harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{E}_{N}\exp\left\{G_{N}(x^{N}(\eta))-G_{N}(x^{N}(0))\right\}\le\mathbb{E}_{N}\left[\exp\left\{a\sum_{i=1}^{N}\left|x_{i}^{N}(\eta)-x_{i}^{N}(0)\right|\right\}\right] \le\left\{\left(1+a\sqrt{\frac{2\eta}{\pi}}\right)\exp\frac{a^{2}\eta}{2}\right\}^{N}
-C_1(1-\eta)^{\alpha_0}\leq\partial_\xi w+\partial_\tau w\leq \frac{(\delta b)^{\frac{1}{\alpha_0}}}{K}(1-\eta)^{\alpha_0}in [0,T^*]\times[0,X]\times[0,1] , & -C_1(1-\eta)^{\alpha_0}\leq\partial_\tau w\leq b\delta(1-\eta)^{\alpha_0}in [0,T^*]\times[0,X]\times[0,1].
\nabla\theta_{1}^{\varepsilon}=\nabla_x\theta^{0}+\left(\nabla_{y}\bar{\theta}\right)^{\varepsilon}\nabla_{x}\theta^{0}+\varepsilon\bar{\theta}^{\varepsilon}\nabla_{x}\nabla\theta^{0}+\varepsilon\left(\nabla_{x}\bar{\theta}\right)^{\varepsilon}\nabla_{x}\theta^{0}.
\sup_{\|f\|_{1,\tau}\leq1}\int_{\mathbb{H}^{n}}\frac{1}{(\rho(\xi))^{\beta}}\left(\exp(\alpha|f(\xi)|^{Q'})-\sum_{k=0}^{Q-2}\frac{\alpha^{k}|f(\xi)|^{kQ'}}{k!}\right)d\xi<\infty,
\lim_n \int\int(F_n(x)-F_n(y))^2J^f(dxdy) =2\lim_n\int\int_{\left\lbrace x<y\right\rbrace}\left(\int_x^y\varphi_n(t)F'(t)dt\right)^2J^f(dxdy) &=\int\int(F(x)-F(y))^2J^f(dxdy).
\|D_c u\|(X)\leq \liminf_{t\to 0^+}\int_X |DT_t u|\,d\mu \leq\liminf_{t\to 0^+}\liminf_{j\to\infty}\int_X |DT_t f_j|\,d\mu &\leq\liminf_{t\to 0^+}\liminf_{j\to\infty}e^{-Kt}\int_X T_t |Df_j|\,d\mu &\leq\liminf_{t\to 0^+}\liminf_{j\to\infty}e^{-Kt}\int_X |Df_j|\,d\mu = \|D_c u\|(X),
\sqrt 2 \biggl ( B_{\underline{{i}}}^{( 1 )}\alpha_{I J}+ i B_{\underline{{i}}}^{( 2 )}\beta_{I J}\biggr ) \epsilon^{J}= e^{- \phi + 3 U}\bigl ( \partial_{\underline{{i}}}\overline{{V}}\bigr ) \gamma^{0}\epsilon_{I}\, .
B^{T}_{n,k-1,k-2}\phi_u(\eta)= \sum_{\substack{\tau\in\binom{[n]}{k-1}, \eta\subset\tau}}\phi_u(\tau) = \sum_{\substack{\tau\in\binom{[n]}{k-1}, \eta\subset\tau,\, u\notin\tau}}\phi(\tau\cup\{u\}) =\sum_{\substack{\sigma\in\binom{[n]}{k}, \eta\cup\{u\}\subset\sigma}}\phi(\sigma)= B^T_{n,k,k-1}\phi(\eta\cup\{u\})=0.
k_{0}^{m}(y,t) = \begin{cases}\displaystyle 2 \left( \frac{m}{2\pi}\right)^{\frac{d+1}{2}}\frac{te^{mt}K_{\frac{d+1}{2}}\big(m(|y|^{2}+t^{2})^{\frac{1}{2}}\big )}{(|y|^{2}+t^{2})^{\frac{d+1}{4}}}, m>0, \displaystyle \frac{\Gamma (\tfrac{d+1}{2})}{\pi^{\frac{d+1}{2}}}\frac{t}{(|y|^{2}+t^{2})^{\frac{d+1}{2}}}, m=0. \end{cases}
\widetilde{S}_{min}{f}=-\frac{d^2f}{dx^2}, \mathcal{D}(\widetilde{S}_{min})=\left\{f\in{W}_2^2(\mathbb{R}\backslash\{0\}) : \begin{array}{c}f_s(0)=f_r(0)=0 f_s'(0)=(q, f)\end{array}\right\}.
\begin{array}{c}{[ s_{\alpha \beta}, s_{\gamma \delta}] = \epsilon_{\alpha \delta}s_{\beta \gamma}+ \epsilon_{\beta \delta}s_{\alpha \gamma}+ \epsilon_{\alpha \gamma}s_{\beta \delta}+ \epsilon_{\beta \gamma}s_{\alpha \delta},}\\{{}}\\{s_{\alpha \beta}= s_{\alpha}{}^{\gamma}\epsilon_{\gamma \beta}= s_{\beta \alpha}.}\\ \end{array}
III = \beta_n\Delta_n \underline{\mathcal M}_n + \beta_n\Delta_n e^{-\beta_n \Delta_n}\mathcal V_n^{(1)}\mathcal J_n^{(1)}+\beta_n\Delta_ne^{-\beta_n \Delta_n}\mathcal V_n^{(2)}\mathcal J_n^{(2)}
\mathbf{A}^0=(g^0)^{-1}(V_1+\overline{g\mathbf{A}}),\mathcal{V}^0=\overline{\mathcal{V}}+\overline{\langle g\mathbf{A},\mathbf{A}\rangle}-\langle g^0\mathbf{A}^0,\mathbf{A}^0\rangle -W .
- \int d^{1 0}x \ \sqrt{- g}e^{- 2 \phi}R \, = \, \int d^{1 0}x \ \sqrt{- g}e^{- 2 \phi}\biggl ( \omega_{a}{}^{a c}\omega_{b}{}^{b c}+ \omega_{a}{}^{b c}\omega_{b}{}^{c a}+ 4 ( \partial_{a}\phi ) \omega_{b}{}^{b a}\biggr ) \ ,
\mathbf E[C_{k,k-1}(n+1)] = \Big(1+\frac{2p^{k-1}+(k-1)p^k}n\Big)\mathbf E[C_{k,k-1}(n)] + \frac 1n O\big(n^{kp^{k-1}}\big) \overset{\ref{co:rec}}=O\Big(n^{\max\{2p^{k-1}+(k-1)p^k,kp^{k-1}\}+\varepsilon}\Big)
\begin{array}{c}\displaystyle p_n(X) = \sigma(0;0,3,\ldots,3) + \sigma(0;0,-1,1,-1,\ldots,1,-1) + \displaystyle + \sigma(0;0,\ldots,0,-2,0,\ldots,0) + \sigma(0;0,\ldots,0,-2,0,\ldots,0,-2,0,\ldots,0).\end{array}
u_{1}=f(t)y+g(t)+W(z), & u_{2}=f(t)x+g(t)+Q(z), & p=-\frac{1}{2}f(t)^{2}(x^{2}+y^{2})-f^{\prime}(t)xy-[g^{\prime}(t)+f(t)g(t)](x+y), & z=f(t)(x+y)+g(t),
\varphi_t(u) =\begin{cases}u^2, |u| \le \frac{t}{2}, (t - u)^2, \frac{t}{2}\le |u| \le t, 0, |u| \ge t,\end{cases}\psi_t(u) =\begin{cases}1, |u| \le \frac{t}{2}, 2 - \frac{2}{t}|u|, \frac{t}{2}\le |u| \le t, 0, |u| \ge t,\end{cases}
\varepsilon^{-1}_n+\varepsilon^{-1}_m=\frac{1}{K}\left(d(\mathbb{I}_Y^{(1)}, N_{\alpha^{(n)}})+d(\mathbb{I}_Y^{(1)}, N_{\alpha^{(m)}})\right)\geq\frac{1}{K}d(N_{\alpha^{(n)}}, N_{\alpha^{(m)}})=\frac{1}{K}d(\mathbb{I}_Y^{(1)}, N_{\alpha^{(n)}-\alpha^{(m)}}),
W_2(a_{\epsilon})\>=\>{(2-\epsilon)(1-\epsilon)\over 2 \epsilon}\int_0^\infty{dy \over [s(y)]^{1-\epsilon}}\int_0^1{du \over u^\epsilon}\>[\epsilon+(1-\epsilon)\> u] \>\exp\left [ - (y/a_{\epsilon})^{1/\epsilon}{u \over s(y)}\right ]
\frac{\partial \mathcal{Z}_1}{\partial g_{ik}}=\frac{\partial \Phi_1^c}{\partial g_{ik}}+(F_A'(\omega_B)-1)\frac{\partial \Phi_2^c}{\partial g_{ik}}&=\Big(\big(F_A'(\omega_B)-1\big)\big(F_B'(\omega_A^c)-1\big)-1\Big)\frac{\partial \omega_A^c}{\partial g_{ik}}+\big(F_A'(\omega_B^c)-F_A'(\omega_B)\big)\frac{\partial \omega_B^c}{\partial g_{ik}}.
\pi_{e_{k}a}(r^{-1}m) =\displaystyle \sum_{s \in L(k), a_{1}\in_{k}T}\pi_{e_{k}a}r^{-1}\xi_{sa_{1}}\pi_{sa_{1}}(m) &=\displaystyle \sum_{s,u \in L(k), a_{1}\in_{k}T}\pi_{e_{k}a}\xi_{ua_{1}}\left(u^{\ast}(r^{-1}s)\pi_{sa_{1}}(m)\right) &=\displaystyle \sum_{s \in L(k)}e_{k}^{\ast}(r^{-1}s)\pi_{sa}(m) &=\pi_{ra}(m)
E_k \triangleright [E_\xi, E_{\xi^\prime}^*]_q = (E_k \triangleright E_\xi) E_{\xi^\prime}^* + (K_k \triangleright E_\xi) (E_k \triangleright E_{\xi^\prime}^*) - q^{-(\xi, \xi^\prime)}(E_k \triangleright E_{\xi^\prime}^*) E_\xi - q^{-(\xi, \xi^\prime)}(K_k \triangleright E_{\xi^\prime}^*) (E_k \triangleright E_\xi).
(M^0)_{ij}=\frac{1}{\det F}\begin{vmatrix}\big(z_1^0\big)^{n-1}\cdots \big(z_n^0\big)^{n-1}\vdots \cdots \vdots \big(z_1^0\big)^{n+1-i}\cdots \big(z_n^0\big)^{n+1-i}\vdots \cdots \vdots 1 \cdots 1 \end{vmatrix},
\mathfrak{i}_{\mathrm{p}}\left( t\right) :=\underset{(\eta,l^{-1})\rightarrow (0,0)}{\lim}\left\{\left( \eta^{2}\left\vert \Lambda_{l}\right\vert \right)^{-1}\mathfrak{I}_{\mathrm{p}}^{(\omega ,\eta\mathbf{A}_{l})}\left( t\right) \right\}\ .
\prod_{k=1}^m\|f_k\|_{\dot{B}^{p_k,\lambda_k}_{\omega_k}\left(\mathbb R^d\right)}=\; \prod_{k=1}^m\left(\frac{d+\alpha_k}{\omega(S_d)}\right)^{\lambda_k}\left(1+\lambda_k p_k\right)^{-1/p_k}\leq\; \left(\frac{d+\alpha}{\omega(S_d)}\right)^{\lambda}\left(1+\lambda p\right)^{-1/p}.
S_{\rm eff}= \int\! d^{2}x\, \Bigl({1\over 2}\,\partial_{\alpha}\chi\partial_{\alpha}\chi +{1\over 2}\,\partial_{\alpha}\chi'\partial_{\alpha}\chi' +{v_{N-1}\over 2}\,\chi'^{2}-{\tilde\delta v\over 2}\,\left(\chi^{2}+ \chi'^{2}\right)+{\pi\tilde V\over N}\, \left( \chi^{2}+ \chi'^{2}\right)^{2}\Bigr).
C=-sA^{L'}\;\; , \;\;\hspace{1em}\bar{{\cal P}}=-isA_{0}\;\; , \;\; \hspace{1em}\mbox{\boldmath $\pi$}^L=\frac{\nabla}{\nabla^2}(s{\cal P}+\rho^{(2)})\;\; , \;\; \hspace{1em}\pi_0=is\bar{C}\;\; ,
\tau_i \le C \begin{cases}m_i^{-2}, \mbox{if}5\le n\le 7, \max\{\Theta_i m_i^{-2}\log m_i,m_i^{-2}\}, \mbox{if}n=8, \max\{\Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\}, \mbox{if}n\ge 9.\end{cases}
\dfrac{S^{(n+2)}(0)}{3(n+3)!}=n\left( n^{2}+9n+2\right) -5n(n+2)\left(n+5\right) +5(n+1)(n+2)^{2}+\dfrac{1}{6}(n-1)(n+6)\left( n+5\right) (n+4)=\allowbreak \dfrac{1}{6}np(n)
S [ \, H \ , B \, ] = \int d^{4}x \left[ \,{\frac{1}{1 2}}H_{\lambda \mu \nu}\, H^{\lambda \mu \nu}-{\frac{1}{6}}H^{\lambda \mu \nu}\partial_{[ \, \lambda}B_{\mu \nu \, ]}+{\frac{m^{2}}{4}}B_{\mu \nu}\, B^{\mu \nu}\, \right]
\prod_{n = 1 \atop n \equiv 1 - j ( \mathrm{m o d}2 )}^{\infty}( 1 + q^{n}) = \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q )_{r}( - q )_{r}}= \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q^{2}; q^{2})_{r}},{}j = 0 , 1 .
\|\omega_z\|_{p/2,v}=\bigg(\int_0^\infty\min(t^p,x^p)t^{-p}w(t)\,dt\bigg)^{2/p}=x^2\bigg(\int_0^\infty\omega_x(t)^{p/2}w(t)\,dt\bigg)^{2/p}.
f^{\prime \prime}=\left( \frac{B_{2}^{\prime}}{B_{2}}\frac{q_{1}}{q_{2}}-\frac{q_{1}^{\prime}}{q_{2}}\right) f+\frac{1}{2}\left( \frac{B_{2}^{\prime}}{B_{2}}-\frac{q_{2}}{q_{3}}-\frac{N^{\prime}}{N}\right) f^{\prime}.
\int^{\eta_n(s)-\delta}_0 \psi_{n,2}^2(u,s) du = \int^{\eta_n(s)-\delta}_0 [ (s- \eta_n(u))^{\alpha}- (\eta_n(s)- \eta_n(u) )^\alpha]^2du \le C n^{-2}\delta^{2\alpha-2}.
\Delta^{\mp , a , p}( x , x^{\prime}) = \int_{- \infty}^{+ \infty}d z \int_{- \infty}^{+ \infty}\frac{d p_{D}}{2 \pi}e^{- i p_{D}y^{D}}\Delta_{Q}^{\mp , a , p}, \quad y_{\mu}= x_{\mu}- x_{\mu}^{\prime},
( X_{t}^{(0)}) \, =\, ( X_{t}^{\bullet}) \, =\, \frac{\, 1 - e^{-2t}\,}{2}\, , ( X_{t}^{(1)}) =( X_{t}^{\dagger}) \, =\, t \,e^{-2t}( I_{0}(2t) + I_{1}(2t) ) \,.
\nabla X_a \nabla \o= \omega_a \frac{a \nabla |x|}{|x| log |x|}\biggl( \textrm{log}|x| \nabla \alpha + \alpha \frac{\nabla |x|}{|x|}\biggl)=\o_a \biggl[\frac{a \nabla |x| \nabla \alpha}{|x|}+ \frac{a \alpha}{|x|^2 \textrm{log}|x|}\biggl].
\det A_n = \prod_{i=1}^n t_i = \prod_{i=1}^n \prod_{j=1}^m \frac{t_i(j,x)}{t_i(j-1,x)}= \prod_{i=1}^n \prod_{j=1}^m \frac{(x+i-j)(x+2i+j-2)}{(x+2i-j)(i+j-1)}. \square
\tilde{A}^{(1)}_1(x) \; = \;- \; \frac{1}{e}\; \frac{x_2}{x_1^2 + x_2^2 + \lambda^2}\;\left[ 1 + \frac{x_1^2}{(x_1^2+\lambda^2)^{\frac{3}{2}}}\right]
f(x) = \int_{\partial M}G^\omega_{\partial M}(x,z) \Lambda^\omega_{g,F}f(z) d\mu_h(y), = \int_{\partial M}N^\omega_{\partial M}(x,z) \Lambda^\omega_{g,F}f(z) d\mu_h(z) + \int_{\partial M}e^{\phi(z)}\frac{u_j(x)u_j(z)}{\lambda_j-\omega^2}\Lambda_{g,F}^\omega f(z) d\mu_h(z).
\left|\left(h_3(t+\delta)-h_3(t)\right)/\delta-d(t)\right| \leq \left| \sum_{k=3}^{k_0}p_{\beta}\left( \left( q^{*k}\right)^{\prime}(t_k)-\left( q^{*k}\right)^{\prime}(t)\right)\right| +\left| \sum_{k\geq k_0}\left(\left(q^{*k}\right)^{\prime}(t_k)-(q^{*k})^{\prime}(t)\right)\right|
\begin{array}{cc}1 + d a_{11}( a_{11}x_1 + a_{12}x_2 )^{d-1}+ d a_{22}( a_{21}x_1 + a_{22}x_2 )^{d-1}+ d^2 (a_{11}a_{22}- a_{12}a_{21}) ( a_{11}x_1 + a_{12}x_2 )^{2d-2}( a_{21}x_1 + a_{22}x_2 )^{2d-2}.\end{array}
\partial_{\mu}\pi_{i}^{j}\to \partial_{\mu}\pi_{i}^{j}- \left( \begin{array}{c c}{O}&{- m_{1}A_{\mu}+ m_{2}\gamma_{5}B_{\mu}}\\{m_{1}A_{\mu}+ m_{2}\gamma_{5}B_{\mu}}&{O}\\ \end{array}\right)_{i}^{j}
b(n) = n \sum \limits_{\substack{m_1,...,m_n \geq 0 m_1 + 2m_2 + ... + nm_n = n}}(-1)^{m_1 + m_2 + ... + m_n}\cdot \frac{(m_1 + m_2 + ... + m_n - 1)!}{m_1!m_2! \cdots m_n!}\cdot \sigma_1^{m_1}\cdots \sigma_n^{m_n}.
\beta_{k 0}( t \to \infty ) = \sqrt{\frac{\omega_{k}}{\tilde{\omega}}}\frac{( \tilde{\omega}- \omega_{k}) ( 2 \omega_{k}^{2}- 2 \tilde{\omega}^{2}+ i \pi g \omega_{k})}{\left[ 4 \left( \omega_{k}^{2}- \tilde{\omega}^{2}\right)^{2}+ \pi^{2}g^{2}\omega_{k}^{2}\right]}\sqrt{g \Delta \omega}\mathrm{e}^{i \omega_{k}t}\; .
P_{\sigma}^{G G}= \alpha^{2}x \sigma \left( \frac{2 ( K^{\prime})^{2}}{x^{2}}e^{2 \gamma \psi}+ ( H^{\prime})^{2}+ ( \psi^{\prime})^{2}\right) \ , P_{H}^{G G}= 2 \sigma H K^{2}\ .
\frac{d^{2}X^{\mu}}{dT^{2}}=\frac{d^{2}x^{\nu}}{dT^{2}}\frac{\partial X^{\mu}}{\partial x^{\nu}}+\frac{dx^{\nu}}{dT}\frac{dx^{\alpha}}{dT}\frac{\partial^{2}X^{\mu}}{\partial x^{\nu}\partial x^{\alpha}}
D ( \vec{V}, \vec{\sigma}) = \left( \begin{array}{c | c}{{\gamma}}&{{0}}\\ \hline{{\displaystyle - \frac{\vec{V}}{c}\gamma^{- 1}}}&{{I + \displaystyle \frac{\vec{V}\otimes \vec{V}^{\mathrm{T}}}{c^{2}\gamma \left[ \gamma + \sqrt{\gamma^{2}+ \displaystyle \frac{\vec{V}}{c}^{2}}\right]}- \frac{\vec{V}\otimes \vec{\sigma}^{\mathrm{T}}}{c^{2}\gamma \gamma_{0}}}}\\ \end{array}\right) .
\hat{q}_3 \equiv \frac{q_3}{\sqrt{2\vert eH \vert}}_,~~~ \hat{m}_{\mu}\equiv\frac{m(0,\mu)}{\sqrt{2\vert eH \vert}}_, ~~~\hat{\mu}\equiv \frac{\mu}{\sqrt{2\vert eH \vert}}_, ~~~\hat{Q}_{1,2}\equiv\frac{Q_{1,2}}{\sqrt{2\vert eH \vert}}_.
\begin{array}{r c l}{{{\cal L}}}&{{=}}&{{\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi - m^{2}\phi^{\dagger}\phi - V ( | \phi | )}}\\{{}}&{{}}&{{}}\\{{}}&{{=}}&{{\frac{1}{2}( \partial_{\mu}\phi_{1})^{2}- \frac{1}{2}m^{2}\phi_{1}^{2}+ \frac{1}{2}( \partial_{\mu}\phi_{2})^{2}- \frac{1}{2}m^{2}\phi_{2}^{2}- V \left( \sqrt{\phi_{1}^{2}+ \phi_{2}^{2}}\right) \ ,}}\\ \end{array}
\sum_{r=1}^{m}(-\delta)^{r-1}[r-1]!\sum_{1\le c_{1}<\cdots <c_{r}\le m}q^{\sum_{p=1}^{r}(c_{p}-m)}\prod_{p=1}^{r}\left( z_{c_{a}}\!\!\!\prod_{c_{a}<i<c_{a+1}}\!\!\!(q^{p}+\delta [p] z_{i})\right)=\sum_{i=1}^{m}z_{i},
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2\frac{2^{\alpha}}{\pi}\Re \int_{0}^{1}\xi^{\frac{\alpha}{2}-\frac{1}{2}}(1-\xi)^{-\frac{1}{2}}\left(\sqrt{1-\xi}+i\sqrt{\xi}\right)^{2p}{\rm d}\xi
\mu_{k}=\delta_{b_{1}^{-1}D_1}\ast\delta_{b_{1}^{-1}b_{2}^{-1}D_2}\ast\delta_{b_{1}^{-1}b_{2}^{-1}b_{3}^{-1}D_3}\ast\cdots\ast\delta_{b_{1}^{-1}b_{2}^{-1}\cdots b_{k}^{-1}D_k},
g(f(xy)) = g((fx)y) - g((fy)x), (f(xy))g = ((fx)y)g - ((fy)x)g, f((gx)y) = (f(gx))y - (fy)(gx), f((xg)y) = (f(xg))y - (fy)(xg), f(x(gy)) = (fx)(gy) - (f(gy))x, f(x(yg)) = (fx)(yg) - (f(yg))x.
c^{0}= \left[ ( 1 - \mu^{2}k^{2}r^{4}) \left( 1 - \mu^{2}/ k^{2}\right) \right]^{\frac{1}{4}\left( 1 - \frac{\mu}{k}\right)}\sqrt{\left( 1 + \mu / k \right) ( 1 + \mu k r^{2})}.
w\,\partial_w \big|_{w = z_i}\, f(w) = \Biggl( \, \sum_{j(\neq i)}^N \frac{z_i}{z_{ij}}\Biggr) f(z_i) + \sum_{j (\neq i)}^N \frac{z_i}{z_{ij}}\, \frac{\varphi_i(z_i)}{\varphi_j(z_j)}\, f(z_j)
\int_Q F_m^{*n}(x)dx =\frac{1}{(2m+1)^d}\int_{(2m+1)Q}f^{*n}(t)dt &\le\frac{1}{(2m+1)^d}2^d \big((2m+1)^d|Q|)\big)^{n\beta/d}\log N &=\frac{2^d k \log m}{(2m+1)^{d-n\beta}}|Q|^{n\beta/d}\le |Q|^{n\beta/d}
\lambda_1-\lambda_2 \in L^{\varphi(p)j_0}({M^*}^{j_0}\lambda_0+\mathbb{Z}^2)\subset L^{\varphi(p)j_0}({M^*}^{j_0}\mathcal{Z}_{D}^2+\mathbb{Z}^2) &\subset\bigcup_{j=1}^{p^2-1}L^{\varphi(p)j}({M^*}^{j}\mathcal{Z}_{D}^2+\mathbb{Z}^2)\subset \mathcal{Z}(\hat{\mu}_{M,D}).
\prod_{k=1}^{p+1}F_N(x_k,t) = \left[\frac{(z-p)t}{pN}+1\right]\prod_{k=1}^{p}F_N(y_k,t)-\left(\frac{t}{pN}\right)\frac{d}{dt}\left[\prod_{k=1}^{p}F_N(y_k,t)\right]
C^{(4)}(x_1,x_2,x_3)=\frac{1}{4}\Bigg(L_1^2+L_2^2+L_3^2 \hphantom{C^{(4)}(x_1,x_2,x_3)=\frac{1}{4}\Bigg(}{}+\big(x_1^2+x_2^2+x_3^2\big) \left(\frac{k_1^2-1/4}{x_1^2}+\frac{k_2^2-1/4}{x_2^2}+\frac{k_3^2-1/4}{x_3^2}\right)-\frac{3}{4}\Bigg).
{\cal U}( f , h , u ) = \frac{( u^{2}- 1 )^{2}}{2 e^{2}r^{4}}+ \frac{\eta^{2}u^{2}h^{2}}{r^{2}}+ \frac{\eta^{2}f^{2}}{r^{2}}+ \frac{\lambda \eta^{4}}4 ( h^{2}- 1 )^{2}+ \frac{\lambda \eta^{4}}4 ( f^{2}- 1 )^{2}\ ,
\frac{\partial}{\partial x^{\nu}}T_{\mu}^{\ \nu}=\frac{\partial T_{\mu}^{\ 0}}{\partial x_{0}}+\frac{\partial T_{\mu}^{\q}}{\partial x_{q}}\qquad\left(\mu,\nu=0,1,2,3;p,q=1,2,3\right) ,
d^{*}(y)= -t +cy + r \sqrt{1+y^2}\frac{d}{dy}d^{*}(y)= c + \dfrac{ry}{\sqrt{1+y^2}}\frac{d^2}{dy^2}d^{*}(y)= \dfrac{r}{(1+y^2)^\frac{3}{2}}.
\frac{i\omega}{\omega_{c}^{\prime}}\rightarrow\frac{1}{Q_{1}(\frac{i\omega}{\omega_{01}}+\frac{\omega_{01}}{i\omega})}+\frac{1}{Q_{2}(\frac{i\omega}{\omega_{02}}+\frac{\omega_{02}}{i\omega})}+\cdot\cdot\cdot
\|[g_{i,t}(x_{i,t})]_+\|^2 =\|[g_{i,t}(x_{i,t})]_+-[g_{i,t}(x_{j,t})]_++[g_{i,t}(x_{j,t})]_+\|^2 &\ge\frac{1}{2}\|[g_{i,t}(x_{j,t})]_+\|^2-\|[g_{i,t}(x_{i,t})]_+-[g_{i,t}(x_{j,t})]_+\|^2 &\ge\frac{1}{2}\|[g_{i,t}(x_{j,t})]_+\|^2-\|g_{i,t}(x_{i,t})-g_{i,t}(x_{j,t})\|^2 &\ge\frac{1}{2}\|[g_{i,t}(x_{j,t})]_+\|^2-G_2^2\|x_{i,t}-x_{j,t}\|^2,
\begin{array}{rcl}\delta(x_i) = x_i\otimes 1+q^{-h_i}\otimes x_i\ ,\cr\delta(y_i) = y_i\otimes q^{h_i}+1\otimes y_i\ ,\cr\delta(h_i) = h_i\otimes 1+1\otimes h_i\ ,\cr\delta(d) = d\otimes 1+1\otimes d\ ,\end{array}
\Gamma_{3 , 0}^{\L \L_{0}}\left[ \Phi \right] = \int d^{2}\theta \int_{p_{1}, p_{2}}\Phi ( \theta , p_{1}) \Phi ( \theta , p_{2}) \Phi ( \theta , - p_{1}- p_{2}) G^{\L \L_{0}}( p_{1}, p_{2})
\left(\frac{d}{dx}\right)^nA(x)^m\Big|_{x=x_0}=\left(\frac{d}{dx}\right)^n\Bigg(\sum_{l=0}^n\frac{1}{l!}\left(\frac{d}{dy}\right)^lA(y)\Big|_{y=x_0}(x-x_0)^l\Bigg)^m\Bigg|_{x=x_0}&=n!\prod_{j=1\atop order}^m\Bigg(\sum_{l_j=0}^n\frac{1}{l_j!}\left(\frac{d}{dy}\right)^{l_j}A(y)\Big|_{y=x_0}\Bigg)1_{\sum_{j=1}^ml_j=n}.
V(\mathbb{T}) =\left\{v_1,\ldots, v_{d(v)-f(v)}\big | v\in V(G) \right \}\bigcup \left \{e_v, e_u \big | vu=e\in E(G) \right \}E(\mathbb{T}) = \left\{v_ie_v \ \big | \i=1,\ldots, d(v)-f(v); \e=vu\in E(G) \right \}\bigcup & \bigcup \left \{e_ve_u \ \big | \e=vu\in E(G) \right \}.
\widetilde{\Gamma}_{k p}\ = \ \frac{1}{\mu_{k}{-}\bar{\mu}_{p}}T_{k}^{\dagger}T_{p}\quad , \qquad \textrm{i . e .}\qquad \sum_{k = 1}^{m}\Gamma^{\ell k}\widetilde{\Gamma}_{k p}\ = \ \delta_{\p}^{\ell}\quad .
\langle D^2_1 J_1(f;v^1,v^2),(w^1,w^2) \rangle = \iint_{\mathcal{O}_1 \times (0,T)}\eta w^2 dxdt + \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}w^1 w^2 dxdt.
\Gamma_{P V}[ A ] = \left. \frac{d}{d s}\zeta_{P V}( s ) \right|_{s = 0}= \lim_{s \to 0}\pm \frac{i}{\Gamma \left( \frac{s + 1}{2}\right)}\int_{0}^{\infty}\! \! \! \! d t ~ t^{\frac{s - 1}{2}}~ \frac{d G_{\mp}( t , s )}{d s}\mathrm{T r}[{D \! \! \! \! /}\exp ( - t{D \! \! \! \! /}^{~ 2}) ] .
\overline{U}=h_{1}^{-1}(\overline{U})\cup h_{2}^{-1}(\overline{U})\subset \left(h_{1}^{-1}(K(h_{2}))\setminus \mbox{int}(K(h_{1}))\right) \cup \left(K(h_{2})\setminus \mbox{int}(h_{2}^{-1}(K(h_{1})))\right).
(-1)^ks(n,n-k) = \sum_{j=0}^k(-1)^{j+k}\binom{n+j-1}{k+j}\binom{n+k}{k-j}S(j+k,j) &= \frac{(n+k)!}{(2k)!(n-1-k)!}\sum_{j=0}^k(-1)^{j+k}\binom{2k}{k+j}\frac{S(j+k,j)}{n+j}&= \frac{1}{(2k)!}\sum_{j=0}^k(-1)^{j+k}\binom{2k}{k+j}\frac{D(n,k)}{n+j}S(j+k,j)
{k\choose i}{k+i\choose i}{k\choose j}{k+j\choose j}=\sum_{r=0}^i{i+j\choose i}{j\choose i-r}{j+r\choose r}{k\choose j+r}{k+j+r\choose j+r}&=\sum_{s=j}^{i+j}{i+j\choose i}{j\choose s-i}{s\choose j}{k\choose s}{k+s\choose s}.
p(z_1, \dots, z_m) = \sum_{\epsilon_i\in\{0,1\}}z_1^{\epsilon_1}\cdots z_m^{\epsilon_m}\partial_1^{\epsilon_1}\cdots \partial_m^{\epsilon_m}p(w_1,\dots, w_m)\big|_{w_1=\dots=w_m=0}&= \prod_{i=1}^{m}(1+z_i\partial_i) p(w_1,\dots, w_m)\big|_{w_1=\dots=w_m=0}.
\Delta U_{T}= 2 A \int_{0}^{\infty}d z \, \Delta w_{T}( z ) ={\frac{\pi}{4}}A{\frac{( k_{B}T )^{3}}{\hbar^{2}}}\int_{- \infty}^{\infty}d{\cal Z}\left[{\frac{2}{{\cal Z}^{4}}}-{\frac{\coth{\cal Z}}{{\cal Z}^{3}}}-{\frac{1}{{\cal Z}^{2}\sinh^{2}{\cal Z}}}\right] .
10\log 2+ \dfrac{1}{4}\sum_{i=1}^r\left(\dfrac{3p_i-2-p_i^{-e_i}}{(p_i-1)^2}-\dfrac{e_i+3}{p_i^{e_i}(p_i-1)}+\dfrac{4}{p_i^{e_i/2}(p_i-1)}\right)\log p_i,
d_{\Sigma,1}=\max_{\substack{\lambda_1,\lambda_2\in(0,1] \lambda_1(N+\min\{M,K(N-1)\}+\lambda_2N\leq M \lambda_1+\lambda_2\left(1+\frac{N}{M}\right)\leq 1}}KN(\lambda_1+\lambda_2).
R^i\hat{S}(L)\otimes k(y)= R^ip_{2,*}(p_1^*L\otimes \mathcal{P})\otimes k(y)\cong H^i(A\times\{y\},(p_1^*L\otimes \mathcal{P})|_{A\times\{y\}}) = H^i(A,L\otimes P_y).
\theta(\lambda\mu_{n_s^{-1}}(-1))c_{s,-1}\theta(\mu) =\theta(\lambda)\theta(\mu_{n_s^{-1}}(-1))c_{s,-1}\theta(\mu) & = \theta(\lambda)\theta(n_s^{-1}(\mu))\theta(\mu_{n_s^{-1}}(-1))c_{s,-1}& = \theta(n_s^{-1}(\mu))\theta(\lambda\mu_{n_s^{-1}}(-1))c_{s,-1}.
\lambda_{112}\chi_1(g_{(1\,r)})\chi_{(3\,r)}(g_{(1\,q)})=\lambda_{112}\xi^2\chi_{(3\,r)}(g_{(2\,q)}) &\lambda_{112}\chi_{(1\,q)}(g_{(1\,p)})\chi_{(3\,p)}(g_{(2\,q)}) \chi_{(2\,q)}(g_{(1\,q)})=\lambda_{112}\chi_{(3\,p)}(g_{12})
S_{\mu \nu}: = \left( \begin{array}{c}{{G_{\mu \nu}}}\\{{E_{\mu \nu}}}\\ \end{array}\right) = \left( \begin{array}{c}{{g_{\mu \nu}}}\\{{H_{\mu \nu}}}\\ \end{array}\right) + i \left( \begin{array}{c}{{B_{\mu \nu}}}\\{{M_{\mu \nu}}}\\ \end{array}\right) \, ,
\varphi ( x )_{x^{+}= 0}= \int_{0}^{\infty}d k^{+}\int_{- \infty}^{\infty}\frac{d k_{\perp}}{\sqrt{2 k^{+}( 2 \pi )^{2}}}\left[ a ({\bf k}) e^{- i k^{+}x^{-}+ i k_{\perp}x_{\perp}}+ a^{\dagger}({\bf k}) e^{i k^{+}x^{-}- i k_{\perp}x_{\perp}}\right] ,
-\left(\dfrac{\Psi_{\textup{cr}}+\delta}{H_p^3}\dot{v}_p^{\delta}\right)_p-\dfrac{\left(\Psi_{\textup{cr}}\right)_p}{H_p^3}\dot{v}_p^{\delta}+\left(\mu_{\textup{cr}}\rho_p \delta+\nu_{\mathrm{D}}\dfrac{\Psi_{\textup{cr}}+\delta}{H_p}\right)\dot{v}^{\delta}=0.
\mathbb{E}\bigg[ \exp \bigg( \sum_{j = 1}^m 2 \alpha_j \sum_{x < k \le y}\frac{\Re X(k) e^{i k t_j}}{\sqrt{k}e^{i k \sigma}}\bigg) \bigg] = \exp \bigg( \sum_{j_1, j_2 = 1}^m \sum_{x < k \le y}\alpha_{j_1}\alpha_{j_2}\frac{\cos \big( k (t_{j_2}-t_{j_1}) \big)}{k e^{2 i k \sigma}}\bigg).
G^{( N , M )}( z_{1}, S_{M + 1}, \cdots , S_{N}; z_{1}, z_{2}) = ( z_{2}z_{1}^{- 1}+ z_{2}z_{1}){\overline{{G}}}^{( N , M + 1 )}( S_{M + 1}, \cdots , S_{N}; z_{1}, z_{2})
l_{k,n}^{(t_{1},...,t_{n+k})}(x_{1},...,x_{n+k}|\rho)= \sum_{j=0}^{2^{n+k}-1}\rho^{j}\sum_{m=0}^{j}\frac{1}{m!}\left. \frac{d^{m}}{d\rho^{m}}w_{k+n}(x_{1},...,x_{k+n}|\rho)\right\vert_{\rho=0}\times\prod_{s=1}^{k}T_{(j-m)+t_{s}}(x_{s})\prod_{s=1+k}^{n+k}U_{(j-m)+t_{s}}(x_{s}).
\frac{q^{\frac{r(r-1)}{4}}}{(q,q)_{\infty}}\sum_{j=-\infty}^{\infty}(q^{j[j\nu(\nu+1)+r(\nu+1)-\nu]}-q^{(j\nu+r)[j(\nu+1)+1]})\;;\;\;\;\;\; r=1,2,\ldots,\nu-1,
\lim_{t\rightarrow\infty}\left[\sup_{|x|\leq ct}|u(x,t;u_0)-\frac{a}{b}|+\sup_{|x|\leq ct}|v(x,t;u_0)-\frac{a}{b}|\right]=0\forall\,\, 0\leq c < c_{\rm low}^{\ast}.
A(X)=\, X^3 (1+a_1+b+a_1 z)+X^2 (a_1+a_1 k+z+a_1z+b z)\cr +X (1+a_1+k+b k+z+a_1 z+b z+a_1 k z)\cr +a_1+k+a_1 k+b k+a_1 k^2+a_1 z+b z+k z+b k z
M_{X}(H)=\sum_{m=0}^{n-1}H_{m}X^{m+1}-H_{n-1}(X^{n}+\sum_{m=0}^{n-1}a_{m}X^{m})=\sum_{m=1}^{n-1}(H_{m-1}-a_{m}H_{n-1})X^{m}-a_{0}H_{n-1},
\tilde{\eta}^2= 4\tilde{\zeta}^3-g_2(\Lambda)\tilde{\zeta}-g_3(\Lambda), \;\textrm{where}, \tilde{\eta}=\eta (4/r_1)^{1/2},\quad\tilde{\zeta}=\zeta -\tfrac{r_2}{3r_1}, g_2(\Lambda)=60 G_4(\Lambda)= 12(r_2/3r_1)^2+4\quad\textrm{and}, g_3(\Lambda)=140 G_6(\Lambda)=8(r_2/3r_1)^3+4(r_2/3r_1).
( Q^{1})^{2}= 1 + Q^{3}, \quad ( Q^{2})^{2}= 1 + Q^{4}, \quad ( Q^{3})^{2}= 1 + Q^{4}Q^{1}, \, \, \, \, \, ( Q^{4})^{2}= 1 + Q^{2}( Q^{3})^{2},
-\partial\mathbf{r}_{ij}/\partial s_{i}\times\mathbf{r}_{ij}= (\mathbf{a}_{1}\zeta+\mathbf{a}_{2}\zeta^{2}+...)\times(\mathbf{a}_{1}+2\mathbf{a}_{2}\zeta+...) &=(\mathbf{a}_{1}\times\mathbf{a}_{2})\zeta^{2}+O(\zeta^{3}) &= (\mathbf{a}_{1}\times\mathbf{a}_{2})|\zeta|^{2}.
{\cal L}= \frac{1}{2}h_{i j}( y ) \dot{y}^{i}\dot{y}^{j}+ \frac{1}{2}L_{A B}( y ) ( \dot{\xi}^{A}+ w_{i}^{A}( y ) \dot{y}^{i}) ( \dot{\xi}^{B}+ w_{j}^{B}( y ) \dot{y}^{j}) ,
2(2n+1)\binom{2n}{n}\biggm| \binom{6n}{3n}\binom{3n}{n}, &4(2n+1)\binom{2n}{n}\biggm| \sum_{k=0}^n(20k+3){\binom{2k}{k}}^2\binom{4k}{2k}(-1024)^{n-k}.
\int d x \ast \biggl [ e^{i k x}\int_{0}^{1}d \tau^{\prime}~ l^{i^{\prime}}D_{i^{\prime}}\hat{A}_{j^{\prime}}( x + l \tau^{\prime}) P \exp \left( i \int_{0}^{1}\hat{A}_{i}( x + l \tau ) l^{i}d \tau \right) \biggr ] = 0 ,
\|y-y^*\|^4 =\|(y-x)-(y^*-x)\|^4 &\leq 2\|y-x\|^4+2\|y^*-x\|^4+12\|y-x\|^2\|y^*-x\|^2-\|(y-x)+(y^*-x)\|^4 &= 2d^4+2d^4+12d^2d^2-\|y+y^*-2x\|^4\leq 16d^4-16d^4=0.
\rho_{\phi}(t) (x)=\sqrt{({\rm vol}_{g})_{\iota_{t}(x)}((\iota_{t})_{*}e_{1}, \cdots, (\iota_{t})_{*}e_{n}, -\xi_{\iota_{t}(x)}, \phi (\iota_{t})_{*}e_{1}, \cdots, \phi (\iota_{t})_{*}e_{n})}.
\eta_{+}^{A}= e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{{\hat{\eta}_{R}}}\\{{- \hat{\eta}_{L}}}\\ \end{array}\right)^{A}\ , \qquad \eta_{-}^{A}= \epsilon ( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{{\hat{\eta}_{L}}}\\{{\hat{\eta}_{R}}}\\ \end{array}\right)^{A}\ ,
\chi_{0,2}^{n,m}(x,y|\rho)\allowbreak=\sum_{j\geq0}\rho^{j}U_{j+n}(x)U_{j+m}(y)\allowbreak= (U_{n}(x)U_{m}(y)(w_{2}(x,y|\rho)-\rho^{4}) +\rho U_{n+1}(x)U_{m+1}(y)(1-2\rho^{2}+4\rho^{2}(x^{2}+y^{2})-4\rho xy) +\rho^{2}U_{n+2}(x)U_{m+2}(y)(1-4\rho xy)+\rho^{3}U_{n+3}(x)U_{m+3}(y))/w_{2}(x,y|\rho)
q_{V}(a - T_{\mu}x) = \langle a - T_{\mu}x , x^{*}_{V}\rangle= \mu_{t}\langle a - T_{t}x , x^{*}_{V}\rangle \leq \| \mu\| \sup_{t}| \langle a - T_{t}x , x^{*}_{V}\rangle | \leq \sup_{t}q_{V}( a - T_{t}x) \leq q_{V}(a-x),
H_{3,p}=\Big \langle r_{(\alpha,\alpha-1,-1)}, r_{(-\alpha,\alpha-1,1)}, r_{(1,-\alpha,\alpha-1)}\Big \rangle, H_{4,p}=\Big \langle r_{(\alpha,\alpha-1,-1,0)}, r_{(-\alpha,\alpha-1,1,0)}, r_{(1,-\alpha,\alpha-1,0)}, r_{(1,0,-\alpha,\alpha-1)}\Big \rangle,
\prod_{\rho \in \Delta_1^C}G_{r-2}( \frac{z}{(\tilde K_\rho \underline \tau)_1}| \frac{(\tilde K_\rho \underline \tau )_2}{(\tilde K_\rho \underline \tau)_1}, \ldots, \frac{(\tilde K_\rho \underline \tau )_{r}}{(\tilde K_\rho \underline \tau)_1})= \exp \left [ - \frac{2\pi i}{r!}B_{r,r}^C ( z | \underline \tau ) \right ] ,
T_{10}^{\alpha}T_{20}^{\beta}T_{21}^{\gamma}=(-1)^{\frac{\alpha(\alpha-1)}{2}+\frac{\beta(\beta-1)}{2}+\frac{\gamma(\gamma-1)}{2}+\alpha\gamma}Q_{2}^{\alpha+\beta}Q_{1}^{\alpha-\gamma}Q_{0}^{-\alpha-\beta}.