harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\frac{1}{2}\frac{d}{dt}\!\int \!|\nabla^N n|^2\! dx+\!\int \!|\nabla^{N+1}n|^2 dx &=-\!\int \nabla^N (u \cdot \nabla n)\nabla^N n dx+\!\int \!\nabla^{N}(|\nabla n|^2(n+w_0))\!\nabla^N ndx.\end{aligned}
\int_{\Omega}(\overline{U}_{+}){I_{1}(x,t)}\leq \frac{\chi_1\overline{u}}{2d_3}\Big(\int_{\Omega}(\overline{U}_{+})^2+ \int_{\Omega}\left((\lambda w-k\underline{u}-l\underline{v})_-\right)^2\Big).
\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}=\frac{1}{D}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}
\alpha^{d4}_{max}:= \alpha_2 + \alpha_3 + 2\alpha_4 + \alpha_5, \alpha^{d6}_{max}:= \alpha_2 + \alpha_3 + 2\alpha_4 + 2\alpha_5 + 2\alpha_6 + \alpha_7, \alpha^{e7}_{max}:= 2\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + \alpha_7,
N(x,y,a,b,d)=\frac{1}{(2\pi )^{d-2}ab}\sum_{n,n^{\prime}=1}^{\infty}\int d^{d-2}p\frac{\left( \cos ( \frac{2n\pi x}{a})\cos (\frac{2n^{\prime}\pi y}{b})\right)}{\left( \vec{p}^{\,2}+(\frac{n\pi}{a})^{2}+(\frac{n^{\prime}\pi}{b})^{2}+m^{2}\right)}.
\tilde{h}=\sum_{i,j=1}^{\infty}a_{ij}\varphi_i v_j \tilde{g}+\sum_{i,j=1}^{\infty}b_{ij}\varphi_i \cdot dr\odot f\omega_j+\sum_{i,j=1}^{\infty}c_{ij}\varphi_i\cdot f^2h_j,
\frac{\partial^{2}\tau}{\partial x_{i}\partial x_{j}}= 0 , \; ( \theta_{i}- \theta_{j}) \frac{\partial^{2}\alpha^{0}}{\partial x_{i}\partial x_{j}}= 2 [ \frac{\partial \tau}{\partial x_{i}}\frac{\partial \alpha^{0}}{\partial x_{j}}- \frac{\partial \tau}{\partial x_{j}}\frac{\partial \alpha^{0}}{\partial x_{i}}] .
\sigma_2(\textbf{r})=\left(r_1,\ldots,r_s,r_1',\widehat{r_1''},\ldots,r_t',\widehat{r_t''}\right) \left(\prod_{i=1}^sG_i\right)\times\left(\prod_{j=1}^t\left(H'_j\times H'_j\right)\right),
g_{1}(\nabla^{^{M_1}}_{X}Y,V) =-g_{1}(V,\mathcal{A}_{X}\phi \mathcal{B}Y+\mathcal{V}\nabla^{^{M_1}}_{X}\phi \mathcal{B}Y)-g_{1}(V,\mathcal{A}_{X}\omega\mathcal{B}Y+\mathcal{H}\nabla^{^{M_1}}_{X}\omega\mathcal{B}Y) &+g_{2}(\pi_*(CY),\pi_*(\nabla^{^{M_1}}_{X}\varphi V)).
\nu(\beta) = \nu(m-2^n j_1 - 2^{n-1}j_2 - \cdots - 2^{n-k+2}j_{k-1}) = \nu(m - (2^n j_1 - 2^{n-1}j_2 - \cdots - 2^{n-k+2}j_{k-1})) = \nu(m),
dp(t,X(t)) =-[\pi(t)\alpha_0(t)\frac{\partial p}{\partial x}(t,X(t))+\frac{1}{2}\pi^2(t)\beta_0^2(t)\frac{\partial^2 p}{\partial x^2}(t,X(t))+X(t)q(t,X(t))]dt &+q(t,X(t)) dG(t)
y \, = \, m(x;\Theta) + \mbox{error}, \; \mbox{where}\;m(x;\Theta) \, = \, \displaystyle{\max_{1 \leq i \leq k_1}}\, \left( \, ( \, a^i \, )^T x + \alpha_i \, \right) - \displaystyle{\max_{1 \leq i \leq k_2}}\, \left( \, ( \, b^i \, )^T x + \beta_i \, \right),
\frac{\partial}{\partial\hat{z}^j}=\frac{\partial\frac{1}{a}}{\partial\hat{z}^j}\hat{z}^1\frac{\partial}{\partial z^1}+\sum_{p=2}^n\frac{\partial z^p}{\partial\hat{z}^j}\frac{\partial}{\partial z^p},j=2,\dots,n.
\Delta u(t)=U(t,0)\Delta u_0 +\int_0^tU(t,s)[2i(D_j\nabla \tilde{b}^j(s)+\nabla \tilde{b}^j(s)D_j+\nabla \tilde{c}(s))\nabla u &+i(D_j\Delta \tilde{b}^j(s)+\Delta \tilde{b}^j(s)D_j+\Delta \tilde{c}(s))u+\Delta f(s)]ds.
\hat{A}_{(d)\, \mu_{1}\cdots\mu_{d}}=c a^{d}\frac{\epsilon_{\mu_{1}\cdots\mu_{d}}}{\sqrt{|g|}}\, , \Rightarrow\hat{F}_{(\hat{d})\, \underline{y}\mu_{1}\cdots\mu_{d}}=c d a^{d}\log^{\prime}{a}\, \frac{\epsilon_{\mu_{1}\cdots\mu_{d}}}{\sqrt{|g|}}\, ,
{{R}_{00}}=\frac{1}{4r}\left( 4{\nu}'+r{{{{\nu}'}}^{2}}+2r{\nu}'' \right){{e}^{\nu (t,r)}},{{R}_{11}}=-\frac{1}{4}\left({{{{\nu}'}}^{2}}+2{\nu}'' \right){{R}_{22}}=\frac{1}{2}r{\nu}',{{R}_{33}}=\frac{1}{2}r{\nu}'{{\sin}^{2}}\theta ,\, R=\frac{1}{2r}\left( 4{\nu}'+r{{{{\nu}'}}^{2}}+2r{\nu}'' \right),
C\Phi_7(q)=\prod_{n=1}^{\infty}\frac{1}{1-q^{7 n}}+49\cdot q\prod_{n=1}^{\infty}\frac{(1-q^{7 n})^3}{(1-q^n)^4}+343\cdot q^2\prod_{n=1}^{\infty}\frac{(1-q^{7 n})^7}{(1-q^n)^8}.
c_{F,j}= \frac{1}{\Delta}a_{(F(t)t^j, 0, \dots, 0)}= \sum_{i=0}^d \mu_i\,\frac{1}{\Delta}a_{(t^{i+j}, 0, \dots, 0)}= \sum_{i=0}^d \mu_i s_{(i+j-(n-1), 0,\dots,0)}& = \sum_{i=0}^d \mu_i c_{i+j-(n-1)}\underset{(*)}{=}\sum_{i=1}^d \mu_i c_{i+j-(n-1)}\ ,
\left| \frac{1}{\sqrt{m}}- y_m \right| = \frac{1}{|\frac1{\sqrt{m}}+ y_m|}\left|\frac{1}{m}- y_m^2 \right| \leq \sqrt{m}\left| \frac{1}{m}- \left(1-\sum_{j \le{m-1}}y_j^2 \right) \right| \cr \leq 3 \sum_{j \le m-1}\left| y_j - \frac{1}{\sqrt{m}}\right|< 3(m-1) \frac{1}{300 m \cdot x\sqrt{m}}. \cr < \frac{1}{100 x \sqrt{m}}.
V ( y_{i}, \theta ) = - V_{p + 1}( i )^{p}\int_{0}^{\infty}\frac{d t}{t}( 8 \pi^{2}\alpha^{\prime}\, t )^{- \frac{p + 1}{2}}e^{- \frac{y^{2}}{2 \pi \alpha^{\prime}}t}1 6 t^{4}\simeq y^{p - 7}\, .
g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{1}{2}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu})=0
\lambda=\frac{3}{8}M_{iikk}^{(2)}-\frac{1}{4}M_{ikik}^{(2)},\mu=\frac{1}{4}M_{ikik}^{(2)}-\frac{1}{8}M_{iikk}^{(2)}v=\frac{1}{4}\epsilon_{ij}M_{ikjk}^{(2)}
\frac{\partial Q}{\partial t}\left(x,t\right)=\intop_{0}^{a\left(x,t\right)}2\pi r\frac{\partial u}{\partial t}\left(x,r,t\right)dr+2\pi a\left(x,t\right)\frac{\partial a}{\partial t}\left(x,t\right)u\left(x,a\left(x,t\right),t\right).
P_{i}= \int d^{3}x \: \left\{F_{0 k}F_{i k}-{\frac{1}{2}}D_{0}\lambda \: D_{i}\phi -{\frac{1}{2}}D_{0}\phi \: D_{i}\lambda + i D_{i}\psi_{j}\: \chi_{j}+ i D_{i}\eta \: \psi_{0}- A_{i}\: G \right\},
\begin{aligned}[\hat{Q}_1,\hat{P}_1]=[\hat{Q}_2,\hat{P}_2]=i\hbar\mathbb{I}, &[\hat{Q}_1,\hat{Q}_2]=i\vartheta\mathbb{I},\;\hbox{and}\;[\hat{P}_1,\hat{P}_2]=i\mathcal{B}\mathbb{I}.\end{aligned}
\int_0^{\infty}v^4 g^T(x,v) \mathrm{d}v = \int_0^{\infty}\int_0^1 v^3 \frac{e^{-(1-y)/\kappa}}{\kappa |v| (1-e^{-1/\kappa|v|})}\rho_g(x+y) \left( \alpha\mathcal{M}_{T(y+x)}+ (1-\alpha) \mathcal{M}_{\tau(y+x)}\right) \mathrm{d}y \mathrm{d}v.
G_{i_{1}j_{1}...i_{N}j_{N}}=-\langle c_{i_{1}}\overline{c}_{j_{1}}\cdot\cdot\cdot c_{i_{N}}\overline{c}_{j_{N}}\rangle=\frac{-1}{Z[0]}\frac{\delta^{N}Z[J]}{\delta J_{j_{1}i_{1}}\cdot\cdot\cdot\delta J_{j_{N}i_{N}}}|_{J=0}
\begin{aligned}D_{\varpi}(P\parallel Q) = \log \sum_{i=1}^n{p_{i}e^{\varpi \frac{p_i}{q_i}}}-\varpi \ge \log e^{\varpi (\sum_{i=1}^n p_i ( \frac{q_i}{p_i}) )^{-1}}-\varpi = \log e^{\varpi}-\varpi = 0. \end{aligned}
\int_{M}\left( R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right) \left(\gamma ,\gamma \right) d\mu +\int_{M}\left\vert \gamma_{1,1}\right\vert^{2}d\mu +\frac{3}{4}\int_{M}Qud\mu +\frac{1}{2}\int_{M}\left( Pu\right)ud\mu =0.
f(x)=\sum_{i=1}^{n}P^{\mu}(y_{i}\in \cdot|X_{1}=x)\begin{pmatrix}g_{i}(c_{1}) \vdots g_{i}(c_{K})\end{pmatrix}&=\begin{pmatrix}QH_{x}T(x|:)A \cdots T(x|:)T^{n-2}A\end{pmatrix}\alpha
p_{X^{\rho m}|S^m}(x^{\rho m}|s^m)=\mathbb{I}(x^{\rho m}=f^{(m,{\rho m})}(s^m)), &p_{\hat{S}^m_i|Y^{\rho m}_i}(\hat{s}^m_i|y^{\rho m}_i)=\mathbb{I}(\hat{s}^m_i=g^{({\rho m},m)}_i(y^{\rho m}_i)),i=1,2,
h_{t}(\exp{H}) \asymp t^{-n/2}\left( \underset{\alpha \in \Sigma_{0}^{+}}{\prod}(1+\langle \alpha ,H\rangle )(1+t+\langle \alpha ,H\rangle)^{\frac{m_{\alpha}+m_{2\alpha}}{2}-1}\right) & \times e^{-\left\Vert \rho \right\Vert^{2}t-\langle \rho ,H\rangle-\left\Vert H\right\Vert^{2}/(4t)},
{\rm det}S = \left(\sum_{k_1,\ldots,k_d=0}^\infty \frac{E[H_{k_1}(X_1)^2\ldots H_{k_d}(X_d)^2]}{k_1!\ldots k_d!}\prod_{i=1}^d \sqrt{S_{ii}}(1-S_{ii})^{k_i}\right)^{-2}.
({\cal M})_{i_1}^{j_1}({\cal M})_{i_2}^{j_2}\cdots ({\cal M})_{i_{F-n}}^{j_{F-n}}\epsilon^{i_1 i_2 \cdots i_{F-n}i_{F-n+1}\cdots i_F}\epsilon^{j_1 j_2 \cdots j_{F-n}j_{F-n+1}\cdots j_F}= B^{i_{F-n+1}\cdots i_F}\bar{B}_{j_{F-n+1}\cdots j_F}.
\begin{dcases}\partial_t f + \partial_{\theta}(v[f] f) = 0, (\theta, \omega) \in \mathbb T \times \mathbb R,~~t > 0, v[f](\theta, \omega, t) = \omega - K \int_{\mathbb T}\sin(\theta-\theta_*) \rho(\theta_*, t) ~ d\theta_*, \rho(\theta, t) := \int_{\mathbb R}f(\theta, \omega, t) ~ d\omega,\end{dcases}
1 =l(a_1,b_1)<l(a_2,b_1)<\cdots<l(a_m,b_1)=(b_1+1)|N|=|N| <b_2|N|+1 =l(a_1,b_2)<l(a_2,b_2)<\cdots<l(a_m,b_2)=(b_2+1)|N| <b_3|N|+1 =l(a_1,b_3)<l(a_2,b_3)<\cdots<l(a_m,b_3)=(b_3+1)|N| & \,\ \vdots <b_n|N|+1 =l(a_1,b_n)<l(a_2,b_n)<\cdots<l(a_m,b_n)=(b_n+1)|N|.
\mathrm{P}\left( \bar{E}_{m,1}\right) = \mathrm{P}\left(\log\left(1+_{2,m'}\right)<R_{1,m}, \right. \log\left(1+_{2,n'}\right)>R_{1,n}, \left.\log\left(1+_{2,n}\right)>R_{2,n}, \forall ~n\in \{1, \cdots, m-1\}\right).
+ B_{\mu}{\frac{{\frac{( p^{2}\alpha + q a^{2})}{4}}\partial^{2}+ q \alpha ( \alpha + q )}{({\frac{p + a}{2}})^{2}\partial^{2}+ ( q + \alpha )^{2}}}\epsilon^{\mu \nu \lambda}B_{\nu \lambda}-{\frac{1}{2 g}}B_{\mu}B^{\mu}.
| h_{3}\rangle = ({\bf P}^{\bar{z}})^{k}\sum_{n = 0}^{k}\rho^{n}\frac{\Gamma ( \frac{D}{2}+ k - n - 2 )}{2^{n}n ! \Gamma ( \frac{D}{2}+ k - 2 )}({\bf M}^{\bar{z}j}{\bf M}^{\bar{z}j})^{n}V_{0}| 0 \rangle_{1}| 0 \rangle_{2}| 0 \rangle_{3}\, ,
A = \int d t d x \left[ 2 N i \bar{Z}_{1}\dot{Z}_{1}- 2 N^{2}( \partial \bar{Z}_{1}\partial Z_{1}- ( \bar{Z}_{1}\partial Z_{1}) ( \partial \bar{Z}_{1}Z_{1}) ) + \lambda ( \bar{Z}_{1}Z_{1}- 1 ) \right]
c_k(n)=c_k(n-1)+\sum_{a=0}^{k-1}g_{n-a}[a]c_{k-a-1}(n-a-1),c_0(n)=1, g_m[k]=c_{k+1}(m+k)-c_{k+1}(m+k-1)-\sum_{a=0}^{k-1}g_{m+k-a}[a]c_{k-a}(m+k-a), g_{m}[0]:=x_m.
=g_2^2g^4{\Lambda^2\over M^4}\left(\lambda_1^2-\lambda_2^2\right)-g^6{m_0^2\over M^4}\left(\lambda_1^2-\lambda_2^2\right)-g^2ln{m_0^2+\lambda_1^2\over m_0^2+\lambda_2^2}+...
\mathbf{E}_{x'}\Big(\sup_{t_1 \le s \le t}\big|z(s) - \bar{z}(s)\big|^2\Big) \le \, 3 \mathbf{E}_{x'}\Big(\big|z(t_1) - \bar{z}(t_1)\big|^2\Big) e^{L_1(t-t_1)}\le \, 3 \mathbf{E}_{x'}\Big(\sup_{0 \le s \le t_1}\big|\xi(x(s)) - z(s)\big|^2\Big) e^{L_1(t-t_1)}\,,
\lim_{n \rightarrow + \infty}\vartheta_p(s_n) = + \infty \Longleftrightarrow \lim_{n \rightarrow + \infty}{\left\vert s_n\right\vert}_p = 0 & \Longleftrightarrow \lim_{n \rightarrow + \infty}s_n = 0 ~~~~~~~~~~ ( p) & \Longleftrightarrow \sum_{k = 1}^{+ \infty}r_k = 0 ~~~~~~~~~~~~~ ( p) .
{\mathcal L}_{e f f}\; = \; \frac{h \rho_{0}q}{2}\epsilon_{k l}[ \dot{x}_{k}^{( 1 )}x_{l}^{( 1 )}- \dot{x}_{k}^{( 2 )}x_{l}^{( 2 )}] \, - \, \frac{h^{2}\rho_{0}q^{2}}{2 \pi m}\, \ln | \frac{x^{( 1 )}- x^{( 2 )}}{\xi}| \; ,
L_{D}= \int_{\cal M}{\frac{1}{4 \pi}}\mathrm{T r}\bigg [ \widetilde{d}\bigg ({\frac{k}{2}}\widetilde{d}g^{- 1}d_{0}g -{\frac{k}{2}}\widetilde{d}h^{- 1}d_{0}h + \widetilde{d}\alpha d_{0}h h^{- 1}\bigg ) -{\frac{k}{6}}( h^{- 1}d h )^{3}+{\frac{k}{6}}( g^{- 1}d g )^{3}\bigg ] .
\upsilon = \left( \frac{n_1-1}{2}+ \nu_1, \frac{n_1-3}{2}+\nu_1, \dots, -\frac{n_1-1}{2}+ \nu_1 , \frac{n_2-1}{2}+\nu_2, \dots, - \frac{n_2-1}{2}+ \nu_2, \dots \right).
X_{L}^{9}+ X_{R}^{9}\stackrel{T}{\longrightarrow}X_{L}^{9}- X_{R}^{9}\stackrel{\Omega}{\longrightarrow}X_{R}^{9}- X_{L}^{9}\stackrel{T^{- 1}}{\longrightarrow}- X_{R}^{9}- X_{L}^{9}
K_{h}(\hat{U}_{\mathbf{i}})=K_{h}(U_{\mathbf{i}})+h_{\mathbf{n}}^{-1}(\hat{U}_{\mathbf{i}}-U_{\mathbf{i}})^{T}\dot{K}_{h}(U_{\mathbf{i}})\bigl[1+\mathrm{o}_{p}(1)\bigr]=K_{h}(U_{\mathbf{i}})+\mathrm{o}_{p}(1).
\gamma = \sqrt{\frac{- c_{g h}- c_{M}- r + 2 4 ( | \rho | | \bar{\rho}| - \rho \cdot \bar{\rho})}{4 8 | \bar{\rho}|^{2}}}- \sqrt{\frac{- c_{g h}- c_{M}- r - 2 4 ( | \rho | | \bar{\rho}| + \rho \cdot \bar{\rho})}{4 8 | \bar{\rho}|^{2}}}~ ,
{K}^{(1)}_i(Y)= \sum_{j \in I_p}\sum_{l \in I_p}K_{ijl}^{(1)}Y_{j}Y_{l}, \tilde{K}_k^{(1)}(Y)= \sum_{j\in I_p}\sum_{l \in I_p}\tilde K_{kjl}^{(1)}Y_{j}Y_{l},
|x_k - y_k| < (3/5)^{(1/2 + 1/2^{n+1})(k - s_l + 2K_n + 20)}\cdot |x_{s_l + 2K_n + 20}- y_{s_l + 2K_n + 20}| < &< (3/5)^{(1/2 + 1/2^{n+1})(k - s_l + 2K_n + 20)}\cdot (3/5)^{(s_l + 2K_n + 20)/2}\cdot |x_0 - y_0| = &= (3/5)^{k/2 + 1/2^{n+1}(k - s_l + 2K_n + 20)}|x_0 - y_0|.
P = \sum_a (-1)^a \binom{a}{2}\binom{|S|-|U'|}{a-|U'|}Q = \sum_b (-1)^b \binom{r-|S|-|U|+|U'|}{b}P' = \sum_a (-1)^a \binom{|S|-|U'|}{a-|U'|}Q' = \sum_b (-1)^b \binom{b}{2}\binom{r-|S|-|U|+|U'|}{b}.
u(x,t)= e^{t(\Delta-I)}u_0-\chi\int_{0}^{t}e^{(t-s)(\Delta-I)}\nabla\cdot (u(\cdot,s)\nabla v(\cdot,s))ds &+\int_0^t e^{(t-s)(\Delta-I)}((1+a^*(\cdot,s))u-b^*(\cdot,s)u^2(\cdot,s)))ds.
\mathfrak{B}_{1}= (x^{2}- 1, xy, y^{2}, yz); \mathfrak{B}_{2}= (y^{2}- 1, xz, yz, z^{2}); \mathfrak{B}_{3}= (x^{2}, xy, xz, yz, z^{2}); \mathfrak{B}_{4}= (x^{4}, xy, y^{2}, xz, yz, z^{2}); \mathfrak{B}_{5}= (x^{2}, y^{2}, xz, yz, z^{2}).
[(g, \alpha) , (g, \beta)] =~ \mathcal{L}_{\Pi^\sharp (g, \alpha)}(g, \beta) - i_{\Pi^\sharp (g, \beta)}d (g, \alpha) =~ \mathcal{L}_{(g, \Pi^\sharp \alpha)}(g, \beta) - i_{(g, \Pi^\sharp \beta)}d (g, \alpha) = (g, [\alpha, \beta]).
\limsup_{n\to \infty}\frac{n}{b_n^{2}}\log J_n \leq \limsup_{n\to \infty}\frac{n}{b_n^{2}}\log e^{- \frac{(b_n')^2}{n}\big( \frac{y^2}{2 \sigma^2}- \eta \big)}& = - \lim_{n\to \infty}\Big( \frac{b_n'}{b_n}\Big)^2 \Big( \frac{y^2}{2 \sigma^2}- \eta \Big) = - \Big( \frac{y^2}{2 \sigma^2}- \eta \Big).
\Sigma = \{\lambda \in \mathfrak{a}\setminus \{0\}\mid \mathfrak{k}_{\lambda}\neq \{0\}\}, W = \{\alpha \in \mathfrak{a}\setminus \{0\}\mid V^{\perp}_{\alpha}(\mathfrak{k}_{1}\cap \mathfrak{m}_{2}) \neq \{0\}\}, \tilde{\Sigma}= \Sigma \cup W.
\chi(\sqrt{\smash[b]{\Delta_g}}-\lambda) = \frac{1}{2\pi}\int e^{it(\sqrt{\smash[b]{\Delta_g}}-\lambda)}\widehat{\chi}(t)\,dt & = \frac{1}{\pi}\int_{-\infty}^\infty e^{-it\lambda}\cos(t\sqrt{\smash[b]{\Delta_g}})\widehat{\chi}(t)\,dt +\tilde{\chi}(\sqrt{\smash[b]{\Delta_g}}+\lambda)
c'_e(k,l;s|a,c|q)={(q a^2/c^2;q^2)_k (q^3 s/c^2;q^2)_k (q^2 s^2/c^4;q^2)_k \over (q^2;q^2)_k(q s/c^2;q^2)_k(q^3 s^2/a^2 c^2;q^2)_k}(q^2x^2/a^2)^k &\times{(c^2/q^2;q)_l (s/q;q)_{2k+l}\over (q;q)_l (q^2 s/c^2;q)_{2k+l}}{1-q^{2k+2l-1}s\over 1-q^{-1}s}(q^2x^2/c^2)^l.
\Gamma\Bigl({D\over 2}-1\Bigr){{\dot x_a\cdot\dot x_b\over{\Bigl[{(x_a - x_b)}^2\Bigr]}^{{D\over 2}-1}}\quad}-{1-\alpha\over 4}\Gamma\Bigl({{D\over 2}-2}\Bigr){\partial\over\partial\tau_a}{\partial\over\partial\tau_b}{\Bigl[{(x_a-x_b)}^2\Bigr]}^{2-{D\over2}}.
0 = \frac{\partial{\cal L}}{\partial W_{\mu}^{I}}= 2 \partial_{\nu}\frac{\partial{\cal L}}{\partial{\cal F}_{\mu \nu}^{I}}= 2 \partial_{\nu}\left( \frac{\partial{\cal L}}{\partial{\cal F}_{\mu \nu}^{+ I}}+ \frac{\partial{\cal L}}{\partial{\cal F}_{\mu \nu}^{- I}}\right) \ ,
H = \frac{t}{2 a}\sum_{x}\left( e^{i A}S^{+}( x + 1 ) S^{-}( x ) + e^{- i A}S^{-}( x + 1 ) S^{+}( x ) \right) + \frac{e_{L}^{2}a}{2}\sum_{x , y}S^{3}( x ) V ( x - y ) S^{3}( y ) + e_{L}^{2}E^{2}/ 2 N
d_1(\lambda_{i_1})= (1\otimes \lceil y_{i_1}\rceil-\lceil y_{i_1}\rceil\otimes 1), \forall \lambda_{i_1}\in \Lambda_1 &d_k(\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes \lceil y_{i_j}\rceil+(-1)^k\lceil y_{i_j}\rceil\otimes 1), \forall \lambda_{i_1\cdots i_k}\in \Lambda_k,
\lim_{n\rightarrow \infty}\int_{0}^{\pi/2}\frac{\phi^{1-\frac{1}{n}}(\theta)\,\sin{\theta}}{\left(\pi-\theta\right)^{2}\,\phi^{2}\left(\pi-\theta\right)}\,d \theta\,=\,\int_{0}^{\pi/2}\frac{\phi(\theta)\,\sin{\theta}}{\left(\pi-\theta\right)^{2}\,\phi^{2}\left(\pi-\theta\right)}\,d \theta\,\lesssim\,1.
\sum_{k=1}^2 (s_{b+1,k}- s_{b,k}) = \frac{b+3}{b+2}x_{b+1}- \frac{b+2}{b+1}x_b &= \frac{b+3}{b+2}\left(x_b + \frac{1}{b}\right) - \frac{b+2}{b+1}x_b &= \frac{1}{b}+ \frac{1}{b(b+2)}- \frac{x_b}{(b+1)(b+2)}&> \frac{1}{b}- \frac{x_b}{(b+1)(b+2)}> \frac{1}{b}- \frac{x_b}{b(b-1)}.
I_{5}= \frac{- 1 6 \pi^{\frac{D}{2}}\Gamma ( \frac{D}{2}- 1 )}{( 4 - D )^{2}}\left[ ( n_{0}^{2}+ i \eta^{\prime})^{2 - \frac{D}{2}}+ ( - n_{0}^{2}+ i \eta^{\prime})^{2 - \frac{D}{2}}\right] , \eta^{\prime}> 0 .
E_{i n}= \frac{1}{2 R}( 0 . 0 0 8 8 7 3 + \frac{0 . 0 0 1 0 1 0}{\varepsilon}) , \hspace{2 c m}E_{o u t}= \frac{- 1}{2 R}( 0 . 0 0 3 2 3 4 + \frac{0 . 0 0 1 0 1 0}{\varepsilon}) .
\mathsf{f}= \mathsf{f}(y, \mathsf{m}) = 2 \mathsf{m}\pi \frac{\mathsf{H}(\|y\|)}{\mathsf{H}(b)}, \mathsf{H}(r) = \int_a^r \frac{dz}{z^{n+1}H(z,z^2)}.
\cdot\cdot\cdot=\sqrt[3]{|z_{0}|}e^{i(\frac{\pi}{3}+\frac{\epsilon}{3})}=\sqrt[3]{|z_{0}|}e^{i\frac{\pi}{3}}e^{i\frac{\epsilon}{3}}=\sqrt[3]{z_{0}}e^{i\frac{\epsilon}{3}}=g(z_{0})e^{i\frac{\epsilon}{3}}
0 = -\cos(\eta_1+\frac{\pi}{3}) - \cos(\theta_2+\frac{2\pi}{3}) + \cos \eta_2 + \cos(\theta_4 + \frac{\pi}{3}) - \cos(\eta_3 - \frac{\pi}{3}) - \cos \theta_3, 0 = -\sin(\eta_1+\frac{\pi}{3}) - \sin(\theta_2+\frac{2\pi}{3}) + \sin \eta_2 + \sin(\theta_4 + \frac{\pi}{3}) - \sin(\eta_3 - \frac{\pi}{3}) - \sin \theta_3.
\mathbb{E}[ G_{n-2}^{j}(z) Z^{j}(n-2,1)] ~ \leq e^{\frac{2+z}{2}}\left(\frac{n-3}{j-1}\right)^{\frac{2+z}{2}}\left( \mathbb{E}[ G_{j-1}^{j}(z)] + \frac{e^{\frac{1}{2}}}{2}(j-1)^{-\frac{3+z}{2}}\sum_{k=j-1}^{n-4}\left( \frac{1}{k}\right)^{\frac{3+z}{2}}\right)+ \frac{1}{2}e^{\frac{1}{2}}\left(\frac{1}{(n-3)(j-1)}\right)^{\frac{1}{2}}
|\hat{F}_2^l(\beta_1)-\hat{F}_2^l(\beta_2)|_0 \le |\hat{F}_2^{l+1}(\beta_1)-\hat{F}_2^{l+1}(\beta_2)|_0 +c \beta_1^{-\frac{1}{2}}c_0^{-1}M^{-l-1}\alpha^{-4}&\le |\hat{F}_2^{N_+}(\beta_1)-\hat{F}_2^{N_+}(\beta_2)|_0+c \beta_1^{-\frac{1}{2}}c_0^{-1}\sum_{j=l}^{N_+-1}M^{-j-1}\alpha^{-4}&\le c\beta_1^{-\frac{1}{2}}c_0^{-1}M^{-l-1}\alpha^{-4}.
\frac{w_j+v}{1+\overline{w}_jv}=w_j+(1-r_j^2)\frac{v}{1+\overline{w}_jv}&=w_j+(1-r_j^2)(v-\overline{w}_jv^2+\overline{w}_j^2v^3-\overline{w}_j^3v^4+\overline{w}_j^4v^5-\cdots).
f(t) =\omega_0^{(\Lambda,N)}(A(\phi)\tilde{B}^{(\Lambda)}(t;\phi);\phi) =\frac{1}{q}\sum_{m=1}^q\sum_{n\ge 1}\langle\Phi_{0,m}^{(N)}(\phi),A(\phi)\Phi_n^{(N)}(\phi)\rangle\langle \Phi_n^{(N)}(\phi),B(\phi)\Phi_{0,m}^{(N)}(\phi)\rangle \qquad\times \exp[i(E_n^{(N)}(\phi)-E_{0,m}^{(N)}(\phi))t].
\left\langle{^0 t},{( l^{\alpha \beta})^{\rho}}_{\mu}x^{\mu}\partial_{\rho}w x^{\alpha_{1}}x^{\alpha_{2}}\right\rangle ={( l^{\alpha \beta})^{\alpha_{1}}}_{\sigma}c^{\sigma \alpha_{2}}+{( l^{\alpha \beta})^{\alpha_{2}}}_{\sigma}c^{\sigma \alpha_{1}}.
|x-a|x|^2|^{\frac{2k-n}{2k}}=\displaystyle|x|^{\frac{2k-n}{2k}}+\frac{n-2k}{2k}a\cdot x|x|^{\frac{2k-n}{2k}}+O''(|a|^2|x|^{\frac{6k-n}{2k}})
g_2:=\frac{i}{\pi}\sum_{j=1}^N \lambda_j\Big(\frac{2z_{tt}+i-\ddot{z}_j}{(z(\alpha,t)-z_j(t))^2}-2\frac{(z_t-\dot{z}_j(t))^2}{(z(\alpha,t)-z_j(t))^3}\Big).
\left( \widehat{p}- \widehat{q}\right) F_{1}\left( s^{\prime}, t ; \left( p - q \right)^{2}, \left( k - q \right)^{2}\right) + F_{2}\left( s^{\prime}, t ; \left( p - q \right)^{2}, \left( k - q \right)^{2}\right) .
Q ( \theta , \varphi , u ) = - \frac{1}{4 \pi}\int d^{2}{\bar{\cal S}}\int_{0}^{\infty}d{\bar{r}}( \log{\bar{r}}) \frac{\partial}{\partial{\bar{r}}}\Bigl [{\bar{r}}^{3}{\cal Y}({\bar{r}},{\bar{\theta}},{\bar{\varphi}},{\bar{u}}) \Bigl |_{{\bar{u}}= f^{*}}\Bigr ] + \; O [ X \times \Re ] \; , \quad
\frac{D^q_j}{\gamma_{j,q}}=\prod_{k=1}^{n^-}\left(\frac{\vartheta_k-\gamma_{j,q}}{\vartheta_k}\right)^{n_k}\prod_{k=1,k \neq j}^{N}\left(\frac{\gamma_{k,q}}{\gamma_{k,q}-\gamma_{j,q}}\right),\ \for \ 1\leq j \leq N.
(\hat d_{2}^{i+1}\circ \frak h_{ab}^{i+1}+\frak h_{ab}^{i+1}\circ \hat d_{1}^{i+1})_{\alpha'_2\alpha_1}=(\frak n_b^{i+1}- \frak n_a^{i+1})_{\alpha'_2\alpha_1}.
\hat{H_{q}}= \frac{\hbar^{2}( I - I_{3})}{2 I I_{3}}J_{3}^{\prime 2}+ \frac{\hbar^{2}}{2 I}\left( \frac{\gamma}{\sinh \gamma}\hat{J}_{+}^{\prime}\hat{J}_{-}^{\prime}+ [ \hat{J}_{3}^{\prime}] [ \hat{J}_{3}^{\prime}+ 1 ] \right) .
\begin{aligned}4\epsilon^{-1}\,T\left(\|m_0f_1\|^2_{L^\infty(0,T;L^2(I))}+ \|u_1\|^2_{X}+ \|u_2\|^2_X + \frac{\epsilon^2}{4}\,\right) \leq \frac{1}{2}.\end{aligned}
\frac{d}{dt}\nabla \sigma_t =(1-q)\, \sigma_t^{-q}\, \left[(1-t)\, \nabla u\, u^{q-1}+t\, \nabla v\, v^{q-1}\right]\,\frac{d}{dt}\sigma_t &+\sigma_t^{1-q}\, \left[\nabla v\, v^{q-1}-\nabla u\, u^{q-1}\right].
\sum_{n=0}^{\infty}p_{-(25r+k)}(5n+5-t)q^{n}\equiv m\left(25r+5s+t,5r+s+1\right)\dfrac{E_{5}^{5r+s-t+6}}{E_{1}^{30r+6s+6}}\equiv m\left(25r+5s+t,5r+s+1\right)\dfrac{E_{5}^{5-r-t}}{E_{1}^{s+1}}.
A'=\begin{cases}\displaystyle{A_k^{-1}\left(\prod_{j \in I}A_j^{[ b_{jk}]_+}+ \prod_{j \in I}A_j^{[-b_{jk}]_+}\right)}\mbox{if $i=k$}, A_i \mbox{if $i \neq k$}, \end{cases}
\begin{aligned}\left(\frac{\partial u_h}{\partial t}, \phi_h\right) + a_h(u_h,\phi_h) + \sqrt{\epsilon}\,b_h(w_h,\phi_h) + \left(\rho_h u_h, \phi_h\right) &= \left(\rho_h V_h, \phi_h\right) \forall\,\, \phi_h \in X_h,\end{aligned}
{\rm Tr}\; S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)^{2n+1+2\epsilon}&={\rm Tr}\;\left(\begin{matrix}(\mathcal{P}_{\rm D}-\mathcal{U}^\ast \mathcal{P}_{\rm D}\mathcal{U})^{2n+1+2\epsilon}0 \cr0 -(\mathcal{P}_{\rm D}-\mathcal{U}\mathcal{P}_{\rm D}\mathcal{U}^\ast)^{2n+1+2\epsilon}\cr\end{matrix}\right).
E_2:=\Biggl\{\sum_{j=1}^{\chi-1}(R_j-R_j')\le \sqrt{\omega \chi}\Biggr\}\bigcap\bigcup_{\chi\le j\le k}\!\!\bigl\{|R_j|+|R_j^\prime|\le2 j^{-1/2}(\log j)^{\alpha}\bigr\},
|g(x,t)| \leq\int_{5I_Q}\frac{|\partial_u^M\varphi(x,u)-\partial_u^M\varphi(x,t)|}{|u-t|^{\frac1{2s}-M+1}}du+\|\partial_t^M\varphi\|_\infty\int_{(5I_Q)^c}\frac{du}{|u-s|^{\frac1{2s}-M+1}}=T_1+T_2.
K\varphi\triangleq\begin{cases}C_{H}\cdot I_{0^{+}}^{1}\left(t^{H-\frac{1}{2}}\cdot I_{0^{+}}^{H-\frac{1}{2}}\left(s^{\frac{1}{2}-H}\varphi(s)\right)(t)\right), H>\frac{1}{2}; C_{H}\cdot I_{0^{+}}^{2H}\left(t^{\frac{1}{2}-H}\cdot I_{0^{+}}^{\frac{1}{2}-H}\left(s^{H-\frac{1}{2}}\varphi(s)\right)(t)\right), H\leq\frac{1}{2},\end{cases}
\bar f(s) = s\xi'(s)-\xi'(s) +1-\xi(s) +\frac{[\xi'(1)-\xi'(s)](1-q)}{z_2}\ -\frac{(1-q)(1+z_2)(\xi'(1)-\xi'(q))}{z_2^2}\log\frac{1+z_2}{1+\frac{z_2(\xi'(s)-\xi'(q))}{\xi'(1)-\xi'(q)}}.
\hat U_{k+1}= \left( \sqrt{a_k}+ \sqrt{b_k}\right)^2 &= \left( \alpha_k \sqrt{c^2+2\delta^2 G^2+\tilde M\delta^2 \hat U_k}+ \sqrt{\left(1-2m\alpha_k+\tilde M\alpha_k^2\right)\hat U_k+2G^2\alpha_k^2}\right)^2
k_R = \frac{1}{Nn}\sum_{\alpha =1}^n\sum_{\bf i}\frac{\partial^2V}{\partial (\varphi^\alpha_{\bf i})^2}= 2Jd -m^2 + \lambda\frac{n+2}{Nn}\sum_{\alpha =1}^n\sum_{\bf i}(\varphi^\alpha_{\bf i})^2~.
R ( \theta ) = \left( \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\frac{1}{( \theta + 1 )}}}&{{\frac{\theta}{( \theta + 1 )}}}&{{0}}\\{{0}}&{{\frac{\theta}{( \theta + 1 )}}}&{{\frac{1}{( \theta + 1 )}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) .
I[\mu-\nu] = \iint f(\theta - \phi) d \mu(\theta) d\mu (\phi) + \iint f(\theta - \phi) d \nu(\theta) d\nu (\phi) - \iint f(\theta - \phi) d \mu(\theta) d\nu (\phi) - \iint f(\theta - \phi) d \nu(\theta) d\mu (\phi) \in [-\infty, \infty)
\sum_{n=1}^{\infty}(n)_{p,\lambda}\bigg(\frac{1}{1-x}-\sum_{l=0}^{n}x^{l}\bigg)=\sum_{n=1}^{\infty}(n)_{p,\lambda}\frac{x^{n+1}}{1-x}&=\frac{x}{1-x}\sum_{n=1}^{\infty}(n)_{p,\lambda}x^{n}=\frac{x}{(1-x)^2}F_{p,\lambda}\Big(\frac{x}{1-x}\Big).
\|A_{x_n}\cdots A_{x_1}\|^s \leq Ce^{nP(\mathsf{A},s)}\mu([x_1\cdots x_n]) &=Ce^{nP(\mathsf{A},s)}\mu([y_1\cdots y_n])\leq C^2\|A_{y_n}\cdots A_{y_1}\|^s
\frac{\partial}{\partial t}h_i^j= -\nabla^j\nabla_iH^{-\alpha}-H^{-\alpha}h_i^kh_k^j+H^{-\alpha}\delta_i^j = -\nabla^j\nabla_iH^{-\alpha}-H^{-\alpha}\mathring{h}_i^k\mathring{h}_k^j-H^{1-\alpha}\mathring{h}_i^j+H^{-\alpha}(1-\frac{H^2}4)\delta_i^j,
P_{\alpha}^{\beta}=\gamma^{\beta\gamma}P_{\gamma\alpha},P_{\alpha\beta}=\lambda_{\alpha\beta,\gamma}^{\gamma}-\lambda_{\gamma\alpha,\beta}^{\gamma}+\lambda_{\alpha\beta}^{\gamma}\lambda_{\gamma\mu}^{\mu}-\lambda_{\alpha\mu}^{\gamma}\lambda_{\beta\gamma}^{\mu}.
\left( x_{1},\alpha^{m_{1}}\right) \left( x_{2},\alpha^{m_{2}}\right)=\left( x_{1}\alpha^{-m_{1}}x_{2},\alpha^{m_{1}+m_{2}}\right), \ \ \left( x_{1},\alpha^{m_{1}}\right) ,\left( x_{2},\alpha^{m_{2}}\right)\in F.
t(x)\vert_{\mathcal{U}_i \setminus S}=(R_i(x) \cdot E_i^\sigma) \vert_{\mathcal{U}_i \setminus S}(x) s(x)\vert_{\mathcal{U}_i \setminus S}=(R_i'(x) \cdot F_i^\sigma) \vert_{\mathcal{U}_i \setminus S}(x),
\delta{\cal I}_{U ( 1 )}= - \lim_{\beta \rightarrow 0}2^{N / 2 - 1}\int [ d X ] \; \frac{e^{- ( X + X )^{2}/ 2 \beta}}{( 2 \pi \beta )^{( D - 1 ) / 2}}= - \frac{2^{N / 2 - 1}}{2^{D - 1}}= - \frac{1}{4},
\lim_{\varepsilon \to +0}\int_{0}^{\infty}\Bigg( h(u) - C - \sum_{i=1}^{r}\frac{C_{i}}{u^{\alpha - \alpha_{i}}}\Bigg) \Phi(v - u) e^{- \varepsilon u}u^{\alpha -1}du &= \int_{\mathbb{R}}\left( h(u) - C - \sum_{i=1}^{r}\frac{C_{i}}{u^{\alpha - \alpha_{i}}}\right) \Phi(v - u) u^{\alpha -1}du .
\int_0^t \|H(\tau) U_{Q}(\tau,t)Q(t)U(t,s)x\|^p d\tau = \int_0^s \|H(\tau) U_{Q}(\tau,t)Q(t)U(t,s)x\|^p d\tau + \int_s^t \|H(\tau) U_{Q}(\tau,t)Q(t)U(t,s)x\|^p d\tau = \int_0^s \|H(\tau) U_{Q}(\tau,s)Q(s)x\|^p d\tau + \int_s^t \|H(\tau) U(\tau,s)Q(s)x\|^p d\tau.
\sigma_{j_1,j_2}= \sum_{\pi \in \mathcal{P}_2(2k_1,2k_2)}f^{-}_I(\pi) + (\kappa-1) \sum_{\pi \in \mathcal{P}_{2,4}(2k_1,2k_2)}(f^-_{II}(\pi)+f^+_{II}(\pi)),