harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\overline \rho}(t) = \gamma \frac{\cos ( \tfrac{a\pi}2 )}{\pi a}(t_2-t)^{\tfrac{1+a}2}t^{- \tfrac{1-a}2},{\overline C}= \frac{1+a}{2a}\gamma t_2, t_2 = \bigg[ \frac\gamma{2 \pi}\frac{\cos ( \tfrac{a\pi}2 )}{a^2 \Gamma (a)}\Gamma \Big( \frac{1+a}2 \Big)^2 \bigg]^{- \tfrac1{1+a}}.
\varphi_{p}(s,x):=\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}},\,\,\,\,(s)>\frac{1}{2},\,\,\,\,x>0,
|S| \lesssim \, \frac{1}{|w| \,\,| \{\vec{v}\,:\, \widetilde{b}(\vec{v}) <|w| \}|^2}\le \, \frac{(\sqrt{3})^{2n+1}}{\sqrt{\delta^2+ \widetilde{b}(\vec{y}-\vec{y'})^2+ w^2}\left| \left\{\vec{v}\,:\, \tilde{b}(\vec{v}) < \sqrt{\delta^2+ \widetilde{b}(\vec{y}-\vec{y'})^2+ w^2}\right\}\right|^2}.
u_h = \sum^{\infty}_{n=-\infty}n\pi e^{in\pi x}\sum_{k\ne n^2}\frac{e^{-ki\pi^2 t}}{n^2-k}a_k +\sum^{\infty}_{n=-\infty}n\pi e^{-i(n\pi )^2 t+in\pi x}\left ( \sum_{k\ne n^2}\frac{a_k}{k-n^2}\right )\!.
\xi_0:= -a\log(\lambda a)+a-\frac{1}{\lambda},\:\:\:\:\xi_k:=(-1)^k\left(\lambda^{-k}\left(\frac{a}{k}-\frac{1/\lambda}{k+1}\right)-a^{k+1}\left(\frac{1}{k}-\frac{1}{k+1}\right)\right),
\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}\phi(x)|^{p}dx=\frac{Q(q-p)-a_{2}pq}{a_{1}pq-Q(q-p)}\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}\phi(x)|^{p}dx
\begin{array}{c}{{u = u_{0}+ \frac{p^{+}}2 \left[ ( \sigma^{+}+ f ( \sigma^{+}) ) + ( \sigma^{-}- g ( \sigma^{-}) ) \right]}}\\{{v = v_{0}+ \frac{p^{-}}2 \left[ ( \sigma^{+}- f ( \sigma^{+}) ) + ( \sigma^{-}+ g ( \sigma^{-}) ) \right]}}\\ \end{array}
\beta_{n,p,q}(x)=\Biggl(\dfrac{[2]_{p,q}\alpha+(p\frac{x}{b_n}+1-\frac{x}{b_n})_{p,q}^{n+m}}{[2]_{p,q}([n+1]_{p,q}+\beta)}b_n+\Biggl(\dfrac{(p+2q-1)[n+m]_{p,q}}{[2]_{p,q}([n+1]_{p,q}+\beta)}-1\Biggl)x \Biggl).
C_{d,0,n}= \{(f, c_1, \ldots, c_1, c_2, \ldots, c_2) \in B^{crit}_{d,k}\;\vert\; c_1 f^n(c_1) = c_1\}C_{d,1,m}= \{(f, c_1, \ldots, c_1, c_2, \ldots, c_2) \in B^{crit}_{d,k}\;\vert\; c_2 f^m(c_2) = c_2\}.
\lambda_0\left\langle a,b\right\rangle_{X,X^\ast}=\left\langle(Bc)*a,b\right\rangle_{X,X^\ast}-\left\langle B\left(c*a\right),b\right\rangle_{X,X^\ast}&=\left\langle(Bc)*a,b\right\rangle_{X,X^\ast}-\left\langle c*a,B^\ast b\right\rangle_{X,X^\ast}.
\sigma_{(a',i')}^{(s+1)n_1}= \sigma_{\sigma_{(a',i')}((a',i'))}^{n_1}= \sigma_{\sigma_{\varphi((a,i))}(\varphi((a,i)))}^{n_1}= \varphi\sigma_{\sigma_{(a,i)}((a,i))}^{n_1}\varphi^{-1}=\varphi\sigma_{(a,i)}^{(r+1)n_1}\varphi^{-1}=\sigma_{(a',i')}^{(r+1)n_1}
\xi_\mathbf{e}(\pi)^\rho= \operatorname{diag}( \pi^{e_1+e_2}, \pi^{e_2}, \pi^{e_1+e_3}, \pi^{e_3}, \pi^{e_1-e_2+2e_3}, \pi^{-e_2+2e_3}, \pi^{e_1+e_2+e_3}, \pi^{e_1+2e_3})
+ 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 )
\left\|\mathcal{U}\left(t\right)x_0\right\|^2 \leq M_2\sum^\infty_{n=1}\bigl|e^{\lambda_nt}\left<x_0,z_n\right> \bigr|^2 &\leq M_2e^{-2\varepsilon t}\sum^\infty_{n=1}\left|\left<x_0,z_n\right> \right|^2\leq\frac{M_2}{M_1}e^{-2\varepsilon t}\left\|x_0\right\|^2 ,
\sum_{n=k^{\ell_0}+1}^{N}f(n) \leq \sum_{\ell=\ell_0}^{\ell_1}k^{\ell+1}f(k^{\ell}) \leq k^{\ell_0+1}f(k^{\ell_0}) + k^2 \sum_{\ell=\ell_0}^{\ell_1-1}k^\ell f(k^{\ell+1}).
\begin{aligned}\|S^*(1-P_R)\widetilde{\mathcal{K}}^{j,k}v\|_{L_t^{\theta_{\widetilde{p}}}(L^{\widetilde{p},2}_{j,k})}\lesssim_{\widetilde{p},p,q}\|S^*(1-P_R)v\|_{L_t^{\theta_{\widetilde{p}}}(L^{\widetilde{p},2}_{j,k})}&\lesssim_{\widetilde{p},p,q}1/R\left\|\sum_{l}\mathcal{K}_{\mathcal{I}_l}v\right\|_{L_t^{\theta_q'}(L^{q',2}_D)}.\end{aligned}
\bar{\partial}(v_1\,v_2\,\cdots\,v_m) \ = \ \sum_{i=1}^m (-1)^{(|v_1|+\cdots+|v_i|)(|v_{i+1}|+\cdots+|v_m|)}(v_{i+1}\,...\,v_m\, v_1\,\cdots\,v_{i-1}) \otimes v_i.
\frac{d(\sigma)}{d\tau^{|\sigma|-1}}\le \frac{d(\sigma')}{d\tau^{j-1}}\le 2|F|^{1-t}\tau^{1-j}\le 2\gamma|F|^{1-t}f^{1-j}\le 2\gamma|F|^{1-t}f^{1-t-k}= 2\gamma\,,
\sum_{i=1}^{n+1}\sigma_i^2-\sum_{i=1}^{n+1}\sigma_i\sigma_{i+1}=2\sum_{i=1}^{n+1}\mu_i\sigma_i. \footnote{Conventionally, we set $\sigma_{\ell}=\sigma_i$ if $\ell\equiv i \ (\mathrm{mod}\n+1)$. Therefore, $\sigma_{n+2}=\sigma_1$.}
\| R_{21}\|_{L^\infty}\lesssim{}t^{-\frac54}\| \mathcal{N}_2 (U(-1/t)v_1, U(-1/t)v_2)\|_{H^1}\lesssim{}t^{-\frac54}\|U(-1/t)v_1\|_{H^1}^2 \|U(-1/t)v_2\|_{H^1}=t^{-\frac54}\|v_1\|_{H^1}^2 \|v_2\|_{H^1}
\mu_t=\sum_{i\geqslant 0}\mu_it^i,{\mu_{V}}_{t}=\sum_{i\geqslant 0}{\mu_{V}}_{i}t^i,l_t=\sum_{i\geqslant 0}l_it^i,r_t=\sum_{i\geqslant 0}r_it^i,T_t=\sum_{i\geqslant 0}T_it^i,
U ( t , \vec{x}) = \left( P e^{- i g \int_{0}^{t}d t^{\prime}\, A_{0}( t^{\prime}, \vec{x})}\right) \left( P e^{i g \int_{0}^{\beta}d t^{\prime}\, A_{0}( t^{\prime}, \vec{x})}\right)^{{\frac{t}{\beta}}}
V(r,u) = - (8 \pi^2 \alpha^{\prime})^{-1/2}\int_0^{\infty}dl ~ e^{-r^2 l/2\pi\alpha^{\prime}}{{l^{1/2}{\rm tanh}(u)}\over{{\rm Sin}(ul)}}\left [ 12 + 4{\rm Cos}(2ul) - 16{\rm Cos}(ul) \right ]\quad .
\|(\upsilon_N\ast|\phi_0|^2)\|_{L^2}\le \|\upsilon_N\|_{L^1}\|~|\phi_0|^2~\|_{L^2}= \|\upsilon_N\|_{L^1}\|\phi_0\|_{L^4}\le \|\upsilon_N\|_{L^1}\|\phi_0\|_{\mathfrak{h}^1_V},
\mathbf{A}^{\top^{2}}:=\left(\mathbf{A}^{\top}\right)^{\top},\ \mathbf{A}^{\top^{3}}:=\left(\mathbf{A}^{\top^{2}}\right)^{\top},\:\cdots,\:\mathbf{A}^{\top^{m}}:=\left(\mathbf{A}^{\top^{\left(m-1\right)}}\right)^{\top}=\mathbf{A}.
{\cal A}= \left( - \frac{3}{4}( \nabla^{2}X^{\mu}\nabla^{2}X^{\nu}G_{\mu \nu}- 4 g^{\alpha \beta}P_{\alpha \beta}) - \frac{1}{2}R_{( 2 )}- \frac{1}{6}g^{\alpha \gamma}g^{\beta \delta}W_{\alpha \beta \gamma \delta}\right) \phi - \frac{5}{6}\nabla^{2}X^{\mu}D_{\mu}\phi .
I \left(W_{k:K}; W_{k-1}|W_{1:k-2}, Z \right) =H\left(W_{k-1}|W_{1:k-2}, Z\right) -H\left(W_{k-1}|W_{1:k-2}, W_{k:K},Z\right) &=\left(L-Lr \right)-\left(L-Lr \right) &=0
\sup_{x \in B \times Y}|\hat{\eta}_i(x)|_{\hat{g}(x)}+ \sup_{x=(z,y) \in B \times Y}\Biggl( \sup_{x' \in B \times Y}\frac{|\hat\eta_i(x)-\mathbf{P}^{\hat{g}_{z,i}}_{x'x}(\hat\eta_i(x'))|_{\hat{g}(x)}}{d^{{g}_{i}}(x,x')^\alpha}\Biggr) \leq C.
G_{p o l a r}( x , y ; k ) = - 4 i \pi \left( S ( x ) \gamma^{5}\right) \otimes \left( S ( y ) \gamma^{5}\right) \frac{\sin \left[ k x / 2 \right] \sin \left[ k y / 2 \right]}{( k^{2}- e^{2}/ \pi + i \epsilon )}\; ,
(yS_0)^*(p_{-1}+y\bar c) = (yS_0)^*(\frac{E[y\bar c]}{E[y]}y) &= (\frac{y}{E[y]}E[y]S_0)^*(\frac{y}{E[y]}E[y\bar c]) &= \frac{y}{E[y]}(E[y]S_0)^*(E[y\bar c])
q^{\frac{( 1 - c_{j}) t_{j}- ( 1 + c_{i}) t_{i}}{2}}\, \tilde{\lambda}_{j}\cdot \sigma_{q}^{x}\gamma_{i}+ \, \, q^{2 H + \frac{( c_{j}- 1 ) t_{j}+ ( 1 + c_{i}) t_{i}}{2}}\,
\frac{p'(\eta_3)-p'(\eta_1+\eta_2+\eta_3)}{p'(\eta_1)-p'(\eta_3)}-\frac{p'(\eta_1)-p'(\eta_3)}{p'(\eta_3)-p'(\eta_1+\eta_2+\eta_3)}=:\frac{\mathcal{A}}{\mathcal{B}},
\begin{aligned}| \pi_x\circ \widetilde{F}(z^o_{n,k}, u^o_{n,k}) - \pi_x\circ \widetilde{F}_{w_{n,k}}(z^o_{n,k}, x^o_{n,k})| = | \pi_x\circ \widetilde{F}(z^o_{n,k}, u^o_{n,k}) - \pi_x\circ \widetilde{F}(z^o_{n,k}, x^o_{n,k})| <\sup|\frac{\partial \pi_x\circ \widetilde{F}}{\partial x}| |u^o_{n,k}-x^o_{n,k}|.\end{aligned}
\mathcal{E}_{\rm left}(x) =\exp\bigg[- \int_0^x \frac{(\alpha+\beta+2)y-(\beta-\alpha)}{2(1-y^2)}\,dy\bigg]=(1-x)^{(\alpha+1)/2}(1+x)^{(\beta+1)/2}\mathcal{E}_{\rm right}( x) ={\exp}\bigg[- \int_0^x -\frac{(\alpha+\beta)y-(\beta-\alpha)}{2(1-y^2)}\,dy\bigg]= (1-x)^{-\alpha/2}(1+x)^{-\beta/2}.
\int_{t-1}^{t}\left| \frac{\widetilde{B(t-\tau)}}{(t-\tau)}\right| ~d\tau\leq \int_{t-1}^{t}\sum_{m=0}^{n_{\tilde{r}}-1}\frac{1}{p^{m+1}}M(\lambda_{\tilde{r}},m+1) (t-\tau)^{m}e^{(t-\tau)\alpha}d \tau,
b ( \vec{\lambda}, m , l , w ) = \sum_{\vec{t}, \; \sum t_{i}= 0}\, \det_{\frac{w}{2}\times \frac{w}{2}}\, \left| \Biggl ({m \atop ( m / 2 ) + \frac{l + w}{2}\, t_{i}+ \lambda_{j}- \lambda_{i}}\Biggr ) \right| ,
6\widetilde{\textbf{c}}_{q,p-1}^{(n)}-3(n+1-q)\widetilde{\textbf{c}}_{q,p}^{(n)}-2\textbf{c}_{q,p-1}^{(n)}+(n+1-q)\textbf{c}_{q,p}^{(n)}= 5\widetilde{\textbf{c}}_{q,p-1}^{(n)}+3 \frac{\binom{n}{q-1}}{(n!)^2}P_{p+1}(q) &-\textbf{c}_{q,p-1}^{(n)}+(-1)^p \frac{\binom{n}{q-1}}{(n!)^2}Q_{p+1}(q).
\bigg| \sum_{j=1}^m \sum_{l=k_{j-1}}^{k_j-1}\int_{\mathbb{R}^4}F(y,x') G(x,x')F(y,y')G(x,y') begin{align*}-1ex]{\vartheta}_{2^{k_j}}(x'-x-y)\psi_{2^l}(y'-x-y) \,dx dy dx' dy' \bigg| \lesssim 1.
(\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}+\frac{1}{r_{4}})^{2}=2(\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}+\frac{1}{r_{4}^{2}}).
\phi_1 := \phi_{4C}(\tau, z) = 2 \zeta^{\pm 1}- 4 + (-4 \zeta^{\pm 2}+ 8 \zeta^{\pm 1}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}+ 14 \zeta^{\pm 1}- 16) q^2 + O(q^3);
Q_4(n) < 2 \cdot 3.44 e^{\pi \sqrt{\frac{2n}{3}}}+ 3.44^2 \sum_{k=1}^{n-1}e^{\pi(\sqrt{\frac{2n}{3}}+\sqrt{\frac{2(n-k)}{3}})}&< 12.08 \cdot n \cdot e^{2\pi \sqrt{\frac{n}{3}}}.
W_{82,1}=1+560x^{14}+60724x^{16}+233545x^{18}+\cdots, W_{82,2}=1+(3280+2\alpha)x^{14}+(36244-2\alpha+128\beta)x^{16}&\quad+(506153-26\alpha-896\beta)x^{18}+\cdots, W_{82,3}=1+(3280+2\alpha)x^{14}+(36244-2\alpha+128\beta)x^{16}&\quad+(514345-26\alpha-896\beta)x^{18}+\cdots,
\tau(1,2)=(\begin{matrix}-1&0\\ 0&1\end{matrix}),\tau(1,3)=(\begin{matrix}\frac{1}{2}&\frac{3}{2}\\ \frac{1}{2}&-\frac{1}{2}\end{matrix}),\tau(2,3)=(\begin{matrix}\frac{1}{2}&-\frac{3}{2}\\ -\frac{1}{2}&-\frac{1}{2}\end{matrix}).
\frac{1}{2}\sum_{s=1}^{2n-1}\frac{q^{s(2n-s)+2n}}{y^{s}z^{2n-s}}+\frac{q^{n^2+2n}}{2y^nz^n}\cdot \frac{J_2^8}{J_1^2J_4^2}\cdot \frac{j(yz;q^2)}{j(qy;q^2)j(-y;q^2)j(-z;q^2)j(qz;q^2)}
S^{\pm}=\left( \begin{array}{lll}\partial_{1}X^{0}... \partial_{1}X^{p}\partial_{2}X^{0}... \partial_{2}X^{p}... ... ... G^{0\nu}(P_{\nu}\pm Tn_{\nu}) ... G^{p\nu}(P_{\nu}\pm Tn_{\nu})\end{array}\right) \ ,
\P\big\{Z_{1}^{x,y}Z_{1}^{y,z}=0 \big\}&~\geq \P\big\{h_{1}((x,0))=h_{1}((y,0))=(x+\lceil\tfrac{y-x}{2}\rceil,\lceil\tfrac{y-x}{2}\rceil),h_{1}((z,0))=(z,\lceil\tfrac{y-x}{2}\rceil) \big\}&~\geq (1-p)^{3(\lceil\tfrac{y-x}{2}\rceil +1)^{2}-5}p^{2}\geq (1-p)^{3(m_{0}+1)^{2}-5}p^{2}.
(\frac{k_{x}^{2}}{n_{o}^{2}}+\frac{k_{y}^{2}}{n_{o}^{2}}+\frac{k_{z}^{2}}{n_{o}^{2}}-\frac{\omega^{2}}{c^{2}})(\frac{k_{x}^{2}}{n_{e}^{2}}+\frac{k_{y}^{2}}{n_{e}^{2}}+\frac{k_{z}^{2}}{n_{o}^{2}}-\frac{\omega^{2}}{c^{2}})=0
\phi_1(0,s) =\phi_2(s,0)=0\quad \phi_1(1,s) =\phi_2(s,1)=1\quad \partial_x \phi_1(x,y) \geq 1-C|K|^{1/2}\tau(x,y)\in [0,1]\times \{0,1\}, \partial_y \phi_2(x,y) \geq 1-C|K|^{1/2}\tau (x,y)\in \{0,1\}\times [0,1].
{\cal{L}}_{\psi}= \frac{1}{2}\, \overline{{{\Psi}}}^{a}\left[ \sqrt{- \left( D^{2}\right)^{a c}+ \left( F_{\rho \sigma}^{b}S^{\rho \sigma}\right) \left( t^{b}\right)^{a c}}\right] \Psi^{c}
{\cal P}_{3}( y ) \| = \frac{1}{r^{n}}\int_{- \infty}^{u}d{\bar{u}}\log \Bigl ( m \frac{u -{\bar{u}}}{2}\Bigr ) \frac{d}{d{\bar{u}}}A ({\bar{u}}) + O \Bigl ( \frac{1}{r^{n + 1}}\Bigr ) \quad .
\sum_{b\in V_n}\epsilon_p^{\langle b,y\rangle}\widehat{f}(b) = \sum_{b\in V_n}\epsilon_p^{\langle b,y\rangle}\sum_{x\in V_n}\epsilon_p^{f(x)-\langle b,x\rangle}=\sum_{x\in V_n}\epsilon_p^{f(x)}\sum_{b\in V_n}\epsilon_p^{\langle b,(y-x)\rangle}= p^n\epsilon_p^{f(y)},
|k_b|\leq|k_bj_b|<\sum_{j\in\mathbb{Z}_*}\frac{|l_jj|}{n-\frac12}+ \frac{|k|}{100n\sum_{b=1}^n|j_b|}\leq\frac{|k|}{2n-1}+\frac{|k|}{100n\sum_{b=1}^n|j_b|}.
\sum_{n=1}^\infty \frac{1}{(n^2+p^2)^k}=\frac{\sqrt{\pi}\Gamma(k-\frac{1}{2})}{2\Gamma(k)p^{2k-1}}-\frac{1}{2p^{2k}}+\left( \frac{\pi}{p}\right)^k\frac{e^{-2\pi p}}{\Gamma(k)},
\forall\:0\le t<n,\quad\left(\mathbf{1}_{n\times\cdots\times n}-\boldsymbol{\Delta}\right)\circ\mbox{Prod}_{\boldsymbol{\Delta}^{(t)}}\left(\mathbf{X}^{(1)},\,\mathbf{X}^{(2)},\,\cdots,\mathbf{X}^{(m)}\right)=
v(s-(\epsilon-ub)') = v(s-\epsilon-ub'-u'b) &= v(u'b) &= v(u'b'(b^{\dagger})^{-1}) &= v(u')-\psi\textstyle\int\gamma + \gamma &> s^2\gamma-s\gamma+\gamma &= -\textstyle\int s\gamma + \gamma.
\widetilde{Z}( R ( S ) , \langle \cdot , \cdot \rangle ) = \det ( \widetilde{T}^{2})^{- 1 / 4}\prod_{k = 1}^{N}\det ( \widetilde{T}_{k}^{*}\widetilde{T}_{k})^{\frac{1}{2}( - 1 )^{k - 1}}
\bigg| \sum_{j=1}^m \int_{\mathbb{R}^4}F(x+u,y)G(x,y+u)F(x+v,y)G(x,y+v) begin{align*}-1.5ex]\vartheta_{2^{k_j}}(u)(\varphi_{2^{k_j}}-\varphi_{2^{k_{j-1}}})(v) \,dxdydudv \bigg| \lesssim_\lambda 1.
{\rm Tutte}\big(K_{n_1,n_2,n_3}^{\ell_1,\ell_2,\ell_3}, 1,1\big) = N (\ell_{12}n_2\!+\ell_{13}n_3)^{n_{1}-1}(\ell_{12}n_1\!+\ell_{13}n_3)^{n_{2}-1}(\ell_{13}n_1\!+ \ell_{23}n_2)^{n_{3}-1}.
\langle r^{2}(t)\rangle=v^{2}(0)\tau^{2}(1-e^{-t/\tau})^{2}-\frac{3k_{B}T}{m}\tau^{2}(1-e^{-t/\tau})(3-e^{-t/\tau})+\frac{6k_{B}T}{m}\tau t.
g_{\mu\nu}=[\begin{matrix}(1-\frac{2GM}{rc^{2}})&0&0&0\\ 0&-(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\ 0&0&-r^{2}&0\\ 0&0&0&-r^{2}sin^{2}\theta\end{matrix}],
D_{m ~ e}^{~ ~ n ~ f}= \biggl ( R - 2 \Lambda \biggr ) \biggl ( \delta_{m}^{n}\delta_{e}^{f}- \delta_{m}^{f}\delta_{e}^{n}\biggr ) + 2 R_{~ ~ m e}^{f n}+ \frac{1}{2}\biggl ( R_{m}^{~ f}\delta_{e}^{n}- R_{~ m}^{n}\delta_{e}^{f}+ R_{e}^{~ n}\delta_{m}^{f}- R_{~ e}^{f}\delta_{m}^{n}\biggr )
S = \frac{1}{\beta^{2}}\int d^{4}x d^{2}\theta d^{2}\bar{\theta}t r \left[ - \frac{1}{2}( e^{- V}D^{\alpha}e^{V}) \bar{D}^{2}( e^{- V}D_{\alpha}e^{V}) - e^{- V}\bar{\Sigma}e^{V}\Sigma \right] .
\frac{1}{( k^{2}+ m^{2}) ( k^{2}+ \alpha m^{2})}~ = ~ \frac{1}{( 1 - \alpha ) m^{2}}\left[ \frac{1}{( k^{2}+ \alpha m^{2})}~ - ~ \frac{1}{( k^{2}+ m^{2})}\right]
2W^{+}_{ijkl}W^{+}_{ijpq,t}= \,\, 2\lambda\, \omega_{kl}\Big(\lambda_{t}\omega_{pq}+(\lambda-\mu)c_{t}\eta_{pq}+(\nu-\lambda)b_{t}\theta_{pq}\Big) &+2\mu\,\eta_{kl}\Big(\mu_{t}\eta_{pq}+(\lambda-\mu)c_{t}\omega_{pq}+(\mu-\nu)a_{t}\theta_{pq}\Big) &+2\nu\,\theta_{kl}\Big(\nu_{t}\theta_{pq}+(\nu-\lambda)b_{t}\omega_{pq}+(\mu-\nu)a_{t}\eta_{pq}\Big)\,.
V^{2IBP,\theta}_{n,k}(\gamma) = \frac{1}{k!}\bigg(\frac{\gamma}{\left(\theta+1\right)_{n-1\uparrow}}\bigg)^{k}\exp\bigg(-\sum_{i=1}^{n}\gamma\frac{\left(\theta\right)_{i-1\uparrow}}{\left(1+\theta\right)_{i-1\uparrow}}\bigg) & = \frac{1}{k!}\bigg(\frac{\gamma}{(\theta+1)_{n-1\uparrow}}\bigg)^{k}\exp\bigg(-\gamma\sum_{i=1}^{n}\frac{\theta}{\theta+i-1}\bigg),
{{d^2t}\over{d\tilde{\lambda}^2}}=\sqrt{h}\left[{h^{\prime}\over 2}\left({{dt}\over{d\tilde{\lambda}}}\right)^2-y\sum_j\left({{dx^j}\over{d\tilde{\lambda}}}\right)^2-{h^{\prime}\over{h}}\right]{{dt}\over{d\tilde{\lambda}}},
S_{\mu_{\beta}}\Phi(\varphi):=\frac{\langle\!\langle\Phi,e^{\langle\cdot,\varphi\rangle}\rangle\!\rangle_{\mu_{\beta}}}{\mathbb{E}\big(e^{\langle\cdot,\varphi\rangle}\big)}=\frac{1}{E_{\beta}(\frac{1}{2}\langle\varphi,\varphi\rangle)}\langle\!\langle\Phi,e^{\langle\cdot,\varphi\rangle}\rangle\!\rangle_{\mu_{\beta}}.
A_{a h}^{- 3 / 2}\, \left( \begin{array}{c c}{{\cos \left( \frac{\sqrt{7}}{2}\ln ( A_{a h}) \right)}}&{{\sin \left( \frac{\sqrt{7}}{2}\ln ( A_{a h}) \right)}}\\{{- \sin \left( \frac{\sqrt{7}}{2}\ln ( A_{a h}) \right)}}&{{\cos \left( \frac{\sqrt{7}}{2}\ln ( A_{a h}) \right)}}\\ \end{array}\right) \,{\cal A}^{- 1}\, \left( \begin{array}{c}{{a}}\\{{b}}\\ \end{array}\right) \, .
\varphi_{M}(n)= \sum\limits_{0\leq q \leq n}\dim_{\mathbb{C}}(P_{q}[x_{1},...,x_{k}]) = \sum\limits_{0\leq q\leq n}\binom{k+q-1}{q}= \binom{k+n}{n}= \frac{n^k}{k!}+O(n^{k-1}).
x\partial_x c(x,\lambda ) = -\frac{d}{d\lambda}c(x,\lambda ) = \kappa x^4 \lim_{y\rightarrow\infty}\lim_{y'\rightarrow\infty}e^{4\phi (y')}\partial_{y'}\int\limits_{-\lambda}^{y}dy_1\int\limits_{y_1}^{\infty}dy_2\int\limits_{-\lambda}^{y_2}dy_3 \; F'_{\lambda}\,,
\tilde{f}^{(1)}_{ij}\tilde{g}^+_j= \sum_{s\geq 1}\sum_{j_1<\ldots<j_s=j}(-1)^{s-1}\left([z^{-1}]T^+_{ij_1}(z)\right) \tilde{e}^{(0)}_{j_1 j_2}\cdots \tilde{e}^{(0)}_{j_{s-1}j_s}.
\mathbb{E}\left[_T(\widetilde{\mu}^{j,N},\widehat{\mu}^{1,N})\right]=\mathbb{E}\left[ \sum_{t=0}^T(\widetilde{\mu}^{j,N}_t,\widehat{\mu}^{1,N}_t)\right]= \sum_{t=0}^T\mathbb{E}\left[ (\widetilde{\mu}^{j,N}_t,\widehat{\mu}^{1,N}_t)\right],
t_1 =\min\!\big\{3,2,3+2-0\big\}=2, t_2 =\min\!\big\{4,3,4+3-\max\{3+2,2+2\}\big\}=2, t_3 =\min\!\big\{5,8,5+8-\max\{3+4,2+4,4+2,3+2\}\big\}=5, t_4 =\min\!\big\{7,9,7+9-\max\{3+6,2+6,4+4,3+4,5+2,8+2\}\big\}=6.
B = \begin{pmatrix}1 0 0 1 0 0 0 1 -1 \end{pmatrix}, B^* = \begin{pmatrix}1 \frac{(s-1)(t-1)}{s+t}0 0 0 \frac{(s-1)(t-1)}{s+t}0 0 -1 \end{pmatrix}.
\tilde K_{a,b}(z;m+2) = a \tilde K_{a+1,b}(z;m) + b\tilde K_{a,b+1}(z;m) \tilde H_{a,b,c}(u,v;m+2) = a\tilde H_{a+1,b,c}(u,v;m) +b\tilde H_{a,b+1,c}(u,v;m) + c\tilde H_{a,b,c+1}(u,v;m).
\sum_{n=1}^{\infty}~e^{-M\sqrt{n^2\beta^2 + \tilde p_{2}^{2}}}=-\frac{1}{2}~e^{-M\tilde p_2}+ \sqrt{2\pi}~[~\frac{1}{2}~F(0) + \sum_{n=1}^{\infty}F(2\pi n)] ~~~,
P=\begin{array}{r}\begin{matrix}R N S\end{matrix}\begin{matrix}R N S\end{matrix}\begin{bmatrix}\frac{1}{2}\frac{1}{4}\frac{1}{4}\frac{1}{2}0 \frac{1}{2}\frac{1}{4}\frac{1}{4}\frac{1}{2}\end{bmatrix}\ \end{array}
J^i_{k|l}y^k y^l = J^s_k \hat{\mathbb{G}}^i_{sl}y^ly^k - J^i_s \hat{\mathbb{G}}^s_{kl}y^ly^k &=J^s_k \hat{\mathbb{G}}^i_s y^k - J^i_s \hat{\mathbb{G}}^s_k y^k = \hat{\mathbb{G}}^i_s u^s - 2 J^i_s \hat{\mathbb{G}}^s.
\sum_{j=0}^{n}\left( -1\right)^{j}Y_{j}\left( \lambda \right) Y_{n-j}\left(\lambda \right) =\frac{4\left( -1\right)^{n}n!}{\left( \lambda -1\right)^{2}}\left( \frac{\lambda^{2}}{\lambda -1}\right)^{n}\sum_{j=0}^{n}\left(-1\right)^{j}\frac{1}{\left( \begin{array}{c}n j\end{array}\right)}
\varphi(s,t) =U_{\varphi_0}(s,t)+H_0^\varphi (s,t) &=U_{\varphi_0}(s,t)+\int^t_0\int_{-\infty}^{\infty}K(s-\xi,t-\tau)h(\varphi)(\xi,\tau)d\xi d\tau
C\Big[ i\pi\epsilon (p_0)e^{-{m^2\over2\mu^2}}\Bigg(\delta(p^2)+\theta(p^2){m\over2\mu^2}{1\over\sqrt{p^2}}I_1\Big({m\sqrt{p^2}\over\mu^2}\Big)\Bigg)e^{-p^2/2\mu^2}\Big].
p^1=\left[\begin{array}{ccccc}1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 \end{array}\right],p^2=\left[\begin{array}{ccccc}1 1 1 1 2 2 2 2 2 1 3 3 3 3 3 \end{array}\right],p^3=\left[\begin{array}{ccccc}1 1 1 1 3 2 2 2 2 2 3 3 3 3 1 \end{array}\right],
\Big|\int_\Omega (\mu_1-\mu_2) e^{i\tilde\xi_+\cdot x-\eta_+}a_1\rho\cdot \Big(-a_2^*\rho^*-\frac{1}{2}b_2^*\overline{\rho}^*\Big)\,dx\Big| = o(1), \Big|\int_\Omega (\mu_1-\mu_2) e^{i\tilde\xi_-\cdot x-\eta_-}a_1^*\rho^*\cdot \Big(-a_2\rho-\frac{1}{2}b_2\overline{\rho}\Big)\,dx\Big| = o(1)
\begin{cases}A(0)-\sqrt{\varepsilon}\,\frac{\partial}{\partial x}A(0)=F_A(0),A(1)+\sqrt{\varepsilon}\,\frac{\partial}{\partial x}A(1)=0, begin{align*}5pt]B(0)-\sqrt{\varepsilon}\,\frac{\partial}{\partial x}B(0)=0,\B(1)+\sqrt{\varepsilon}\,\frac{\partial}{\partial x}B(1)=F_B(1).\end{cases}
\Delta\bar{C}_t= \big(\mu_1(\beta-1)-\mu_0(\alpha-1)\big)+H(\alpha)-H(\beta)+\log\Big(\frac{1+2^{\mu_1+\Delta\bar{C}_{t-1}}}{1+2^{\mu_0+\Delta{C}_{t-1}}}\Big),~\Delta\bar{C}_{-1}=0,~t\in\mathbb{N}_0^n.
\frac{\partial}{\partial y_1^{n_1}\partial y_2^{n_2}\dots \partial y_j^{n_j}}\, \frac{Y(y_1,y_2,\dots,y_j,\overbrace{z,\dots,z}^{m-j},y_{m+1},\dots,y_c)}{n_1!n_2!\dots n_j!\,z^{m-j}}\Big|_{y_1=\dots=y_j=0}.
\partial_x (u\times (u\times g(u)))\cdot \partial_x u =u\times (\partial_x u\times g(u))\cdot \partial_x u+u\times (u\times A\partial_x u)\cdot \partial_x u &= [\partial_x u (u\cdot g(u))-g(u)(\partial_x u\cdot u)]\cdot \partial_x u+u\times (u\times A\partial_x u)\cdot \partial_x u &=|\partial_x u|^2(u\cdot g(u))-A\partial_x u\cdot \partial_x u\, ,
P^+A^k+SP^+B^k=O_{L^2}(\varepsilon)+P^+\Big(\frac{\zeta_{1,\varepsilon}f^k}{a^*}\Big), &\overline{P^+B^k}-S\overline{P^+A^k}=O_{L^2}(\varepsilon)+\overline{P^-\Big(\frac{\zeta_{1,\varepsilon}f^k}{a^*}\Big)}.
\Omega(x)=\dfrac{1}{2\pi}\left(\,\int\limits_{-\pi}^\pi\left(\dfrac{\cos(x\xi)}{A(\xi)}-\dfrac{1}{\xi^2}\right)d\xi -\dfrac{2}{\pi}\right) - 2x \int\limits_0^1\dfrac{h(\rho)-h(0)}{\rho^2}\,d\rho - 2x\left(\int\limits_1^\infty\dfrac{h(\rho)}{\rho^2}\,d\rho-h(0)\right).
\frac{\partial{\cal V}}{\partial \varphi}={\textstyle\frac{2}{n}}\frac{1}{{\rm Tr}({\cal M}^2)}\,{\cal M}_{mn}\frac{\partial{\cal V}}{\partial{\cal M}_{mn}}\, ,\,\,\,\,\Rightarrow{\cal M}^{mn}{\cal M}_{pq}\frac{\partial{\cal V}}{\partial{\cal M}_{pq}}-{\rm Tr}({\cal M}^2) \frac{\partial{\cal V}}{\partial{\cal M}_{mn}}= 0\, .
(I-xK_u^2)^{-1}(h)-(I-xH_u^2)^{-1}(h) =(I-xK_u^2)^{-1}\left[ (I-xH_u^2)-(I-xK_u^2) \right] (I-xH_u^2)^{-1}(h) &= -x (h\vert (I-xH_u^2)^{-1}(u)) (I-xK_u^2)^{-1}(u).
\left(\begin{matrix}u_{n}(E,\theta) u_{n-1}(E,\theta) \end{matrix}\right)=B(E;\theta+n\alpha)\begin{pmatrix}\lambda(E)^n * 0 \lambda(E)^{-n}\end{pmatrix}B(E;\theta)^{-1}\left(\begin{matrix}u_{0}(E,\theta) u_{-1}(E,\theta) \end{matrix}\right).
\mathbf{Pr}\left(\hat{x}_{i,min,\gamma}> x_i + \epsilon\right)\geq \left[1-\gamma^*\left(1-\gamma^*\right)^{K}\right]^M = \left[1-\frac{1}{K+1}\left(1-\frac{1}{K+1}\right)^K\right]^M
V_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}= - i g f^{a_{1}a_{2}a_{3}}\left[ \eta_{\mu_{1}\mu_{2}}( k_{1}- k_{2})_{\mu_{3}}+ \eta_{\mu_{2}\mu_{3}}( k_{2}- k_{3})_{\mu_{1}}+ \eta_{\mu_{3}\mu_{1}}( k_{3}- k_{1})_{\mu_{2}}\right]
Z(C/\mathbb{F}_q,T)=1 + (N_1)T + \bigg(\frac{N_2}{2}+\frac{N_1^2}{2}\bigg)T^2 + \bigg(\frac{N_3}{3}+\frac{N_1N_2}{2}+\frac{N_1^3}{6}\bigg)T^3 + \ldots.
\f_2 = - e^2 \int \frac{d^4k}{(2\pi)^4}\frac{d^4q}{(2\pi)^4}\frac{(k+p+q)^2+q^2-k^2-p^2}{k^2\Big(1 + (-1)^n k^{2n}/\Lambda^{2n}\Big)(k+q)^2 (k+p+q)^2 q^2 (q+p)^2}
(F^{\bullet, Rla})^{Rla}= \varinjlim_{h,h'\to \infty}(F^{RG_{(h')}-an})^{RG_{(h)}-an}= \varinjlim_{h\to \infty}(F^{RG_{(h)}-an})^{RG_{(h)}-an}= \varinjlim_{h\to \infty}F^{RG_{(h)}-an}= F^{Rla},
G(q) :=\sum_{n=0}^\infty\frac{q^{n^2}}{(1-q)\cdots(1-q^n)}=\prod_{n=0}^\infty\frac{1}{(1-q^{5n+1})(1-q^{5n+4})}H(q) :=\sum_{n=0}^\infty\frac{q^{n^2+n}}{(1-q)\cdots(1-q^n)}=\prod_{n=0}^\infty\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}
|f_{n_k}(\kappa_n(\xi)) - f_{n_k}(\kappa_{n_k}(\xi))| \ \leq \ \sum_{m=1}^{n_k-n}|f_{n_k}(\kappa_{n+m-1}(\xi))-f_{n_k}(\kappa_{n+m}(\xi))| & \leq \C\Big (\frac{\lambda}{r^\alpha}\Big)^{n/2}:= \varepsilon(n).
A_{0}= ( 4 \pi t )^{d / 2}K ( t ) \, \sum_{i}\mathrm{t r}\left( T_{i}\right) \, e^{- m_{i}^{2}t}\left( \delta_{i 1}\mathrm{t r}P \, + \delta_{i 2}+ \delta_{i 3}+ \delta_{i 1}\xi^{d / 2}\right)
\pi_{[a,i]}\cdot \pi_{[i+1,b]}= \binom{b-a+1}{i-a+1}\pi_{[a-1,b-1]}+\binom{b-a}{i-a}\pi_{[a,b-1]}y_i +\binom{b-a}{i-a+1}\pi_{[a,b-1]}y_{b}.
1=q_{r}^{2}+q_{i}^{2}+q_{j}^{2}+q_{k}^{2}=q_{r}^{2}(1+\frac{q_{i}^{2}}{q_{r}^{2}}+\frac{q_{j}^{2}}{q_{r}^{2}}+\frac{q_{k}^{2}}{q_{r}^{2}})=q_{r}^{2}(1+g_{i}^{2}+g_{j}^{2}+g_{k}^{2})
\sum_{\stackrel{m_{1}, m_{2}, \ldots , m_{\nu - 2}= 0}{m_{i}\equiv V_{i , r}^{\pm}( m o d 2 )}}^{\infty}\frac{q^{\frac{\vec{m}^{t}C \vec{m}}{4}}}{( q , q )_{m_{\nu - 2}}}\prod_{i = 1}^{\nu - 3}\left[ \begin{array}{c}{{\frac{1}{2}( K_{\nu - 2}\cdot \vec{m})_{i}+ \frac{1}{2}\delta_{i , b^{\pm}}}}\\{{m_{i}}}\\ \end{array}\right]_{q}=
T_{12}= \left(\begin{array}{rrr}1 1 0 0 1 0 0 0 1\end{array}\right),T_{13}= \left(\begin{array}{rrr}1 0 1 0 1 0 0 0 1\end{array}\right),T_{21}= \left(\begin{array}{rrr}1 0 0 1 1 0 0 0 1\end{array}\right),
\overrightarrow{C D}> = \vert \overrightarrow{A B}\vert = \frac{\overrightarrow{A B}\cdot \overrightarrow{C D}}{\vert \overrightarrow{A B}/ / \overrightarrow{C B}\vert}= \frac{\overrightarrow{A B}\cdot \overrightarrow{C D}}{\vert \overrightarrow{C D}\vert}= \frac{1 5}{5 \sqrt{2}}
\mbox{Det}\left [{\cal D}\right ]=\left ( p_0^2 - p_1^2 - p_2^2 - p_3^2 -m^2 \right)^2 \frac{( p_0^2 -p_1^2 -p_2^2 -p_3^2 +3m^2 +4mp_0 )^2}{16m^4 (p_0 +m)^4}\quad.
{\cal R}\equiv \left( \begin{array}{c c c c}{{R^{( 1 )}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{R^{( 3 )}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{R^{( 2 ) \; - 1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{{\cal P}R^{( 4 )}{\cal P}}}\\ \end{array}\right) \quad . \qquad
\bar{\rho}_S(z_S) = \frac{e^{-\sum_{i=1}^k\lambda_iF_i(z_S) -\frac{1}{k_BT}H_S(z_S)}}{\mathrm{Tr}\left[e^{-\sum_{i=1}^k\lambda_iF_i(z_S) -\frac{1}{k_BT}H_S(z_S)}\right]},
( \mathcal Q_0 +\ell R_b(\ell, \omega) )^{-1}= ( I + \ell \mathcal Q_0^{-1}R_b(\ell, \omega))^{-1}\mathcal Q_0^{-1}, &= \sum_{k=0}^\infty \ell^k \bigr( - \mathcal Q_0^{-1}R_b(\ell, \omega) \bigr)^{k}\mathcal Q_0^{-1}= \mathcal Q_0^{-1}+ \mathcal O(\ell) ,
Y_\alpha \bullet (\sum_v p_v f_v) =\sum_v \tfrac{p_v}{v(x_{-\alpha})}f_v+\sum_v \tfrac{p_v}{vs_\alpha(x_\alpha)}f_{vs_\alpha}=\sum_v (\tfrac{p_v}{x_{-v(\alpha)}}+\tfrac{p_{vs_\alpha}}{x_{v(\alpha)}})f_v =\sum_v (\kappa_{v(\alpha)}p_v+\Delta_{v(\alpha)}(p_v))f_v,
\mathbb{E}\left[e^{\alpha N_{s,t}^{(n)}}\right] =\mathbb{E}\left[e^{\alpha\xi_{s,t}}\right]\mathbb{E}\left[\exp\left(\alpha\max_{1\leq i\leq n}\left(\omega_{t_{i}}-\omega_{s}\right)\right)\right] \leq\exp\left(\frac{\alpha^{2}}{2}\gamma_{H}(t-s)^{2H}\right)\mathbb{E}\left[\exp\left(\alpha\sup_{s\leq u\leq t}\left(\omega_{u}-\omega_{s}\right)\right)\right].