harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
t_A(a,c_1,\dots,c_{n-1})=\begin{cases}0, \textrm{if}c_1,\dots,c_{n-1}\in A_0; t(a,c_1,\dots,c_{n-1}), \textrm{otherwise}; \end{cases}t_B(a,c_1,\dots,c_{n-1})=\begin{cases}t(a,c_1,\dots,c_{n-1}), \textrm{if}c_1,\dots,c_{n-1}\in A_0; 0, \textrm{otherwise}; \end{cases}.
\begin{aligned}\sum_{i=3}^{k-1}\binom{n-i}{k-3}-(k-\frac{7}{2})\binom{n-3}{k-3}=\sum_{i=3}^{k-1}\left(\binom{n-i}{k-3}-\binom{n-3}{k-3}\right)+\frac{1}{2}\binom{n-3}{k-3}&=\frac{1}{2}\binom{n-3}{k-3}-\sum_{i=4}^{k-1}\sum_{j=4}^{i}\binom{n-j}{k-4}&>\frac{1}{2}\binom{n-3}{k-3}-\frac{(k-3)(k-4)}{2}\binom{n-4}{k-4}>0.\end{aligned}
G_{-m+\frac{q-1}{2}}=\frac{G_{\frac{q-1}{2}}G_{-2m}T^m(4)}{G_{-m}}, G_{md+\frac{q-1}{2}}=\frac{G_{\frac{q-1}{2}}G_{2md}T^{md}(4^{-1})}{G_{md}}, G_{-m(d-1)+\frac{q-1}{2}}=\frac{G_{\frac{q-1}{2}}G_{-2m(d-1)}T^{m(d-1)}(4)}{G_{-m(d-1)}}.
\textstyle h(\alpha,s) = 1 - sg(s) \textstyle = - \sum_{j}(-1)^j{\alpha \choose j+1}\frac{(s+j)_{j+1}}{(s-\alpha+1+j)_{j+1}}= \sum_{j}(-1)^j{\alpha \choose j}\frac{(s-1+j)_{j}}{(s-\alpha+j)_{j}}= 0.
T_{E N}^{( 2 )}( \hat{r}\rightarrow \infty ) \approx [ c_{1}\frac{\Gamma ( c ) \Gamma ( b - a )}{\Gamma ( b ) \Gamma ( c - a )}+ c_{2}\frac{\Gamma ( 2 - c ) \Gamma ( b - a )}{\Gamma ( b - c + 1 ) \Gamma ( 1 - a )}] e^{- ( 1 + 2 i \lambda ) \log \hat{r}}.
\zeta_{A}(s)^{2}=(\sum_{h}|h|^{-s})(\sum_{g}|g|^{-s})=\sum_{f}(\sum_{hg=f}1)|f|^{-s}=\sum_{f}d(f)|f|^{-s}=D_{d}(s)=\sum_{n=0}^{\infty}D(n)u^{n}
(\underbrace{u_{00}x_0^- + u_{01}X(v_{11}u_{10}x_0^- + v_{12}y_2^+)}_{\in V_0^+}+ \underbrace{x_0^-}_{\in V_0^-}, \underbrace{y_2^+}_{\in V_2^+}+ \underbrace{v_{21}Y(u_{10}x_0^- + u_{11}v_{12}y_2^+) + v_{22}y_2^+}_{\in V_2^-}) &= (w_{00}x_0^- + w_{02}y_2^+ + x_0^-, y_2^+ + w_{20}x_0^- + w_{22}y_2^+)
\mathcal{E}_1\Big(\frac{|\xi-\mathrm{z}|^2}{4\mathrm{t}}\Big) = -\gamma - \ln \frac{|\xi-\mathrm{z}|^2}{4\mathrm{t}}+ \mathrm{e}_{1}\Big(\frac{|\xi-\mathrm{z}|^2}{4\mathrm{t}}\Big),
p(t,g) =e^{i\mu g}\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)]*(\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]) &=e^{i\mu g}[e^tp(0,e^t\cdot)e^{-i\mu e^t \cdot}]\ast(\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]).
u\in\mathcal{W}\ : langle \mathcal{A}(u)+\lambda\mathcal{B}(u),v\rangle_{\mathcal{W}}=\langle \mathcal{F}(u),v\rangle_{\mathcal{W}}+\langle \mathcal{G}(u),v\rangle_{\mathcal{W}}v\in\mathcal{W},
\bar{\Delta}\Delta = f^{- 1}1_{[ 2 ] \times [ 2 ]}= \left( \! \begin{array}{c c}{f_{1}^{- 1}}&{0_{[ 2 ] \times [ 2 ]}}\\{0_{[ 2 ] \times [ 2 ]}}&{f_{2}^{- 1}}\\ \end{array}\! \right) ,
\partial_{s}\left(v-\breve{v}\right)-a\left(y,\,s\right)\partial_{yy}^{2}\left(v-\breve{v}\right)-\frac{1}{y}\,b\left(y,\,s\right)\partial_{z}\left(v-\breve{v}\right)-\frac{1}{y^{2}}\,c\left(y,\,s\right)\left(v-\breve{v}\right)=\frac{1}{y^{2}}\,f\left(y,\,s\right)
T_{\lambda \mu \nu}^{AVV}=-\int \frac{d^4k}{\left( 2\pi \right)^4}tr\left\{\gamma_\mu \left[ (\not{k}+\not{k}_1)- m\right]^{-1}\gamma_\nu \left[ (\not{k}+\not{k}_2)- m\right]^{-1}\gamma_\lambda \gamma_5\left[ (\not{k}+\not{k}_3)- m\right]^{-1}\right\},
{\cal Z}_{c}\simeq \exp \left\{- \frac{g^{2}}{2 4}\int d^{4}x \int d^{4}y \left[ 2 \Sigma_{\mu \nu}( x ) \Sigma_{\mu \nu}( y ) \hat{D}\left( ( x - y )^{2}\right) + j_{\mu}( x ) j_{\mu}( y ) \int_{( x - y )^{2}}^{+ \infty}d \lambda \hat{D}_{1}( \lambda ) \right] \right\},
| \widehat{g}(x) |^2 \leq \sum_{n \geq 2}^{\log\lambda^{\frac{1}{2}}}\sum_{\substack{|\mathcal I|=|\mathcal J|=2^{-n}\mathcal I \sim \mathcal J}}| \widehat{g_{\mathcal I}}(x) \widehat{g_{\mathcal J}}(x) | + 2\sum_{\mathcal I \in \mathfrak I_{E}}| \widehat{g_{\mathcal I}}(x) |^{2}
v_{P}=[\dot{T}(t)]p=[\begin{matrix}v_{P}\\ 0\end{matrix}]=(\frac{d}{dt}[\begin{matrix}A(t)&d(t)\\ 0&1\end{matrix}])[\begin{matrix}p\\ 1\end{matrix}]=[\begin{matrix}\dot{A}(t)&\dot{d}(t)\\ 0&0\end{matrix}][\begin{matrix}p\\ 1\end{matrix}].
b_{0}+\overset{\infty}{\underset{m=1}{\mathbf{K}}}(a_{m}/b_{m})=b_{0}+\frac{a_{1}b_{2}}{a_{2}+b_{1}b_{2}-\frac{a_{2}a_{3}\frac{b_{4}}{b_{2}}}{a_{4}+b_{3}b_{4}+a_{3}\frac{b_{4}}{b_{2}}-\frac{a_{4}a_{5}\frac{b_{6}}{b_{4}}}{a_{6}+b_{5}b_{6}+a_{5}\frac{b_{6}}{b_{4}-\frac{a_{6}a_{7}\frac{b_{8}}{b_{6}}}{\ddots}}}}}
\alpha_{H_i}:= \sup_{u: \hat{c}_i^T u\leq 1}\{\beta~|~H_i^*\big(\nabla H_i(u)\big)\geq \beta H_i(u)\}&= \sup_{u: \hat{c}_i^T u\leq 1}\{\beta~|~\langle \nabla H_i(u),u\rangle\geq (1+\beta)H_i(u)\}&= \inf_{u: \hat{c}_i^T u\leq 1}\frac{\langle \nabla H_i(u),u\rangle}{H_i(u)}-1
S_{\mathrm{e f f}}[ A_{0}] = \int d^{2}x \left( Z ( m , e_{N}{\tilde{A}}_{0}/{{T}}) \frac{1}{2 T}\vec{\nabla}A_{0}\cdot \vec{\nabla}A_{0}\, - \, N \, V ( m , e_{N}A_{0}/{T}) \right) \, ,
\frac{\partial x}{\partial\lambda_{1}}\times\frac{\partial x}{\partial\lambda_{2}}=(\frac{\partial x}{\partial q^{i}}\frac{\partial q^{i}}{\partial\lambda_{1}})\times(\frac{\partial x}{\partial q^{j}}\frac{\partial q^{j}}{\partial\lambda_{2}})=E_{kmp}(h_{ki}\frac{\partial q^{i}}{\partial\lambda_{1}})(h_{mj}\frac{\partial q^{j}}{\partial\lambda_{2}})b_{p}
\sum_{j=1}^m \sum_{l=k_{j-1}}^{k_j-1}\int_{\mathbb{R}^4}\Big( \int_{\mathbb{R}}F(y,x')F(y,y') \omega_{2^l}(y-q) \,dy \Big) \Big( \int_{\mathbb{R}}G(x,x')G(x,y') \omega_{2^l}(x-p) \,dx \Big) \vartheta_{2^{k_j}}(x'-p-q) \psi_{2^l}(y'-p-q) \,dx' dy' dp dq .
T_{z z}^{\Psi}= T_{z z}, T_{z \bar{z}}^{\Psi}={\cal O}_{0}\Psi T_{\bar{z}\bar{z}}^{\Psi}= T_{\bar{z}\bar{z}}, T_{\bar{z}z}^{\Psi}= \overline{{\cal O}}_{0}\Psi .
|\psi_{\alpha}^{(\pm)}\rangle=|\phi_{\alpha}\rangle+\int d\beta\frac{T_{\beta\alpha}^{(\pm)}|\phi_{\beta}\rangle}{E_{\alpha}-E_{\beta}\pm i\epsilon},T_{\beta\alpha}^{(\pm)}=\langle\phi_{\beta}|V|\psi_{\alpha}^{(\pm)}\rangle.
B^+_\rho(0) = \{\lambda(0)+r\cos(\theta)\sin(\xi){\bf n}(0)+r\sin(\theta)\sin(\xi){\bf b}(0),\; (r,\theta,\xi)\in [0,\rho)\times[0,2\pi)\times(\pi,2\pi)\}, B^+_\rho(L) = \{\lambda(L)+r\cos(\theta)\sin(\xi){\bf n}(0)+r\sin(\theta)\sin(\xi){\bf b}(0),\; (r,\theta,\xi)\in [0,\rho)\times[0,2\pi)\times(0,\pi)\}.
\alpha_{n+k,-k}= \sum_{j=0}^{k+1}\prod_{l=0}^m (n+j+\eta_l) \prod_{\begin{subarray}{c}i=0 i \neq j \end{subarray}}^{k+1}\frac{1}{j-i}& = \sum_{j=0}^{k+1}\frac{(-1)^{k+1-j}}{j! (k+1-j)!}\prod_{l=0}^m (n+j+\eta_l) .
\hat{X}_{q}=\hat{U}_{q}\hat{\Sigma}_{q}^{1/2}\hat{V}_{q}^{T},\hat{K}_{q}=\hat{U}_{q}\hat{\Sigma}_{q}\hat{U}_{q}^{T},\hat{C}_{q}=\hat{V}_{q}\hat{\Sigma}_{q}\hat{V}_{q}^{T}.
\left(E(X_1^t)\right)^{'}= \lim\limits_{\theta \to 0}\frac{E(X_1^{t+\theta}) - E(X_1^t)}{\theta}= \lim\limits_{\theta \to 0}\int \dfrac{x^\theta - 1}{\theta}x^t dF(x) = \int x^t\log x dF(x) = E\left(X_1^t\log X_1\right).
\mathcal{N}':=\bigg\{2\pi\sqrt{\frac{j^{2}+l^{2}+jl}{3}};\;j,l\in \mathbb{N}^{\ast}~{}~ j>l ~{}~ &~~~~~~~~j^2+jl+l^2\neq m^2+mn+n^2, \forall m,n\in \mathbb{N}^*\backslash \{j\}\bigg\}. ~~~~
t = DG_2(q) = \sum_{n \ge 1}n \left(\sum_{d|n}d\right) q^n, g(t) = (DG_2)^{<-1>}(t) = \sum_{n \ge 1}\left. \frac{d^{n-1}}{dq^{n-1}}\left( \frac{q}{DG_2(q)}\right)^n \right|_{q=0}\frac{t^n}{n!}.
{1 \over 2\pi}\sum_{l=1/2}^{\infty}2le^{-{l^{2}\over \rho^{2}}t}={\rho^{2}\over 4\pi t}+{\rho^{2}\over (4\pi t)^{3 \over 2}}{\rm P}\int_{-\infty}^{\infty}dx({x \over 2\rho}{\rm cosec}{x \over 2\rho}-1)e^{-{x^{2}\over 4t}},
R ( a_{j}, a_{j}) = \sum_{k \neq j}( - 1 )^{k}\,{\frac{( q_{j}p_{k}- p_{j}q_{k})^{2}}{a_{j}- a_{k}}}\, + p_{j}^{2}- a_{j}q_{j}^{2}- 2 p_{j}q_{j}u + 2 q_{j}^{2}v \, .
A(N) =\sum_{P<p\leq2P}(\log p)\left(\left(N+1-\left[p^{c}\right]\right)^{\gamma}-\left(N-\left[p^{c}\right]\right)^{\gamma}\right) &=\gamma\sum_{P<p\leq2P}(\log p)\left(\left(N-[p^c]\right)^{\gamma-1}+\O(N^{\gamma-2})\right)\asymp N^{2\gamma-1}.
\left|\sum_{\substack{(\delta_1,\ldots,\delta_K,j_1,\ldots,j_K, n_1,\ldots,n_K)\in\mathcal{A}_{J_1,J_2}}}\frac{1}{2^K}\frac{1}{N^{K/2+1}}\prod_{k=1}^Ka'_{j_k}\right|=o(1)
\sum_{\ell = 0}^{\infty}{\frac{( - 1 )^{\ell}}{\ell + a}}\left( \begin{array}{c}{- 1 / 2}{\ell}\end{array}\right) ={\frac{\Gamma ({\frac{1}{2}}) \Gamma ( a )}{\Gamma ( a +{\frac{1}{2}})}}.
d^{(n,m)}(e^{-i\frac{\omega}{2^{m+1}}}) = - e^{-i\frac{\omega}{2^{m+1}}}A^{(n,m)}\bigl(- e^{i\frac{\omega}{2^{m+1}}}\bigr) \rho_\phi^{(n,m+1)}(\omega+2^{m+1}\pi)\,.
n_0\left\{(N+2V_k)^2-2N(N+2V_k)+2N^2/3\right\}+n_{1/2}(4V_k^2-4NV_k+2N^2/3) &=n_0\left\{(N+2V_k')^2-2N(N+2V_k')+2N^2/3\right\}+n_{1/2}(4V_k'^2-4NV_k+2N^2/3),
\mathbf{R}_k = \sum_{i=1}^{K}D_{ki}\mathbf{X}(\tau_k) - \frac{t_f-t_0}{2}\mathbf{f}(\mathbf{X}(\tau_k),\mathbf{\mathbf{U}}(\tau_k),\tau_k;t_0,t_f) = \mathbf{0}, (k = 1,...,K)
w_{n}( z ) = \left. \frac{1}{( n - 1 ) !}\frac{\partial^{n - 1}}{\partial w^{n - 1}}S ( z , w ) \right|_{w = 0}= \frac{1}{z^{n}}+ \sum_{m = 1}^{\infty}B_{n m}z^{m - 1}\; ,
\Vert (B_\varepsilon (\vartheta )-\widehat{\zeta}Q_0^\varepsilon )^{-1}-(B^0(\vartheta )-\widehat{\zeta}\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant C_4\varepsilon c(\phi)^2 , 0 < \vartheta \leqslant 1,0< \varepsilon \leqslant \vartheta^{-1},\widehat{\zeta}=e^{i\phi},\phi \in (0,2\pi),
H_{t,n}(\chi)=\alpha\chi\!\left(\frac{t+2^{r/2}\omega}{2}\right)2^{e-3}\begin{cases}3 e\ge3, r>e, 1+2(-1)^e e\ge 3, r=e, 1-2(-1)^d e\ge3, r=e-1, 2\nmid e, 2 e=2, r\geq e, -1 e=2, r<e, 4 e=1, 0 \end{cases}
= \frac{1}{2}\hat{f}^{\mu}\hat{f}_{\mu}- \epsilon^{\alpha \beta \gamma}\hat{f}_{\alpha}\partial_{\beta}( \hat{A}_{\gamma}+ \hat{a}_{\gamma}) + \frac{m}{2}\epsilon^{\alpha \beta \gamma}( \hat{A}_{\alpha}\partial_{\beta}\hat{A}_{\gamma}+ 2 \hat{a}_{\alpha}\partial_{\beta}\hat{A}_{\gamma}) + ~ O ( \theta^{2}) .
\int_0^t ds \int \,dV_j\,C_{j+1}^{N,\delta}f_{j+1}^N\varphi &=\int_0^t ds \int \,dV_j\,C_{j+1}^\delta f_{j+1}^N\varphi \longrightarrow \int_0^t ds \int \,dV_j\,C_{j+1}^\delta f_{j+1}\varphi
m\{x:x\notin\cup J_{m}^{*},|Tb(x)|\ge\lambda\}\le\lambda^{-1}\int_{(\cup J_{m}^{*})^{c}}|Tb(x)|dx\le\lambda^{-1}\sum_{n}\int_{(J_{n}^{*})^{c}}|Tb_{n}(x)|dx.
2 \int d^{2}x \, \partial_{+}X^{\mu}\cdot \partial_{-}X^{\nu}\cdot \partial_{[ \mu}B_{\nu ]}^{\prime}( X ) = \int d^{2}x \, \left[ \partial_{+}\left( B_{\mu}^{\prime}\partial_{-}X^{\mu}\right) - \partial_{-}\left( B_{\mu}^{\prime}\partial_{+}X^{\mu}\right) \right] \ .
det[\begin{matrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\ 1&1&1&1&0\end{matrix}]=0.
\begin{array}{cl}\langle \bar R(X,Y)Z,V\rangle= \langle\nabla^{\perp}_X\sigma(Y,Z),V\rangle-\langle\sigma(\nabla_XY,Z),V\rangle-\langle\sigma(Y,\nabla_XZ),V\rangle &-\langle\nabla^{\perp}_Y\sigma(X,Z),V\rangle+\langle\sigma(\nabla_YX,Z),V\rangle+\langle\sigma(X,\nabla_YZ),V\rangle.\end{array}
\big[\tilde{\phi}(\tilde{A}/ f \tilde{A})\big]_{\tilde{A}}=\det_{\mathbb{F}_q(z)[X]}(X-\tilde{\phi}_{\theta}\mid \tilde{A}/f\tilde{A}\otimes_{\mathbb{F}_q(z)}\mathbb{F}_q(z)[X])_{|X=\theta}.
\left(\frac{2n+1}{n+1}y-(y+\sqrt{y^2-1})\right)P_n(y) \leq \frac{n}{n+1}(y-\sqrt{y^2-1})P_n(y) &\leq \frac{n}{n+1}(y-\sqrt{y^2-1})(y+\sqrt{y^2-1})P_{n-1}(y) &=\frac{n}{n+1}P_{n-1}(y),
\phi\big(s+h,t+h,\omega,x_0\big) = \psi\big(s + h,t + h,\omega,\chi_{s+h}(t+h,\omega,x_0)\big) &= \psi\big(s ,t ,\theta_h\omega,\chi_{s}(t, \theta_h\omega,x_0)\big) = \phi\big(s,t,\theta_h \omega,x_0\big)
\int_{U(G)}\langle x, \xi\rangle^2{\rm d}x =c(G, S)^2|G|^{-1}\int_{G}\langle S^{-1}x, \xi\rangle^2{\rm d}x &=c(G, S)^2|G|^{-1}\langle M(S^{-1}\xi),S^{-1}\xi \rangle &=c(G, S)^2|G|^{-1}|\xi|^2.
\prod_{u=1}^N dv_u^ldv_u^{l*}=\prod_{k=1}^{l-1}d\xi_k^ld\xi_k^{l*}\cdot2^{N-l-1}(\xi_l^l)^{2N-2l+1}\,d\xi_l^l\,d^{2N-2l+1}\hat\Omega_l\ .
\underset{t\to 0}{\lim}\ \mu_{\mathcal O}|_{U}(\psi\circ g_t)=\mu_{\mathcal O}|_{U}(\psi(N)\Bbb I_U)=\mu_{\mathcal O}|_{U}(\Bbb I_U)\psi(N)=\mu_{\mathcal O}|_{U}(U)\psi(N_m),
\lambda_{U, t}^s(r, y) = \lambda_t \Big(1 - \exp\Big(-\pi (r + y)^2 \lambda_t (\frac{P_t}{P_k})^{2 / \alpha}\Big)\Big), \lambda_{U}^s(r, y) = \sum_{t \in \mathcal{K}}\lambda_{U, t}^s(r, y),
{\frac{\epsilon_{j}( \theta )}{T}}= \phi_{3}( \theta - \theta^{\prime}) \star \ln \left( 1 + e^{\epsilon_{j}( \theta^{\prime}) / T}\right) - \sum_{l = 0}^{t - 3}\left( \delta_{j , l + 1}+ \delta_{j , l - 1}\right) \phi_{3}( \theta - \theta^{\prime}) \star \ln \left( 1 + e^{- \epsilon_{l}( \theta^{\prime}) / T}\right)
\lim_{n\to \infty}\frac{1}{n}\log|Df^{n}(x)| =\lim_{n\to \infty}\frac{1}{n}\log \prod_{j=0}^{n-1}|Df(f^j(x))| &= \lim_{n\to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\log |Df(f^j(x))| = \int \log |Df(x)| \d\mu= L.
\Lambda_4(t) \leq \frac{1}{2}|\sqrt{\rho}u_{tt}(t)|^2_{2}+Cc^{8}_1(1+|v_t|^2_{D^1_0})|\nabla u_t|^2_{2}&+Cc^4_1(1+|P_{tt}|^2_2+|\rho_{tt}|^2_2+|H_{tt}|^2_2)+Cc^{18}_1|v_t|^2_{D^1_0}.
C_{1}\sum_{k = - \infty}^{\infty}\overline{v}^{T}_{k}v_{k}\leq \Big \Vert \sum_{k = - \infty}^{\infty}\overline{v}^{T}_{k}F (\cdot - k) \Big \vert L_{2}(\mathbb{R}) \Big \Vert^{2}\leq C_{2}\sum_{k = - \infty}^{\infty}\overline{v}^{T}_{k}v_{k}
\langle V ({\bf x}) V ({\bf y}) \rangle = g M^{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\, \frac{e^{i{\bf p}\cdot ({\bf x}-{\bf y})}}{p^{2}+ M^{2}}= \frac{g M^{2}}{4 \pi}\int_{0}^{\infty}\frac{d s}{s}\, e^{- M^{2}s - ({\bf x}-{\bf y})^{2}/ 4 s}\; .
\begin{pmatrix}R J^* J S\end{pmatrix}=\begin{pmatrix}I J^*S^{-1}0 I\end{pmatrix}\begin{pmatrix}R- J^*S^{-1}J 0 0 S\end{pmatrix}\begin{pmatrix}I 0 S^{-1}J I\end{pmatrix}.
S_{R_n}^{\beta+\delta}- \frac{C_{\beta,\delta}}{C_{\beta, \delta +1}}S_{R_n}^{\beta+\delta+1}=C_{\beta,\delta}R_n^{-(\beta+\delta)}\int^{R_n}_0 (R_n - t)^{\beta -1}t^\delta\big(S_t^{\delta}-S_t^{\delta+1}\big)d t &\ \ \ + C_{\beta,\delta}R_n^{-(\beta+\delta)}\int^{R_n}_0 (R_n - t)^{\beta -1}t^\delta(1-R_n^{-1}t)S_t^{\delta+1}d t &=:I_{R_n}+II_{R_n}.
V_{<l;\xi}= - W_{<l}(x) V_{<k}^{-1}W_{<l}^{-1}(y)W_{<l; \xi}(y) W_{<l}(y) V_{<k}W^{-1}_{<l}(x) + W_{<l}(x) V_{<k;\xi}W^{-1}_{<l}(y) +W_{<l;\xi}(x)
\|f(u^{n-1})-f(U^{n-1}_h)\|_{L^2}= \|f'(\xi) ( u^{n-1}-U^{n-1}_h)\|_{L^2}\leq C\| u^{n-1}-U^{n-1}_h\|_{L^2},~~\xi \in (u^{n-1}, U_h^{n-1}).
\|^{c}_{0}{D}^{\alpha}_{t_{n-\sigma}}R_hu - \,^{c}_{0}{D}^{\alpha}_{t_{n-\sigma}}u\| \; \le \; C \, h^2 \, \| \Delta \,^{c}_{0}{D}^{\alpha}_{t_{n-\sigma}}u \| \; \le \; C \, h^2.
e^{i W}= \int d g_{\mu \nu}~ d \bar{\Gamma}_{~ \mu \nu}^{\sigma}~ d \bar{c}^{\mu}~ d c^{\nu}~ d \bar{\chi}^{\mu}~ d \chi^{\nu}~ d \bar{\eta}^{\mu}~ d \eta^{\sigma \lambda \nu}e^{i S_{q u a n}}\left( d e t \omega_{\mu \nu}\right)^{\frac{1}{2}}\left( d e t \zeta_{\mu \nu}\right)^{\frac{1}{2}}\left( d e t \varsigma_{\mu \nu}\right)^{\frac{1}{2}}
\mathbb{E}\left(Y_{\beta}(t)\right) =\frac{t^{\beta}}{\Gamma(\beta+1)}, \operatorname{Var}\left(Y_{\beta}(t)\right) =\left(\frac{2}{\Gamma(2\beta+1)}-\frac{1}{\Gamma^{2}(\beta+1)}\right)t^{2\beta}.
d(\alpha\wedge\beta\wedge{\chi}_{\mathcal{F}})=d\alpha\wedge\beta\wedge \chi_{\mathcal{F}}+(-1)^{s}\alpha\wedge(d\beta)\wedge \chi_{\mathcal{F}}+(-1)^{s+t}\alpha\wedge\beta\wedge d \chi_{\mathcal{F}}.
|\eta|^2 = O_0^4(\eta) + \nu^2 + \rho^2, (w_* \cdot \delta E(\eta))^2 = O^3_3(\eta) + O_4^4(\eta) + \left( 2 \partial_{xx}\theta_* \nu + 2 \partial_x \theta_* \partial_x \nu \right)^2
e^{2 \pi i \langle{m_1, (Ak_1-Ak_2)}\rangle/N}\sum_{m \in \mathfrak{D}^*}e^{2 \pi i \langle{m, (Ak_1-Ak_2)}\rangle/N}= \sum_{m \in \mathfrak{D}^*}e^{2 \pi i \langle{(m+m_1), (Ak_1-Ak_2)}\rangle/N}.
\mathcal{F}(t_0,t_1,t_2,Q) = \sum_{d \geq 0}\sum_{n=0}^\infty \sum_{\alpha_0 + \alpha_1 + \alpha_2 = n}\frac{t_0^{\alpha_0}}{\alpha_0!}\frac{t_1^{\alpha_1}}{\alpha_1!}\frac{t_2^{\alpha_2}}{\alpha_2!}\left\langle T_0^{(\alpha_0)}, T_1^{(\alpha_1)},T_2^{(\alpha_2)}\right\rangle^\textnormal{coh}_{0,n,d[l]}
L_E(c_{k,j}^q) = \ln \left(\frac{\sum\limits_{s \in S_{q,0}}p(x_{k,j}= s \vert \hat{x}_{k,j})}{\sum\limits_{s\in S_{q,1}}p(x_{k,j}= s \vert \hat{x}_{k,j})}\right) - \ln\left(\frac{\sum\limits_{s\in S_{q,0}}p(x_{k,j}= s)}{\sum\limits_{s\in S_{q,1}}p(x_{k,j}= s)}\right)
\nabla_x{X}_{t}^{M_{{\mathcal{P}}}}=I_d+\int_{0}^{t}\int_{B_{M_{{\ \mathcal{P}}}}(s)}\nabla_xc(z,{X}_{s-}^{M_{{\mathcal{P}}}})\nabla_x{X}_{s-}^{M_{{\mathcal{P}}}}dN(z,s)+\int_{0}^{t}\nabla_x b({X}_{s}^{M_{{\mathcal{P}}}})\nabla_x{X}_s^{M_{{\mathcal{P}}}}ds,
\left[\begin{array}{cc}k [m]_n s [m]_{n-1}\end{array}\right]=\sqrt{\frac{\prod_{s^\prime \neq s}^{n-1}(p_{kn}-p_{s^\prime, n-1})\prod_{k^\prime \neq k}^n (p_{s, n-1}-p_{k^\prime n}+1)}{\prod_{k^\prime \neq k}^n (p_{kn}-p_{k^\prime n})\prod_{s^\prime \neq s}^{n-1}(p_{s, n-1}-p_{s^\prime, n-1}+1)}}
(1-x)^{-1}f^{\prime}(x)= (1+x)^{\beta-1}\int_{0}^{x}\frac{(1-t)^{-1}(1+t)^{\beta/2}}{(1-t)^{-1}(1+t)^{\beta/2}}\left( (1-t)^{-1}(1+t)^{\beta+1}f^{\prime}(t)\right)^{\prime}\,dt & \quad\quad+(1+x)^{-\beta-1}f^{\prime}(0)\,.
D_f(g^0, g) := J_f(g) - [J_f(g^0) + \langle J_f'(g^0), g - g^0 \rangle], \Pr_{{\cal G}}(-\tilde x, g^0, \tau) := \arg\min\limits_{g \in{\cal G}}\left\{\langle - \tilde x, g \rangle + J_f(g)+ \tau D_f(g^0, g)\right\},
= -i\sum_{k = 1}^{n}\{f({\bf{x}}_{k})a(\sigma_{k})\nabla^{k}_{s}+ \frac{1}{2}[\nabla^{k}_{s}f({\bf{x}}_{k})]a(\sigma_{k}) \}\varphi_{n}({\bf{x}}_{1}\sigma_{1},{\bf{x}}_{2}\sigma_{2}, ... ,{\bf{x}}_{n}\sigma_{n})
\begin{array}{l}A_1=c D_x-c \displaystyle\frac{u_{xt}}{u_{zt}}D_z,A_2=-b D_y+b \displaystyle\frac{u_{yt}}{u_{zt}}D_z, begin{align*}5mm]B_1= -D_x+\displaystyle\frac{u_{xz}}{u_{zt}}D_t,B_2= -D_y+\displaystyle\frac{u_{yz}}{u_{zt}}D_t.\end{array}
p_\eta(\rho,\eta) = \frac{2p\left[1 + \frac{1}{2}\alpha(1 - \alpha)\left(\frac{5}{2}+ \frac{T_i}{T}\right)\right]}{3(1 + \alpha)\left[ 1 + \alpha(1 - \alpha)\left(\frac{5}{4}+ \frac{T_i}{T}+ \frac{T_i^2}{3T^2}\right)\right]}> 0.
e_1\cdot e_2=e_3,e_2\cdot e_1= -e_3, & \varepsilon(i,j)=(-1)^{ij}, & \begin{array}[t]{l}\alpha(e_1)=-2e_1, \alpha(e_2)=e_3, \alpha(e_3)=e_2-e_3.\end{array}
P_{j , n , s}^{S P F ~ j^{\prime}, n^{\prime}, s^{\prime}}= P_{j}^{~ j^{\prime}}P_{n}^{~ n^{\prime}}P_{s}^{~ s^{\prime}}+ ( - 1 )^{( 2 j - n + s ) / 2}P_{j}^{~ k / 2 - j^{\prime}}P_{n}^{~ n^{\prime}+ k + 2}P_{s}^{~ s^{\prime}+ 2}~ ,
h_{k}( \underbrace{x_{A}, \l , x_{A}}_{\mathrm{m t i m e s ,}}, \underbrace{x_{B}, \l , x_{B}}_{\mathrm{m t i m e s}}) = \sum_{r = 0}^{k}{\binom{r + m - 1}{r}}{\binom{k - r + m - 1}{k - r}}x_{1}^{r}x_{2}^{k - r}.
\sum_{n=0}^{\infty}\tilde{c}(5n+4)q^{n}=35\dfrac{E_{5}^{2}E_{10}^{2}}{E_{1}^{5}E_{2}}+700q \dfrac{E_{5}^{4}E_{10}^{4}}{E_{1}^{7}E_{2}^{3}}+6875 q^2 \dfrac{E_{5}^{6}E_{10}^{6}}{E_{1}^{9}E_{2}^{5}}&+31250 q^3 \dfrac{E_{5}^{8}E_{10}^{8}}{E_{1}^{11}E_{2}^{7}}+78125 q^4 \dfrac{E_{5}^{10}E_{10}^{10}}{E_{1}^{13}E_{2}^{9}}.
e^{1}=\frac{e_{2}\times e_{3}}{e_{1}\cdot(e_{2}\times e_{3})};e^{2}=\frac{e_{3}\times e_{1}}{e_{2}\cdot(e_{3}\times e_{1})};e^{3}=\frac{e_{1}\times e_{2}}{e_{3}\cdot(e_{1}\times e_{2})}.
\Pi_{r}^{\alpha}= \prod_{A}( \Phi_{r}^{A})^{n_{\alpha}^{A}}= e^{\sum_{I}q_{I}^{\alpha}g^{I}/ 2}\Pi^{\alpha}, \Phi_{r}^{A}= e^{\sum_{I}q_{I}^{A}g^{I}/ 2}\Phi^{A},
\tilde{\phi}_{1}(\textbf{z}) = I \left\lbrace \textbf{z}\in \mathcal{C}_{1}\cap \mathcal{C}_{0}, p(\textbf{z},1) \dfrac{p}{\mu{(\mathcal{C}_{1})}}> p(\textbf{z},0) \dfrac{1-p}{\mu{(\mathcal{C}_{0})}}\right\rbrace &+ I \left\lbrace \textbf{z}\in \mathcal{C}_{1}- \mathcal{C}_{0}, p(\textbf{z},1)>0 \right\rbrace.
D\left[\begin{array}{c}i_1 j_1\end{array}\right]D\left[\begin{array}{c}i_2 j_2\end{array}\right]=D\left[\begin{array}{c}i_1 j_2\end{array}\right]D\left[\begin{array}{c}i_2 j_1\end{array}\right]+DD\left[\begin{array}{cc}i_1 i_2 j_1 j_2\end{array}\right],
\leq 3K((1-\theta)^2\zeta h+1)E(\int_{t_n}^{t_n+\zeta h}(|x(t)-x(t_n)|^2+|x(qt)-x(qt_n)|^2){\rm d}t) &+3K\theta^2\zeta hE(\int_{t_n}^{t_n+\zeta h}(|x(t)-x(t_{n+1})|^2+|x(qt)-x(qt_{n+1})|^2){\rm d}t) \leq C_2h^2,
\mathcal{F}_{gh}=\mathrm{span}\left\{\prod_{i=1}^kc_{-n_i}\prod_{j=1}^lb_{-m_j}|0\rangle_{gh}\ |\k,l\geq 0\ ,\ 0<n_1<\dots<n_k,\ 0<m_1\dots<m_l\right\}\;.
(v-v')^2 \rightarrow \frac{(v-v')^2}{\Omega(v)\Omega(v')}\, ,\quad \vec{x}_\alpha \rightarrow \frac{\vec{x}_\alpha}{\Omega(v)}\, \quad \vec{x}'_\alpha \rightarrow\frac{\vec{x}'_\alpha}{\Omega(v')}\, .
\mathbb Q:=\left[\begin{array}{ccc}0 p/(p+1) 1/( p+1) 0 p/(p+1) 1/( p+1) 0 p/(p+1) 1/(p+1) \end{array}\right],\hskip 1cm q=[0\,\p/(p+1)\,\ 1/(p+1)].
J(t) \sim t^{(p+q)(r+s)}\cdot (\frac{2\pi}{t^2})^{ps+rq-rs}\cdot (-2\sqrt{2}\pi t)^{2r(q-s)+2s(p-r)}\cdot \mathrm{det}( A)^{-\frac{1}{2}}\cdot \exp(-t^2\pi h(0))\cdot (-\frac{1}{2}i)^{ps+rq-rs}= (-i)^{ps+rq-rs}2^{ps+rq-5rs}\pi^{2ps+2rq-4rs}t^{(p+q)(r+s)-2rs}\exp(-(r+s)\pi t^2).
\|S((2\rho)^\theta)\tilde{u}(\tau))\|_{L^\infty({\bf R}^N)}\ge CG(0,(2\rho)^\theta+\rho^\theta)\int_{B(z,\rho)}u(y,\tau)\,dy \qquad\ge C[(2\rho)^\theta+\rho^\theta)]^{-\frac{N}{\theta}}G(0,1)\int_{B(z,\rho)}u(y,\tau)\,dy \qquad\ge C\rho^{-N}\int_{B(z,\rho)}u(y,\tau)\,dy
\MoveEqLeft E\hat{V}_{d-1}(Z\cap A) - \lim_{a\to 0}E\hat{V}_{d-1}(Z\cap A)= ae^{-\gamma EV_d(K)}\sum_{l=1}^{2^{n^d}-2}w_l^{(d-1)}\bigg(\gamma E Q(K,B_l,W_l){}+ \frac{\gamma^2}{2}\sum_{S\subseteq B_l}(-1)^{|S|}\bigg(E\int_{S^{d-1}}h(\check{S}\cup \check{W}_l,u) S_{d-1}(K,du)\bigg)^2 \bigg)+ O(a^2).
S(x)\equiv s_i(x) = \frac{y_i (1-\phi)^3 + (r_i y_i + h_i d_i)\phi (1- \phi)^2 + (r_i y_{i+1}- h_i d_{i+1}) \phi^2 (1-\phi) +y_{i+1}\phi^3}{1 + (r_i -3) \phi(1-\phi)},
V ( \Phi ) = - \frac{1}{4}m_{0}^{2}\Phi^{2}+ \frac{1}{4}g_{0}\Phi^{4}+ \frac{3}{8}\frac{m_{0}^{4}}{g_{0}}= \frac{1}{4 !}g_{0}\left( \Phi^{2}- v_{0}^{2}\right)^{2},
A^{rstu}={}\frac{5}{2}\left(\sum_{\alpha=1}^K q_\alpha^r q_\alpha^s \right) \left(\sum_{\alpha=1}^K q_\alpha^t q_\alpha^u \right)- \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s q_\alpha^t q_\alpha^u \geq{}\frac{5}{2}\sum_{\alpha=1}^K q_\alpha^r q_\alpha^s q_\alpha^t q_\alpha^u - \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s q_\alpha^t q_\alpha^u \geq{}0.
S=\frac{1}{2}\int d^4x \int_{-1/\alpha}^{1/\alpha}\left(-e^{-2\sigma (z)}\delta^{IJ}\partial_I \Phi \partial_J \Phi + \Phi \partial_z \left(e^{-4\sigma (z)}\partial_z \Phi \right) + M_5^2 e^{-4\sigma (z)}\Phi^2 \right)
(\ast) = w_{ij}R^\varphi_{ik}R^\varphi_{kj}+ w_j R^\varphi_{ik}R^\varphi_{kj,i}+ w_j R^\varphi_{ik,i}R^\varphi_{kj}& \phantom{=\;}+ w_{ti}R_{ijkt}R^\varphi_{jk}+ w_t R_{ijkt,i}R^\varphi_{jk}+ w_t R_{ijkt}R^\varphi_{jk,i}\, .
f(x) =\left[\begin{array}{cc}D_{1}G_{1}D_{1}H_{1}I_{2^p-2}\end{array}\right]\cdots\left[\begin{array}{cc}D_{p-1}G_{p-1}D_{p-1}H_{p-1}I_{2^{p-1}}\end{array}\right]\left[\begin{array}{c}D_pG_p D_pH_p\end{array}\right]x,
m_{0}^{2}+ \frac{4 \lambda \mu^{4 - D}}{( \pi )^{D / 2}}\Gamma ( \frac{D}{2}- 1 ) \left[ ( \beta_{c}^{2 - D}+ L_{c}^{2 - D}) \zeta ( D - 2 ) + 2 E_{2}( ( D - 2 ) / 2 ; \beta_{c}, L_{c}) \right] = 0 .
\begin{gathered}E_{\sqrt{c}}-F_{\sqrt{c}}= \frac{1-c}{2\sqrt{c}}H - \frac{1+c}{2\sqrt{c}}H_{\sqrt{c}}, E_{\sqrt{c}}+F_{\sqrt{c}}= \frac{1-c}{4\sqrt{c}}[H,H_{\sqrt{c}}],\end{gathered}
\left[\begin{array}{ccc}0 0 0 1/r 0 k/r 1/(mk+r) r/(mk+r) 0\end{array}\right]\left[\begin{array}{c}x_1 x_2 x_3\end{array}\right]=\left[\begin{array}{c}1-1/1 1-1/r 1-1/(mr+k)\end{array}\right]
\partial_{t}^{2}\widehat{v}(t,\xi)+ a(t) B(1+2\xi_{2}) \widehat{v}(t,\xi)=0, \widehat{v}(t,\xi)=\widehat{v}_{0}(\xi), \; \partial_{t}\widehat{v}(t,\xi)=\widehat{v}_{1}(\xi),
0 < \tilde{\mu}\left(\left(C_{s_{\operatorname{stag}}}\times \{a'\}\right)\cap E_j\right) \le \tilde{\mu}\left(T^{-i}\left(\Omega \times \{a\}\right)\cap E_j\right) &= \tilde{\mu}\left(T^{-i}\left((\Omega \times \{a\}) \cap E_j\right)\right) = \tilde{\mu}\left((\Omega \times \{a\})\cap E_j\right) &= \tilde{\mu}\left( E_{j,a}\times \{a\}\right),
K^{\frac{1}{4}}\xi + Z = \langle\eta,\xi\rangle\xi -\lambda_1\langle\eta,\xi\rangle\langle \eta,E_1\rangle E_1 -\lambda_2\langle\eta,\xi\rangle\langle\eta,E_2\rangle E_2 = = \langle\eta,\xi\rangle\xi + \langle\eta,E_1\rangle E_1 + \langle\eta,E_2\rangle E_2.
Z_{i,-1,0}:= N_{i,0,0}+U_{i,0,0}, i=1,\ldots,m, Z_{i,j,k}:= \Phi^{-1}\circ G_{N_{i,j,k}, 1/2}(N_{i,j+1,2k}+U_{i,j,k}), (i,j,k) \in \Lambda, \ 0\leq j\leq \overline J.
+\sum_{i_{1},i_2,i_3,i_{4}=1}^{m}G_0^{(i_{4})}G_0^{(i_{3})}G_0^{(i_{2})}\Sigma_{i_{1}}\hat I_{0000_{\tau_{p+1},\tau_p}}^{(i_{4}i_{3}i_{2}i_{1})}+{\bf u}_{p+1,p}+{\bf v}_{p+1,p},
\sum_{l = 0}^{\infty}( 2 l + 1 ) [ s_{l}^{\prime}( \lambda r ) e_{l}( \lambda \rho ) ]^{2}= \frac{1}{2 r \rho}\int_{r - \rho}^{r + \rho}\left( \frac{1}{\lambda}\frac{\partial{\cal D}}{\partial r}\right)^{2}R d R .
B_{2}^{f}= - \frac{V}{1 6 \pi^{2}}\frac{2^{1 - 2 \nu}}{\nu \Gamma ( \nu )^{2}}\int_{0}^{\infty}d t t^{2 \nu + 3}\frac{K_{\nu}^{\prime}( t )}{I_{\nu}^{\prime}( t )}> 0 ,
|\Phi_{t}(\mathbf{x},\mathbf{y})| \leq \int \Big|g_t(\mathbf{x},\mathbf{z})(\Phi^{\{1\}}_t*g_t)(\mathbf{z},\mathbf{y})\Big|\,dw(\mathbf{z}) \leq \int_{d(\mathbf{x},\mathbf{y}) \leq 2d(\mathbf{x},\mathbf{z})}+\int_{d(\mathbf{x},\mathbf{y}) \leq 2d(\mathbf{y},\mathbf{z})}=I_1+I_2.