty}{[n]_q-1} \left(\frac{q^n}{x}h_n(\frac{x}{q};q)-q^{n-1}[n]_q h_{n-1}(\frac{x}{q};q) \right), \end{align}