harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\zeta^{(T)}\left(z,\beta, \sigma \right)=\sum_{n=0}^{\infty}\sum_{l=1}^{\infty}D^{T}_{n}(\Lambda^{T}_{n,l}(\sigma))^{-z}+\sum_{n=1}^{\infty}\bar{D}^{T}_{n}(\Lambda^{T}_{n,0}(\sigma))^{-z}~~~,
W^{[ 2 , 3 ]}( \Phi^{1}, \Phi^{2}) ={\frac{( \Phi^{1})^{6}}{6}}- ( \Phi^{1})^{4}\Phi^{2}+{\frac{2}{3}}( \Phi^{1})^{2}( \Phi^{2})^{2}- ( \Phi^{2})^{3}- \mu \Phi^{1}.
\Gamma(x,1) =P\Big(\theta_n=(O,1)n\geq 0\Big|\theta_0=(x,1)\Big) &\leq P\Big(\theta_n(1)=On\geq 0\Big|\theta_0=(x,1)\Big) &=P\Big(S_n=On\geq 0\Big|S_0=x\Big)=\widetilde{\Gamma}(x)
\mu_{N_1}^{\sigma}(s) (dx) := \frac{1}{Z}\exp \left( \sigma \sum_{i=1}^{N_1}x_i -H_{N_1}(x) + s \cdot \frac{1}{2}\sum_{(i, j) \in I_{N_1,N_2}}|M_{ij}| x_i^2 \right) dx.
I_{i+1,j+1}^t=\frac{I_{i+1,j}^t+V_{i+1,j}^t}{I_{i,j}^t+V_{i,j}^t}I_{i,j}^t,V_{i,j}^{t+1}=\frac{I_{i+1,j}^t+V_{i+1,j}^t}{I_{i,j}^t+V_{i,j}^t}V_{i,j}^t.
log_{e}(T)\approx log_{e}(\frac{\mu_{x}}{\mu_{y}})+\frac{X}{\mu_{x}}-\frac{Y}{\mu_{y}}\sim log_{e}(\frac{\mu_{x}}{\mu_{y}})+\mathbb{N}(0,\frac{\sigma_{x}^{2}}{\mu_{x}^{2}}+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}).
b_{p,\gamma}(f_{k+1},g_{k+1}) = b_{p,\gamma}(f_k+df_{k+1},g_k+dg_{k+1}) \leq b_{p,\gamma}(f_k,g_k)+ \left\langle \frac{\partial b_{p,\gamma}(f_k,g_k)}{\partial x},df_{k+1}\right\rangle+ \left\langle \frac{\partial b_{p,\gamma}(f_k,g_k)}{\partial y},dg_{k+1}\right\rangle,
\left[2\partial_{-}F_{+i}(x^{+},\vec{x}),F_{+-}(x^{+},\vec{y}) \right] = - i\partial_i^x(\Delta_\perp)^2 [\Delta_{2-2\delta}]^{-1}(x_\perp- y_\perp) \delta({x^-}- y^-)
\begin{array}{ll}g_M(x_1,\dots,x_k) = |R_1|^{1-k}\left(\sum_{a_2,\dots,a_k\in R_1}f(x_1+\sum_{i=2}^k a_i x_i,x_2,\dots,x_k) \right. \left. - \sum_{a_2,\dots,a_k\in R_1}f(ex_1+\sum_{i=2}^k a_i x_i,x_2,\dots,x_k)\right). \end{array}
R_M(y_1) = (1 + y_3 + y_3 y_1)^{-1}y_2^{-1}(1 + y_1 + y_1 y_2), R_M(y_1^-) = (1 + y_1 + y_1 y_2)^{-1}y_1 y_1^- (1 + y_2 + y_2 y_3), R_M(y_1^+) = (1 + y_2 + y_2 y_3)^{-1}y_2 y_1^+ (1 + y_3 + y_3 y_1).
y_{\mathbf{r}}^{(m)}=y_{\mathbf{s}}^{(m)}+O\Bigl( \frac{T_k}{k}(Y_{\mathbf{r}}+Y_{\mathbf{s}})\frac{\phi(a_m W B)}{a_m W B}(\log{A}+(\log\log{R})^2)\Bigr) &=y_{\mathbf{s}}^{(m)}+O\Bigl(\frac{T_k}{k}(Y_{\mathbf{r}}+Y_{\mathbf{s}})\frac{\phi(a_m W B)}{a_m W B}(\log{A})(\log\log{R})^2\Bigr).
\begin{aligned}\left|\zeta_n(x)-\int_M\zeta_n\mathrm{d}\mu-\sum_{j=1}^m A_{n,j}G(x_{n,j}^{(1)},x)\right| \le{C}\sum_{j=1}^m e^{-\frac{\lambda_{n,j}^{(1)}}{2}}\left(\frac{1_{U_{2r_0}(x_{n,j}^{(1)})}(x)}{(|T_j(x)-T_j(x_{n,j}^{(1)})|)}+1_{M\setminus U_{2r_0}(x_{n,j}^{(1)})}(x)\right),\end{aligned}
\varepsilon = d\left(f\circ \pi^{(k)}_j,f\circ \pi^{(k)}_t\right) \leq d\left(f\circ \pi^{(k)}_j,f_{n_{\nu,i}}\circ \pi^{(k)}_t\right)+ d\left(f_{n_{\nu,i}}\circ \pi^{(k)}_t,f\circ \pi^{(k)}_t\right) <d\left(f\circ \pi^{(k)}_j,f_{n_{\nu,i}}\circ \pi^{(k)}_t\right)+ \frac{\varepsilon}{2},
\Delta(A_{ij}x^ix^j) = (A_{ij,k}x^ix^j+A_{ij}(x^i\delta_{jk}+x^j\delta_{ik}))_{,k}= (\Delta A_{ij})x^ix^j+2A_{ij,k}(x^i\delta_{jk}+x^j\delta_{ik})+2\sigma_1(A)+O(r^3) = (\Delta A_{ij})x^ix^j+4\sigma_1(A)_{,i}x^i+2\sigma_1(A)+O(r^3).
\displaystyle \int_{\mathcal{B}_N}D^G(X)^{1/2}\widehat{f}(k^{-1}u^{-1}l^{-1}Xluk) \kappa^Y_{N,X,Q}(lk)dldudkdX = -\int_{\mathcal{D}_N}D^G(X)^{1/2}\widehat{f}(k^{-1}u^{-1}l^{-1}Xluk) \kappa^Y_{N,X,Q}(lk)dldudkdX
A=\left(\begin{array}{ccc}|K_{1}| a_0^{K_1}|K_1| b_0^{K_1}|K_1| |K_{2}| a_0^{K_2}|K_2| b_0^{K_2}|K_2| \vdots \vdots \vdots |K_{N}| a_0^{K_N}|K_N| b_0^{K_N}|K_N| \end{array}\right)+O(h^{2+\alpha}),
\tilde{z}_{\pm}= - Q_{\pm}^{- 1}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{z}}_{\pm}}- z_{\pm}, \tilde{p}_{\pm}= \displaystyle \displaystyle \frac{\partial}{\partial z_{\pm}}- Q_{\pm}^{- 1}\displaystyle \frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{z}}_{\pm}}.
W [{\psi}, A ] = \int_{- \infty}^{\infty}d x^{+}\int_{0}^{L}d x^{-}({\psi}_{+}^{\star}i{\partial}^{-}{\psi}_{+}- \frac{e}{4}j^{+}A_{+}) - \frac{1}{2}\int_{- \infty}^{\infty}d x^{+}A_{+}( 0 , x^{+}) Q^{+}
\lim\limits_{j\to+\infty}\varepsilon_j\dot{\widehat{u}}_{\tau_j,\varepsilon_j}(t^-_j)=\lim\limits_{j\to+\infty}\varepsilon_j\dot{\widehat{u}}_{\tau_j,\varepsilon_j}(t^+_j)=0\,.
\frac{\partial L}{\partial w_{ij}}=\frac{\partial L}{\partial w_{ji}},\frac{\partial L}{\partial w_{ijk}}=\frac{\partial L}{\partial w_{jik}},\frac{\partial L}{\partial w_{ijkl}}=\frac{\partial L}{\partial w_{jikl}}=\frac{\partial L}{\partial w_{ijlk}}.
f_{k+1}= (x+\sum_{j=1}^k (-1)^j\alpha^j(z))\cdot(x+\sum_{j=1}^{k-1}(-1)^j\alpha^j(z)) = x - \alpha(x) + \sum_{j=1}^k (-1)^j\alpha^j(z) - \sum_{j=1}^k (-1)^j\alpha^{j+1}(z) + \alpha(x) + \sum_{j=1}^{k-1}(-1)^j\alpha^{j+1}(z) = x + \sum_{j=1}^k (-1)^j\alpha^j(z) - (-1)^k\alpha^{k+1}(z) = x + \sum_{j=1}^{k+1}(-1)^j\alpha^j(z).
\begin{aligned}\bar{\nabla}_{S}S{}=\frac{1}{2}B, \bar{\nabla}_{X}X{}=\frac{1+2\lVert T\rVert^{2}}{2(1+\lVert T\rVert^{2})}B-\frac{1}{1+\lVert T\rVert^{2}}JT.\end{aligned}
\widetilde{\beta}^+(p)\geq\limsup_{N \to \infty}\frac{\log \langle|X|^p \rangle(L_N)}{p \log L_N}\geq\limsup_{N \to \infty}\frac{\log\Big(C_p L_N^{-2\frac{1 - \eta}{\eta}\nu_N}L_N^p \Big)}{p \log L_N}=1,
\begin{pmatrix}-\vartheta \overline\mu 0 \overline \gamma begin{align*}0.1cm]\mu \overline \vartheta \overline{\gamma}0 begin{align*}0.1cm]0 -\gamma \vartheta \mu begin{align*}0.1cm]-\gamma 0 \overline \mu -\overline{\vartheta}\end{pmatrix}, \mbox{with $\mu,\vartheta,\gamma \in \mathbb C$.}
\left({\lambda_i}^k{\lambda_j}^{-k}\right)^m \ = \ \left( (a^it)^k (a^j t)^{-k}\right)^m & = \ \left( a^i (t a^i t^{-1}) \cdots (t^{k-2}a^i t^{-(k-2)}) (t^{k-1}a^{i-j}t^{-(k-1)}) (t^{k-2}a^{-j}t^{-(k-2)}) \cdots (ta^{-j}t^{-1}) a^{-j}\right)^m
I_1(x) \leq e^{\alpha x}\left( \P(B>e^x/2)+\P(e^{\delta x}<B\leq e^x/2, AR+B>e^x)+\P(A>e^{\alpha x/\gamma'})\right. &\left.+\P(B\leq e^{\delta x},A\leq e^{\alpha x/\gamma'},AR\leq e^x\leq AR+B)\right) &= K_1+K_2+K_3+K_4.
|(F_m P_{1,m})*\cdots*(F_m P_{n,m})-(F_m g_1)*\cdots*(F_m g_n)\|_{C^{\rho_n,\psi}}&\le C m^{nkd-L}n \prod_{j=1}^n (1+\|g_j\|_\infty) \le m^{-1}.
\frac{1}{2i}\{\rho^2 p,\tilde s\}= \alpha^2(r,\tau,\Phi,\Psi) \tau^2 (r-r_{a,b}(\tau,\Phi,\Psi))^2 +\beta^2(r,\tau,\Phi,\Psi)\xi^2 \ \ \{p = 0\}
\sum_{0\le i\le r, \, i \, \rm{even}}\binom{cn}{r- i}\binom{(1-c)n}{i}= \frac{n^r}{2r!}( 1+ (2c-1)^r ) - O(n^{r-1}), \sum_{0\le i\le r, \, i \,\rm{odd}}\binom{cn}{r- i}\binom{(1-c)n}{i}= \frac{n^r}{2r!}( 1- (2c-1)^r ) - O(n^{r-1}).
\frac{\partial}{\partial t}\Big(x\Psi_B^{{\bf f}}(t_1,\ldots,t_{m-1},t)^{-1}\Big) = -xV_B^{{\bf f}}(t_1,\ldots,t_{m-1},t)\Psi_B^{{\bf f}}(t_1,\ldots,t_{m-1},t)^{-1}\,.
\prod_{i=1}^k e^{-V_i(x_i)}\le \rho \left( \sum_{i, j=1, i < j}^{k}\langle x_i, x_j \rangle \right), \hskip 2mm \hskip 2mm x_i, x_j \in \mathbb{R}^n \hskip 2mm \hskip 2mm \langle x_i, x_j \rangle \geq 0.
\mathfrak{S}^{r,t}_{g,d}:= \left\{[C,x_1,\dots,x_n] : \exists \, \ell \in G^r_{d}(C) \,\,\mbox{with}\,\, h^0\left( C,\ell \otimes \mathcal{O}_C\left({}- x_1-\cdots -x_n \right) \right) \geq r+1-t \right\}.
I ( \phi ) = \frac{1}{2}\int_{\cal G}d^{d + 1}y \partial_{\mu}( \sqrt{g}g^{\mu \nu}\phi \partial_{\nu}\phi ) = \lim_{\epsilon \rightarrow 0}\frac{1}{2}\int_{\bar{\cal G}\cap T_{\epsilon}}d^{d}{\bf y}\sqrt{h}\phi{\bf n}\cdot \nabla \phi
\partial_{t}^{2}\left( \begin{array}{c}{\varphi}\\{{\pi_{\chi}/ \kappa}}\\ \end{array}\right) = \left( \begin{array}{c c}{\nabla^{2}- \mu^{2}-{\frac{\kappa^{2}}{2}}}&{\kappa^{2}}\\{\mu^{2}+{\frac{\kappa^{2}}{4}}}&{\nabla^{2}- \mu^{2}-{\frac{\kappa^{2}}{2}}}\\ \end{array}\right) \left( \begin{array}{c}{\varphi}\\{{\pi_{\chi}/ \kappa}}\\ \end{array}\right)
\begin{aligned}D^{Z}=d^{Z}+(d^{Z})^{*}, \nabla^{\Omega^{\bullet}(Z,F|_{Z}), u}=\frac{1}{2}(\nabla^{\Omega^{\bullet}(Z,F|_{Z})}+(\nabla^{\Omega^{\bullet}(Z,F|_{Z})})^{*}).\end{aligned}
Z = \int D\varphi^i J(\varphi)\exp\left\{i\int dx\,\frac{1}{2}\partial_\mu\varphi^ig_{ij}\partial_\mu\varphi^j\right\}\equiv\int D\varphi^i J(\varphi) \exp\left\{iS_{\rm ch}(\varphi)\right\}.
(d\vec{a})(d\vec{b})^{-1}=-(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{a}_{0}...\vec{a}_{n})}_{\vec{b}})^{-1}(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{b}_{0}...\vec{b}_{m})}_{\vec{a}})
\alpha_{ij}(z,x,f) = \eta_i\circ p_i\circ\alpha\circ \eta_j\circ p_j(z,x,f) = \left( 0,\langle w_{ij},z \rangle x_i,0 \right), \beta_{ij}(z,x,f) = \eta_i\circ p_i\circ\beta\circ \eta_j\circ p_j(z,x,f) = \left(0,0,\langle v_{ij},z \rangle f_i\right),
\Omega(m, n) \times \Omega(m + n, g) = \Omega(m, n )\times \big(\Omega(m, g) \times\Omega(n, g) \big)= &= \big( \Omega(m, n )\times \Omega(m, g) \big) \times\Omega(n, g) = \Omega(m, n + g)\times \Omega(n, g)\,.
x(t^av^k) = g(t^a)x(v^k) + x(t^a)v^k = \gamma^a t^a ( \alpha_{22}[k]_\omega v^k) + \phi [a]_\gamma t^{a-1}v^k = \left( \frac{\gamma^a-\gamma^a\mu^k q^{k(1-d)}}{1-\gamma}+ [a]_\gamma \right)\phi^{(1)}t^a v^k = \left( \frac{1-q^{k(1-d)}}{1-\gamma}\right)\phi^{(1)}t^a v^k.
P(0<\mathcal{G}\leq c_1 \log |a|) =\sum_{\gamma:0\rightarrow a, g(\gamma)\leq c_1 \log |a|}p(\gamma) &\leq\sum_{\gamma:0\rightarrow a, g(\gamma)\leq c_1 \log |a|}s(\gamma) &=P^{RW}(0<\mathcal{G}\leq c_1 \log |a|),
\lim\limits_{\delta\rightarrow 0}\delta(1+\frac{2}{a(\delta)})^{\frac{1}{2}}=\lim\limits_{\delta\rightarrow 0}(\frac{2\delta^2}{a(\delta)})^{\frac{1}{2}}=(\frac{4n(n-1)}{\min S_{\hat{g}}})^{\frac{1}{2}}
{\rm I}_{1}^{s}(m) \equiv - \frac{1}{2}\sum_{k \in \cal Z}{\rm sign}(\varepsilon_k) |\lambda \varepsilon_{k, \rm R}|^{-s}( |\lambda \varepsilon_{k+m, \rm R}|^{-s}-|\lambda \varepsilon_{k-m, \rm R}|^{-s}).
X \Box Y (a_0, \dots , a_{p+q}) = \sum_{{\tiny \begin{array}{c}i_0 < \dots < i_q i_{q+1}< \dots < i_{q+p}\end{array}}}\epsilon_a(i_0, \dots , i_{p+q}) X(Y(a_{i_0}, \dots , a_{i_q}), a_{i_{q+1}}, \dots , a_{i_{q+p}}).
\MoveEqLeft \lim_{\theta \to -\infty}\mathbb{P}(\mathit{SINR}>\theta) = \lim_{\theta \to -\infty}\sum_{n=1}^\infty f_{N_i}(n) \mathbb{P}(\mathit{SINR}>\theta \mid n) = \sum_{n=1}^\infty f_{N_i}(n) = 1 - f_{N_i}(0).
\prod_{r=s+1}^{n}\Big(1-\frac{b}{r}\Big)=\begin{cases}0, s<\lfloor{b}\rfloor, \: b=\lfloor{b}\rfloor, O\big(\frac{\lfloor{b}\rfloor}{n}\big)^{\lfloor{b}\rfloor}, s<\lfloor{b}\rfloor, \: b\neq\lfloor{b}\rfloor, \big(1+O\big(\frac{n-s}{sn}\big)\big)\big(\frac{s}{n}\big)^b, s\geq\lfloor{b}\rfloor.\end{cases}
\zeta(s) = \pi^{\frac{s}{2}}\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\frac{\zeta(1-s)}{\Gamma\left(\frac{s}{2}\right)}s\rightarrow 0\;:\; \zeta(0) = \pi^{-\frac{1}{2}}\Gamma\left(\frac{1}{2}\right)\lim_{s\rightarrow 0}\frac{\zeta(1-s)}{\Gamma\left(\frac{s}{2}\right)}\;.
\sum_{t=3}^{T}\sum_{s=1}^{t-2}\lambda^{t-s-2}\sum_{j=1}^n\|\epsilon^z_{j,s}\| =\sum_{t=1}^{T-2}\sum_{j=1}^n\|\epsilon^z_{j,t}\|\sum_{s=0}^{T-t-2}\lambda^s\le\frac{1}{1-\lambda}\sum_{t=1}^{T-2}\sum_{j=1}^n\|\epsilon^z_{j,t}\|.
\ \tau \phi\big(\exp(L+r\log A_j-r\log C_j)\big) + (1-\tau)\phi\big(\exp(L+r\log B_j-r\log C_j)\big) \leq \ \phi\big(\exp(L+r\log(\tau A_j+(1-\tau) B_j)-r\log C_j)\big) = \ \phi\big(\exp(L)\big).
N_{n m}^{r s}= \frac{1}{n m}\oint \frac{d z}{2 \pi i}z^{- n}h_{r}^{\prime}( z ) \oint \frac{d w}{2 \pi i}w^{- m}h_{s}^{\prime}( w ) \frac{1}{( h_{r}( z ) - h_{s}( w ) )^{2}}.
S ={\frac{\pi^{3}R^{5}}{2 \kappa_{1 0}^{2}}}\left[ \; \int_{0}^{1}\! d u \, d^{4}x \, \sqrt{- g}\left({\cal R}- 2 \Lambda \right) + 2 \int \! d^{4}x \, \sqrt{- h}K + a \int \! d^{4}x \, \sqrt{- h}\; \right] \, .
I = \frac{1}{1 6 \pi ( 8 \pi^{6})}\int d^{1 0}x \sqrt{- g^{( 1 0 )}}e^{2 \phi^{( 1 0 )}}\left( R^{( 1 0 )}+ 4 ( D \phi^{( 1 0 )})^{2}- \frac{1}{1 2}H^{( 1 0 ) 2}\right) ,
t_{j}= \frac{\operatorname{t a n}( \frac{\pi}{4}- \frac{\pi j}{2 H})}{2 \operatorname{c o s}^{2}( \frac{\pi j}{2 H})}\prod_{i = 0}^{j - 1}\frac{\operatorname{t a n}^{2}( \frac{\pi ( j - 1 - 2 i + w_{r})}{2 H})}{\operatorname{t a n}^{2}( \frac{\pi ( i + 1 )}{2 H})}.
g_1[3]= q_{14}\big((x_2-x_1)(x_3-x_1)+q_{23}-q_{12}\big)+ q_{24}\big(q_{13}-q_{12}\big), g_2[3]= q_{15}\big((x_3-x_1)(x_4-x_1)+q_{24}+q_{34}-q_{12}-q_{13}\big) \hphantom{g_1[3]=}{}+q_{25}\big((x_3-x_2)(x_4-x_2) +q_{14}+q_{34}-q_{12}-q_{23}\big) +q_{35}\big(q_{14}+q_{24}-q_{13}-q_{23}\big).
(-A)^{-\alpha}=\frac{\sin(\pi \alpha)}{\pi}\int_{0}^{\infty}t^{-\alpha}(tI-A)^{-1}dt, (-A)^{\alpha}=\frac{\sin(\pi \alpha)}{\pi}\int_{0}^{\infty}t^{\alpha-1}(-A)(tI-A)^{-1}dt,
\lim_{t\rightarrow 0^+}-it\cdot\omega_2(it) = \lim_{t\rightarrow 0^+}\frac{t|\lambda|^2}{s(|\lambda|,t)-t}=\lim_{t\rightarrow 0^+}\frac{|\lambda|^2}{s(|\lambda|,t)/t-1}=\lambda_1(\mu)^2-|\lambda|^2.
\sigma^1{}_{\dot ab}:=\begin{pmatrix}0 1 1 0\end{pmatrix},\sigma^2{}_{\dot ab}:=\begin{pmatrix}0 -i i 0\end{pmatrix},\sigma^3{}_{\dot ab}:=\begin{pmatrix}1 0 0 -1\end{pmatrix},\sigma^4{}_{\dot ab}:=\begin{pmatrix}1 0 0 1\end{pmatrix}.
G^{( 2 )}( x , x^{\prime}) = \frac 1{2 \pi^{2}}\frac 1{\sqrt{2 r r^{\prime}}}\int_{\rho}^{\infty}d t \frac 1{\sqrt{\cosh t - \cosh \rho}}\frac{\sinh ( t / \alpha )}{\cosh ( t / \alpha ) - \cos ( \gamma )}\ .
\lim_{R\to +\infty}\lim_{k \to +\infty}\int_{\mathbb{R}^{N}}\big[\Phi_{R}\mathrm{H}_{u}(x,u_k,v_k)u_k+ \Phi_{R}\mathrm{H}_{v}(x,u_k,v_k)v_k\big]\,dx=0.
\int dt\;{\cal L}=\int dt\;\left[\,\frac{1}{2}D_tX^a_iD_tX^a_i-\frac{1}{4}(\epsilon^{abc}X^b_iX^c_j)^2+\frac{i}{2}\tilde\psi^aD_t\psi^a+\frac{1}{2}X^a_iK^a_i \right],
\frac{d}{dz}\left[{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z) \right] = \frac{d}{dz}\left[\sum_{n=0}^{\infty}(a)_n \frac{{}^{M\!L\!\!}B_{p}(b+n,\;c-b)}{B(b,\;c-b)}\frac{z^n}{n!}\right] = \sum_{n=1}^{\infty}(a)_n \frac{{}^{M\!L\!\!}B_{p}(b+n,\;c-b)}{B(b,\;c-b)}\frac{z^{n-1}}{(n-1)!}.
2 \mathrm{I m}\; S_{\mathrm{A B C D}}^{( I A )}= \frac{1 6 \pi^{2}}{g^{2}}+ 2 \pi^{2}\rho^{2}v^{2}- \frac{9 6 \pi^{2}\rho^{4}}{l^{4}}+ O ( \rho^{6}/ l^{6}) \; .
\mathcal{N}_1 :=\mathbf{n}\left(e^{-q\tau_0^-}(w_p-v_q)(X(\tau_0^-));\tau_0^-<\tau_b^+,\tau_0^->0\right), \mathcal{N}_2 := \mathbf{n}\left(e^{-q\tau_0^-}(w_p-v_q)(X(\tau_0^-));\tau_0^-<\tau_b^+,\tau_0^-=0\right).
F(\infty)=-k_M(-1)^{\frac{k_{\rm id}-2}{2}-m_{\rm id}}\langle \mu_{\underline{m}},\mu_{-\underline{m}}\rangle_V\sum_{i\geq \frac{k_c}{2}-m_c}\binom{k_c-1}{i}\binom{k_{\rm id}-1}{\lambda-\bar m+i}.
\int_D v ({\Delta}^2q)\{\rm d}x=\int_{\partial D}\left(Q_{n}({{\bf D}^2 q})+\frac{\partial U_{nt}({{\bf D}^2 q})}{\partial t}\right)v\{\rm d}s-\int_{\partial D}U_{nn}({{\bf D}^2 q})\frac{\partial v}{\partial n}\{\rm d}s.
P = \frac{(T - \lambda_{1}I)(T - \lambda_2I)}{(1 - \lambda_1)(1 - \lambda_2)}, Q = \frac{(T - I)(T - \lambda_2I)}{(\lambda_1 - 1)(\lambda_1 - \lambda_2)}, R = \frac{(T - I)(T - \lambda_1 I)}{(\lambda_2 - 1)(\lambda_2 - \lambda_1)},
\mathbf{P}(\tau_{x_n}>n) =\mathbf{P}(x_n+\min_{k\le n}S_k>0)=\mathbf{P}(u+\min_{k\le n}S_k/B_n>0) &\to \mathbf{P}(u+\min_{t\le1}W(t)>0)=\mathbf{P}(\tau_{x_n}^{bm}>B_n).
\beta: \mathbb{C}[s_{1}, \dots, s_{d}] \to \frac{\mathbb{C}[s_{1}, \dots, s_{d}]}{(\{s_{t}- s_{j}\mid t, j \in S_{k}, 1 \leq k \leq r\})}= \mathbb{C}[s_{1}, \dots, s_{r}].
((\lambda_{i_0j_0}-\lambda_{j_0k_0}+\lambda_{i_0k_0}) \mu_{j_0}^{i_0j_0k_0}- 2\lambda_{i_0j_0}\,\mu_{k_0}^{i_0k_0j_0}) \wedge dF_{i_0}\wedge dF_{j_0}\wedge dF_{k_0}= 0.
f|S = \sum a_{\underline{i}}\phi_{1}^{i_1}\phi_{2}^{i_2}\phi_{3}^{i_{3}}\phi_{4}^{i_4}= \sum a_{\underline{i}}(x \widetilde{\phi}_{3}+ x' \widetilde{\phi}_{4})^{i_{1}}(y \widetilde{\phi}_{3}+ y' \widetilde{\phi}_{4})^{i_{2}}\widetilde{\phi}_{3}^{i_3}\widetilde{\phi}_{4}^{i_4}.
D(t) \geq \left[\int_{0}^{t}a_0ds-\frac{(\int_{0}^{t}e^{s-t}a_0ds)^2}{\int_{0}^{t}e^{2(s-t)}a_0ds}\right]+\left[\int_{0}^{t}a_1N(s)ds-\frac{(\int_{0}^{t}e^{s-t}a_1N(s)ds)^2}{\int_{0}^{t}e^{2(s-t)}a_1N(s)ds}\right] &\geq \left[\int_{0}^{t}a_0ds-\frac{(\int_{0}^{t}e^{s-t}a_0ds)^2}{\int_{0}^{t}e^{2(s-t)}a_0ds}\right]=a_0(t-2\frac{e^t-1}{e^t+1}).
\hat\kappa_{1}\ = \c_n p \bigg[ \frac{n+1}{n}+\frac{n-1}{n}\frac{{\hat\varsigma}_n^2 - 2p}{p(p+2)}- \frac{2(n-1)}{n^2}\frac{p^2+\hat\varsigma_n^2}{p(p+2)}\bigg] begin{align*}0.2cm] \hat\kappa_{2}\ = \c_n\bigg[\frac{2(n-1)}{n}\frac{p^2+\hat\varsigma_n^2}{p(p+2)}- 1\bigg].
\begin{array}{lll}\noindent (1-\lambda_3)b_3+2(1-\lambda_3)a_3+\lambda_2b_3a_1-\lambda_2b_3a_3-\lambda_2a_3^2+\lambda_2a_1a_3\ & \ =(\lambda_2-\lambda_3)(1-\lambda_3).\end{array}
\begin{array}{c}{{L = \left[ 1 + \frac 1 2 h \right] i \bar{\psi}\gamma^{a}\left[ \eta_{a}^{\mu}+ h_{a}^{\mu}\right] \left[ \partial_{\mu}+ \Omega_{\mu}\right] \psi + + \left[ 1 + \frac 1 2 h \right] m \bar{\psi}\psi}}\\ \end{array}
\prod_{k=1}^{\ell}q_{\nu_k}(\lambda^{i+})-\prod_{k=1}^{\ell}q_{\nu_k}(\lambda)= \sum\limits_{(*)}\prod_{k\in U}q_{\nu_k}(\lambda){\prod_{k'\in V}\left(q_{\nu_{k'}}(\lambda^{i+})-q_{\nu_{k'}}(\lambda)\right)},
\phi_{ij}= \frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\bigg[ A_{ij}(n){\rm e}^{-i\pi n x^{-}/L}+A_{ji}^{\dag}(n){\rm e}^{i\pi nx^{-}/L}\bigg],
\sum_{h=0}^\infty \frac{A(p^{2h+1},1)}{(p+1)p^{sh-1}}=\frac{p}{p+1}\frac{1+p^sA(p,1)}{p^s+A(1,p)}\sum_{h=0}^\infty \frac{A(p^{2h},1)}{p^{sh}},
F_N(a,b;t) = \frac{1-tq^N}{1-t}+ \frac{(1-tq^N)}{(1-t)}\sum_{n=1}^{N}(-1)^n\begin{bmatrix}N n\end{bmatrix}\frac{(b/a)_n(q)_n(at)^nq^{n(n+1)/2}}{(bq)_n(tq)_n}&= \frac{1-tq^N}{1-t}+ \frac{(1-tq^N)(b-a)t}{(1-t)(1-bq)(1-tq)}\sum_{n=0}^{N-1}(-1)^{n}\frac{(q)_{N}(bq/a)_{n}(at)^{n}q^{(n+2)(n+1)/2}}{(q)_{N-n-1}(bq^2)_{n}(tq^2)_{n}}.
\begin{array}{ccc}S\left(a_{i},b_{i},\bar{p},\omega_{i};t\right) = \left(a_{1}\cos\omega_{1}t+b_{1}\sin\omega_{1}t+\bar{p}\right)\mathbf{1}_{\left[0,T_{0}\right)}\left(t\right)+ \left(a_{2}\cos\omega_{2}t+b_{2}\sin\omega_{2}t+\bar{p}\right)\mathbf{1}_{\left[T_{0},T\right)}\left(t\right),\end{array}
w_{i,k}^n = \left( t^n_{-i, k+Q^n_i(0)-R^n_i(t^n_{i,k}-)-Q^n_{-i}(0)+R^n_{-i}(t^n_{i,k}-)}- t^n_{i,k}\right)1_{\{Q^n_{-i}(t^n_{i,k}-)=0\}}.
\bigotimes_{i=1,}^{2q-1}(U_i^*U_{i+1}L) = \bigotimes_{i=1,}^{2q-1}\left((U_i^*)^{r_{i+1}}(U_{i+1})L^{r_{i+1}}\right) &=P(U^*_1)^{r_2+1}(U_3^*)^{r_4+1}\ldots (U_{2q-1}^*)^{r_{2q}+1}U_{2q+1}^*
\left( \frac{i \epsilon_{\mu \lambda \alpha}p_{\lambda}}{p^{2}}\right) \left( \frac{i \alpha \epsilon_{\alpha \sigma \beta}p_{\sigma}}{\sqrt{p^{2}}}\right) \left( \frac{i \epsilon_{\beta \kappa \nu}p_{\kappa}}{p^{2}}\right) = \frac{i \alpha \epsilon_{\mu \lambda \nu}p_{\lambda}}{\left( p^{2}\right)^{3 / 2}}.
\left.\begin{array}{ll}S(u,v)= \displaystyle\dfrac{a_2}{2m_1}\int_{\mathbb{R}^N}|\nabla u|^2dx+\dfrac{a_1}{4m_2}\int_{\mathbb{R}^N}|\nabla v|^2dx &\displaystyle+a_2\omega_1\int_{\mathbb{R}^N}|u|^2dx+\frac{a_1}{2}\omega_2\int_{\mathbb{R}^N}|v|^2dx+a_1a_2\int_{\mathbb{R}^N}vu^2dx.\end{array}\right.
\begin{pmatrix}\partial_{t^1}F^{(2)}\dots \partial_{t^n}F^{(2)}\end{pmatrix}= \Phi^{-1}\begin{pmatrix}n-1 -2F_{1,1}^{(1)}\dots -2F_{1,n-1}^{(1)}\end{pmatrix}F^{(2)}-\Phi^{-1}\begin{pmatrix}0 F_{1,1,e}^{(1)}g^{ef}F_f^{(1)}\dots F_{1,n-1,e}^{(1)}g^{ef}F_f^{(1)}\end{pmatrix}.
\begin{array}{l}\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big(\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot z)]\Big) +\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big(2[\alpha(h)\cdot[x,y],\alpha^2(x)]\Big)=0.\end{array}
\begin{aligned}{}\bigg|\psi(\tau\overline{x'},\tau^{2}y_{2}\cdots,\tau^{k}y_{k})-\psi(\tau\overline{x'},\tau^{2}\overline{y}_{2}\cdots,\tau^{k}\overline{y}_{k}) \bigg|\leq C_{3}\tilde{\delta}\tau^{1+\alpha}d(p,\overline{p})\leq C_{3}\tau\omega({\tau})\tilde{\delta}d(p,\overline{p}).\end{aligned}
G_1(s) =(p-1)(n+p+q+1)P_q^{n-1,p-q}+p\Big((n+p+q-1)P_{q-1}^{n,p-q}&-(n+p)P_{q-1}^{n+1,p-q}+p(p-1)P_{q-2}^{n+1,p-q}\Big).
\frac{\big[\mathbb{A}/f\mathbb{A}\big]_{\mathbb{A}}}{\big[\varphi(\mathbb{A}/f\mathbb{A})\big]_{\mathbb{A}}}=\frac{f}{f + c(f)p_1\prod_{i=1}^nf(z_i)+ c(f)p_2\prod_{i=1}^nf(z_i)^2 + \dots +c(f)\prod_{i=1}^nf(z_i)^{r_0}}&=\frac{1}{1 + c(f)p_1f^{-1}\prod_{i=1}^nf(z_i)+ \dots +c(f)f^{r_0-1}f^{-r_0}\prod_{i=1}^nf(z_i)^{r_0}}&=D^{\varphi}_f(f^{-1})^{-1}.
T(Y(\xi,s) )- T(Y(\xi_0,t)) -\langle \nabla T(Y(\xi_0,t)), Y(\xi,s) - Y(\xi_0,t)\rangle &-\frac{1}{2}\langle R_{\xi,t}(\xi_0,t)(Y(\xi,s) - Y(\xi_0,t)),Y(\xi,s) - Y(\xi_0,t)\rangle = o(| Y(\xi,s) - Y(\xi_0,t)|^2).
A_{k}^{2}\mathcal{L}^{n-2}=[A_{k}]_{A_{k}}\mathcal{L}_{A_{k}}^{n-2}=\sum_{h_{1}+...+h_{r}=n-2}\frac{(n-2)!}{h_{1}!...h_{r}!}a_{1k}^{h_{1}}...a_{kk}^{h_{k}+1}...a_{rk}^{h_{r}}l_{k}
1 + \frac{p-1}{p}\left[ \sum_{i=1}^{a_{1}}p^{\max \{2i - a_{3}, 0\}}+ \sum_{i=a_{1}+1}^{a_{2}}p^{\max \{-i, i - a_{3}\}}+ \sum_{i=a_{2}+1}^{a_{3}}p^{\max \{-2i, -a_{3}\}}\right].
E_{\varepsilon,\delta_j}(t_j,x_j,u^{\varepsilon,\delta_j}(t_j,x_j),J[u^{\varepsilon,\delta_j}](t_j,x_j)+K_{(0,t_j)}[u^{\varepsilon,\delta_j}](t_j,x_j),p_j,X_j) &=E_*(t'_j,x'_j,u^{\varepsilon,\delta_j}(t_j,x_j),J[u^{\varepsilon,\delta_j}](t_j,x_j)+K_{(0,t_j)}[u^{\varepsilon,\delta_j}](t_j,x_j),p_j,X_j).
\gamma:=\begin{cases}\frac{u_1^2(\cos (E_0\tau)-\cos (E \tau ))^2}{2 E_0 E\sin (E_0 \tau ) \sin (E \tau )-\left(E_0^2+E^2\right) (1-\cos (E_0 \tau ) \cos (E \tau))}E\neq E_0 \frac{2u_1^2\tau^2 \sin^2(E_0 \tau)}{1 + 2 E_0^2 \tau^2 - \cos(2 E_0 \tau)}E=E_0. \end{cases}
\mathcal{F}_S(s)= \frac{1}{\ell}\mathcal{F}_U(s) +\frac{1}{\ell^2}\sum_{j=0}^{\ell-1}\sum_{k=1}^{\ell-1}\left(\sum_{r=0}^{\ell - 1}\xi_\ell^{-rk \deg{v_0}}\right) \mathcal{M}_{j,k}(s,v_0,\mathrm{split}),
\tilde{\pi}_1(abc) =[\tilde{f}_{11}(abc),\tilde{g}_{12}(1)]-[\tilde{f}_{11}(1),\tilde{g}_{12}(abc)]=[\tilde{f}_{11}(ab),\tilde{g}_{12}(c)]-[\tilde{f}_{11}(a),\tilde{g}_{12}(bc)]=0,
\gamma^{(t)}=\left[-\frac{(\mathbf{G}^{(t)}\mathbf{x}^{(t)}-\mathbf{b}^{(t)})^T (\hat{\mathbf{x}}^{(t)}-\mathbf{x}^{(t)})+\mu^{(t)}(\bigl\Vert\hat{\mathbf{x}}^{(t)}\bigr\Vert_{1}-\bigl\Vert\mathbf{x}^{(t)}\bigr\Vert_{1})}{(\hat{\mathbf{x}}^{(t)}-\mathbf{x}^{(t)})^T\mathbf{G}^{(t)}(\hat{\mathbf{x}}^{(t)}-\mathbf{x}^{(t)})}\right]^{1}_{0}
\psi^{1}( z ; x ) = \left( \begin{array}{c r}{{0}}\\{{e^{- \phi ( z )}G_{+}( z - \omega )}}\\ \end{array}\right) , ~ \psi^{2}( z ; x ) = \left( \begin{array}{c r}{{e^{\phi ( z )}G_{-}( z + \omega )}}\\{{0}}\\ \end{array}\right) ,
\frac{\partial}{\partial z}y^{-2}(cNz^2-bz+a)=-i\sqrt{N}y^{-2}p_z(\lambda), &\frac{\partial}{\partial z}R(\lambda,z) =-\frac{i}{2\sqrt{N}}y^{-2}p_z(\lambda)(cN\bar{z}^2-b\bar{z}+a), & y^{-2}(cNz^2-bz+a)(cN\bar{z}^2-b\bar{z}+a) = 2N R(\lambda,z).
\frac{A\psi_j}{r^{n-2}_{j}}=\frac{(A\psi_j)\varphi(|x_j|)^{\alpha(n-2)}}{(A\psi_j)^{\lambda\alpha(n-2)}}=\left(\frac{\varphi(|x_j|)^{n-2}}{(A\psi_j)^{\lambda(n-2)-\frac{1}{\alpha}=n}}\right)^\alpha .
\Phi_{n_j}^{(j)}(\varphi_j ) = (\sin 3\varphi_j )^{\frac{1}{2}+ a_j}C_{n_j}^{(\frac{1}{2}+a_j)}(\cos 3\varphi_j ), 0\leq \varphi \leq \frac{\pi}{3}, n_j=0,1,2,.., a_j =\frac{1}{2}\sqrt{1+2\lambda_j}\ .
P_\varepsilon(t)(\omega) = \int_0^t \sigma(s, \omega+\varepsilon h, X(s)(\omega+\varepsilon h)) \dot{h}(s) ds &+\int_0^t \Big[ b(s, \omega+\varepsilon h, X(s)(\omega + \varepsilon h)) - b(s, \omega, X(s)(\omega))\Big] ds &+\int_0^t \Big[ \sigma(s, \omega+\varepsilon h, X(s)(\omega + \varepsilon h)) - \sigma(s, \omega, X(s)(\omega))\Big] dW(s).
\left( \bar{D}_{0}^{- 1}\right)_{\Theta \Phi}^{a b}( x , y ) = - i \delta^{a b}\left( \begin{array}{c c}{{\cal O}_{\mu \nu}( x )}&{x_{\mu}}\\{x_{\nu}}&{0}\\ \end{array}\right) \delta ( x - y ) .
2^{a_{\min}}/3^{k+1}-1+2^{k+1}/3^{k+1}>0\Longrightarrow 2^{a_{\min}}>3^{k+1}-2^{k+1}\Longrightarrow &\Longrightarrow a_{\min}>\log_{2}\left(3^{k+1}-2^{k+1}\right) \Longrightarrow a_{\min}=\left\lceil \log_{2}\left(3^{k+1}-2^{k+1}\right) \right\rceil.
\iota(\Bbbk,\{2\}^{c},3) =(-1)^{c}x_{2c+3}\cdot\iota(\Bbbk)\quad(\Bbbk\in W_{2,3}'), \theta(\Bbbk,3,\{2\}^{c}) =(-1)^{c}x_{2c+2}\cdot s_{1}(\theta(\Bbbk))\quad(\Bbbk\in W_{2,3}).
\frac{2x}{a^{2}}-\frac{2yy^{\prime}}{b^{2}}=0\Rightarrow y^{\prime}=\frac{x}{y}\frac{b^{2}}{a^{2}}\Rightarrow y=\frac{x_{0}}{y_{0}}\frac{b^{2}}{a^{2}}(x-x_{0})+y_{0}.
\hat{H}(k_x, k_y, k_z) =\left(\begin{array}{cc}\sin k_y 2 + \cos k_x + \cos k_y + e^{ik_z}2 + \cos k_x + \cos k_y + e^{-ik_z}- \sin k_y\end{array}\right) &=\hat{H}_{\mathrm{loc}}(a(k_x, k_y), b(k_x, k_y), k_z),
\begin{array}{l l}{e^{2 \varphi}}{= 1 - \frac{\mu^{2}}{r^{2}}\operatorname{c o s h}^{2}\! \alpha}{e^{- 2 \lambda}}{= 1 - \frac{\mu^{2}}{r^{2}}\operatorname{c o s h}^{2}\! \beta ,}\end{array}
f(\vec{u})=\underbrace{\sum_{p=0}^{d}\sum_{j,k\geq0}^{j+k=p}c_{jk}\frac{u^{j}v^{k}}{j!k!}}_{}+\underbrace{\mathcal{O}(\left\Vert \vec{u}\right\Vert^{d+1})}_{}.
\begin{array}{ll}\langle{\psi+ \chi}, L(\Lambda, \psi +\chi, g) ({\psi + \chi}) \rangle = \langle{\psi}, L(\Lambda, \psi, g){\psi}\rangle + \langle{\chi}, L(\Lambda, \chi, g){\chi}\rangle + S^{int}_\Lambda[\psi, \chi] \end{array}