harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{p}\|(u_k,v_k)\|^p-\frac{1}{q}\int_\Omega(\lambda|u_k|^q+\mu|v_k|^q)dx-\frac{2}{\alpha+\beta}\int_\Omega|u_k|^\alpha|v_k|^\beta dx = c+o_k(1), \|(u_k,v_k)\|^p- \int_\Omega(\lambda|u_k|^q+\mu|v_k|^q)dx-2\int_\Omega|u_k|^\alpha|v_k|^\beta dx = o_k(\|(u_k,v_k)\|),
A^{\infty}_{dr,2}f(\eta )=f(\eta )\int_{{\Bbb R}^d\times [0,1]\times [0,\infty )}\Bigg(\frac{\sum_{S\subset\eta^g_{y,w}}H(g,\hat{g},S,y,\zeta ,w)}{\prod_{(x,\kappa ,u)\in\eta^g_{y,w}}g(x,\kappa ,u)}-1\Bigg)dy\nu^1(w,d\zeta )\nu^2(dw),
\psi^{(n)}(z)=\frac{d^{n+1}}{dz^{n+1}}ln\Gamma(z)=-\gamma\delta_{n0}-\frac{(-1)^{n}n!}{z^{n+1}}+\sum_{k=1}^{\infty}(\frac{1}{k}\delta_{n0}-\frac{(-1)^{n}n!}{(k+z)^{n+1}})
\{Q_{( \alpha )}, Q_{( \beta )}\}= g [ \mathcal{C}\Gamma_{s}( \hat{M}^{\hat{a}\hat{b}}) ]_{\alpha \beta}\hat{M}_{\hat{a}\hat{b}}= - i ( \mathcal{C}\gamma^{a})_{\alpha \beta}P_{a}-{\textstyle \frac{g}{2}}( \mathcal{C}\gamma^{a b})_{\alpha \beta}M_{a b}\, ,
\lfloor x\rfloor-\sum_{i}\lfloor\frac{x}{p_{i}}\rfloor+\sum_{i<j}\lfloor\frac{x}{p_{i}p_{j}}\rfloor-\sum_{i<j<k}\lfloor\frac{x}{p_{i}p_{j}p_{k}}\rfloor+\cdot\cdot\cdot
\mathcal{B}^-_I = \left\{\begin{aligned}\left(\tau,v_{s+k+1},\omega_{k+1}\right)\in\mathcal{A}^- \textnormal{such that}& \inf_{i\in \left\{1,\dots,s,s+1,\dots,s+k\right\}\backslash \left\{i_{k+1}\right\}}\left|\left(x_{i_{k+1}}^\prime - x_i^\prime\right) - \tau \left(v_{i_{k+1}}^\prime - v_i^\prime\right)\right| \leq y \end{aligned}\right\}
\dim_{\rm LB}C = \liminf_{n\in\mathbb{N}}\frac{n\log 2}{-\sum_{i=1}^{n}\log\lambda_{i}}\qquad\dim_{\rm B}C = \limsup_{n\in\mathbb{N}}\frac{n\log 2}{-\sum_{i=1}^{n}\log\lambda_{i}}.
R^s_1+R^s_2 \leq \left[\underbrace{\log\bigg(1+\frac{\eta_1 p_1|h_{11}|^{2}}{\sigma^{2}_1+\eta_1(\varrho^{2}_1+p_2|h_{12}|^{2})}\bigg)}_{R_1}+ \underbrace{\log\bigg(1+\frac{\eta_2 p_2|h_{22}|^{2}}{\sigma^{2}_2+\eta_2(\varrho^{2}_2+p_1|h_{21}|^{2})}\bigg)}_{R_2}-\sum_{k=1}^{2}R_{_k}\right]^{+},
\beta_k^2(0,\varphi_0,\lambda_0)=\frac1{\sqrt{\pi}}\lim_{r\to 0^+}r^{-k/2}\int_0^{2\pi}\varphi_0(r(\cos t,\sin t)) e^{-i\frac t2}\sin\big(\tfrac k2 t\big)\,dt.
\begin{aligned}\sigma: V \longrightarrow V_1, (x_0, x_1, x_1^\prime, x_2) \longmapsto x_1 - x_1^\prime, \end{aligned}\qquad\qquad \begin{aligned}\delta: V_1 \longrightarrow V, x \longmapsto (0, x, x, 0).\end{aligned}
\begin{array}{r c l}{{I_{1}( \rho^{2}, m )}}&{{=}}&{{\displaystyle \mu^{4 - n}\int \frac{d^{n}k}{( 2 \pi )^{n}}( k^{2}+ m^{2})^{- 2}\cos ( k \cdot \rho ) ,}}\\ \end{array}
\nabla_{j}= \left( \begin{array}{c c}{{i \theta_{j k}^{- 1}\hat{x}^{k}}}&{{0}}\\{{0}}&{{\partial_{j}}}\\ \end{array}\right) + i \left( \begin{array}{c c}{{( D_{j})_{i}^{l}}}&{{0}}\\{{0}}&{{( \hat{D}_{j})_{\beta}^{\alpha}}}\\ \end{array}\right)
\Delta_{1 2}^{i j}= - [ ( \pi . \Lambda ) g^{i j}- \pi^{i}\Lambda^{0 j}+{\frac{e}{m}}\epsilon^{j b c}F_{~ b}^{i}S_{c}] = - \Delta_{2 1}^{j i}~ ~ ; ~ ~ \pi . \Lambda = \pi_{a}\Lambda^{0 a}
\nabla_{X_2^v}Y_2^{v}=(\nabla{\hskip -0.2cm^{^{^{2}}}}_{X_2}Y_2)^v -\frac{f_1^hg_2(X_2,Y_2)^v}{1+c^2(f_2^v)^2b_1^h}(gradf_1)^h
\|b\|_{\widetilde{L}^{\infty}_t({B}^s_{2,2})}\leq \|b_0\|_{{B}^s_{2,2}}+C_3\|u\|_{\widetilde{L}^{1}_t(B^{s+1}_{2,2})}\|b\|_{\widetilde{L}^{\infty}_t(B^s_{2,2})}.
\sum_{i=1}^n \|\mu(|x_i|^q)\|_{E^{(1/q)}}=\sum_{i=1}^n \|\mu^q(x_i)\|_{E^{(1/q)}}\le \left\|\sum_{i=1}^n \mu^q(x_i)\right\|_{E^{(1/q)}}\le \left\| \mu^{q/p}\left(\sum_{i=1}^n |x_i|^p\right)\right\|_{E^{(1/q)}}= \left\| \mu \left(\sum_{i=1}^n |x_i|^p\right)^{q/p}\right\|_{E^{(1/q)}}.
D_{\mu \nu}^{p r o p}= \left[ - g_{\mu \nu}^{\perp}+ \frac{n_{\mu}\partial_{\nu}^{\perp}+ n_{\nu}\partial_{\mu}^{\perp}}{( n \partial )}- \frac{n_{\mu}n_{\nu}\partial_{\perp}^{2}}{( n \partial )^{2}}\right] \frac{1}{\partial_{\mu}^{2}+ i \varepsilon}.
\psi_{1}= f_{1}( x_{1}+ \frac{2 i \gamma_{1}\hbar}{\theta}x_{2}) e^{- \frac{3 x_{1}^{2}}{8 \gamma_{1}\hbar}- \frac{\gamma_{1}\hbar x_{2}^{2}}{2 \theta^{2}}+ \frac{i x_{1}x_{2}}{2 \theta}},
\int_{0}^{\infty}d y \, y^{1 - s}\frac{d}{d y}t^{2 ( \rho - 1 )}= ( 1 - \rho ) \, \frac{\Gamma \left( \frac{\displaystyle 3 - s}{\displaystyle 2}\right) \Gamma \left( \rho - \frac{\displaystyle 3 - s}{\displaystyle 2}\right)}{\Gamma ( \rho )}, \quad 3 - 2 \mathrm{~ R e ~}\rho < \mathrm{~ R e ~}s < 3 .
\mu \left( A^{-T}(\delta k/\sqrt{d})\langle\delta\cdot B_1(\ell / \sqrt{d})\rangle \right) = \left|\det \left( A^{-T}( \delta \cdot k / \sqrt{d}) \right) \right| \cdot \mu\left(\delta \cdot B_1(\ell / \sqrt{d})\right) = \frac{\mu(B_1(0)) \cdot \delta^{d}}{w(\delta \cdot k / \sqrt{d})}.
\gamma_{1}=(b_{1}\sqrt{a_{n}^{2}+a_{1}^{2}},\dots,b_{n-1}\sqrt{a_{n}^{2}+a_{n-1}^{2}},a_{n}b_{n}), &\gamma_{2}=(a_{1}b_{1}\sqrt{a_{n}^{2}+a_{1}^{2}},\dots, a_{n-1}b_{n-1}\sqrt{a_{n}^{2}+a_{n-1}^{2}},-|\beta|).
D_{0}= D_{1}=\ \ldots \ = D_{n_{0}-1}= 0 \ , \D_{n_{0}}= (-1)^{\tfrac{n_{0}(n_{0}+1)}{2}}s_{n_{0}}^{n_{0}+1}= (-1)^{\tfrac{n_{0}(n_{0}+1)}{2}}\Delta_{0}= t_{n_{0}}\ .
uv=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u=x\cdot x\delta R_v\mathbb{L}_u=x\cdot xR_v\pi\mathbb{L}_u
I = \frac{1}{2}\int_{X}\varepsilon^{\mu \nu \rho \sigma}\left( i \Sigma_{\mu \nu}^{a b}R_{\rho \sigma a b}+ \frac{1}{2}\varepsilon_{a b c d}\Sigma_{\mu \nu}^{a b}R_{\rho \sigma}^{c d}\right) d^{4}x .
d e g [ ( T^{+})^{n_{+}}( T^{0})^{n_{0}}( J )^{\overline{{{n}}}}( T^{-})^{n_{-}}{Q_{1}}^{m_{1}}{Q_{2}}^{m_{2}}{\overline{{{Q}}}_{1}}^{\overline{{{m}}}_{1}}{\overline{{{Q}}}_{2}}^{\overline{{{m}}}_{2}}] \ =
\hat{H}(t) = \sum_{\alpha}\Omega^{(D)}_{\alpha}(t)\Bigl(\hat{a}^{\dagger}_{\alpha}\hat{a}_{\alpha}+ \frac{1}{2}\Bigr) + \Omega_{\alpha}(t) \Bigl(\hat{a}^{\dagger 2}_{\alpha}+\hat{a}^2_{\alpha}\Bigr) \equiv \sum_{\alpha}\hat{H}_{\alpha}(t).
\displaystyle \sum_{l=1}^{m}\dfrac{(-1)^{l+m}(r)_{l-1}\genfrac{\{}{\}}{0pt}{}{m}{l}}{(r+l-2)!}\displaystyle \sum_{k=1}^{r+l-1}\begin{bmatrix}r+l-1 k\end{bmatrix}B_n^{(-k)}=\displaystyle \sum_{l=1}^{m}(-1)^{l+m}(r)_{l-1}\genfrac{\{}{\}}{0pt}{}{m}{l}(r+l-1)(r+l)^n &=\displaystyle \sum_{l=1}^{m}(-1)^{l+m}(r)_{l}\genfrac{\{}{\}}{0pt}{}{m}{l}(r+l)^n.
R_{1}( a , b , d ) = - \frac{1}{\left( 2 \pi \right)^{d - 2}a b}\sum_{n^{\prime}= 1}^{\infty}\int d^{d - 2}p \frac{1}{( \vec{p}^{\, 2}+ m^{2}+ ( \frac{n^{\prime}\pi}{b})^{2})}
B(j,\ell) =B^+(j,\ell)\cap{\cal F}(B^+(j,\ell)), E^+(j,\ell) =B^+(j,\ell)\cap \lim\sup_{i\to \infty}\Phi^{-i}B^+(j,\ell)=B(j,\ell)\cap (\cap_{u=1}^\infty(\cup_{i=u}^\infty \Phi^{-i}B(j,\ell))), E(j,\ell) =E^+(j,\ell)\cap{\cal F}(E^+(j,\ell)){\cal E}=\cup_j(\cup_\ell E(j,\ell)).
Z(G_{7,4}\vee B^{(2)}\vee G_{7,4}) \geq Z(C(\dot1,2) \vee B^{(2)}\vee C(\dot1,2)) = 4Z(B^{(2)})+2Z(B_u)+2Z(B_v)+B_{u,v}\geq 48+2Z(B_v)+2Z(B_u)+1.
J_{\pm}= \int_0^{\vert a\vert^{3/2}}\frac{4\pi\imath\vert \beta\vert^2\vert\lambda\vert^2 \vert a\vert^2 (a^* \dot{a}- a\dot{a}^* )}{(1+\vert\beta\vert^2 \vert a\vert^2 + 4\vert\lambda\vert^2\vert a\vert^2 \vert b\vert^2 )^2}d\vert b\vert^2.
l^{\dot{I}}_{\dot{J}}l^{\dot{K}}_{\dot{L}}= \delta^{\dot{K}}_{\dot{J}}l^{\dot{I}}_{\dot{L}}+ \sum_{\dot{J}_1 J_2 = \dot{J}}\delta^{\dot{K}}_{\dot{J}_1}l^{\dot{I}}_{\dot{L}J_2}+ \sum_{\dot{K}_1 K_2 = \dot{K}}\delta^{\dot{K}_1}_{\dot{J}}l^{\dot{I}K_2}_{\dot{L}}.
\frac{\pi}{4}=(\prod_{p\equiv1(mod4)}\frac{p}{p-1})(\prod_{p\equiv3(mod4)}\frac{p}{p+1})=\frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot\frac{13}{12}\cdot\cdot\cdot,
\begin{aligned}\Phi_{n,\alpha}(z) \,=\, z \Phi_{n,\alpha}(\partial) \,=\, \alpha z^n + \partial\end{aligned}\begin{aligned}\Phi'_{n,\alpha}(z) \,=\, z + \alpha \partial^n \Phi'_{n,\alpha}(\partial) \,=\, \partial\end{aligned}
p_j(u^\prime)=\begin{cases}1, \; \textnormal{if}j=1, 0, \; \textnormal{if}j=2, 1, \; \textnormal{if}3 \leq j \leq 2k-2 p_j(w^{**}-v_d)=-1, 0, \; \textnormal{if}3 \leq j \leq 2k-2 p_j(w^{**}-v_d)=10, 0, \; \textnormal{if}j=2k-1 \end{cases}
P_{0}= \int \! d \,^{3}x T_{e^{+}e^{-}}^{0 0}= \int \! d \,^{3}x [ \overline{{{\Psi}}}( i \gamma_{i}\partial_{i}- e \gamma_{i}A_{i}+ m ) \Psi + e \overline{{{\Psi}}}\gamma_{0}A_{0}\Psi ] = \int \! d \,^{3}x \overline{{{\Psi}}}i \gamma_{0}\partial_{0}\Psi \ ,
\frac{c_j^{kl}(A)}{\bar{a}}= \int_Y \frac{r}{\bar{a}}A e_j\cdot \nabla v^{kl}= \int_Y r_B B e_j \cdot \nabla\left(v^{kl}_B+ \bar{b}_{kl}w \right) = c_j^{kl}(B)+\bar{b}_{kl}\int_Y r_B B e_j\cdot \nabla w
\begin{array}{c}\Delta \tilde u(x) = \phi''(\delta)|\nabla \delta|^2+ \phi'(\delta) \Delta \delta = -\mu \frac{\phi(\delta)}{\delta^2}- M \epsilon (2 \beta_- + \epsilon - 1)\delta^{\beta_- +\epsilon-2}+\phi'(\delta) \Delta \delta\,.\end{array}
\sum_{d=1}^{m}sin^{2}(\frac{\pi jd}{m+1})=\sum_{d=1}^{m}\frac{1}{2}(1-cos(\frac{2\pi jd}{m+1}))=\frac{1}{2}(m-\sum_{d=1}^{m}cos(d\frac{2\pi j}{m+1}))
\pi_\beta d_0u_0 =\left(\sum_{k=0}^{l+1}(-1)^{l}R_{l+1-k}^{g\upsilon_0\beta'\tau_{l+1-k}}\pi_{\beta'\sigma_k}-\sum_{i=1}^l (-1)^{i}\pi_{\beta'\delta_i}\right) u_0 &=\sum_{k=0}^{l+1}(-1)^{l}R_{l+1-k}^{g\upsilon_0\beta'\tau_{l+1-k}}\pi_{\upsilon_0\beta'\sigma_k}-\sum_{i=0}^l (-1)^{i}\pi_{\upsilon_0\beta'\delta_i}
\psi ( \theta^{a}) = \sum_{i = 0 , 1 , . . , 3 , 5 , . . , d}\quad \sum_{\{a_{1}< a_{2}< . . . < a_{i}\}\in \{0 , 1 , . . , 3 , 5 , . . , d \}}\alpha_{a_{1}, a_{2}, . . . , a_{i}}\theta^{a_{1}}\theta^{a_{2}}\cdots \theta^{a_{i}}.
\Psi ( r , \theta , t ) = \sqrt{{\frac{i}{r}}}{\frac{1}{\sqrt{2 \pi k_{0}}}}{\frac{\sin{\frac{\pi}{\alpha}}}{\cos{\frac{\pi}{\alpha}}+ \cos \theta}}e^{-{\frac{b^{2}}{4 \xi^{2}}}}\Psi_{\mathrm{{f r e e}}}( r , \alpha^{2}t ) ,
{\rm d}\big(\xi\widehat\theta^2|v^\varepsilon|^2\big)=\big(\xi_t\widehat\theta^2+2\xi\widehat\theta\widehat\theta_t\big)|v^\varepsilon|^2{\rm d}t+2\xi\widehat\theta^2v^\varepsilon{\rm d}v^\varepsilon+ \xi\widehat\theta^2({\rm d}v^\varepsilon)^2.
\left| r^0 (u) - r^0 (0) + \frac{1}{2 \sqrt{2 \pi}}u^2 \right| = \left| r^0 (x) - r^0 (0) - (r^0)' (0) u - \frac{1}{2}(r^0)'' (0) u^2 \right| \lesssim | u |^3
eT_{\mu\nu}={\tilde{\overline\Psi}_{L}}{\gamma_\mu}i\left(D_\nu -\frac{1}{2}\Gamma^\alpha_{\alpha\nu}\right){\tilde\Psi}_{L}-\frac{i}{2}g_{\mu\nu}\partial_\alpha({\tilde{\overline\Psi}}_{L}\gamma^\alpha{\tilde\Psi_{L}}).
\frac{1}{2 4 \pi^{2}}\epsilon^{\mu \nu \lambda \rho}\mathrm{T r}\left( U \partial_{\nu}U^{\dagger}U \partial_{\lambda}U^{\dagger}U \partial_{\rho}U^{\dagger}\right) \frac{\partial f}{\partial \alpha}\partial_{\mu}\alpha \; .
\begin{array}{l}{\bf X}^{a-1}_{a,a-2}{\bf D}_{a-1}^{'}+c_{1a}(u_{+}){\bf U}_{a-1}{\bf X}^{'a-1}_{a,a-2}=c_{1a}(u_{-}){\bf X}^{'a-1}_{a,a-2}{\bf D}_{a-1}\mbox{}+c_{1a}(u_{+})c_{1a}(u_{-}){\bf U}_{a-1}^{'}{\bf X}^{a-1}_{a,a-2}+c_{3a}(u^{'},u){\bf D}_{a+1}^{'}{\bf X}^{a-1}_{a,a-2}\end{array}
\Pi_{k,m}:=Q_{k,m}\times(0,r(Q_{k,m})), \widetilde{\Pi}_{k,m}:=\widetilde{Q}_{k,m}\times(0,r(Q_{k,m})), \widehat{\gamma}_{k,m}:=\gamma_{\Pi_{k,m}}, \gamma_{k,m}:= (r(Q^{s}_{\alpha}))^{n+1}\widehat{\gamma}_{k,m}.
c_{\mu\mu'}= \frac{1}{a_{\mu\mu}}\left( \delta_{\mu\mu'}- \sum_{\kappa=2}^{\mu-1}a_{\mu\kappa}c_{\kappa\mu'}\right)\,, = (\mu-2)! \left( \delta_{\mu\mu'}- \sum_{\kappa=2}^{\mu-1}a_{\mu\kappa}c_{\kappa\mu'}\right)\,.
+i\{gg^{\mu\nu}\delta(x-y)+{1\over{e^2}}N^{\mu\beta}(x)N^{\nu\lambda}(y)({a\over{a^2\partial^2+4\alpha^2}}g_{\beta\lambda}\delta(x-y)+{{2\alpha}\over{\partial^2(a^2\partial^2+4\alpha^2)}}\epsilon_{\beta\lambda\sigma}\partial^\sigma\delta(x-y))\},
\frac{d}{dt}\left(\int_I a(0,t,x)\rho_3^2 |y_x|^2 dx \right) + \int_I \rho_3^2 |y_t|^2 dx &\leq C \left(\int_I \rho^2 |G|^2 dx + \int_{\mathcal{O}}\rho_1^2 |f|^2 dx + \int_I \rho_0^2 |y|^2 dx + \sum_{i=1}^2 \int_I \rho_0^2 |p^i|^2 dx + \int_I \rho_2^2 |y_x|^2 dx\right).
\begin{aligned}g_{1\bar 1k\bar k}= \chi_{1\bar 1k\bar k}+ u_{1\bar 1k\bar k}+W_{1\bar 1k\bar k}&= \chi_{1\bar 1k\bar k}+ u_{k\bar k1\bar 1}+ W_{1\bar 1k\bar k}&= \chi_{1\bar 1k\bar k}-\chi_{k\bar k1\bar 1}+ g_{k\bar k1\bar 1}- W_{k\bar k1\bar 1}+W_{1\bar 1k\bar k},\end{aligned}
\sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}\bigg( \int_{x_k}^{\infty}u \bigg)^{\frac{p}{p-q}}V_r(0,x_k)^{\frac{pq}{p-q}}\approx \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}\bigg( \int_{x_k}^{\infty}u \bigg)^{\frac{p}{p-q}}V_r(x_{k-1},x_k)^{\frac{pq}{p-q}}
\gamma(s,r_0(\sigma) \otimes \eta,\psi) \mathrel{\mathop:}= \varepsilon(s,(r_0 \circ \sigma) \otimes \eta,\psi) \, \dfrac{L(1-s,(r_0 \circ \sigma^\vee) \otimes \eta^{-1})}{L(s,(r_0 \circ \sigma) \otimes \eta)}\,,
S_- =-\partial_z, S_0 =z\partial_z+s, S_+=z^2\partial_z+2s\, z\,, \bar S_- =-\partial_{\bar z}, \bar S_0 ={\bar z}\partial_{\bar z}+\bar s, \bar S_+=\bar z^2\partial_{\bar z}+2\bar s\, \bar z\,.
\begin{array}{r c l}{{\cal R}( u , c )}&{=}&{\displaystyle \frac{u - i c P}{u + i c}}\\{}&{=}&{\displaystyle \frac{1}{u + i c}\left[ \begin{array}{c c c c}{u - i c}&{0}&{0}&{0}\\{0}&{u}&{- i c}&{0}\\{0}&{- i c}&{u}&{0}\\{0}&{0}&{0}&{u - i c}\\ \end{array}\right] ,}\\ \end{array}
B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\frac{182}{3}x^{12}+\frac{572}{3}x^{10}-429x^{8}+\frac{1820}{3}x^{6}-\frac{1382}{3}x^{4}+140x^{2}-\frac{3617}{510}
Z ({\cal A}, G , q ) = \mathrm{e}^{F ({\cal A}, G , q )}= \sum_{N = 1}^{\infty}Z_{N}({\cal A}, G ) q^{N}= \prod_{n = - \infty}^{+ \infty}Z^{( \mathrm{c o v )}}( G , \mathrm{e}^{-{\cal A}v ( n )}q ) ~ .
\sum_{m=0}^{\infty}\sum_{\ell=0}^{\min\{2p,2m\}}A_{m,\ell}=\sum_{\ell=0}^{2p}\sum_{m=\lceil\ell/2\rceil}^{p}A_{m,\ell}+\sum_{\ell=0}^{2p}\sum_{m=p+1}^{\infty}A_{m,\ell}=\sum_{\ell=0}^{2p}\sum_{m=\lceil\ell/2\rceil}^{\infty}A_{m,\ell}
A_{\mu}^{( I )}: = \sum_{b = 1}^{N}U_{I b}a_{\mu}^{( b )}\; \; \stackrel{U^{T}= U^{- 1}}{\Longleftarrow \! \! = \! \! = \! \! = \! \! = \! \! = \! \! \! \Longrightarrow}\; \; a_{\mu}^{( b )}= \sum_{I = 1}^{N}U_{I b}A_{\mu}^{( I )}\; .
I = -\nu{\Gamma(-\nu) \over \Gamma(\nu)}\int{d^dk \over (2\pi)^d}\tilde{A}_i({\bf k}) \left[ (\delta_{ij}-{k_i k_j \over |{\bf k}|^2}) +{\bar{c}\over c}{k_i k_j \over |{\bf k}|^2}\right] \left({|{\bf k}| \over 2}\right)^{2\nu}\tilde{A}_j(-{\bf k}).
s_1(x_1) = - x_1, s_2(x_1, x_2) = \frac{1}{3}(3x_1 - x_2), s_3(x_1, x_2, x_3) = - \frac{1}{2}(2x_1 - x_2), s_4(x_1, \dots, x_4) = \frac{1}{90}(90x_1 - 55x_2 + x_4), s_5(x_1, \dots, x_5) = - \frac{1}{36}(36x_1 - 25x_2 + x_4),
\displaystyle\frac{\partial E}{\partial\theta_j}=\Im\left(\frac{z_j(1-z_j)f^{\prime\prime}(z_j)-(n+\alpha-1-\beta-(n-\alpha-\beta-1)z_j)f^{\prime}(z_j)}{f^{\prime}(z_j)(1-z_j)}\right)=0.
\begin{aligned}\mathcal{A}(p) =2^n\int_{\partial X}dV_{\hat{g}^p}=2^n\int_{\partial X}e^{-n(s_p-s_E)}dV_{\hat{g}^E}&\leq2^n\int_{\partial X}dV_{\hat{g}^E}=\mathcal{A}(E).\end{aligned}
\int_{X}\psi(u - \phi) \theta^{n}_{v}= \int_{X}\psi(\tilde{u}- \phi + C) \theta^{n}_{\tilde{v}}\leq \int_{X}\psi(\tilde{u}- \phi)\theta^{n}_{\tilde{v}}+ \int_{X}\psi(C) \theta^{n}_{\tilde{v}}< \infty
\left( Y^{\mu}\right) ( \tau , \sigma^{1}, \sigma^{2}) = \left( \begin{array}{c}{\tau}\\{Y^{1}( \tau , \sigma^{1}, \sigma^{2})}\\{Y^{2}( \tau , \sigma^{1}, \sigma^{2})}\\{Y^{3}( \tau , \sigma^{1}, \sigma^{2})}\\ \end{array}\right) ,
\mbox{tr}\Big((\mathbf{E}_{\mu_z}A)(\mathbf{E}_{\mu_z}A^2)^{\frac{1}{2}}\Big) \ge \mathbf{E}_{\mu_z}\mbox{tr}\Big((\mathbf{E}_{\mu_z}A)A(\cdot)\Big) = \mbox{tr}\Big(\big(\mathbf{E}_{\mu_z}A\big)^2\Big)\,,
D_{\lambda,k}(f_r) =D_{\sigma,k-1}(h_r) &\leqslant \frac{4^{k-2}(2-r)r^{2k-2}}{(1+r)^{2k-4}}D_{\sigma,k-1}(h) &\leqslant \frac{4^{k-1}(2-r)r^{2k}}{(1+r)^{2k-2}}D_{\sigma,k-1}(g)=\frac{4^{k-1}(2-r)r^{2k}}{(1+r)^{2k-2}}D_{\lambda,k}(f).
\Bigl\| \big|\frac1K{}^{\vee}{\mathfrak{m}_{m,m',l,l'}}(\frac{\cdot}{K}) \big| \ast_{y_1}\phi \Bigr\|_{L^1(w_{[0,K]})}\le C_M \int \phi(y_1) \, d\mu_{[-K,K]}(y_1),\;\;\;d\mu_{[-K,K]}(y_1):= \bigl( 1+ \bigl| \frac{y_1}{K}\bigr| \bigr)^{-M}\, dy_1.
|[b,M](f)(x)| =|b(x)M(f)(x)-M(bf)(x)| =\bigg|\sup_{Q\ni{x}}\frac1{|Q|}\int_Q b(x)|f(y)|dy -\sup_{Q\ni{x}}\frac1{|Q|}\int_Q b(y)|f(y)|dy\bigg| \le \sup_{Q\ni{x}}\frac1{|Q|}\int_Q |b(x)-b(y)||f(y)|dy = M_b(f)(x).
\begin{array}{c}{{\big [ \hat{\partial}_{\mu}, \hat{\phi}\big ] \ = \ \int \mathrm{d}^{D}x \ \hat{\partial}_{\mu}\phi ( x ) \hat{\Delta}( x ) ,}}\\{{\big [ \hat{\partial}_{\mu}, \hat{\Delta}( x ) \big ] \ =}}\\{{h a t{\partial}_{\mu}\hat{\Delta}( x ) .}}\\ \end{array}
\begin{array}{ll}w_{yy}= w_t + (\tilde{u}w)_x + (\tilde{v}w)_y - 2(\eta w)_y- \partial_y\zeta \int\limits_y^{+\infty}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y}+ \zeta w - \partial_y\tilde{f}.\end{array}
\int d^{4}z_{1}d^{4}z_{2}\langle \Phi ( x_{1}) \Phi ( x_{2}) \Phi^{*}( y_{1}) \Phi^{*}( y_{2}) : ( \Phi \Phi^{*})^{2}: ( z_{1}) : ( \Phi \Phi^{*})^{2}: ( z_{2}) \rangle
\sum_{m=m_0}^\infty \frac{1}{m^s}\left((c_2 u(m))^{-s+1}- (c_3 v(m))^{-s+1}\right) &\leq\sum_{m=m_0}^\infty\frac{1}{m^s}\left(f(m)^{-s+1}- g(m)^{-s+1}\right) &\leq\sum_{m=m_0}^\infty\frac{1}{m^s}\left((c_1 u(m))^{-s+1}- (c_4 v(m))^{-s+1}\right).
\langle \psi^{-}|{\mathsf s}^{-}\rangle \langle^{+}{\mathsf s}| \phi^{+}\rangle = \langle \psi^{o u t}|{\mathsf s}\rangle \langle{\mathsf s}| \phi^{i n}\rangle \in{\cal S}\cap{\cal H}_{-}^{2}\, ,
2\textup{Re}\left(\frac{e^{iuv}}{\exp{\left(ae^{-iu}\right)}-1}\right)=2\textup{Re}\left(\frac{\cos(uv)+i\sin(uv)}{e^{a\cos(u)}\left(\cos(a\sin(u))-i\sin(a\sin(u))\right)-1}\right).
X_j(t) \approx \int_{\tau_j}^t \mu \cdot \frac{s}{\mu}e^{s(q_j - 1)(u - \tau_j)}\cdot e^{sq_j(t - u)}\: du = s e^{sq_j(t - \tau_j)}\int_{\tau_j}^t e^{-s(u - \tau_j)}\: du \approx e^{sq_j(t - \tau_j)}.
\alpha^{2}= \left( 1 - \left({\frac{a}{r}}\right)^{3}\right)^{- 1}, \quad \beta^{2}= \frac 1 9 r^{2}\left( 1 - \left({\frac{a}{r}}\right)^{3}\right) , \quad \gamma^{2}= \frac{\displaystyle r^{2}}{1 2}
\mathtt p\leftrightarrow (\mathtt q+\mathtt r)=(\mathtt p\leftrightarrow\mathtt q)+(\mathtt p\leftrightarrow\mathtt r),\ (\mathtt q+\mathtt r)\leftrightarrow\mathtt p=(\mathtt q\leftrightarrow\mathtt p)+(\mathtt r\leftrightarrow\mathtt p),
\langle x\otimes g, (y\otimes g) a\rangle = \langle x\otimes g, y\beta_g(a)\otimes g\rangle = \beta_{g^{-1}}(\langle x, y\beta_g(a)\rangle) &= \beta_{g^{-1}}(\langle x, y\rangle \beta_g(a)) &= \beta_{g^{-1}}(\langle x, y\rangle) a &= \langle x\otimes g, y\otimes g\rangle a.
\binom{\sum_{i=1}^{n}c_iN_i}{2}=\sum_{i=1}^{n}\binom{c_iN_i}{2}+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k &=\sum_{i=1}^{n}\left(\sum_{k_1+\cdots+k_{N_i}=2}\binom{c_i}{k_1}\cdots\binom{c_i}{k_{N_i}}\right)+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k &=\sum_{i=1}^{n}N_i\binom{c_i}{2}+\sum_{i=1}^{n}c_i^2\binom{N_i}{2}+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k.
\|B\mathrm{e}_{\mu,\nu}\|^2 =(1-N^{-1})^{-2}(1-\delta_{t(\mu),t(\nu)})\sum_{m\neq t(\mu)}\sum_{\ell=0}^{|\nu|-1}\sum_{|\gamma|=|\nu|-1, \, |\gamma\vee \nu|=\ell}N^{2\ell-2|\nu|}&= (1-\delta_{t(\mu),t(\nu)})\frac{N}{N-1}\sum_{\ell=0}^{|\nu|-1}N^{\ell-|\nu|}=(1-\delta_{t(\mu),t(\nu)})\frac{N}{(N-1)^2}(1-N^{-|\nu|}).
\varpi_{j*}\circ \tilde{\tau}_j^s(1_{Z_j})= \phi_j\left( t_j^{\lambda_j+s}\frac{\prod_{i=1}^{j-1}( 1-\bar t_i/\bar t_j)}{\prod_{i=1}^{\gamma_j}(1 - t_i/\bar t_j)}\right)
|g(u, v)|= |\alpha_{i_0}u^{i_0}+\sum_{i=i_0+1}^{\infty}\alpha_iu^i+\sum\limits_{j=j_0}^{\infty}\beta_jv^j| = q^{-e_1}|\pi^{-e_1}\alpha_{i_0}u^{i_0}+\pi^{-e_1}\big(\sum_{i=i_0+1}^{\infty}\alpha_iu^i+\sum\limits_{j=j_0}^{\infty}\beta_jv^j\big)| = q^{-e_1}.
\sum_{\substack{p \mid mn \Rightarrow T_{j-1}\leq p < T_j \Omega(n) = \Omega(n) = r}}\frac{r!^2 g(n) g(m)}{[n,m]}\cdot \frac{(m,n)^{z_1 + z_2}}{m^{z_1}n^{z_2}}
\sigma_{N - 2 , 3}= \left( \begin{array}{c c c c c c c c c c}{1}&{0}&{0}&{0}&{\ldots}&{0}&{0}&{0}&{0}&{0}\\{0}&{\star}&{\star}&{\star}&{\ldots}&{\star}&{1}&{0}&{0}&{0}\\ \end{array}\right)
\mathcal{H}_{_{v_{1},...,v_{m},b,d,\mu_{1},...,\mu_{m},1}}^{\prime}(z)=\prod_{j=1}^{m}\left( \frac{D_{\lambda}^{n,\gamma}\varphi_{v_{j,b,d}}(z)}{z}\right)^{1/\mu_{j}}
H_{2\mathrm{-mode}}=\; E_0+E^w_N+\mathcal{T}\big(a^\dagger_1a_2+a^\dagger_2a_1\big)-\mu\mathcal{N}_\perp+\frac{\lambda U}{N-1}\left(\mathcal{N}_1-\mathcal{N}_2\right)^2 &+\frac{2\lambda}{N-1}w_{1122}\mathcal{N}^2_-+\frac{\lambda}{4(N-1)}(w_{1111}-2w_{1122}+w_{1212})\mathcal{N}_\perp^2.
t_\lambda^\Lambda{t_\mu^\Lambda}^*\phi\big(t_\eta^{\Lambda^i}{t_\rho^{\Lambda^i}}^*\big)=t_\lambda^\Lambda{t_\mu^\Lambda}^*t_\eta^\Lambda{t_\rho^\Lambda}^* =\sum_{(\alpha,\beta)\in \Lambda^{\min}(\mu,\eta)}t_{\lambda\alpha}^\Lambda{t_{\rho\beta}^\Lambda}^*\in X_n
\langle \psi \gamma,\psi^{a_1}\alpha_1,\psi^{a_2}\alpha_2\rangle_{0}=\sum_{\mu=0}^{n+m}\sum_{\nu=0}^{n+m}\langle \gamma,\gamma_\mu\rangle_{0}g^{\mu \nu}\langle \gamma_\nu,\psi^{a_1}\alpha_1,\psi^{a_2}\alpha_2\rangle_{0}.
\frac{1}{2\pi i}\,\intop_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(z)\,\Gamma(s-z)\,\left(\frac{t^{2}}{x^{2}}\right)^{-z}dz=\frac{\Gamma(s)\,x^{2s}}{\left(x^{2}+t^{2}\right)^{s}}-\frac{\Gamma(s)\,x^{2s}}{t^{2s}},\,\,\,\,\,0<(s)<\frac{1}{2},
\lambda_{\phi}(t) =\frac{\int_{\mathbb{R}^d}\left(\frac12|\nabla\phi|^2+V|\phi|^2+\beta|\phi|^4-\Omega\overline{\phi}L_z\phi\right)\mathrm{d}\mathbf{x}-\|\dot{\phi}(\cdot,t)\|^2}{\|\phi(\cdot,t)\|^2}.
X_{i}^{\pm}(u_{1})X_{i}^{\pm}(u_{2})X_{j}^{\pm}(v)-2X_{i}^{\pm}(u_{1})X_{j}^{\pm}(v)X_{i}^{\pm}(u_{2})+X_{j}^{\pm}(v)X_{i}^{\pm}(u_{1})X_{i}^{\pm}(u_{2}) &+\{u_{1}\leftrightarrow u_{2}\}=0 if~ |i-j|=1
h(x,y) \,\, = \,\,\frac{1}{2}\bigl(x+\sqrt{-1}\cdot y \bigr)^5 \, + \,\frac{1}{2}\bigl(x-\sqrt{-1}\cdot y \bigr)^5 \, + \, y^5 \,\,= \, \, x^5 - 10 x^3 y^2 + 5 x y^4 + y^5.
\bold{H}_{k,1}=\bold{R}_{k}^{\frac{1}{2}}\bold{X}_{k}\bold{T}^{\frac{1}{2}}_{S,k},~\bold{H}_{k,2}=\bold{R}_{S}^{\frac{1}{2}}\bold{Y}\bold{T}^{\frac{1}{2}}
g_{\lambda,\mu}\left( \left( \begin{array}{cr}p_1 -\bar{q}_1 q_1 r_1 \end{array}\right), \left(\begin{array}{cr}p_2 -\bar{q}_2 q_2 r_2 \end{array}\right) \right)={\rm Re}(\lambda \bar{p}_1 p_2 + \bar{q}_1 q_2 + \mu \bar{r}_1 r_2)\, ,
\lim_{r \downarrow 0}\sup_{Q \in \Lambda_D}\Delta_{\{rB\}}(Q) = \sup_{Q \in \Lambda_D}\lim_{r \downarrow 0}\Delta_{\{rB\}}(Q) = \sup_{Q \in \Lambda_D}(\rho(|Q|) + (1 - \rho(|Q|)) \Delta_{\{B\}}) &= \Delta_{\{B\}}+ (1 - \Delta_{\{B\}}) \sup_{Q \in \Lambda_D}\rho(|Q|)= \Delta_{\{B\}}+ (1 - \Delta_{\{B\}}) \Delta_D,
S = \int d t \left\{P_{a x}\dot{X}^{a x}- \left[ \frac{g^{2}}{2}X^{a x}X^{a x}+ \frac{1}{4 g^{2}}( \partial_{[ a}P_{b ] x})^{2}\right] + \lambda_{x}X_{, a}^{a x}\right\}
- \frac{1}{4}| \nabla_W d|^2- \left(\cos \frac{d(x)}{2}\right)^2 |\nabla_W R|^2 + | \nabla_W \left((1-\eta) R \right)|^2 &= -\frac{\cos^4R(x) \cos^4 \frac{d(x)}{2}}{1-\sin^2 R(x) \cos^2 \frac{d(x)}{2}}\left| \nabla \frac{\tan\frac{d(x)}{2}}{\cos R(x)}\right|^2.
\begin{array}{c}{{S = - \Sigma_{n}m_{n}\int \sqrt{- g_{\mu \nu}U_{n}^{\mu}U_{n}^{\nu}}d \tau_{n}+ \frac{1}{2}\Sigma_{n}\int \lambda_{n}( \tau_{n}) [ g_{\mu \nu}U_{n}^{\mu}U_{n}^{\nu}+ 1 ] d \tau_{n}}}\\{{+ \alpha \Sigma_{n}m_{n}\int \phi [ x_{n}( \tau_{n}) ] d \tau_{n}- \frac{1}{2 \beta}\int [ \nabla^{\mu}\phi \nabla_{\mu}\phi - \Lambda^{2}e^{2 \phi}] \sqrt{g}d^{4}x}}\\ \end{array}
i_1 <... < i_{k+1}, \; \; \; j_1 <... < j_{b-k}, \; \; \; j_1=1 \{i_1 ,..., i_{k+1}\}\cup \{j_1 ,... , j_{b-k}\}=\{1,...,b+1\}\{i_1 ,..., i_{k+1}\}\cap \{j_1 ,... , j_{b-k}\}= \emptyset.
\max_{\mathbf{x}_i}\U_i = \sum_{m\in\mathcal{M}}\sum_{j\in\mathcal{N}_i}p_j^mrs_{m}F_{i,j}^m - w_d\sum_{m\in\mathcal{M}}p_i^ms_m\bar{D}_{i}^{m}s.t.~ \sum_{m\in\mathcal{M}}x_i^ms_m\leq c_i ~ \mathbf{x}_i \in [0,1]^{1\times M}
\left(\sigma^2-\lambda^2\right)L^{(2n-2k-2)}=\overline{A_{n-k}}A_{n-k}, \;A_{n-k}=\sigma^{n-p-k-1}\left(\sigma^2-\lambda^2\right)\cdot\prod\limits_{i=1}^{p-1}\left(\sigma-\lambda_i\right).
V ( \varphi_{0}) = 4 \sum_{s = 1}^{\infty}\int \frac{d^{4}p}{( 2 \pi )^{4}}\frac{( - 1 )^{s}}{2 s}\frac{( M + g \varphi_{0})^{2 s}}{( p_{E}^{2})^{s}}.
\int_a^R f_\infty(r)^2 \sinh^{d-1}r \d r = [ f_\infty(r)^2 u(r)]_a^R - \int_a^R 2 f_\infty(r) f_\infty'(r) u(r) d r &= f_\infty(R)^2 \int_0^R \sinh^{d-1}r \d r - \int_a^R 2 f_\infty(r) f_\infty'(r) u(r) d r .
V_{p^{+}, p^{-}}^{r , s}\left( z \right) = e^{i X^{-}p^{+}}\, \, e^{- \frac 1 2 X_{2}\left( 1 - 2 i s \right)}\, F_{r , s}\left( e^{- X_{2}}p^{-}p^{+}\right) e^{i X^{+}p^{-}}
\mathcal{D}_2 = \frac{1}{2N}\int dxdy\v_N(x-y) a^\ast_x(u)a^\ast_y(u)a_x(\bar u)a_y(\bar u) &+ \frac{1}{2N}\int dxdy\v_N(x-y)a^\ast_x(c)a^\ast_y(c)a_x(\bar c)a_y(\bar c) -\frac{1}{2N}\int dxdy\v_N(x-y)a_x^\ast a_y^\ast a_x a_y .
\frac{\alpha}{\sqrt{nF_j}s_j}= \big(1+ n \mu_{j,3}\alpha^{-2}\theta_j^0 -\alpha^2/nF_j\big)^{-1/2}= 1- \frac{n\mu_{j,3}\theta_j^0}{2\alpha^2}+O\Big(\frac{\alpha^2}{nF_j}\Big).
\sum_{x_{1}\in H}\cdots\sum_{x_{n+1}\in H,\ [x_{1},\ldots ,x_{n+1}]\in H \cap Z^{\otimes}(G)}\vert C^{\otimes}_{G}([x_{1},\ldots ,x_{n+1}])\vert = &\vert H \vert^{n+1}d_{n}^{\otimes}(\overline{H})\vert G \vert .