harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{{\bf m}\in \mathbb{N}_{\geq 0}^{n(n-1)/2}}\frac{q^{C({\bf m})}\cdot F}{\displaystyle\prod_{1\leq i<j\leq n}(q)_{m_{i,j}}}=\sum_{k_1,\cdots,k_{n-1}\geq 0}\frac{q^{k_1^2/2+ \cdots + k_{n-1}^2/2-k_1 k_2 -\cdots-k_{n-2}k_{n-1}}x_1^{k_1}\cdots x_{n-1}^{k_{n-1}}}{(q)_{k_1}\cdots (q)_{k_{n-1}}},
\langle y_{n},S y_{n}\rangle =\langle y_{n-1},S y_{n-1}\rangle +h\displaystyle \sum_{i=1}^{s}b_{i}\langle y_{n-1},S f(Y_{i})\rangle &+h\displaystyle \sum_{j=1}^{s}b_{j}\langle f(Y_{j}),S y_{n-1}\rangle+h^{2}\displaystyle \sum_{i,j=1}^{s}b_{i}b_{j}\langle f(Y_{i}),S f(Y_{j})\rangle.
{{R}_{{\rm{DS}}}}\approx \frac{1}{M_{\rm R}}\sum\limits_{n = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left( 1 + \frac{{{\left(\sum\limits_{m = 1}^{{M_{\rm{R}}}}\left|[{{{\bf{\Xi}}_{{\rm{DS}},m}}}]_{n,n}\right|\right)^2}E}}{{{N_{\rm{R}}}{M_{\rm{R}}}{\sigma^2}}}\right)}.
\lim_{\delta \rightarrow 0}\limsup_{n\rightarrow \infty}\left|E\left[ \exp \left( i\mu n^{\alpha+\frac 12}\int_0^{t-\delta}(t-s)^\alpha \xi^n_s dW_s \right) \right] -E \left[ \exp \left(i \mu Y^{n,1}_t \right) \right] \right| =0.
\beta_{2n}= -(p+\beta) \; , \beta_{2n+1}= \beta \; , n \geq 0\, , \chi_{2n,2n}= \alpha_{2n+1}= \alpha_{1}\; , \chi_{2n+1,2n+1}= \alpha_{2n+2}= \alpha_{2}\;\; , n \geq 0\, , & \chi_{n,n-1}= \gamma_{n}= (-1)^{n}\gamma , \, n \geq 1\, , \chi_{n,\nu}= 0 \; , 0 \leq \nu < n-1 ,\, n \geq 1 \, .
\mathbb{P}[\mathbf{D}\mathbf{x}= \mathbf{e}_i | wt(\mathbf{x})=k] = \mathbb{P}[\sum_{j=1}^k d^{e_j}_{i}= 1] \prod_{\substack{l'=1 l' \neq i}}^m \mathbb{P}[\sum_{j=1}^k d^{e_j}_{l'}= 0].
{}\sum_{u=1}^{N}\left (t_{u}-t_{u-1}\right )({\bf a}^{(u)})^{\top}\bar{{\bf z}}^{(u)}=\sum_{u=1}^{N}\left (t_{u}-t_{u-1}\right )({\bf c}^{(u)})^{\top}\bar{{\bf w}}^{(u)}.
|\nabla_\xi (II)| = \left|\nabla_\xi \ln\left(1 + \frac{|x + \xi\sqrt{\theta(x)}|^2}{(4 + 4\alpha)\theta_0|\ln \theta_0|}\right) \right| &\leq \frac{C\sqrt{\theta(x)}}{|x + \xi\sqrt{\theta(x)}|}\leq \frac{C}{|\frac{K_0}{4}\sqrt{|\ln \theta(x)|}+ \xi |}\leq \frac{C}{\sqrt{|\ln \theta(x)|}}.
\frac{d}{dt}(\Phi^{\bar{u}}(t,\tau)) = \left( \frac{\partial f}{\partial x}(x^{(\bar{u},x_0)}(t),\bar{u}(t)) + \Lambda^{\bar{u}}(t)\frac{\partial g}{\partial x}(x^{(\bar{u},x_0)}(t),\bar{u}(t))\right) \Phi^{\bar{u}}(t,\tau),\Phi^{\bar{u}}(\tau,\tau) = I
u_1(x_1,x_2,t)=-x_1\int_0^1\int_0^1\frac{(y_2-x_2)\rho_0(y_1\exp\{\int_0^tI(y_2,s)ds\})}{(y_1^2+(x_2-y_2)^2)^2}\int_0^t \exp\left\{-\int_s^t I(y_2,r)dr\right\}ds dy_1 dy_2.
\big[ d^{-2}x,\big[ d^{-1}y,cz\big] \big] +\big[ d^{-2}y,\big[d^{-1}z,cx\big] \big] +\big[ d^{-2}z,\big[ d^{-1}x,cy\big] \big]=0, \ \x,y,z\in L.
A(s) =\frac{6(n-1)^6(n-2)^2(n^2 + 6n-12)}{n^3}s^4 &+ \frac{(n-1)^2}{n^2}(24 n^7 -116 n^6-88 n^5 + 1732 n^4 - 4920 n^3 +6744 n^2 -4800 n +1440) s^3 &+ \frac{n-2}{n^2}(36 n^8 -252 n^7 + 506 n^6 + 392 n^5-3386 n^4 + 6504 n^3 -6480 n^2 + 3456n -768) s^2 &+ (n-2)^2(24 n^5 -156 n^4 + 316 n^3 -224 n^2 + 32 n) s &+ 2n^2(n-2)^3(3n-4)(n-4),
\int\limits_{O_v^{\times 2}}\chi(ac \ (F^{(m)}(x,y)))\ |F^{(m)}(x,y)|^s\ |dxdy|= \frac{U_0(q^{-s}, \chi)}{1-q^{-1-s}}+ \sum_{\{\theta \in O_v| f_0(1,\theta^a)= 0\}}J_\theta(s,m,\chi),
G\left(s_2,K\right)+F\left(s_2,N\right) =G\left(s_3,K\right)+F\left(s_3,N\right) =\frac{-1}{\left(s_2-\frac{1}{2}\right)^2-\frac{1}{4}}=\frac{-1}{\left(s_3-\frac{1}{2}\right)^2-\frac{1}{4}}
\mathcal K_{j,k}^{p,q}= \frac{\Gamma(j-\frac12)}{\Gamma(p)\Gamma(p+q+j+k-\frac32)}\sum_{\ell=1}^{q}\frac{\Gamma(j+k+\ell-\frac32)\Gamma(p+q-\ell)}{\Gamma(j+\ell-\frac12)\Gamma(q-\ell+1)}.
R = C_{1}\xi^{( \omega^{\prime}- i m^{\prime}) / 2}( 1 - \xi )^{1 / 2 + \nu}F \bigg ( \frac 1 2 + \nu + \omega^{\prime}, \frac 1 2 + \nu - i m^{\prime}, 1 + 2 \nu ; 1 - \xi \bigg ) .
V_C(p,0) = V_C(0,p) = V_0 e^{\mu \alpha_C}\int\limits_\mu^\infty dz\,\frac{\exp (-\alpha_C z)}{z^2+p^2}=V_0\mu e^{\mu \alpha_C}\int\limits_0^1dy\,\frac{\exp (-\alpha_C\mu /y)}{\mu^2+(p y)^2}.
\frac{\partial \mathcal{L}(\mathbf{p}_,\mu)}{p_{_k}}=\frac{-\alpha_k\hat{\lambda}^{2}_{_k}}{\left(\hat{\lambda}^{2}_{_k}p_{_k}+ \beta_k\right)\left(\hat{\lambda}^{2}_{_k}p_{_k}+ \beta_k+\alpha_k\right)}+\mu=0.
\displaystyle \int_{0}^{T}\Phi_{t}dW_{t}=\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{i=0}^{n-1}\Phi(B_{t_{i}},t_{i}) (B_{t_{i+1}}-B_{t_{i}})
\lambda_{1}E_{1}^{1}+\lambda_{2}E_{2}^{2}+\cdots+\lambda_{N}E_{N}^{N}=\begin{pmatrix}\lambda_{1}0 \cdots 0 0 \lambda_{2}\cdots 0 \vdots \vdots \vdots \vdots 0 0 \cdots \lambda_{N}\end{pmatrix}
\sum_{\ell=0}^{N-1}\big|\mathcal{D}_{N,\ell}^{h,2,2}\big|\le \frac{C(1+|x|_{L^p})^{K}}{\tau^{2\kappa}}\bigl(1+t_{N-1}^{\frac{1}{2}-\frac{1}{2q}-\kappa-\beta}\bigr)\Bigl(|(-A)^{-\beta}h|_{L^{2q}}+\frac{\Delta t}{t_{N-1}}|h|_{L^{2q}}\Bigr).
W'(f(x),g(x))=f_{n+1}'(x)g_{n}(x)+f_{n+1}(x)g_{n}'(x)-f_{n}'(x)g_{n+1}(x)-f_{n}(x)g_{n+1}'(x) =AW_{n}(f'(x),f(x))-A^{-1}W_{n}(g'(x),g(x)) =\frac{1-x^{2}}{x^{2}}\left(A\sum_{j=n+1}^{\infty}f_{j}^{2}(x)+A^{-1}\sum_{j=-\infty}^{n}g_{j}^{2}(x)\right)\!\!.
(1+w_{N}(\underline{\xi}))\Big(\frac{1}{N}+\underline{\psi}_{N,1}^{\ast}\Big)=(1+\widehat{w}_{N}(\underline{\xi}))\Big(\frac{1}{N}+\overline{\psi}_{N,1}^{\ast}\Big)=\frac{N+1}{N}.
\tau_1=\inf \{t\in [0,T]:2^{\alpha-1}C_t^{\alpha}(C_T+(||x||_{H^2}+1) +(||x||_{H^2}+1)^{\alpha})^{\alpha-1}(t^{\theta}+t^{\alpha}) & +C_tt^{\alpha}(||x||_{H^2}+1)^{\alpha-1}>\frac{1}{2}\}\wedge T,
C_{H}(X) =\{f\in C^{\ast}(X)\mid f (\ast)_{s}^f s\in \mathcal{H}\}, C_{P}(X) =\{f\in C^{\ast}(X)\mid f (\ast)_{s}^f s\in \mathcal{P}\}, C_{L}(X) =\{f\in C^{\ast}(X)\mid f (\ast)_{s}^f s\in \mathcal{L}\}.
\begin{array}{ll}d(\parallel\!\!\Psi\!\!\parallel_F \!F^2)= 0, begin{align*}7pt] \Omega^{A}\wedge F^2=0,(\Omega^{A})^{0,2}=(\Omega^{A})^{2,0}=0, begin{align*}6pt] dT-\frac{\alpha'}{4}\left({\rm tr}\, \Omega\wedge\Omega -{\rm tr}\, \Omega^A\wedge \Omega^A\right)=0.\end{array}
\sum_{n=0}^{\infty}p_{-2}\left(5^{2j-1}(5n+4)+\dfrac{7\times5^{2j-1}+1}{12}\right)q^{5n+4}= \dfrac{1}{qE_{5}^{2}}\sum_{l=1}^{\infty}a(2j-1,l)T^{l}\left(\sum_{k=1}^{\infty}m(6l,k)T^{-k}\right).
\begin{aligned}T =(\mu+\nu)\bar{g}(\cdot,X)\otimes\bar{g}(\cdot,X)+\nu\bar{g}=\sec^2(\xi(x))\left[\nu(\xi(x)) x^*g_1+\mu(\xi(x))\left(e^{-\xi(x)\sqrt{n-1}}\right)^2dt^2\right], \end{aligned}
\eta^2_* := \sum_{K\in \mathcal{T}_h}h_K^2 \|{\rm div}p_h -u_h +f\|^2_K+ \sum_{e\in{\cal E}_h^i}h_e \|[p_h]\|^2_e + \sum_{e\in{\cal E}_h^2}h_e \|p_h\cdot n-g\|^2_e, \zeta^2_* := \sum_{e\in{\cal E}_h^0}h_e^{-1}\|[u_h]\|^2_e
\dot{\rho}=Re(\lambda_l)\rho+\sum_{m=1}^M Re(\beta_m)\rho^{2m+1}+\varepsilon r \sin(\theta-\phi), \dot{\theta}=Im(\lambda_l)+\sum_{m=1}^M \mbox{Im}(\beta_m)\rho^{2m}+\frac{\varepsilon}{\rho}\left( r\cos(\theta-\phi)\right), \dot{\phi}=\Omega.
-n!C(l,h) = \sum_{h\le q \le\max(l,h-1)}(-1)^{h+n}{n\choose h}{n-h\choose n-q}\sum_{0\le k \le n}(-1)^kk{h\choose q-k}= \sum_{h\le q \le\max(l,h-1)}(-1)^{h+n}{n\choose h}{n-h\choose n-q}\sum_{0\le K \le q}(-1)^{q+K}(q-K){h\choose K}= \,0,
\widetilde{\eta}_{\sigma \tau}= \epsilon_{\sigma}^{\mu *}\left( k \right) \cdot \epsilon_{\tau , \mu}\left( k \right) \equiv \left( \begin{array}{c c c c}{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right) .
A(1,x) = \begin{cases}\ \begin{bmatrix}\cos(\pi\alpha) -\sin(\pi\alpha) \sin(\pi\alpha) \cos(\pi\alpha)\end{bmatrix}x \in [0,1-\eta), begin{align*}20pt]\ \begin{bmatrix}1 0 0 -1\end{bmatrix}x \in [1-\eta,1).\end{cases}
\kappa=\frac{\partial\psi}{2\partial\theta_{1}\partial\theta_{1}}+\frac{\partial\psi}{2\partial\theta_{2}\partial\theta_{2}},\gamma_{1}\equiv\frac{\partial\psi}{2\partial\theta_{1}\partial\theta_{1}}-\frac{\partial\psi}{2\partial\theta_{2}\partial\theta_{2}},\gamma_{2}\equiv\frac{\partial\psi}{\partial\theta_{1}\partial\theta_{2}}
I\left(f_{\omega_{\pi_\nu},s}^\circ,v^\circ_{\pi_{\nu}}\right)(I_m)=\int \limits_{Z_n(k_\nu)\backslash T^-}\mu(\underline{t})c_{v_{\pi_{\nu}}^\circ,\check{v}_{\pi_{\nu}}^{\circ}}(\underline{t}){v}^\circ_{{\pi}_{\nu}}\left|\det \underline{t}\right|^{ms+\frac{m}{2}}d\underline{t},
P \bigg( |V_\chi^S| \ge \frac{\rho}{4 \sqrt{r}\phi(q)}\bigg) \le \bigg(\frac{16r\phi(q)^2}{\rho^2}\bigg) \sigma^2 (V_\chi^S) = \bigg(\frac{16r\phi(q)^2}{\rho^2}\bigg) \cdot 4 \sum_{\substack{\gamma\in\Gamma^S(\chi) \gamma>T}}\frac1{\frac14+\gamma^2}< 1,
\mathbf{C}_w^{-1}-(\mathbf{C}_s+\mathbf{C}_w)^{-1}=\frac{\sigma_{\theta}^2}{1+\sigma_{\theta}^2g(\mathbf{a})}\mathbf{C}_w^{-1}\mathbf{H}\mathbf{a}\mathbf{a}^H\mathbf{H}^H\mathbf{C}_w^{-1}\;,
S^{t}\, \delta_{1}= - \delta_{1}\ , \S^{t}\, \delta_{2}= - \delta_{2}+ \delta_{1}^{2}\ , \S^{t}\, \delta_{3}= - \delta_{3}+ 4 \, \delta_{1}\, \delta_{2}- 2 \, \delta_{1}^{3}\ , \ \ldots
V ( \psi \otimes \bar{v}, z ) \; V ( \chi \otimes \bar{w}, \zeta ) \; \left( \Omega \otimes \overline{{{\Omega}}}\right) = \sum_{\bar{w}^{\prime}, \bar{v}^{\prime}}{R \;}_{\bar{v}^{\prime}\; \bar{w}^{\prime}}^{\bar{v}\; \bar{w}}\; \; V ( \chi \otimes \bar{w}^{\prime}, \zeta ) \; V ( \psi \otimes \bar{v}^{\prime}, z ) \; \left( \Omega \otimes \overline{{{\Omega}}}\right) ,
\frac{e^{x}}{x}\mathcal{L}^{-1}\left( \left( 2\pi\right)^{3/2}e^{\frac{1}{2}t^{2}}m_{\Sigma}\left( t/\kappa\right) \right)(\sqrt{2x}) =\frac{1}{x}{\displaystyle\iint\limits_{\sqrt{2x}T_N}}e^{-\frac{1}{2}\left( y^{2}+z^{2}\right)}\, d y \, d z.
\mathcal{T}(k) \cap h \mathcal{G}(A) h^{-1}= \left\lbrace \mathrm{diag}(t, \overline{t}t^{-1}, \overline{t}^{-1}): t \in \mathrm{R}_{\ell/k}(\mathbb{G}_{m,\ell}) \right\rbrace \cap h \mathcal{G}(A) h^{-1}.
(S f)(y,x_0,\dots,x_{d-1})=\int \frac{\prod_{i=0}^{d-1}dp_i}{(2\pi y)^{d/2}}\, f(y,p_0,\dots,p_d)\, e^{\frac{i}{y}\sum_{i=0}^{d-1}p_i x_i}
\big\{\alpha\beta^*-\delta_{\alpha, \beta}e_{t(\alpha)}\; |\; \alpha, \beta\in Q_1 \mbox{with}s(\alpha)=s(\beta)\}\cup \{\sum_{\{\alpha\in Q_1\; |\; s(\alpha)=i\}}\alpha^*\alpha-e_i\; |\; i\in Q_0^{\rm re}\big\}.
R\left({\mathcal{H}}_{0}\right) =\left(1-{P}_{\mathrm{f}}\right) \frac{{T}_{\mathrm{f}}-{T}_{\mathrm{s}}}{{T}_{\mathrm{f}}}P\left({\mathcal{H}}_{0}\right) C\left({\mathcal{H}}_{0}\right).
\mathcal{R}(p_1[x_d^{\ast}],\ldots,p_d[x_d^{\ast}])\!=\!0 \!\!\Longleftrightarrow\!\quad\! \exists (x_1^{\ast},\ldots,x_{d-1}^{\ast})\in\mathbb{C}^{d-1}p_1(\underline{x}^{\ast})\!=\!\cdots\!=\!p_d(\underline{x}^{\ast})\!=\!0,
E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\widetilde{\mathbf{L}}(\beta)^n\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] -E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\pi_{S^{\beta+}}\bigl(\widetilde{\mathbf{L}}^n\bigr)\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] \to0
c_r\frac{m+1}{k+1}\mathbb{P}(\sum_{i=1}^{k+1}(X_i-1)\geq m+1)+c_r(k+1)^{-1}\sum_{j\geq m+2}^{}\mathbb{P}(\sum_{i=1}^{k+1}(X_i-1)\geq j).
-u_i \cdot u_j = \cosh(t_i-t_j) +\frac12 (s_i-s_j)^2 e^{t_i+t_j}+ (\bar\psi_i-\bar\psi_j)(\psi_i-\psi_j)e^{t_i+t_j}z_i = \cosh t_i + (\frac12 s_i^2 +\bar\psi_i\psi_i)e^{t_i}.
\frac{1}{N}\sum_{i=0}^{k^*}(j - i) X_i(t) \leq \sum_{i=0}^{k^*}\frac{(j - i) X_i(t)}{X_j(t)}&\leq C_8 \kappa(t) 2^{j - k^*}\bigg( \frac{\mu}{s}\bigg)^{(j - k^*)(j - k^* - 1)/6k_N}\sum_{i=0}^{k^*}(j - i) k_N^{-(k^*-i)}e^{-s(j-i)(t - \gamma_j)}.
\frac{\partial F}{\partial a_{i}}= \oint_{\beta_{i}}\lambda , \ \ \, \frac{\partial F}{\partial T_{n}}= - 2 \pi i \, m b o x{r e s}\Big ( z^{- n}\lambda \Big ) , \ \ \, \frac{\partial F}{\partial T_{0}}= - 2 \pi i \, \int_{z_{*}= 0}^{z = 0}\lambda ,
{\tilde{x}^{\alpha \beta}}{}^{\prime}={\tilde{x}^{\alpha \beta}}- \frac{i}{4}\big [ \varepsilon^{\alpha}{\tilde{\tau}}^{\beta}+ \xi^{\alpha}\tilde{\psi}^{\beta}( \tilde{x},{\tilde{\tau}},{\tilde{\omega}}) + \alpha \leftrightarrow \beta \big ] ,
\prod_{i,j=1}^N\frac{(q^{s_i-a_j+1};q)_{\infty}}{\sin\pi(s_i-a_j)}\prod_{1\le i<j\le N}\frac{\sin\pi(s_j-s_i)\sin\pi(a_j-a_i)(q^{s_j}-q^{s_i})(q^{a_j}-q^{a_i})}{(q^{a_i-a_j};q)_{\infty}(q^{a_j-a_i};q)_{\infty}}
L^\ast_{\tilde{a}}g_{(\mu_1,\mu_2)}= -\{d(x^0+a^0)\}^2+e^{-2\mu_1(x^0+a^0)}\{d(e^{\mu_1a^0}x^1+a^1)\}^2 &+e^{-2\mu_2(x^0+a^0)}\{d(e^{\mu_2a^0}x^2+a^2)\}^2 = -(dx^0)^2+e^{-2\mu_1x^0}(dx^1)^2+e^{-2\mu_2x^0}(dx^2)^2.
\lefteqn{2h^{\varphi}\left(C^{\varphi}\omega(X)^{\varphi}(Y'),Z'\right)=h^{\varphi}\left((\nabla^{\varphi}_X C^{\varphi})(Y'),Z'\right) +}&+ h^{\varphi}\left((\nabla_{\varphi^{-1}(Y')}^{\varphi}C^{\varphi})(\varphi(X))),Z'\right) -h^{\varphi}\left((\nabla_{\varphi^{-1}(Z')}^{\varphi}C^{\varphi})(\varphi(X))),Y'\right).
\frac{1}{2}\sum_{j=1}^{n}\psi\left(\frac{j}{2}\right) \sim{n\over 2}\log n, \frac{1}{4}\sum_{j=1}^{n}\psi^{(1)}\left(\frac{j}{2}\right) ={1\over 2}\log{n}+c_1 + o(1),
\alpha_t=\sum_{n\in\mathbb N}\eta_n^2\frac{\omega_n\varphi_n g h_{01}}{\psi_{nt}}\ \ \ \ \alpha=\sum_{n\in\mathbb N}\eta_n^2\frac{\omega_n\varphi_n g h_{01}}{\psi_{1n}}\ \ \mathbb T.
\sum_{w\in \widehat W^\natural}det(w)ch\,M^W(w.\widehat\nu_h)=\left(\sum_{w\in \widehat W^\natural}det(w)ch\,M^W(w.\widehat\nu_0)\right)e^{(0,\tfrac{2h^2+(h^\vee-1)h}{2(k+h^\vee)}-h)}.
g_1(s) = \frac{g(s)-g(0)-(g(s_b)-g(0))s/s_b}{s(s-s_b)}&=\frac{\sum^{+\infty}_{n=1}{\frac{d^n g}{d s^n}(0)}s^{n-1}/{n!}-\sum^{+\infty}_{n=1}{\frac{d^n g}{d s^n}(0)}s_b^{n-1}/{n!}}{s-s_b}
CT_{\vec{t}}\,\prod_{i=0}^{c-1}(1+t_i)^a\left(1+t_i^{-1}\right)^b\prod_{i\neq j}^{0,c-1}\left(1-t_jt_i^{-1}\right)^m =\prod_{\ell=0}^{c-1}\frac{(a+b+\ell m)!\,((\ell+1)m)!}{(a+\ell m)!\,(b+\ell m)!\, m!}.
B_{\vec\tau, \vec s}^{(1)}(r)=\frac{s_{2}}{2\pi i}\left(T^{-1}\partial_{s_m}T\right)_{21}(0)=B_{\vec\tau, \vec s}^{(1,1)}(r)+B_{\vec\tau, \vec s}^{(1,2)}(r)+B_{\vec\tau, \vec s}^{(1,3)}(r),
{\cal A}= N \int_{z_{-}}^{z_{1}}\! d z{\frac{1 - z}{z ( 1 + z / N )^{2}}}{\frac{\bigl [ 1 - 1 / N - z / N^{2}\bigr ] \bigl [ 1 + z / ( N ( 1 - z ) ) \bigr ]}{\displaystyle \sqrt{1 - N^{2}( 1 + z / N )^{2 N}/ ( z^{2}r^{2 N})}}}\cdotp
H(t)=I_{\varepsilon}(tu)=\frac{1}{2}t^{2}\|u\|_{\varepsilon}^{2}+\frac{q\omega^{2}}{2\varepsilon^{n}}t^{2}\int_{M}\psi(tu)u^{2}d\mu_{g}-\frac{t^{p}}{p}.
(dB_{t}, 0)= \theta^{\ell}_{g_{t}}(dg_{t}) & = \left( d\overline{g}_{t},d\widehat{g}_{1}(t) - \frac{1}{2}\langle U^{(1)}\overline{g}_{t}, d\overline{g}_{t}\rangle, \ldots, d \widehat{g}_{n}(t) - \frac{1}{2}\langle U^{(n)}\overline{g}_{t}, d\overline{g}_{t}\rangle \right),
y_1=x_1, \, y_2=\frac{1}{\sqrt{\mu}}x_3, \, y_3=x_2; &[y_1,y_2]=-\frac{1+w}{\sqrt{\mu}}y_1, \,[y_2,y_3]=\frac{1-w}{\sqrt{\mu}}y_3, \,[y_3,y_1]=0.
\Psi\longrightarrow \Psi^{\prime}= e^{\mathrm{i}\chi}\Psi, \mathbf{A}\longrightarrow \mathbf{A}^{\prime}= \mathbf{A}+ \nabla \chi ,\phi \longrightarrow \phi^{\prime}= \phi - \frac{\partial \chi}{\partial t},
(-\Delta)^{s}_{p}u + a(x)\vert u\vert^{p-2}u +(-\Delta)^{s}_{q}u + b(x)\vert u\vert^{q-2}u +\mu(x)\vert u\vert^{r-2}u \quad= \lambda h(x)\vert u \vert^{m-2}u,x\in{\mathbb{R}}^{N},
( 1 + \frac{1}{2}) ( 1 - \frac{1}{2}) ( 1 + \frac{1}{3}) ( 1 - \frac{1}{3}) ( 1 + \frac{1}{4}) ( 1 - \frac{1}{4}) \cdots ( 1 + \frac{1}{2 0 1 3}) ( 1 - \frac{1}{2 0 1 3})
\Gamma_{l , s}^{\alpha , \alpha_{0}}( 0 ) = \Gamma_{l , s}^{\infty , \alpha_{0}}( 0 ) - \int_{\alpha}^{\infty}d \alpha^{\prime}\underbrace{\partial_{\alpha^{\prime}}\Gamma_{l , s}^{\alpha^{\prime}, \alpha_{0}}( 0 )}_{\mathrm{r . h . s . o f ( \ref{f g 5})}}
\breve{T}_{2}+T_{1}=\alpha_{30}\int\zeta_{20}(x_{t})H_{p}(\gamma_{10}(x_{t}))[y_{2t}-\hat{\gamma}_{1}(x_{t})]F_{0}(dw) & =\alpha_{30}\int\zeta_{20}(x_{t})H_{p}(\gamma_{10}(x_{t}))[\gamma_{10}(x_{t})-\hat{\gamma}_{1}(x_{t})]F_{0}(dw).
S ( a ; b_{1}, \ldots , b_{k - 1}) = \sum_{n = 1}^{\infty}\frac{( \mathrm{s i g n}a )^{n}}{n^{| a |}}\prod_{i = 1}^{k - 1}\sum_{m_{i}= 1}^{n - 1}\frac{( \mathrm{s i g n}b_{i})^{m_{i}}}{m_{i}^{| b_{i}|}}.
\left| \sum_{n>N}a_{n}\sum_{\lambda \in (W), \atop |\lambda|< 1/R}(\lambda R)^{n}\right| \leq \sum_{n>N}a_{n}\sum_{\lambda \in (W), \atop |\lambda|< 1/R}(|\lambda| R)^{n}& < 2 \epsilon \sum_{n>N}(\rho R)^{n}= \frac{2 \epsilon \rho R}{1- \rho R}(\rho R)^{N}.
W^{\prime}(r)^{2}=\frac{1}{V^{2}(r)}\left( \Delta_{L}E^{2}-V\left(m^{2}+\frac{4j^{2}}{r^{2}}\right) -\frac{4\Delta}{r^{4}}aj_{R}E-\frac{4g^{2}a^{2}}{r^{6}}j_{R}^{2}\right) \,,
\sum\limits_{n=1}^{N}\left( \|\delta_t e^{n-\frac{1}{2}}\|^2-c_2\Lambda_4^{n-\frac{1}{2}}\right) \leq \sum\limits_{n=1}^{N}\left(\frac{2\mu_2}{\mu_1}\|e^{n-\frac{1}{2}}\|\| \delta_te^{n-\frac{1}{2}}\|+c_2\Lambda_2^{n-\frac{1}{2}}+c_2\Lambda_3^{n-\frac{1}{2}}+c_2\Lambda_5^{n-\frac{1}{2}}\right).
(-\tilde{g})^{-1/2}\partial_{\mu}\left(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi \right)-\frac{\alpha M^{4}}{M_{p}(\zeta +b)}e^{-2\alpha\phi/M_{p}}+\frac{\alpha m}{M_{p}}F(\zeta)\overline{\Psi}^{\prime}\Psi^{\prime}=0.
(1+a) \alpha^2 \tau_0 \tau_1((\sigma^2 \sigma_1)^2 - (\sigma \sigma_1)^2) - (a+b) \alpha \tau_0 ((\sigma^2 \sigma_1)^2 - \sigma^2) + b((\sigma^2 \sigma_1)^2 - 1) = 0.
\psi_r^2e(i)=\begin{cases}0 \textit{if}i_r= i_{r+1}; e(i) \textit{if}i_r\neq i_{r+1}\pm 1; (y_{r+1}-y_r)e(i) \textit{if}i_r\rightarrow i_{r+1}; (y_r-y_{r+1})e(i) \textit{if}i_r\leftarrow i_{r+1}; (y_{r+1}-y_r)(y_r-y_{r+1})e(i) \textit{if}i_r\leftrightarrow i_{r+1}.\end{cases}
L_{B}= -{\frac{1}{2}}\left( \, \partial_{\mu}\phi \, \right)^{2}+{\frac{m}{4 !}}\epsilon^{\mu \nu \rho \sigma}\, \phi \, \partial_{\, [ \, \mu}A_{\nu \rho \sigma \, ]}-{\frac{1}{2 \cdot 4 !}}F^{2}{}_{\mu \nu \rho \sigma}( A )
{\cal L}_{1}= \psi_{\uparrow}^{\dagger}( i \partial_{t}- \epsilon ( p ) ) \psi_{\uparrow}+ \psi_{\downarrow}^{\dagger}( i \partial_{t}- \epsilon ( p ) ) \psi_{\downarrow}+ \sqrt{\lambda g}( \phi^{*}\psi_{\downarrow}\psi_{\uparrow}+ \phi \psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}) + g \phi^{*}\phi \, \, \, \, .
\delta_{w_0}^{-2}(G/P) \mu(s,\tau,\tilde{w}_0) = \delta_{w_0}^{-2}(G/P) \prod_i \mu(s,\tau_i,\tilde{w}_i) \prod_{j<i}\mu(s,\tau_{i,j},\tilde{w}_{i,j}),
f\left(x\right) =\int_{0}^{x}\varphi\left(y\right)F\left(x-y\right)dy+\int_{x}^{a}\varphi\left(y\right)F\left(x-y\right)dy f'\left(x\right) =\int_{0}^{x}\varphi\left(y\right)F'\left(x-y\right)dy+\int_{x}^{a}\varphi\left(y\right)F'\left(x-y\right)dy \quad+\underset{=0}{\underbrace{\varphi\left(x\right)F\left(0\right)-\varphi\left(x\right)F\left(0\right)}}.
\begin{aligned}\int_{\mathbb{T}^\infty}\left(\sup\limits_{0<r<1}I_{w,r}\right)dm_\infty(w) =\sup\limits_{0<r<1}\int_{\mathbb{T}^\infty}\int_{\mathbb{T}}\log^{+}|F_{w}(r\lambda)|dm_{1}(\lambda)dm_\infty(w) &=\sup\limits_{0<r<1}\int_{\mathbb{T}^\infty}\int_{\mathbb{T}}\log^{+}|F_{[r]}(\lambda\star w)|dm_{1}(\lambda)dm_\infty(w).\end{aligned}
V(T) = \sum_{n=1}^{\infty}(v_{ij}+ (n-1)^2v_{jj})(H_{jj}^{n-1}-H_{jj}^{n}) & + \sum_{n=1}^{\infty}(\tau_{ij}+ (n-1)\tau_{jj})^2(1-H_{jj}^{n-1}+H_{jj}^n)(H_{jj}^{n-1}-H_{jj}^{n}) &- 2\sum_{n=1}^{\infty}\sum_{m=1}^{n-1}(\tau_{ij}+ (n-1)\tau_{jj}) (H_{jj}^{n-1}-H_{jj}^{n}) (\tau_{ij}+ (m-1)\tau_{jj})(H_{jj}^{m-1}-H_{jj}^{m})
C_{1}\Big(\sum_{i=1}^{N}\lambda_{k_{i}}\Big)^{\frac{1}{2}}\leq\sum_{i=1}^{N}\lambda_{k_{i}}^{\frac{1}{2}}\leq C_{2}\Big(\sum_{i=1}^{N}\lambda_{k_{i}}\Big)^{\frac{1}{2}}.
\lim_{\rho_0\rightarrow\infty}D : =\lim_{\rho_0\rightarrow\infty}\left(\nu+|l|-\beta + 2\beta\,{A_1\over B_1}\,{\Phi(A_1+1,B_1+1;\beta) \over\Phi(A_1,B_1;\beta)}\right) = 0\,.
\int_0^te^{-(t-s)A}\Big(\frac{d}{dt}V_{m}^{(2)}(s)\Big)\,ds =(-1)^{m}\sum_{k=1}^{m}\begin{pmatrix}m k\end{pmatrix}\frac{(-tA)^{k}}{k!}e^{-tA}u_1 &=-V_{m+1}^{(1)}(t)+(-1)^{m+1}e^{-tA}u_1.
\mathrm{P}\left( \bar{E}_{m,1}\right) = \mathrm{P}\left( x_m< \frac{\frac{\epsilon_{1,m}}{\rho}}{\alpha^2_m - \beta_{m}^2\epsilon_{1,m}}\right) \times \prod^{m-1}_{n=1}\mathrm{P}\left( x_n> \frac{\frac{\epsilon_{1,n}}{\rho}}{\alpha^2_n - \beta_{n}^2\epsilon_{1,n}}, x_n>\frac{\epsilon_{2,n}}{\rho \beta_n^2}\right).
\sum_{i}\left[ \Psi \right]_{i}\left( a_{\alpha i}\Delta p_{\alpha}+ b_{\alpha i}^{\mu}\partial_{\mu}\left( \Delta p_{\alpha}\right) \right) \equiv - \sum_{i}\partial_{\mu}\left\{\frac{\partial L}{\partial \left( \partial_{\mu}\psi_{i}\right)}\left( a_{\alpha i}\Delta p_{\alpha}+ b_{\alpha i}^{\mu}\partial_{\mu}\left( \Delta p_{\alpha}\right) \right) \right\},
\begin{cases}\eta_0 \xrightarrow{D_3}\mu_1 \xrightarrow{D_{23}}\cdots \xrightarrow{D_{23}}\mu_s \xleftarrow{D_1}\xi_0 \tau(K)>0 \xi_0 \xrightarrow{D_{12}}\eta_0 \tau(K) =0 \xi_0 \xrightarrow{D_{123}}\mu_1 \xrightarrow{D_{23}}\cdots \xrightarrow{D_{23}}\mu_s \xrightarrow{D_2}\eta_0 \tau(K) < 0.\end{cases}
{t\over 2M}= \ln{(r_\infty/2M-1)^{1/2}+ \tan{(\eta/2)}\over (r_\infty/2M-1)^{1/2}- \tan{(\eta/2)}}+ \left({r_\infty\over 2M}-1\right)^{1/2}\left[\eta + \left({r_\infty\over 4M}\right)(\eta +\sin \eta)\right]
\lim \frac{1}{n}\mathbb{E}_{\omega_0}\left[\log\Vert A^n(\omega)v_n \Vert \right] &= \lim \frac{1}{n}\mathbb{E}_{\omega_0}\left[\log\Vert A^n(\omega) \Vert \right] + \lim \frac{1}{n}\mathbb{E}_{\omega_0}\left[\log \frac{\Vert A^n(\omega)v_n \Vert}{\Vert A^n(\omega) \Vert}\right] &= L_1(A,K) + 0 = L_1(A,K)
\hspace{1 . 4 i n}\times e^{- \overline{{\Lambda}}_{1}}( ( - 1 )^{n - 1}q z )^{- \partial_{\overline{{\Lambda_{1}}}}+ \frac{n - i - 1}{n}}q^{- i}( - 1 )^{i n + \frac{1}{2}i ( i + 1 )}
E\big[(s\mathcal{T}(\nu))^2\big] = \frac{s^2\nu}{\nu-2}, \big(s\mathcal{T}(\nu)\big) =\frac{K_{\nu/2}(\sqrt{\nu}ts)(\sqrt{\nu}ts)^{\nu/2}}{2^{\nu/2-1}\Gamma(\nu/2)},\ \forall t\ge 0.
\widetilde{(\xi^0)}_l = e^{i\beta_l^+\tau}\widetilde{(\xi^{0,+})}_l - e^{i\beta_l^-\tau}\widetilde{(\xi^{0,-})}_l , \widetilde{(\xi^n)}_l = e^{i\beta_l^+\tau}\widetilde{(\xi^{n,+})}_l - e^{i\beta_l^-\tau}\widetilde{(\xi^{n,-})}_l - e^{\frac{i\tau}{\alpha}}(\widetilde{(\xi^{n-1,+})}_l - \widetilde{(\xi^{n-1,-})}_l),n\ge 1,
w^t_2 (x) = -\int \frac 1 t \bigg(\frac{\partial_Nv}{N\cdot \nu^0}\, \mathcal{H}^{n-1}\restriction_{\Gamma^0}- \frac 1 t \Delta h^0 \, \chi_{\Omega^0\setminus \Omega^t}\bigg) (dy) P(x-y).
\begin{tabular}{llllllll}$N\ \D_{p}-\mathrm{branes}:$ 0 1\thinspace \thinspace 2 3$\cdots p$ \quad $-$ \quad $-$ \quad $-$ ,$\quad \mathrm{with}\B_{12}$ $\quad O_{p+2}-\mathrm{plane}:$ 0 1$\,-$ 3$\cdots p$ $p+1$ $p+2$ $p+3$ .\end{tabular}
\sum_{1 \le i \le r'}\left(d(v,u_i)-1\right)+\sum_{r'-r+1 \le i \le r}\left(d(u_i,v)-1\right) \ge \sum_{1 \le i \le r'}(i-1) +\sum_{r'-r+1 \le i \le r}(r-(i-1)-1) &=\frac{r'(r'-1)}{2}+\frac{(2r-r')(2r-r'-1)}{2}&=\frac{(r'^2+(2r-r')^2)}{2}-r & \ge r^2-r>(r-1)^2.
p_2(x) = \, x^8-4\left((16(\mu^2+\nu^2)-16\mu-4)(\mu^2+\nu^2)+16\nu^2+4\mu+1\right)x^6+ \, + \left((128\mu^4-128\mu^3-32\mu^2+32\mu+6)+(-768\mu^2+384\mu+32)\nu^2+128\nu^4\right)x^4 \, - 4\left((16(\mu^2+\nu^2)-16\mu-4)(\mu^2+\nu^2)+16\nu^2+4\nu+1)\right)x^2+1.
\psi_1(u,v,w)= \alpha(w/u)+u(1+\beta u), \psi_2(u,v,w)=\alpha (w/v)+u(1+\beta u), \psi_3(u,v,w)= \alpha (w/v)+v(1+\beta v)\psi_4(v,w)=\alpha (w/u)+v(1+\beta v).
{\cal F}_0 = -\frac{{\rm ln}Z_0}{\beta V}= \frac{1}{\beta V}\sum_n\sum_{\bf p}{\rm ln}\left(\frac{\beta M^2}{\rho}\right) + 2T\int\frac{d^2p}{(2\pi)^2}\left[\beta\omega+ ln(1-e^{-\beta\omega})\right]\;,
\begin{aligned}Q_0= (1+\alpha)\Phi(\frac{\partial G}{\partial \lambda_1}\lambda_1^2+\frac{\partial G}{\partial \lambda_2}\lambda_2^2)+\kappa(1-\alpha)\Phi(\frac{\partial G}{\partial \lambda_1}+\frac{\partial G}{\partial \lambda_2}),\end{aligned}
\Phi_{\mathrm{trial},s}:= \frac{\big(\mathbb{U}_\mathrm{left}\mathbb{U}_\mathrm{right}\Omega\big)_{s}}{\sqrt{\sum_{s=0}^N\left\|\big(\mathbb{U}_\mathrm{left}\mathbb{U}_\mathrm{right}\Omega\big)_{s}\right\|^2}}\in\mathcal{F}_\perp.
A=max \left\{{a+2\choose 2}-{d_1+1\choose 2},0\right\}A=max \left\{{a+2\choose 2}-{d_1+1\choose 2}-{d_2+1\choose 2}+{d_1+d_2-a\choose 2},0\right\}A=max \Big\{{a+2\choose 2}-{d_1+1\choose 2}-{d_2+1\choose 2}-{d_3+1\choose 2}+{d_1+d_2-a\choose 2}+ &+{d_2+d_3-a\choose 2}+{d_3+d_1-a\choose 2}-{d_1+d_2+d_3-2a-1\choose 2},0\Big\}
W_{(m)1/4, ir}(y)= \Big(\frac{\Gamma(-2ir)}{\Gamma(1/2-\mu-(m)1/4)}y^{1/2+ir}+\frac{\Gamma(2ir)}{\Gamma(1/2+2ir-(m)1/4)}y^{1/2-ir}\Big)(1+O(t^{-1})).
\bigg|\bigg(\bigcup_iW_i\bigg)\bigcap \{x \in K : |Bf_1(x)| > \lambda/2\}\bigg| ~=~ \bigg|\bigcup_iW_i\cap \{x \in K : |Bf_1(x)| > \lambda/2\}\bigg| ~\leq~ \bigg|\bigcup_iW_i\bigg| ~ \leq ~\sum\limits_i |W_i| ~ \leq~ \|f\|_{L^1(K)}~ \lambda^{-1}.
\displaystyle\frac{2\nu(\lambda_{2}+\lambda_{1})}{\lambda_{2}}b_{1}^{2}=-2c\nu\lambda_{1}(\lambda_{2}-\lambda_{1})+2\nu(1-\nu^{2})-\displaystyle\frac{4\nu^{3}(\lambda_{2}-\lambda_{1})}{\lambda_{2}}-\displaystyle\frac{4c\nu^{3}(1-\nu^{2})}{\lambda_{2}(\lambda_{2}-\lambda_{1})}.
\delta ( \Delta \beta ) = - \frac{2 i}{\lambda^{2}}\int_{2 \Delta \alpha}^{\pi - 2 \bar{\alpha}}d \eta \, \operatorname{l o g}\left[ \frac{e^{i \eta + 2 \Delta \beta}- 1}{e^{i \eta}- e^{2 \Delta \beta}}\right] .