harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{N\to\infty}\frac{1}{N}\log\P\left(N^{-1}\max_{i \in [N]}d_i\geq\beta\right)= \lim_{N\to\infty}\sup_{x\in[0,1]}\frac{1}{N}\log\P\left(N^{-1}d_{i_x}\geq\beta\right) &= \sup_{x\in[0,1]}\lim_{N\to\infty}\frac{1}{N}\log\P\left(N^{-1}d_{i_x}\geq\beta\right)= - \inf_{x\in[0,1]}J_r(x,\beta) = -\widehat{\psi}_r(\beta).
\|Vf\|_{L^q(\mu)}= \lim_{i \to \infty}\|V_if\|_{L^q(\mu)}\le \limsup_{i \to \infty}\|V_i\|_{L^q(\mu) \to L^q(\mu)}\|f\|_{L^q(\mu)}&\lesssim \limsup_{i \to \infty}(1+V_{\textup{loc}, q}^i) \|f\|_{L^q(\mu)}&\le (1 + V_{\textup{loc}, q}) \|f\|_{L^q(\mu)}.
\Theta^{(1)}_{k\bar{l}}:= h^{i\bar{j}}\Theta_{i\bar{j}k\bar{l}}; \; \; \; \; \; \; \; \; \; \; \Theta^{(2)}_{i\bar{j}}:= h^{k\bar{l}}\Theta_{i\bar{j}k\bar{l}}; \; \; \; \; \; \; \; \; \; \; \Theta^{(3)}_{i\bar{l}}:= h^{k\bar{j}}\Theta_{i\bar{j}k\bar{l}}\; .
\sum_{k=2n}^r\frac{1}{\sigma_\Gamma(k)}\gg\frac{n^2}{\deg(\Gamma)\beta_\Gamma(n)}\sum_{i=n}^{\lfloor r/2\rfloor}\frac{1}{2i}\gg\frac{n^2}{\deg(\Gamma)\beta_\Gamma(n)}(\log r-\log n)
\left( \begin{array}{c}{{n^{\prime}}}\\{{m^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\alpha}}&{{\beta}}\\{{\gamma}}&{{\delta}}\\ \end{array}\right) \left( \begin{array}{c}{{n}}\\{{m}}\\ \end{array}\right) \; .
p = \frac{{{A_3}}}{{{A_1}}}- \frac{1}{3}{\left({\frac{{{A_2}}}{{{A_1}}}}\right)^2}, q = \frac{2}{{27}}{\left({\frac{{{A_2}}}{{{A_1}}}}\right)^3}- \frac{{{A_2}{A_3}}}{{3A_1^2}}+ \frac{{{A_4}}}{{{A_1}}},
\varphi_3'(\varepsilon,b,c,\eta,n,\varphi) :=\min\{\varphi_1(\varepsilon/3,b,n,\varphi),\varphi_1(\eta(\min\{\varepsilon/3b,2\}) c/2,b,n,\varphi), &\qquad\qquad\qquad\varphi_1(c/2,b,n,\varphi),\lambda_0/2\}
T_{2}=(\frac{1.02\times10^{10}}{2}[3\cdot5\times10^{-12}+\frac{5\cdot5\times10^{-12}}{1+(3.2\times10^{-5})^{2}}+\frac{2\cdot5\times10^{-12}}{1+4\cdot(3.2\times10^{-5})^{2}}])^{-1}
\Theta = \frac{1}{\sqrt{2}}\left( \begin{matrix}{\theta_{1}+ i \theta_{2}}\\{\theta_{1}- i \theta_{2}}\\ \end{matrix}\right) , \quad \bar{\Theta}= \frac{1}{\sqrt{2}}\left( \begin{matrix}{\theta_{1}- i \theta_{2}}\\{\theta_{1}+ i \theta_{2}}\\ \end{matrix}\right)^{T}{\cal C},
G_D(z,z') = \frac{1}{4\pi}\bigg( \frac{1}{|z - z'|}- \frac{1}{|z^* - z'|}\bigg), G_N(z,z') = \frac{1}{4\pi}\bigg( \frac{1}{|z-z'|}+ \frac{1}{|z^*-z'|}\bigg),
\begin{array}{rl}\underset{\mathbf{u}}{}-2\mathrm{Re}\left\{\left(\mathbf{u}^{(t)}\right)^H\mathrm{Diag}(\mathbf{y})^H\mathbf{F}^{H}\mathbf{B}\mathbf{F}\mathrm{Diag}(\mathbf{y})\mathbf{u}\right\}|u_n| = 1,n=0,1,\dots,N_c-1,\end{array}
\begin{cases}A_{2}+A_{3}+A_{4}=q^{2m}+q^m+1 &(q^{2}-1)A_{2}+(q^{3}-1)A_{3}+(q^{4}-1)A_{4}=(q^{m+2}-1)(q^m+1) &(q^{2}-1)(q-1)A_{2}+(q^{3}-1)(q^{2}-1)A_{3}+(q^{4}-1)(q^{3}-1)A_{4}=(q^{m+2}-1)(q^{m+1}-1)\end{cases}
S_j(\lambda) t_j S_i(\lambda') - S_j(\lambda') t_j S_i(\lambda) ={\prod_{k\neq i,j}(\lambda - \lambda_k) \prod_{k\neq i,j}(\lambda' - \lambda_k)\over \prod_{k\neq j}(\lambda_j - \lambda_k) \prod_{k\neq i}(\lambda_i - \lambda_k)}(\lambda_i - \lambda_j) (\lambda - \lambda')
d\mathbb{K}_4^{Ch}(\lambda) =\sqrt{-1}[\mathbf{T}^1\wedge\overline{\varphi^1}-\overline{\mathbf{T}^1}\wedge\varphi^1+\mathbf{T}^2\wedge\overline{\varphi^2}~~~~~-\overline{\mathbf{T}^2}\wedge\varphi^2 +\lambda^2(\overline{\Psi_2^1}\wedge\varphi^3-\Psi_2^1\wedge\overline{\varphi^3})].
\alpha_{\mathrm{e f f}}^{\prime}= \frac{\alpha^{\prime}}{\epsilon}, \qquad u = \frac{r}{\epsilon \alpha_{\mathrm{e f f}}^{\prime}}, \qquad G_{o ( 1 )}^{2}= \frac{g_{s}}{\epsilon}, \qquad Q_{5}= \alpha_{\mathrm{e f f}}^{\prime}N
1 \geq\int_X\Phi_2(\frac{C_Tf_n(x)}{\parallel C_Tf_n\parallel_{\Phi_2}})d\mu &=\int_{A_n}f_0(x)\Phi_2(\frac{\Phi_1^{-1}(\frac{1}{\mu(A_n)})}{\parallel C_Tf_n\parallel_{\Phi_2}})d\mu &=f_0(A_n)\Phi_2(\frac{\Phi_1^{-1}(\frac{1}{\mu(A_n)})}{\parallel C_Tf_n\parallel_{\Phi_2}})\mu(A_n).
{\cal G}_{\omega}( r , r^{\prime}, \theta ) = \sum_{n = 0}^{\infty}r^{1 - D / 2}\left[ b_{n}J_{n - 1 +{\frac{D}{2}}}( | \omega | r ) + c_{n}J_{- n + 1 -{\frac{D}{2}}}( | \omega | r ) \right] C_{n}^{( - 1 + D / 2 )}( z ) \quad ( r^{\prime}< r < a ) .
\begin{aligned}r(x',\xi') :=\frac{\sqrt{1-|\xi'|_g^2}-\aleph\sqrt{c^{2}-|\xi'|_g^2}}{\aleph \sqrt{c^2-|\xi'|_g^2}+\sqrt{1-|\xi'|_g^2}}r_N(q) :=\frac{\sum_{j=1}^N\log |r(\beta^j(q))|^2}{N}l(q_1,q_2) :=|\pi_x(q_1)-\pi_x(q_2)| l_N(q) :=\frac{\sum_{j=1}^{N}l(\beta^{j-1}(q),\beta^j(q))}{N}\end{aligned}
H_{Q}( p ) = 4 \pi d_{Q}( R ) \int \frac{d^{4}l}{{( 2 \pi )}^{4}}\left\{\frac{( l + p )^{2}- m_{Q}^{2}}{{[ ( l + p )^{2}- m_{Q}^{2}]}^{2}+ \varepsilon^{2}}+ ( p \to - p ) \right\}\delta ( l^{2}- m_{Q}^{2}) \sin^{2}\theta ( l^{0}, \mu_{Q}) ,
\pi_*\big(\theta^{i + 1}\cdot [C]\big) = p_*\Bigg(\bigg(\frac{1}{2}K + [x_0] + \frac{1}{2}\psi\bigg)^{i + 1}\Bigg) & = \sum_{\substack{j + k + l = i + 1 j, k, l \geq 0}}\frac{(i + 1)!}{j!k!l!}\frac{1}{2^{j + l}}p_*\big(K^j \cdot [x_0]^k \cdot \psi^l\big).
U_{N}( \vec{r}^{N}) = \mathrm{e x p}[ i \{\sum_{L = 1}^{N - 1}\alpha_{L}( \vec{r}^{N}) \hat{J}_{L}( [ \vec{r}^{N}] ) + \sum_{j = 1}^{k - 1}\sum_{k = 2}^{N}\sum_{i = 1}^{2}\alpha_{j k}^{( i )}( \vec{r}^{N}) \hat{J}_{j k}^{( i )}( [ \vec{r}^{N}] ) \}]
- \frac{dM(u)}{du}= \frac{1}{4\pi}\int d^2{\cal S}\Bigl[\Bigl( \frac{\partial}{\partial u}C_1 \Bigr)^2 +\Bigl( \frac{\partial}{\partial u}C_2 \Bigr)^2 \Bigr] +\frac{1}{8\pi}\int d^2{\cal S}\:\frac{1}{4}\: r^2 R^{\mu\nu}\nabla_{\mu}v \nabla_{\nu}v\biggl|_{{\cal I}^+}
N^+Nv=[N^+,N]v=Yv=2v, &N^+N^2v=[N^+,N]Nv+NN^+Nv=YNv+2Nv=2Nv. &(N^++N)v=Nv,(N^++N)^2v=N^+Nv+N^2v=2v+N^2v, &(N^++N)^3v=2Nv+N^+N^2v=4Nv.
( v - v^{\prime})^{2}\rightarrow \frac{( v - v^{\prime})^{2}}{\Omega ( v ) \Omega ( v^{\prime})}, \quad \vec{x}_{\alpha}\rightarrow \frac{\vec{x}_{\alpha}}{\Omega ( v )}\quad \vec{x}_{\alpha}^{\prime}\rightarrow \frac{\vec{x}_{\alpha}^{\prime}}{\Omega ( v^{\prime})}.
\sum_{j = 0}^k{G_j^2}= \frac{3}{2}G_0^2 + \frac{1}{2}G_1^2 - \frac{1}{2}G_2^2 - \frac{3}{2}G_{k + 1}^2 - \frac{1}{2}G_{k + 2}^2 + \frac{1}{2}G_{k + 3}^2
= \left \{(u(\tau , \tau -t , \theta_{-\tau}\omega, u_0), v(\tau , \tau -t , \theta_{- \tau}\omega, v_0) + \varepsilon y(\omega) u(\tau , \tau -t , \theta_{-\tau}\omega, u_0) ) : (u,v) \mbox{is a solution of $\eqref{pde1}$-$\eqref{pde3}$}\right \}.
\left\vert \langle \alpha_{r+1}^{(r+1,0)},\alpha_{r+1}^{(r+1,0)}\rangle \right\vert^2=\frac{F^{2r+2}}{2^{2r+4}}\left(\Vert \alpha^f_{r+1}\Vert^4-4^r(K_r^{\perp})^2\right).
Q_\xi(S;\varphi):=\inf_{x\in S}\operatorname{Pr}\Big(|\langle x,\varphi\rangle|\geq\xi\Big),W_M(S;\varphi):=\mathbb{E}\sup_{x\in S}\bigg\langle x,\frac{1}{\sqrt{M}}\sum_{i=1}^M\epsilon_i\varphi_i\bigg\rangle,
\int_\Omega \nabla u_n . \nabla T_1(u_n) T_1^{\gamma-1}(u_n) = \int_\Omega h_n\left(u_n+\frac{1}{n}\right)f_nT_1^{\gamma}(u_n) +\int_\Omega T_1^{\gamma}(u_n)\mu_n & \leq C.
\hat{\phi}_{1}^{\prime}= \frac{\phi}{\sqrt{2 \mathrm{T r}\phi^{2}}}, \ \ \hat{\phi}_{2}^{\prime}= \frac{1}{\sqrt{2 [ - 2 ( \mathrm{T r}\hat{\phi}_{1}^{\prime}{}^{3})^{2}+ \mathrm{T r}\hat{\phi}_{1}^{\prime}{}^{4}]}}[ - 2 ( \mathrm{T r}\hat{\phi}_{1}^{\prime}{}^{3}) \hat{\phi}_{1}^{\prime}+ \hat{\phi}_{1}^{\prime}{}^{2}]
x_{1}^{3}-6sx_{1}+-6x_{1}x_{2}^{2}+\sum\limits_{i=2}^{i=n}a_{i}x_{i}^{2}+2\left( \sum\limits_{i=2}^{i=n}a_{i}\right) sa_{i}\neq 0\text{pour tout}i\sum\limits_{i=2}^{i=n}a_{i}\neq 0
1 + 2 \sum_{m=1}^\infty \frac{B_{2m}}{(2m)!}\omega_m(n) x^{2m}= \frac{1}{\Phi_n(1)}\prod_{d \mid n}\left(\sinh \left(d \sinh^{-1}\left(\frac{x}{2}\right) \right)\right)^{\mu(n/d)}&= \frac{1}{\Phi_n(1)}\prod_{d \mid n}V_d (x)^{\mu(n/d)}.
S_{BF}[e,\omega,\phi] = \int d^{2}x \,\left[- \phi^{0}\left(d\omega - \frac{\Lambda}{2}e^{a}\wedge e^{b}\epsilon_{ab}\right) + \Lambda \delta_{ab}\phi^{a}\left( d e^{b}+ \omega^{b}_{\,\, c}\wedge e^{c}\right) \right].
\sqrt{g}g^{1 j}\partial_{j}X_{\mu}- \frac{1}{2}\varepsilon^{1 j k}\partial_{j}X^{\nu}\partial_{k}X^{\rho}A_{\mu \nu \rho}\mid_{\sigma_{1}= 0 , \pi}= 0 , ~ ~ \mu , \nu = 0 , 1 , . . . , p
a_{n}^{\prime\prime\prime}:= \sum_{k=1}^n (-1)^{k+1}\left( s_{n,k(3k-1)/2}^{(-1)}+ s_{n,k(3k+1)/2}^{(-1)}\right) \quad\longmapsto \{1,1,2,3,6,7,14,17,27,34,55,63,\ldots\}.
\lim_{\delta\rightarrow 0+}\sup_{n\in\mathbb{N}}\mathbb{E}\int_0^{T-\delta}\Vert Y_n^M (t+\delta) - Y_n^M (t) \Vert_{H}^{\alpha}ds \leq C \bigg( \lim_{\delta\rightarrow 0+}\sup_{n\in\mathbb{N}}\mathbb{E}\int_0^{T-\delta}\Vert Y_n^M (t+\delta) - Y_n^M (t) \Vert_{H}^{2}ds \bigg)^{\frac{\alpha}{2}}= 0.
|\Lambda(x,y)| =\lim_{m\rightarrow\infty}\left|\frac{1}{3^l}\right|^{m}\left|\Lambda\left(\frac{x}{3^{m}},\ \frac{y}{3^{m}}\right)\right| &\leq\lim_{m\rightarrow\infty}\left|\frac{1}{3^l}\right|^{m}G\left(\frac{x}{3^{m}},\ \frac{y}{3^{m}}\right) &=0.
I_{1}(h) =\int_{|\gamma(t)-1|\leq Ch}|B_{1}(\gamma(t))|^2|w_{1}(\gamma(t))|^2dt I_{-1}(h) =\int_{|\gamma(t)+1|\leq Ch}|B_{-1}(\gamma(t))|^2|w_{-1}(\gamma(t))|^2dt.
\begin{array}{r c l}{{0}}&{{=}}&{{g \left[ \mathsf{M}\mathsf{P}\, + \, \mathsf{N}\mathsf{Q}\right] \, ,}}\\{{}}&{{}}&{{}}\\{{0}}&{{=}}&{{{\cal B}_{+}{\cal B}_{-}\, ,}}\\ \end{array}
S_{s , \tiny \mathrm{d i v}}^{C}[ g , \beta_{H}, \epsilon ] ={\frac{1}{\pi}}\int_{\Sigma}\left\{{\frac{1}{3 6 0 \epsilon^{2}}}-{\frac{1}{1 4 4 0}}\left[ 2{\cal R}-{\cal Q}+ 3 0 \left( \frac 1 6 - \xi \right) R - 3 0 m^{2}\right] \ln{\frac{\epsilon^{2}}{l^{2}}}\right\},
\left|\int_{M}\nabla_{g}a(x)\nabla_{g}Z_{\delta_{\varepsilon}(t),\eta}^{k}\Phi_{\delta_{\varepsilon}(t),\eta}d\mu_{g}\right|\le\|\Phi_{\delta_{\varepsilon}(t),\eta}\|_{L^{2}}\|\nabla_{g}a(x)\nabla_{g}Z_{\delta_{\varepsilon}(t),\eta}^{k}\|_{L^{2}}=o(|\varepsilon|).
\lim_{c \rightarrow \infty}\int_{\mathbb{R}^n}\nabla u_c \nabla u_{\infty}+ 2m \mu u_c u_{\infty}dx &\qquad= \int_{\mathbb{R}^n}\nabla u_{\infty}\cdot \nabla u_{\infty}+ 2m \mu u_{\infty}^2 dx = 2m \int_{\mathbb{R}^n}u_{\infty}^{p}dx.
v = \frac{1}{d e t D}\sqrt{( e^{\phi}\vec{\pi}. \vec{\pi}+ 1 ) ( e^{- \phi}\vec{B}. \vec{B}+ 1 ) - [ ( \vec{\pi}. \vec{B})^{2}+ 2 \psi e^{\phi}( \vec{\pi}. \vec{B}) - \psi^{2}e^{\phi}\vec{B}. \vec{B}]}.
\int d^{10}\hat{x}\sqrt{|\hat{\jmath}|}\, \left\{{\textstyle\frac{1}{4\cdot 5!}}\left(\hat{G}^{+}_{(5)}\right)^{2}+{\textstyle\frac{1}{4\cdot 5!}}\left(\hat{G}^{-}_{(5)}\right)^{2}+{\textstyle\frac{1}{4!\cdot5!}}\hat{\epsilon}\left[\hat{G}^{+}_{(5)}\hat{\cal H}\hat{C}^{-}_{(2)}+\hat{G}^{-}_{(5)}\hat{\cal H}\hat{C}^{+}_{(2)}\right]\right\}\, .
b^{T,skew}_{h}(u_{h},v_{h}):=\frac{1}{2}\left[\int_{T}\vec{b}\cdot \Pi_{k-1}(\nabla u_{h}) \, \Pi_{k}(v_{h}) \, dT -\int_{T}\vec{b}\cdot \Pi_{k-1}(\nabla v_{h})\Pi_{k}(u_{h}) \, dT \right]
B(u) = \prod_{i=1}^{r}(\mu_i u+ \sum_{y=1}^{\mu_i-1}\sum_{b=2}^{\mu_i-y+1}\binom{\mu_i-y-1}{b-2}yu^{b}) &= \prod_{i=1}^{r}(\mu_i u+ \sum_{y=1}^{\mu_i-1}yu^2 (1+u)^{\mu_i-y-1}) &=\prod_{i=1}^{r}((1+u)^{\mu_i}-1),
\sqrt{n(x)}=\frac{\mathsf{l}(x)}{\mathsf{L}(\sqrt{\mathsf{a}(x)},\sqrt{\mathsf{h}(x)})}&=\frac{\sqrt{x}\,\mathsf{L}\!\left(\sqrt{x},\frac{1}{\sqrt{x}}\right)}{\sqrt[4]{\mathsf{a}(x)\mathsf{h}(x)}\,\mathsf{L}\!\left(\sqrt[4]{\frac{\mathsf{a}(x)}{\mathsf{h}(x)}},\sqrt[4]{\frac{\mathsf{h}(x)}{\mathsf{a}(x)}}\right)}>\sqrt[4]{x}>1.
\binom{k}{\ell_1}\binom{k-\ell_1}{\ell_2}\dots \binom{k-\ell_1-\dots- \ell_{r-1}}{\ell_r}&=\frac{k!}{\ell_1!(k-\ell_1)!}\cdot\frac{(k-\ell_1)!}{\ell_2!(k-\ell_2)!}\cdots \frac{(k-\ell_1-\dots-\ell_{r-1})!}{\ell_r!(k-\ell_1-\dots-\ell_r)!}&=\frac{k!}{\ell_1!\dots \ell_r!},
\det\big(M_{n}(\lambda,b,\Omega)\big) = \Big(\Omega_{n}(\lambda)-\Omega-b\Lambda_{1}(\lambda,b)\Big)\Big(\Lambda_{1}(\lambda,b)-b\big[\Omega_{n}(\lambda b)+\Omega\big]\Big)+b\Lambda_{n}^{2}(\lambda,b) & = b\Omega^{2}-B_{n}(\lambda,b)\Omega+C_{n}(\lambda,b),
N_{k,\ell}= \sum_{d = 1}^{\ell - k}\binom{\ell - 1 - k}{d-1}2^d = 2\sum_{d = 0}^{\ell -1 - k}\binom{\ell - 1 - k}{d}2^d = 2 (1 + 2)^{\ell - 1 - k}.
d s_{d e S}^{2}= -{\frac{\ell^{2}}{4}}\left({\frac{d \rho}{\rho}}\right)^{2}+{\frac{1}{\rho}}\left[ \left( 1 +{\frac{\ell^{2}\rho}{4}}\right)^{2}d \vec{n}\cdot d \vec{n}\right] = -{\frac{\ell^{2}}{4}}{\frac{d \rho^{2}}{\rho^{2}}}+{\frac{1}{\rho}}\, g_{a b}( x , \rho ) \, d x^{a}d x^{b}\, ,
\gamma (s,\tau,(r_0 \circ \rho_n) \boxtimes{\rm sim}^{-1},\psi) =\prod_{i=1}^{d}\gamma(s + 2s_i, \tau_i, (r_0 \circ \rho_{n_i}) \boxtimes{\rm sim}^{-1},\psi) \prod_{i<j}\gamma(s + s_i + s_j,\pi_i \times (\pi_j \cdot \eta),\psi).
\begin{array}{ll}h^{(1)}:=-\Delta+U_{\rm trap}^{(1)}+c_1V^{(1)}*|u_0|^2+c_2V^{(12)}*|v_0|^2-\mu^{(1)}, h^{(2)}:=-\Delta+U_{\rm trap}^{(2)}+c_2V^{(2)}*|v_0|^2+c_1V^{(12)}*|u_0|^2-\mu^{(2)}\end{array}
\int_0^1S_1(F_{\eta_1}^{-1}(x))S_2(F_{\eta_2}^{-1}(x))\,dx = \mathbb{E}[S_1(F_{\eta_1}^{-1}(X))S_2(F_{\eta_2}^{-1}(X))] \leq \int f\,d\nu_1 + \int f^*\,d\nu_2.
0 = \delta \psi_{\mu}= \partial_{\mu}\epsilon + \frac 1 4 \omega_{\mu}^{a b}\Gamma_{a b}\epsilon + \frac 1{2 8 8}e_{\mu}{}^{a}\left( \Gamma_{a}{}^{b c d e}- 8 \delta_{a}^{[ b}\Gamma^{c d e ]}\right) F_{b c d e}\epsilon .
\sum_{n\ge0}\frac{(q)_nq^{n^2}}{(q^{n+1})_{n}}=\sum_{n\ge0}q^{n^2}(1-q^{2n+1})\sum_{i\ge0}(-1)^iq^{i(3i+1)/2}(1-q^{2i+1})\sum_{|j|\le i}(-1)^jq^{-j(j-1)/2+nj}.
\frac{(\neg a_{1}\wedge\neg a_{2}\wedge\cdot\cdot\cdot)\rightarrow c,c\rightarrow(b_{1}\vee b_{2}\vee\cdot\cdot\cdot)}{(\neg a_{1}\wedge\neg a_{2}\wedge\cdot\cdot\cdot)\rightarrow(b_{1}\vee b_{2}\vee\cdot\cdot\cdot)}
< ( O^{\dagger}O )_{c l}>_{P C P}= \langle \Psi | O^{\dagger}O | \Psi \rangle = \langle \Psi | ({\cal P}_{o}^{\psi}( 1 , \xi ) )^{\dagger}{\cal P}_{o}^{\psi}( 1 , \xi ) | \Psi \rangle = < O_{c l}^{*}O_{c l}>_{P C P}.
w' = (s_4 s_3 s_2 s_1) (s_4 s_3 s_2 s_3 s_4 s_3 s_2 s_3 s_2) = (s_1 s_3 s_4 s_3) (s_2 s_1 s_3 s_2 s_4 s_3 s_4 s_2 s_3).
\left| \left. q\left( T\right) \right\rangle \hspace{-0.03in}\right\rangle=\prod_{k=-\infty}^{\infty}e^{iG_{k}^{1}\left( T\right)}\prod_{s=-\infty}^{\infty}e^{iG_{s}^{2}\left( T\right)}\left| \left. q\right\rangle \hspace{-0.03in}\right\rangle ,
\mathcal J_1(m) = \int_{\mathcal B_1^+(m)}\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh + \int_{\mathcal B_1^-(m)}(c,d)_p\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh.
\int_0^\infty P_t^{\lambda ,1}(x,y)dy =\left(\int_0^{x/2}+\int_{x/2}^{3x/2}+\int_{3x/2}^\infty \right)P_t^{\lambda ,1}(x,y)dy &=:Q_t^1(x)+Q_t^2(x)+Q_t^3(x),x\in (0,\infty ).
D ( x , x^{\prime}) = \sum_{n = - \infty}^{\infty}\int{\frac{d^{4}k}{( 2 \pi )^{4}}}\int_{0}^{\infty}{\frac{q d q}{2 \pi \alpha}}{\frac{e^{i k_{\mu}( x^{\mu}- x^{\mu})}e^{i n ( \theta - \theta^{\prime}) / \alpha}}{k_{4}^{2}+ q^{2}}}J_{\left| n / \alpha \right|}( q r ) J_{\left| n / \alpha \right|}( q r^{\prime}) ,
g_m(w) = \frac{2}{\pi^2 w}\dfrac{1}{J_m^2(\sqrt{w})+Y_m^2(\sqrt{w})}, L_{\frac{d}{2}}(w) = \frac{1}{(2\pi)^{\frac{d}{2}}}w^{\frac{d}{2}}K_{\frac{d}{2}}(w),
\int_{-\infty}^{+\infty}z^{\ell}p^{(k)}(z)\,\mathrm{d}z =\begin{cases}0 (-1)^k\,k! \end{cases}begin{align*}1ex] \int_0^{+\infty}z^{\ell}p^{(k)}(z)\,\mathrm{d}z =\begin{cases}(-1)^{\ell-1}\ell!p^{(k-\ell-1)}(0) \displaystyle (-1)^k\,\frac{k!}{2}\end{cases}
S=d+i\theta\cdot\sigma^\mu\cdot\bar{\theta}\,\partial_\mu d+\frac{1}{4}\theta^2\bar{\theta}^2\partial^2 d+\sqrt{2}\theta\cdot\delta-\frac{i}{\sqrt{2}}\theta^2\partial_\mu\delta\cdot\sigma^\mu\cdot\bar{\theta}+\theta^2 Z
\varphi(y) \langle e^{-t L_1^{(2)}}1_x,1_y\rangle =\langle e^{-t L_2^{(2)}}U1_x,U 1_y\rangle &= \langle e^{-t L_2^{(2)}}U 1_y, U1_x \rangle &= \varphi(x) \langle e^{-t L_1^{(2)}}1_y, 1_x\rangle &= \varphi(x) \langle e^{-t L_1^{(2)}}1_x, 1_y\rangle.
D_{2}\pi^{\alpha_{1}}= \frac 1 2 C_{\; \; \beta_{1}\gamma_{1}}^{\alpha_{1}}\pi^{\beta_{1}}\eta_{2}^{\gamma_{1}}+ \frac 1 2 C_{\; \; \beta_{1}\gamma_{3}}^{\alpha_{1}}\left( \pi^{\beta_{1}}\eta_{2}^{\gamma_{3}}- \eta_{2}^{\beta_{1}}\pi^{\gamma_{3}}\right) ,
||d(v,u)||_{L^2(B^l)}=\lim_{j \rightarrow \infty}||d(u_{n_j}^l,u_{n_j}^{\Lambda})||_{L^2(B^l)}\leq \lim_{j \rightarrow \infty}||d(u_{n_j}^l,u_{n_j}^{\Lambda})||_{L^2(\Sigma)}=0
\pi^{\mu}\rho ( \tilde{\eta}_{o}) U ({\textstyle{\frac{1}{2}}}\tilde{\eta}_{o},{\textstyle{\frac{1}{2}}}\tilde{\eta}_{o}, d - \tilde{\eta}_{o}) = \frac{\Gamma ( \tilde{\eta}_{o}) \Gamma ( \mu - \frac{1}{2}\tilde{\eta}_{o})}{\Gamma ( \mu - \tilde{\eta}_{o}) \Gamma ( \frac{1}{2}\tilde{\eta}_{o})}.
\mathcal{I}_1=\int_\Omega U\,u^2\, dx-\left(\int_\Omega |u|^{2p/(p+1)}\right)^{(p+1)/p}\qquad\mbox{and}\qquad\mathcal{I}_2=\|u-u_0\|^2_{W^{1,2}_0(\Omega)}.
U^{(2)}(R,T)=-T^2\left[\frac{\partial}{\partial T}\frac{F^{(2)}(R,T)}{T}\right]=-8\pi^2 R T^3\mathop{{\sum}'}\limits_{n=0}^{\infty}\frac{n^2}{\sqrt{\Omega_n^2+\omega_0^2}}~\frac{1}{\exp(2R\sqrt{\Omega_n^2+\omega_0^2}) -1}.
p(x_1, \dots, x_{\lfloor wM\rfloor}, y_1, \dots, y_M) = \left(\sum_{m=1}^{\lfloor wM\rfloor}\frac{x_m}{a^m}+ \sum_{m=\lfloor wM\rfloor +1}^M \frac{s(y_m)}{a^m}+ \sum_{m=M+1}^\infty \frac{\xi}{a^m}, \sum_{m=1}^M \frac{y_m}{b^m}+ \sum_{m=M+1}^\infty \frac{\eta}{b^m}\right).
ds^2\, :=\, B_{\rm H}[\sigma(r-r_{\rm H})]^{2(1-\alpha)}\, \bar{g}_{\mu\nu}dx^\mu dx^\nu+ \frac{dr^2}{B_{\rm H}(r-r_{\rm H})^2}+r_{\rm H}^2 d\Omega_{n-1}^2.
W^{\left( \mathrm{i}\right)}{\left( x_{1}, x_{2}, x_{3}, x_{4}\right)}= \left\langle \Omega , \phi_{{i}_{1}}\left( x_{{i}_{1}}\right) \phi_{{i}_{2}}\left( x_{{i}_{2}}\right) \phi_{{i}_{3}}\left( x_{{i}_{3}}\right) \phi_{{i}_{4}}\left( x_{{i}_{4}}\right) \Omega \right\rangle ,
B^j_{\Delta 1}= \frac{i}{2}\int [123] \left[\sum_{l=1}^2 \left(p^-_l \frac{\partial}{\partial p^j_l}+ 2p^j_l \frac{\partial}{\partial p^+_l}\right) r_\Delta (12) \right] \tilde{\delta}\;a_1^\dagger a_2^\dagger a_3 - h.c.\; ,
\Phi =\left\{\phi\left(x,y\right)=-a\left(x^{2}+y^{2}\right)+b_{1}x+b_{2}y,a\geq0,b_{1},b_{2}\in\mathbb{R}\right\}, \Psi =\left\{\psi\left(x\right)=cx,c\in\mathbb{R}\right\}.
G_k(z;q) := \sum_{n,m \geq 0}p_k(m,n) z^m q^n & = \frac{1}{(zq;q)_\infty}\sum_{r, s \geq 0}\frac{(-1)^r z^{kr + (k+1)s}\,q^{\textstyle \frac{(k+1)k(r+s)^2}{2}+ \frac{(k+1)(s+1)s}{2}}}{\left(q^k; q^k\right)_r \left(q^{k+1}; q^{k+1}\right)_s}.
\mathcal{E}_{k,m}= \underset{\substack{1 \leq \lambda \leq b \lambda \mid b}}\sum a_f( a \lambda^2 ) \underset{\substack{1 \leq d \leq b d \lambda \mid b}}\sum \mu(d) E_{k, a\lambda^2 d^2}\mid U_{\frac{b}{\lambda d}}.
|K(y_{0}, \cdots, y_{j}, \cdots, y_{m}) - K(y_{0}, \cdots, y'_{j}, \cdots, y_{m})| \leq \frac{A|y_{j}-y_{j'}|^{\epsilon}}{(\sum_{k,l=0}^{m}|y_{k}-y_{l}|)^{mn+\epsilon}},
\gamma_p = \overline{\gamma}_p\; \delta^{-2}~~\mbox{and}~~ \rho_{\mathrm{p}}\mathrm{c}_{\mathrm{p}}\sim 1, ~~\mbox{such that}~~ \gamma_m <\sqrt{\overline{\gamma}_p\; \rho_{\mathrm{p}}\mathrm{c}_{\mathrm{p}}}, ~~ \delta \ll 1.
\lambda_{1}^{2}= \dot{\lambda}+ \lambda \frac{q_{2}q_{3}}{q_{2}^{2}}, ~ \lambda_{1}^{1}= \ddot{\lambda}+ \frac{\lambda}{q_{2}^{2}}\left[ q_{2}{\dot{q}}_{3}- \frac{q_{2}q_{3}}{q_{2}^{2}}( 2 q_{2}{\dot{q}}_{2}+ q_{2}q_{3}) \right] .
N^{- 1 / 2}\left| r^1_{\lambda}(u) - r^1_{\lambda}(0) + \frac{1}{2 \sqrt{2 \pi}}\frac{m_{3, \lambda}}{\sigma^3_{\lambda}}u \right| = N^{- 1 / 2}| r^1_{\lambda}(u) - r^1_{\lambda}(0) - (r^1_{\lambda})' (0) u | \lesssim N^{- 1 / 2}\| (r^1_{\lambda})'' \|_{\infty}u^2,
\Big |\int_{\mathbb R}\varphi_k^3 \xi_{n_k}\nu_{n_k}^2dx-(\lambda-\theta) \Big | \leqq \Big |\int_{\mathbb R}\chi_k^3 \xi_{n_k}\nu_{n_k}^2dx\Big |+\epsilon\leqq \|{\bf{w}}_{k}\|^3_{1\times 1}+ \epsilon=O(\epsilon).
K_n(x)= \big((n+\frac{k}{2}-1)x+1\big)A_{n-1,0}(x) & + \sum_{j=1}^{n-3}\big((n-j-1)x+j+1\big)A_{n-1,j}(x) & + \big(x+n+\frac{k}{2}-1\big)A_{n-1,n-2}(x).
\frac{1}{t^{2}}\sum_{p \le q}( 2 - \delta_{p q}) \left( \frac{d ( p + q ) + 2 ( p - q ) - 2 p q}{4}- \frac{d ( d - 2 )}{8}\right) \left( \frac{( m^{[ p ]})^{2}}{V_{d}~ p !}\right) \left( \frac{( m^{[ q ]})^{2}}{V_{d}~ q !}\right)
\uppercase\expandafter{\romannumeral2}\leq \exp\left({n\tau(\int_{\Omega}\log X dP - \frac{\delta}{2})}\right)\left(E(\left(X_1^{-s_0}\right))^n (1 + \varepsilon)^{2n}\right)^{\frac{1}{N_2}}= \exp\left({n\tau(\int_{\Omega}\log X dP -\frac{\delta}{2}+ \frac{\log E(X_1^{-s_0})}{s_0}+ \frac{2}{s_0}\log (1 + \varepsilon))}\right).
N_B = i (2\pi)^{p+1}g^N \delta ( \sum_{i=1}^{N}{\bf p}_i ) (4\pi^2 \alpha^{\prime}l )^{-(p+1)/2}{\rm det}({\bf 1}+{\bf B}) \prod_{i=1}^N \int d s_i{\sqrt{g}}\quad .
\left(AP\right)_h~~~~~~~~~~\left(\begin{array}{ccc}A_{xa}\frac{1}{L_z}C_a 0 C_f A_{xf}+A_z B_l 0 B_c 0\end{array}\right)\left(\begin{array}{c}\alpha \tilde{\beta}\gamma \end{array}\right)=\left(\begin{array}{c}F_{\overline{u}}F_{u^\prime}0\end{array}\right),
\Psi(t) = -iM S_{\pm\xi}(-t) P_\mu \psi_{\mp}^\mu(t) + i S_{\pm\xi}(-t) P_\mu ( \Theta P_\mu \phi^\mu \cdot \Theta P_\mu \psi_{\mp}^\mu)(t) - M_{\mathfrak K_1}S_{\pm\xi}(-t) P_\mu^2 \psi_\pm^\mu(t)
\nu = d2^{\alpha/2-1}\frac{\Gamma\bigl(\frac{\alpha}{2}+ \frac{d}{2}\bigr)}{\Gamma\bigl(1+\frac{d}{2}\bigr)}\frac{(\rho/2)^{\alpha/2}\Gamma\bigl(\frac{\rho-\alpha}{2}\bigr)}{\Gamma\bigl(\frac{\rho}{2}\bigr)}.
\sigma := \alpha+\beta+1, &F_{i}(h) := \frac{(n-h)(h+1-k)}{(n-l-h)(h+1)}\left[1-H(h)-\frac{(k+l-i)(i+k+l+\sigma)}{(n+l+\alpha-h)(k-h-1)}\right], &G(h) := \frac{(n-h-1)_2(h+1-k)_2}{(n-l-h-1)_2(h+1)_2}H(h)
\begin{cases}- k_{\varepsilon}\Delta \Psi_{\varepsilon}+ |\nabla \Psi_{\varepsilon}|^2 - \tilde{\lambda}_0 \underline{m}= 0 \Omega \Psi_{\varepsilon}= -k_{\varepsilon}\log(w)(\frac{\mathbf{x}-\mathbf{q}}{k_{\varepsilon}}) \partial\Omega\end{cases}
\tilde{\gamma}_{i_{1}\ldots i_{p - 2 k - 2}}^{( 1 )}\equiv \pi_{i_{1}\ldots i_{p - 2 k - 2}}- \pi_{i_{1}\ldots i_{p - 2 k - 2}}^{( 1 )}\approx 0 , \; k = 0 , \ldots , c ,
\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{y_ix_jx_l}\widetilde{m}dy=-\int_{\mathcal{Y}^d}\big(\widetilde{m}(\Lambda_i+\widetilde{w}_{y_i})\big )_{y_i}\widetilde{w}_{x_jx_l}dy=0.
\pi\sqrt{2}=\frac{1}{8}\sum_{k=0}^{\infty}\frac{1}{64^{k}}(\frac{32}{12k+1}+\frac{8}{2k+3}+\frac{8}{12k+5}-\frac{4}{12k+7}-\frac{1}{12k+9}-\frac{1}{12k+11})
C=\begin{bmatrix}c(i,r) c(i+2r,3r) \cdot \cdot c(i,3r) c(i+2r,r) \cdot \cdot \cdot \cdot c(i+3r,3r) c(i+3r,r) \cdot \cdot c(i+r,r) c(i+r,3r)\end{bmatrix}
\psi_{Q}^{\lambda_{Q}}({\bf y}) :=\psi_{Q}({\bf x}^0 +\lambda_{Q}{\bf y})\,, V_{Q}^{\lambda_{Q}}({\bf y}) :=\lambda_{Q}V_{Q}({\bf x}^0+\lambda_{Q}{\bf y})\,, H_{Q}^{\lambda_{Q}}({\bf y}) :=\big(\nabla_{\bf x}F_{Q}\big) ({\bf x}^0+\lambda_{Q}{\bf y})\,, K_{Q}^{\lambda_{Q}}({\bf y}) :=K_{Q}({\bf x}^{0}+\lambda_{Q}{\bf y})
(u-c)\partial_q \varrho = (u-c) \left(\partial_x + \frac{h_q}{h_p}\partial_p\right) \varrho = (u-c) \left(\partial_x + h_q \partial_y\right) \varrho & = \left((u-c) \partial_x + v \partial_y \right) \varrho = 0,
\begin{array}{lll}d(x_{2i -1}) = \lambda^*(c_i) = c_{i}((\xi_{k}\otimes \mathbb{C}) \oplus \gamma_{n-k}) = \sum_{j=0}^{\infty}c_{j}(\xi_{k}\otimes \mathbb{C}) c_{i-j}(\gamma_{n-k}) = \sum_{j=0}^{[i/2]}p_{j}c_{i-2j}= c_i + \sum_{j=1}^{[i/2]}p_{j}c_{i-2j}\end{array}
\frac{4\pi}{y_{p}\big(e^{4\pi\slash y_{p}}-1\big)}=\frac{4\pi}{y_{p}\bigg(\displaystyle\sum_{n=1}^{\infty}\frac{(4\pi)^{n}}{n!\,y_{p}^{n}}\bigg)}=1+O\bigg(\frac{1}{y_{p}}\bigg);
I_{i j}^{( 1 ) \mu}={\frac{i}{2}}\Psi^{\lambda}\Psi^{\sigma}( f_{i \sigma}^{\nu}D_{\nu}f_{j \lambda}^{\mu}+ f_{j \sigma}^{\nu}D_{\nu}f_{i \lambda}^{\mu}+{\frac{1}{2}}f_{i}^{\mu \rho}c_{j \lambda \sigma \rho}+ f_{j}^{\mu \rho}c_{i \lambda \sigma \rho})
t_{j}=\frac{\tan (\frac{\pi}{4}-\frac{\pi j}{2H})}{2\cos^{2}(\frac{\pi j}{2H})}\prod_{i=0}^{j-1}\frac{\tan^{2}(\frac{\pi (j-1-2i+w_{r})}{2H})}{\tan^{2}(\frac{\pi (i+1)}{2H})}.
_{i}&\approx \textstyle \frac{p r^{2}_{ii}}{pr^{2}_{ii}\kappa^{2}_{T}+p\sum^{m}_{j=1}\left|\left((\mathbf{R}^{-1})^{\mathcal{H}}\mathbf{\Delta H}^{\mathcal{H}}\mathbf{Q}\mathbf{R}\right)_{ij}\right|^{2}(1+\kappa^{2}_{T})+p\kappa^{2}_{R}m+N_{0}}.
g_{m-1,\ell,s,m}=g_{m-1,\ell,s+1,m-1}+g_{m-1,\ell-1,s-1,m-1}=0,(\ell,s)\neq (1,1) g_{m-1,1,1,m}=g_{m-1,1,2,m-1}+g_{m-1,0,0,m-1}=1.
\begin{aligned}\theta(ab) = \varphi(ab) - \varepsilon_m(ab) = \varphi(a)\varphi(b) - \varepsilon_m(a)\varepsilon_m(b) = (\theta(a)+\varepsilon_m(a))(\theta(b)+\varepsilon_m(b)) - \varepsilon_m(a)\varepsilon_m(b) = \varepsilon_m(a) \theta(b) + \varepsilon_m(b) \theta(a), \end{aligned}
{\frac{1}{p^{2}+ \mu^{2}- i \varepsilon}}+{\frac{g^{2}}{M^{2}}}\int_{m_{0}^{2}}^{\infty}{\frac{d m^{2}}{\left( p^{2}+ m^{2}- i \varepsilon \right) \left[ \left( 1 - g^{2}{\frac{m^{2}}{M^{2}}}\right)^{2}+ g^{4}\right]}}
\langle\tilde v_\lambda |B(y_{n}|t)\cdots B(y_{1}|t)v_{0\cdots 0}\rangle =\frac{\Pi (y)}{\Pi(t_{\lambda})}\sum_{\mathcal{T}\in \operatorname{SVT}(\lambda^{\vee})}\beta^{|\mathcal{T}|-|\lambda^{\vee}|}\prod_{\substack{\langle i,j\rangle \in\lambda^{\vee}r\in \mathcal{T}(i,j)}}y_{r}\ominus t_{r+j-i}^{\prime}