harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{n=1}^\infty \lambda_\pi(n,r)e\left(\frac{an}{q}\right)g(n) = q\sum_\pm\sum_{n_1|qr}\sum_{n_2=1}^\infty \frac{\lambda_\pi(n_1,n_2)}{n_1n_2}S(r\bar a, \pm n_2; qr/n_1)G_\pm\left(\frac{n_1^2n_2}{q^3r}\right).
\sum_{i=1}^j b_i \leq \sum_{i=1}^j a_i = a_k+a_{\ell}+ \sum_{\substack{i=1 i \neq k,\ell}}^j a_i = c_k+c_{\ell}+ \sum_{\substack{i=1 i \neq k,\ell}}^j c_i = \sum_{i=1}^j c_i .
|\mathrm{Cal}(\phi)| & =|\mathrm{Cal}(\phi_{F^1})|+\cdots +|\mathrm{Cal}(\phi_{F^k})| &\leq \mathrm{Vol}(U,\omega)||F^1||+\cdots + \mathrm{Vol}(U,\omega)||F^k|| = \mathrm{Vol}(U,\omega)(||F^1||+\cdots+||F^k||).
B^t_i(x,\delta) =\left\{y\in\xi^{i-1}(x):\mathrm{dist}(E^{i}(x)\cap\xi^i(x), E^{i}(x)\cap\xi^i(y))<\delta\right\}, B^t_d(x,\delta) =B^T_d(x,\delta)\cap\xi^{d-1}(x),
\mathrm{det}(\lambda{\bf I}- \hat{\bf M}) = (\lambda - u_1) \bigg[ (\lambda - u_1)^4 - \frac{2}{45}(101 \theta_{11}+ 16 \theta_{22}) (\lambda-u_1)^2 & -\frac{96}{25}\frac{q_1}{\rho}(\lambda - u_1) + \frac{1}{15}(53\theta_{11}^2 - 16 \theta_{11}\theta_{22}+ 8 \theta_{22}^2) \bigg].
0 = \sigma_{-1}(N^\omega_{\partial M})(t,\xi')\sigma_0(\Lambda_{g,F}^\omega)(t,\xi')+\sigma_{-2}(N^\omega_{\partial M})(t,\xi')\sigma_1(\Lambda_{g,F}^\omega)(t,\xi') &+\nabla_{\xi'}\sigma_{-1}(N^\omega_{\partial M}) \cdot D_{t'}\sigma_1(\Lambda_{g,F}^\omega).
\prod_{i=0}^{\infty}\sum_{k=0}^{b-1}f(k,i) = \prod_{i=0}^{\infty}\sum_{k=0}^{b-1}\frac{(xw^{b^i})^{k}}{k!}= \prod_{i=0}^{\infty}\exp\left(xw^{b^i}\right) \frac{\Gamma(b,xw^{b^i})}{(b-1)!}= \exp\left(\sum_{i=0}^{\infty}xw^{b^i}\right) \prod_{i=0}^{\infty}\left(\frac{\Gamma(b,xw^{b^i})}{(b-1)!}\right).
{{E}_{\mu}^{[\alpha,\gamma],\rho,\varepsilon}=\left\{x\in\mathbb{R}^d : \ \underline\dim (\mu,x)\in[\alpha,\gamma] \forall r\leq \rho, \ \mu(B(x,r))\leq r^{\underline{\dim}(\mu,x)-\varepsilon}\right\}},
I_{T}(\boldsymbol{\zeta},\psi )\doteq \inf_{\boldsymbol{\varphi}\in \mathcal{S}_{T}(\boldsymbol{\zeta},\psi)}\left\{\sum_{k=0}^{\infty}\int_{[0,T]\times \lbrack 0,1]}\ell (\varphi_{k}(s,y))\,ds\,dy\right\}.
0\neq a-a' =\sum_{i=1}^{\ell}h^{(i)}_{\tilde{\lambda}_i(q)d_i}(n_i)=\sum_{i=1}^{\ell}\frac{h^{(i)}\left(r^{(i)}_{\tilde{\lambda}_i(q)d_i}+\tilde{\lambda}_i(q)d_in_i\right)}{\lambda_i\left(\tilde{\lambda}_i(q)d_i\right)}=\sum_{i=1}^{\ell}\frac{h^{(i)}\Big(r^{(i)}_{\tilde{\lambda}_i(q)d_i}+\tilde{\lambda}_i(q)d_in_i\Big)}{\Lambda(q)\lambda_i(d_i)}
(x,y,z) =-(x,y,z)+(y,x,z)+(x,z,y) =-(xy)z+x(yz)+(yx)z-y(xz)+(xz)y-x(zy) =-(xy)z+(yx)z+x(yz)-x(zy)+(xz)y-y(xz) =-[x,y]z+x[y,z]+[xz,y].
\# \left\{1 \leq m \neq n \leq N: |x_m - x_n| \leq s \right\}= \left\langle \left(\sum_{i=1}^{N}{\delta_{x_i}}\right) * \chi_{\left[-s,s\right]}, \sum_{i=1}^{N}{\delta_{x_i}}\right\rangle - N.
{\cal E}_{f,\theta}= \inf_{\{\psi: \|\psi\|_2= 1\}}\left(\frac{1}{2}\int_{{\mathbb R}^d}|\nabla \psi|^2 dx-\|f\|_1\int_{{\mathbb R}^d}\int_{{\mathbb R}^d}\frac{\psi(x)^2 \psi(y)^2}{|x-y|^{\theta}}\,dx\,dy\right).
_{b^{-}}^{C}D_{x}^{q}f(x) =(-1)^{n}{}_{b^{-}}\mathbf{I}_{x}^{n-q}f^{(n)}(x) & =\frac{(-1)^{n}}{\Gamma(n-q)}\int_{x}^{b}\frac{f^{(n)}(\tau)d\tau}{(\tau-x)^{1-n+q}},
\lambda =\frac{k(k-1)(k-2)(m-1)!(m-2)!}{(m+1)(m^2-2) |K_\Delta|}&=\frac{105\cdot 104\cdot 103\cdot 37!\cdot 36!}{39\cdot (38^2-2)\cdot 2^9\cdot 3!\cdot 4!\cdot 5!^2}&\approx 9.6\cdot 10^{76}.
\|A^{-1}(f(u_N)-f(u_M))\|^2 = \|(u_N-u_M)-E_{M,N}\|^2 \geq \frac{\|f\|_{}+\gamma}{2\gamma}\|u_N-u_M\|^2-\frac{3\gamma-\|f\|_{}}{\gamma-\|f\|_{}}\|E_{M,N}\|^2.
p^{\nu_{m+1}}_{1}(m+1,t) =I_t^{\nu_{m+1}}\left(-\lambda_{m+1}p^{\nu_{m+1}}_{0}(m+1,t)+\lambda_{m}p^{\nu_{m}}_{0}(m,t)\right)=0, p^{\nu_{m+1}}_{2}(m+1,t) =I_t^{\nu_{m+1}}\left(-\lambda_{m+1}p^{\nu_{m+1}}_{1}(m+1,t)+\lambda_{m}p^{\nu_{m}}_{1}(m,t)\right)=0.
\lefteqn{\lambda_{(\vec{x},\mu)}\lambda_{(\vec{y},\mu')}(\vec{z},\eta)}&=\lambda_{(\vec{x},\mu)}(f^{q(\vec{y},\mu')}(\vec{z}),\eta+b(\vec{y},f^{q(\vec{y},\mu')}(\vec{z}))) &=(f^{q(\vec{x},\mu)+q(\vec{y},\mu')}(\vec{z}),\eta+b(\vec{y},f^{q(\vec{y},\mu')}(\vec{z}))+b(\vec{x},f^{q(\vec{x},\mu)+q(\vec{y},\mu')}(\vec{z}))).
N\left(\frac{v_k}{|\lambda_k|},\frac{v_{ik}}{|\lambda_k|},\frac{v_{ek}}{|\lambda_k|},\lambda_k\right) = \sup_{x\in\Omega}\left(|v_k(x)|+|\lambda_k|^{-1/2}\left(|\nabla v_k(x)|+|\nabla v_{ik}(x)|+|\nabla v_{ek}(x)|\right)\right) = 1.
\begin{array}{| c | c | | c c c c |}\hline{{}}&{{}}&{{1}}&{{i}}&{{j}}&{{k}}\\ \hline{{\hline 1}}&{{e_{4}}}&{{2}}&{{2}}&{{2}}&{{2}}\\{{e_{1}}}&{{e_{5}}}&{{2}}&{{2}}&{{2}}&{{2}}\\{{e_{2}}}&{{e_{6}}}&{{2}}&{{2}}&{{2}}&{{2}}\\{{e_{3}}}&{{e_{7}}}&{{}}&{{}}&{{}}&{{0}}\\ \hline{{}}\\ \end{array}
\Delta t_{0}^{i j}~ = ~{\bf 1}\otimes t_{0}^{i j}~ + ~ t_{0}^{i j}\otimes{\bf 1}~ , \Delta t_{n}^{i j}~ = ~{\bf 1}\otimes t_{n}^{i j}~ + ~ t_{n}^{i j}\otimes{\bf 1}~ + ~ h \, \sum_{p + q = n - 1}\, \sum_{k = 1}^{N}~ t_{p}^{i k}\otimes t_{q}^{k j}~ ,
A_1(n):=\{(\lambda,(a^b))\in A(n) \mid a+b-1 \not \in \lambda \}; A_2(n):=\{(\lambda,(a^b))\in A(n) \mid a+b-1 \in\lambda \}; A_3(n):=\{(\lambda,(a^b))\in A(n) \mid \lambda=(a+b-1) \}.
S_{\rm YM}= -\,\frac{1}{4 g^{2}}\int d^4 x \,F_{\alpha \beta}^a F^{\alpha \beta}_a + \frac{\theta_{\rm YM}}{32 \pi^2}\int d^4 x \,F_{\alpha \beta}^a{\widetilde{F}}^{\alpha \beta}_a
\begin{array}{rcl}T_i(f_i)=-t_i^{-1}e_i~, begin{align*}3mm] T_i(f_j)=\displaystyle\sum_{m=0}^{-a_{ij}}(-1)^{m-a_{ij}}q_i^{m}\frac{f_i^{m}}{[m]_{q_i}!}f_j\frac{f_i^{-a_{ij}-m}}{[-a_{ij}-m]_{q_i}!}~,~~~{\rm if~}i\not=j~;\end{array}
\frac{\sum\limits_{\mathbf{a}\in \mathbb{B}^{N\times 1}\atop ||\mathbf{a}||_{0}=\lambda N}|\hat{\mathcal{U}}_{\rm{ac}}^{\mathbf{a}}(\mathbf{I})-\mathcal{U}^{\mathbf{a}}_{\rm{ac}}|}{\binom{N}{\lambda N}}= (1-\lambda) \sum_{u=1}^{N}\prod_{l=1}^{L}(1-(1-\lambda)^{\frac{w_{c}}{r}-1})^{\mathbf{S}[l,u]}.
A_{\mu}^{a}\left( x \right) = \sum_{\vec{k}}\left( \sum_{\sigma = 1 , 2}A_{\vec{k}, \sigma}^{a}f_{k , \mu}^{\sigma}\left( x \right) + A_{\vec{k}}^{L , a}f_{k , L , \mu}\left( x \right) + B_{\vec{k}}^{a}f_{k , S , \mu}\left( x \right) \right) + h . c . ,
\frac{C_{j}}{C_{o}}=1-e^{-\frac{nt}{\overline{t}}}[1+\frac{nt}{\overline{t}}+\frac{1}{2!}(\frac{nt}{\overline{t}})^{2}+...+\frac{1}{(j-1)!}(\frac{nt}{\overline{t}})^{j-1}]
\,f_{s}=x_s s=1,\cdots, k-1,s\neq m-\delta_{m,k}\hbox{or}l-\delta_{l,k}, &\, f_{m-\delta_{m,k}}=x_mx_k, f_{l-\delta_{l,k}}=x_lx_k, &\,f_{s}=x_{s+1}s=k,\cdots, n-2, f_{n-1}=1, f_{n}=x_l,
d\pi(v)=\sum_{i,j,k\neq r,l\neq r}(a_{ijkl}\zeta_{i}\zeta_{j}- a_{ijkr}\zeta_{i}\zeta_{j}\zeta_{k}- a_{ijrl}\zeta_{i}\zeta_{j}\zeta_{l}) \frac{\partial}{\partial \zeta_{k}}\wedge \frac{\partial}{\partial \zeta_{l}}
\frac{m_2}{m}\cdot K \leq \sum_{i=m-m_2+1}^{m}k_i = \sum_{\ell=0}^{m_2-1}k_{(m-m_2+1)+\ell}&\leq \frac{\log(n_{m-m_2+1}+1)}{\log(2)}\cdot \sum_{\ell=0}^{m_2-1}\delta^{\ell}&= \frac{\log(n_{m-m_2+1}+1)}{\log(2)}\cdot \frac{\delta^{m_2}-1}{\delta-1}.
Q+R \ = \ \sum_{i=m-t+2}^m \left[R(i)+\sum_{j:i\notin e_j}Q_{e_j}\right] & \ \leq\sum_{i=m-t+2}^m \left[R(i)+\sum_{j:i\notin e_j}Q_{e_{j|i}}\right] \ = \ (t-1)P.
z(1-z)\partial_{z}^2 g(z) + (1 + 2 \kappa )(1-z)\partial_{z}g(z) + \left[ \frac{1}{z}\left(\kappa^2 + \frac{\omega^2 \ell^4}{4 r_{H}^2}\right) - \kappa^2 \right]g(z) = 0,
\nabla_i\chi= \bar{g}( s_{\kappa}(u)\partial_r, h_i^k\partial_{x^k}), \nabla_j\nabla_i\chi= \bar{g}( s_{\kappa}(u)\partial_r,g^{kl}\nabla_lh_{ij}\partial_{x^k})+h_{ij}s'_{\kappa}(u)-h_i^kh_{kj}\chi.
\begin{array}{l l}{A = A^{1}+ i A^{2},}&{A^{*}= A^{1}- i A^{2}}\\{\gamma = \gamma^{1}+ i \gamma^{2},}&{\gamma^{*}= \gamma^{1}- i \gamma^{2}}\\{\gamma_{\pm}= \gamma_{0}\pm \gamma_{3}}&{}\\ \end{array}
h\left(\mathbf{Y}^{T_E}_{[1:\ell]}\right) ~\leq~ \sum_{k=1}^\ell T_E \log \Big(2\pi e \big({\Lambda}P + 1 \big)\Big) ~=~\ell{T_E}\log \Big(2\pi e \big({\Lambda}P + 1 \big)\Big) ,
\begin{array}{l}{{c_{1 a}f_{-}( u_{+}){\bf U}_{a - 1}{\bf U}_{a - 1}^{'}+ c_{2 a}f_{+}( u_{-}){\bf U}_{a - 1}{\bf D}_{a + 1}^{'}+ f_{-}{\bf X}_{a , a - 2}^{a - 1}{\bf X}_{a - 2 , a}^{' a - 1}}}\\{{\mathrm{}= c_{2 a}f_{-}( u_{+}){\bf D}_{a + 1}^{'}{\bf D}_{a + 1}+ c_{1 a}f_{+}( u_{-}){\bf U}_{a - 1}^{'}{\bf D}_{a + 1}+ f_{-}{\bf X}_{a , a + 2}^{' a + 1}{\bf X}_{a + 2 , a}^{a + 1}}}\\ \end{array}
\Delta V_{P F}={\frac{\hbar^{2}}{8 M}}\Big [ g^{a b}\Gamma_{a}\Gamma_{b}+ 2 ( g^{a b}\Gamma_{b})_{, b}+{g^{a b}}_{, a b}\Big ] +{\frac{\hbar^{2}}{8 M}}\Big ( 2 h^{a c}{h^{b c}}_{, a b}-{h^{a c}}_{, a}{h^{b c}}_{, b}-{h^{a c}}_{, b}{h^{b c}}_{, a}\Big ) \enspace .
\int_{Q}q(x)(\phi_{2}\overline{\phi}_{1})(x,t)(b_{2,\lambda}^{\sharp}\overline{b_{1}^{\sharp}}_{\lambda})(x,t)\,dx\,dt=\displaystyle\int_{\Sigma}(N_{A'_{1},q_{1}}-N_{A'_{2},q_{2}})(f) \overline{v}(x,t)\,d\sigma\,dt+I_{\lambda},
\widetilde{g}(\zeta) = -\frac{1}{\zeta^j}\cdot(\zeta-1)^{j-p-1+k-l}\cdot\frac{\prod_{b=1}^l (\zeta-\beta_b)}{\prod_{a=1}^k (\zeta-\alpha_a)}\cdot\prod_{i=1}^p (\zeta-1/z_i).
\deg_{in}(i) - \deg_{out}(i) = \left(\sum_{i=1}^{\frac n2-1}W_i\right) - \left(\sum_{i=1}^{n/2}Z_i\right) = \sum_{i=1}^{n/2}\left( W_i - Z_i \right) + Z_{\frac{n}2},
\Delta_{1}(t,\varepsilon)=\int_{0}^{+\infty}\Big(\Psi_{1}(\sigma,\varepsilon)\Gamma_{1}(\sigma+t,\varepsilon) +(1/\varepsilon)\Psi_{2}(\sigma,\varepsilon)\Gamma_{2}(\sigma+t,\varepsilon)\Big)d\sigma,\t\ge 0,
\psi((ab^i)^p)=\big( a^{i(e_2+\dots + e_{p-1})}b^i a^{ie_1}, a^{i(e_3+\dots + e_{p-1})}b^i a^{i(e_1+e_2)}, \dots, a^{i(e_1+\dots + e_{p-1})}b^i \big).
H_2\leq3^q\int_0^{2\pi}\!\!\!\!\int_\gamma^{R}\frac{dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^{2q-1}}=3^q\int_0^{2\pi}\!\!\!\!\int_\gamma^{R}\frac{dr d\theta}{|\gamma+r\sin\theta|^\tau r^{2q-1}}.
\mathrm{dist}\left(q_n\theta,a+b\mathbb{Z}\right)^{\frac{1}{1+q_n}}=\left|q_n(\theta-a_0) -p_n\right|^{\frac{1}{1+q_n}}<\left(\frac{1}{q_{n+1}}\right)^{\frac{1}{1+q_n}}<\left(\frac{1}{2Kq_{n}}\right)^{\frac{1}{1+q_{n-1}}}
T_{m}(1+2^{n}t)-(1+2^{n}t) = \sum_{i=0}^{k}c_{2i+1}(1+2^{n}t)^{2i+1}-(1+2^{n}t) = (1+2^{n}t)\left(c_{1}-1+\sum_{i=1}^{k}c_{2i+1}(1+2^{n}t)^{2i}\right).
|3 \Phi(\xi_1,\eta_1,\xi,\eta) -\tau + \xi^3 +\eta^3| =|3 \xi_1 \xi (\xi_1+\xi) + 3 \eta_1 \eta (\eta_1+\eta)-\tau + \xi^3 +\eta^3| & = | (\tau_1 - \xi_1^3 -\eta_1^3) - (\tau_1+\tau - (\xi_1+ \xi)^3 - (\eta_1 + \eta)^3)| & \lesssim \max(L_1, L_2).
\begin{array}{l}{{\displaystyle \dot{x}= - \frac{1}{2}x^{2}+ x y + \frac{1}{2 N}y^{2}+ \frac{1}{2}W_{1}( z )}\vspace{2 m m}}\\{{\displaystyle \dot{y}= \frac{N}{2}x^{2}+ \frac{1}{2}y^{2}- \frac{1}{2}W_{2}( z )}\vspace{2 m m}}\\{\dot{z}= - z ( y + N x )}\\ \end{array}
\frac{dU}{dt}(t)\psi= \sum_{j=1}^r\frac{df_j}{dt}(t){\rm Ad}_{\prod_{k=1}^{j-1}\exp({\rm i}f_k(t)\widehat H_k)}({\rm i}\widehat H_j)U(t)\psi,\forall \psi\in \mathcal{H},
\log\mathbb{E}_{N}\left[\exp\left\{\int_{0}^{T}\sum_{i=1}^{N}\left[\nabla G(t,\,x_{i}^{N}(t))-2G^{2}(t,\,x_{i}^{N}(t))\right]dt\right\}\right] =\sum_{i=1}^{N}\log\mathbb{E}_{N}\left[\exp\left\{\int_{0}^{T}\left[\nabla G(t,\,x_{i}^{N}(t))-2G^{2}(t,\,x_{i}^{N}(t))\right]dt\right\}\right].
\Delta S = - \frac{1}{8}\left({\frac{1}{4 \pi}}\right)^{3 / 2}\int d^{3}x{\frac{\left( \vec{\nabla}B \right)^{2}}{B^{3 / 2}}}\int_{0}^{\infty}{\frac{d s}{s^{1 / 2}}}e^{- m^{2}s / B}\left({\frac{d}{d s}}\right)^{3}\left[ s c o t h ( s ) \right]
-\frac{\partial L_{d}D_{\alpha}(p\|q)}{\partial q_{j}}= \frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]p^{\alpha}_{j}q^{-\alpha}_{j}&-\frac{1}{\alpha}\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]
\Theta^\beta(f_0) =\Theta_{C,R}^\beta(f_0) &:= \Big\{f\in \Theta: \ \frac{1}{32}f_0 \leq f \leq 32 f_0\ \n \int_0^1 \frac{(f(x) -f_0(x))^4}{f_0(x)^3}dx\leq C n^{\frac{1-2\beta}{2\beta+1}}\int_0^1 f_0(x)^{-\frac{2\beta+3}{2\beta+1}}dx \Big\},
d s^{2}= M d{\bf x}^{2}+{\frac{1 6 \pi^{2}}{M e^{4}}}d \chi^{2}+ \frac{4 \pi \kappa}{e^{2}}\left[ d \rho^{2}+{\frac{\rho^{2}}{4}}( \sigma_{1}^{2}+ \sigma_{2}^{2}+ \sigma_{3}^{2}) \right] .
S_2= \frac{p^w}{p^{m-w-1}(p-1)}\frac{p^2-1}{3p}\sum_{k=0}^{m-w-1}p^kp^{m-w-k-1}(p-1) = p^w\frac{p^2-1}{3p}(m-w)\leq p^{\min\{w,m\}}m\frac{p^2-1}{3p}.
T_{11}=-\pi i(cz+d)\sum\limits_{j=1}^{c}\sum\limits_{\mu=0}^{k-1}f(-d(j+\left[ R_{1}\right] ))B_{2}\left( \frac{c\mu+j-\left\{R_{1}\right\}}{ck}\right) & \quad\times\sum\limits_{v=0}^{k-1}f^{\ast}(-a(\left[ R_{2}+d(j-\left\{R_{1}\right\})/c\right] -v+d\mu)) & =-\frac{2\pi i}{c}(cz+d)B_{0}\left( B\right) P_{2}(-R_{1},A_{d}).
j_{r}=\frac{-(1+e)v_{r}\cdot\hat{n}}{{m_{1}}^{-1}+{m_{2}}^{-1}+({I_{1}}^{-1}(r_{1}\times\hat{n})\times r_{1}+{I_{2}}^{-1}(r_{2}\times\hat{n})\times r_{2})\cdot\hat{n}}
C_{\varphi}^{*}= \big(C_{\varphi^{-1}}^{*}\big)^{-1}= \Big(M_{1/h}C_{\varphi}M^{*}_{1/g}\Big)^{-1}= M^{*}_{g}C_{\varphi^{-1}}M_{h}\ \ \mathcal{K}.
\lim_{n \to \infty}\left|\int_{\mathcal{X}}\tilde{f}_{n}(x)\pi_{n-}^{\mu}(dx)-\int_{\mathcal{X}}\tilde{f}_{n}(x)\pi_{n-}^{\nu}(dx) \right|+\frac{2}{3}\epsilon = \lim_{n \to \infty}\bigg|\int_{\mathcal{Y}}g_{n}(y_{n})P^{\mu}(dy_{n}|y_{[0,n-1]}) & -\int_{\mathcal{Y}}g_{n}(y_{n})P^{\nu}(dy_{n}|y_{0,n-1]}) \bigg| +\frac{2}{3}\epsilon
\Phi_2(x_1,x_2) = -x_1^2x_2^2 + x_1^3 + x_2^3 + 1488 x_1^2x_2 +1488 x_1x_2^2 +40773375x_1x_2 &- 162000x_1^2-162000x_2^2+ 8748000000x_1+8748000000x_2 &-157464000000000
S_{\mathrm{L}}( \tau ) = Q \Psi_{\mathrm{L}}+ 2 \pi i \tau k = \bar{Q}\bar{\Psi}_{\mathrm{L}}+ 2 \pi i \bar{\tau}k , \qquad \tau = \frac{\theta}{2 \pi}+ \frac{4 \pi i}{e^{2}},
ip = a (\, i_{1}\, \cdots \, i_{t}\,)\circ (\, p_{1}\, \cdots \, p_{t}\,)^{T}a^{-1}= \sum_{j=1}^{t}ai_{j}p_{j}a^{-1}= \sum_{j=1}^{t}ae_{j}a^{-1}= \sum_{j=1}^{t}g_{j}= e,
\theta_{\mathrm{e v}}( \tau , z ) = \sum_{n \in{\bf Z}}q^{\frac{1}{4}( 2 n )^{2}}\zeta^{2 n}\, , \qquad \theta_{\mathrm{o d}}( \tau , z ) = \sum_{n \in{\bf Z}}q^{\frac{1}{4}( 2 n + 1 )^{2}}\zeta^{2 n + 1}
||\sqrt{\omega}\lambda(t) \mathcal{F}(\sqrt{\cdot}L_{1}(u)(t,\cdot \lambda(t)))(\omega \lambda(t)^{2})||_{L^{2}(\rho(\omega \lambda(t)^{2})d\omega)}^{2}=\frac{1}{\lambda(t)^{2}}||L(L_{1}(u)(t,\cdot \lambda(t)))||_{L^{2}(R dR)}^{2}
\zeta_{l_0,t}-D_{l_0}\Delta \zeta_{l_0}=f_{l_0}(v)-f_{l_0}(u) &\geq f_{l_0}(\cdots,u_{l_0-1},v_{l_0},u_{l_0+1},\cdots)-f_{l_0}(u) &\geq-M\zeta_{l_0}\ \ \ (M:=\sup_{w\in[p^-,p^+]}|DF(w)|).
\widehat{Q(g,f)}(\eta) = \int_{\S^{d-1}}b\left(\frac{\eta}{|\eta|}\cdot\sigma\right) \left[ \hat{g}(\eta^-) \hat{f}(\eta^+) - \hat{g}(0) \hat{f}(\eta) \right] \, \mathrm{d}\sigma, \eta^{\pm}= \frac{\eta\pm |\eta| \sigma}{2},
\left( 1-2st+2t^{2}\right)^{-2\alpha-\beta}\left( 1-\frac{4\left(1-s^{2}\right) t^{2}}{\left( 1-2st+2t^{2}\right)^{2}}\right)^{-\alpha-1}& =\sum_{\ell=0}^{\infty}\frac{\left( \alpha+1\right)_{\ell}}{\ell!}2^{2\ell}\left( 1-s^{2}\right)^{\ell}t^{2\ell}\left( 1-2st+2t^{2}\right)^{-2\alpha-\beta-2\ell}.
\xi_0 (q_0) = \phi_0^{-1}(1) [ \xi (q_0,1) - \widetilde{\xi}(q_0,1) ] , \ \xi'_0 (q_0) = \phi_0^{-1}(1) [ \xi_q (q_0,1) - \widetilde{\xi}_q (q_0,1) ] .
P = \sum_{k=1}^{n^4}P_k \leq \sum_{k=1}^{n^4}n^{3k}\frac{n^{k 4d}}{n^{4k}}< \sum_{k=1}^{\infty}n^{(4d - 1)k}< \frac{1}{1 - n^{4d - 1}}- 1
V(x_{-}) \equiv - \frac{1}{2f}\frac{d^2f}{dx_{-}^2}+\frac{3}{4}\left(\frac{1}{f}\frac{df}{dx_{-}}\right)^2- \frac{1}{4{\hbar}^2c^2}f^2 + \frac{4m^2c^4P_{cm}^2}{{\hbar}^2(f^2 - 4c^2P_{cm}^2)}
\mathcal{X}_{j , u , \theta}( \Omega ) = \left| \int_{0}^{\tau}d t \, \left\langle \mathbf{\Phi}_{j , u , \theta}^{A}( x , t )^{\dagger}x \mathbf{\Phi}_{j , u , \theta}^{A}( x , t ) \right\rangle \exp i \Omega t \right| \, \, .
L_{1 / 2}= \frac{1}{2}h_{i j}( y ) \dot{y}^{i}\dot{y}^{j}+ \frac{1}{2}L_{\alpha \beta}( y ) ( \dot{\psi}^{\alpha}+ w_{i}^{\alpha}( y ) \dot{y}^{i}) ( \dot{\psi}^{\beta}+ w_{j}^{\beta}( y ) \dot{y}^{j}) .
v_{s}v_{t}=\lim_{i}\lim_{j}\psi(\mu_j\delta_{s}\mu_i\delta_{t})=\lim_{i}\lim_{j}\psi(\alpha_{s}(\mu_i)u(s,t)\mu_j\delta_{st}) &=\lim_{i}\pi(\alpha_{s}(\mu_i)u(s,t))v_{st}=\overline{\pi}(u(s,t))v_{st}.
f_{\varphi_{\omega_{\pi_{\nu}}},s}\left( t\left(I_m,xg_\nu\right)\right)=\sum_{j=1}^{N}c^\prime_{j}\left((x^*)^{-1}\right)f^{(j)}_{\varphi_{\omega_{\pi_{\nu}}},s}\left(\tilde{\varepsilon}t\left(I_m,g_\nu\right)\right),
\mu^{n+1}_{t}-\Delta\mu^{n+1}+\varepsilon\mathrm{div}(\mu^{n}\Theta_{p}(t,x,\mu^{n},Dw^{n}))+\varepsilon\bar{m}\mathrm{div}(\Theta_{p}(t,x,\mu^{n},Dw^{n}))=0,
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^p(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^p(\mathbb{R}^n)}.
\bar z a_p^\dagger b_{p-k}+ \alpha_k \alpha_{p-k}z b_{p-k}a_{-p}= \bar z a_p^\dagger b_{p-k}+ \alpha_k \alpha_{p-k}z a_{-p}b_{p-k}+ \alpha_k \alpha_{p-k}z [ b_{p-k}, a_{-p}] ,
\frac{1}{1}\frac{y\cdot r}{x},\frac{1}{3}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}},\frac{1}{5}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}},\cdot\cdot\cdot
\widehat{H}= \frac{2 \pi}{L}\left( \widehat{L_{0}}+ \widehat{\overline{{{L}}}_{0}}- \frac{c}{1 2}\widehat{I}+ g \frac{L^{2 - 2 \Delta_{V}}}{\left( 2 \pi \right)^{1 - 2 \Delta_{V}}}\widehat{B}\right) \: ,
h_{m}=q^{\left( m\right)}\left( \vartheta_{m}~\left\vert \underline{\theta}\right. \right) ~,~~~~\vartheta_{m}\equiv \frac{i~\log \left[ \rho_{m}\left( \underline{\theta}\right) ~\dot{\theta}_{m}+\gamma_{m}\left( \underline{\theta}\right) \right]}{2~m-N-1}~,
\lim_{p\rightarrow\infty}\frac{2}{7p(p-1)}\sum_{m=0}^{6}(16f^{2}+4f-(6f+2)((m,m)_{7}+2(m,m+1)_{7}+(m+1,m+1)_{7})- &((m,m)_{7}+2(m,m+1)_{7}+(m+1,m+1)_{7})^{2}) &=\lim_{f\rightarrow\infty}\frac{2}{7(7f+1)7f}\sum_{m=0}^{6}(16f^{2}+4f-(6f+2)(\frac{4f}{7}+o(f))-(\frac{4f}{7}+o(f))^{2}) &=\frac{1200}{2401}.
\tilde S_2 \tilde U = S_2S_1S_1^*U^{-2}+S_1^2S_2^*U^{-2}&= S_2U^{-2}S_1S_1^*+S_1US_2S_2^*U^{-2}&= S_2U^{-2}US_2S_1^*+S_1UU^{-2}S_2S_2^* &= S_2U^*S_2S_1^*+S_1U^*S_2S_2^* &= U^{-2}S_2S_2S_1^*+U^{-2}S_1S_2S_2^* &= U^{-2}\tilde S_1\; .
\begin{cases}\psi (z_1)+\psi (z_2)=q^{-1}(\varphi (z_1))+q^{-1}(\varphi (z_2))= \alpha (\varphi (z_1)+\varphi (z_2))+2\beta =2\beta \cr ={q^{-1}}^{\tau}(\varphi (z_2))+{q^{-1}}^{\tau}(\varphi (z_1)) =\alpha^{\tau}(\varphi (z_1)+\varphi (z_2))+2\beta^{\tau}=2\beta^{\tau}\end{cases}
\|X\|_{S_{1/3}}= \min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{d\times d},W\in\mathbb{R}^{n\times d}:X=UVW^{T}}\|U\|_{*}\|V\|_{*}\|W\|_{*}= \min_{U,V,W:X=UVW^{T}}\!\left(\frac{\|U\|_{*}+\|V\|_{*}+\|W\|_{*}}{3}\right)^{3}.
\begin{pmatrix}Z_k Q_k \end{pmatrix}(t_2) = S_k(t_2,t_1) \begin{pmatrix}Z_k Q_k \end{pmatrix}(t_1) + \int_{t_1}^{t_2}S_k(t_2, t) \begin{pmatrix}c_k^{+}Z_{k+1}+ c_{k}^{-}Z_{k-1}d_k^{+}Z_{k+1}+ d_{k}^{-}Z_{k-1}\end{pmatrix}dt,
d(b_i, b_j) =d(b_i, e_u)+d(e_u, e_v)+d(e_v, b_j) &= d(f_u(b_i), f_u(e_u))+d(e_u, e_v)+d(f_v(e_v), f_v(b_j)) &= d(f(b_i), e_u)+d(e_u, e_v)+d(e_v, f(b_j)) &=d(f(b_i), f(b_j)).
a_{1}= x_1x_2^{2}x_3^{2}x_4^{3}x_5^{15},\a_{2}= x_1x_2^{2}x_3^{2}x_4^{7}x_5^{11},\a_{3}= x_1x_2^{3}x_3^{3}x_4^{6}x_5^{10},\a_{4}= x_1x_2^{2}x_3^{3}x_4^{6}x_5^{11}.
(M_0/2)\rho_n\epsilon_n\geq Q_1\sqrt{\frac{2^{\sum_{l=1}^dJ_{n,l}(\boldsymbol{\alpha})}\log{n}}{n}}+\sqrt{2Q_2C_{I}}2^{\sum_{l=1}^dJ_{n,l}(\boldsymbol{\alpha})/2}\epsilon_n,
S_0 :=(x+L)\cap \overline{B_r(x)}, \Sigma_x := \Sigma \cap \overline{B_r(x)}, \tau_0 \colon S_0 \to S_0 ; \ \z\mapsto z, \delta_0 <\left( 48(3C_1(m)+ 2) \right)^{-1}
\rho_{Q E D}= \frac{\exp [ - \beta \int d^{3}x ( T_{Q E D}^{0 0}( x ) - \mu e \overline{{\Psi}}\gamma^{0}\Psi ) ]}{{\mathrm{t r}}\exp [ - \beta \int d^{3}x ( T_{Q E D}^{0 0}( x ) - \mu e \overline{{\Psi}}\gamma^{0}\Psi ) ]}\ ,
\lim_{\epsilon \to 0}\int_0^{\epsilon}d \sigma \int d^3 p~ \frac{e^{-i k \times p \sigma}}{p^2 (k-p)^2}\left\{(k \times p) (k_\mu l_\nu - k_\nu l_\mu) - l^2 (k_\mu p_\nu - k_\nu p_\mu) \right\}= 0.
x:=\frac{a_{2}c_{1}}{a_{0}c_{0}},y:=\frac{a_{1}b_{2}}{a_{0}b_{0}},z:=\frac{b_{1}c_{2}}{b_{0}c_{0}},u:=-\frac{a_{1}b_{1}c_{1}}{a_{0}b_{0}c_{0}},v:=-\frac{a_{2}b_{2}c_{2}}{a_{0}b_{0}c_{0}}
\phi(s)=\int_0^s\phi'(\tau)\,d\tau =\int_0^s\sinh^{n-1}(\tau)\cosh^{d-1}(\tau)\,d\tau\geq\frac{1}{\cosh(s)}\int_0^s\sinh^{n-1}(\tau)\cosh(\tau)\,d\tau &=\frac{1}{n\cosh(s)}\sinh^n(s),
S_i = \frac{1}{n^2}\sum_{k=1}^n \left\{(D_{k}-D_{k+1})(\sqrt{n}f_{1,k}+f_{3,k})+ D_{k+1}f_{2,k}\right\}f_{1,k}= \frac{G_{k+1}-G_{k-1}}{8\sqrt{\beta}x_{k}^{3/2}},f_{2,k}=\frac{1}{4\beta x_{k}^2}G_{k}(G_{k+1}-G_{k-1}) f_{3,k}= \frac{1}{8\beta x_{k}^2}(G_{k+1}G_k -G_{k}G_{k-1}+\gamma_k(G_{k+1}-G_{k})+\gamma_{k-1}(G_{k}-G_{k-1})).
\langle x^{L(0)}w_{(1)},{\cal Y}(x^{-L(0)}w'_{(3)},x_{0})x^{-L(0)}w_{(2)}\rangle_{W'_1}=\langle w_{(1)},{\cal Y}(w'_{(3)}, xx_{0})w_{(2)}\rangle_{W'_1}
[D_t,Op_h^w(\Gamma)]= i h^{\frac{1}{2}}Op_h^w\left(\gamma'(\frac{x+p'(\xi)}{\sqrt{h}})p''(\xi)\xi\right) - \frac{ih}{2}Op_h^w\left(\gamma'(\frac{x+p'(\xi)}{\sqrt{h}})(\frac{x+p'(\xi)}{\sqrt{h}})\right)\,,
\Delta S = ( S_{I I}+ S_{I I I}) - S_{I}= - \, M^{2}\tau_{f}~ \frac{2}{\pi r^{2}}\left( \frac{1}{\sigma_{0}}+ \frac{1}{2 \pi - \sigma_{0}}\right){\sin}^{2}\, \frac{r \sigma_{0}}{2}
\begin{aligned}\sum_{{\substack{i_1,i_2\geq0 i_1+i_2=n}}}\frac{n!}{i_{1}!i_{2}!}B_{N,i_1}(x_1)B_{N,i_2}(x_2) &=\frac1N(N-n)B_{N,n}(x)+\frac nN(x-1)B_{N,n-1}(x) &\quad(x=x_1+x_2 n\geq1),\end{aligned}
\small\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}\boldsymbol{W}_j=\left[{\begin{array}{*{20}{c}}{{0}}{{0}}\cdots{{w}}{{w}}0 0 0 0 \ddots 0 0 0 0{{w}}0\end{array}}\right],
\sum_{n\ge 1, n\neq k}\frac{1}{\gamma^4_{\nu,n}-\gamma^4_{\nu,k}}=\frac{1}{8\gamma^4_{\nu,k}}\left[2\nu+5+\sum_{n\ge 1}\frac{4\gamma^4_{\nu,k}}{\gamma'^4_{\nu,n}-\gamma^4_{\nu,k}}\right].
u_i^{\u}=\frac{1}{\sqrt{1+t\bar{t}}}\left( \begin{array}{cc}t 1 -1 \bar{t}\end{array}\right), \u_u^{\i}=\frac{1}{\sqrt{1+t\bar{t}}}\left( \begin{array}{cc}\bar{t}-1 1 t\end{array}\right),
I_{1}= \frac{\pi^{3 / 2}}{2}\biggl ( \frac{\theta p}{2}\biggr )^{3 / 2}[ \nu^{- 3 / 2}K_{3 / 2}( \nu \theta p ) - \frac{\theta p}{3}\nu^{- 1 / 2}K_{1 / 2}( \nu \theta p ) ] ,
\begin{array}{r l l}{{\mathrm{I K , ~ T L}}}&{{:}}&{{\omega_{7}= 0}}\\{{\mathrm{~ F Z}}}&{{:}}&{{\mathrm{~ v e r t e x ~ w e i g h t s ~ b e i n g ~ i n v a r i a n t ~ u n d e r ~ r e v e r s a l ~ o f ~ a l l ~ a r r o w s ~ a n d ~}n = 2 .}}\\ \end{array}
\times\left\langle \prod_{b=1}^N \prod_{j=1}^{n_b}\left[ : e^{+i 2\sqrt{\pi}\varphi^{(b)}(x_{j}^{(b)})}:_{Q^{0,S}}:: e^{-i 2\sqrt{\pi}\varphi^{(b)}(y_{j}^{(b)})}:_{Q^{0,S}}\right] \right\rangle_{Q^{0,S}}\; \leq
(f_{1}+\ldots+f_{n})^{\ast}(x^{\ast}) \leq\left( f_{1}^{\ast}\square\ldots\square f_{n}^{\ast}\right) (x^{\ast}) & :=\inf\left\{f_{1}^{\ast}(x_{1}^{\ast})+\ldots+f_{n}^{\ast}(x_{n}^{\ast})\mid x_{1}^{\ast}+\ldots+x_{n}^{\ast}=x^{\ast}\right\}.
h_a(t)=\frac{e^{at}}{\prod_{k=1}^rM_{f_k}(a)}\int_{\mathbb{R}^{r-1}}f_1(e^{x_1})f_2(e^{x_2-x_1})\dots f_r(e^{t-x_{r-1}})\mathrm{d}x_1,\dots,x_{r-1}.
h_n=\frac{aq(1-q)(-acq^2)^nq^{\binom{n}{2}}(q,a^{-1}c,ac^{-1}q,abq;q)_\infty(q,aq,bq,cq,abc^{-1}q;q)_n}{(1-abq^{2n+1})(aq,bq,cq,abc^{-1}q;q)_\infty(abq;q)_n(abq^{n+1};q)_n^2}
S={1\over 8\pi}\int dx dy\left[\sum_{i=1}^2(\partial_x\phi_i)^2+(\partial_y\phi_i)^2-\Lambda'\left(e^{i{\beta\over \sqrt{2}}(\phi_1+\sqrt{3}\phi_2)}+e^{i{\beta\over \sqrt{2}}(\phi_1-\sqrt{3}\phi_2)}+e^{-i\sqrt{2}\beta\phi_1}\right)\right]
h_{H}(f^{nk}(P))= -c_{1}h_{Z_{1}}(p^{-1}f^{k(n-1)}(P))+h_{E}(p^{-1}f^{k(n-1)}(P)) &+\rho_{k}c_{1}h_{D_{1}}(f^{k(n-1)}(P))+c_{1}h_{{E_{1}'}}(f^{k(n-1)}(P)) \leq \rho_{k}c_{1}h_{D_{1}}(f^{k(n-1)}(P))+C \sqrt[]{h_{H}(f^{k(n-1)}(P))+\gamma}&+c_{1}C \sqrt[]{h_{H}(f^{k(n-1)}(P))}
\beta_{[\pm 1]}(n,k) \geq \frac{\binom{n}{k}}{2\binom{\frac{n}{2}}{\lceil\frac{k}{2}\rceil}\binom{\frac{n}{2}}{\lfloor\frac{k}{2}\rfloor}}= \Omega( \sqrt{\frac{k (n-k)}{n}}).
M_{w}= \begin{pmatrix}0 L_{1,2}L_{1,3}L_{1,4}L_{1,5}-L_{1,2}0 L_{2,3}L_{2,4}L_{2,5}-L_{1,3}-L_{2,3}0 L_{3,4}L_{3,5}-L_{1,4}-L_{2,4}- L_{3,4}0 L_{4,5}-L_{1,5}-L_{2,5}-L_{3,5}- L_{4,5}0 \end{pmatrix}
|N(x) \cap (S_0 \setminus (S \cup T))| = |N(x) \cap S_0| - |N(x) \cap S_0 \cap (S \cup T)| &= 2^{2\nu-4}- |N(x) \cap S_0 \cap S| - |N(x) \cap S_0 \cap T| &= 2^{2\nu-4}- |N(x) \cap S| &= 2^{2\nu-4}- 4.
(\nabla_{e_k}S)(e_i,e_j) = e_k(S(e_i,e_j))- S(\nabla_{e_k}e_i,e_j)-S(e_i,\nabla_{e_k}e_j) = S(\nabla_{e_k}^{M}e_i -\nabla_{e_k}e_i,e_j)-S(e_i,\nabla_{e_k}^{M}e_j-\nabla_{e_k}e_j) = h_{\alpha k i}S_{\alpha j}+ h_{\beta k j}S_{\beta i}.
P_{\mu}^{5}( k ) = - \frac{m}{\pi}\epsilon_{\mu \nu}k^{\nu}\frac{1}{k^{2}\sqrt{1 - \frac{4 m^{2}}{k^{2}}}}\ln \left( \frac{\sqrt{1 - \frac{4 m^{2}}{k^{2}}}+ 1}{\sqrt{1 - \frac{4 m^{2}}{k^{2}}}- 1}\right) .
A_{p}^{m}=\frac{\pi^{n/2}\Gamma(p(m+\alpha)+\frac{n}{2}+1)\Gamma(m+\alpha+\frac{n}{p}+1)}{\Gamma(p(m+\alpha)+1)\Gamma(m+\alpha+\frac{n}{2}+\frac{n}{p}+1)}\left(\frac{\Gamma(pm-n+1)}{\Gamma(pm-\frac{n}{2}+1)}\right)^{1/p}.