harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{MN}= 1 + \sum_{d=1}^\infty \frac{z^d}{d!d!}N^{2d}\sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N) s_\lambda(b_1,\dots,b_N) \frac{(\dim \mathsf{V}^\lambda)^2}{(\dim \mathsf{W}_M^\lambda) (\dim \mathsf{W}_N^\lambda)}.
E_1= \begin{bmatrix}1 0 0 0 0 0 0 0 1\end{bmatrix},A_1= \begin{bmatrix}-1 0 4\pi 0 -1 0 -4\pi 0 -4 \end{bmatrix}&E_2=\begin{bmatrix}1 0 0 0 0 1 0 0 1\end{bmatrix},A_2= \begin{bmatrix}-k_2k_3 0 0 0 -1 0 -4k_1k_2k_3 -1 -4k_3\end{bmatrix}
E(\Sigma, X, T_0) = \frac{1}{8\pi}\int_{\widehat{\Sigma}}\widehat{H}d{\widehat{\Sigma}}+ \frac{1}{8\pi}\int_\Sigma \left[\sqrt{1+|\nabla\tau|^2}\cosh\theta|{H}| + \nabla \tau \cdot \nabla \theta -\alpha_H ( \nabla \tau) \right]d\Sigma,
\Big\vert I_u(0,1) - I_u(x,1) \Big\vert = \Big\vert \frac{D_u(0,1)}{H_u(x,1)}\left( \frac{H_u(x,1)}{H_u(0,1)}- \frac{D_u(x,1)}{D_u(0,1)}\right) \Big\vert \stackrel{\eqref{e:H limitato}}{\leq}C.
P\{\max_{0\le s\le t}\mathcal{T}(s)=0\ |\V(0) = -c_2,\N(t) = 2k+1\}= \Bigl(\frac{c_2}{c_1+c_2}\Bigr)^{k+1}\sum_{j=0}^k A_j^{(k)}\Bigl(\frac{c_1}{c_1+c_2}\Bigr)^{j}
\frac{d^{2}x^{a}}{d \hat{s}^{2}}+ \{_{m n}^{a}\}_{h a t}\frac{d x^{m}}{d \hat{s}}\frac{d x^{n}}{d \hat{s}}- \frac{\psi_{, n}}{2}( \frac{d x^{n}}{d \hat{s}}\frac{d x^{a}}{d \hat{s}}- \hat{g}^{n a}) = 0 ,
\psi_n(x) \propto \begin{cases}(1+\lambda x^2)^{-\beta/(2\lambda)}C_n^{(-\beta/\lambda)}({\rm i}\sqrt{\lambda}\, x) begin{align*}0.2cm] (1-|\lambda| x^2)^{\beta/(2|\lambda|)}C_n^{(\beta/|\lambda|)}(\sqrt{|\lambda|}\, x) \end{cases}
\sigma_{( a_{i}^{\dag}a_{k})}^{-}= <{\bf A}_{-}\mid \frac{1}{2}( a_{i}^{\dag}a_{k}+ a_{k}a_{i}^{\dag}) \mid{\bf A_{-}}> = \alpha_{i}^{*}\alpha_{k}c o t h \mid{\bf A}\mid^{2}+ \frac{1}{2}\delta_{i k}. \nonumber \,
E_1' \colon y^2= (x^2-1)\left(x^2-\left(\frac{1-\mu\sqrt{\lambda}}{1+\mu\sqrt{\lambda}}\right)^2\right), E_2' \colon y^2= (x^2-1)\left(x^2-\left(\frac{1-\sqrt{\lambda}}{1+\sqrt{\lambda}}\right)^2\right).
g_1 = -4z^2, g_2 = g_1 I_1, g_3 =16 xy (x+z)(y+z), g_4 = g_1 I_1^2, g_5 = g_3 I_1, begin{align*}1mm]I_2 = - \frac{4xy(x+z)(y+z)}{z^2}, I_3 = I_1^2, I_4 = I_1 I_2.
F(r,R) = \frac{r}{(1+r)^2}\left(\frac{r^2-1}{2}\sin(2x) + R'(x)(\cos^2(x) + r \sin^{2}(x)) + (r-1)\sin(2x)R(x) + R(x)R'(x)\right) + \frac{1}{4\pi}\sum_{i=1}^{2}F_{i}(r,R) & = F_{0}(r,R) + \frac{1}{4\pi}F_{1}(r,R) + \frac{1}{4\pi}F_{2}(r,R)
((x_1^{p^en},\dots,x_{i-1}^{p^en}):_{R}\, x_i^{p^en}) =((x_1^{p^en},\dots,x_{i-1}^{p^en}):_{R}\, x_i) &\subseteq ((x_1^{p^em},\dots,x_{i-1}^{p^em}):_{R}x_i) &=((x_1^{p^em},\dots,x_{i-1}^{p^em}):_{R}x_i^{p^en-p^em})
B_{8}= \left( \begin{array}{c c}{{P}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) , B_{9}= \left( \begin{array}{c c}{{Q}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) , B_{7}= \left( \begin{array}{c c}{{b I}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) , B_{1}= \left( \begin{array}{c c}{{I v t}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) .
(g_{31}y_1+g_{32}y_2+g_{33}y_3)\{(g_{11}+g_{21}+g_{31})y_1+(g_{22}+g_{32})y_2+(g_{23}+g_{33})y_3\}^{a-1}=g_{31}(g_{11}+g_{21}+g_{31})^{a-1}y_1^a+g_{32}(g_{22}+g_{32})^{a-1}y_2(y_1+y_2)^{a-1}
\|x\|\ =\ \left\|x-P_A(x) + \sum_{n\in A}e_n^*(x)e_n\right\| \ \leqslant\ \sup_{\delta}\|x-P_A(x)+1_{\delta A}\| \ \leqslant\ \mathbf C_{\lambda, pl}\|x-P_A(x) + 1_{\varepsilon B}\|.
\frac{d \phi_1}{dx}=\pm\frac{\partial W}{\partial \phi_1}=\pm\left( 2 \phi_1^2+ \sigma \phi_2^2-\frac{1}{2}\right) \hspace{1cm}\frac{d \phi_2}{dx}=\pm\frac{\partial W}{\partial \phi_2}=\pm 2\sigma \phi_1 \phi_2 \qquad .
\textrm{Pr}(X_\beta(t+h)=j\mid X_\beta(t)=i) = \begin{cases}( \bar F_j(\rho)- \bar F_i(\rho))_+h+o(h)\ , \quad \textrm{if}~ j\in N(i)\ ; 1-\sum_{j\in N(i)}(\bar F_j(\rho)-\bar F_i(\rho))_+h+o(h)\ , \textrm{if}~ j=i\ ; 0\ , \textrm{otherwise}\ ,\end{cases}
\mu(1_{\{\}}(1-e^{-\langle l,\chi\rangle}))=\ln(\prod\limits_{x\in S}\frac{\chi(x)-L^x_x}{-L^x_x}) &\mu(1_{\{\}}(1-e^{-\langle l,\chi\rangle}))=\ln(I+M_{\sqrt{\chi}}VM_{\sqrt{\chi}})+\ln(\prod\limits_{x\in S}\frac{-L^x_x}{\chi(x)-L^x_x}).
|a(t_{j})-a(t_{j-1})|^2 = \Big| \int_{t_{j-1}}^{t_j}t a'(t)\frac{dt}{t}\Big|^2 \leq (t_j-{t_{j-1}}) \int_{t_{j-1}}^{t_{j}}(ta'(t))^2 \frac{dt}{t^2}\leq 2^i\int_{t_{j-1}}^{t_{j}}(ta'(t))^2 \frac{dt}{t^2}.
\tilde{E}_{i}(h,\theta)= \int\limits_{y_{i}+2n_{i-1}s}^{y_{i}}\big(W(U_{h,\theta}(ih-,y))-W(U_{h,\theta_i}(ih-,y))\big)dy &+\int\limits_{y_{i}+(2n_{i-1}+d_i)s}^{y_{i}+2n_{i-1}s}W(U_{h,\theta}(ih-,y))dy-\int\limits_{y_{i}+2(n_{i-1}-1)s}^{y_{i}+2n_{i-1}s}W(U_{h,\theta_i}(ih-,y))dy.
\mathbf{CK}_{x_2}^{\mu_2}={}_0F_{1}\left(\genfrac{}{}{0pt}{}{-}{\mu_2+1/2}\;\Big\rvert\;-\left(\frac{x_2\,\sigma_1 T_1}{2}\right)^2\right)-\frac{\sigma_2\,x_2\,(\sigma_1 T_1)}{2\mu_2+1}\;{}_0F_{1}\left(\genfrac{}{}{0pt}{}{-}{\mu_2+3/2}\;\Big\rvert\;-\left(\frac{x_2\,\sigma_1 T_1}{2}\right)^2\right).
R_n(t;z) \prod_{k=0}^{n-1}\frac{1 + \mathcal{R}e(\rho_{k}(t)\alpha_{k-1})}{1+\overline{\rho_{k}(t) \alpha_{k-1}}}= zS_{n-1}(z) - \rho_{n-1}(t)S_{n-1}^{\ast}(z), n \geq 1,
{\rm Prob}(S_B(x) \le s)\ =\ \lim_{k\to\infty}\lim_{\epsilon\to 0}\frac{\sum_{l=0}^{k-1}\mu(\exp(\log B^lX)\exp(U_{\epsilon}([0,\log s)X)))}{\mu(\exp(U_{\epsilon}([0,k\log B)X)))}.
\begin{array}{ccc}(Au)_1 L^k = (u_{+}k b_1) \theta_{1}L^{k-1}(Au)_2 L^k = (u_{+}k b_2) \theta_{2}L^{k-1}\vdots (Au)_{d-1}L^k = ( u_{+}k b_{d-1})\theta_{d-1}L^{k-1}.\end{array}
\lambda_{p,n}^{k,v}:= n \cdot Cov \left(\frac{\hat{\gamma}_p(k/n;B^H)}{\gamma_p(k/n;B^H)},\frac{\hat{\gamma}_p(v/n;B^H)}{\gamma_p(v/n;B^H)}\right) &= \frac{n}{\gamma_p(k/n;B^H) \gamma_p(v/n;B^H)}\cdot Cov \left(\hat{\gamma}_p(k/n;B^H),\hat{\gamma}_p(v/n;B^H) \right) , k,v = 1, 2, \ldots, m.
\frac{d}{dt}|D^\alpha H|^r_r\leq \Big(\sum_{j=1}^{3}|\partial_{x_j}A_j|_\infty+|B|_\infty\Big)|D^\alpha H|^r_r+|\Theta_1|_r|D^\alpha H|^{r-1}_r+|\Theta_2|_r|D^\alpha H|^{r-1}_r.
\sum_{n\geq L}E_M(t,n)|C_n| =\sum_{n= L}^{M_--1}E_M(t,n)|C_n|+\sum_{n=M_-}^{M_+-1}E_M(t,n)|C_n|+\sum_{n= M_+}^{\infty}E_M(t,n)|C_n| &=:H_1+H_2+H_3\,,
\left. \begin{array}{l}{{\delta_{a}\bigl ( [ S_{a}]^{- 1}[ S_{2}] \bigl ) = \delta_{a}\bigl ( ( D_{2}) ( \gamma_{2}) ( D_{1})^{- 1}\bigl ) = 0 ,}}\\{{\delta_{b}\bigl ( [ S_{b}] [ S_{1}] \bigl ) = \delta_{b}\bigl ( ( D_{1}) ( \gamma_{1}) \bigl ) = 0 ,}}\\ \end{array}\right.
\left\vert tu_{k}+z\right\vert_{c,2^{\ast}}^{2^{\ast}}=\int_{\Theta\smallsetminus\theta}c(x)\left\vert tu_{k}+z\right\vert^{2^{\ast}}+\int_{\theta}c(x)\left\vert z\right\vert^{2^{\ast}}& \geq t^{2^{\ast}}\int_{\Theta}c(x)u_{k}^{2^{\ast}}+2^{\ast}t^{2^{\ast}-1}\int_{\Theta}c(x)u_{k}^{2^{\ast}-1}z+2^{\ast}A\left\Vert z\right\Vert_{a}^{2^{\ast}}.
0 < \frac{r - \lambda x}{xp(x)}\int_0^{x}p(t)\,\mathrm{d}t \le \frac{r - \lambda x}{xp(x)}\int_0^{x}\frac{r - \lambda t}{r - \lambda x}p(t)\,\mathrm{d}t = \frac{1}{xp(x)}\int_0^{x}(t p(t))' \,\mathrm{d}t =1.
\#\{i:|\lambda^T_i| \leq \lambda\}= \frac{\omega_d}{(2\pi)^d}\left [\operatorname{Vol}(\mathcal O) + \int_{\mathcal O}n^{\frac d 2}(x) dx \right ] \lambda^{\frac{d}{2}}+ o(\lambda^{\frac d 2}), \lambda \rightarrow \infty,
\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{\lambda^i}\mathcal P^i\left(h(\cdot)\frac{1}{\lambda^{n-i}}\mathcal P^{(n-i)}(\cdot,M)\right)(x) \xrightarrow[L^1(M,\mu)]{n\to\infty}\eta(x)\int_M h(y)\eta(y)\mu(\d y),
d\mathcal{L}(0,0,\phi)|_{(g_c,0,f_c)}=(2\triangle\phi-2\langle\nabla\phi,\nabla f_c\rangle,-\int_M \phi e^{-f_c}dV_{g_c})\in C^{0,\alpha}_{g_c}(M)\times \mathbb{R}
\bigl\|\sum_j\bigl\langle G_R(\Lambda_{\psi}(q_j)\otimes p_j^*\zeta_2),\pi_R(\rho_2)^*\zeta_1\otimes\rho_1\bigr\rangle\bigr\| &=\left\|\left\langle V\left(\sum_j\Lambda_{\psi}(p_j)\otimes q_j^*\zeta_1\right),V\bigl(\pi_R(\rho_1)^*\zeta_2\otimes\rho_2\bigr)\right\rangle\right\|\le\varepsilon\bigl\|\pi_R(\rho_1)^*\zeta_2\otimes\rho_2\bigr\|.
\sum_{k\not=0}\exp(-2Td^3(k,0)) \min\{d^{-1}(k,0),\ell^2d(k,0)\}\lesssim\min\{(T^\frac{1}{3})^{-\frac{3}{2}},\ell^2(T^\frac{1}{3})^{-\frac{7}{2}}\}.
\operatorname{O}_{V^{\perp}}(V,q):=\big\{\varphi\in\operatorname{O}(V,q)\big|\varphi|_{V^{\perp}}=\operatorname{Id}_{V^{\perp}}\big\}=\{\varphi\in\operatorname{O}(V,q)|\varphi(v)=v\ \forall v \in V^{\perp}\}.
\Upsilon(l_{1},l_{2},l_{3};\mu) ={\sum_{k}}\frac{(-1)^{k}}{R(l_{1},l_{2},l_{3};k)}& +\frac{1}{\mu}\ \frac{(l_{1}+l_{2}+l_{3}+1)}{2}\{\sum_{k}}\frac{(-1)^{k}(2k-(l_{1}+l_{2}+l_{3}))}{R(l_{1},l_{2},l_{3};k)}& +O(\mu^{-2})
\Big(\int_{0}^1{\int_{\Omega}{\Big\Vert \Big( \sum_{k=1}^{L}{(\mathfrak{M}_{{\sigma},2^{t_k}}h_{k}^{\omega,t})^q}\Big)^{{1}/{q}}\Big\Vert_{L^p}^p}d\lambda}dt\Big)^{{1}/{p}}\lesssim_{p,q}1
\epsilon_{\mu}^{1}(p)=\frac{1}{\sqrt{2}}(\frac{-ip_{1}p_{2}+E^{2}+p_{3}E-p_{1}^{2}}{E(E+p_{3})},\frac{-p_{1}p_{2}+iE^{2}+ip_{3}E-ip_{2}^{2}}{E(E+p_{3})},\frac{-p_{1}-ip_{2}}{E},0),
P \coloneqq1+\sum_{n=1}^{\infty}\sum_{\lambda\vdash n}L_{\lambda}x^{n}\prod_{i=1}^{\infty}z_{i}^{N_{i}(\lambda)}=\sum_{m_{1},m_{2},\dots}\prod_{i=1}^{\infty}h_{m_{i}}[L_{i}](z_{i}x^{i})^{m_{i}}=\prod_{i=1}^{\infty}\sum_{m_{i}=0}^{\infty}h_{m_{i}}[L_{i}](z_{i}x^{i})^{m_{i}}
\dot{x}(t) = y(t), \dot{y}(t) = \begin{cases}-\frac{k}{m}\,x(t) - \frac{b}{m}\,y(t) + \frac{F}{m}, x(t-\mu) < 0, -\frac{k}{m}\,x(t) - \frac{b}{m}\,y(t) - \frac{F}{m}, x(t-\mu) > 0.\end{cases}
\Gamma_{1;f}\leq \frac{\beta^2}{2\beta-1}\Big[1+\frac{(\beta-1)^2}{\Big(2\beta-\frac{7}{3}\Big)-\frac{2}{3}\sqrt{2(3\beta-4)}}\Big] \lambda_{1;f}^2 &=\frac{\beta^2}{2\beta-1}\Big[1+\frac{1}{3}\Big(\frac{(3\beta-4)+1}{\sqrt{2(3\beta-4)}-1}\Big)^2\Big]\lambda_{1;f}^2.
\Gamma(E)=\frac{e^{2}n_{r}^{3}}{2\pi\hbar mc^{3}\epsilon_{0}}\frac{1}{N_{p}}\sum_{l}^{N_{p}}\Delta E_{1,0}(R_{l})^{2}|f_{1,0}(R_{l})|g(E-\Delta E_{1,0}(R_{l}),\delta)
S^+ = \left( \begin{array}{cccc}0 a_1 0 0 0 0 a_2 0 0 0 0 a_1 0 0 0 0 \end{array}\right) , \quad S^- = (S^+)^t \,, \quad S^3 = \left( \begin{array}{cccc}\frac 32 0 0 0 0 \frac 12 0 0 0 0 - \frac 12 0 0 0 0 -\frac 32 \end{array}\right) ,
\left. \begin{array}{l}\left.\phi'\right|{\mbox{\raisebox{-1.2ex}{{\tiny{$\!0^{-}$}}}}}=-\left.\phi'\right|{\mbox{\raisebox{-1.2ex}{{\tiny{$\!0^{+}$}}}}}= -\frac{1}{4}\sqrt{\alpha}\sigma, \left.a'\right|{\mbox{\raisebox{-1.2ex}{{\tiny{$\!0^{-}$}}}}}=-\left. a'\right|{\mbox{\raisebox{-1.2ex}{{\tiny{$\!0^{+}$}}}}}= \frac{1}{4}\sigma . \end{array}\right.
\langle \mathcal{H}\rangle_{\Psi}\geq b\frac{K_L^2}{\ell^2}\langle n_+^H\rangle_{\Psi}+ \frac{1}{2}\langle \mathcal{Q}_4^{}\rangle_{\Psi}+ \frac{1}{2}\rho N \widehat{g}(0) \times \begin{dcases}\Big(1 - C \sqrt{\rho a^3}\frac{K_{L}}{K_{\ell}}\Big), d=3, \Big(1 - C \widehat{g}(0) \Big), d=2,\end{dcases}
(ev_-)^*F^{j'}(\psi_--\chi)= (ev_-)^*F^{j'}(-\chi) + \psi_-(ev_-)^*\displaystyle\frac{\partial}{\partial t}\left.F^{j'}(t)\right|_{t=-\chi}+ \frac{(\psi_-)^2}{2!}(ev_-)^*\frac{\partial^2}{\partial z^2}\left.F^{j'}(t)\right|_{t=-\chi}+\dotsc .
\omega^{(k,i)}(\xi)e^{(k,i)}(\xi)=\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi),
\mathbb{P}[X^n=\mathbf{x}_1, Y^n=\mathbf{y}_1|X^n\in\mathcal{A}, Y^n\in \mathcal{B}] =\mathbb{P}[X^n=\mathbf{x}_2, Y^n=\mathbf{y}_2|X^n\in\mathcal{A}, Y^n\in \mathcal{B}] &=\frac{p_{11}^{n/2}p_{22}^{n/2}}{2(p_{11}^{n/2}p_{22}^{n/2}+p_{12}^{n/2}p_{21}^{n/2})}
|u'_0(r)-u_{\delta}'(r)| \le\frac{1}{r^{N-2}}\int_0^{r_0}\eta_0r^{N-2}dr+\delta\left\|\frac{2(N-3)r}{1+\delta r^2}u_{\delta}(r)\right\|_{C^0[0,r_0]}&=\frac{\eta_0r_0}{N-1}+\delta\left\|\frac{2(N-3)r}{1+\delta r^2}u_{\delta}(r)\right\|_{C^0[0,r_0]}.
\Re{( \sum_{j = 1}^{\Delta m / 2}\tilde{z}_{j})}= 0 \, , \quad \Im{( \sum_{j = 1}^{\Delta m / 2}\tilde{z}_{j})}= \frac{1}{2}\Im{\left( \sum_{k = 1}^{m^{+}}z_{k}- \sum_{l = m^{+}+ 1}^{m}z_{l}\right)}\, .
h\left(x\right) =e^{-x}\int_{0}^{x}\varphi\left(y\right)e^{y}dy+e^{x}\int_{x}^{1}\varphi\left(y\right)e^{-y}dy h'\left(x\right) =-e^{-x}\int_{0}^{x}\varphi\left(y\right)e^{y}dy+e^{x}\int_{x}^{1}\varphi\left(y\right)e^{-y}dy.
(d^{\ast}\phi ,G(d^{\ast}\phi )) =(2\pi R)^{d - 4}\sum_{{\bf l}\in{\bf Z}^{2}}R^{4}\prod_{k = 1,2}l_{k}^{- 2}|\exp [iR^{- 1}L_{k}l_{k}] - 1|^{2}-(2\pi R)^{d}\biggl( \frac{L_{1}L_{2}}{(2\pi R)^{2}}\biggr)^{2}.
E_{\gamma \eta, 1}(-\vartheta t^{\gamma \eta}) = \frac{\sin \gamma \pi}{\pi}\int_0^\infty \frac{r^{\gamma-1}}{r^{2\gamma}+2r^\gamma \cos \gamma \pi+1}E_{\eta,1}\left(-r \vartheta^{1/\gamma}t^\eta\right) \mathrm dr,
\Sigma(s)\ll\frac{(1+|r|)^AX^{\theta}}{X^{1/4}}\left(\sum_{n=1}^{[2|c|]}f\left(\frac{n}{2|c|}\right)+\sum_{n=[2|c|]+1}^{\infty}f\left(\frac{n}{2|c|}\right)\right).
s_{\nu,M}^{(j+1)}(\epsilon) = \left(\frac{\sqrt{2}\left(|a_{2M}(\nu)|+|a_{2M+1}(\nu)|/s_{\nu,M}^{(j)}(\epsilon) \right)}{\sqrt{\pi}\epsilon}\right)^{\frac{1}{2M+\frac{1}{2}}}, j\geq 0.
\varepsilon_{\mu_{1}\ldots \mu_{d}}= e^{- 1}e_{\mu_{1}}^{a_{1}}\ldots e_{\mu_{d}}^{a_{d}}\varepsilon_{a_{1}\ldots a_{d}}\, , \qquad \varepsilon^{\mu_{1}\ldots \mu_{d}}= e e_{a_{1}}^{\mu_{1}}\ldots e_{a_{d}}^{\mu_{d}}\varepsilon^{a_{1}\ldots a_{d}}\, .
\begin{aligned}a_{ij}(z) =\delta_{ij}\alpha_{ii}(z) +\tilde \alpha_{ij}(z)\sqrt{x_ix_j},\quad\forall\, z=(x,y)\in\bar S_{n,m}, a_{i,n+l}(z)=a_{n+l,i}(z) = \frac{1}{2}c_{il}(z)\sqrt{x_i},\quad\forall\, z=(x,y)\in\bar S_{n,m},\end{aligned}
\gamma^{0}= \left( \begin{array}{l r}{0}&{\hspace *{2 e x}1}\\{1}&{0}\\ \end{array}\right) , \gamma^{1}= \left( \begin{array}{l r}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) , \gamma^{5}= \gamma^{0}\gamma^{1}= \left( \begin{array}{l r}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) ,
\textbf{S}_1(x_1) + 2 \{\textbf{S}_2(\textbf{x})\}_{12}+ \textbf{S}_1(x_2) = \textbf{S}_1(y_1) + 2 \{\textbf{S}_2(\textbf{y})\}_{12}+ \textbf{S}_1(y_2).
{}_2F_1\bigg[\begin{matrix}\alpha \frac12 &1\end{matrix}\bigg|\,z\bigg]=\bigg(1-\frac z2\bigg)^{-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12+\frac12\alpha &1\end{matrix}\bigg|\,\frac{z^2}{(z-2)^2}\bigg].
\iota_{(1,n-1),\ldots,(n-2,n-1)}\prod_{i=1}^{n-1}(y_i-y_n)^{\Gamma_{in}}&=\iota_{(\sigma(1),n-1),\ldots,(\sigma(n-2),n-1)}\prod_{i=1}^{n-1}(y_i-y_n)^{\Gamma_{in}},
T\phi (\Pi_M^{\sharp}\big({\bigwedge}^{n-1}T(\phi^{-1}(C))^0 \big)) = T\phi ( \Pi_M^{\sharp}\big((T \phi)^{*}{\bigwedge}^{n-1}(TC)^0 \big)) = \Pi_N^{\sharp}({\bigwedge}^{n-1}(TC)^0) \subseteq TC.
\begin{array}{rcl}\check{S}^{K,L}\check{S}^{L,K}= \mbox{id.}, (\check{S}^{V_{z_1}, V^{*}_{z_{2}}})^{i^* k}_{jl^*}= (\check{S}^{V_{z_2}, V_{z_1}})^{kl}_{ij}, (\check{S}^{V^{*}_{z_1}, V_{z_{2}}})_{i^* j}^{kl^*}= (\check{S}^{V_{z_2}, V_{z_1 +nw}})^{ik}_{jl}.\end{array}
P_n(x)=\frac{1}{2^n\,n!}\,\frac{d^{n}}{dx^{n}}\big\{(x^2-1)^{n}\big\}=\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}^2(x-1)^k(x+1)^{n-k}\,.
\widetilde{\psi}_{n}^{(n^{2})}\big(x;\omega ,N,-\Vert \psi_{n}\Vert^{2}\big)=1/\Psi_{n}^{(n)}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big)\propto z^{(N+1/2) /2}\exp ( z/2 ) \frac{\mathit{L}_{n}^{N}(z)}{Q_{n}^{N}(z)}
\begin{aligned}\dot{R}_{E}(\theta,0) =\lim_{P\to0}\frac{\mathbb{E}_{\widehat{\mathbf{H}}}\left\{\dot{\mathcal{I}}(P)e^{-\theta T\mathcal{I}(P)}\right\}}{N\mathbb{E}_{\widehat{\mathbf{H}}}\left\{e^{-\theta T\mathcal{I}(P)}\right\}},\end{aligned}
\int_0^s e^{\mu_3s_1}(T-t)^{2{\sf d}_1}\sigma^2 ds_1 < C \left[ \tfrac{1}{\mu_3}e^{\mu_3s_1}(T-t)^{2{\sf d}_1}\sigma^2 \right]_{s_1=0}^{s_1=s}\lesssim \tfrac{1}{\mu_3}e^{\mu_3s}(T-t)^{2{\sf d}_1}\sigma^2.
N_n=\{a\in P_n |\{\rm each\word\in}\F_a\{\rm is\nondecreasing}\ \}begin{align*}5pt]D_n=\{a\in P_n |\F_a=(w_1,w_2,\ldots,w_k)\{\rm and}\w_i\cap w_j=\emptyset\}begin{align*}5pt]\mathfrak{S}_n=\{a\in P_n |\a\{\rm is\a\permutation\of\}[n]\}begin{align*}5pt]C_n=\{a\in P_n |\a=a_1a_2\cdots a_n\{\rm with}\a_1\geq a_2\geq\cdots\geq a_n\}
b = \begin{psmallmatrix}0 0 0 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}0 0 1 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}0 1 1 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}1 1 1 & 1 \end{psmallmatrix}
A=\underset{i=0}{\overset{n-1}{\sum}}m\left( x_{i},x_{i+1}\right) f\left(\xi_{i}\right) -\underset{i=0}{\overset{n-1}{\sum}}w(\xi_{i})h_{i}\left(\xi_{i}-\frac{x_{i}+x_{i+1}}{2}\right) f^{{\acute{}}}\left( \xi_{i}\right)
s_v^\Lambda+\sum_{\substack{\emptyset \neq G\subseteq E \mu\in \mathrm{MCE}(G)}}(-1)^{|G|}s_\mu^{\Lambda}{s_\mu^{\Lambda}}^*=\sum_{\substack{G\subseteq E\cup F G\cap F\neq \emptyset \mu\in \mathrm{MCE}(G)}}(-1)^{(|G|+1)}s_\mu^{\Lambda}{s_\mu^{\Lambda}}^*.
\sum\limits_{0 < |u-s| \leq N^{1+ \delta}}\mathbb{P}(A_{u}\cap A_{s}) = \sum\limits_{k=1}^{N^{1+\delta}}(t(N)-k+1) \mathbb{P}(\sigma^{\eta}(k) \in V, \sigma(0) \in V) & \leq \sum\limits_{k=1}^{N^{1+\delta}}t(N) \nu(V) = N^{1+\delta}t(N) \nu(V).
N^*(1) =\int_{[x,1)}\![S^*(y)-S^*(x)]m^*(y)\,dy=\int_{[x,1)}\!\left[\frac{1}{M(x)}-\frac{1}{M(y)}\right]s(y)M^2(y)\,dy &=\int_{[x,1)}\!s(y) \left[\frac{M^2(y)}{M(x)}- M(y)\right]\,dy &\geq \int_x^1\!s(y)[M(y)- M(x)]\,dy=\Sigma(1)=\infty
x(Q_1) = \,-4(13t^2-458t+1021)^2(t^2+4t-149)^2(t^2-290t+2017)(t^2+58t-347)\times &\times(17t^2-226t-367)(t^2-14t-83), x(Q_2) = \, -9(17t^2-226t-367)^3(t^2-14t-83)^3(t^2-290t+2017)(t^2+58t-347), x(Q_3) = \,(445t^8-26680t^7+298540t^6+7487800t^5-100559026t^4-816359048t^3+14922867436t^2 &-55316709112t+68821828189)(17t^2-226t-367)^2(t^2-14t-83)^2.
s(\boldsymbol{x}^*(t),\boldsymbol{u}^*(t),t)= \min_{\boldsymbol{x}\in \mathbb{X}, \boldsymbol{u}\in \mathbb{U}}s(\boldsymbol{x}(t),\boldsymbol{u}(t),t), \forall t \in [t_0,t_f)
t_{ij}(a): =e_{ij}(a)-e_{ji}(\bar{a}), u_{kl}(a): =e_{m+k,m+l}(a)-e_{m+n+l,m+n+k}(\bar{a}), v_{kl}(a): =e_{m+k,m+n+l}(a)+e_{m+l,m+n+k}(\bar{a}), w_{kl}(a): =e_{m+n+k,m+l}(a)+e_{m+n+l,m+k}(\bar{a}), f_{ik}(a): =e_{i,m+k}(a)+e_{m+n+k,i}(\rho(\bar{a})), g_{ki}(a): =e_{m+k,i}(a)-e_{i,m+n+k}(\rho(\bar{a})),
(\frac{\partial}{\partial t}-\Delta)\eta_{\gamma\overline\delta}= R_{\beta\overline\alpha\gamma\overline\delta}\eta_{\alpha\overline\beta}-\frac{1}{2}(R_{\gamma\overline{p}}\eta_{p\overline\delta}+R_{p\overline\delta}\eta_{\gamma\overline{p}})
\begin{cases}dQ_t=P_t\,dt, dP_t=- \beta\, Q_t\,dt-A(Q_t)\,dt-B\ast\rho_t(Q_t)\,dt+\lambda\,Z_t\,dt, dZ_t=-\lambda\,P_t\,dt-\alpha\, Z_t\,dt+\sqrt{2}\,dW_t,\end{cases}
\tilde{R}_{s;k}:=[\tilde{R}^k]_s= \begin{pmatrix}0 \langle R\mathbf{r}\rangle_{s;k}R_{s;k}\langle \mathbf{r}^\dag R\mathbf{r}\rangle_{s;k}\langle \mathbf{r}^\dag R\rangle_{s;k}0 \end{pmatrix},
\Psi_{2n}-\Phi_{2n}=-T_{(2n)}(p)(\vec{p}\cdot \xi )\vec{\xi}\cdot \left[\frac{\partial \vec{A}(x)}{\partial x^{0}}+\vec{\partial}A_{0}(x)\right]=+T_{(2n)}(p)(\vec{p}\cdot \xi )\vec{\xi}\cdot \vec{E}(x).
\sum_{\nu = 1}^N{\frac{1}{{\nu^s (\nu + \mu )^s}}}= \sum_{i = 0}^{s - 1}{\binom{s+i-1}i\frac{{(-1)^iH_{N,s - i}+ (-1)^s\left[H_{N+\mu,s - i}- H_{\mu,s - i}\right]}}{{\mu^{s + i}}}}&= ( - 1)^{s - 1}\sum_{i = 1}^s{\binom{2s-i-1}{s-1}\frac{{H_{\mu ,i}- H_{N + \mu ,i}- ( - 1)^i H_{N,i}}}{{\mu^{2s - i}}}}\,.
\partial_s u + \Delta u= \frac{f'(u)}{f(u)}\left(m+\frac{|\widetilde{\nabla}u|_{\widetilde{g}}^2}{f(u)^2-|\widetilde{\nabla}u|_{\widetilde{g}}^2}\right) +\mathcal{H}\, \frac{f(u)}{\sqrt{f(u)^2-|\widetilde{\nabla}u|_{\widetilde{g}}^2}}, u(0, \cdot)=u_0,
f^{(2)}(\boldsymbol\beta) \triangleq -\log p^{(2)}(\mathbf R;\boldsymbol\beta)-L\log\pi = \log|\boldsymbol\Sigma^{(2)}_{\boldsymbol\beta}|+\left(\boldsymbol\Sigma_{\boldsymbol\beta}^{(2)-1}\widehat{\mathbf \Sigma}_{\mathbf R}\right).
\rho_{0}^{\mathrm{\tiny{b o s}}}( \omega , k ) = \frac{\pi k ( \omega^{2}- k^{2})^{2}}{3 2}\left[ 1 + \frac{2 T}{k}\log \left( \frac{1 - e^{- \frac{\omega + k}{2 T}}}{1 - e^{- \frac{\omega + k}{2 T}}}\right) \right] .
\times \prod_{I = 2}^{N}\; \Bigg \langle \prod_{b = 1}^{N}\; \prod_{j = 1}^{n_{b}}: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( x_{j}^{( b )})}:_{M^{( I )}}\; \prod_{j = 1}^{m_{b}}: e^{+ i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( y_{j}^{( b )})}:_{M^{( I )}}\Bigg \rangle_{K^{( I )}}
H(X^{(i)}|\bar{X}|^{2n},\mathfrak{c}I_m) = e^{-\frac{\mathfrak{c}}{2}\Delta}X^{(i)}|\bar{X}|^{2n}= \sum_{k = 1}^{n}b_k(\mathfrak{c}) X^{(i)}|\bar{X}|^{2k}\;,
\ddot \mu_{k}(0)=-\frac{1}{3}\frac{\left< D_{\phi\phi\phi}^3 \Phi(0,\mu^*_{k})[\phi^*_{k},\phi^*_{k},\phi^*_{k}],\phi^*_{k}\right>_{L_2}}{\left< D_{\phi \mu}^2 F(0,\mu^*_{k})\phi^*_{k},\phi^*_{k}\right>_{L^2}},
\mathbf{k}_{\hat{f}}(\hat{\check{x}},\hat{R},\hat{\mathbf{z}},\hat{\mathbf{S}}) =-\hat{R}\hat{f}(\hat{\check{x}})+(I_n-\hat{R}\hat{\mathbf{S}})\hat{\mathbf{z}}&=-\Delta^{-1}((V^T\otimes W^{-*})f(\check{x}) &\qquad-(\Delta-I_n\otimes (W({A}-G\mathbf{\check{X}}) W^{-1})^* & \qquad-(V^{-1}(A-G\mathbf{\check{X}}) V)^T\otimes I_n) \hat{\mathbf{z}}),
\phi_1 := 18 \frac{\eta(\tau)^3 \eta(9 \tau)^3}{\eta(3\tau)^2}\phi_{-2, 1}(\tau, z) = (18 \zeta^{\pm 1}- 36)q + (-36 \zeta^{\pm 2}+ 90 \zeta^{\pm 1}- 108) q^2 + O(q^3)
\tilde{J}^p_4[H_0]_a:=\frac{r^p}{r^2\left(1-\frac{2M}{r}\right)}T_{ab}[rH_0]L^b-\frac{Mr^p}{r^3\left(1-\frac{2M}{r}\right)}|rH_0|^2L_a-\frac{1}{2}\frac{r^p}{r^2\left(1-\frac{2M}{r}\right)}V_0 |rH_0|^2L_a.
p\approx\frac{p_{n+2}p_{n}-p_{n+1}^{2}}{p_{n+2}-2p_{n+1}+p_{n}}=\frac{p_{n}^{2}+p_{n}p_{n+2}+2p_{n}p_{n+1}-2p_{n}p_{n+1}-p_{n}^{2}-p_{n+1}^{2}}{p_{n+2}-2p_{n+1}+p_{n}}
[\begin{matrix}C\\CA\end{matrix}]=[\begin{matrix}[\begin{matrix}1&0\end{matrix}]\\ [\begin{matrix}1&0\end{matrix}][\begin{matrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{matrix}]\end{matrix}]=[\begin{matrix}1&0\\ 0&1\end{matrix}]
\sum_{I_{n_k}(a_1,\ldots,a_{n_k})\cap F\neq\emptyset}|I_{n_k}(a_1,\ldots,a_{n_k})|^{s}\leq K_2^{sn_k}\prod_{j=1}^k \widehat{G}(m(j),n(j),b,1/3,s) \leq K_2^{sn_k}\cdot 6^k \cdot \widehat{C}^{n_k-k-1}\cdot 3^{-k}\prod_{j=1}^k e^{(1-ds)(\log m(j))^{1/b}}.
X(\vec{H};j,k):=\|\partial_t^j\vec{H}\|_k, X_0(\vec{H};j,k):=X(\vec{H};j,k)|_{t=0}, &Z(\vec{H};n):=\sum_{j+k=n}X(\vec{H};j,k), Z_0(\vec{H};n):=Z(\vec{H};n)|_{t=0}, &W(\vec{H}; n):=\sum_{j+k\leq n}X(\vec{H};j,k)=\sum_{\nu\leq n}Z(\vec{H};\nu), W_0(\vec{H};n):=W(\vec{H};n)|_{t=0}.
W_{1}( \alpha ) = - \frac{1}{2}\int [ 1 2 3 ] f_{\lambda}\left[ \frac{\tilde{\delta}( p_{3 \hat{\alpha}}- p_{1 \hat{\alpha}2 \hat{\alpha}})}{p_{3 \hat{\alpha}}^{-}- p_{1 \hat{\alpha}2 \hat{\alpha}}^{-}}- \frac{\tilde{\delta}( p_{3}- p_{1 2})}{p_{3}^{-}- p_{1 2}^{-}}\right] a_{\lambda 1}^{\dagger}a_{\lambda 2}^{\dagger}a_{\lambda 3}- h . c . \; .
\displaystyle \mathbb{E}^{0}(YL_T S_t) = \displaystyle\mathbb{E}^{0}(Y)+ \mathbb{E}^{0}\left( \int_{0}^{t\wedge T}\mathbb{E}^{0}(Y/\mathcal{F}_{u})S_u\rho_u L_u h(X_u) du \right) =\displaystyle \mathbb{E}^{0}(Y)+ \mathbb{E}^{0}\displaystyle \left( \int_{0}^{t}YS_u\rho_u L_u h(X_u) du \right).
\sum_{d \mid v}\mu(d) \frac{N}{dw}\sum_{|h| \leq W^{1+\epsilon}d/N}\widehat{G}\left(\frac{hN}{dw}\right) \sum_{(a,w)=1}e \left(\frac{ad \overline{m\ell_1\ell_3 v}}{w}+ \frac{ah}{w}\right).
{\frac{3}{2}}{\frac{\sigma^{\prime \prime}}{\sigma}}+{\frac{3}{4}}{\frac{\sigma^{\prime}\gamma^{\prime}}{\sigma \gamma}}-{\frac{1}{4}}\left({\frac{\gamma^{\prime}}{\gamma}}\right)^{2}+{\frac{1}{2}}{\frac{\gamma^{\prime \prime}}{\gamma}}= -{\frac{\Lambda_{0}}{4 M^{4}}}-{\frac{V_{0}^{i}}{4 M^{4}}}\delta ( \rho - \rho_{i})
\sqrt{a_k^2+b_k^2}= \left[(1+\alpha_{1,k})^2 + (2k\pi + \alpha_{2,k})^2\right]^{\frac{1}{2}}= \left[4k^2\pi^2 + O(k)\right]^{\frac{1}{2}}= 2|k\pi| + O(1), \ \ \forall |k|\geq k_0.
I_{4}\equiv \det \left[ \left( \begin{array}{c}{{\vec{\tilde{q}}}}\\{{\vec{\tilde{p}}}}\\ \end{array}\right) \left( \begin{array}{c c}{{\vec{\tilde{q}}\ \ \}}\\{{v e c{\tilde{p}}}}\\ \end{array}\right) \right] = \left( \vec{\tilde{q}}\cdot \vec{\tilde{q}}\right)^{2}\left( \vec{\tilde{p}}\cdot \vec{\tilde{p}}\right)^{2}- \left( \vec{\tilde{q}}\cdot \vec{\tilde{p}}\right)^{2}\, ,
\sum_{\mu^{(1)}\vdash q_1}\sum_{\mu^{(2)}\vdash q_2}\cdots\sum_{\mu^{(k)}\vdash q_k}\frac{\varepsilon_{{\cup_i \mu^{(i)}}}}{z_{\cup_i \mu^{(i)}}}\prod_{j=1}^{n}\frac{\left(\sum_{i=1}^{k}m_j(\mu^{(i)})\right)!}{\prod_{i=1}^{k}m_j(\mu^{(i)})!}
\begin{aligned}\mathcal H(k,d,l)= \left\{F\in\binom{[n]}{k}:[d-1]\subseteq F,\, F\cap[d,d+l-1]\neq\emptyset\right\}&\cup\left\{F\in\binom{[n]}{k}:|[d-1]\cap F|=d-2,\, [d,d+l-1]\subseteq F\right\}.\end{aligned}
\begin{aligned}\overline{F}(k,t+1;p,n) = p_{k}^{}\overline{F}(k-1,t;p,n) + (1 - p_{k}^{}) \overline{F}(k,t;p,n) &= \frac{(n-k)p}{n}\overline{F}(k-1,t;p,n) + \frac{n - (n-k)p}{n}\overline{F}(k,t;p,n), \end{aligned}
[L,\partial_B]=[L,\bar\partial_B] =[\Lambda,\partial_B^*]=[\Lambda,\bar\partial_B^*]=0, &[L,\partial_B^*]=-i\bar\partial_T,\ [L,\bar\partial_B^*]=i\partial_T,\ [\Lambda,\partial_B]=-i\bar\partial_T^*,\ [\Lambda,\bar\partial_B]=i\partial_T^*.
\zeta_{2}\left(-n_{1},-n_{2},z_{1}+1,z_{2}\right)=\zeta_{2}\left(-n_{1},-n_{2},z_{1},z_{2}\right)+\frac{\left(-1\right)^{n_{1}+n_{2}}}{n_{2}+1}z_{1}^{n_{1}}B_{n_{2}+1}\left(z_{1}+z_{2}\right).