harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
m \simeq \frac{\alpha}{\pi^{2}}\int d q_{3}d q_{4}\int_{0}^{\infty}d r^{2}\mathrm{e}^{- r^{2}}\frac{1}{2 r^{2}+ L^{2}( q_{3}^{2}+ q_{4}^{2})}\frac{m}{m^{2}+ ( q_{3}^{2}+ q_{4}^{2})}
f_{Y}(y)=\frac{\delta^{r\sigma^{-1}}}{\chi(\sigma^{-1})}I^{n,m}_{q,p}\Bigg[\delta^{r\sigma^{-1}}y\left|\begin{matrix}(1-b_j+r\sigma^{-1}B_j-\sigma^{-1}B_j,-r\sigma^{-1}B_j,\beta_j)_{1,q}(1-a_i+r\sigma^{-1}A_i-\sigma^{-1}A_i,-r\sigma^{-1}A_i,\alpha_i)_{1,p}\end{matrix}\right.\Bigg],\ \y>0,
\left[\begin{matrix}J_i P_{k}+J_{i-1}P_{k}J_i Q_{k}+J_{i-1}P_{k}J_i Q_{k}+J_{i-1}P_{k}J_i P_{k}+J_{i-1}P_{k}\end{matrix}\right]=\left[\begin{matrix}2^{i-1}P_{k}J_i Q_{k}+J_{i-1}P_{k}J_i Q_{k}+J_{i-1}P_{k}2^{i-1}P_{k}\end{matrix}\right]
|f(x,t)| = \big|\int_{| \tau - |\xi|^2| \ \le \frac{1}{100N^2},\; 1\le |\xi|\le 1 + \frac1{10N^2}}e^{2\pi i (x\cdot \xi + t \tau)}\, d\xi d\tau\big| &= \big|\int_{\xi: 1\le |\xi| \le 1+ \frac1{10N^2}}e^{2\pi ix\cdot \xi}\big(\int_{\tau: |\tau-|\xi|^2|\le \frac{1}{100N^2}}e^{2\pi i t(\tau - 1)}\, d \tau \big)\, d\xi\big|.
m(\mathfrak{L};r,s):=\frac{1}{2}\int_{M\langle r\rangle}\log \frac{1}{|\mu|_{\mathfrak{h}}^{2}}\sigma-\frac{1}{2}\int_{M\langle s\rangle}\log \frac{1}{|\mu|_{\mathfrak{h}}^{2}}\sigma.
\mathcal{E}^\alpha_{w,T}:=\Big\{f\in C([0,T]:\dot{H}^{N_0+\alpha,p_1}):\,\|f\|_{\mathcal{E}^\alpha_{w,T}}:=\sup_{t\in[0,T]}\|f(t)\|_{\mathcal{E}^\alpha_{w}}\Big\},
e T_{\mu \nu}={\tilde{\overline{{\Psi}}}_{L_{0}}}{\gamma_{\mu}}i \left( D_{\nu}- \frac{1}{2}\Gamma_{\alpha \nu}^{\alpha}\right){\tilde{\Psi}}_{L_{0}}- \frac{i}{2}g_{\mu \nu}\partial_{\alpha}({\tilde{\overline{{\Psi}}}}_{L_{0}}\gamma^{\alpha}{\tilde{\Psi}_{L_{0}}}) . \nonumber \,
F\circ G\,(a_1, \ldots, a_{p+q-1})= \sum_{1\leq i \leq p}(-1 )^{(i-1)(q-1)}F(a_1, \ldots a_{i-1}, G(a_i, \ldots a_{i+q-1}), a_{i+q}, \ldots , a_{p+q-1}),
\delta V ( R ) = - \frac{2 ( D - 2 ) \beta^{2}}{M^{4}}\int_{0}^{\infty}d y y^{3}\bar{\Omega}\frac{\left( e^{- R y}- e^{- R \bar{\Omega}}\right)^{2}}{\left( 1 - e^{- 2 R y}\right) \left( 1 - e^{- 2 R \bar{\Omega}}\right)}< 0 .
e^{-\theta j(K-k)c}< \sum_{s=1}^S w_s e^{-\theta r_s}\Rightarrow & \Lambda_S(-\theta, \mathbf{H})>\Lambda_S (-\theta, \mathbf{F}) \forall \theta \in [\theta^*(\mathbf{\Phi}), \theta^*(\mathbf{D})].
\lambda_{(0) \alpha}^A = \frac{1}{2}{\overline F}_{\mu\nu}\sigma_\alpha^{\mu\nu\,\beta}\frac{1}{\sqrt{\rho_0}}\left( \rho_0\eta_\beta^A + (x-x_0)_\mu \sigma^\mu_{\beta{\dot\beta}}{\overline \xi}^{\dot \beta A}\right) \, .
f_{-}^{\prime\prime}(0):=\lim_{t\searrow0,v\in\partial f(-t)}\frac{1}{-t}[v-f^{\prime}(0)]\mbox{and}f_{+}^{\prime\prime}(0):=\lim_{t\searrow0,v\in\partial f(t)}\frac{1}{t}[v-f^{\prime}(0)].
A(p, p)e^{\alpha q}*B(p,p)e^{\beta q}=A(p_0,\bar p_0)e^{\alpha(q_0+\bar q_o)}*B(p_0,\bar p_0)e^{\beta(q_0+\bar q_o)}|_{p_0=\bar p_0=p, q_0+\bar q_0=q}.
S = \frac i{4 \pi \alpha^{\prime}}\int \left\{g_{i \bar{\jmath}}( \partial x^{i}\bar{\partial}x^{\bar{\jmath}}+ \bar{\partial}x^{i}\partial x^{\bar{\jmath}}) - i B_{i \bar{\jmath}}( \partial x^{i}\bar{\partial}x^{\bar{\jmath}}- \bar{\partial}x^{i}\partial x^{\bar{\jmath}}) \right\}\, d^{2}z .
Q_{+}:=\tfrac1r ( (2 ( \gamma + k (\gamma-1))\bar{\rho}^{\gamma-2}\bar{\rho}_{r}d + (2 (1+k) \bar{\rho}^{\gamma-1}) d_r) + &+((\gamma+k(\gamma-1))((\gamma-2)\bar{\rho}^{\gamma-3}\bar{\rho}_r^2 + \bar{\rho}^{\gamma-2}\bar{\rho}_{r r})d +(2+2k) (\gamma-1) \bar{\rho}^{\gamma-2}\bar{\rho}_r d_r + (1+k) \bar{\rho}^{\gamma-1}d_{r r}.
S^{(\mathrm{q})}_{\mathrm{BI-YM}}= \sqrt{\,{N\, V\over 4g_0^2}\, \mathbf{F}_{cd\, \mu\nu}^{(0)}\,\mathbf{F}^{(0)}{}^{cd\,\mu\nu}\,}\equiv \sqrt{\,{N\, V\over 4g_0^2}\mathrm{Tr}\, \mathbf{F}_{\mu\nu}^{(0)}\, \mathbf{F}^{(0)}{}^{\mu\nu}\,}
{\|}\,{(\cal{A}}_1)^n\,{\|}=(ig)^n\int{\sf D}(1,\cdots,n)\,\psi^{{\alpha_1}}_{(1)i_1}(1)\;\psi^{{\alpha_2}}_{(1)i_2}(2)\;\cdots\psi^{{\alpha_n}}_{(1)i_n}(n)\;\Pi_{i_1}^{\alpha_1}(1)\;\Pi_{i_2}^{\alpha_2}(2)\;\cdots\;\Pi_{i_n}^{\alpha_n}(n)\;,
\sup_{\substack{\psi\in\Psi_{Q,b}\psi\circ L = \psi_1 +\psi_2}}- \beta^c (0,\psi) =\sup_{\substack{\psi\in \Psi_{Q,b}\psi\circ L = \psi_1 +\psi_2}}-\left(f+\psi_1 \right)^{*}_{\Phi_A}\left( -\psi_2\right)-g^{*}_{\Psi_{Q,b}}\left(\psi\right).
0=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\sin{(\alpha_{j}-\alpha_{i})}}{(1-\cos{(\alpha_{j}-\alpha_{i})})^{\frac{3}{2}}(2-\sigma r^{2}(1-\cos{(\alpha_{j}-\alpha_{i})})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\},
\int_{B_R(\xi)}U_{\delta,\xi}^q=\begin{cases}O(\delta^{q\frac{n-2}{2}}) 0<q<\frac{n}{n-2}, O(\delta^\frac{n}{2}|\log \delta|) q=\frac{n}{n-2}, O(\delta^{n-q\frac{n-2}{2}}) \frac{n}{n-2}<q<\infty, \end{cases}
\langle m^2 \rangle ={\tilde\Sigma\over\tilde\rho}={\left( e^{2h}- 1 \right)^2 \left( 3 + (2+\sqrt{7})e^{2h}\right)^2 \over \left(1 + (-1 + \sqrt{7})e^{2h}+ (2+\sqrt{7})e^{4h}\right)^2}\ ,
E={1\over 2}m_f\dot{x}^2+V(x);\ \ \ \V(x)=E[1-{{1+{E\over{2m_f}}H_F}\over{(1+{E\over{m_f}}H_F)^2}}]+{{J^2_{\parallel}}\over{2m_fx^2}}{1\over{(1+{E\over{m_f}}H_F)^2}}.
X(s)-X^{\epsilon}(s) =\int_0^s f(u,X_u)-f(u,X_u^{\epsilon}) du+ \int_0^s g(u,X_u)-g(u,X_u^{\epsilon})dW(u) &+\int_0^s \int_{\mathbb R_0^d}(h_0(u,X_u)-h_0(u,X_u^{\epsilon}))\lambda(z)+ h_0(u,X^{\epsilon}_{u})\lambda_{\varepsilon}(z) \tilde{N}(du,dz) &- \int_0^s h_0(u,X_{u}^{\epsilon})\Lambda_p(\epsilon) dB(u), X_0-X_0^{{\epsilon}}= 0.
\left( \begin{array}{c}P_\lambda-z \gamma_1 \end{array}\right)u= \left( \begin{array}{c}v v_0 \end{array}\right)\;\;\Rightarrow\;\;\|u\|_{B_{z,\lambda,r}}\leqslant C(\|v\|_{L^2_r}+\langle\lambda-\Re z\rangle^{1/4}|v_0|).
\begin{array}{r c l r c l}{{A_{\mu}^{\prime}}}&{{=}}&{{A_{\mu}\, ,}}&{{( k^{2})^{\prime}}}&{{=}}&{{k^{2}\, ,}}\\{{}}&{{}}&{{}}\\{{B_{\mu}^{\prime}}}&{{=}}&{{B_{\mu}- \beta V_{\mu}+{\textstyle \frac{1}{2}}\beta^{2}A_{\mu}\, , \hspace{1 . 2 c m}}}&{{\ell^{\prime}}}&{{=}}&{{\ell + \beta \, ,}}\\{{}}&{{}}&{{}}\\{{V_{\mu}^{\prime}}}&{{=}}&{{V_{\mu}- \beta A_{\mu}\, .}}&{{}}&{{}}&{{}}\\ \end{array}
\frac{\xi|\xi|^{-1}\varphi_t(\xi) \chi(\xi-\eta,\eta) - \eta|\eta|^{-1}\varphi_t(\eta) \chi(\eta-\xi,\xi)}{|\xi|^{3/2}-|\xi-\eta|^{3/2}- |\eta|^{3/2}}= O \big( |\eta|^{-3/2}\mathbf{1}_{[2^2,\infty]}(|\eta|/|\xi-\eta|) \mathbf{1}_{[(1+t)^{-2},0]}(|\xi|) \big) ,
S_{\^{1}T^{\ast}TG}=\left\{\left( g,\xi ,T_{e}^{\ast}R_{g}\frac{\delta\bar{L}}{\delta g}+ad_{\xi}^{\ast}\frac{\delta \bar{L}}{\delta \xi},\frac{\delta \bar{L}}{\delta \xi}\right) \in \^{1}T^{\ast}TG:\left(g,\xi \right) \in G\circledS \mathfrak{g}\right\}
S_1(z, q, N) =\frac{(1-zq^2)(1-zq^{2N+1})}{(1-zq)(1-zq^{2N+2})}\sum_{n=0}^{N}\frac{(q;q^2)_n(q^{-2N};q^2)_nq^{2n}}{(zq^3;q^2)_n(q^{-2N}/z;q^2)_n}&=\sum_{n=0}^{N}\frac{(q;q^2)_n(zq^2;q^2)_{N-n}(q^2;q^2)_N(zq^2)^n}{(zq;q^2)_{n+1}(zq^2;q^2)_{N+1}(q^2;q^2)_{N-n}},
R_{c}\simeq \frac{1}{4}\ \bigg [ \frac{8 \, ( J_{x}^{2}- J_{z}^{2})}{J_{x}( J_{x}- J_{y})}\bigg ]^{\frac{1}{2 - 2 \eta}}\, \quad \qquad \Big ( \eta = \frac{1}{\pi}\, \operatorname{a r c c o s}( J_{z}/ J_{x}) \, \Big )
f^{\#}(t,k) = f^{\#}(0,k) \pm i\int_{0}^{t}\iiint e^{is(-k^2+\ell^2-m^2+n^2)}f^{\#}(s,\ell)\overline{f^{\#}(s,m)}f^{\#}(s,n)\mu^{\#}(k,\ell,m,n)\,dndmd\ell ds
b_n^{(\theta)}= \frac{1}{x_1 - x_2}\big( T_{q, x_2}b_n^{(\theta - 1)}- T_{q, x_1}b_{n-1}^{(\theta - 1)}\big), n = 1, 2, \dots, \theta-1, b_0^{(\theta)}= \frac{1}{x_1 - x_2}T_{q, x_2}b_0^{(\theta-1)}, b_{\theta}^{(\theta)}= \frac{1}{x_2 - x_1}T_{q, x_1}b_{\theta-1}^{(\theta-1)}.
W^{x_0}_{1+s}(r,u)=\frac{1}{r^n+1}\int_{B_r(x_0)}|\nabla u|^2 \,|x_n|^a\, dx - \frac{1+s}{r^{n+2}}\int_{\partial B_r(x_0)}u^2\,|x_n|^a\, d\mathcal{H}^{n-1}.
\left|\widetilde{(e_M^{q+1})}_l\right|^2 \le 6\biggl( \left|\widetilde{(e_M^{0,+})}_l\right|^2 + \left|\widetilde{(e_M^{0,-})}_l\right|^2 \ + (q+1) \sum_{k=0}^q \left(\left|\widetilde{(\xi^{k,+})}_l\right|^2 + \left|\widetilde{(\xi^{k,-})}_l\right|^2 + \left|\widetilde{(\eta^{k,+})}_l\right|^2 + \left|\widetilde{(\eta^{k,-})}_l\right|^2\right) \biggr).
\big|\{m<0:f(m,\sigma(x))\ge -n(x)\}\big| - \big| \{m<0 : f(m,\sigma(x))\ge 0\}\big| = \big|\{m<0:0>f(m,\sigma(x))\ge -n(x)\}\big|, \big|\{n\ge 0:f(n,\sigma(x))< 0\}\big| - \big| \{n\ge 0: f(n,\sigma(x))< -n(x)\}\big| = \big|\{n\ge 0:0>f(n,\sigma(x))\ge -n(x)\}\big|.
R_n(x, y) = \ \textup{P.V.}\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{du}{u^2}\ \times (2\pi)^{-\frac{n+1}{2}}\int_{-\infty}^{\infty}\cdots \int_{-\infty}^{\infty}e^{-\frac{\|\textbf{c}\|^2}{2\sigma^2}}\left(\cos(y G_n(x))-\cos(y G_n(x))\cos(u G_n'(x))\right)\,dc_0dc_1\cdots dc_n,
\lim_{s\rightarrow0}\Gamma(s)\left( (cz+d)^{-s}G(Vz,s;A,B)-G(z,s;B_{-b},A_{-c})\right) & =2\pi is\left( d,c;B_{b},A_{c}\right) -\frac{\pi i}{c(cz+d)}B_{0}(B)B_{2}(A)-2\pi i\frac{cz+d}{c}B_{0}(A)P_{2}\left( 0,B_{b}\right).
\gamma ((\mathbb{R}^2, \|\cdot\|_{|\Omega|})) = \sup\{\langle |C|,|D|\rangle :\|C\|_{((\mathbb{R}^2,\|\|_{|\Omega|})\rightarrow(\mathbb{R}^2,\|\cdot\|_{|\Omega|})^*)}\leq 1,\|D\|_{((\mathbb{R}^2,\|\cdot\|_{|\Omega|})^*\rightarrow(\mathbb{R}^2,\|\cdot\|_{|\Omega|}))}\leq 1,C\geq 0,D\geq 0\}.
P_{out}= 1-\textrm{Pr}\bigg[\bar{\gamma_1}\ge 1,\bar{\gamma_2}\ge 1,...,\bar{\gamma_i}\ge 1,...,\bar{\gamma_N}\ge 1\bigg] &= 1-\textrm{Pr}\bigg[{\underset{i \in \{1,...,N\}}{\textrm{min}}}\{\bar{\gamma_i}\}\ge 1\bigg].
X^{I}= \left( \begin{array}{c c c c}{x_{1}^{I}}&{0}&{\dots}&{0}\\{0}&{x_{2}^{I}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{x_{N}^{I}}\\ \end{array}\right) , \I = 1 , \dots , 2 6 \ ,
| (\bar N_j^{(m)*}- \bar N_j^{(n)*})(t) | \leq \sum_{\ell:|\ell-\bar i(t)| > L - k}|\alpha_{j\ell}| |\bar N_\ell^{(n)*}(t)| &\lesssim \sum_{\ell: |\ell-\bar i(t)|> L -k}\frac{q^{|\ell-\bar i_n(t)|}}{\lambda(\bar I_n(t))}\lesssim \frac{q^L}{\lambda(\bar I(t))}\leq \varepsilon.
z^{(s+1)(p-1)}= \sum_{k=0}^\infty\binom{s+1}{k}(z^{p-1}-1)^k=\sum_{k=0}^\infty\binom{s}{k}(z^{p-1}-1)^k+\sum_{k=1}^\infty\binom{s}{k-1}(z^{p-1}-1)^k = z^{s(p-1)}+(z^{p-1}-1)\cdot z^{s(p-1)}=z^{p-1}\cdot z^{s(p-1)},
\begin{array}{lcl}C( \Lambda ) E_{- \alpha}(\mu) \omega = \frac{2}{\Lambda - \mu}\left( E_{- \alpha}( \Lambda ) h_\alpha (\mu) \right. & \left.- E_{- \alpha}(\mu) h_\alpha ( \Lambda ) \right) \omega +E_{- \alpha}( \Lambda ) C( \Lambda ) \omega\end{array}
\Theta_{n-1}([\xi_n(x),(1,1)]_f) =[\xi_n(x),(1,1)]_g [1,(1,\phi^*_{n-1}(\xi_n(x)))]_g &=[1,(p_n(x),g_n(x))]_g [1,(1,\phi*_{n-1}(\xi_n(x)))]_g
25\sum_{k = 1}^n{( - 1)^{k - 1}F_{mk}{}^4}= \sum_{k = 1}^n{( - 1)^{k - 1}L_{4mk}}& + 4\sum_{k = 1}^n{( - 1)^{k(m - 1)}L_{2mk}}+ 3(( - 1)^{n - 1}+ 1)\,.
\mathcal{F}(1+|x|^2)^{-n}=\frac{2^{\frac{1}{2}-n}\pi^{\frac{1}{2}}}{(n-1)!}e^{-|\xi|}, &\mathcal{F}(1+|x|^2)^{-(n+1)}=\frac{2^{-\frac{1}{2}-n}\pi^{\frac{1}{2}}}{n!}(1+|\xi|)e^{-|\xi|}.
P_{\mu \nu}= \tilde{\delta}_{\mu \nu}- \frac{\tilde{p}_{\mu}\tilde{p}_{\nu}}{\tilde{p}^{2}}, \qquad Q_{\mu \nu}= \frac{p^{2}}{\tilde{p}^{2}}\overline{{{u}}}_{\mu}\overline{{{u}}}_{\nu}
\begin{aligned}S_{n,p}\left(\frac{n-p}{p-1}\right)^{-\frac{n-1}{n}}\left( \int_{B_{R}}\frac{|u(x)|^{p^{*}}}{\left[\log_{{\frac{n-1}{p-1}}}\frac{R}{|x|}\right]^{\frac{p(n-1)}{n-p}}}dx \right)^{\frac{1}{p^{*}}}\le\Biggl( \int_{{B_{R}}}|L_p u(x)|^{p}dx\Biggr)^{{\frac{1}{p}}}\end{aligned}
c_1(\langle \mathcal{K}_1,\phi^*\mathcal{L}_0,\dots,\phi^*\mathcal{L}_{n-1}\rangle(\mathfrak{X}/S))=\deg(\mathcal{K}_1)\cdot c_1(\langle \mathcal{L}_0,\dots,\mathcal{L}_{n-1}\rangle(\mathcal{Y}/S)).
F(t)=\frac{C_{n}}{C_{o}}=1-e^{-\frac{nt}{\overline{t}}}[1+\frac{nt}{\overline{t}}+\frac{1}{2!}(\frac{nt}{\overline{t}})^{2}+...+\frac{1}{(n-1)!}(\frac{nt}{\overline{t}})^{n-1}]
\!\!R^c(u,k) = \frac{1}{v_ut_u^c}\int_{r_{u,k}(\boldsymbol{x})}^{r_{u,k}(\boldsymbol{x}')}\!\!\!\!\!\!w\log\left(1+ \frac{\beta P_t\psi r_{u,k}^{-\alpha}}{wN_0}\right)dr_{u,k},
\delta ~ \psi_{1}= \xi^{z}\partial_{z}\psi_{1}- ( \partial_{z}\xi^{z}) \psi_{1}+ \xi^{z z}( \partial_{z}^{2}\psi_{1}) -{\frac{1}{2}}( \partial_{z}\xi^{z z}) ( \partial_{z}\psi_{1}) + ( ({\frac{1}{6}}\partial_{z}^{2}-{\frac{2}{3}}\tilde{u}_{2}) \xi^{z z}) \psi_{1}\quad ,
\begin{aligned}g_-^{-1}t^n g_+^{-1}=(g_-)_\alpha \, t^{-n}_\alpha \,(g_+)_\alpha=(g_-)_\alpha \,(\alpha)^{-n}\, (g_+)_\alpha =(g_-)_\alpha\,\alpha_-^{-n}\,t^n \,\alpha_+^{-n}\, (g_+)_\alpha = \eta_- t^n\eta_+,\end{aligned}
\overline{V}_t \le e^{-c_1\tilde{\gamma}t}\overline{V}_0+\frac{c_2}{c_1}(1-e^{-c_1\tilde{\gamma}t})+e^{-c_1\tilde{\gamma}t}\int_{0}^{t}\frac{\tau}{N}\tilde{\gamma}e^{c_1\tilde{\gamma}s}\sum_{i=1}^N{dB_{i,s}}^Te_{i,s}.
\int^{T}_{0}\int_{\Omega}\big( \rho u\cdot\partial_t\phi+(\rho u\otimes u):\nabla_x\phi+\rho^\gamma{\rm{div}}_x\phi-\mathbb{S}(\nabla_x u):\nabla_x\phi &+(j-nu)\cdot\phi\big)\,dxdt+\int_{\Omega}\rho_0u_0\cdot \phi(0,x)\,dx=0,
f= \ |\psi|^2+\operatorname{div}\!\left[\psi^*\Bbb{J}\left({\mathcal{A}}-i\hbar\nabla\right)\psi\right] = \ |\psi|^2+\operatorname{div}(|\psi|^2\Bbb{J}{\mathcal{A}})+i\hbar\{\psi,\psi^*\}\,.
f(\{x_{i,r}\}_{i\in I}^{1\leq r\leq k_{i}})=0 x_{i,s_{1}}=v_{i}^{2}x_{i,s_{2}}=\cdots=v_{i}^{-2a_{ij}}x_{i,s_{1-a_{ij}}}=v_{i}^{-a_{ij}}x_{j,r}
P e^{i \oint_{\sigma_{z}}A}= U ( z ) P e^{i \oint_{\sigma_{z}}A^{U}}U^{- 1}( z ) = U ( z ) P e^{i \oint_{\sigma_{z}}d \sigma^{\mu}\int_{0}^{1}d s s \sigma^{\nu}U^{- 1}( s \sigma ) F_{\nu \mu}( s \sigma ) U ( s \sigma )}U^{- 1}( z ) .
\frac{d}{dt}\|x_\nu^{(\frac{N-1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+3)}(t)\|^2 \leq-\alpha\nu\|x_\nu^{(\frac{N-1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+3)}(t)\|^2+\nu\|x_\nu^{(\frac{N-1}{2}-1)}(t)-x_\nu^{(\frac{N+1}{2}+4)}(t)\|^2 &+\nu\|x_\nu^{(\frac{N+1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+2)}(t)\|^2+\frac{1}{\nu}M_{T_1,T_2}^{\frac{N-1}{2},\frac{N+1}{2}+3,2-\alpha}(\omega)
\begin{aligned}{E_{{\rm{th}}}}\Big{|}_{N_{\rm R}=3}-{E_{{\rm{th}}}}\Big{|}_{N_{\rm R}=2}= \frac{{{\sigma^2}{L_{\rm{R}}}\left({\kappa + 1}\right)}}{{2{N_{\rm{T}}}\kappa{C^2}}}\left({\sqrt{9 + 3\pi \left({K - 1}\right)}-{K\pi}+ \pi - 3}\right) \mathop \le \limits^{\left( a \right)}0, \end{aligned}
K_{12}=\frac{-2a(a-b-c)-(a-b+c)(a+b-c)}{4}K_{13}=\frac{2b(a-b+c)+(a-b-c)(a+b-c)}{4}K_{23}=\frac{2c(a+b-c)+(a-b+c)(a-b-c)}{4}
\begin{aligned}\nabla_{j+i-1, j+i}(\overrightarrow{W}^\ell_{j+1})^2 = \ell^{-1}\overrightarrow{W}^\ell_{j+1}\nabla_{j+i-1, j+i}Z_{i,j}- \ell^{-2}(\nabla_{j+i-1, j+i}Z_{i,j})^2 \end{aligned}
\sum_{s_i\in S_k}\left(N-\xi_j^{(f)}\right)\leq \sum_{i=1}^m V_i^{(f)}\left(\zeta_i^{(f)}\right)<<\sum_{i=1}^m V_i\left(\zeta_j^{(f)}\right)<\sum_{i=1}^m V_i^{opt}(N)
\vartheta((\mathrm{W},\mathrm{X})+\lambda) =\sum_{\alpha\in X_\xi}\Big(\sum_\beta a_{\alpha\beta}b(\lambda,\beta)\mathrm{W}^{\beta}\Big)b(\lambda,\alpha)\mathrm{X}^\alpha &=\sum_{\alpha\in X_\xi}\sigma_\alpha(\vartheta)(\mathrm{W}+c^t(\lambda))b(\lambda,\alpha)\mathrm{X}^\alpha.
\varphi^{(p,q)}\bigl(\rho e^{i\sigma}\bigr)=e^{i(p-q)\sigma}\frac{(-1)^{q+\nu+1}}{q}(1-\rho^2)\rho^m\sum_{j=0}^{\nu}(-1)^j\frac{(j+\nu+m+1)!}{j!(j+m)!(\nu-j)!}\rho^{2j}.
\sum\limits_{\left\{i,j\right\}\in E(G)}\left( x_{i}+x_{j}\right)^{2}=\sum\limits_{\left\{i,j\right\}\in E(G^{\prime})}\left( x_{i}+x_{j}\right)^{2}+\sum\limits_{\left\{i,j\right\}\in E(B)}\left(x_{i}+x_{j}\right)^{2}+\sum\limits_{\left\{i,j\right\}\in E(A,B)}\left(x_{i}+x_{j}\right)^{2}
g(n+1)-h(n+1) =\frac{8(3n^2+5n+1)}{(n+1)^2}-\frac{128(n-1)}{ng(n)}-\frac{16(n^3+2n^2+n+1)}{n(n+1)^2}begin{align*}5pt] =\frac{8(n^3+n^2-n-2)}{n(n+1)^2}-\frac{128(n-1)}{ng(n)}begin{align*}5pt] =\frac{8(n^3+n^2-n-2)g(n)-128(n-1)(n+1)^2}{n(n+1)^2g(n)}.
Z_{\rm orb.}={\rm Tr}_{\rm untw'd}\left({(1+R_{25})\over2}q^{L_0-{1\over 24}}{\bar q}^{{\bar L}_0-{1\over 24}}\right)+{\rm Tr}_{\rm tw'd}\left({(1+R_{25})\over2}q^{L_0-{1\over 24}}{\bar q}^{{\bar L}_0-{1\over24}}\right)
- p_{2}\eta^{2}- q^{2}( \partial_{i}\Pi_{i}+{\frac{\mu}{2}}\epsilon_{i j}\partial_{i}A_{j}) - \bar{C}_{1}\{\eta^{1}, \bar{\chi}_{1}\}C^{1}- \bar{C}_{2}\{\eta^{2}, \bar{\chi}_{2}\}C^{2}.
\int_{B_r(y)}\left(\sum_{i=1}^N \sum_{l=1}^Q \langle D_iu_l, D_iu_l\rangle + \langle A_{u_l}(D_iu_l,D_iu_l), u_l\rangle\right) = \int_{\partial B_r(y)}\sum_{l=1}^Q \langle u_l, \frac{\partial u_l}{\partial r}\rangle,
\Delta_{k-1}\left(\frac{w-t x_1}{w-x_1}\cdots\frac{w-t x_{k-1}}{w- x_{k-1}}\frac{g(x)}{w-x_k}\right) =\frac{1}{x_{k-1}-x_k}(\sigma_{k-1,k}-1)\left(\frac{w-t x_1}{w-x_1}\cdots\frac{w-tx_{k-1}}{w-x_{k-1}}\frac{g(x)}{w-x_k}\right) &=t\times\frac{w-tx_1}{w-x_1}\cdots\frac{w-tx_{k-2}}{w-x_{k-2}}\frac{g(x)}{(w-x_{k-1})(w-x_k)}.
{1\over (\omega_{l_1}+\omega_{l_2})^3}={\omega_{l_1}^3-\omega_{l_2}^3-3 \omega_{l_1}^2\omega_{l_2}+3\omega_{l_1}\omega_{l_2}^2\over \left[({\pi l_1\over a})^2- ({\pi l_2\over a})^2\right]^3},
L = X_{-}(Z-Z')(Z-Z'-1) - P_{1-}N_-(Z-Z'-1) + P_{0-}N_-^2, U = P_{0+}(Z-Z')(Z-Z'-1)+P_{1+}N_-(Z-Z'-1)+X_+N_-^2, E_{+}= P_{1+}(Z-Z'+2)+2N_-X_+, E_{-}= P_{1-}(Z-Z'+2)-2N_-P_{0-}.
P_1(a_1,...,a_n) =(a_2,a_1,...,a_n), P_2(a_1,...,a_n) =(a_2,...,a_n,a_1), H(a_1,...,a_n) =(a_1^{-1},a_2,...,a_n), T(a_1,...,a_n) =(a_1a_2,a_2,...,a_n).
R^{(a)}(\xi, t|\xi|) = \eta(|\xi|^2) \hat{\phi}(\xi) t|\xi|^2 \frac{\hat{\psi}_\xi(t|\xi|)-\hat{\psi}_\xi(0)}{t|\xi|}= \eta(|\xi|^2)\hat{\phi}(\xi) t|\xi|^2 h_\xi( t|\xi|),
\bar{R}^{(d)}_T(0)\leq \sum_{n=0}^{N-1}R^{(d)}_{2^n}(0)+R^{(d)}_C(0) \leq (N+1)R^{(d)}_{2^N}(0) \leq (N+1)R^{(d)}_{T}(0) \leq 2\log_2(2T)T\mathcal{F}(d,\lfloor T^{\frac{1}{d}}\rfloor)r^{(1)}_{\lfloor T^{\frac{1}{d}}\rfloor}(0)
\beta ( p , k ) = \int_{- \infty}^{\infty}d x^{\prime}\frac{q}{2}\frac{A_{p}( 0 , x^{\prime})}{i q \pi \sqrt{\omega_{p}\omega_{k}}}\left\{\dot{\phi}_{1 k}^{+ \ast}( t , x ){\frac{\partial t}{\partial t^{\prime}}}+ \left( \partial_{x}\phi_{1 k}^{+ \ast}( t , x ) \right){\frac{\partial x}{\partial t^{\prime}}}- i \omega_{p}\phi_{1 k}^{+ \ast}( t , x ) \right\}_{t^{\prime}= 0}.
\|h-h_1\|_{L^p(\Omega)}= \|G^TG-G_1^TG_1\|_{L^p(\Omega)}\leq \|G_2\|_{L^{\frac{2p}{2-p}}(\Omega)}(\|G\|_{L^2(\Omega)}+2\|G_2\|_{L^2(\Omega)}) \leq C\|\tau(u)\|_{L^p(\Omega)}(\|\nabla u\|_{L^2(\Omega)}+\|\tau(u)\|_{L^p(\Omega)}) \leq C\|\tau(u)\|_{L^p(\Omega)}(1+\|\tau(u)\|_{L^p(\Omega}).
f(z) =(f_1(z), \ldots, f_p(z), f_{p+1}, \ldots, f_n(z)), T_1(r) := \sum_{j=1}^p T\left(r, e^{f_j}\right), T_2(r) := \frac1{4\pi}\int_{|z|=r}\sum_{j+1}^n |f_j(z)|^2 \, d\theta- \frac1{4\pi}\int_{|z|=1}\sum_{j+1}^n |f_j(z)|^2 \, d\theta, T_{\exp_Af}(r) =T_1(r) + T_2(r).
I m \, I_{1_{\beta}}\approx \frac{- g^{2}}{8}\int \frac{d^{3}l}{( 2 \pi )^{2}E_{1}E_{2}}2 \delta ( E_{1}- E_{2}) \Bigg ( \frac{2}{e^{\beta E_{1}}- 1}+ \frac{2}{( e^{\beta E_{1}}- 1 )^{2}}\Bigg ) .
\nabla f(x^*)d_x = -\left(\sum\limits_{i=1}^m\lambda_i^*\nabla g_i(x^*)+\sum\limits_{i=1}^p\mu_i^*\nabla h_i(x^*) + \sum\limits_{i=1}^n \gamma_i^*e_i^T\right)d_x & = -\nabla\ell(x^*,\lambda^*,\mu^*,\gamma^*))d_x = 0.
(\left<A_n\right>,\left<A_{n+1}\right>) = \sqrt{2^{2n-1}pq\left( \frac{1}{2^{\frac{2^n}{\ln 2}}}\right)^2}&= \sqrt{pq 2^{2n-1-\frac{2^{n+1}}{\ln 2}}}&\le \sqrt{pq 2^{-1-\frac{2}{\ln 2}-n^2\ln2}},
\|\theta\| = \sup_{u \in L^2(G),\ \|u\|_2 = 1}\|\theta u\|_2 =\sup_{u \in L^2(G),\ \|u\|_2 = 1}\|\theta \sum_{i\in \mathcal{I}}u_i\|_2 &\le \sup_{u \in L^2(G),\ \|u\|_2 = 1}\sqrt{\sum_{i \in \mathcal{I}}\|\theta\|^{\prime \,2}\|u_i\|_2^2}= \|\theta\|'.
\lambda^\alpha_a(f_0)\, h_\alpha + \frac{1}{2}\sum_{\alpha\leq\beta}\lambda^{\alpha\beta}_a(f_0)(\partial_\beta h_\alpha+\partial_\alpha h_\beta) = \frac{1}{2}\sum_{\alpha\leq\beta}\lambda^{\alpha\beta}_a(f_0)\,(\delta g_0)_{\alpha\beta}.
\int_{0}^{r_0}|u(r)|^{q(r)}r^{\beta-1}dr = \int_0^{r_0}|u(r)|^{\frac{2\beta}{\beta-2}}|u(r)|^{f(r)}r^{\beta -1}dr &\leq \int_0^{r_0}|u(r)|^{\frac{2\beta}{\beta-2}}\Big( \frac{a r^{2-\beta}}{\beta -2}\Big)^{f(r)/2}r^{\beta-1}dr.
\rho(x)-\Pi E =-p_{2}= \frac{ET_{2}^{2}}{\Pi}\frac{1}{1+\left[1+\frac{E^{2}T_{2}^{2}}{\Pi^{2}(E^{2}-1)}\right]{\rm sinh}^{2}\left(\frac{\sqrt{E^{2}-1}}{T_{0}}x\right)}.
\varphi^{(i)}=\begin{cases}1 (i=0) \nu_n^{-i+1}(\varphi_{T}) \circ \nu_n^{-i+2}(\varphi_{\nu_n^{-}T}) \circ \cdots \circ \nu_n^{-}(\varphi_{\nu_n^{-i+2}T}) \circ \varphi_{\nu_n^{-i+1}T}(i \geq 1).\end{cases}
0\mkern3.5mu\ge\mkern2.5mus_{\mkern-1mu+}y'_-\mkern-1.2mu(s_{\mkern-1mu+})\mkern1.5mu=\mkern1.5mu-1 + \mkern-1mu{{\textstyle{\sum\limits_{i=1}^{n_{{\textsc{g}}}}1}}}\mkern1.5mu=\mkern2.5mun_{{\textsc{g}}}-1.
\sum_{\{\gamma\}\in\mathcal{S}_1'(N)}\left(e^{\frac{1}{2}\ell_{\gamma}}+1\right)^{-1}+\sum_{\{\alpha,\beta\}\in\mathcal{S}_2'(N)}\left(e^{\frac{1}{2}(\ell_{\alpha}+\ell_{\beta})}+1\right)^{-1}=\frac{1}{2}.
d s^{2}: = B_{\mathrm{H}}[ \sigma ( r - r_{\mathrm{H}}) ]^{2 ( 1 - \alpha )}\bar{g}_{\mu \nu}d x^{\mu}d x^{\nu}+ \frac{d r^{2}}{B_{\mathrm{H}}( r - r_{\mathrm{H}})^{2}}+ r_{\mathrm{H}}^{2}d \Omega_{n - 1}^{2}.
q(h) = \exp(h \Omega) q(0) + h \phi_1( h \Omega) \big( \dot{q}(0) - \Omega q(0) \big) + \tfrac{h^2}{m}\phi_2(h \Omega) F(q(0)) + \mathcal{O}(h^3) = q(0) + \tfrac{h}{m}\phi_1(h\Omega) p(0) + \tfrac{h^2}{m}\phi_2(h \Omega) F(q(0)) + \mathcal{O}(h^3).
\sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ even}}}^m (-1)^i{m \choose i}(m+i) \sum\limits_j{\frac{m+i}{2}-1\brace j-1}x^{2j-2}\quad+ \sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ odd}}}^m (-1)^i{m \choose i}(m+i) x^{m+i-1}= 0.
ds_+(j',u)(0,\xi)=\tfrac{d}{dl_1}s_+(j',u_{l_1})|_{l_1=0}= -\frac{d r_{j'}(z)\circ (d\phi_u (z))^{-1}(\xi^{s,t}(z))}{dr_{j'}(z)\circ (d\phi_u(z))^{-1}(\partial_s)},
F \equiv \frac{\xi^{2}m^{4}( \xi m^{2}+ M^{2})}{( \xi m^{2}- M^{2})^{3}}\log ( \sqrt{\xi}m / T ) - \frac{M^{2}( M^{4}+ 4 \xi^{2}m^{4}- 3 \xi m^{2}M^{2})}{( \xi m^{2}- M^{2})^{3}}\log ( M / T ) .
S_{(P,Q)}=T_{(P,Q)}{\Omega}_{p}{\int}_{0}^{t_{+}}dt{\rho}^{p}\sqrt{1+{\dot{\rho}}^{2}}-\frac{Q{\Phi}}{p+1}{\Omega}_{p}{\int}_{0}^{t_{+}}dt{\rho}^{p+1}
ES_{n+1}^{2l}=EX_1^{2l}+2l \sum_{1\le i\le j \le n}E Z_i X_{j+1}+\sum_{j=1}^n \sum_{r=2}^{2l}{2l\choose r}ES_j^{2l-r}X_{j+1}^r &=2l \sum_{1\le i \le n}E Z_i W_i+\sum_{j=1}^n \sum_{r=2}^{2l}{2l\choose r}ES_j^{2l-r}X_{j+1}^r
[a,b] \overset{\hphantom{(2.7)}}{=}\partial_1(a) \cdot \partial_2(b) - \partial_2(a) \cdot \partial_1(b) \overset{\eqref{eq:qadm1}}{=}-\eth_2(\partial_1(a) \cdot b) + \eth_1(\partial_2(a) \cdot b) + \eth_2(\partial_1(a)) \cdot b - \eth_1(\partial_2(a)) \cdot b.
|t| \leq \frac{1}{p}\left( \sum_{(i,j)\in \Delta \setminus \{(k,\ell)\}}|u_{i,j}| H^{i+j}+|v|H^{k+\ell}\right) \ll_d\frac{(H/\sqrt{N})^{d(\delta-1)}H^d}{p}\ll_d \frac{(H/\sqrt{N})^{d(\delta-1)}H^d}{H^{\left( (d-1/2) \delta +1/2 \right)}}=\frac{H^{(\delta-1)/2}}{N^{d(\delta-1)/2}}.
W^{M}= \frac 1 6 \sum_{\pm}\left\{c_{0}^{M}+ c_{2}^{M}\sinh^{2}( r_{\pm}/ \ell ) + c_{4}^{M}\sinh^{4}( r_{\pm}/ \ell ) \right\}+ \frac{1}{8}\sum_{n = 1}^{\infty}\frac{1}{n ( \sinh n L )^{4}}.
q_0^{(1)}(\ell,\omega) [g,h] = - \ell^2 \cdot \left(\frac{2\pi^2}{\alpha^3}\right) \int_{\{|\xi| \le 1\}}\frac{1}{|\xi|^2- \omega +\pi^2/\alpha^2}\cdot \hat g(\ell \xi) \; \overline{h(\ell \xi)}\ \mathrm{d}\xi + \mathcal O(\ell^2) .
q^{\alpha^{2}/2}S_{n}\left(xq^{\alpha-1/2};q\right)=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\left(xe^{iu};q\right)_{n}}{\left(q;q\right)_{n}}\exp\left(\frac{u^{2}}{\log q^{2}}+i\alpha u\right)du
\sum_{m=-(s-1)}^{s-1}\frac{\zeta_{0,m}}{\zeta}=\frac{\sin(\pi s p/p')}{\sin(\pi p/p')}\ , \sum_{n=-(r-1)}^{r-1}\frac{\eta_{0,n}}{\eta}=\frac{\sin(\pi rp'/p)}{\sin(\pi p'/p)}\ ,
f^L(\tilde{\vec x}) =\int_0^{\|\tilde{\vec x}\|_\infty}f(V_t^+(\tilde{\vec x}),V_t^-(\tilde{\vec x}))d t \&=\int_0^\alpha f(V^+_0,V^-_0)dt+\int_\alpha^{\|x\|_\infty+\alpha}f(V_t^+(\tilde{\vec x}),V_t^-(\tilde{\vec x}))d t &=\alpha f(V^+_0,V^-_0)+\int_0^{\|x\|_\infty}f(V_t^+({\vec x}),V_t^-({\vec x}))dt &=\alpha f(V^+_0,V^-_0)+f^L(\vec x).
v_n (s,\omega(n);\alpha,\lambda_j) := \frac{e(\lambda_j n)\omega(n)}{(n+\alpha)^{s}},\omega(n) \in \gamma, &\underline{v_n}(s,\omega(n)) := \bigl( v_n(s,\omega(n);\alpha,\lambda_1),\ldots, v_n (s,\omega(n);\alpha,\lambda_m) \bigr).
S_{z}^{D}= \int d^{4}x \left( \int d^{4}\theta K ( z , z^{\dagger}) \int d^{2}\theta l z \left\{w - U ( A_{D}) + \frac{\Lambda^{4}z l}{w}f \left( \frac{A_{D}^{2}}{w}, \frac{\Lambda^{4}}{w^{2}}\right) \right\}+ h . c . \right)
\mathbf{D}_{i,j}=\left(\begin{array}{ccccccc}0 c_{1}c_{1}\ddots \ddots & \ddots 0 c_{s-1}& c_{s-1}-2e c_{s}& c_{s}0 \ddots & \ddots \ddots c_{m-1}& c_{m-1}0\end{array}\right).
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E)=\Bigr(\sum\limits^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi}\Bigr)\lim_{t\rightarrow0^+}\int_X\sum^n_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}\alpha^+_\ell(x)-\mathrm{Tr\,}\alpha^-_\ell(x)\Bigr)dv_X(x).
\mathfrak{S}_{\delta}^{\mu_2}= - \frac{1}{2\pi}\sum_{L}\big(\mathfrak{S}_{\delta, \flat}^{\mu_2}(L) + \overline{\mathfrak{S}_{\delta, \flat}^{\mu_2}(L)}\big)+ \sum_{L}\mathfrak{S}_{\delta, \dagger}^{\mu_2}(L),