harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q(t) = (1-t)\det\begin{bmatrix}s'_{0}s'_{1}\ldots s'_{m-1}1 s'_{1}s'_{2}\ldots s'_{m}t \vdots \vdots \vdots \vdots \vdots s'_{m}s'_{m+1}\ldots s'_{2m-1}t^{m}\end{bmatrix},
\frac{d{f_2(q,z_2)}}{d q}= [-\xi'(1)-\xi'(q)-(1-q)\xi''(q)] \Big( \frac{1+z_2}{z_2^2}\log(1+z_2)-\frac1{z_2}\Big)+\xi''(q)(1-q) -(1-q)[\xi'(1)-\xi'(q)]\Big(\frac{2+z_2}{z_2^3}\log(1+z_2)-\frac2{z_2^2}\Big) \frac{d z_2}{d q}=0.
E =\frac14(\rho_1^4+\rho_2^4-\sigma^4) &\qquad+\frac12\frac{\rho_1^2(\rho_1^2-\sigma^2)^2+\rho_2^2(\sigma^2-\rho_2^2)^2+2\rho_1\rho_2(\rho_1^2-\sigma^2)(\sigma^2-\rho_2^2)\cos(\varphi_1-\varphi_2)}{(\rho_1^2-\rho_2^2)^2}&=\frac14+\frac12|p|^2\ .
Z[{\overline{A}}]_{0}=\exp\left[-\int\!\!d^{^{D}}\!\!x_{_{E}}\left\{{m^{2}\over2g^{2}}+{1\over2(2\pi)^{{D\over2}}}\lim_{s\rightarrow0}\int_{0}^{\infty}\!\!d\tau\,\tau^{s-{D\over2}-1}{\mathrm{e}}^{-\tau m^{2}}G(\tau F)\right\}\right],
\mathcal{P}={\displaystyle\iiiint}\psi\left( k_{1},k_{2},t\right) \psi\left( k_{3},k_{4},t\right) \psi^{\ast}\left( k_{1},k_{4},t\right) \psi^{\ast}\left( k_{3},k_{2},t\right)dk_{1}dk_{2}dk_{3}dk_{4},
\int_{0^-}^{\infty}e^{-s x}dG_1(x)=\frac{\mathbb E\left[e^{-s \overline{X}_{e(q)}}\right]}{\mathbb E\left[e^{-s \overline{X}_{e(p+q)}}\right]}=e^{\int_{0}^{\infty}(e^{-s x}-1) \Pi_1(dx)}, \ \s\geq 0,
W_{\mathrm{X}}\tilde{\Sigma}= 0 \, \, \, , \, \, \, W_{\mathrm{i s o}}^{a}\tilde{\Sigma}= 0 \, \, \, , \, \, \, W_{\mathrm{c h i r a l}}^{a}\tilde{\Sigma}= 0 \, \, \, ,
\sphericalangle\big(P(y,r_{x,k}),P(y,r_{x,k+i})\big) \leq \sum \limits_{l=0}^{i-1}\sphericalangle\big(P(y,r_{x,k+l}),P(y,r_{x,k+l+1})\big) \leq \tilde C C_1(m) \sum \limits_{l=0}^{i-1}(\delta_{x,k+l+1}+2 C \delta_{x,k+l}) \xrightarrow[k \to \infty]{}0,
\det\left( 1-2i\alpha~\boldsymbol{\hat{n}\cdot J}\right) ={\displaystyle\prod\limits_{m=0}^{2j}}\left( 1-2i\alpha\left( j-m\right) \right) ={\displaystyle\prod\limits_{n=1}^{\left\lfloor j+1/2\right\rfloor}}\left( 1+4\alpha^{2}\left( j+1-n\right)^{2}\right) \ .
S =\left\lfloor\frac{4n+2k-2}{m}\right\rfloor+3\left\lfloor\frac{k}{m}\right\rfloor+\left\lfloor\frac{2n}{m}\right\rfloor- 3\left\lfloor\frac{2k}{m}\right\rfloor-\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1}{m}\right\rfloor &\quad-2\left\lfloor\frac{n-k}{m}\right\rfloor-\left\lfloor\frac{2n+k-1}{m}\right\rfloor.
\frac{d}{dt}\begin{bmatrix}\tilde{y}(t) \phi_0(t) \vdots \phi_{N-1}(t) \end{bmatrix}=A_N\begin{bmatrix}\tilde{y}(t) \phi_0(t) \vdots \phi_{N-1}(t) \end{bmatrix},\;\;\begin{bmatrix}y(0) \phi_0(0) \vdots \phi_{N-1}(0) \end{bmatrix}=\begin{bmatrix}y_0 e_1\end{bmatrix}
\left| v \mapsto \sum_{n=1}^\infty \gamma^n \left( \rho_{[\xi]_n}(v) \right) \cdot g' \left( \rho_{[\xi]_n}(v) \right) \right|_\alpha \leqslant \frac{| g' |_{\mathrm{pw}, \alpha}+ C_2}{1 - \| \gamma \|_\infty}.
D^k_h(s) := \frac{(2\pi)^{s + \frac{k-1}{2}}}{\Gamma(s + \frac{k-1}{2})}\langle P_h^k(\cdot, s), V\rangle = \sum_{m \in \mathbb{Z}}\frac{r_{2k+1}(m^2 + h)}{(2m^2 + h)^{s + \frac{k-1}{2}}}- \mathfrak{E}_h^k(s).
\lambda_{1,m}\int_{-\pi/2}^{\pi/2}\cos^2(x) = \frac{(n-1)(n-3)}{4}\sum_{i+j=m-1}a_i\int_{-\pi/2}^{\pi/2}x^{2i}y_{1,j}\cos(x)-\sum_{\substack{i+j=m i,j <m}}\lambda_{1,i}\int_{-\pi/2}^{\pi/2}y_{1,j}\cos(x)
p(X(t)|Y_{t}) =p[X(t)|(y(t),Y_{t-1})]=\frac{p(y(t)|X(t)|Y_{t-1})p(X(t)|Y_{t-1})}{p(y(t)|Y_{t-1})}&=\frac{p(y(t)|X(t))}{p(y(t)|Y_{t-1})}p(X(t)|Y_{t-1})
\nabla^{\prime}(\frac{1}{|r-r^{\prime}|})\equiv(e_{x}\frac{\partial}{\partial x^{\prime}}+e_{y}\frac{\partial}{\partial y^{\prime}}+e_{z}\frac{\partial}{\partial z^{\prime}})(\frac{1}{|r-r^{\prime}|})=\frac{r-r^{\prime}}{|r-r^{\prime}|^{3}}
{\cal L}= \sum_{k = 1}^{2}[ \overline{{{\psi}}}_{k}{\gamma}^{\mu}( i{\hbar}c{\partial}_{\mu}- e_{k}A_{\mu}){\psi}_{k}- m_{k}c^{2}\overline{{{\psi}}}_{k}{\psi}_{k}] - \frac{1}{4}F_{\mu \nu}F^{\mu \nu},
\mathcal{P}_s = \mathcal{X}_s^\top \mathcal{J}\mathcal{X}_s^{}\mathcal{J}= \begin{bmatrix}I 0 0 - I \end{bmatrix}, \mathcal{D}_s = \begin{bmatrix}\hat{A}_s 0 0 \hat{S}\end{bmatrix}, \mathcal{X}_s = \mathcal{D}_s^{1/2}\begin{bmatrix}I \hat{A}^{-\top}B^\top 0 I \end{bmatrix}.
\begin{cases}\partial_t f(t,m,x) + \frac{1}{2}\partial_{xx}^2 f(t,m,x) \ = \ 0, \qquad\;\, \forall\,(t,m,x)\in[0,T[\times\overline Q, f(T,m,x) \ = \m, \forall\,(m,x)\in\overline Q.\end{cases}
\{x : 0 \le c_{k}(x) \le \tilde r_{k}, 0 \le c_{k+1}(x) \le \tilde r_{k+1}\}&= \bigcup_{i=1}^{(\tilde r_k +1) q_{k-1}}\Big( I_{k+1}(i) \cup I_{k+1}(i + q_k) \cup \dots \cup I_{k+1}(i + \tilde r_{k+1}q_k) \Big) &= \bigcup_{i=1}^{(\tilde r_k +1)q_{k-1}}\big( i\theta - \| q_k \theta\|,\i\theta + \tilde r_{k+1}\|q_k \theta\| + \|q_{k+1}\theta\| \big),
p_{k}=\left(\frac{(L_{\nabla h}-\sigma)L^2_{\nabla g}}{2}\right)\frac{\lambda^2_{k-1}}{\lambda_k}+\frac{1}{2}\left(\frac{\sigma}{\lambda_k}-L_{\nabla g}\right)-p_{k-1}, M_{1,k}=p_{k-1}-\frac{\alpha_k+\sigma L_{\nabla g}\lambda_{k-1}}{2\lambda_k}-\frac{(L_{\nabla h}-\sigma)L^2_{\nabla g}\lambda_{k-1}^2}{2\lambda_k},
\mathfrak{b}^0[\mathbf{u},\mathbf{u}]\geqslant c_* \Vert \mathbf{D}\mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)}-(c_0+c_4)\Vert \mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)},\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\int_{\sum_{i=1}^{d}i^{-2\alpha-1}x_{i}^2 \leq d^{-2\alpha}}\frac{\exp\{-\sum_{i=1}^{d}x_{i}^2 /2 \}}{(2\pi)^{d/2}}\mathrm{d}x\geq \exp\{-c_{3}d\}
(\xi_1'-\xi_2')(y_\delta) \le \frac{p}{p-1}\xi_2(y_\delta)^{(p-1)/p}- \frac{p}{p-1}\xi_1(y_\delta)^{(p-1)/p}& = \frac{p}{p-1}\xi_2(y_\delta)^{(p-1)/p}- \frac{p}{p-1}\left( \xi_2(y_\delta) + \delta \right)^{(p-1)/p}< 0\ ,
w_i= \frac{\Gamma(m+i)(\mu K)^i m^m}{\Gamma(m)\Gamma(i+1)}\times\underset{I_2}{\underbrace{\frac{1}{\pi}\int_{0}^{\pi}\frac{\left( 1+\Delta \cos \theta \right)^i}{\left( \mu K\left( 1+\Delta \cos \theta \right)+m \right)^{m+i}}d\theta}}.
{\cal A}_{D_2}(k_1, \epsilon_1; k_2, \epsilon_2) = g^{2}_c e^{-\lambda}\int^{1}_{0}dy \langle c(z) V_{\epsilon_1}(z,\bar{z})c(w) \tilde{c}(\bar{w}) V_{\epsilon_2}(w,\bar{w}) \rangle |_{z=iy, w=i}~,
\begin{dcases}\partial_t \rho_{00}+ \nabla \cdot m_{00}= 0 \partial_t m_{00}+ \nabla \cdot \big(\frac{m_{00}\otimes m_{00}}{\rho_{00}}\big) + \nabla p_1(\rho_{00}) + \rho_{00}\nabla h_2^\prime(n_{00}) = 0 n_{00}= \rho_{00}\ , \end{dcases}
\left( \begin{array}{c c}{{\varphi}_{o u t}^{\prime}}\\{{{\varphi}^{\prime}}_{o u t}^{*}}\\ \end{array}\right) = \left( \begin{array}{c c}{- 1}&{0}\\{0}&{- 1}\\ \end{array}\right) \left( \begin{array}{c c}{\varphi_{i n}}\\{\varphi_{i n}^{*}}\\ \end{array}\right) ,
\Re \sum_{n=1}^{\infty}{\widehat{z}_n}^{j}\leq \frac{1}{(\sigma_0-1)^{2j}}- \frac{1}{(\sigma_0-\omega_0)^{2j}}+\Re \left[\frac{1}{\{(\sigma_0-1)+it_0\}^{2j}}-\frac{1}{\{(\sigma_0-\omega_0)+it_0\}^{2j}}\right].
M^{m+n}=\left(Q\setminus\coprod\limits_{i=1}^{+\infty}(B_{\frac45r_i}(o_i)\times_{g_i}S^{n-1})\right)\cup_{\textrm{Id}}\coprod\limits_{i=1}^{+\infty}P_i &=\left((\mathbb{R}^{m+1}\setminus\coprod\limits_{i=1}^{+\infty}D_i^{m+1})\times S^{n-1}\right)\cup_{\textrm{Id}}\coprod\limits_{i=1}^{+\infty}(S^m\times D^n)_i,
0 \leq \left\|h-\sum_{j\in \mathbb{J}}\Psi_j^*A_jh\right\|^2 = \left\langle h-\sum_{j\in \mathbb{J}}c_jA_j^*A_jh, h-\sum_{k\in \mathbb{J}}c_kA_k^*A_kh\right\rangle =\|h\|^2-2\sum_{j\in \mathbb{J}}c_j\|A_jh\|^2+\sum_{j\in \mathbb{J}}c^2_j\|A_jh \|^2=\|h\|^2-\sum_{j\in \mathbb{J}}(2c_j-c_j^2)\|A_jh \|^2,
H(Y|X,F)=H\left(\sum\limits_{k=1}^{Q}\omega_k\mathcal{CN}(z;0,\sigma_k^{2}) \right) \geqslant \sum\limits_{k=1}^{Q}\omega_k H\left(\mathcal{CN}(z;0,\sigma_k^{2}) \right) &=\frac{1}{2}+\sum\limits_{k=1}^{Q}\omega_k\log_2 (2\pi e\sigma_k).
\hat{M}:= S^1 \times \left(S^2 \setminus{\textstyle{\coprod\limits_{i=0}^n D^2_i}}\right),\;\;\;\;\;\;\partial_i \hat{M}:= -\partial (S^1 \times D^2_i),\;\;i \in \{0, \ldots, n\}, begin{align*}-3pt]
\theta =\left( \begin{array}{c}-\frac{\sqrt{2\omega_{o}}}{\mu +\lambda +\mu_{o}}d_{4}U(a,x) d_{2}U(a+1,x) -\frac{\sqrt{2\omega_{o}}}{\mu +\lambda +\mu_{o}}d_{2}U(a,x) d_{4}U(a+1,x)\end{array}\right) .
U_{M , N}( a ) ={\frac{2 \pi}{L}}\left( M -{\frac{1}{2}}-{\frac{g L a}{2 \pi}}\right)^{2}-{\frac{2 \pi}{L}}\left( N -{\frac{g L a}{2 \pi}}\right)^{2}+{\frac{\pi}{4 L}}+ V^{[ N = 2 ]}.
m_{00}=g_{32}\!-\!g_{20}\!-\!g_{13}\!-\!g_{29}\!-\!g_{9}\!-\!g_{24}\!+\!g_{1}\!-\!g_{4}, m_{11}=g_{1}\!+\!g_{9}\!+\!g_{20}\!+\!g_{24}\!+\!g_{13}\!-\!g_{29}\!+\!g_{32}\!-\!g_{4}, m_{22}=g_{20}\!+\!g_{29}\!+\!g_{4}\!+\!g_{13}\!-\!g_{9}\!-\!g_{24}\!+\!g_{1}\!+\!g_{32}, m_{33}=g_{9}\!+\!g_{24}\!+\!g_{29}\!-\!g_{13}\!+\!g_{1}\!+\!g_{4}\!-\!g_{20}\!+\!g_{32},
\{S^{p}, T^{r}\}= \left( \sum_{k_{p}= 0}^{r}. . . \sum_{k_{2}= 0}^{k_{3}}\sum_{k_{1}= 0}^{k_{2}}\alpha^{k_{1}}\alpha^{k_{2}}. . . \alpha^{k_{p}}\right) S^{p}T^{r}.
\frac{\lambda}{1+\lambda}I_{1+\lambda}(P_V, P_{Y|V}) = & \log \Bigg(\bigg(\frac{P_V(1)}{2^{1+\lambda}}\bigg)^{\frac{1}{1+\lambda}}+ \bigg(1-\frac{2^{1+\lambda}-1}{2^{1+\lambda}}P_V(1)\bigg)^{\frac1{1+\lambda}}\Bigg)
\nabla_{i}e^{0}=\coth(r)e^{i}, \nabla^{\ast}\nabla \alpha =-\alpha_{0}^{\prime\prime}e^{0}-\alpha_{i}^{\prime\prime}e^{i}-(m-1)\coth(r)(\alpha_{0}^{\prime}e^{0}+\alpha_{i}^{\prime}e^{i}) +\mathcal{D}\alpha
H_{}^{i,(l)}= \nabla^2 \bar f_i(w^{i,(l)}) +\sum_{j = 1}^{m_i}\! \lambda^{i,(l)}_j \nabla^2 \bar G_j^i(w^{i,(l)})+ \sum_{j=1}^{m_i}\frac{\lambda^{i,(l)}_j}{s^{i,(l)}}\nabla \bar G_j^i(w^{i,(l)})\left(\nabla \bar G_j^i(w^{i,(l)})\right)^T,
G(n) := \left.\prod_{i \ge 0}\left( \prod_{s=-2n+4i+1}^{-n+2i}(2n+s)\prod_{s=n-2i}^{2n-4i-2}(2n+s) \right) \middle/ \prod_{i=1}^{n-1}(2i+1)^{n-i}\right. = \left.\prod_{i \ge 0}\left( \prod_{s=4i+1}^{n+2i}s \prod_{s=3n-2i}^{4n-4i-2}s \right) \middle/ \prod_{i=1}^{n-1}(2i+1)^{n-i}\right..
- \Delta u_k \leq f^* \vee (-\Delta u_{k-1}) \leq f^* \vee \left( f^* \vee (-\Delta u_{k-2}) \right) = f^* \vee (-\Delta u_{k-2}) \leq \cdots \leq f^* \vee (-\Delta u_0)\mbox{a.e.~in}\Omega.
logM=\frac{1}{2\pi i}\int_{\Sigma}\frac{log2}{\zeta-z}d\zeta=\frac{log2}{2\pi i}\int_{-1-z}^{1-z}\frac{1}{\zeta}d\zeta=\frac{log2}{2\pi i}log\frac{z-1}{z+1},
\begin{aligned}\frac{\partial y}{\partial t}= \Re h(u_{[<\lambda^{\sigma}]}) \partial_xu_{[<\lambda^{\sigma}]}- \Re \int_{-\infty}^x h_1(u_{[<\lambda^{\sigma}]}) (\partial_x u_{[<\lambda^{\sigma}]})^2\, dx'.\end{aligned}
\sum_{y_{t}}\Big(\log\Big(\frac{r_t(x_t|y^{t-1}_{t-M},y_t)}{{\pi}_{t}(x_t|y^{t-1}_{t-J})}\Big)+C_{t+1}(y^t_{t+1-J})\Big)q_t(y_t|y^{t-1}_{t-M},x_t)-s\gamma_t(x_t,y^{t-1}_{t-N})=1-\lambda_t(y^{t-1}_{t-J}),~\forall{x_t}\in{\cal X}_t.
\lim_{u\to\infty}u^{(\theta/\alpha-1 )}\int_{0}^{u}\mathbf{P}_{0,i}(I>s){\rm d}s=v_i(\theta)\sum_{j=\pm1}\frac{\pi_j^\theta}{\mu_{\theta}|\alpha-\theta|v_j(\theta)}\mathbf{E}_{0,j}[I^{\theta/\alpha-1}],
t_{j}=\frac{1}{2\cos^{2}(\frac{\pi j}{2H})}\prod_{i=1}^{j}\frac{\tan (\frac{\pi (2w_{r}+2i-1)}{4H})\tan (\frac{\pi (2w_{r}-2i+1)}{4H})}{\tan^{2}\left(\frac{\pi i}{2H}\right)}.
h = \left( \begin{array}{c c}{1}&{0}\\{X}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{e^{- \frac{1}{2}\tilde{\Phi}}}&{0}\\{0}&{e^{\frac{1}{2}\tilde{\Phi}}}\\ \end{array}\right) \left( \begin{array}{c c}{1}&{Y}\\{0}&{1}\\ \end{array}\right) .
\frac{a_{o}^{\prime}}{a_{o}}\left( \frac{a_{o}^{\prime}}{a_{o}}+ \frac{n_{o}^{\prime}}{n_{o}}\right) - \frac{b_{o}^{2}}{n_{o}^{2}}\left( \frac{\dot{a}_{o}}{a_{o}}\left( \frac{\dot{a}_{o}}{a_{o}}- \frac{\dot{n}_{o}}{n_{o}}\right) + \frac{\ddot{a}_{o}}{a_{o}}\right) - k \frac{b_{o}^{2}}{a_{o}^{2}}= - \frac{1}{6 M^{3}}\Lambda b_{o}^{2}+ \frac{1}{6 M^{3}}T_{5 5},
(wt_{sp}(C), wt_{RT}(C))=\begin{cases}(3,2) \mbox{$C$ is given in Example \ref{09_10_2020_ex_2}(i)}, (2,1) \mbox{$C$ is given in Examples \ref{09_10_2020_ex_2}(ii) and \ref{09_10_2020_ex_3}(i)}, (4,4) \mbox{$C$ is given in Example \ref{09_10_2020_ex_3}(ii)}. \end{cases}
-i\bar{\kappa}\Gamma^{a}\kappa ={\textstyle\frac{32}{R_{5}}}\tilde{\mathcal{C}}[\Gamma_{s}(\hat{M}^{\hat{a}\hat{b}})k_{(\hat{a}\hat{b})}+\Gamma_{s}(\tilde{M}^{\tilde{a}\tilde{b}})k_{(\tilde{a}\tilde{b})}]\, .
4\int_{\mathcal{H}}\int_{\mathcal{H}}\psi(\|x-y\|^{2})d\eta_{t}(x)d\eta_{t}(y) = \int_{\mathcal{H}}\int_{\mathcal{H}}\psi(\|x-y\|^{2})d[\delta_{tv_{\mu}}- \delta_{-tv_{\mu}}](x)d[\delta_{tv_{\mu}}- \delta_{-tv_{\mu}}](y) &=2\psi(0) - 2\psi(4t^{2}\|v_{\mu}\|^{2})\geq 0.
T_{\mu \nu}\sp{\underline{a}}\,=\partial_{\mu}E^{\underline{a}}_{\nu}-\partial_{\nu}E^{\underline{a}}_{\mu}= \,\,-{\lambda\over 2}\,\epsilon_{\mu\nu\rho}\sum\limits_{\beta}\xi_{\beta}\sp{\underline{a}}\,\dot x\sp{\rho,\beta}\,\delta(\vec{x}-\vec{x}\sp{\beta}).
\sum_{\lvert 2m^2 + h - X \rvert \leq X}r_{2k+1}(m^2 + h) = \delta_{[h = a^2]}\bigg( 2 R'_h X^{\frac{1}{2}}\log X -4 R'_h X^{\frac{1}{2}}\bigg) + 2R_{1/2, h}^{1/2}X^{\frac{1}{2}}+ O(X^{\frac{1}{2}-\frac{1}{44}+ \epsilon}),
a^{(n)}(t;x,y) := \left(a^{1,(n)}(t),\sum_ja^{2,(n)}_j(t)f_j(y),a^{3,(n)}(t),\sum_ia^{4,(n)}_i(t)f_i(x),\sum_{ij}a^{5,(n)}_{ij}(t)f_i(x)f_j(y),\sum_ia^{6,(n)}_i(t)f_i(x)\right)
\gamma^I_J s^L = \delta^L_J s^I + \sum_{L_1 L_2 = L}\delta^{L_1}_J s^{I L_2}+ \sum_{L_1 L_2 = L}\delta^{L_2}_J s^{L_1 I}+ \sum_{L_1 L_2 L_3 = L}\delta^{L_2}_J s^{L_1 I L_3}.
\begin{array}{rcl}{\cal L}_{2G}= \displaystyle-\frac{1}{4}Z_1Z_1+\frac{1}{2}m_Z^2Z_{\mu}Z^{\mu}-\frac{1}{4}A_1A_1,{\cal L}_{3G}= \displaystyle\frac{1}{2}ig\left[(c^2-s^2)Z_1Z_2+cs(A_1A_2+A_1Z_2+Z_1M)\right],{\cal L}_{4G}= \displaystyle\frac{1}{4}g^2\left[(c^6+s^6)Z_2Z_2+c^2s^2(A_2A_2+2Z_2A_2+MM)+2cs(c^2-s^2)Z_2M\right],\end{array}
\varepsilon ( x )_{\mid s t a t i c}= \frac{1}{2}\phi{'}^{2}+ U ( \phi ) = 2 U ( \phi ) = \phi{'}^{2}= \frac{\mu^{4}}{4 \lambda}\frac{1}{\cosh^{4}[ \frac{\mu}{\sqrt{2}}( x - x_{0}) ]},
\left(\int_\Omega f_\alpha (\tau_\alpha)\, d \mu(\alpha)\right)^2 =(\operatorname{Tra}(S_{f,\tau}))^2=\left(\sum_{k=1}^d \lambda_k\right)^2\leq d \sum_{k=1}^d \lambda_k^2 =d\operatorname{Tra}(S^2_{f,\tau})=d\int_\Omega\int_\Omega f_\beta (\tau_\alpha)f_\alpha (\tau_\beta)\, d \mu(\alpha)\, d \mu(\beta).
\hat{Z}_l=\sum_{1\leq i<j}^{n_l}L_{ij}^2-\bigg(\sum_{i=1}^{l}r_i^2 \bigg)\bigg(\sum_{i=1}^{l}\frac{1}{r_i^2}f_i(\Omega_i) \bigg),~~~l=2,\cdots,N.
\sum_{r,s=1}^n \dot{h_{rs}}\frac{\partial \xi_j^a (z,0)}{\partial z_r}\frac{\partial \xi_j^b(z,0)}{\partial z_s}-\dot{g}_{ab}^j-\sum_{c=1}^n \dot{\xi}_j^c\frac{\partial g_{ab}(\xi_j,0)}{\partial \xi_j^c}+\sum_{c=1}^n (g_{cb}^j(\xi_j,0)\frac{\partial \dot{\xi}_j^a}{\partial \xi_j^c}+g_{ac}^j(\xi_j,0)\frac{\partial \dot{\xi}_j^b}{\partial \xi_j^c})=0
Q^{x A}= 0 , \quad \bar{Q}_{A}^{l}= \left( \begin{array}{c c c | c c}{{\bar{a}_{1}}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\bar{a}_{r}}}&{{}}&{{}}\\ \hline{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{}}\\ \end{array}\right) .
\|v_{1}\|_{L^{2}(\Omega_{1})}^{2}\leq \|f_{1}\|_{L^{2}(\Omega_{1})}^{2}+|\mu|^{2}\|u_{1}\|_{L^{2}(\Omega_{1})}^{2}&\|v_{2}\|_{L^{2}(\Omega_{2})}^{2}\leq \|f_{2}\|_{L^{2}(\Omega_{2})}^{2}+|\mu|^{2}\|u_{2}\|_{L^{2}(\Omega_{2})}^{2}.
\{\{a,b\}',\gamma\}'+\{\{b,\gamma\}',a\}'+\{\{\gamma,a\}',b\}'= 0, \{\{a,\gamma\}',\eta\}'+\{\{\gamma,\eta\}',a\}'+\{\{\eta,a\}',\gamma\}'= 0, \{\{\gamma,\eta\}',\zeta\}'+\{\{\eta,\zeta\}',\gamma\}'+\{\{\zeta,\gamma\}',\eta\}'= 0.
P\left(\max_{\theta\in\mathcal{G}_n}|Z_n(\theta)|\leq q_n^Z(1-\alpha-\varepsilon_n)\right)=1-\alpha-\varepsilon_n\leq P(T_n^*\leq z|\mathcal{F}^X)(\omega)-\varepsilon_n\leq P\left(\max_{\theta\in\mathcal{G}_n}|Z_n(\theta)|\leq z\right),
|\mu^{-(p_k,q_k,0)}-\mu_3|\leq C' (C'\tilde\Delta(p_k,q_k))^{|P_k|L_{k}}, & \tilde\Delta(p_k,q_k)=\min\left\{|\mu^{(p,q,r)}-1|\colon 0<|r|+|q|<3(q_k+1),\right. &\qquad\qquad \left. (p,q,r)\neq 0,\pm( p_k,q_k,-1), \pm2(p_k,q_k,-1)\right\}.
\partial_t \widetilde{U}- \mathcal{L}_{\frac{\sigma}{1+\lambda}}\widetilde{U}= \sum_{j=1}^m \partial_j B(U) (\partial_t U_j - \mathcal{L}_\sigma U_j) & + \frac{\sigma \lambda}{1 + \lambda}\sum_{j=1}^m \partial_j B(U) \Delta U_j - \frac{\sigma}{1+ \lambda}\sum_{j,k=1}^m \partial_{j,k}B(U)\langle \nabla U_j, \nabla U_k \rangle.
L(z, w) = \frac12\biggl(1- (1-2z)\sqrt{1-\frac{4w}{(1-2z)^2}}\,\biggr) = \frac12\biggl(1-\sum_{k\ge 0}\binom{1/2}{k}\frac{(-1)^k4^kw^k}{(1-2z)^{2k-1}}\biggr) = z + \sum_{k\ge 1}C_{k-1}\frac{w^k}{(1-2z)^{2k-1}}= z + \sum_{\substack{k\ge 1 n\ge 0}}C_{k-1}\binom{n+2k-2}{n}2^n w^k z^{n}.
d s^{2}= 2 \frac{( c^{3}+ L^{3})^{\frac 1 3}}L \left\{d x^{i}d \bar{x}^{i}- \frac{c^{3}}{L ( c^{3}+ L^{3})}\bar{x}^{i}d x^{i}x^{j}d \bar{x}^{j}\right\},
\langle L_{c_*}\hat{u}_0,\hat{u}_0 \rangle_{L^2(\mathbb{R})}\geq A \| \hat{u}_0 \|_{H^1(\mathbb{R})}^2\mbox{\rm if}\;\; \langle u_{c_*}, \hat{u}_0 \rangle_{L^2(\mathbb{R})}=\langle \partial_{\xi}^{-1}\partial_c u_{c}|_{c = c_*}, \hat{u}_0 \rangle_{L^2(\mathbb{R})}= 0
\begin{aligned}\lim_{n \to \infty}\frac{1}{n}\log\!\Bigg[ \binom{w n}{\lambda n}S_{d+1,k}\big(n - \lambda n (d+1), (w - \lambda) n\big) \Bigg] = \; w H\Big(\frac{\lambda}{w}\Big) + (w - \lambda) \log \sum_{i=d+2}^{k+1}\rho_{w,\lambda}^{i - \frac{1-\lambda(d+1)}{w - \lambda}},\end{aligned}
B_1: = \bigg(\int_0^{\infty}\bigg(\int_x^{\infty}[\Psi_1]^q(t) w(t)\,dt\bigg)^{r / p}[\Psi_1]^q(x) \Psi[v;p]^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}W^{r / p}(x) \bigg[\sup_{x \le \tau < \infty}\Psi_1(\tau)\Psi[v;p]^{-1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
\overline{{{\Psi}}}_{L}\partial \! \! \! / \Psi_{L}\to \overline{{{\Psi}}}_{L}e^{- i \beta ( \hat{C}_{+}^{\dagger}- \hat{C}_{-}^{\dagger})}e^{i \beta ( \hat{C}_{+}- \hat{C}_{-})}\partial \! \! \! / \Psi_{L},
\triangle ( T ) = - \frac{1 7 2 8 ( j^{\prime})^{6}}{( 4 8 \pi^{2})^{3}j^{4}( j - j ( i ) )^{3}}= \eta^{2 4}( T ) = \frac{1}{( 2 \pi i )^{6}}\frac{( j^{\prime})^{6}}{j^{4}( j - j ( i ) )^{3}},
\frac{3}{2}\sum_{k=1}^\infty \mu_k \bigg\|\frac{P_k f}{|x|^2}\bigg\|_{2}^2 \leq \|\Delta_r f \|_{2}^2 + \bigg(1 - \frac{1}{\mathbf{C}_3}\bigg) \sum_{k=1}^\infty \mu_k^2 \bigg\|\frac{P_k f}{|x|^2}\bigg\|_{2}^2 & - 2\bigg\langle \sum_{j=1}^3 L_j^2 \bigg(\partial_r f - \frac{f}{2|x|}\bigg), \partial_r f - \frac{f}{2|x|}\bigg\rangle.
M=-\frac{\gamma}{2}\int r^{2}{\mathcal{V}}d^{2}x=-\frac{\gamma}{4}\int_{{\bf{R}}^{2}_{0}}\partial_{i}q_{i}d^{2}x = \frac{\gamma}{4}\sum_{s=1}^{N}\int_{C^{s}}q_{i}n_{i}^{s}\, dl.
\begin{aligned}\nabla_n \varPhi (x_0) \leq \, - \eta (x_0) \nabla_{nn}(x_0) + \nabla_n H [u] (x_0) + C \leq \, \Big(- \eta (x_0) + \int_0^1 H_{p_n}[u^t] (x_0) dt\Big) \nabla_{nn}u (x_0) + C. \end{aligned}
\left|\frac{\|Q_{jr}-Q_{2r}\|-\|Q_{1r}-Q_{jr}\|}{\|Q_{1r}-Q_{2r}\|}\right|=\frac{|\|Q_{jr}-Q_{2r}\|-\|Q_{1r}-Q_{jr}\||}{\|Q_{1r}-Q_{2r}\|}\leq\frac{\|Q_{1r}-Q_{2r}\|}{\|Q_{1r}-Q_{2r}\|}=1,
L^{\mbox{\tiny \sc Time}}f(\eta) = \lim_{\epsilon \downarrow 0}\frac{f(\eta^{\mbox{\tiny \sc Time}}+ \epsilon, \eta^{\mbox{\tiny $\mathcal{D}$}}) - f(\eta^{\mbox{\tiny \sc Time}}, \eta^{\mbox{\tiny $\mathcal{D}$}})}{\epsilon}
\sup_{x\in\Omega}\left(\int_{\Omega\,\cap\,B_{H_0}(x,1)}|e^{-h(y,t)}u(y,t)|^2\,dy\right)^{\frac{1}{2}}\qquad\le C_2t^{-\frac{N}{4}}\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}e^{-H_0(y)^2}|\phi(y)|\,dy
\frac{\partial F(m,r,t,s)}{\partial r}=\frac{\partial{\rm I}}{\partial r}+ \frac{\partial{\rm II}}{\partial r}+\frac{\partial{\rm III}}{\partial r}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\psi_{k}^{+}( x ) = \left( \begin{matrix}{\eta_{k}^{S}}\\{\tilde{\eta}_{k}^{A}}\\ \end{matrix}\right) \mathrm{a n d}\psi_{k}^{-}( x ) = \left( \begin{matrix}{\eta_{k}^{A}}\\{\tilde{\eta}_{k}^{S}}\\ \end{matrix}\right)
\exp\left\{iNT[\sigma]\right\}\equiv\int\left[\prod_{n,x}d\psi'_r(x)\right]\exp\left\{i\int d^dx\;\left[-\overline{\psi}'_r\gamma^\mu\partial_\mu\psi'_r+\sigma_\ell(x)\,j'_\ell(x)\right]\right\}\;.
J\Delta_{\omega_U|\omega}^{1/2}\pi(A)\Omega =\pi(A)^\ast\pi(U)J\pi(U)J\Omega &=J\pi(U)J\pi(A)^\ast\pi(U)\Omega &=J\pi(U)JJ\Delta_\omega^{1/2}\pi(U)^\ast\pi(A)\Omega,
\sum_{n=0}^\infty \frac{q^{n^2+n}}{(-q;q)_n^2\left(1+q^{n+1}\right)}=\frac{2}{(q;q)_\infty}\sum_{n=0}^\infty \frac{(-1)^nq^{\frac{3n(n+1)}{2}}\left(1-q^{2n+1}\right)}{\left(1+q^n\right)\left(1+q^{n+1}\right)}.
\hat{\mathbf{h}}^{(i)}= \left(N_c\check{\mathbf{F}}^{H}\mathbf{S}^{H}\mathbf{S}\check{\mathbf{F}}\right)^{-1}\check{\mathbf{F}}^{H}\mathbf{S}^{H}\left(\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{(i-1)}\right)^{H}\mathbf{r}.
{\cal L}^{-1}(\mbox{e}^{- \lambda^2\|\cdot\|^2}) (x) = \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{\mbox{e}^{- \lambda^2\|y\|^2}}{\|x-y\|}d y= \frac{1}{2 \|\lambda x\|}\int_{0}^{\|\lambda x\|}\mbox{e}^{-t^2}\, dt= \frac{\sqrt{\pi}}{4 \lambda \|x\|}\mbox{erf}(\|\lambda x\|).
\gamma_{( o )_{E 8}}\gamma_{( o )_{E 8}}= \gamma_{( o )_{E 8}}( \gamma_{( o )_{1 6}}+ \gamma_{( s )_{1 6}}) \rightarrow \mathrm{T r u n c a t i o n}\rightarrow \gamma_{( v )_{8}}- \gamma_{( s )_{8}}\,
G_{\pm}( \alpha_{\pm}) = e^{\alpha_{\pm}\bar{T}_{\pm}}= 1 + \alpha_{\pm}\bar{T}_{\pm}= \left( \begin{array}{c c c}{1}&{\frac{\alpha_{\pm}}{\sqrt{2}}}&{\mp i \frac{\alpha_{\pm}}{\sqrt{2}}}\\{0}&{1}&{0}\\{0}&{0}&{1}\\ \end{array}\right)
\tilde{\Gamma}_{\mu \nu}^{a b c , r}= ( G^{- 1})_{\nu \alpha}^{r}( y ) ( G^{- 1})^{r}( z ) < \tilde{z}_{2}\tilde{g}t^{a e d}B_{\mu}^{e}( x ) c^{d}( x ) \bar{c}^{c}( z ) B_{\nu}^{b}( y ) >^{r}
G (\Theta(x),y)=\log|x-\Theta(y)|+\alpha\log|x-{\bf 1}+\overline{\Theta(y)}| &\quad-\frac{2\beta(1+\alpha)}{1+a_0}\sum_{k=0}^\infty(\alpha\beta)^k\log|x+{\bf 2k+1}+\overline{\Theta(y)}|-\log|x|-\alpha\log|x-{\bf 1}| &\quad+\frac{2\beta(1+\alpha)}{1+a_0}\sum_{k=0}^\infty(\alpha\beta)^k\log|x+{\bf 2k+1}|+\frac{(1-\beta)(1+\alpha)}{1-\alpha\beta}\log|y|;
h_{-}= D_{l}(e^{-izr}+ \alpha_{-}e^{izr}+ \beta_{-}e^{-zr})\equiv \tilde{h}_{-}+ \beta_{-}g_{+}, g_{+}= D_{l}e^{-zr}, h_{+}= D_{l}(e^{zr}+ \alpha_{+}e^{-zr}+ \beta_{+}e^{izr})\equiv \tilde{h}_{+}+ \beta_{+}g_{-}, g_{-}= D_{l}e^{izr}
\frac{3 S^{\prime \prime}}{S}- 1 2 \alpha \frac{S^{\prime \prime}{S^{\prime}}^{2}}{S^{3}}+ 1 2 \alpha \frac{k S^{\prime \prime}}{S^{3}}+ 1 2 \alpha \frac{S^{\prime \prime}\dot{S}^{2}}{N^{2}S^{3}}+ . . = - \rho \delta ( w )
e_h(t) =F_h(t)u_0+\int_0^t F_h(t-s)\widehat{\xi}(s)ds+\int_0^t F_h(t-s)b(\widehat{u}(s))ds &+\int_0^t E_h(t-s)P_h[b(\widehat{u}(s))-f(\widehat{u}_h(s))]ds.
\widehat{F}_n(x_1, ...,x_{n-1}) := \sum \limits_{\substack{m_1,...,m_{n-1}\geq 0 m_1 + 2m_2 + ... + (n-1)m_{n-1}= n}}(-1)^{m_1 + ... + m_{n-1}}\cdot \frac{(m_1 + ... + m_{n-1}- 1)!}{m_1!\cdots m_{n-1}!}\cdot x_1^{m_1}\cdots x_{n-1}^{m_{n-1}}.
\sum_{i=0}^{n}(-1)^i\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot A\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right)=0 (x_{1}, \ldots, x_{n+1}\in R)
\|\tilde T_j F\|\Big\|\frac{1}{n}\sum_{i=1}^nf_i\tilde T_j\epsilon_i\Big\| \leq C\sqrt{\frac{1}{n}\frac{\tilde\lambda_j}{\tilde g_j}\operatorname{tr}(\tilde T_j\Sigma_\epsilon \tilde T_j)}\bigvee\sqrt{\frac{t}{n}\frac{\tilde\lambda_j}{\tilde g_j}\|\tilde T_j\Sigma_\epsilon \tilde T_j\|_{\infty}}.
\oint{\rm d}u_1\oint{\rm d}u_2\oint{\rm d}u_3 \frac{(u_1-u_2)^2}{(u_1-u_3)^2(u_2-u_3)^4}\prod_{i=1}^4 \frac{(z_i-u_1)^{1/2}(z_i-u_2)^1}{(z_i-u_3)^1}\,.
\left\lbrace\begin{array}{l}\alpha^2 AC + \alpha\gamma BC + \gamma^2 C^2-\beta^2 A^2-\beta\delta AB-\delta^2 AC = 0, \alpha^2 AB + \alpha\gamma B^2 + \gamma^2 BC - 2\alpha\beta A^2 - \alpha\delta AB - \beta\gamma AB - 2\gamma\delta AC = 0.\end{array}\right.
\phi_{L}^{a b}= \frac 1 2 ( \phi^{a b}+ \frac 1 2 \epsilon^{a b c d}\phi^{c d}) \quad \quad \phi_{R}^{a b}= \frac 1 2 ( \phi^{a b}- \frac 1 2 \epsilon^{a b c d}\phi^{c d})
\begin{aligned}\int_\Omega |\nabla \omega^{t}_\sigma(x)|^2\,dx = \int_{B(x_0,\tau)\setminus B(x_0,\sigma\tau)}\frac{t^2}{(1-\sigma)^2\tau^2}dx = \frac{t^2}{(1-\sigma)^2\tau^2}(|B(x_0,\tau)|-|B(x_0,\sigma\tau)|) = \frac{t^2 \omega_n\tau^{n-2}(1-\sigma^n)}{(1-\sigma)^2}.\end{aligned}
D_{1}= \tau_{1}^{+}\tau_{1}^{-}\ , \D_{2}= \tau_{2}^{+}\tau_{2}^{-}, \D_{3}= \tau_{1}^{+}\tau_{2}^{+}\ , \D_{4}= 2 \cos{\alpha}\left( \tau_{1}^{+}\tau_{2}^{-}+ \tau_{1}^{-}\tau_{2}^{+}\right) - \left[ \tau_{1 2}, \tau_{2 1}\right]_{+}\ll{e t s a - c a s}
D_{r}^{+ +}[ ( 1 2 )^{2}( 3 4 )^{2}{\cal F}] = D_{r}^{+ +}[ ( 1 4 )^{2}( 2 3 )^{2}{\cal F}] = D_{r}^{+ +}[ ( 1 2 ) ( 2 3 ) ( 3 4 ) ( 4 1 ){\cal F}] = 0 \; .
\left( \begin{array}{c c}{\hat{K}}&{g}\\{- g}&{\hat{F}}\\ \end{array}\right) \longrightarrow \left( \begin{array}{c c}{\Lambda \hat{K}\Lambda^{t}}&{\Lambda g \Lambda^{t}}\\{- \Lambda g \Lambda^{t}}&{\Lambda \hat{F}\Lambda^{t}}\\ \end{array}\right) .
\mu_{t}= \frac{2}{3}\frac{u_t}{u}\chi, \chi_{t}= \frac{2}{3}\frac{u_t}{u}\mu -\frac{1}{3}\nu, \nu_{t}= \frac{2}{3}\left(\frac{u_t}{u}\right)_t\mu +\frac{2}{3}\left(t - \frac{u^2_t}{u^2}\right)\chi,