harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d^2(x_2(\gamma_1+\gamma_2)) &=y_1(2\gamma_1^2+3\gamma_1\gamma_2+\gamma_2^2)+y_2(\gamma_1^2+3\gamma_1\gamma_2+2\gamma_2^2) &=y_1(\gamma_1^2+2\gamma_1\gamma_2)+y_2(\gamma_2^2+2\gamma_1\gamma_2) &=d^4(x_4)
\hat{\theta}_{e}\, = \, \theta \, ({\frac{P_{0}}{2}}+ \mu_{e}) \, \theta \, ({\frac{P_{0}}{2}}- \mu_{e}) \, \theta \, ( \, \big [{\frac{P_{0}}{2}}+ \mu_{e}\big ]^{2}- h_{1 e}^{2}+ m_{1}^{2}\, ) \, ,
\frac{d}{dt}(g^{-1}Fg)=-g^{-1}\dot{g}g^{-1}Fg+g^{-1}\dot{F}g+g^{-1}F\dot{g}=g^{-1}(-\dot{g}g^{-1}F+\dot{F}+F\dot{g}g^{-1})g=0.
\alpha_j = x_{\mathcal{N}_{j-1}+1}x_{\mathcal{N}_{j-1}+2}\cdots x_{\mathcal{N}_j}y_j &= (y_j+ \beta_{11}^jy_1+ \beta_{21}^jy_2+ \cdots + \beta_{(j-1)1}^jy_{j-1})(y_j+ \beta_{12}^jy_1+ \beta_{22}^jy_2+ & \cdots + \beta_{(j-1)2}^jy_{j-1}) \cdots (y_j+ \beta_{1n_j}^jy_1+ \beta_{2n_j}^jy_2+ \cdots + \beta_{(j-1)n_j}^jy_{j-1})y_j,
\mathbf{b}_j'=T\mathbf{b}_j\quad\textrm{with}~T=\left[\begin{matrix}(\mathfrak{g}(1+a)^{-1})^{\psi_1}\cdots 0 \vdots \ddots \vdots 0 \cdots (\mathfrak{g}(1+a)^{-1})^{\psi_g}\end{matrix}\right].
L_{v_0}(f)\; =\;\lim_{r\downarrow 0}W_r(\overline{v_0},f)\;=\;a_0^{(f)}\,\lim_{r\downarrow 0}W_r(\overline{v_0},v_0)+a_\infty^{(f)}\,\lim_{r\downarrow 0}W_r(\overline{v_0},v_\infty) \qquad+\lim_{r\downarrow 0}b_\infty^{(f)}(r)\,W_r(\overline{v_0},v_0)+\lim_{r\downarrow 0}b_0^{(f)}(r)\,W_r(\overline{v_0},v_\infty) \;=\;a_\infty^{(f)}\,W_0^{\infty}\,.
\gamma=\frac{\lambda P-|A^{3/2}u|^2}{G^2-P}, &\beta=\frac{\lambda P-|A^{3/2}u|^2}{\lambda e-E}, \alpha=\frac{P}{E}-\frac{\lambda P-|A^{3/2}u|^2}{G^2-P}-\frac{e}{E}\frac{\lambda P-|A^{3/2}u|^2}{\lambda e-E}.
\langle \tilde{D}_{-}\tilde{A}_{k}| \dot{\tilde{A}}_{k}\rangle = i \frac{1}{2}\sum_{p > 0}\big [ ( a_{k}^{p})^{\dagger}\dot{a}_{k}^{p}- ( \dot{a}_{k}^{p})^{\dagger}a_{k}^{p}\big ] ,
d\rho_i=\sum_{j\in N(i)}\omega_{ij}(S_i-S_j)\theta_{ij}(\rho) \circ dW_t; & d S_i+(\frac 12\sum_{j\in N(i)}\omega_{ij}(S_i-S_j)^2 \frac{\partial \theta_{ij}}{\partial \rho_i}+\frac 18 \frac{\partial}{\partial \rho_i}I(\rho))\circ dW_t+(\mathbb V_i+\sum_{j\in N(i)}\mathbb W_{ij}\rho_j)dt=0,
0 \le \Big( \lambda_{k,1}(v,\sigma) - \lambda_{k,1}(\omega) \Big) \int_{0}^t V_\omega \phi \Phi & = \int_{0}^t (V_\omega \Phi')' \, \phi -(V_\omega \phi')' \, \Phi &= V_\omega(t) \Phi'(t) \phi(t) - V_\omega(t) \phi'(t) \Phi(t)
\frac{1}{K}\sum_{k=1}^K G_{k,K}= \frac{1}{K}\sum_{k=1}^K \left[ \Phi( \rho_k^{**}) - \Phi( \rho_k^{*}) \right] \leq \frac{1}{K}\sum_{k=1}^K \left[ \Phi( \rho_k ) - \Phi( \rho_{k-1}) \right] &= \frac{\Phi( \rho_K ) - \Phi( \rho_0 )}{K}= \frac{\Phi( \infty ) - \Phi( 0 )}{K}= \frac{1}{K},
\phi = A(x-y) \Sigma_{lm}\int d\omega ( f^{b}_{\omega lm}b^{b}_{\omega lm}+\bar{f}^{b}_{\omega lm}b^{b \dagger}_{\omega lm}+f^{a}_{\omega lm}b^{a}_{\omega lm}+ \bar{f}^{a}_{\omega lm}b^{a \dagger}_{\omega lm}),
\mbox{$\Psi(t,\widehat x,\widehat \eta)=\langle\widehat x,\widehat \eta\rangle$ on $\widehat\Sigma_+$}, \mbox{$d_{\widehat x,\widehat\eta}(\Psi-\langle\widehat x,\widehat\eta\rangle)=0$ on $\widehat\Sigma_+$}.
\mathcal{L}_0=-\sum_n \left(\frac{n+\omega}{R}\right)^2 \bar A^{(n)}A^{(n)}-\sum_{m,n}\bar A^{(m)}\left[\delta(0)\ \frac{W^2}{\pi R}+i\,\frac{W}{\pi}\ \frac{m-n}{R^2}\right] A^{(n)}\, .
\mathbb{P}(N_{-s}^{\dagger}(0,T)>0) =1-\mathbb{E}\left[\exp\left(-\int_{0}^{T}\lambda_{s}^{\dagger}(t)dt\right)\right] &\leq\mathbb{E}\left[\int_{0}^{T}\lambda^{\dagger}_{-s}(t)dt\right] &=\int_{0}^{T}\sum_{n=1}^{\infty}\mathbb{E}[\lambda_{-s}^{\dagger,n}(t)]dt,
\langle \kappa^{\prime}, s | L_{0}| \kappa , s \rangle = \left[ \frac{s}{2}+ \frac{i \kappa}{4}\right] \, \delta ( \kappa - 2 i - \kappa^{\prime}) + \left[ \frac{s}{2}- \frac{i \kappa}{4}\right] \, \delta ( \kappa + 2 i - \kappa^{\prime}) .
\Lambda(f \otimes \mathrm{Ad}(g), k) = \frac{4 L(1,\pi,\mathrm{ad})L(1,\tau,\mathrm{ad})}{\langle \mathbf h, \mathbf h \rangle\langle \mathbf g, \mathbf g\rangle\langle \pmb{\phi}, \pmb{\phi}\rangle}\left( \prod_v \mathcal I_v^{\sharp}(\mathbf h, \mathbf g, \pmb{\phi})^{-1}\right) \mathcal Q(\mathbf h, \mathbf g, \pmb{\phi}).
0\leftarrow I_\lambda(\omega_n) =I_\lambda(\omega_n)-\frac{1}{2}I'_\lambda(\omega_n)\omega_n &=\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega_n(x,0)|^2dx+\left(\frac{1}{2}-\frac{1}{2_\alpha^*}\right)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx &\geq 0,\ \n \rightarrow \infty,
\frac{\partial}{\partial R_{{\lambda}{\tau}}}\int d^{D}p \, \, \, e^{- \frac{1}{2}p \cdot R \cdot p}= - \frac{1}{2}( 2 \pi )^{D / 2}\left[ \frac{\partial}{\partial R_{{\lambda}{\tau}}}( \det R ) \right] ( \det R )^{- 3 / 2}
|F_m(\beta_1)-F_m(\beta_2)|_l &\le |J_m(\beta_1)-J_m(\beta_2)|_l &\quad+\sum_{n=m+1}^{N(\beta_2)}2^{2n}D_{et}^{\frac{n-m}{2}}\Bigg(|J_n(\beta_1)-J_n(\beta_2)|_l+D(\beta_1,\beta_2)\sum_{a=1}^2\|J_n(\beta_a)\|_{l,0}&\qquad\qquad\qquad\qquad\qquad+\frac{2\pi}{\beta_1}\sum_{a=1}^2\|J_n(\beta_a)\|_{l,1}\Bigg).
(I)' \hspace{7mm}\Psi_{\bf D}=\left(\begin{array}{c}a_1 +{\rm j}b_1 a_2 +{\rm j}b_2 a_3 +{\rm j}b_3 a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm}\leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C}=\left(\begin{array}{c}a_3+{\rm i}a_1 a_4+{\rm i}a_2 b_3+{\rm i}b_1 b_4+{\rm i}b_2\end{array}\right)
S_{ee}= S_e\left(1\,+\,m_{e\mu}^2 \,S_\mu S_e \,+\, m_{e\mu}^4 \,S_\mu S_e S_\mu S_e \,+\, ...\right) = S_e\left(1 - m_{e\mu}^2 \,S_\mu S_e \right)^{-1}\,,
\tau(ev_{(X_1,...,X_n)}^{\sigma h}(Q(a))b)=\tau(bS^{(m,i_m)}_{j_m}a_{m-1}...a_1S^{(1,i_1)}_{j_1}a[ev_{(X_1,...,X_n)}^{\sigma h}(\partial_n^{|n|}P))]\#(S^{(1,i_1)},...,S^{(m,i_m)}),
S_{D B I}\sim \int d x_{4}^{2}\, \, d \Omega_{2}\, \, e^{- \Phi_{D}}\sqrt{g}\left[ 1 + \frac{1}{2}g^{\mu \nu}g_{m n}\partial_{\mu}x^{m}\partial_{\nu}x^{n}- \frac{1}{2}g^{\mu \tau}g^{\nu \rho}F_{\mu \nu}F_{\tau \rho}\right] .
b_{- \omega + 2}= \frac{1}{2^{2 \omega}\pi^{\omega}}( 2 - \frac{2 \omega}{1 2}+ \frac{1}{1 2}[ 1 - \alpha^{\omega - 2}] ) N_{c}G_{\mu \nu}^{a}( x ) G_{\mu \nu}^{a}( x ) ,
n^{p}= \sum_{l = 0}^{p}( - 1 )^{p - l}\left( \begin{array}{c}{p}\\{l}\\ \end{array}\right) \left( n + \gamma m +{\frac{d - 1}{2}}\right)^{l}\left( \gamma m +{\frac{d - 1}{2}}\right)^{p - l}
\| f\|_{\Psi_q}=\left\| \Psi_q^{-1}(\Phi(|f|))\frac{|f|}{\Psi_q^{-1}(\Phi(|f|))}\right\|_{\Psi_q}\leq \left\| \Psi_q^{-1}(\Phi(|f|))\right\|_{\Psi_q}\frac{\|f\|_\infty}{\Psi_q^{-1}(\Phi(\|f\|_\infty))}.
A_{N}^{\ast}(w_{1,r}) = A_{N}^{\ast}(a_{1},\{1\}_{b_{1}-1},w_{2,r}) = \sum_{n=1}^{N}\binom{N}{n}\frac{(-1)^{k-1}}{n^{k}}\zeta^{\ast}_{n}(\{1\}_{b_1-1},w_{2,r}) &=\sum_{i_1=1}^{N}\frac{1}{i_1}\dots \sum_{i_{k}=1}^{i_{k-1}}\frac1{i_{k}}\sum_{n=1}^{i_{k}}\binom{i_{k}}{n}(-1)^{n-1}\zeta^{\ast}_{n}(\{1\}_{b_1-1},w_{2,r}).
y^{2}\left( 1 - \epsilon \right)^{2}{\frac{d^{2}u_{k}}{d y^{2}}}+ 2 y \epsilon \left( \epsilon - \eta \right){\frac{d u_{k}}{d y}}+ \left[ y^{2}- F \left( \epsilon , \eta , \xi \right) \right] u_{k}= 0 ,
f =\frac{kv-f_1(d_1-d_2)}{d_2}v(2d_2-k d_2+2k) = 2f_1 d_1 + (4-2f_1) d_2 \frac{e}{kd_2}\biggl(4-(k-2)(d_2-2)\biggr) =\Phi(f_1,d_1,d_2),
J_N(z_0, u) = \mathbf{E}_{z_0}^u\!\left[\int_0^{5.0}\Big(|z_t^{(1), u, N}- 2 z_t^{(2), u, N}| + |z_t^{(1), u, N}- 1.5|\Big)\, dt\right], u \in \mathcal{U}_{\sigma, 0}\,,
d_K ( \alpha_{\rm HJM}) = d_K \bigg( \frac{1}{z_6}e^{-2 z_6 \cdot}\bigg) \leq \bigg\| \frac{1}{z_6}e^{-2 z_6 \cdot}- \frac{1}{z_6}e^{-z_7 \cdot}\bigg\| = \frac{1}{z_6}\| e^{-2 z_6 \cdot}- e^{-z_7 \cdot}\|,
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}D^{(m)}_n(x,y;a,b,c)\frac{t^n}{n!}\frac{u^m}{m!}=\frac{2e^{\left(\frac{y\ln c+\ln a}{\ln a +\ln b}\right)u}e^{\left(\frac{x\ln c+\ln a}{\ln a +\ln b}\right)t}e^{t+u}\left(1-e^{-t}\right)}{(e^t+1)(e^t+e^u-e^{t+u})}.
\ln J = \lambda \frac{-1}{16\pi^{2}}\int d^{4}x E B \frac{1-\cosh t(E\pm B)}{\sinh t E \sinh t B}= \lambda \frac{1}{16\pi^2}\int d^4 x \frac{1}{2}(E\pm B)^2 + O(1/M^4).
\sum_{s=1}^{n-1}\left(\langle \varepsilon_s - \varepsilon_{s+1}, \lambda \rangle +\langle \delta_s - \delta_{s+1}, \lambda \rangle\right) = \sum_{s=1}^{n-1}\sum_{i > j}\left(\langle \varepsilon_s - \varepsilon_{s+1},\alpha_{i,j}+ \beta_{i,j}\rangle+ \langle \delta_s - \delta_{s+1},\alpha_{i,j}+ \beta_{i,j}\rangle\right) \geq 0.
\int_{({\mathbb R}^d)^{k-1}}\varphi^{(j)}(K_1\cap K_2^{x_2}\cap \dots \cap K_k^{x_k})\, \lambda^{k-1}(d(x_2,\dots ,x_k)) & = \sum_{{m_1,\dots,m_k=j}\atop{m_1+\dots +m_k=(k-1)d+j}}^d\varphi^{(j)}_{m_1,\dots,m_k}(K_1,\dots,K_k).
\sum_{i=1}^{\infty}\big[P(T(t)\ge i)\big]^{1/k}=\sum_{i=1}^{\infty}\big[ i^{\,p-1}P(T(t)\ge i)\big]^{1/k}{\strut i}^{\,-(p-1)/k}&\le \bigg[ \sum_{i=1}^{\infty}i^{\,p-1}P(T(t)\ge i)\bigg]^{\tfrac{1}{k}}\ \bigg[ \sum_{i=1}^{\infty}{\strut}i^{\,-\tfrac{p-1}{k-1}}\bigg]^{\tfrac{k-1}{k}}&\le \big({\|T(t)\|}_p\big)^{p/k}.
\begin{array}{l}y^0=\displaystyle a\frac{1+xx'}{1-xx'}~,~~~~~~iy^{n+1}=a\frac{2x^{n+1}}{1-xx'}~, y^i=a\displaystyle\frac{2x^i}{1-xx'}~,~~~~~~(i=1,~2,~\cdots,~n)~.\end{array}
U(T+S)z =(I+T^\dagger S)^{-1}T^\dagger (T+S)z =(I+T^\dagger S)^{-1}T^\dagger (T+TT^\dagger S)z =(I+T^\dagger S)^{-1}T^\dagger T(I+T^\dagger S )z =(I+T^\dagger S)^{-1}P_{N(T)^{\bot}}(I+T^\dagger S )z =(I+T^\dagger S)^{-1}(I+T^\dagger S )z =z=P_{R(U)}z.
TZ^{p}_{rat}(X) :=TZ^{M}_{p,rat}(D^{\mathrm{perf}}(X) = \mathrm{Ker}\{Z^{M}_{p,rat}(D^{\mathrm{perf}}(X[\varepsilon])) \xrightarrow{\varepsilon =0}Z^{M}_{p,rat}(D^{\mathrm{perf}}(X)\}.
\omega_2(t) \triangleq \mathbb{E}[e^{\imath tT}]=\mathbb{E}\big[e^{\imath t \sum_{i\in\mathcal{K}'}Z_i}\big]=\mathbb{E}[e^{\imath tZ}]^{K'}\stackrel{(a)}{=}\bigg[\int_{-\infty}^{\infty}\big(\cos (tz)+\imath\sin (tz)\big)f_Z(z)\mathrm{d}z\bigg]^{K'}&\stackrel{(b)}{=}\bigg[2\int_{0}^{\infty}\cos (tz)f_Z(z)\mathrm{d}z\bigg]^{K'},\ \forall t\ge 0,
I_{(Z_-,D_{Z,-})}(y,z)=ze^{t_-/z}\sum_{d\in\mathbb K_{-}}\tilde y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right) \left(\prod_{0<a\leq v_2\cdot d}(v_2+az)\right)\textbf{1}_{[-d]}I_{D_{Z,-},d},
\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}(h(x))\frac{\partial h_{n+1}}{\partial x_{i}}(x)=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}(h(x))\frac{\partial g}{\partial x_{i}}(x)=Df(h(x))Dg(x)^{T}
\begin{cases}-\Delta v + \frac{\lambda}{|\Omega|}\int_{\partial \Omega}b(x) g(t + v) = \lambda Q f(x, t + v) \mbox{in $\Omega$}, \dfrac{\partial v}{\partial \mathbf{n}}= \lambda b(x) g(t + v) \mbox{on $\partial \Omega$}, \end{cases}
{\cal H}= \frac{1}{2}( \vec{E}^{a}\vec{E}^{a}+ \vec{B}^{a}\vec{B}^{a}) + \frac{1}{2}( \vec{\cal E}^{\, 2}+ \vec{\cal B}^{\, 2}) + \pi^{\dagger}\pi + ( \vec{D}\varphi )^{\dagger}( \vec{D}\varphi ) + V ( \varphi^{\dagger}\varphi )
g_1(\omega,y) = |v(0)||1-e^{i\omega_0}|^{-1}|e^{i\omega r(y)}-e^{i\omega_0 r(y)}|, & g_2(\omega) =2|v(0)||(1-e^{i\omega})^{-1}-(1-e^{i\omega_0})^{-1}|.
(I-U_A^*U_A)Y_0 =(I-U_A^*U_A)\big[D+(I-U_A^*U_A)Y_0\big] &=(I-U_A^*U_A)\big[D^*+Y_0^*(I-U_A^*U_A)\big] &=(I-U_A^*U_A)D^*+Z_0.
\lim_{z \to 0}\left| \ln |\ln z^2 |\right|^{-1}\mathfrak{n}(\mathcal{A}_0 (z), a ) =\lim_{z \to 0}\left| \ln |\ln z^2 |\right|^{-1}\Bigl( \mathfrak{n}(T^+ (z), a ) + \mathfrak{n}(T^- (z), a )\Bigr) = \frac 8{3\pi a}.
s^{I}( z , \vec{x}) = \frac{\Gamma ( 2 k )}{A \pi^{3}\Gamma ( 2 k - 3 )}\int d \vec{x}^{\prime}\left( \frac{z}{z^{2}+ | \vec{x}- \vec{x}^{\prime}|}\right)^{2 k}s_{0}^{I}( \vec{x}^{\prime}) .
\left\{\nabla_{\bf r^{\prime}}^{2}+ \frac{2 M}{\hbar^{2}}\left[ E - V ({\bf r^{\prime}}) \right] \right\}{\mathcal G}_{D}({\bf r^{\prime \prime}},{\bf r^{\prime}}; E ) = \delta ({\bf r^{\prime \prime}}-{\bf r^{\prime}}) ,
\displaystyle\frac \delta{\delta x^i}=\displaystyle\frac \partial{\partial x^i}-\underset{(1)}{N_i^j}\displaystyle\frac \partial{\partial y^{(1)j}}-\cdots -\underset{(k)}{N_i^j}\displaystyle\frac \partial{\partial y^{(k)j}}\, .
\Psi_{M-1}^\alpha(t) =\int_{t_{M-1}}^t \Big[\int_{t_{M-1}}^t\int_\tau^s \lambda_\alpha e^{-\lambda_\alpha(t-u)}dud\tau+\int_t^{t_M}e^{-\lambda_\alpha(t-s)}d\tau\Big]^2ds &+ \int_t^{t_M}\Big[\int_{t_{M-1}}^t e^{-\lambda_\alpha(t-\tau)}d\tau \Big]^2ds :=\Psi_{M-1,1}^\alpha(t)+\Psi_{M-1,2}^\alpha(t).
\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{f(s)}{s-z}ds=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(s)\hat{g_{z}}(s)ds=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(s)g_{z}(s)ds=V_{y}Pf(x).
\tilde{C}( \bar{k}, \bar{p}) = \left( \begin{array}{c c c}{{- 2 i \bar{k}\, \delta ( \bar{k}+ \bar{p})}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- \delta ( 0 )}}\\{{0}}&{{\delta ( 0 )}}&{{0}}\\ \end{array}\right) \, .
\Gamma_{\phi^{-}}^{b a}( p ) = \frac{i G}{2 \Delta_{C}( p )}\left( \left\{[ 1 + G L^{2 2}( p ) ] \delta^{b 1}- G L^{2 1}( p ) \delta^{b 2}\right\}\delta^{a 1}- \left\{[ 1 - G L^{1 1}( p ) ] \delta^{b 2}+ G L^{1 2}( p ) \delta^{b 1}\right\}\delta^{a 2}\right) ,
f(G_{i+1}) -f(G_i) &\geq \frac{1}{K-i}(f(G_i\oplus \bar{O}_{K-i})-f(G_i)) &\geq \frac{1}{K-i}\left(\frac{K-i}{K}f(O_K)+\frac{i}{K}f(G_i)-f(G_i)\right) &= \frac{1}{K}(f(O_K)-f(G_i)),
\frac{{\partial [\frac{{\sigma_k (\lambda )}}{{\sigma_l (\lambda )}}]}}{{\partial \lambda_1}}\geqslant c_1\sum\limits_{i = 1}^n{\frac{{\partial [\frac{{\sigma_k (\lambda )}}{{\sigma_l (\lambda )}}]}}{{\partial \lambda_i}}}, \forall 0 \leq l < k,
( \bar{\sigma}_{\nu \mu})_{u}^{~ v}= \left( \begin{array}{l l}{0_{[ N - 2 ] \times [ N - 2 ]}}{0_{[ N - 2 ] \times [ 2 ]}}{0_{[ 2 ] \times [ N - 2 ]}}{( \bar{\sigma}_{\nu \mu})_{\dot{\alpha}}^{~ \dot{\beta}}}\end{array}\right) ~ ~ ,
\big| \big(\partial_{x_1}^{\gamma}f\big)(x_1)\big| \le \sum_{\underset{\gamma_1+\cdots+\gamma_s=\gamma}{1\le m\le|\gamma|}}\tilde{c}_{m,\gamma}|x_1|^{-|\gamma_1|}\cdot\ldots\cdot|x_1|^{-|\gamma_s|}=C|x_1|^{-|\gamma|}\,.
\sum_{n=0}^\infty \frac{q^{n^2}(-q;q)^2_n}{(q;q)_{2n}}=\frac{1}{J_{1}}f_{4,4,3}(-q^3,-q^2;q) =\frac{1}{4}\frac{\overline{J}_{0,3}}{J_1}\mu(q^3) -\frac{1}{2}\frac{\overline{J}_{1,3}}{J_1}\phi(q)+\frac{1}{J_1}\Theta(q),
\xi(z) = \int_1^z \frac{\rho_{\alpha, \varepsilon}(s) - \rho(\cos \theta)}{(s^2-1)^{1/2}}ds + \rho(\cos \theta) \int_1^z \frac{1}{(s^2-1)^{1/2}}ds & = \int_1^z \frac{\rho_{\alpha, \varepsilon}(s) - \rho(\cos \theta)}{(s^2-1)^{1/2}}ds + \rho(\cos \theta) \log \varphi(z).
\sum_{\imath=1}^{\infty}\frac{\varsigma(\imath)\Big(\sum_{\jmath=1}^{\imath}\varsigma(\jmath)\varrho(\jmath)\Big)^p}{\Big(\sum_{\jmath=1}^{\imath}\varsigma(\jmath)\Big)^\alpha}\leq \Big(\frac{p}{\alpha-1}\Big)^p\sum_{\imath=1}^{\infty}\varsigma(\imath)\varrho^p(\imath)\Big(\sum_{\jmath=1}^{\imath}\varsigma(\jmath)\Big)^{p-\alpha}, p\geq\alpha>1,
\begin{aligned}\frac{H_{2n}(-zq^{1/2},k+3/2|\q)}{(q;q)_{2n}}&=\sum_{n\ge s_1\ge s_2\ge\cdots\ge s_{k+1}\ge 0}\frac{q^{s_1^2+\cdots s_{k+1}^2}(qz,1/z;q)_{s_{k+1}}}{(q)_{n-s_1}(q)_{s_1-s_2}\cdots (q)_{s_{k}-s_{k+1}}(q)_{2s_{k+1}}}\end{aligned}
\mathbb{P}_{n_{l_q}}\left( \left| \log\left(\frac{d\tilde{\mathbb{Q}}_{n_{l_{q}}}}{d\mathbb{P}_{n_{l_{q}}}}\right) - \sum_{k=1}^{m_{n_{l_q}}}\frac{2\mu_{k}(C_{n_{l_q},k}- (n-1)\mathbb{I}_{k=2})-\mu_{k}^{2}}{4k}\right| \ge \epsilon \right)\le \delta.
g_{m_1}h_{n_1}g_{m_2}h_{n_2}g_{m_3}h_{n_3}\cdots g_{m_k}h_{n_k}g_{m_{k+1}}= [g_{m_1},h_{n_1}]\cdot [h_{n_1},g_{m_1}g_{m_2}]\cdot [g_{m_1}g_{m_2},h_{n_1}h_{n_2}]\cdot [h_{n_1}h_{n_2},g_{m_1}g_{m_2}g_{m_3}] &\cdot [g_{m_1}g_{m_2}g_{m_3},h_{n_1}h_{n_2}h_{n_3}]\cdots [h_{n_1}\cdots h_{n_k}, g_{m_1}\cdots g_{m_{k+1}}].
2 \cdot \operatorname{Tr}(f[F,g][F,h]M) = f(0)(g_{0,1}h_{1,2}- g_{0,3}h_{3,2})+ f(2)(g_{2,3}h_{3,0}- g_{2,1}h_{1,0}) \ - f(1)(g_{1,0}h_{0,3}- g_{1,2}h_{2,3}) - f(3)( g_{3,2}h_{2,1}- g_{3,0}h_{0,1}).
a_1(u,y) = (k_1-k_2)\circ\psi(u,y), a_2(u,y) = (k_1+k_2)\circ \psi(u,y), b_1(u) = \beta_1 \circ \psi(u,0), b_2(u) = \beta_2 \circ \psi(u,0), \beta_1(x) = (2\alpha(x)-1),\beta_2(x) = \bar \zeta(x)\alpha(x) + \bar \eta(x)(1-\alpha(x)).
M_{A D M}^{2}={\frac{1}{2}}{\vec{\tilde{\alpha}}^{T}}\mu_{+}{\vec{\tilde{\alpha}}}+{{\frac{1}{2}}}e^{- 4 \Phi_{\infty}^{\prime}}{\vec{\beta}}^{T}\mu_{+}{\vec{\beta}}+ e^{- 2 \Phi_{\infty}^{\prime}}\left[ ({\vec{\beta}}^{T}\mu_{+}{\vec{\beta}}) ({\vec{\alpha}}^{T}\mu_{+}{\vec{\alpha}}) - ({\vec{\beta}}^{T}\mu_{+}{\vec{\alpha}})^{2}\right]^{\frac{1}{2}},
T^2 \log^4T \le \bigg( \sum_{\substack{T<\gamma \le 2T Z'(\gamma) \, Z(\gamma+\frac{2\pi\kappa}{\log T}) <0}}\!\!\!\! 1 \bigg) \bigg( \sum_{T<\gamma \le 2T}Z'(\gamma)^2 \, Z\Big(\gamma+\frac{2\pi\kappa}{\log T}\Big)^2\, \Big|M\Big(\frac12+i\gamma+\frac{2\pi i\eta}{\log T},P\Big)\Big|^4 \bigg),
\mathbf B_{\lambda -\frac{n}{2}+1}F(\mathbf x) = -(t+x_0)^{-\lambda-2}x_n (\Delta f) \left(\frac{x_1}{t+x_0}, \dots,\frac{x_n}{t+x_0}\right) \hphantom{\mathbf B_{\lambda -\frac{n}{2}+1}F(\mathbf x) =}{}-2\left(\lambda-\frac{n}{2}+1\right)(t+x_0)^{-\lambda}(t+x_0)^{-1}\frac{\partial f}{\partial \xi_n}\left(\frac{x_1}{t+x_0}, \dots,\frac{x_n}{t+x_0}\right).
X_{L}( a , b , c ; q ) = \displaystyle \sum_{\begin{array}{c}{{\{\scriptstyle \mu |{\mu_{0}= a , \mu_{L}= b , \mu_{L + 1}= c}\}}}\\ \end{array}}\! \! \! \! \! \! q^{\sum_{k = 0}^{L - 1}( k + 1 ) H ( \mu_{k}, \mu_{k + 1}, \mu_{k + 2})},
G_{r}^{( 2 )}( x ) = \sum_{n_{0}, n_{1}, \ldots , n_{\nu - 2}= 0}^{\infty}x^{\frac{r - 1}{2}+ \sum_{i = 0}^{\nu - 2}n_{i}( \nu - 1 - i )}\times \prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{n_{i}+ \tilde{n}_{i}+ \tilde{V}_{i , r}}\\{n_{i}}\\ \end{array}\right] .
J_\varepsilon(v):=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2\alpha}}\,dx\,dy+f_\varepsilon \left (\mathcal{L}^n(\{u>0\}\cap D^c) \right ),
\langle \tilde{D}_{u}v, w\rangle = \langle D^{(1)}_{u}v , w\rangle -\frac{1}{8}\left( \eta (\mathcal J v, u, w)+ \eta ( w, u, \mathcal J v) \right) +\frac{1}{8}\left( \eta (v, u, \mathcal J w) +\eta (\mathcal J w, u, v)\right) ,
m(c,\gamma_k) \leq \frac{I(\psi_k,c,\gamma_k)}{K(\psi_k)^{\frac{2}{3}}}&= \frac{I(\psi_k,c,0) + \gamma_k\int|\partial_x^{-1}\psi_k|^2dx}{K(\psi_k)^{\frac{2}{3}}}&\leq \frac{I(\psi_k,c,0) + 1/k}{K(\psi_k)^{\frac{2}{3}}}.
\left(\left(\frac{1}{4}-\frac{1}{4i}+\frac{1}{16i^2}+o(1)\right)+\left(\frac{1}{2}-\frac{1}{8i^2}+o(1)\right)+\left(\frac{1}{4i}+o(1)\right)+\left(\frac{1}{16i^2}+o(1)\right)\right)\binom{t}{2}&=\left(\frac{3}{4}+o(1)\right)\binom{t}{2},
\hat{\Phi}_{d}= \left( \begin{array}{c c c}{\Phi^{\prime}}&{0}\\{0}&{\Phi^{\prime}}\\ \end{array}\right) ; \hat{\Phi}_{n d}= \left( \begin{array}{c c c}{0}&{\Phi^{\prime \prime}}\\{- \Phi^{\prime \prime}}&{0}\\ \end{array}\right) ;
\|\varphi^*-\mathcal{L}^{\lambda}_L\varphi^*\|_2=\left(\sum_{\ell=1,~|c_{\ell}|\leq\lambda\mu_{\ell}}^d|c_{\ell}|^2+\sum_{\ell=1,~|c_{\ell}|>\lambda\mu_{\ell}}^d|\lambda\mu_{\ell}|^2\right)^{1/2}.
Z_{B}( \beta ) = \left[ \det \left( -{\frac{d^{2}}{d t^{2}}}+ m^{2}\right) \right]^{- 1 / 2}= \exp \biggl \{-{\frac{1}{2}}\mathrm{T r}\ln \left( -{\frac{d^{2}}{d t^{2}}}+ m^{2}\right) \biggr \}.
\int_{N_r}|\mathfrak{u}_n|^2 = 16\int_0^r\int_0^{2\pi}\int_0^{2\pi}|\mathfrak{u}_n(r,s,t)|^2rdsdtdr &\leq 64\pi^2 r^2\int_{N_r}|\nabla_{e_2}\mathfrak{u}_n|^2.
\lim_{n\to+\infty}2S(I)\int_{0}^{(b_1/a_1-1)\ln n}\frac{D_n((1+z/\ln n) G_n(x)/2)}{D_n(G_n(x)/2)}\frac{(1+z/\ln n)a_1}{\sqrt{4-(1+z/\ln n)^2a_1^2}}dz = 2S(I)\int_{0}^{+\infty}e^{-2z}\frac{a_1}{\sqrt{4-a_1^2}}dz=\frac{S(I)a_1}{\sqrt{4-a_1^2}}=\frac{S(I)\sqrt{4-a^2}}{|a|},
\Gamma ( \phi , \phi_{a}^{*}, \bar{\phi}) = \frac{\hbar}{i}\ln Z ( J , \phi_{a}^{*}, \bar{\phi}) - J_{A}\phi^{A}, \phi^{A}= \frac{\hbar}{i}\frac{\delta \ln Z ( J , \phi_{a}^{*}, \bar{\phi})}{\delta J_{A}},
B_{X}= \gamma+ \log C_{X}- \sum_{[P]}\sum_{n \geq 2}\dfrac{1}{n}R_{X}^{n \ell(P)}= \gamma+ \log C_{X}+ \prod_{[P]}\left(\, \log \left( 1- R_{X}^{\ell(P)}\right)+ R_{X}^{\ell(P)}\right).
\hat{h}_{\mu \nu}= a^{2}\chi_{\mu \nu}+ 2 a_{0}^{2}a^{2}\left( \int \frac{d z}{a^{4}}\right) \partial_{\mu}\partial_{\nu}\xi_{z}- 2 a^{2}\left( \int \frac{d z}{a^{2}}\right) \partial_{\mu}\partial_{\nu}\xi_{z}- \frac{U_{B}}{2}a^{2}\xi_{z}\eta_{\mu \nu}
\sup_{\gamma \in \mathcal{P}_{\mathrm{D}}}I_{\mathrm{D}, \lambda}[\gamma] = \sup_{\gamma \in \mathcal{P}_{\mathrm{D}}}\sup_{\pi \in \Pi(\mu_1, \mu_2, \gamma)}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda \, d \pi.
\frac{\bar{B}( p^{2})}{m}= P P \Big [{\frac{e^{-{\frac{m^{2}}{2 \mu^{2}}}}}{p^{2}}}+ \int_{0}^{\infty}d \lambda^{2}{\frac{\bar{\sigma}( \lambda^{2})}{p^{2}- \lambda^{2}}}\Big ] .
\frac{|T_1|+|T_2|+\cdots+|T_N|}{|lI^2|}= \frac{N|T|}{|lI^2|}&\leq \frac{kN|T|}{|S_1|+|S_2|+\cdots+|S_N|}&\leq \frac{kN|T|}{A^*(r_1)+A^*(r_2)+\cdots+A^*(r_N)}&\leq \frac{k|T|}{A^*(2k-1)}=\delta_L^k(T),
\mathrm{d e t}_{r e n}[ 1 - K ( \not{\! \! B}) ] = \mathrm{d e t}_{3}[ 1 - K ( \not{\! \! B}) ] \exp \Big ( - \frac{1}{2}\mathrm{T r}_{r e n}\big [ K^{2}( \not{\! \! B}) \big ] \Big ) .
I -\tilde{X}^2-\tilde{Y}^2 = \begin{pmatrix}0 -(x_{12}(X_{11}+ x_{22})+ y_{12}(Y_{11}+y_{22}))e t(x_{12}\alpha + y_{12}\beta)e * 1-[x_{12}^2 +x_{22}^2 +y_{12}^2 + y_{22}^2 +t^2 (\alpha^2+\beta^2)] t[(x_{22}+ a)\alpha+(y_{22}+ b)\beta] * * 1-t^2(\alpha^2+\beta^2) - a^2-b^2 \end{pmatrix}.
H ={\frac{1}{2 m}}\int d x \mathrm{T r}| \nabla \phi |^{2}+{\frac{T_{0}^{2}}{2 m N_{c}}}\int d x d y V ( x - y ) \mathrm{T r}[ \phi^{\dagger}( x ) \phi^{\dagger}( y ) \phi ( y ) \phi ( x ) ] \; .
(\tilde{Q}_i)_{j,k}:=([\tilde{Q}]_{i+1,i})_{j,k}=\big(\phi_j+c_i\lambda_j^{-1}\big)\delta_{j,k}-|g|\lambda_j^{m-i-1}\lambda_k^i(\lambda_j^m-\lambda_k^m)^{-1}(1-\delta_{j,k}).
\left\|\mathbb{I}_{L}\right\|^{2}_{2}:= \int^{1}_{0}\left(\mathbb{I}_{L}\left(\alpha\right)\right)^{2}d \alpha= \mathbb{P}\left(R_{L}\right)\left(1-\mathbb{P}\left(R_{L}\right)\right) \leq \mathbb{P}\left(R_{L}\right).
\sum_{|\alpha| \leq m}(\mathcal{E}_k^{\alpha}) = \sum_{|\alpha|\leq m}\sum_{j=1}^{+\infty}\mu_j||\partial^{\alpha}\psi_j||_{L^2}^2 = \sum_{j=1}^{+\infty}\mu_j ||\psi_j||_{H^m}^2 = \sum_{i}\mu_j = (ii^*) < + \infty
T(X,Y)= u\, g\left(\nabla_X U,Y\right)-(\nabla_Xu)g(U,Y)+\bar{\varphi}(\nabla_XU,U,Y) &(u^2-|U|^2)\bar{T}(X,Y)+2\bar{T}(X,U)g(U,Y)-2u\,\bar{\varphi}(U,\bar{T}(X),Y),
f_1 :=\frac{1}{2}[R_{10}+R_{11}], \ \ f_3:=\frac{1}{2}[2R_1+R_2+R_4+2R_5+2R_6+2R_7+2R_8+R_9+R_{12}], f_2 :=[R_{9}+R_{10}], \ \ f_4:=\frac{1}{2}[2R_1+2R_2+2R_3+2R_4+2R_5+R_6+R_8+R_9+R_{12}].
\mathbb{A}(d, h_1(d))=d^2 \mbox{Arccos}\left(\frac{d^2 + R_1^2 -R_0^2}{2 R_1 d}\right)- \frac{d^2 + R_1^2 - R_0^2}{2 R_1}\sqrt{d^2 -\left(\frac{d^2 + R_1^2 - R_0^2}{2 R_1}\right)^2}
\begin{array}{r c l c r c l}{\mathrm{T r}[ T_{a}, \cdots ]}&{=}&{0 ,}&{{\qquad}}&{\mathrm{T r}[ \Sigma_{\alpha}, \cdots ]}&{=}&{0 ,}\\{\mathrm{S t r}[ T_{a}, \cdots ]}&{=}&{0 ,}&{{\qquad}}&{\mathrm{S t r}\{\Sigma_{\alpha}, \cdots \}}&{=}&{0 ,}\\ \end{array}
f(\alpha)(\overline{\mu}_Y-\underline{\mu}_Y)+\alpha(\overline{\mu}_X-\underline{\mu}_X)=(\overline{h}-\underline{h}) &f(-\alpha)(\overline{\mu}_Y-\underline{\mu}_Y)+\alpha(\overline{\mu}_X-\underline{\mu}_X)=(\overline{h}-\underline{h}).
R_{( 0 )}( r ) = C_{\mathrm{i n}}e^{- i \frac{\omega \ell^{2}}{2 r_{H}}\ln \left( \frac{r - r_{H}}{r + r_{H}}\right)}+ C_{\mathrm{o u t}}e^{i \frac{\omega \ell^{2}}{2 r_{H}}\ln \left( \frac{r - r_{H}}{r + r_{H}}\right)}.
\begin{cases}dY_1(t)=\left(-a(Y_1(t))+\frac{\partial \log\theta(Y_1(t),Y_2(t))}{\partial y_1}\right)dt+dW_1(t), dY_2(t)=\left(-a(Y_2(t))+\frac{\partial \log\theta(Y_1(t),Y_2(t))}{\partial y_2}\right)dt+dW_2(t), Y_1(0)=Y_2(0)=x, Y_1(t)<Y_2(t) \mbox{for a.a.}t>0,\end{cases}
A_{C}^{N S N S}={\frac{G_{C}}{4}}\int_{0}^{\infty}{\frac{d t}{2 t}}( 8 \pi^{2}t )^{- 5}{\frac{f_{3}( e^{- \pi t})^{8}}{f_{1}( e^{- \pi t})^{8}}}.
\delta S=[-mu_{\mu}\delta x^{\mu}]_{t_{1}}^{t_{2}}+m\int_{t_{1}}^{t_{2}}\delta x^{\mu}\frac{du_{\mu}}{ds}ds=-mu_{\mu}\delta x^{\mu}=\frac{\partial S}{\partial x^{\mu}}\delta x^{\mu}=-p_{\mu}\delta x^{\mu},
\rho(x,x^{\prime},t)=\frac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}dk\int_{-\infty}^{+\infty}dwe^{-ikw/\hbar}F_{CSL}(k,x-x^{\prime},t)\rho^{QM}(x+w,x^{\prime}+w,t),
{\bf{Cov}}\left[V_{\varepsilon}(\psi), \,V_{\varepsilon}(\xi)\right] = \begin{cases}-2 \, \log|e^{2\pi i\psi}-e^{2\pi i\xi}|, \, |\xi-\psi|\gg \varepsilon, 2\left(\kappa-\log\varepsilon\right), \psi=\xi,\end{cases}& + O(\varepsilon),
{1\over N_k^\tau}\leq{1 \over q_k^{(\eta-\epsilon)\tau}}={1 \over q_k^{(\eta-\epsilon)({1 \over \eta}+\delta)}}={1 \over q_k^{1+\delta\eta-\epsilon({1 \over \eta}+\delta)}}={1 \over q_k^{1+\epsilon_1}}.
L(E,s)=\prod_{v\nmid N}^{}\left(1-\frac{a_v}{q_v^{s}}+\frac{q_v}{{q_v}^{2s}}\right)^{-1}\times\prod_{v\mid N}^{}\left(1-\frac{1}{q_v^{s}}\right)^{-1}