harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S = - \int d^{2}x \, \sqrt{g}\left[{\frac{1}{2}}g^{\mu \nu}\nabla_{\mu}\Phi \nabla_{\nu}\Phi + C R \Phi + V ( \Phi ) +{\frac{1}{4}}f ( \Phi ) g^{\mu \alpha}g^{\nu \beta}F_{\mu \nu}F_{\alpha \beta}\right] .
\begin{array}{l}{\psi_{a}( z ) \psi^{b}( w ) = \frac{\delta_{a}^{b}}{( z - w )}+ \cdots}{\psi_{a b}( z ) \psi^{c d}( w ) = \frac{\Omega_{a b}^{d c}}{( z - w )}+ \cdots}\end{array}
\gamma B_{\mu \nu}^{A}= \partial_{\mu}C_{\nu}^{A}- \partial_{\nu}C_{\mu}^{A}, \; \gamma C_{\mu}^{A}= \partial_{\mu}\rho^{A}, \; \gamma \rho^{A}= 0 , \; \gamma B_{A}^{* \mu \nu}= 0 , \; \gamma C_{A}^{* \mu}= 0 , \; \gamma \rho_{A}^{*}= 0 .
\underline D_\mu^\psi(\eta) = \liminf_{t\to\infty}\frac{\mu(B(\eta,e^{-t}))}{\psi(e^{-t})}\asymp_\times \liminf_{t\to\infty}\frac{e^{-\delta t}e^{b(\eta_t)(k(\eta_t) - \delta)}}{e^{-\delta t}e^{\Psi(t)}}= \exp\liminf_{t\to\infty}[b(\eta_t)(k(\eta_t) - \delta) - \Psi(t)].
\sum_{j=1}^N (w_j(\partial_{z_j}g) ({\bf w}) +\overline{w_j}(\partial_{\overline{z_j}}g) ({\bf w}))=\left. \frac{d}{d h}g(h{\bf w}) \right|_{h=1}=g({\bf w})\left. \frac{d}{d h}h^p \right|_{h=1}= p g({\bf w}).
q_{k}^{(s,t)}:=\frac{H_{k-1}^{(s,t)}H_{k}^{(s+1,t)}}{H_{k}^{(s,t)}H_{k-1}^{(s+1,t)}}, &e_k^{(s,t)}:=\frac{H_{k+1}^{(s,t)}H_{k-1}^{(s+1,t)}}{H_{k}^{(s,t)}H_{k}^{(s+1,t)}}.
\rho:=\rho^{k+1},\mathbf{u}:=\mathbf{u}^{k+1},w:=\rho^k,\mathbf{v}:=\mathbf{u}^k, \tilde{\rho}:= \rho^{k+1}-\rho^k,\tilde{\mathbf{u}}:= \mathbf{u}^{k+1}-\mathbf{u}^k,\tilde{w}:= \rho^{k}-\rho^{k-1},\tilde{\mathbf{v}}:= \mathbf{u}^{k}-\mathbf{u}^{k-1},
\begin{array}{c}t\longmapsto \left\vert \frac{\phi^{\prime}\left(t\right)}{\phi \left( t\right)}\right\vert \underset{t\rightarrow +\infty}{\lim}\left\vert \frac{\phi^{\prime}\left( t\right)}{\phi \left( t\right)}\right\vert =0.\end{array}
\sum_{r = 0}^{k - 1}\frac{( - 1 )^{r}\Gamma ( \mu )}{2^{r + \lambda}r ! \Gamma ( \mu - r ) ( r + \lambda )}+ \sum_{r = 0}^{s - 1}\frac{( - 1 )^{r}\Gamma ( \lambda )}{2^{r + \mu}r ! \Gamma ( \lambda - r ) ( r + \mu )}
u_t+(u\cdot\nabla) u-(B\cdot\nabla) B+\nabla P= \ \nu\Delta u, B_t+(u\cdot\nabla) B-(B\cdot\nabla) u +\nabla\times ((\nabla\times B)\times B)= \ \mu\Delta B, \nabla\cdot u=0, \ \ \nabla\cdot B= \ 0,
a_h(u,v) := \sum_{k=1}^N a_e^{(k)}(u,v) \left\langle F, v \right\rangle:=\sum_{k=1}^N \int_{\Omega^{(k)}}f v^{(k)}dx, a^{(k)}_e(u,v) := a^{(k)}(u,v) + s^{(k)}(u,v) + p^{(k)}(u,v),
P(T, x) = \prod_{j=1}^{8}\mathcal{G}(u_j, x) = x^5 \ \frac{x^2-2}{x}\ \frac{x^2-3}{x}\ \frac{x^5-7x^3 +11x}{(x^2-2)(x^2-3)}= x^4 (x^4 - 7x^2 + 11) .
\sum_{\ell \in A \cap [k,n+k-N]}\sum_{i=0}^{n-1}\psi_\ell(x_i,k+i) \leq \sum_{\ell \in A \cap [k,n+k-N]}N\chi_{[\omega]}(\sigma^{\ell-k}x) &\leq N\sum_{\ell =k}^{n+k-N}\chi_{[\omega]}(\sigma^{\ell-k}x) &= N\sum_{i=0}^{n-N}\chi_{[\omega]}(\sigma^{i}x),
\mathcal{J}:=\sum\limits_{a_m\neq a_i}\prod_{i=1}^{m-1}\{2+h\chi_4(a_m-a_i)+\overline{h}\chi_4^3(a_m-a_i)\}&=2^{m-1}(q-m+1) &+\sum\limits_{a_m\neq a_i}[(3^{m-1}-1)\chi_4]
\textbf{err}^{(2)}:= \mathcal{O}\Bigg(\Big\Vert \gamma^{\textbf{int}}_{0}\mathrm{U}_{\mathrm{i}}\Big\Vert_{\mathrm{H}^{\frac{1}{2},\frac{1}{4}}\Big(\partial \Omega \times \mathbb{R}\Big)}\Big\Vert \gamma^{\textbf{int}}_{1,\mathrm{y}}\Phi^{\textbf{e}}(\xi,t;\mathrm{y},\cdot)\Big\Vert_{\mathrm{L}^2\Big(\partial \Omega \times (0,\mathrm{t})\Big)}\Bigg).
F_{B}^{b o s o n i c}={\frac{\Gamma ( 2 - 2 k_{0}^{2}) \Gamma ( \vec{k}\cdot \vec{p}+ k_{0}^{2}- 1 )}{\Gamma ( \vec{k}\cdot \vec{p}- k_{0}^{2}+ 2 )}}\bigg [ \vec{k}\cdot \vec{p}\epsilon_{i j}^{1}\epsilon_{i j}^{2}- 2 \epsilon_{i j}^{1}\epsilon_{i l}^{2}p^{j}k^{l}( 1 - k_{0}^{2}) \bigg ]
\sum_{p}h_{jk}^{p^{\ast}}\mathcal{L}H^{p^{\ast}}=\sum_{p}h_{jk}^{p^{\ast}}H^{p^{\ast}}-\sum_{i,l,p,q}h_{jk}^{p^{\ast}}h_{il}^{p^{\ast}}h_{il}^{q^{\ast}}H^{q^{\ast}},
\mathcal{B}_{t}(f,g): =e^{t^{\frac{1}{\alpha}}\Lambda_{1}}(e^{-t^{\frac{1}{\alpha}}\Lambda_{1}}fe^{-t^{\frac{1}{\alpha}}\Lambda_{1}}g) =\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot(\xi+\eta)}e^{t^{\frac{1}{\alpha}}(|\xi+\eta|_{1}-|\xi|_{1}-|\eta|_{1})}\hat{f}(\xi)\hat{g}(\eta)d\xi d\eta.
G_{\xi, \sigma}(x)\asymp\begin{cases}\begin{rcases}|x|^{\sigma-n},\,\,\, 0<\sigma<n \log\left(\frac{1}{|x|}\right),\,\,\, \sigma=n 1, \sigma>n \end{rcases}, \,\,\,\,|x|<1, |x|^{\frac{\sigma-l-1}{2}-|\Sigma^{+}_0|}\varphi_{0}(x) e^{-\xi|x|},\,\,\xi>0,\,\,\sigma>0,\,\,|x|\geq1.\,\,\, \end{cases}
(-\Delta)^{s}\Big\{\frac{1}{\left(1+x^2\right)^{\gamma}}\Big\}=\frac{A_s^{\gamma}}{(1+x^2)^{s+\gamma}}\,{}_{2}F_{1}\Big(\!-s, s+\gamma;\frac{1}{2};\frac{x^2}{1+x^2}\Big),
\vec{f}(\vec{x}) =\sum_{0\leq |m| \leq l}y_{lm}(r) \vec{\Upsilon}_{lm}+\sum_{1 \leq l, |m| \leq l}\psi_{lm}(r) \vec{\Psi}_{lm}+\sum_{1 \leq l, |m| \leq l}\phi_{lm}(r) \vec{\Phi}_{lm}.
\frac{d^2}{ds^2}\bigg|_{s=0}\widetilde{Vol}(\Sigma_s)= -\int_{\Sigma}\tilde{g}(X,\tilde{L}(X))d\tilde{v}=-\int_{\Sigma}e^{-\frac f{n}}\varphi \tilde{L}(e^{-\frac f{n}}\varphi)e^{-f}dv,
\begin{array}{rl}\int_0^{+\infty}e^{-u}\big( 1+(\frac{\eta}{h})^\gamma u\big )^{\frac{1}{\gamma}}\,du =\displaystyle{\int_0^{+\infty}e^{-u}\,du+\frac{1}{\gamma}\frac{\eta^\gamma}{h^\gamma}\int_0^{+\infty}e^{-u}u\,du}+ o(\frac{1}{h^{\gamma}}) & & = \displaystyle{1+\frac{1}{\gamma}\frac{\eta^\gamma}{h^\gamma}+o(\frac{1}{h^{\gamma}})}.\end{array}
\tau = -\frac{1}{B}\ln \left( \frac{1}{\left\| V \right\|_{C^1(M)}}\frac{\left\| y^{\frac{1-2s}{2}}_{n+1}w \right\|_{L^2(B_{r}^+)}}{\left\|y^{\frac{1-2s}{2}}_{n+1}w \right\|_{L^2(B_{2r}^+)}}\right) + C\left\| V \right\|_{C^1(M)}^{\frac{1}{2s}}.
\theta(\mathsf{N}_{m}^{n})_{j}^{i}=q^{(2\rho,\lambda_{m}-\lambda_{n})}q^{(2\rho,\lambda_{i}-\lambda_{j})}(\mathsf{N}_{m}^{n})_{j}^{i}=q^{(2\rho,\lambda_{m}-\lambda_{n})}\pi(K_{2\rho})_{i}^{i}(\mathsf{N}_{m}^{n})_{j}^{i}\pi(K_{2\rho}^{-1})_{j}^{j}.
\int_X \|\mathcal G(u) - m_N^\mathcal G(u) \|^{2}\mu_0(\mathrm{d}u) = \int_X \sum_{j=1}^J (\mathcal G^j(u) - m_N^{\mathcal G^j}(u))^2 \mu_0(\mathrm{d}u) \leq C h_U^{2\nu + K}\sum_{j=1}^J \| \mathcal G^j\|^2_{H^{\nu + K/2}(X)},
\Phi_{4,0}=-\frac{5}{4}V_{\Phi}-\frac{4 e^{\Phi_{\infty}}V_s}{3 P}+\frac{21}{2}M_1 V_1-\frac{16 M_2 V_1}{3 P e^{\Phi_{\infty}}}+\frac{8 M_2 V_2}{3 P^2 e^{2 \Phi_{\infty}}}-\frac{11 M_1 V_2}{2 P e^{\Phi_{\infty}}}+ O(M_i^4), begin{align*}2mm]
\begin{array}{c l r}{{\sum_{i = 0}^{r + 4}q_{i}^{a}s_{i}}}&{{=}}&{{0}}\\{{\sum_{i = 0}^{r + 4}q_{i}^{a}n_{i}}}&{{=}}&{{0}}\\{{\sum_{i = 0}^{r + 4}q_{i}^{a}m_{i}}}&{{=}}&{{0 .}}\\ \end{array}
\sum_{\nu \in v\Lambda^n}S_\nu S_\nu^*(f \sqrt{d\mu}) = \sum_{\nu\in v\Lambda^n}S_\nu( f \circ \sigma_\nu) \sqrt{d(\mu \circ \sigma_\nu)}&= \sum_{\nu \in v\Lambda^n}f \circ \sigma_\nu \circ \sigma^n \sqrt{d(\mu \circ \sigma_\nu \circ \sigma_\nu^{-1})}&= \sum_{\nu \in v\Lambda^n}\chi_{Z(\nu)}\cdot f \sqrt{d\mu},
D = \left( \begin{array}{c c}{{v^{- 1}\partial_{1}}}&{{u^{- 1}\partial_{2}}}\\{{u \partial_{4}}}&{{v \partial_{3}}}\\ \end{array}\right) \; , \quad \bar{D}= \hat{V}^{\epsilon \, - 1}D = \left( \begin{array}{c c}{{v \partial_{3}}}&{{- q^{- 2}u^{- 1}\partial_{2}}}\\{{- q^{- 2}u \partial_{4}}}&{{v^{- 1}\partial_{1}}}\\ \end{array}\right) \; ,
\| Q_n \varphi - Q \varphi\|\leq\sum_{\ell=n}^\infty \|Q_\ell \varphi - Q_{\ell+1}\varphi \|\leq\sum_{\ell=n}^\infty C_0^{- q_{\ell+1}}\leq\sum_{j=q_{n+1}}^\infty C_0^{-j}=\frac{C_0^{-q_{n+1}}}{1-C_0^{-1}}
\bigg\|\sum_{i=n+1}^{\infty}\frac{x_{2i}w_{2i-1}\cdots w_{2i-2n}}{x_{2n}w_{2n-1}\cdots w_{0}}e_{2i-2n}\bigg\|^{2}=\sum_{i=n+1}^{\infty}\bigg|\frac{x_{2i}w_{2i-1}\cdots w_{2i-2n}}{x_{2n}w_{2n-1}\cdots w_{0}}\bigg|^{2}\leq \sum_{i=n+1}^{\infty}\bigg|\frac{x_{2i}}{x_{2n}}\bigg|^{2}\bigg(\frac{w_{2i-1}}{w_{0}}\bigg)^{2}.
{\cal A}_{3 3}= \frac{3 g^{2}}{4}\sum_{\pi ( p_{1}, \ldots p_{6})}\bar{u}( p_{\pi_{1}}) \gamma_{\mu}u ( p_{\pi_{4}}) \bar{u}( p_{\pi_{2}}) \gamma^{\mu}\frac{i}{p_{\pi_{1}}\! \! \! \! \! \! \! / + p_{\pi_{2}}\! \! \! \! \! \! \! / - p_{\pi_{4}}\! \! \! \! \! \! \! / - m}\gamma_{\nu}u ( p_{\pi_{5}}) \bar{u}( p_{\pi_{3}}) \gamma^{\nu}u ( p_{\pi_{6}})
b :C_k(\mathfrak{A})\to C_{k-1}(\mathfrak{A}), b (a_0\otimes a_1\otimes \cdots\otimes a_k):=\sum_{j=0}^{k-1}(-1)^{j}a_0\otimes \cdots\otimes a_ja_{j+1}\otimes\cdots \otimes \cdots \otimes a_k+ &\qquad\qquad\qquad\qquad+(-1)^k a_ka_0\otimes a_1\otimes \cdots\otimes a_{k-1}.
\begin{array}{cccc}\Theta_n : G \to \bigoplus_{i=1}^n \mathfrak{sl}_2(\mathbb{Z}_\ell) & (g_1,\ldots,g_n) \mapsto \left( g_1 -\frac{1}{2}\operatorname{tr}(g_1), \ldots, g_n -\frac{1}{2}\operatorname{tr}(g_n) \right),\end{array}
\begin{aligned}p_1(x)p_2(x) = 1 \times 1 + (-2x_1) \times 1 + x_1^2 \times 1 + (1-2x_1 +x_1^2)\times (-2x_2) + (1-2x_1 +x_1^2) \times x_2^2, &(1-2x_1 +x_1^2) (- 2x_2) = 1 \times(-2x_2) + (-2x_1) \times (-2x_2) + x^2(-2x_2), &(1-2x_1 +x_1^2) x_2^2 = 1\times x_2^2 + (-2x_1)\times x_2^2 + x_1^2 \times x_2^2.\end{aligned}
\frac{(-1)^j}{j!}(N_-^j\Psi^-)(b_Z) =2(j+1)(P_{0-}\vec\Phi)(b_Z)_{j+1}\qquad+(m-2j)(P_{1-}\vec\Phi)(b_Z)_j \qquad-2(m-j+1)(X_-\vec\Phi)(b_Z)_{j-1}.
\sum_{\lambda\in\mathcal{P}}q^{|\lambda|}x^{\#\mathcal{H}_{2}(\lambda)}b^{BG(\lambda)}\prod_{h\in\mathcal{H}_{2}(\lambda)}\dfrac{1}{h^{2}}=\exp\left(\dfrac{xq^{2}}{2}\right)\sum_{j=-\infty}^{+\infty}b^{j}q^{j(2j-1)},
\Big{|}\int_{-1+ \beta b(\cdot)}^0u(\cdot,z)\: \mathrm{d}z \Big{|}_{H^k}^2 \leq M(k) \big{(}\|\nabla_{X,z}^{\mu}u\|^2_{H^{k,0}(\mathcal{S}_b)}+ \sum\limits_{j=1}^k\|\partial_z^{j}u\|_{H^{k-j,0}(\mathcal{S}_b)}^2\big{)}.
\langle \delta j^\mu (X) \rangle \simeq \frac{1}{2}\int \frac{d^{\, 3}p}{(2 \pi)^3}\left[ \hat{\cal P}_+^\mu Z_+ \frac{- i \Sigma_{off}^{(+)}}{|Im \Sigma_R^{(+)}|}- \hat{\cal P}_-^\mu Z_- \frac{- i \Sigma_{off}^{(-)}}{|Im \Sigma_R^{(-)}|}\right] ,
\|u\|^2_{L^2_\delta(\Omega)}=\int_\Omega u^2\rho^{-2\delta-n}dV_{g}\leq \left[\int_{\Omega}|u|^q dV_{g}\right]^{\frac{2}{q}}\left[\int_{\Omega}\rho^{-\frac{2\delta q}{q-2}-\frac{nq}{q-2}}dV_{g}\right]^{\frac{q-2}{q}}
a_{2s}= \left[\theta(x)+\left(\theta(x)v^{1s}_{x_1 x_1}+ s v^{1s}_{x_2 x_2}\right)\right]^{-1}=\left[\theta(x)+\left(\tfrac{\theta(x) \psi_{x_1 x_1}}{C_0 s}+ \tfrac{\psi_{x_2 x_2}}{C_0}\right)\right]^{-1}
{\tilde{\delta}}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] = \int ~{\cal D}\lambda_{a}~ \exp ~ i \int \lambda_{a}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] d t \nonumber
P_sf(A^*z)^\alpha \le P_s\big[f^\alpha\big](A(\Lambda z) +c) e^{\frac12 \frac{\alpha \alpha'}{e^{2s}-1}| A^*z - A(\Lambda z) - c |^2}e^{-\frac12 \frac{\alpha}{e^{2s}-1}| A^*z - A(\Lambda z) - c |^2}&= P_s\big[f^\alpha\big](A(\Lambda z) +c) e^{\frac12 \frac{\alpha}{(e^{2s}-1)(\alpha-1)}| A^*z - A(\Lambda z) - c |^2}.
\lim_{n \rightarrow +\infty}\frac{1}{n^{1+\frac{1}{\theta}}}K_n\left(\frac{x}{n^{1+\frac{1}{\theta}}}, \frac{y}{n^{1+\frac{1}{\theta}}}\right) = \mathbb K_{\alpha,\theta}^{\textrm{MB}}(x,y),x\in\mathbb C,\y>0,
E(y) = 20\pi \alpha_3 \varepsilon^{\frac{1}{2}}w_{\mu_i^{\prime},\zeta_i^{\prime}}(y)^4\Bigl(-\mu_i^{\frac{1}{2}}g_\lambda(\zeta_i)+\sum_{j\not=i}\mu_j^{\frac{1}{2}}G_\lambda(\zeta_i,\zeta_j)\Bigr) & + O( w_{\mu_i^{\prime},\zeta_i^{\prime}}(y)^3 \varepsilon^2) + O(\varepsilon^5) , y \in B_{\delta/\varepsilon}(\zeta_i')),
h(k,\ell+1) =\left(\frac{\ell}{\ell+1}-\frac{f(k)}{\ell+1}\right)h(k,\ell)+\frac{f(k)}{\ell+1}h(k-1,\ell) &\leq \left(\frac{\ell}{\ell+1}-\frac{f(k)}{\ell+1}\right)\frac{C}{\ell}+\frac{f(k)}{\ell+1}\frac{C}{\ell}=\frac{C}{\ell+1},
J_{k}g\left( \alpha\right) =\sum_{m_{k-1}\left( \alpha\right)\leq\beta\leq m_{k}\left( \alpha\right)}g\left( A\beta\right) , J_{\infty}g\left( \alpha\right) =g\left( \alpha\right) +\sum_{m_{N\left( \alpha\right)}\left( \alpha\right) \leq\beta\leq\alpha}g\left( A\beta\right)
F(x) =\int_0^x f(y)dy,c^\ast = a^\ast b + \int_0^1 F(y) \, dy, F(x,y) = \big[c^\ast - F(y) \big] \textbf{1}_{[0, x]}(y) + x \Big( F(y) - \int_0^1 F(z) dz - a^\ast b \Big)\textbf{1}_{[0, 1]}(y),
\sigma^2(y_0)=(x_2,x_3,x_4,x_5,x_0,x_1,\ldots,x_5,x_0,\ldots):=y_1 &\sigma^2(y_1)=(x_4,x_5,x_0,x_1,\ldots,x_5,x_0,\ldots):=y_2 &\sigma^2(y_2)=(x_0,x_1,\ldots,x_5,x_0,\ldots)=y_0
\|TX\|_2^2 =\sum_{i=1}^{\lfloor n/2\rfloor}\biggl(\sum_{j=n-\lfloor n/2\rfloor+1}^n M_{\sigma(i)\sigma(j)}x_{\sigma(j)}\biggr)^2 &\geq c^2\sum_{i=1}^{\lfloor n/2\rfloor}\biggl(\sum_{j=1}^n M_{\sigma(i)\sigma(j)}x_{\sigma(j)}\biggr)^2\chi_i &= c^2\sum_{i=1}^{\lfloor n/2\rfloor}\biggl(\sum_{j=1}^n M_{\sigma(i)j}x_{j}\biggr)^2\chi_i,
\delta ( t ) = \delta ( t_{0}) + \frac{k}{6 H^{2}a^{2}( t_{0})}+ \frac{\dot{\delta}( t_{0})}{3 H}- \frac{k}{2 H^{2}a^{2}( t_{0})}e^{- 2 H ( t - t_{0})}+ \left[ \frac{k}{3 H^{2}a^{2}( t_{0})}- \frac{\dot{\delta}( t_{0})}{3 H}\right] e^{- 3 H ( t - t_{0})}
V_0= \{\mathbf{z}\in \mathbb{C}^{n,1}\setminus \{\mathbf{0}\}: \langle\mathbf{z},\mathbf{z}\rangle=0\}, V_+= \{\mathbf{z}\in \mathbb{C}^{n,1}: \langle\mathbf{z},\mathbf{z}\rangle>0\}, V_-= \{\mathbf{z}\in \mathbb{C}^{n,1}: \langle\mathbf{z},\mathbf{z}\rangle<0\}.
\mathrm{\frac{1}{2}}( S_{m}, S_{m})^{a}- V_{m}^{a}S = i \hbar \Delta^{a}S_{m}\qquad \Longleftrightarrow \qquad \bar{\Delta}_{m}^{a}\mathrm{e x p}\{( i / \hbar ) S_{m}\}= 0 ,
\sum_{n=0}^{\infty}\frac{\left(q;q\right)_{n}q^{n^{2}}c^{n}S_{n}\left(xq^{-n};q\right)S_{n}\left(yq^{-n};q\right)}{\left(cxyq;q\right)_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}c^{n}A_{q}\left(cxq^{n}\right)A_{q}\left(cyq^{n}\right)}{\left(q;q\right)_{n}\left(cxyq;q\right)_{\infty}}.
\smash{\sum_{\substack{a \in \overline{E}\mathrm{head}(a) = i}}\epsilon(a) aa^*}(x) = (\pi(z_1) \to i)(i \to \pi(z_2))(x) + (\pi(z_2) \to i)(i \to \pi(z_1))(x) = (\pi(z_1) \to i)(z_1) + (\pi(z_2) \to i)(z_2) = -x + x = 0.
\sum_{n=1}^\infty \left[ \left(\frac{(-1)^{n+1}2 \pi n \sinh(\mu r_0)\sin(n \pi x /r_0)}{\sinh(\mu x) (n^2 \pi^2 + \mu^2 r_0^2)}\right) \exp\left(-\frac{(n^2 \pi^2/r_0^2 + \mu^2 )t}{2}\right) \right].
\mathcal{E}(p,0) =1-\frac{\sigma_{k-1}(p,1) \sigma_{\ell-1}(p,1)}{p^2}= 1 - (1- (1-p^{-1})^{k-1}))(1- (1-p^{-1})^{\ell-1})) = (1-p^{-1})^{k-1}+(1-p^{-1})^{\ell-1}- (1-p^{-1})^{k+\ell-2}
\left\Vert \left( \left( D^{l}F_{t}\right) -\left( D^{l}F_{s}\right) \right)\left( x_{t}\right) \right\Vert \leq \omega \left( s,t\right)^{\theta -\frac{l+1}{p}}\forall 0\leq s\leq t\leq Tl=0,1,\dots ,\left[ p\right] -1
(\varphi j_P^-(E_{o_{+,P}}^P(w)))(E_{o_+}(\lambda n^{-1})) = \varphi(j_P^-(E_{o_{+,P}}^P(w))(E_{o_+}(\lambda n^{-1}))) & = \varphi(E_{o_+}(\lambda n^{-1})j_{P'}^+(E^{P'}_{o_{+,P'}}(nwn^{-1}))) & = \varphi(E_{o_+}(\lambda n^{-1}))(n\sigma)(E^{P'}_{o_{+,P'}}(nwn^{-1})) & = \varphi(E_{o_+}(\lambda n^{-1}))\sigma(E^{P}_{o_{+,P}}(w)).
\sum\limits_{i=1}^m\lambda_i^k g_i(x^*)+\sum\limits_{i=1}^p\mu_i^k h_i(x^*)+\sum\limits_{i=1}^n\gamma_i^k x_i^*=0, \sum\limits_{i=1}^m\lambda_i^* g_i(x^k)+\sum\limits_{i=1}^p\mu_i^* h_i(x^k)+\sum\limits_{i=1}^n\gamma_i^* x_i^k=0,
\mathrm{t r}\, \mathbf{1}_{8}= 8 , \; \; \; \; \mathrm{t r}\, \mathbf{R}= 2 = \mathrm{t r}\, \mathbf{R}^{2}, \; \; \; \; \mathrm{t r}\, \mathbf{T}= 0 ,
\begin{array}{l c l}{{F_{1}\, = \, d D_{0},}}&{{\quad}}&{{F_{2}\, = \, d D_{1},}}\\{{F_{3}\, = \, d D_{2}- B_{2}\wedge d D_{0},}}&{{}}&{{F_{4}\, = \, d D_{3}- B_{2}\wedge d D_{1},}}\\{{F_{5}\, = \, d D_{4}- B_{2}\wedge d D_{2}+{\frac{1}{2}}B_{2}\wedge B_{2}\wedge d D_{0}.}}&{{}}&{{~}}\\ \end{array}
dX_t ={\varepsilon}^{-1}b_X(X_t, Y_t) dt +{\varepsilon}^{-1/2}\sigma_X(X_t, Y_t) dB^X_t, X_0 = x_0, dY_t = b_Y(X_t, Y_t)dt + \sigma_Y(Y_t) dB^Y_t, Y_0 = y_0
({\, \lower 0 . 9 p t \mathrm{\hrule \mathrm{\vrule ~ h e i g h t ~ 0 . 2 ~ c m ~ \hspace{~ 0 . 2 ~ c m}~ \vrule ~ h e i g h t ~ 0 . 2 c m}\hrule}\,}- m^{2}) F_{\alpha \beta}- 2 ( \partial_{\beta}F_{\alpha \mu , \mu}- \partial_{\alpha}F_{\beta \mu , \mu}) = 0 \quad ,
S = \int d^{4}x \sqrt{- g}[ \frac{\mathcal{R}}{1 6 \pi G}- \frac{1}{4 \pi}\frac{1}{2}( ( \partial_{\mu}\phi ) ( \partial^{\mu}\phi ) + ( \frac{g \phi}{2})^{2}A_{\mu}A^{\mu}+ V ( \phi ) + \xi \phi^{2}\mathcal{R}) - \frac{1}{4 \pi}\frac{1}{4}\mid F \mid^{2}] .
\frac{(2 c e \theta \log n)^{k^{\ast}}}{k^\ast (2 c \theta \log n)^{k^{\ast}}}=\frac{e^{k^{\ast}}}{k^{\ast}}\asymp \frac{n^{-c + 2c \theta}}{\log n}=\frac{n^{2\sqrt{ab}}}{\log n}
\begin{array}{l}-{\left(1-{2M\over r}\right)}^{-1}\partial_t^{\,2}{f\over r}+{\left(1-{2M\over r}\right)}^{-1}r^{-2}\partial_{r^*}\left(\left(1-{2M\over r}\right)r^2{\left(1-{2M\over r}\right)}^{-1}\partial_{r^*}{f\over r}\right) -\frac{l\left(l+1\right)}{r^2}{f\over r}-m^2{f\over r}=0\;,\end{array}
\sigma^1 = \left ( \begin{array}[c]{cc}0 1 1 0 \end{array}\right ), \quad\sigma^2 = \left ( \begin{array}[c]{cc}0 -i i 0 \end{array}\right ), \quad\sigma^3 = \left ( \begin{array}[c]{cc}1 0 0 -1 \end{array}\right )
D(s, S_f \times \overline{S_g}) = \sum_{n \geq 1}\frac{S_f(n) \overline{S_g(n)}}{n^{s + k - 1}}= \sum_{n \geq 1}\frac{1}{n^{s + k - 1}}\sum_{m = 1}^n a(m) \sum_{h = 1}^n \overline{b(h)}.
\left| \frac{1}{N(N+1)^2}\sum_{m, n = 0}^{N-1}(N + 1 - m) a_n \cdot b_m \cdot c_{n+m}\right|^2\leq \sup_t \left| \frac{1}{N}\sum_{m'=1}^{2(N-1)}c_{m'}e^{2\pi i m' t}\right|^2
S L ( 2 ,{\bf R})_{S}: \left( \begin{matrix}{\vec{\hat{\alpha}}}\\{\vec{\hat{\beta}}}\\ \end{matrix}\right) \to \left( \begin{matrix}{a}&{c}\\{b}&{d}\\ \end{matrix}\right) \left( \begin{matrix}{\vec{\hat{\alpha}}}\\{\vec{\hat{\beta}}}\\ \end{matrix}\right) .
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{nj}\binom kj\binom{k-j}s\frac{{G_{m - (n + 1)k - (n - 1)j + 2s}}}{{F_n^j}}}}= \frac{{G_m}}{{F_n^k}},n\ne 0\,,
g(\delta_1) =-q^{d+1}(q^3-(\alpha d+8d+4\alpha+22)q^2-(8d+4)q-4d) &\hphantom{={}}\mathrel{+}(d+5)q^6+(d+4)q^5+(d^2+8d+16)q^4, g(\delta_2) =-q^{2d+2}((\alpha+3)q-2\alpha^2-12\alpha-18) &\hphantom{={}}\mathrel{+}(\alpha+3)q^{d+5}+(d+\alpha+6)q^{d+4}&\hphantom{={}}\mathrel{+}(5d+10)q^{d+3}+(8d+4)q^{d+2}+4dq^{d+1}+q^6.
\sum_{k=1}^{\overline{r}(n;\theta) + 1}\mathfrak{J}_k^{(1)}(n;\theta) \leq \frac{S_1(\gamma)}{[n\theta(1-\theta)]^{(1+\varepsilon(\gamma))}}+ \frac{S_2(\gamma)+S_3(\gamma)}{[n\theta(1-\theta)]^{(1+\varepsilon(\gamma))}}\log\left(\frac{n}{\theta(1-\theta)}\right)\ .
\mathbf{P}((A_1,\ldots,A_N),s) :=\lim_{n \to \infty}\frac{1}{n}\log \sum_{i_1,\ldots,i_n=1}^N \varphi^s(A_{i_1}\cdots A_{i_n}) &=\mathbf{P}\left(\sum_{i=1}^N\delta_{A_i},s\right)=s\log N+\mathbf{P}\left(\frac{1}{N}\sum_{i=1}^N\delta_{A_i},s\right).
r^{S_0 \oplus S_1}_{(1, i)}= R_i + \beta \sum_{j \in N}p_{ij}f^{S_0 \oplus S_1}_{(1, j)}- f^{S_0 \oplus S_1}_{(1, i)}= R_i + \beta \sum_{j \in S_1}p_{ij}f^{S_1}_j - f^{S_1}_i = r^{S_1}_i,
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}-U_{z_2}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, &\left(\Omega^{[1]}\,U\right)_{z_1}-\frac{U}{2n}=0, &\Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
F_{n}\left({\vec{z}}\right) = \sum_{{\vec{c}}}D_{{\vec{c}}, n}\theta \left[ \begin{array}{c}{{{\vec{0}}}}\\{{\frac{{\vec{c}}}{\ell}}}\\ \end{array}\right] \left({\vec{z}}- \frac{\tau}{\omega \kappa}\left( n + \frac{\beta}{2 \pi}\right){\hat{e}}+ \frac{\alpha}{2 \pi \omega \kappa}{\hat{e}}, \Omega \right)
K_{N,V}(x,y)=\frac{\gamma_{N-1}^{(N,V)}}{\gamma_{N}^{(N,V)}}\,\frac{p_N^{(N,V)}(x)p_{N-1}^{(N,V)}(y)-p_{N-1}^{(N,V)}(x)p_{N}^{(N,V)}(y)}{x-y}e^{-\frac{N}{2}(V(x)+V(y))}.
T_{\alpha},\quad G_{a}=\frac{1}{12}\sigma^{\alpha\dot{\beta}}_{a}(R^\gamma_{\,\,\dot{\beta}\alpha\gamma}+ R^{\dot\gamma}_{\,\,\alpha\dot{\beta}\dot{\gamma}}),\quad W_{\alpha\beta\gamma}= \sigma^{a}_{(\alpha\dot{\delta}}\sigma^{b\dot{\delta}}_{\beta}T_{ab\gamma)}
\begin{array}{l l}{\varphi^{\prime}- \frac{A \varphi}{r}= 0}\\{\frac{A^{\prime}}{r h}- h \varphi^{2}e^{- 2 g M}+ 2 m M}\\{M^{\prime}+ \frac{G}{g r}= 0 ,}\\ \end{array}
|c_{i,j}| \le \dfrac{4V_{k,l}}{\pi^2}\begin{cases}\Gamma_{0,0}[s](i)\Gamma_{0,0}[r](j), ~~~~~~~\mbox{if}k=2s, l=2r, \Gamma_{0,0}[s](i)\Gamma_{1,-1}[r](j),~~~~~~\mbox{if}k=2s, l=2r+1, \Gamma_{1,-1}[s](i)\Gamma_{0,0}[r](j),~~~~~~\mbox{if}k=2s+1, l=2r, \Gamma_{1,-1}[s](i)\Gamma_{1,-1}[r](j),~~~~~~\mbox{if}k=2s+1, l=2r+1,\end{cases}
\delta_0: f\rightarrow\left( \begin{array}{c}-z f 2\left(\frac{\partial}{\partial\bar{s}}+\frac{1}{2}\chi\right) f \end{array}\right),\ \ \delta_1:\left( \begin{array}{c}g_0 g_1 \end{array}\right)\rightarrow \left(-2\left(\frac{\partial}{\partial\bar{s}}+\frac{1}{2}\chi\right) g_0-z g_1\right).
\{(x,U,\gamma_1,\ldots,\gamma_t)\in K^3\times K[x]_{\le r}\times \mathcal{C}^t \colon &(x,U,\gamma_i)\in \eqref{TX0GammaAlphaSet}\ \textrm{for each}\i=1,\ldots,t;\ \gamma_i\neq\gamma_j\ \textrm{if}\i\neq j\}
[c, \exp(-D)] = \sum_{n \ge 0}\frac{(-1)^n}{n!}[c, D^n] &= \sum_{n \ge 0}\frac{(-1)^{n+i}}{n!}\sum_{i=0}^{n-1}\sum_{i \le p \le n -1}\binom{p}{i}\sum_{\bold{q}= (q_1,\ldots,q_{i+1})}\tau_{\bold{q}}b_{\bold{q}}D^{n-i-1}\,.
\mathrm{Tr}(\mathcal{K}_{\alpha}^{2})=\int_{\mathcal{D}}\int_{\mathcal{D}}\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{2\alpha}}d\mathbf{y}d\mathbf{x}< \infty,0<\alpha <d/2.
\varphi_{1}=\begin{cases}\arccos\sqrt{{\displaystyle \frac{\cos^{2}\varphi-\cos^{2}\angle\left(v,V^{\bot}\right)}{1-\cos^{2}\angle\left(v,V^{\bot}\right)}}}, \angle\left(v,V^{\bot}\right)\in\left\langle 0,\frac{\pi}{2}\right] \varphi, \mbox{else}.\end{cases}
L_{x_3}^1 \big({0,1;W_{(t_{loc},x_1 ,x_2 ,m)}^{1,\infty}(\mathbb{R}_+ \times\mathbb{R}^2 \times \mathbb{R}_+)}\big)=\bigcap\nolimits_{T > 0}{L^1 \big((0,1);W^{1,\infty}((0,T) \times\mathbb{R}^2 \times \mathbb{R}_+)\big)};
\sum_{j=1}^{\infty}\frac{q^{j^2}}{(q)_{j}^{2}}\sum_{n=1}^{j}\frac{q^n}{(1-q^n)^2}=\frac{1}{2}\left.\frac{d^2}{dz^2}\sum_{j=0}^{\infty}\frac{q^{j^2}}{(zq)_j(z^{-1}q)_j}\right|_{z=1}.
\mathbb{P}\Bigl( \sup_{(s,t)\in[0,S]\times[0,1]}Z(s,t)> u \Bigr) = S2^{1-2/\alpha}\alpha^{-1}\mathcal{H}_{\alpha}^2u^{{4}/{\alpha}-2}\Psi(u) \bigl(1+o(1)\bigr);
Q_{K+u}(x,y,z)= \dfrac{\mathfrak{c}(m-4)}{2}\left(a^{\prime}\left(x-\ell_{0}d-\dfrac{d}{\mathfrak{c}}\right)^{2}+b^{\prime}\left(y-\ell_{0}d-\dfrac{d}{\mathfrak{c}}\right)^{2}+c^{\prime}\left(z-\ell_{0}d-\dfrac{d}{\mathfrak{c}}\right)^{2}\right).\textsc{}
H_\lambda D^{r}\Bigl(\frac{S(\lambda,r)}{H_\lambda}\Bigr) = K_r |\lambda|, H_\lambda D^{r+1}\Bigl(\frac{S(\lambda,r)}{H_\lambda}\Bigr) = K_r, H_\lambda D^{r+2}\Bigl(\frac{S(\lambda,r)}{H_\lambda}\Bigr) = 0.
\delta=\sup_{\beta'\in\left(0,1/2-2/d\right)}\frac{4d\beta'\left(1-\kappa\right)}{\left(1+nd\right)\left(d+1+\kappa\right)}>\frac{2\left(d-4\right)\left(1-\kappa\right)}{\left(1+nd\right)\left(d+1+\kappa\right)}-\varsigma
\Lambda_{n}= \left[ \begin{matrix}{{0}}&{{1}}&{{0}}&{{\cdots}}&{{0}}\\{{\vdots}}&{{0}}&{{1}}&{{\ddots}}&{{\vdots}}\\{{\vdots}}&{{{}}}&{{\ddots}}&{{\ddots}}&{{0}}\\{{0}}&{{{}}}&{{{}}}&{{\ddots}}&{{1}}\\{{\lambda}}&{{0}}&{{\cdots}}&{{\cdots}}&{{0}}\\ \end{matrix}\right]
\chi_i^{\mathcal{L}}(x)= m_i u_{i}^{\mathcal{L}}(x)+\ell_{i+1}u_{i+1}^{\mathcal{L}}(x), \chi_{n-1}^{\mathcal{L}}(x)= m_{n-1}u_{n-1}^{\mathcal{L}}(x), & \chi_i^{\mathcal{R}}(x)= m_i u_{i}^{\mathcal{R}}(x)+\ell_{i+1}u_{i+1}^{\mathcal{R}}(x), \chi_{n-1}^{\mathcal{R}}(x)= m_{n-1}u_{n-1}^{\mathcal{R}}(x).
F_N(1) =\frac{1}{N}\sum^{N-1}_{n=1}\frac{1-\cos\phi_n}{(1-\cos\phi_n)^2}=\frac{1}{2N}\sum^{N-1}_{n=1}\frac{1}{\sin^2 \phi_n/2}\,, F_N(2) =\frac{1}{N}\sum^{N-1}_{n=1}\frac{1-\cos2\phi_n}{(1-\cos\phi_n)^2}=\frac{2}{N}\sum^{N-1}_{n=1}\frac{\cos^2\phi_n/2}{\sin^2\phi_n/2}\,.
Tf=\sum_{n\in\mathbb{N}_0}e_n'(f)y_n,\,\,\,\,\,\textnormal{and}\,\,\,\,\,\sum_{n\in\mathbb{N}_0}\Vert e_n' \Vert^r_{E'}\Vert y_n \Vert^r_{F}<\infty.
s_{nN}^{k_{1}k_{2}}=\sqrt{(2k_{1}+1)(2k_{2}+1)}\cdot\sum_{M,M^{\prime}}(\begin{matrix}I&I&k_{1}\\M^{\prime}&-M&N\\ \end{matrix})(\begin{matrix}I&I&k_{2}\\M^{\prime}&-M&N\\ \end{matrix})
\begin{multlined}E_m E_{m-1}E_m E_{m+1}+ E_{m-1}E_m E_{m+1}E_m + E_m E_{m+1}E_m E_{m-1}+ E_{m+1}E_m E_{m-1}E_m - [2] E_m E_{m-1}E_{m+1}E_m = 0,\end{multlined}\begin{multlined}F_m F_{m-1}F_m F_{m+1}+ F_{m-1}F_m F_{m+1}F_m + F_m F_{m+1}F_m F_{m-1}+ F_{m+1}F_m F_{m-1}F_m - [2] F_m F_{m-1}F_{m+1}F_m = 0,\end{multlined}
{n \choose k_1,k_2}\geq \frac{1}{k_1!k_2!}(n-k_1-k_2)^{k_1+k_2}= \frac{n^{k_1+k_2}}{k_1!k_2!}\left(1-\frac{k_1+k_2}{n}\right)^{k_1+k_2}.
{\bf E}_M{\bf E}_v^{\otimes M}\{\bf E}_{y_L}^{\otimes M}\sum_{\gamma_1,...,\gamma_n = 1}^M \prod_{j=1}^n v_{\gamma_j}\exp(2 \pi i q_j \cdot y_{L,\gamma_j}) = \sum_{A \in \mathcal{A}_n}\prod_{a \in A}\left\{m_{|a|}\delta_{*,L}\left( \sum_{l \in a}q_l \right) \right\}
\frac{\partial}{\partial\bar{z}}h_{(r)}=\frac{\partial}{\partial\bar{z}}(h_{(r)}-h_{z,r})= \frac{\partial}{\partial\bar{z}}(\phi_{(r)}*(h-h_{z,r})) = \Big(\frac{\partial}{\partial\bar{z}}\phi_{(r)}\Big)*(h-h_{z,r}).
\tilde{F}_2(i,\hat{v}_1(i),v_2)= \bigg[\sum_{j\in S}{\psi}_{2}^{\hat{v}_1}(j)\pi_{ij}(\hat{v}_1(i),v_2)+c_2(i,\hat{v}_1(i),v_2){\psi}_{2}^{\hat{v}_1}(i)\bigg],~i\in S,