harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\mathbf{x},\mathbf{y}\right)\in\bigcap_{0\le k<n}S_{k}\Leftrightarrow\mathbf{x}=\sum_{0\le i<n}\alpha_{i}\,\mathbf{V}\left[:,i\right],\;\mathbf{y}=\sum_{0\le j<n}\beta_{j}\,\mathbf{U}\left[:,j\right]\;\mbox{s.t.}\;\left\{\alpha_{k}\,\beta_{k}\right\}_{0\le k<n}\subset\mathbb{R}_{\ge0}.
| \kappa , \Omega \rangle ( z ) = \sqrt{A_{1}( \kappa )}\sum_{m = 1}^{\infty}V_{m - 1}^{( 1 )}( \kappa ) \frac{z^{m}}{m}\equiv \sum_{m = 1}^{\infty}\langle \kappa , 1 | m - 1 , 1 \rangle \frac{z^{m}}{\sqrt{m}}.
\kappa = \frac{k}{m_{f}}\; \; ; \; \; L^{2}= \frac{m_{i}^{2}}{m_{f}^{2}}= \frac{\left[ T_{i}^{2}- T_{c}^{2}\right]}{\left[ T_{c}^{2}- T_{f}^{2}\right]}\; \; ; \; \; \tau = m_{f}t \; \; ; \; \; \vec{x}= m_{f}\vec{r}
C_{\mu \alpha}= \overline{{\Psi}}( x ) \Gamma_{\mu}\overline{{\Gamma}}_{\alpha}\Psi ( x ) \Theta_{\mu [ \alpha \beta ]}= \overline{{\Psi}}( x ) \Gamma_{\mu}\overline{{\Gamma}}_{[ \alpha}\overline{{\Gamma}}_{\beta ]}\Psi ( x )
\begin{array}{ll}\mbox{minimize}\sum_{t=1}^m \sigma_{F_t}\big(\begin{bmatrix}y_{1,t}-z_1c_{1,t}\vdots y_{n,t}-z_n c_{n,t}\end{bmatrix}\big)-\sum_{i=1}^n H_i^*(\hat{y}_i)+\sum_{i=1}^n z_i \mbox{subject to}z_i\geq 0 ~\forall i\in[n] \end{array}
\int_{D \setminus D_{t}}\langle A \nabla u_0, \nabla u_0 \rangle - k^2 \int_{D \setminus D_{t}}\Sigma |u_0|^2 - \int_{\partial D_t \setminus \Gamma}A \nabla u_0 \cdot \nu \; \bar u_0 = 0.
\dim(\mathcal{M}(L)) = \dim(\mathcal{M}(L/\gamma_2(L))) + \dim(\gamma_2(L))(\dim (\frac{L}{\gamma_2(L)})-1) - \sum_{i=2}^{c}\dim (ker (\lambda_{i})) &\leq\dim(\mathcal{M}(L/\gamma_2(L))) + \dim(\gamma_2(L))(\dim (\frac{L}{\gamma_2(L)})-1) - \sum_{i=2}^{l}(m+n-r-s-i).
\mathbb{P}(|\mathcal{C}_{\max}(\mathbb{G}(n,m,p))|<n^{2/3}/A)\leq \mathbb{P}(|\mathcal{C}_{\max}(\mathbb{G}(n,m,p))|<T)=\mathbb{P}(t_i-t_{i-1}<T \forall i).
\overline{a}_{n0}=1,\n=0,1,..., a_{n-1,v}\geq a_{nv},\ \textnormal{for}~~ n\geq v+1, a_{nn}=O(\frac{p_{n}}{P_{n}}), \sum_{v=1}^{n-1}a_{vv}\hat{a}_{n,v+1}=O(a_{nn})
\left(\int_\Omega |V|^p\, dx\right)^{1/p}-\int_\Omega V\, V_0^{p-1}\, dx\ge \frac{p'-1}{8}\, \left\|\frac{|V|^{p-2}\, V}{\|V\|^{p-1}_{L^p(\Omega)}}-V_0^{p-1}\right\|^2_{L^{p'}(\Omega)},
Y_s^{n,t,\eta}\ = \G_n(\mathbb W_T^{t,\eta}) + \int_s^T F_n(r,\mathbb W_r^{t,\eta},Y_r^{n,t,\eta},Z_r^{n,t,\eta}) dr - \int_s^T Z_r^{n,t,\eta}dW_r, t \leq s \leq T.
\left( g_{m n}\right)^{D}= \left( \begin{array}{c c}{{g_{m n}+ g_{t s}{\Gamma^{t}}_{m}{\Gamma^{s}}_{n}}}&{{{\Gamma^{t}}_{n}g_{t m}}}\\{{{\Gamma^{t}}_{m}g_{t n}}}&{{g_{m n}}}\\ \end{array}\right) .
\frac{H_2(q\ell)}{H_1(q\ell)+\chi\,q\ell\,H_2(q\ell)}\sim\frac{1}{1+2\chi}\,\left(\frac{q\ell}{2}+\frac{2}{q\ell}\right)+\frac{q\ell}{(1+2\chi)^2}\,\left[\ln(q\ell/2)-\chi+(\Gamma-1/2)\right]+{\cal O}\left((q\ell)^3\right)\,,
\textrm{E}\left[ M^{(_m)}\right] =\sum_{m=0}^{\infty}m\prod_{j=1}^{m}\hat{P}_{V}^{(m+1)}\left(1-\hat{P}_{V}^{(j)}\right), &=\sum_{m=0}^{\infty}\prod_{j=1}^{m+1}\left(1-\hat{P}_{V}^{(j)}\right),
\omega((a,b,0),(c,d,1)) = \omega((a,b),(c,d)) = \exp\left(-\frac{\sqrt{-1}\theta}{2}(a d - b c)\right), \omega((c,d,1), (a,b,0)) = \omega((c,d), (-a-b,a)) = \exp\left(-\frac{\sqrt{-1}\theta}{2}( ca+ad+bd)\right).
U_{1}( \varphi ) = \mu_{1}e^{- 2 b \varphi_{1}}; \quad U_{2}( \varphi ) = \mu_{2}e^{- b \varphi_{1}}\quad U_{3}( \varphi ) = \mu_{3}e^{- b ( \varphi_{1}+ \varphi_{2})}.
{\cal V}\equiv \left( \begin{array}{c}{U_{\alpha}^{T}}{\bar{U}^{\alpha \, T}}\end{array}\right) \equiv \left( \begin{array}{c c}{e_{\alpha}^{A}}{h_{A \alpha}}{g^{\alpha \bar{\beta}}\bar{e}_{\bar{\beta}}^{A}}{g^{\alpha \bar{\beta}}\bar{h}_{A \bar{\beta}}}\end{array}\right) \ ,
\frac{1}{n}\sum_{i=1}^{n}\sum_{t=1}^{T}(l_{i,t}(x_{i,t})-l_{i,t}(y_{t}))\le \frac{1}{n}\sum_{i=1}^n\sum_{t=1}^T\Big(-q_{i,t+1}^\top g_{i,t}(x_{i,t}) +\frac{1}{n}\tilde{\Delta}_t+\Delta_{i,t}(y)\Big).
\|{\mathcal I}_3(\varepsilon;\zeta)\|_{L_2 \to L_2}\leqslant \vert \zeta +1\vert \Vert B_\varepsilon^{1/2}(B_\varepsilon -\zeta Q_0^\varepsilon )^{-1}\Vert_{H^{-1}\rightarrow L_2}&\times \Vert [Q_0^\varepsilon -\overline{Q_0}]\Vert_{H^1\rightarrow H^{-1}}\Vert (B^0-\zeta \overline{Q_0})^{-1}\Vert_{L_2\rightarrow H^1}.
R_{\mathcal{S}^{*}_{R}}(F) = \begin{dcases}\sigma_{0}\ 2\alpha+\beta-2\geqslant 0 \sigma_{0}\ 2\alpha+\beta-2<0\ \X(\alpha,\beta)\leqslant 0 \tilde{\sigma_{0}}\ 2\alpha+\beta-2<0\X(\alpha,\beta)> 0,\end{dcases}
\begin{aligned}\Delta u_1\wedge\Delta u_2\wedge \dots\wedge\Delta u_n =d_0(d_1u_1\wedge\Delta u_2\wedge \dots\wedge\Delta u_n)=-d_1(d_0u_1\wedge\Delta u_2\wedge \dots\wedge\Delta u_n) &=d_0d_1(u_1\Delta u_2\wedge \dots\wedge\Delta u_n)=\Delta(u_1\Delta u_2\wedge \dots\wedge\Delta u_n).\end{aligned}
A= \sum_{x,y\in \Omega_{\sigma}:x\sim y}w_{xy}(u(x)-u(y)) = -\sum_{x,y\in \Omega_{\sigma}:x\sim y}w_{xy}(u(y)-u(x)) = -\sum_{y\in\Omega_{\sigma}}\sum_{x\in \Omega_{\sigma}: x\sim y}w_{xy}(u(y)-u(x))=-A.
\||x'|^{\alpha}u\|_{L^s(D_1)}\le C\||x'|^{\alpha}u\|_{L^s(D_{2}\setminus D_{1})}+ C \big(\||x'|^{\mu}\nabla u\|^a_{L^1(D_{2})}+ \| |x'|^{\mu}v\|^a_{L^1(D_{2}\setminus D_{1})}\big) \||x'|^{\beta}u\|^{1-a}_{L^{q}(D_{2})}.
\nabla \left( \sigma\circ\tilde{u}\right) = \nabla \left( \frac{\tilde{u}}{\left\vert \tilde{u}\right\vert^2}\right) = \frac{\nabla \tilde{u}}{\left\vert \tilde{u}\right\vert^2}- \frac{\nabla{\left( \left\vert \tilde{u}\right\vert^2 \right) \tilde{u}}}{\left\vert \tilde{u}\right\vert^4}
\mathfrak{w}_{n,n}=\frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}\int\limits_{t_{n-1}}^{t}\rho_{\alpha}(t-s)dsdt=\frac{\tau^{\alpha}}{\Gamma(2+\alpha)},1\leq n\leq \mathcal{N}.
\mathfrak{u}(t,r,\theta)=\sum_{l,k}e^{ilt}\left( \begin{array}{c}e^{i(k-\frac{1}{2})\theta}U^+_{k,l}(r) e^{i(k+\frac{1}{2})\theta}U^-_{k,l}(r)\end{array}\right)
det[\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]=\int{\cal D}c^{a}{\cal D}{\bar c}_{a}\exp\left\{-\int{\bar c}_{a}[\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]c^{b}dt\right\}.
H_{m, n}^+(x) = \frac{4 M}{\pi}\int_{u = -\frac{T}{M}}^\infty \int_{\zeta = -\infty}^\infty \cos(x \cosh \zeta) e\Bigl ( \frac{(M u + T) \zeta}{\pi}\Bigr) (M u + T) e^{-u^2}\times{}&\qquad\qquad{}\times V(m^2 n, M u + T) \, d \zeta \, d u + O(T^{-A}).
n a_{1}( \vec{q}\,^{\prime}, \vec{q}) + \sum_{\nu = 1}^{3}( q_{\nu}^{\prime}- q_{\nu}) \frac{\partial a_{1}( \vec{q}\,^{\prime}, \vec{q})}{\partial q_{\nu}^{\prime}}= a_{1}( \vec{q}\,^{\prime}, \vec{q}\,^{\prime}) = - V ( \vec{q}\,^{\prime}) ,
\partial^\square \big( (c+tw) \circ \theta_t^{-1}\big)_{|t=0}= \frac{\mathrm{d}}{\mathrm{d}t}(c+tw)_{|t=0}- \partial_t \theta_t \cdot \nabla_{\Gamma_t}\big((c+tw) \circ \theta_t^{-1}\big)_{|t=0}= w - V\nu \cdot \nabla_\Sigma c= w
f_{\theta, q, 3}(\bar{x}) -f_{\theta, q, 3}(\bar{\bar{x}})=\frac{(\theta-1)(1-\theta-q)(\bar{\bar{x}}-\bar{x})}{(\bar{x}-\mathbf{x}^{\infty})(\bar{\bar{x}}-\mathbf{x}^{\infty})}F_{\theta, q, 3}\left(\bar{x},\bar{\bar{x}}\right)
\Delta_{0}^{t+1}=\frac{1}{M}\sum_{a}\Bigl[\frac{\bigl(y_{a}-Z_{a}^{t}\bigr)^{2}}{\bigl(1+V_{a}^{t}/\Delta_{0}^{t}\bigr)^{2}}+\frac{\Delta_{0}^{t}V_{a}^{t}}{\Delta_{0}^{t}+V_{a}^{t}}\Bigr],
\min\Big\{\sum_{i\in I}e_{ij}: I \subseteq \{1, \dots, n\}, |I| = k\Big\}- \lambda^{(j)}_{n-k+1}= \min\Big\{\sum_{i \in I}e_{ij}: I \subseteq \{1, \dots, n\}, |I| = k-1\Big\}
P_s= Erfc \left(\frac{\rho_I \sqrt{\pi \gamma_0}\sum\limits_{l=0}^{\infty}w_{Il}\frac{\Gamma(0.5+l+\mu)}{\Gamma(l+\mu)}}{2 \sqrt{\mu(1+\kappa) (\frac{c_0 R^{-2 \alpha_d}}{T}-\sigma^2)}}\right)
\vert{\rm Hess}(\tau)\vert=\frac{1}{9\tau^2}\{2+\cosh^4\theta\}+\cosh^2\theta\,{\rm trace}(A^2)+\frac{2}{\tau}H\cosh\theta-\frac{2}{3\tau}\cosh\theta\,g(A\partial_t^\top,\partial_t^\top).
g(x,y) = - \frac{1}{2}\left( 2 \frac{y}{1-\beta x}- \frac{1}{\tilde{A}}\right) \pm \sqrt{\frac{1}{4}\left( 2 \frac{y}{1-\beta x}- \frac{1}{\tilde{A}}\right)^2 - \frac{y^2}{(1-\beta x) (1-\beta y)}}.
\sum_{i=1}^n |c_i| + \sum_{i=1}^n |d_i| + \sum_{j=1}^t |e_j| \leq \sqrt{2n + t}\cdot \sqrt{\sum_i c_i^2 + \sum_i d_i^2 + \sum_j e_j^2}&\leq \sqrt{\frac{k}{4}+2}\cdot \sqrt{\sum_i c_i^2 + \sum_i d_i^2 + \sum_j e_j^2}.
J_1=\int_{|t|< 2|h|}F(t)u(t)\,d^2t,J_2=\int_{|t|> 2|h|}\frac{h(2t-h)(a(t)-a(h))}{t^2(t-h)^2}u(t)\,d^2t, &J_3=\int_{|t|> 2|h|}\frac{u(t)}{t^2}\,d^2t.
\begin{array}{l}{{K ( R_{j + 1}, p_{j + 1}\mid R_{j}, p_{j}, \xi ) = \Sigma_{r_{j}}\Sigma_{r_{j + 1}}J_{r}( Q_{j + 1}, Q_{j + 1}^{\prime}\mid Q_{j}, Q_{j}^{\prime}, \xi ) \exp \frac{2 i}{\hbar}( p_{j}r_{j}- p_{j + 1}r_{j + 1}) .}}\\ \end{array}
<\sigma_0\sigma_1>_1\left({1\over\lambda_1\lambda_2^3}+{1\over \lambda_1^3\lambda_2}\right)={1\over 4}\Gamma_1+{1\over 4}(\Gamma_2+\Gamma_3+\lambda_1\leftrightarrow\lambda_2)={1\over 2}{\lambda_1^2+\lambda_2^2\over \lambda_1^3\lambda_2^3}.
\begin{aligned}C^{-1}\leq \lambda_1^D \leq \frac{d(d+1)\rho^n_{\sup}}{\mu(K\setminus K_{b,n})}, V=1\, \Longrightarrow\, \lambda_1^D \leq \frac{(d+1)^2\rho^2_{\sup}w_{\sup}^2}{w_{\inf}^2},\end{aligned}
(\gamma_{3})^{T}=\alpha(\begin{matrix}0.1907\\ 0.8093\end{matrix})\circ(\begin{matrix}0.6533\\ 0.3467\end{matrix})=\alpha(\begin{matrix}0.1246\\ 0.2806\end{matrix})=(\begin{matrix}0.3075\\ 0.6925\end{matrix})
=\frac{(p-1)^2x}{(1-p^2x)(1-px)}\bigg(\frac{px^2}{1-px^2}- \frac{p^2x^3}{1-p^2x^3}\bigg) - \frac{p(p-1)x}{(1-p^2x)}\bigg(\frac{p^3x^3}{1-p^3x^3}- \frac{p^2x^3}{1-p^2x^3}\bigg).
s^{\frac{n-2}{2}- q}a_{i,s}= s^{\frac{n-2}{2}}\max_{\partial B(p_i,s)}u_i\leq Cd_g(x,p)^{\frac{n-2}{2}}\varphi_i(x) &\leq Ca_{i,s}d_g(x,p_i)^{\frac{n-2}{2}-q}+ C u_i(p_i)^{-\theta}d_g(x,p_i)^{-\frac{n-2}{2}+q}\{ir_i \leq d_g(x,p_i) \leq s\}.
L(t):=\left( \begin{array}{ccc}\frac{1}{\sqrt{1-|s(t)|^2}}0 0 1 \end{array}\right)\,,\quad\tilde \Phi(t,x):=L(t)R(t)(x-r(t))\,, &\tilde\Omega(t):=\tilde\Phi(t,\Omega)\,,\quad\tilde\Gamma(t):=\tilde\Phi(t,\Gamma(t))\,,
\displaystyle{\raisebox{0 . 6 e x}{\scriptsize{*}}}\! S_{j i}( K ) = - \gamma_{0}\left[ \displaystyle{\raisebox{0 . 6 e x}{\scriptsize{*}}}\! S_{\underline{{i}}\underline{{j}}}( K ) \right]^{\dagger}\gamma_{0}\, .
(n\cdot K)^{2}=[\begin{matrix}1&0&0&0\\ 0&n_{x}^{2}&n_{x}n_{y}&n_{x}n_{z}\\ 0&n_{y}n_{x}&n_{y}^{2}&n_{y}n_{z}\\ 0&n_{z}n_{x}&n_{z}n_{y}&n_{z}^{2}\end{matrix}]
\Vert (1+Q)^n \Vert \leq \sum_{k=0}^n \binom nk\Vert Q^k\Vert\leq \sum_{k=0}^{N_1-1}\binom nk\Vert Q^k \Vert + \sum_{k=N_1}^n\binom nk\left( \frac{\varepsilon}{2}\right)^k &\leq \left(1+\frac{\varepsilon}{2}\right)^n + \sum_{k=0}^{N_1-1}n^k\Vert Q \Vert^k\leq \left(1+\frac{\varepsilon}{2}\right)^n + N_1(1+n\Vert Q \Vert)^{N_1}.
\tilde{m}_{0,0}(\xi - \eta, \eta) = - \frac{\xi \cdot (\xi - \eta)}{|\xi |^{2}}\frac{(\xi - \eta) \cdot \eta}{|\xi - \eta| |\eta|}\big( (\xi - \eta) \times \eta \big),\tilde{m}_{0,0}^{1}(\xi-\eta, \eta) = \frac{-\big((\xi-\eta) \times \eta \big)^2}{2|\xi| |\xi-\eta||\eta|},
m_{2}^{*}=\lambda^{2}\, \frac{\zeta}{2}\, \rho \,- \, \Delta t^{2}\, \lambda^{4}\, \frac{\zeta}{2}\, \sigma_{1}\, \big( \frac{1}{2}-\sigma_{2}\big) \frac{\partial^2 \rho}{\partial x^{2}}+{\rm{O}}(\Delta x^{3})
(csgn([\begin{matrix}A^{H}&Q\\F&-A\end{matrix}])-[\begin{matrix}I&0\\ 0&I\end{matrix}])[\begin{matrix}X&-I\\I&0\end{matrix}]=[\begin{matrix}X&-I\\I&0\end{matrix}][\begin{matrix}0&Y\\ 0&-2I\end{matrix}],
e^{-\frac{16d}{19}}+e^{-\frac{72d}{95}}\sum_{m=1}^{d_0}e^{\varphi(m)}\le e^{-\frac{16d}{19}}+ \frac{3}{2}e^{-\frac{3d}{19}}\le e^{-\frac{d}{10}}\Big(e^{-\frac{141}{190}}+\frac{3}{2}\Big)\le2e^{-\frac{d}{10}}.
10XD_{13}^4(q)+(-116 X-404XY^2+528XY-Y^2+Y^3+1280X^2)D_{13}^2(q) &-16X-20Y^5-16000X^2Y-176XY^2+2112X^2+4Y^4 +37824X^2Y^2-3504XY^3+8240XY^4-23040X^3+192XY=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})}.
\displaystyle\int\limits_T h_n(t,x^n_\infty(t))d\mu(t)= \inf\limits_{w \in \textnormal{L}^p({T},X)}\int\limits_T h_n(t,w(t)) d\mu(t) = \int\limits_T \inf\limits_{u\in X}h_n(t,u) d\mu(t),
( - m_{\pi}^{2}+ \partial_{\bot}^{2}) \omega_{\pi}^{( 1 )}= \frac{1}{2 L}\int_{- L}^{L}d x^{-}( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) .
H^{fer}_D = \frac{1}{\sqrt{2}}\int d^3x \xi^{\dagger}(\vec{x}) \left({\cal G}_{(-)}[V_{-}] \ast\xi\right) (\vec{x}) + \frac 1 2 \frac{e^2}{2L}\int d^2x_\perp\Gamma^i (x_\perp) \left({\cal G}_{(\perp)}[{\cal M}^2] \ast\Gamma^i \right) (x_\perp)
H = 1 + ( \sqrt{\alpha^{\prime}})^{{\frac{x - \tilde{d}}{\beta}}}( g_{s})^{{\frac{2 ( k - 1 )}{\beta}}}\left[ g_{f}^{2}\left({\frac{u}{\cal R}}\right)^{\tilde{d}}\left({\frac{d_{p}}{c_{p}}}\right) \right]^{- 1 / \beta}.
J_{\mu}= i \overline{{{\Psi}}}( x ) \Gamma_{\mu}\Psi ( x ) , \hspace{0 . 3 i n}K_{\mu}= \overline{{{\Psi}}}( x ) \Gamma_{\mu}\overline{{{\Gamma}}}_{5}\Psi ( x ) , \hspace{0 . 3 i n}R_{\mu \alpha}= \overline{{{\Psi}}}( x ) \Gamma_{\mu}\overline{{{\Gamma}}}_{5}\overline{{{\Gamma}}}_{\alpha}\Psi ( x ) ,
(\phi(x_{+}(c))x_{+}(c))^{5}= (\gamma_{k,l}(x_{+}(c))qx_{+}(c))^{5}&= q ({}^{x_{+}(c)}q)({}^{x_{+}(c)^{2}}q)({}^{x_{+}(c)^{3}}q)({}^{x_{+}(c)^{4}}q)( \gamma_{k,l}(x_{+}(c))x_{+}(c))^{5}.
\beta_{a}(I)=\beta_{a,b}(I) =\big|\big\{w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big| &=\big|\big\{w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-\big|\big\{v_{\ell+1}>\dots>v_d\big\}\big| &=\big|\big\{w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-(d-\ell)=0,
\Bigl(\sum_{i=0}^m e^{x_iz}-\sum_{j=1}^m e^{y_jz}\Bigr) \Bigl(\sum_{i=0}^m e^{-x_iz}-\sum_{j=1}^m e^{-y_jz}\Bigr) & =\sum_{i=0}^m\sum_{j=0}^m e^{(x_i-x_j)z}+\sum_{i=1}^m\sum_{j=1}^m e^{(y_{i}-y_j)z}-\sum_{i=0}^m\sum_{j=1}^m e^{(x_i-y_j)z}-\sum_{i=0}^m\sum_{j=1}^m e^{(y_j-x_i)z},
R_{\mathbf{s}_{\scriptscriptstyle{A}}}= E \left[ \log_2 \det\left( \mathbf{I}+ \mathbf{P}_{\scriptscriptstyle{A}}\widetilde{\mathbf{H}}_1^H \mathbf{K}^{-1}\widetilde{\mathbf{H}}_1 \mathbf{P}_{\scriptscriptstyle{A}}\right)\right].
\frac{(q^{m + 1 - i})_{j - I'}}{(q)_{j - I'}}= (-1)^{m + 1 - i + \delta(i \leq m)}q^{-\left\{(m - i) (m - i - 1) / 2\right\}- m + i}\frac{(q^{1 - s})_{I_{r}+ j}}{(q^{1 - s})_{s - 1}(q^{i - m})_{1 - s + I_{r}+ j}},
O_{F}= \left( \begin{array}{c c}{- \frac{d^{2}}{d z^{2}}+ \frac{\partial^{2}}{\partial \phi^{2}}V}&{\frac{\partial^{2}}{\partial \chi \partial \phi}V}\\{\frac{\partial^{2}}{\partial \phi \partial \chi}V}&{- \frac{d^{2}}{d z^{2}}+ \frac{\partial^{2}}{\partial \chi^{2}}V}\\ \end{array}\right)_{\vert \phi = \phi ( z ) , \chi = \chi ( z )}
\begin{array}{rcl}\sum_{k=0}^{n}\frac{a_k}{p+(n+l-k-t)r}[x^{p+(n+l-k-t)r}, y^{q+(k+t)s}] = 0 \sum_{k=0}^{n}\frac{a_k}{q+(k+t)s}[x^{p+(n+l-k-t)r}, y^{q+(k+t)s}] = 0.\end{array}
=\hat{U}^{\dagger}i(\hat{a}^{\dagger}\hat{a}\hat{a}-\hat{a}\hat{a}^{\dagger}\hat{a})\hat{U}=\hat{U}^{\dagger}i[\hat{a}^{\dagger},\hat{a}]\hat{a}\hat{U}=-i\hat{U}^{\dagger}\hat{a}\hat{U}
|E_1| ~ =~ |\{x \in K : |Bf_1(x)| > \lambda/2\}| = ~\bigg|\bigg(\bigcup_iW_i\bigg)\bigcap \{x \in K : |Bf_1(x)| > \lambda/2\}\bigg|~ + ~ \bigg|\bigg(\bigcup_iW_i\bigg)^\complement \bigcap \{x \in K : |Bf_1(x)| > \lambda/2\}\bigg|.
\int_{\Omega}\big( k \omega \sigma \, \boldsymbol{y}_k \cdot \boldsymbol{v}_k^{\perp}+ \nu \, \textbf{curl}\, \boldsymbol{y}_k \cdot \textbf{curl}\, \boldsymbol{v}_k \big)\,d\boldsymbol{x}= \int_{\Omega}\boldsymbol{u}_k \cdot \boldsymbol{v}_k\,d\boldsymbol{x},
\left( \not \! \! D^{D}+ m^{D}\right) \, \psi ( n \alpha ) = - \int_{}^{}d^{d}y \, \left[ ( \not \! \partial - m ) \, f ( | y | ) \right] U \left( L_{n \alpha , n \alpha + y}\right) \, \psi ( n \alpha + y ) \; ,
D_{t}U(m,s) = \frac{U(m, s + 1) - U(m,s)}{\Delta t}, D^{+}U(m,s) = \frac{U(m - 1,s) - U(m,s)}{\Delta x}, D^{-}U(m,s) = \frac{U(m,s) - U(m + 1,s)}{\Delta x},
\lim\limits_{r\rightarrow\infty}B_{jr}=\begin{cases}\lim\limits_{r\rightarrow\infty}f'(\|Q_{1r}-Q_{jr}\|)\frac{\|Q_{jr}-Q_{2r}\|-\|Q_{1r}-Q_{jr}\|}{\|Q_{1r}-Q_{2r}\|}\textrm{if}\|Q_{jr}-Q_{2r}\|\neq\|Q_{1r}-Q_{jr}\| 0\textrm{if}\|Q_{jr}-Q_{2r}\|=\|Q_{1r}-Q_{jr}\|\end{cases},
\displaystyle{\left. \Pi_{2 2}( p ) \right|_{\mathrm{G}}}= \displaystyle{\frac{1}{4 \pi}\left[ - \frac{p^{2}}{8}\mathrm{l n}\left( \frac{p^{2}}{\mu^{2}}\right) - \frac{5 3}{6}\frac{( p_{1})^{2}( p_{2})^{2}}{p^{2}}\right] .}
F_{_{( 2 )}}= \left( \begin{array}{c c c}{{f}}&{{0}}&{{M_{_{( 2 )}}}}\\{{0}}&{{0}}&{{R_{_{( 2 )}}}}\\{{- M_{_{( 2 )}}^{^T}}}&{{- R_{_{( 2 )}}^{^T}}}&{{0}}\\ \end{array}\right) \; ,
\liminf_{n\to\infty}\int_{\mathbb{R}^3}\tilde{j}_{m\epsilon_{n}}(w_{n}^{(j)}(x)){\rm d}x\geq\frac{q}{4\pi^2}\int_{\mathbb{R}^3}\tilde{\eta}(x)^{2}{\rm d}x =\frac{9}{8K_{\rm cl}\lambda}\int_{\mathbb{R}^3}Q(x)^{\frac{2}{3}}{\rm d}x,
\Theta \left[ \begin{array}{l l l}{1 / 2}&{1 / 2}&{1 / 2}\\{1 / 2}&{1 / 2}&{0}\\ \end{array}\right] \stackrel{D_{\alpha_{2}^{- 1}\alpha_{3}}}{\longrightarrow}\Theta \left[ \begin{array}{l l l}{1 / 2}&{1 / 2}&{1 / 2}\\{1 / 2}&{0}&{1 / 2}\\ \end{array}\right] ,
[\begin{matrix}1&0&0&0&0\\ 0&2&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\end{matrix}][\begin{matrix}0\\ 1\\ 0\\ 0\\ 0\end{matrix}]=2[\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix}].
\forall \omega \in L^{r}_{p}(M)\cap{\mathcal{H}}_{p}(M)^{\perp},\ \exists u\in L^{t}_{p}(M),\ \Delta u=\omega ,\{\left\Vert{u}\right\Vert}_{L^{t}(M)}\leq c{\left\Vert{\omega}\right\Vert}_{L^{r}(M)}.
\Delta_{Z_1, Z_{-1}}(A) Z_{-1}^* = Z_1 A Z_{-1}^* - A= (Z + e_1 e_n^*) A (Z^* - e_n e_1^*) - A & = Z A Z^* - A + e_1 e_n^* A Z^* - Z A e_n e_1^* - e_1 e_n^* A e_n e_1^* & = - \nabla(A) + \begin{bmatrix}\alpha r c 0_{n-1,n-1}\end{bmatrix}= - \nabla(A) + \nabla(T).
\omega_{\exp(a)H}= \sum_{1\leq j_1,j_2\leq r}\frac{1}{4}d^2_au(l_{j_1},l_{j_2}) \omega_{j_1,j_2}+ \sum_{\alpha\in \Phi_{Q^u}}\frac{-e^{2\alpha}}{2}(d_au-2\chi)(\alpha^{\vee})\omega_{\alpha,\alpha}& + \sum_{\beta \in \Phi_s^+}\frac{d_au(\beta^{\vee})}{\sinh(2\beta(a))}\omega_{\beta,\beta}.
\int_{0}^{v}\lambda_{U}^s(r, y) y dy &= \sum_{t \in \mathcal{K}}\Big(\frac{\lambda_t v^2}{2}- \lambda_t\Big(\frac{\exp(-\pi C_t r^2) - \exp(-\pi C_t (r + v)^2)}{2\pi C_t}&\quad- r\frac{\mathrm{erf}((r + v)\sqrt{\pi C_t}) - \mathrm{erf}(\sqrt{\pi C_t}r)}{2 \sqrt{C_t}}\Big) \Big),
\phi_{A_{\nu}}\left(z\right) =1+\sum_{k=1}^{s-1}c_{ko}z^{ko}=\frac{1-\left(c_{o}z^{o}\right)^{s}}{1-c_{o}z^{o}}=\frac{1-\frac{\mu_{\nu}}{\mu_{w}}\cdot c_{o}^{1-n/p}z^{os}}{1-c_{o}z^{o}}.
\int_{\pi / 2}^{3 \pi / 2}d \theta_{3}\int_{- \pi / 2}^{\pi / 2}d \theta_{2}\int_{- 3 \pi / 2}^{- \pi / 2}d \theta_{1}\left| 3 \sin \frac{\theta_{2}- \theta_{1}}{3}\right|^{- 1}\left| 3 \sin \frac{\theta_{3}- \theta_{1}}{3}\right|^{- 1}\left| 3 \sin \frac{\theta_{3}- \theta_{2}}{3}\right|^{- 1}= \frac{\pi^{2}}{3}.
\mbox{$d=2g_1$ and $b-a=g_1$}\implies \frac12d = b-a begin{align*}1ex]\mbox{$d=3g_1$ and $b-a=g_1$}\implies \frac13d = b-a begin{align*}1ex]\mbox{$d=3g_1$ and $b-a=2g_1$}\implies \frac23d = b-a begin{align*}1ex]\mbox{$d=4g_1$ and $b-a=1g_1$}\implies \frac14d = b-a begin{align*}1ex]\mbox{$d=4g_1$ and $b-a=3g_1$}\implies \frac34d = b-a
Y_{3}^{\mathrm{t r i a n g l e}}( R ) = - \frac{8}{\sqrt 3}\Lambda^{2}L_{3}\left[ R^{2}\Lambda^{2}- R^{4}\Lambda^{4}- \frac{4 \sqrt 3}{4 5}R^{5}\Lambda^{5}+{\cal O}\left( R^{6}\Lambda^{6}\right) \right] .
\|u_\infty\|_{C^{2,\alpha}_{\gamma,\delta}(\{r\leq r_\infty\})}\leq C\left[\|u_\infty\|_{C^{0}_{\gamma,\delta}(\{r\leq r_\infty\})}+\|(\Delta-X)u_\infty\|_{C^{\alpha}_{\gamma+2,\delta}(\{r\leq r_\infty\})}\right]=C\sup_{\{r\leq r_\infty\}}w_0 |u_\infty|<\infty.
B(\gamma)= \frac{1+g(0)}{\gamma}+\frac{c(a+1)}{2}\gamma^{-a}\left(\log\gamma \right)^{-b}F(t)=t^{\min(a,1)}\,, &A(\gamma)=c'\times\begin{cases}a'\gamma^{-(a'+2)}(\log\gamma)^{-b'}a'>0\,, b'\gamma^{-2}(\log\gamma)^{-(b'+1)}a'=0\,,\end{cases}
Z=<\exp\left(\int d\tau\left[ -\frac{1}{2\pi}\xi\dot{\xi}+\frac{\sqrt{\alpha'}}{2\pi}\xi \psi^1u-\frac{1}{8\pi}u^2 (X^1)^2+iA_a(X^a)\partial_{\tau}X^a +\frac{i\alpha'}{2}\theta_{ab}\psi^a\psi^b\right]\right)> \ ,
P(l)=\frac{1}{2l}\sum_{n=1}^\infty \frac{A(n,1)}{n^\frac{3}{2}}(\frac{16}{ln})\sum_{\substack{\alpha \leq Y (\alpha,2ln)=1}}\frac{\mu(\alpha)}{\alpha^2}G_0(ln)\widetilde \Phi_{n}(0),
[X \xleftarrow{p_1}V_1 \xrightarrow{s_1}Y; E_1] + [X \xleftarrow{p_2}V_2 \xrightarrow{s_2}Y; E_2] & := [X \xleftarrow{p_1+p_2}V_1 \sqcup V_2 \xrightarrow{s_1+s_2}Y; E_1 + E_2],
\gamma^2 \beta = \beta \gamma^2 , \gamma^2 \delta = \delta \gamma^2, aa^*a=a \gamma^2, a^* a a^* = \gamma^2 a^*, &\beta^2= -a^* a \gamma - \gamma a^*a+ \gamma^3, a^*a+\beta+\gamma + \delta - \gamma^2/2=0, \delta^2 =\gamma^4/4+\gamma^2+a^* a \beta + \beta a^* a -\gamma^2 \beta
|\psi\rangle\equiv(\begin{matrix}\langle1|\psi\rangle\\ \langle2|\psi\rangle\end{matrix})=(\begin{matrix}c_{1}\\c_{2}\end{matrix})=c_{1}(\begin{matrix}1\\ 0\end{matrix})+c_{2}(\begin{matrix}0\\ 1\end{matrix})=c,
J_{j}\boldsymbol{I}_{jk}^{\pm}=\pm \frac{1}{2}w^{j}w^{i}\boldsymbol{a}_{jk}=-\frac{wj}{2}+\frac{wj}{2}\pm \frac{1}{2}w^{j}d\boldsymbol{x}^{jk}=w^{j}(\boldsymbol{I}_{jk}^{\pm}-\frac{1}{2}),
\left|fl\left(B^{(k+1)}\right) - B^{(k+1)}\right| \le \left| fl \left(B^{(1)}\right) - B^{(1)}\right|\begin{pmatrix}0 I\end{pmatrix}(1+u)^{k}& + u \sum_{i=1}^{k}\left| B^{(i)}\right| \begin{pmatrix}0 I\end{pmatrix}(1+u)^{k-i}+ 3 u \sum_{i = 1}^{k}\left| B^{(i)}\right| \begin{pmatrix}I 0\end{pmatrix}\left| L_i^T \right| (1+u)^{k-i}+ O(u^2),
\lim_{\xi\rightarrow\overline{c_{0}}^{+}}\frac{p_{\mu_{1}\boxtimes\mu_{2}}(\xi)}{\left|\xi-\overline{c_{0}}\right|^{1/2}}=\frac{2\left|a_{1}\right|}{\sqrt{\left|c_{2}\right|}}\cos\left(\frac{\theta}{2}-\frac{\pi}{4}\right),
w(y)=\int_{D\setminus \overline{B}_1}h(z)K_{B_1}(y,z)dz\geq \tilde c_1\int_{D\setminus \overline{B}_1}h(z)K_{B_1}(x_1,z)dz=\tilde c_1 w(x_1)\geq \frac{\tilde c_1}2 h(x_1).
\Box \partial^{s'}_1u + \partial^{s'}_1u + \gamma^{01}\partial_0\partial_1(\partial^{s'}_1u) + \gamma^{11}\partial_1^2 (\partial^{s'}_1u) + \gamma^0\partial_0 (\partial^{s'}_1u) + \gamma^1 \partial_1 (\partial^{s'}_1u) = f^{s'}\, ,
\sum_{n\geq 1}\frac{2q^n}{1+q^n}\frac{(-q;q)_{n-1}}{(q;q)_{n-1}}= \sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(q;q)_n}\frac{2q^{n+1}}{1-q^{2(n+1)}}\frac{(-q^{n+2};q)_\infty}{(q^{n+2};q)_\infty}
{\cal A}(g_1 g_2)^{\alpha \beta}\: = \:{\cal A}(g_2)^{\alpha \beta}\: + \: g_2^*{\cal A}(g_1)^{\alpha \beta}\: - \: \theta(g_1, g_2)^{\alpha}\: + \: \theta(g_1, g_2)^{\beta}\: + \: d \log \Omega^{g_1, g_2}_{\alpha \beta}
\# \big\lbrace z \in \sigma_\mathrm{disc}( H(b_0,e V)) : -r_0 e^2 \leq z < -r e^2 \big\rbrace & = \# \big\lbrace z \in \sigma\left( \big( p_{0}Up_{0}\big)^{\ast}p_{0}Up_{0}\right) : z \geq 2 r b_0 \big\rbrace \big( 1 + o(1) \big).
A = C_{b/2,\gamma}^2 \, \frac{a (1-e^{-2\gamma/b})(1-e^{-\gamma/b})^2}{1-e^{-2 \gamma a}}\, e^{-2 a \gamma}B = C_{b/2,\gamma}^2 \, \frac{a (1-e^{-2\gamma/b})(1+e^{-\gamma/b})^2}{1-e^{-2 \gamma a}}.
\sum_{j = 0}^k{\frac{{G_j^3}}{{2^j}}}= \frac{{26}}{{25}}G_0^3 + \frac{{16}}{{25}}G_1^3 + \frac{2}{{25}}G_2^3 - \frac{2}{{25}}G_3^3 &- \frac{{13}}{{25}}\frac{{G_{k + 1}^3}}{{2^k}}- \frac{8}{{25}}\frac{{G_{k + 2}^3}}{{2^k}}- \frac{1}{{25}}\frac{{G_{k + 3}^3}}{{2^k}}+ \frac{1}{{25}}\frac{{G_{k + 4}^3}}{{2^k}}\,.
|b(t,x)| =\frac{|b(t,x)|}{\langle |b(t,\cdot)|^q\rangle^{\frac{1}{q}}}\langle |b(t,\cdot)|^q\rangle^{\frac{1}{q}}\leq \frac{d}{q}\biggl(\frac{|b(t,x)|^q}{\langle |b(t,\cdot)|^q\rangle}\biggr)^{\frac{1}{d}}+ \frac{2}{p}\bigl( \langle |b(t,\cdot)|^q\rangle^{\frac{1}{q}}\bigr)^{\frac{p}{2}},
\sum_{k=0}^{\infty}kr^{k}=\frac{r}{(1-r)^{2}};\sum_{k=0}^{\infty}k^{2}r^{k}=\frac{r(1+r)}{(1-r)^{3}};\sum_{k=0}^{\infty}k^{3}r^{k}=\frac{r(1+4r+r^{2})}{(1-r)^{4}}
\mathbb{E}(\kappa(G, B))= \sum_{e \in E( G )}\mathbb{E}\bigg( \frac{1}{d'(e) + 1}\bigg)= \sum_{e \in E( G )}\frac{{n \choose 2}}{(2n - 3) m}\bigg( 1 - \frac{{{n \choose 2}- m \choose 2n - 3}}{{{n \choose 2}\choose 2n - 3}}\bigg)
S= \int_{-\infty}^{\infty}dt \int_{-\infty}^{\infty}dx_1 [ \bar{\psi}(x_1,t) ({\gamma}^{\mu}i{\hbar}c{\partial}_{\mu}-mc^2){\psi}(x_1,t) - j_{\mu}^{tot}(x_1,t) A^{\mu}(x_1,t)
\begin{array}{l}\omega^j_{r_1\cdots r_n}\ (j=1,\ldots,\sigma_1),\omega^j_{0r_2\cdots r_n}\ (j=\sigma_1+1,\ldots,\sigma_2),\quad\ldots \omega^j_{0\cdots 0r_{n-1}r_n}\ (j=\sigma_{n-2}+1,\ldots,\sigma_{n-1}),\quad\omega^j_{0\cdots 0r_n}\ (j=\sigma_{n-1}+1,\ldots,\sigma_{n})\end{array}
G\, \bar\psi\gamma^{\mu}\psi\, \bar\psi\gamma_{\mu}\psi ={\pi (\pi -g_{\rm eff}^{2})\over g_{\rm eff}^{2}}\, \Bigl(\bigl( j^{0}\bigr)^{2}-{g_{\rm eff}^{2}\over\pi}\, \bigl( j^{1}\bigr)^{2}\Bigr).