harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q(x) = \frac{2^k}{k!}\frac{1}{(2 \pi i)^4}\oint \ldots \oint \frac{G(z_1, \ldots, z_4) \Delta(z_1^2, \ldots, z_4^2)^2}{\prod_{j=1}^4 z_j^{7}}q^{\frac{x}{2}\sum_{j=1}^4 z_j}\, dz_1 \ldots dz_4,
\Delta(\lambda)=\left(\begin{array}{cccc}\lambda^{M\alpha_1}-a_{11}-a_{12}\ldots -a_{1n}-a_{21}\lambda^{M\alpha_2}-a_{22}\ldots -a_{2n}\vdots \vdots \ddots \vdots -a_{n1}-a_{n2}\ldots \lambda^{M\alpha_n}-a_{nn}\end{array}\right).
\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K}}}\Phi(B_{t_{j}},t_{j}) (B_{t_{j+1}}-B_{t_{j}})= \int_0^T \Phi_{t}d B_{t}.
\tilde{g}^\omega_{\lambda,z}(x) =\det(C_n G^{\omega,\lambda}_{p,n,n}(z)-xI) &=\det(C_n G^\omega_{n,n}(z)-\lambda C_n G^\omega_{n,p}(z)(I+\lambda C_p G^\omega_{p,p}(z))^{-1}C_p G^\omega_{p,n}(z)-xI) &=\frac{p_l^\omega(z,\lambda)x^l+p_{l-1}^\omega(z,\lambda)x^{l-1}+\cdots+p_0^\omega(z,\lambda)}{\det(C_p^{-1}+\lambda G^\omega_{n,n}(z))},
\frac{1}{\rho_{F}(\Omega)}=\Lambda_1(\infty,\Omega)\le\lim_{j \to \infty}\lambda_{2}(p_{j},\Omega)^{\frac1p_{j}}=\overline{\Lambda}\le \Lambda_2(\infty,\Omega)= \frac{1}{\rho_{2,F}(\Omega)}.
\langle q^{\prime \prime}, t^{\prime \prime}| p^{\prime}, t^{\prime}\rangle ={\frac{1}{\sqrt{2 \pi \cos \tt}}}\exp \left( -{\frac{i}{2}}\tan \tt \left[ ( p^{2}+ q^{2}) - 2 p^{\prime}q^{\prime \prime}\csc \tt \right] \right)
[a_1 a_2, b_1, b_2, b_3] = a_1 [a_2, b_1, b_2, b_3] + [a_1, b_1] [a_2, b_2, b_3] + [a_1, b_2][a_2, b_1, b_3] + [a_1, b_3][a_2, b_1, b_2] + [a_1, b_1, b_2] [a_2, b_3] + [a_1, b_1, b_3] [a_2, b_2] + [a_1, b_2, b_3] [a_2, b_1] + [a_1, b_1, b_2, b_3] a_2 .
{\cal L}_{\rm SCS}= - \frac{1}{2}\epsilon^{ij}\left( A_i^{(0)}+ \sum^{\infty}_{n=1}A_i^{(n)}\right) \left( \dot{A}_j^{(0)}+ \sum^{\infty}_{n=1}\dot{A}_j^{(n)}\right) + A_0^{(0)}\epsilon^{ij}\partial_i \left( A_j^{(0)}+ \sum^{\infty}_{n=1}A_j^{(n)}\right)
\begin{cases}Y^{i}(t)=\psi^{i}(t) + \int_t^T g^{i}(t,s,Y^{i}(s),Z^{i}(t,s),Y^{i}(s+\delta_{s})) ds - \int_t^T Z^{i}(t,s) dW(s), \ \t\in [0,T]; Y^{i}(t)=\psi^{i}(t), \ \t\in [T,T+K]. \end{cases}
\stackrel{\wedge}{h}= \frac{1}{2}\{A^{-}, \, A^{+}\}_{q}= \frac{1}{2}\left( q \, A^{-}A^{+}+ q^{- 1}A^{+}A^{-}\right) = 2 \left( \stackrel{\wedge}{T}^{2}- \cosh \frac{\omega}{4}\right)
\begin{cases}v_t +A_1v= 0, (t,x) \in Q_T, begin{align*}5pt]v_x(t,0)=v_x(t,1) =0, t \in (0,T), begin{align*}5pt]v(T,x)= v_T(x) \,\in D(A_1^2),\end{cases}
Y^{(\ell)}[(\partial_\mu z)^{\gamma_1}, \ldots, (\partial_\mu z)^{\gamma_m}; f_1,\ldots, f_m]:=\sum_{\mu\leq \ell}\lambda_\mu \prod_{t=1}^m\left\{\frac{C_W}{n_{\ell}}\sum_{k=1}^{n_\ell}f_{t;k}^{(\ell)}(\partial_\mu z_k^{(\ell)})^{\gamma_t}\right\}
\begin{array}{l l}{{\begin{array}{l l}{{\frac{d}{d r}\sigma ( r ) = 0}}&{{r = 0}}\\{{\sigma ( r ) = 0}}&{{r \rightarrow \infty}}\\ \end{array}}}&{{\begin{array}{l l}{{A ( r ) \neq 0}}&{{r \rightarrow \infty}}\\{{A ( r ) = 1}}&{{r = 0 .}}\\ \end{array}}}\\ \end{array}
\mathcal{I}_{y_1}= \int_0^\infty\, U(y_1)\,U(y_1 + y_3)\,e\left(t\log\left(1+\frac{y_3}{y_1}\right) + \frac{2\sqrt{N}}{p_1q\sqrt{p_2}}\left({\sqrt{my_1}}-{\sqrt{m_1(y_1+y_3)}}\right) \right) dy_1
\int\limits_{O_v^{\times 2}}\chi(ac (F^{(m)}(x,\theta+\pi^{k}y)))\ |F^{(m)}(x,\theta+\pi^{k}y)|^s\ |dxdy| = q^{-(D_{i+1}-d_0)ms-\varepsilon_{i+1}ks}\int\limits_{O_v^{\times 2}}\chi(ac ( g(x,y))\ |g(x,y)|^s\ |dxdy|.
\sum_{b = 0}^{N - 1}\phi_{j}^{( 0 )}( b | u , v , w ) L ( a , b | u , v , w ) \,{\bf p}_{j + 1}= \phi_{j}^{( 1 )}( a | u , v , w ) \,{\bf p}_{j}\quad \mathrm{f o r}\quad a \in{\bf Z}/ N{\bf Z}.
-{\lambda^2\over 32}(F_{\mu\nu}^2)^2+{\lambda^2\over 8}F^{\mu\nu}F_{\nu\alpha}F^{\alpha\beta}F_{\beta\mu}-i{\lambda\over 2}(\overline{\psi}_+\Gamma_{\nu}\partial_{\mu}\psi_+ + \overline{\psi}_-\Gamma_{\mu}\partial_{\nu}\psi_-)F^{\nu\mu}
|D_{\rho}v(\ell)| \leq \sum_{i = 1}^{N_{\rho}}|D_{\rho_{i}}v(\ell_{i})| \leq N_{\rho}^{1/2}\bigg( \sum_{i = 1}^{N_{\rho}}|D_{\rho_{i}}v(\ell_{i})| \bigg)^{1/2}\leq C|\rho|^{1/2}=: C_{0}|\ell - m|^{1/2},
V^{\mathrm{design}}(N_1,N_0,n_1,n_0) =E\left[\mbox{var}\big(\widehat{\theta}\,|\,{\mathbf{R}},N_1,N_0\big)\,|\, N_1,N_0\right]=\frac{S^2_1}{N_1}+\frac{S^2_0}{N_0}-\frac{S^2_{\theta}}{N_0+N_1}.
\left|\sum\log\Gamma\left(b,xw^{b^{i}}\right)-\log\Gamma\left(b\right)\right|=\frac{1}{b}\sum_{i\ge0}x^bw^{b^{i+1}}\le\frac{1}{b}\sum_{i\ge0}w^{b^{i+1}}\le\frac{1}{b}\frac{w^{b}}{1-w}
0\to a_{2,*}(\omega_X\otimes a_1^*P)\otimes P' \to \mathcal{E}xt^1(a_{2,*}(a_1^*P^{\vee}),\mathcal{O}_{A_2})\otimes P' & \to\mathcal{E}xt^3(R^1a_{2,*}(a_1^*P^{\vee}),\mathcal{O}_{A_2})\otimes P'\to 0.
A_{g_0,2}(\L_h) = 2 \, b\left(e^{-2 \pi/\sqrt{3}}\right) = 2 \, c\left(e^{-2 \pi/\sqrt{3}}\right) B_{g_0,2}(\L_h) = 2 \, a\left(e^{-2 \pi/\sqrt{3}}\right)
\begin{array}{rl}\bar \lambda_\gamma \!\!\! = \bar \lambda_\gamma (F, r^{-\gamma}, B(0,1)\setminus \{0\}) begin{align*}2ex] = \sup \{\mu\, : \ \exists\, u \in C(B(0,1)\setminus \{0\})\, ,\u>0 \hbox{in}B(0,1)\setminus \{0\}, \F (D^2 u) + \mu \frac{u}{r^\gamma}\leq 0\}\, , \end{array}
0 = \Biggl[i\frac{\partial}{\partial t}- H_{\epsilon}(t), \Lambda(t)^{-1}\Lambda(t)\Biggr] = \Biggl[i\frac{\partial}{\partial t}- H_{\epsilon}(t), \Lambda(t)^{-1}\Biggr]\Lambda(t) + \Lambda(t)^{-1}\Biggl[i\frac{\partial}{\partial t}- H_{\epsilon}(t), \Lambda(t)\Biggr],
P_\lambda(\lambda)=-\frac{1}{m\,\Delta_p(\{\hat{v}^{-1}\})}\, \sum_{\mu=1}^m (-1)^{\mu}\det\begin{bmatrix}\left[\frac{\lambda^{m-\mu}}{\Gamma(m-\mu+1)}\frac{\exp(-\lambda/\hat{v}_j)}{\hat{v}_j^m}\right]_{j=1,...,p}\left[\hat{v}_j^{-i+1}\right]_{\substack{i,j=1,...,p (i\neq \mu)}}\end{bmatrix}.
\left(Q_h^* \dfrac{a_l}{r_z}Q_h -Q_h^*Q_h\dfrac{A_l}{R_z}\right)\psi=\sum_{\mu\in\{0,\pm1\}^n}\hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}\mathcal B_h(\xi+h^{-1}\mu)\psi(\xi+h^{-1}\mu),
\begin{aligned}\int_{V_{\min}}^{V_{F}}\left(\partial_{t}p+\partial_{v}(hp)-a\partial_{v v}p\right)\phi dv - \left(hp\phi|_{V_{\min}}^{V_R^-}+hp\phi|_{V_R^+}^{V_F}\right)+\left(a\partial_vp\phi|_{V_{\min}}^{V_R^-}+a\partial_vp\phi|_{V_{V_R^+}}^{V_F})+a\partial_vp(V_F)(\phi(V_R)-\phi(V_F)\right)=0.\end{aligned}
( 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 [ i R^{- 1}L_{k}l_{k}] - 1 |^{2}- ( 2 \pi R )^{d}\biggl ( \frac{L_{1}L_{2}}{( 2 \pi R )^{2}}\biggr )^{2}.
\{h:x_i-y_1+1\leq h\leq x_{i}\}\cup \{h:x_i+y_i\leq h\leq 2x_{i}-2\}& \cup \bigcup_{j=1}^{i-1}\left(\{h:x_i-y_{j+1}+1\leq h\leq x_{i}-x_j\}\cup \{h:x_i+y_{j}\leq h\leq x_{i}+x_j-1\}\right)
S[B_{(p+2)}] =\int d^{d}x\ \sqrt{|g|}\ \left[{\textstyle\frac{(-1)^{(p+2)}}{2\cdot (p+3)!}\frac{1}{e_{(p+1)}^{2}}}H_{(p+3)}^{2}+{\textstyle\frac{(-1)^{(p+1)}}{2\cdot (p+2)!}\frac{1}{e_{(p)}^{2}}}B_{(p+2)}^{2}\right]\, ,
( A^{( 1 ) \mathrm{T}}Y )_{n}\approx \sqrt{\frac{2}{\mu}}\frac{( - 1 )^{n + 1}\sqrt{n}}{\pi^{2}y^{2}( 1 - y )}\sum_{m = 1}^{\infty}\frac{\sin^{2}( \pi m y )}{m^{2}- n^{2}/ y^{2}}\frac{\sqrt{\mu + \sqrt{\mu^{2}+ m^{2}}}}{\sqrt{\mu^{2}+ m^{2}}}.
\operatorname*{s u p}_{x}\frac{| f ( x ) - f ( p ) |}{\mathrm{d i s t}( x , p )}\leq \operatorname*{s u p}_{x}\operatorname*{l i m}_{p \to x}\frac{| f ( x ) - f ( p ) |}{\mathrm{d i s t}( p , q )}~ .
\varphi_{i j}\left( \theta + \frac{i \pi}{h}\right) + \varphi_{i j}\left( \theta - \frac{i \pi}{h}\right) = \sum_{k = 1}^{r}I_{i k}\varphi_{k j}\left( \theta \right) - 2 \pi I_{i j}\delta \left( \theta \right) \quad ,
F^{(4)}(u):=\sum_{\substack{j_1+j_2+j_3+j_4=0, at\,\, least\,\, 3\,\, indices\,\, belong \,\,to\,\, S}}F^{(4)}_{j_1 j_2 j_3 j_4}\,u_{j_1}u_{j_2}u_{j_3}u_{j_4},
\sum_{x \in [1, p^{\alpha-1}]}\min\left(f(k), \sum_{i=1}^{r_k}\psi_{y_i}(x) \right) = f(k) \frac{p^{\alpha-1}-3}{2}+ \sum_{x=\frac{p^{\alpha-1}-1}{2}}^{p^{\alpha-1}-1}(-r_k x + p^{\alpha-1}r_k + b -a) + b = f(k)\frac{p^{\alpha-1}-3}{2}+ \frac{p^{\alpha-1}+1}{2}(b-a+p^{\alpha-1}r_k) & - \frac{r_k}{2}\frac{3p^{2\alpha-2}-3}{4}+ b.
\Delta_{k-1}(d\overline{E})-d_\mathcal{D}\Delta_k(\overline{E}) = \frac{(-1)^{k}}{(k+1)!(2\pi i)^k}\int_{(\mathbb{P}^1)^k}C_{k+1}\Big(d_\mathcal{D}\Delta\big({\rm tr}_k\circ\lambda(\overline{E})\big),\log\mid z_1\mid^2,\cdots,\log\mid z_{k}\mid^2\Big).
S_{F,n}=\int d^3x \, \left[{\bar\chi}^{(1)}_n (\not \! d + \omega_n+e \varphi ) \chi^{(1)}_n \,+\,{\bar\chi}^{(2)}_n(x)(\not \! d - \omega_n-e \varphi ) \chi^{(2)}_n(x) \right] \;.
h_{j,\Lambda}^{kl}(X,Y,\Xi,\hat{x}_n,X,\xi) = \sigma(\omega^{-\frac{1}{2}})(Y,\xi_j) R_{\mathrm{od},2}^{kl}(X,\Xi,\hat{x}_{n},X,\xi) \overline{\sigma}(\omega^{-\frac{1}{2}})(X,\xi_{n}).
\sum_{n=0}^{\infty}\hat{c}_n^q(r)\frac{t^n}{n!}=(1+qt)^{\frac{r}{q}}\sum_{k=0}^{\infty}\left(-\,\frac{\ln(1+qt)}{q}\right)^k\:\frac{1}{k!}\:\frac{1}{k+1}&=\frac{q(1+qt)^{\frac{r}{q}}}{\ln(1+qt)}\left(1-(1+qt)^{\frac{-1}{q}}\right).
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=-\frac{\partial p}{\partial x}\frac{1}{\rho}+\nu(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}})+f_{x},
\Lambda_{n,k}(\gamma, \epsilon) := \frac{1}{d_{\gamma}}\cdot u_{n,k,p}(d_{\gamma}) + \frac{1}{|d'_{\gamma}|}\cdot u'_{n,k,p}(d'_{\gamma}), \Lambda'_{n,k}(\gamma, \epsilon) := \frac{1}{d_{\gamma}}\cdot v_{n,k,p}(d_{\gamma}) + \frac{1}{|d'_{\gamma}|}\cdot v'_{n,k,p}(d'_{\gamma}).
V_\alpha(x)_i=-\frac{n^2}{2(n-2)}\ln\left(1+\frac{\vert x\vert^2}{\mu_\alpha^2}\right)\tilde{V}_\alpha(0)_i+\frac{n}{\mu_\alpha^2+\vert x\vert^2}\left\langle x,\tilde{V}_\alpha(0)\right\rangle x_i
\begin{array}{c c}{Z_{-}( z ) = C^{- 1}\bar{Z}_{+}^{t}( z ) \left( \begin{array}{c c}{1}&{0}\\{0}&{\bar{\cal E}}\\ \end{array}\right)}&{\bar{Z}_{-}( z ) = \left( \begin{array}{c c}{1}&{0}\\{0}&{- \bar{\cal E}}\\ \end{array}\right) Z_{+}( z ){}^{t}C}\\ \end{array}
+e^{2\, \alpha^{*}}\, n^{4}\, \Theta (x_{0}-y_{0})\, W_{e}(x,y^{A})^{2}+2\, e^{\alpha +\alpha^{*}}\, n^{4}\, \Theta (x_{0}-y_{0})\,W_{e}(x,y^{A})\, W_{e}(x^{A},y)
\dot{\varrho}\phi\dot{\varrho}^{-1}=(p^2\cdot{\rm id})\dot{s}_{\gamma(1)}\cdots\dot{s}_{\gamma(2d-2)},\quad\quad\dot{\varrho}^{-1}\phi\dot{\varrho}=(p^2\cdot{\rm id})\dot{s}_{\delta(1)}\cdots\dot{s}_{\delta(2d-2)}.
{\textstyle \frac{1}{2}}\left( M_\alpha^{}\otimes M_{-\alpha}^{}\, - \, M_{-\alpha}^{}\otimes M_\alpha^{}\right) \, =~M_\alpha^- \otimes M_\alpha^+ \, - \, M_\alpha^+ \otimes M_\alpha^-~ =~M_\alpha^- \otimes M_\alpha^{}\, - \, M_\alpha^{}\otimes M_\alpha^-
u^\alpha_{x^J}= D_{x^{j_1}}\cdots D_{x^{j_k}}u^\alpha = \bigg(\sum_{\ell=1}^p\> W^\ell_{j_1}\, D_{s^\ell}\bigg) \cdots \bigg(\sum_{\ell=1}^p\> W^\ell_{j_k}\, D_{s^\ell}\bigg) u^\alpha.
\mu_i\left(\mathcal{A}\right)=\int_{W^N}^{}\mu_i(dx) \int_{\mathcal{A}}^{}e^{-\lambda_{N,s}t}\frac{\Delta_N(y)}{\Delta_N(x)}\det\left(p^{(N),s}_t(x_i,y_j)\right)^N_{i,j=1}dy=\int_{W^N}^{}\mu_i(dx)f^{(N)}_{\mathcal{A},t}(x)>0,
(t^2at^{-1}\phi(a)^{-1})a^{-1}=t^2at^{-1}\phi(a^{-1}a_1^{-1}aa^{-1}) &=t^2a_1^{-1}at^{-1}\phi(a^{-1}) &=ta^{-1}tat^{-1}\phi(a^{-1})
\mathrm{d}a\mathrm{d}n=\mathrm{d}a,\mathrm{d}n\mathrm{d}n=\mathrm{d}n,\mathrm{d}n\mathrm{d}a^\star=\mathrm{d}a^\star,\mathrm{d}a\mathrm{d}a^\star=\mathrm{d}t,
\limsup_{\delta \to 0}\limsup_{n \to \infty}\sup_{\tau \in \mathcal{T}^\delta}{{E}}\left[ \sum_{k=0}^\infty (k+2) \left|{\bar{B}}^n_k(\tau+\delta) -{\bar{B}}^n_k(\tau) \right| + \left|{\bar{\eta}}^n(\tau+\delta) -{\bar{\eta}}^n(\tau) \right| \right] & \le (K+3)\gamma(M) M_0 + 5{\bar{U}}_K.
\! \!{\cal F}( \psi^{1}) *{\cal F}( \psi^{2}) = \int \phi_{2}\phi_{3}\phi_{4}( \bar{\psi}_{1}- \phi_{1}) ( \bar{\psi}_{3}- \phi_{3}) ( \bar{\psi}_{4}- \phi_{4}) d \phi = \bar{\psi}_{3}\bar{\psi}_{4}.
Z^\circ(\emptyset, [5^2, 1^{10}])= Z'(\emptyset, [5^2, 1^{10}])-Z'(\emptyset, [5,1^3])Z'(\emptyset, [5, 1^7]) &-Z'(\emptyset, [1^4])Z'(\emptyset, [5^2, 1^6])+Z'(\emptyset, [5,1^3])^2Z'(\emptyset, [1^4]) &+\frac{1}{2}Z(\emptyset,[5^2,1^2])Z'(\emptyset, [1^4])
S_{( d u s t )}^{'}[ T , Z^{r}, M , W_{r};{}^{4}g_{\mu \nu}] = -{\frac{1}{2}}\int d^{4}x \sqrt{{}^{4}g}( \mu n ) \Big ( U_{\mu}\,{}^{4}g^{\mu \nu}U_{\nu}- \epsilon \Big ) ,
P^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}= c_{A}{\cal T}_{A}^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}+ c_{B}{\cal T}_{A}^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}+ c_{C}{\cal T}_{A}^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}+ \sum_{i = 4}^{1 4}c_{i}T_{i}^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}.
f \star g = \exp \Bigg [ \hbar \Bigg ( \frac{\partial}{\partial q}\frac{\partial}{\partial \tilde{p}}- \frac{\partial}{\partial p}\frac{\partial}{\partial \tilde{q}}\Bigg ) \Bigg ] f ({\bf x}) g ({\bf \tilde{x}}) \vert_{{\bf x}={\bf \tilde{x}}},
\mathrm{I m}\left[ \mathrm{L i}_{2}\left( r e^{{\mathrm{\scriptsize{i}}}\theta}\right) \right] = \omega \ln r +{\textstyle{\frac{1}{2}}}\left[ \mathrm{C l}_{2}\left( 2 \theta \right) + \mathrm{C l}_{2}\left( 2 \omega \right) + \mathrm{C l}_{2}\left( 2 \chi \right) \right] ,
{\cal R}_{1 2}^{s h}( x ){\cal R}_{1 3}^{s h}( x y ){\cal R}_{2 3}^{s h}( y ) \, = \,{\cal R}_{2 3}^{s h}( y ){\cal R}_{1 3}^{s h}( x y ){\cal R}_{1 2}^{s h}( x ) \, .
\frac{1}{2}\frac{d}{dt}\|k\|^2+(H_1\dot{D}h|\dot{D}k) +((DH_1)\check{D}h|k)+(b_1\check{D}h|k)+(b_0h|k) + &+(a_{21}\check{D}h|k)+(a_{20}k|k)=(g_2|k).
u_{j\bar{k}\bar{j}}^{a}=u_{j\bar{j}\bar{k}}^{a}+K_{bc\bar{d}}^{a}u_{j}^{b}u_{\bar{k}}^{c}\overline{u_{j}^{d}}-K_{bc\bar{d}}^{a}u_{j}^{b}u_{\bar{j}}^{c}\overline{u_{k}^{d}},
\sum_{j=1}^{n_{S}}\sum_{b_{j}=0}^{a_{j}}\sum_{\beta_{j}}x_{b_{j}}a_{j}=\sum_{h=1}^{n_{P}}\sum_{d_{h}=0}^{c_{h}}\sum_{\gamma_{h}}u_{\gamma_{h}}y_{d_{h}}c_{h},
\tilde{D}_{-}\tilde{D}_{+}\tilde{W}= e^{2 \tilde{W}}\tilde{\Psi}_{L}^{+}\tilde{\Psi}_{R}^{-}, \tilde{D}_{\pm}= \tilde{\partial}_{\pm}+ i \tilde{\eta}^{\pm}\tilde{\partial}_{\pm \pm},
S_{b}( \lambda ) = \exp \Bigl \{\int_{- \infty}^{\infty}e^{- i \omega \lambda}{\frac{\cosh \Bigl ( ( 2 \tilde{\nu}- 3 ){\frac{\omega}{2}}\Bigr )}{\cosh ({\frac{\omega}{2}})}}{\frac{d \omega}{\omega}}\Bigr \}={\frac{\sinh ( \pi \lambda ) + i \sin ( \pi ( \tilde{\nu}- 1 ) )}{\sinh ( \pi \lambda ) - i \sin ( \pi ( \tilde{\nu}- 1 ) )}}\, .
{\Psi}_{B L}^{( w )}\simeq N_{w}[ \Psi_{L}^{( w )}+ e^{- \pi w / \kappa}\Psi_{R}^{( - w )}] , \qquad{\Psi}_{B R}^{( w )}\simeq N_{w}[ \Psi_{R}^{( w )}+ e^{- \pi w / \kappa}\Psi_{L}^{( - w )}] ,
\begin{array}{c}{{u_{k}= u_{0}+ k ( u_{0}- u_{- 1})}}\\{{U_{k}( z ) = U_{0}( z ) + k ( u_{0}- u_{- 1})}}\\{{\bar{W}_{k}= U_{0}( \sigma^{+}) + U_{0}( \sigma^{-}) + ( k + 1 ) ( u_{0}- u_{- 1}) - u_{0}}}\\ \end{array}
E[\mathcal{A}_T] = \int_0^T\!\!\!\int_0^t f(t - s)E\left(\frac{1}{|X_t - X_s|^{\theta}}\right)\,ds\,dt = \int_0^T\!\!\!\int_0^t f(t - s)\int_{{\mathbb R}^d}\frac{1}{|y|^{\theta}}p_{t - s}(y)\,dy\,ds\,dt
F(x+h)-F(x)\leq \lambda_{1}^{(k)}\left(g\left(x_{1}^{(k)}+\frac{h}{\lambda_{1}^{(k)}}\right)-g(x_{1}^{(k)})\right) +\left(\sum_{i=1}^{n+1}\lambda_{i}^{(k)}g(x_{i}^{(k)}) -F(x)\right),
\hat{x}_{k,j}= \frac{\mathbf{h}_{k,j}^H \mathbf{V}_{\xi_{k,j}}^{-1}}{1 + \mathbf{h}_{k,j}^H \mathbf{V}_{\xi_{k,j}}^{-1}\mathbf{h}_{k,j}}\left( \mathbf{y}- \mathbf{H}\mathbf{m}_\mathbf{x}^{prio}+ \mathbf{h}_{k,j}m_{x_{k,j}}^{prio}\right).
\mathrm{I m}\,{\Delta{\cal L}}= \frac{e^{2}a b}{8 \, \pi^{2}}\sum_{n = 1}^{\infty}\frac{1}{n}\coth \left( \frac{n \pi a}{b}\right) \exp \left( - \frac{n \pi m^{2}}{e \, b}\right) \, .
\mathcal{E}_{s,\Lambda}(N) =\frac{2\pi^{\frac{d}{2}}|\Lambda|^{-1}}{\Gamma(\frac{s}{2})(d-s)}N^2+C_{s,d}|\Lambda|^{-s/d}N^{1+\frac{s}{d}}+o(N^{1+\frac{s}{d}}), 0<s<d, \mathcal{E}_{\log,\Lambda}(N) =\frac{2\pi^{\frac{d}{2}}}{d}|\Lambda|^{-1}N(N-1)-\frac{2}{d}N\log N+\left(C_{\log,d}-2\zeta'_{\Lambda}(0)\right)N+o(N).
(Q_i^p(x))_j := \begin{cases}\frac{1}{np_{ij}}(S_i(x))_j p_{ij}\neq 0; 0 \end{cases}\qquad(Q_i^d(y))_j := \begin{cases}\frac{1}{np_{ij}}y_{i,j}p_{ij}\neq 0; 0 \end{cases}
var_{P_{\alpha}}\left[ \psi_{P,a}(\mathbf{V};\mathcal{G})\right]=E_{P_{\alpha}}\left[ \frac{I_{a}(A)}{\pi_{a}^{2}(\mathbf{O}_{min};P_{\alpha})}var_{P_{\alpha}}(Y\mid A=a,\mathbf{O})\right] +2+\alpha^{2}.
\frac{\vec{\delta}_{102}}{|\vec{\delta}_{102}|}\cdot \frac{\vec{\delta}_{304}}{|\vec{\delta}_{304}|}=\frac{1}{\sqrt{2(1+\cos\alpha_{102})}}\frac{1}{\sqrt{2(1+\cos\alpha_{304})}}(-2(1+\cos\alpha_{102})).
E_{2}= \{x\in \mathbb{R}^{n}\setminus \Omega^{*}: |T_{\alpha}(b_{1},g_{2})(x)|> \lambda/4\}, E_{3}= \{x\in \mathbb{R}^{n}\setminus \Omega^{*}: |T_{\alpha}(g_{1},b_{2})(x)|> \lambda/4\}, E_{4}= \{x\in \mathbb{R}^{n}\setminus \Omega^{*}: |T_{\alpha}(b_{1},b_{2})(x)|> \lambda/4\}.
F_T(x)-1-xF_T(x) &=x^2F_T(x)C(x)^2+\frac{x^3C(x)^3(F_T(x)-1)}{1-2x}+\frac{x^3C(x)^3}{1-2x}-\frac{x^3C(x)^3}{(1-x)^2}+\frac{x^3C(x)^4}{1-x}\,.
\frac{s_3}{s_1+s_3-1}\sum_{n \leq at_3}\frac{1}{n^{s_2-1}}\int_{at_3}^\infty \frac{du}{u^{s_1}(u+n)^{s_3+1}}\ll \begin{cases}t_3^{-\sigma_1-\sigma_3}(\sigma_2>2) t_3^{-\sigma_1-\sigma_3}\log t_3 (\sigma_2=2) t_3^{2-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<2), \end{cases}
E=\frac{-q}{4\pi\epsilon_{0}}[\frac{e_{r^{\prime}}}{r^{\prime2}}+\frac{r^{\prime}}{c}\frac{d}{dt}(\frac{e_{r^{\prime}}}{r^{\prime2}})+\frac{1}{c^{2}}\frac{d^{2}}{dt^{2}}e_{r^{\prime}}]
\left( \begin{array}{c}{{\Phi_{1}}}\\{{\Phi_{2}}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\sum_{n}e^{- i n \sigma}\left( \begin{array}{c c}{{i e^{i m H \alpha^{\prime}\tau}}}&{{- i e^{- i m H \alpha^{\prime}\tau}}}\\{{e^{i m H \alpha^{\prime}\tau}}}&{{e^{- i m H \alpha^{\prime}\tau}}}\\ \end{array}\right) \left( \begin{array}{c}{{C_{1 n}}}\\{{C_{2 n}}}\\ \end{array}\right) .
|M_r(n)\setminus E_r(n, \epsilon, \eta, d)| <(N-\frac{1}{\eta}+1) |\tilde{M}_r(N)| 2^{S(n)}m(r)^{(1-\epsilon)({n\choose 2}+\frac{n}{2})}< (N-\frac{1}{\eta}+1)r^{{N \choose 2}}2^{S(n)}m(r)^{(1-\epsilon)({n\choose 2}+\frac{n}{2})}= 2^{S'(n)}m(r)^{(1-\epsilon)({n\choose 2}+\frac{n}{2})}.
(x_1-1)^2+(x_2-2)^2+(x_3+2)^2 \,\to\,\min\limits_{x,u,v}4-x_1-u \,\leq\,0 5-x_1-(x_2-2)^2-(x_3+2)^2-u \,\leq\,0 x_1^2+x_2^2-x_3-v \,\leq\,0 1-(x_1-1)^2-x_2^2-x_3-v \,\leq\,0 x_2-v \,\leq\,0 u\,\leq\,0\,\lor\,v \,\leq\,0
\frac{1}{c_{GN}}=\mathcal{W}(u_n)+o_n(1) &=W_1(v_n(\tfrac{\cdot}{\lambda_n}))+W_2(v_n(\tfrac{\cdot}{\lambda_n}))-W_1(v_n(\tfrac{\cdot}{\lambda_n}))^{1-\theta}W_2(v_n(\tfrac{\cdot}{\lambda_n}))^\theta+o_n(1) &=\cdots=\frac{X+(\lambda_n^{1-\sigma}Y)-X^{1-\theta}(\lambda_n^{1-\sigma}Y)^\theta}{\|v_n\|_{L^4}^4}+o_n(1)
\cosh\tfrac{\ell_\nu}{2}+e^{-\ell_\mu}=2\sinh(\ell_{\mu'})\left(-e^{-\frac{\ell_\mu}{2}}+\sinh\tfrac{\ell_{\mu'}}{2}\right), \cosh\tfrac{\ell_\nu}{2}+\cosh\ell_\mu =2\sinh(\ell_{\mu'})\left(\sinh\tfrac{\ell_\mu}{2}+\sinh\tfrac{\ell_{\mu'}}{2}\right).
A_{i,k}= \Big\{m_k^i(x_k^{\ell}) + \sqrt{\Delta}\sigma_k^{i1}(x) Z_{k+1}^{1}+ \sqrt{\Delta}\big(\sigma_k^{(i,2:q)}(x) \vert u \big) \in \big(v^{i,j_i-}, v^{i,j_i+}\big) \Big\}.
f = a_1x_1y_1z_1w_1 + a_2x_1y_1z_1w_2 + a_3x_1y_1z_2w_1 + a_4x_1y_1z_2w_2 + a_5 x_1y_2z_1w_1 + a_6 x_1y_2z_1w_2 + a_7x_1y_2z_2w_1 + a_8x_1y_2z_2w_2 + a_9 x_2y_1z_1w_1 + a_{10}x_2y_1z_1w_2 + a_{11}x_2y_1z_2w_1 + a_{12}x_2y_1z_2w_2 + a_{13}x_2y_2z_1w_1 + a_{14}x_2y_2z_1w_2 + a_{15}x_2y_2z_2w_1 + a_{16}x_2y_2z_2w_2 ,
f^{i}= f_{\,{\hat{\alpha}}{\hat{\beta}}}^{i}{\hat{e}}^{\hat{\alpha}}\wedge{\hat{e}}^{\hat{\beta}}= 2{\hat{e}}^{5}\wedge{\hat{e}}^{i}+ \varepsilon_{i j k}{\hat{e}}^{j}\wedge{\hat{e}}^{k}
(-1)^{n-1}\det\begin{bmatrix}a_1^0 a_1^{1+(z-n)}a_1^{2+(z-n)}\cdots a_1^{(n-1)+(z-n)}a_2^0 a_2^{1+(z-n)}a_2^{2+(z-n)}\cdots a_2^{(n-1)+(z-n)}\vdots \vdots \vdots \ddots \vdots a_n^0 a_n^{1+(z-n)}a_n^{2+(z-n)}\cdots a_n^{(n-1)+(z-n)}\end{bmatrix}.
\begin{cases}\mathcal{E}=\bigcup_{a \in A}\{x \in X\: | \: a <_{X}x <_{X}ah_{\sf M}\}\mathcal{O}=\bigcup_{a \in A}\{x \in X\: | \: ag_{\sf min}<_{X}x <_{X}aa_{\sf min}.\}\end{cases}
\bigg|c_{q_1}(n_1a+n_2b) S\left( (n_1\overline{b}a+ n_2) \ell_1, d\tfrac{q_0}{a}m ; q_0 \right)\bigg| & \leq \gcd(n_1a+n_2b, q_1)\sigma_0(q_0) \sqrt{q_0}\gcd(n_1\overline{b}a+n_2, \tfrac{q_0}{a}m, q_0)^{\frac{1}{2}}& \leq\sqrt{q}\sigma_0(q_0)\gcd(n_1a+n_2b, q_1)^{\frac{1}{2}}\gcd(n_1\overline{b}a+n_2, q_0)^{\frac{1}{2}}
R = R_{h}+ \frac{e^{- 2 \Phi_{h}}}{8 K R_{h}^{2}}\left( 1 6 R_{h}^{4}- 4 R_{h}^{2}( w_{h}^{2}- 1 )^{2}- ( w_{h}^{3}- 3 w_{h}+ 4 \kappa )^{2}\right) r +{\cal O}( r^{2}) \, .
{\ddot w}+\frac{1}{2g}\frac{\partial g}{\partial w}{\dot w}^2+\biggl(\frac{\dot \lambda}{\lambda}+\frac{1}{g}\frac{\partial g}{\partial \sigma}\dot\sigma\biggr)\dot w+\frac{1}{2g}\frac{w+1}{(w-1)^3}=0,
S_{D 1}= - \int d^{2}\sigma \left( \frac r R \right)^{2}\left( \sqrt{- \det ( g + F )}- 1 \right) + 2 R^{2}\sin^{2}\theta_{1}\sin \theta_{2}\theta_{3}\varepsilon^{\mu \nu}\partial_{\mu}\theta_{1}\partial_{\nu}\theta_{2}\, ,
\nabla^2f(e,e)=-\frac{1}{4}\,\frac{1}{(1+\tfrac{1}{2}\,d^2)^{3/2}}\,g(\nabla h,e)^2+\frac{1}{2}\,\frac{1}{(1+\tfrac{1}{2}\,d^2)^{1/2}}\,\nabla^2h(e,e).
\textrm{Hess}_W\mathcal{F}(\rho)(\sigma_1,\sigma_2)= g_W(\sigma_1, \nabla_{\sigma_2}\nabla_W\mathcal{F}(\rho)) = \sigma_2\big(g_W(\sigma_1,\nabla_W\mathcal{F}(\rho))\big)-g_W(\nabla_{\sigma_2}\sigma_1, \nabla_W\mathcal{F}(\rho)).
\Psi_\lambda(\omega):=\|\omega\|_X^2-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx-\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx.
[E_{\delta_{mj}},E_{\beta_{m-1,k}}^*]\eta\xi\mu= ([E_{\delta_{mj}},E_{\beta_{m-1,k}}^*]\eta_3[E_{\delta_{mj}},E_{\beta_{m-1,k}}^*]^{-1}\eta_4)(\eta_4^{-1}[E_{\delta_{mj}},E_{\beta_{m-1,k}}^*]\eta_4 \xi)\mu
\sum_{j=1}^\infty \Big\|\Big ( q - \sum_{k = 1}^{L}q_k \widetilde{\phi_k}\Big)e_j \Big\|_{H^2}^2 \le \frac{C}{\delta^{2\max m_j}}\Big(\frac{1}{\delta^2}\log\frac{1}{\delta}\Big)^2.
\begin{array}{rclcrclcrclcrcl}\;\! [h,e_2^\pm] = \mp e_2^\pm, [z,e_2^\pm] = \pm e_2^\pm, \;\! [h,e_3^\pm] = \pm e_3^\pm, [z ,e_3^\pm] = \pm e_3^\pm, \;\! [e_1^\pm,e_2^\mp] = 0, [e_1^\pm,e_2^\pm] = - e_3^\pm , \;\! [e_1^\pm,e_3^\pm] = 0, [e_1^\pm,e_3^\mp] = - e_2^\mp.\end{array}
\begin{array}{ccccc}\Phi_1^0 = \phi' ~~ \Phi_2^0 = \phi_M ~~ \Phi_3^0 = \phi_P \Psi_1^0 = -\int_0^\theta \phi_P(z)\,dz ~~ \Psi_2^0 = 1 ~~ \Psi_3^0 = \phi,\end{array}
\begin{cases}c_0 \leq \tilde u+u_s,\tilde v+v_s,\tilde \phi+\phi_s \leq c_1, (\tilde u+u_s)^2 - g(\tilde \phi+\phi_s)\geq c_2^2,\, (\tilde v+v_s)^2 - g(\tilde \phi+\phi_s) \geq c_2^2. \end{cases}
\Delta H_{L}=\frac{g_{s}\mu_{B}}{\hbar m_{e}ec^{2}}\frac{1}{r}\frac{\partial U(r)}{\partial r}L\cdot S\approx\frac{2\mu_{B}}{\hbar m_{e}ec^{2}}\frac{1}{r}\frac{\partial U(r)}{\partial r}L\cdot S.
E_{\exp}^{\left( 1 \right)}= \left. - \frac \mu 2 \frac d{d t}\left( \sum_{n = - \infty}^{\infty}e^{- t \left( \left( \frac{2 n \pi}{L \mu}\right)^{2}+ \left( \frac m \mu \right)^{2}\right)^{\frac 1 2}}\right) \right\rfloor_{t = 0}\, .
f(t) = f(0) + \int_0^t \left[f'(0) + \int_0^s f''(0) d\tau\right] ds &= f(0) + tA + \int_0^t\int_0^s \langle v \nabla^2 u |v\rangle d\tau ds &\geq f(0) + tA + \int_0^t \int_0^s -C_{\phi_0}d\tau ds &= f(0) + tA - \frac{t^2 C_{\phi_0}}2.
H=\int dx[:((\psi_{x}^{\dagger}\psi_{x}+c\rho^{2}+i\kappa\rho(\psi^{\dagger}\psi_{x}-\psi_{x}^{\dagger}\psi)):+\kappa^{2}(\psi^{\dagger}\rho^{2}\psi)],\rho\equiv(\psi^{\dagger}\psi)
\Big{(}\mathcal{D}_tU_h^{\frac12},v_h\Big{)}-\gamma(\nabla \Sigma_h^{\frac12},\nabla v_h)+(\nabla U_h^{\frac12},\nabla v_h) &+\frac12(\mathfrak{F}(U_{h}^{1},u_{H}^{1})+ f(U_{h}^{0}),v_h)=(g^{\frac12},v_h),~\forall v_h\in L_h,
\frac{\sum\limits_{\mathbf{a}\in\mathbb{B}^{N\times 1}\atop||\mathbf{a}||_{0}=\lambda N}|\hat{\mathcal{U}}_{\rm{ac}}^{\mathbf{a}}(\mathbf{I})-\mathcal{U}^{\mathbf{a}}_{\rm{ac}}|}{\binom{N}{\lambda N}}=(1-\lambda)N(1-(1-\lambda)^{\frac{w_{c}}{r}-1})^{w_{c}}.
\begin{aligned}\mathcal{E}_{S}=\{(m_x,m_y)\in \mathbb{Z}^{2}| (\frac{m_x}{L_{S,x}})^2+(\frac{m_y}{L_{S,y}})^2 \le 1 \}, \mathcal{E}_{R}=\{(l_x,l_y)\in \mathbb{Z}^{2}| (\frac{l_x}{L_{rRx}})^2+(\frac{l_y}{L_{R,y}})^2 \le 1 \}.\end{aligned}
\tilde{H}= H_{c}+ H^{( 1 )}+ H^{( 2 )}= \frac{m}{2}[ q^{i}q^{i}+ \frac{2}{\sqrt{\kappa}}\epsilon_{i j}\Phi^{i}q^{j}+ \frac{1}{\kappa}\Phi^{i}\Phi^{i}] .
\int_{M}u_{j\bar{k}}^{a}\overline{u_{\bar{k}}^{a}}f_{\bar{j}}\phi^{2}e^{-f}=-\int_{M}u_{j}^{a}\overline{u_{k\bar{k}}^{a}}f_{\bar{j}}\phi^{2}e^{-f}+\int_{M}u_{j}^{a}\overline{u_{\bar{k}}^{a}}f_{\bar{j}}f_{\bar{k}}\phi^{2}e^{-f}-\int_{M}u_{j}^{a}\overline{u_{\bar{k}}^{a}}f_{\bar{j}}\left( \phi^{2}\right)_{\bar{k}}e^{-f},
I = \sum_r \left( \frac{-1}{4}\right)^{n_1 + n_2 + \cdots + n_N}\frac{\left[ \prod_{i=1}^N (N-i)! \right]^2}{\left[ \prod_{i=1}^N k_i!\right]^2}\left[ \frac{\det(x_i^{2k_j})\det(y_i^{2k_j})}{\Delta(x^2)\Delta(y^2)}\right] .
C_{2}(\varepsilon)^{p}(\sqrt{np})^{p}+\sum_{k=1}^{\infty}C_{2}(\varepsilon)^{p}(\sqrt{np})^{p}(2^{k})^{p/2}\exp(-c2^{\frac{2k}{4+\varepsilon}})\le C_{3}(\varepsilon)^{p}(\sqrt{np})^{p}.