harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma^{0}= \left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) , \quad \gamma^{1}= \left( \begin{matrix}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{matrix}\right) , \quad \gamma^{5}= \gamma^{0}\gamma^{1}= \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) .
L_{S C S}^{\pm}= S T r \int_{X}d^{3}x \varepsilon^{i j k}\bigg ({{\bf A}^{\pm}}_{i}\partial_{j}{{\bf A}^{\pm}}_{k}+ \frac{2}{3}{{\bf A}^{\pm}}_{i}{{\bf A}^{\pm}}_{j}{{\bf A}^{\pm}}_{k}\bigg ) ,
Y(t):=\left(\|w_0\|_2^2+C\|f\|_{L^2(0,t; H^{-1}(\Omega))}^2\right)e^{CNt}, &N:=1+|h|_\infty^2\|u_s\|_\infty^2+\|\widetilde U_{\varepsilon_0}\|_{L^\infty(0,\infty; L^\infty(\Omega))}^2,
( M_{\mathrm{P V}})^{3 T_{G}- \sum T ( R_{i})}\left( \frac{1}{g^{2}}\right)^{T_{G}}\! \exp \left( - \frac{8 \pi^{2}}{g^{2}}\right) \prod_{i}\left( Z_{i}\right)^{- T ( R_{i})}= \mathrm{c o n s t}.
|\Omega|^2=\left(\int_{\Omega}u_0^{-p/2}(x) u_0^{p/2}(x)dx\right)^2\le \int_\Omega u_0^{-p}(x) dx\cdot\int_\Omega u_0^p(x)dx\le M_1^*\int_\Omega u_0^p(x)dx.
\widetilde{\nabla}_XY =\sum_{i=1}^k(-1)^{k-i}X[i]\wedge \widetilde{\nabla}_{X_i}Y &=\sum_{i=1}^{k}\sum_{j=1}^l (-1)^{k-i}(-1)^{j-1}X[i]\wedge \widetilde{\nabla}_{X_i}Y_j\wedge Y[j] &=\sum_{i=1}^{k}\sum_{j=1}^l (-1)^{i+j}\widetilde{\nabla}_{X_i}Y_j\wedge X[i]\wedge Y[j].
[ \phi_{R}, P^{-}] ={\frac{g^{2}L}{4 \pi^{2}}}\sum_{n}\left[ \phi_{R},{\frac{1}{\left( n + C_{0}^{3}\right)^{2}}}\right] \left\{C_{n}^{+}, C_{- n}^{-}\right\}+ \dots \; ,
\mathbf{V}(\mathbf{E}\mathbf{x}_{k+1})-\mathbf{V}(\mathbf{E}\mathbf{x}_{k})\leq \sum_{i,j=1}^2\|u_{i,k}^j\|^2-\|y_{i,k}^j\|^2 \leq \|u_{1,k}^2\|^2+\|u_{2,k}^2\|^2-\|y_{1,k}^2\|^2-\|y_{2,k}^2\|^2 =\|\mathbf{u}_k\|^2-\|\mathbf{y}_k\|^2,
E ( \sigma , \sigma^{\prime}: \gamma ) \equiv \bar{\epsilon}( \sigma - \sigma^{\prime}: \gamma ) + \bar{\epsilon}( - \sigma + \sigma^{\prime}: \gamma ) + e^{- 2 \beta_{0}}\bar{\epsilon}( \sigma + \sigma^{\prime}: \gamma ) + e^{2 \beta_{0}}\bar{\epsilon}( - \sigma - \sigma^{\prime}: \gamma ) .
\int_{\mathbb{R}^m}u^2dV_g =\int_{\mathbb{R}^m}u^2e^{-\frac{m|x|^2}{m-2}}dV_{g_0}&\le \frac{2(m-2)}{m}\int_{\mathbb{R}^m}|\nabla u|_{g_0}^2e^{-\frac{m|x|^2}{m-2}}dV_{g_0}&\le \frac{2(m-2)}{m}\int_{\mathbb{R}^m}|\nabla u|_{g_0}^2e^{\frac{|x|^2}{2(m-2)}}e^{-\frac{m|x|^2}{m-2}}dV_{g_0}&=\frac{2(m-2)}{m}\int_{\mathbb{R}^m}|\nabla u|_g^2dV_g.
w(z) = ic - I + \frac{1}{2\pi}\langle h_b, P_r(\theta - \cdot)\rangle - \frac{1}{2\pi}\iint_{|\zeta|<1}\left( \frac{f(\zeta)}{\zeta}\,\frac{\zeta + z}{\zeta - z}+ \frac{\overline{f(\zeta)}}{\overline{\zeta}}\,\frac{1+z\overline{\zeta}}{1-z\overline{\zeta}}\right)\,d\xi\,d\eta,
\begin{array}{l}\Vert \nabla_x\epsilon_f \Vert^2+\tau^2\Vert \epsilon_f \Vert^2\leq e^{-\tau^2T}\Vert u(\cdot, T) \Vert \Vert \epsilon_f \Vert+\Vert\rho\Vert_{C^0(\partial{D})}\Vert \epsilon_f \Vert_{L^2(\partial{D})}^2(= Q).\end{array}
\frac{C_{2,n}}{(k-4)!}= - [2(n-1) a_k + b_{k-1}] m_{(2,\dot{0})}- [2(n-1) a_{k-1}+ b_{k-2}] m_{(1,\dot{0})}- 2a_k m_{(1^2,\dot{0})}- \frac{2n(n-1)}{k(k-1)}(2k-3) a_{k-2}- \frac{2n}{k-1}b_{k-3},
U\otimes H(0)\equiv \begin{pmatrix}0 0 \dots 0 b_{1}0 \dots 0 0 0 \dots 0 0 b_{2}\dots 0 \vdots \vdots \ddots \vdots \vdots \vdots \ddots \vdots 0 0 \dots 0 0 0 \dots b_{q}\end{pmatrix},\quad\mbox{where $b_{1},b_{2},\dots,b_{q}\in[0,1)$.}
|u_{\xi \xi}(x, t)| \leq (n-1) \max\limits_{(\Omega \times [0, T)) \times \mathbb{S}^{n-1}}u_{\xi \xi}(x,t) \leq (n-1)[ \max\limits_{(\Omega \times [0, T]) \times \mathbb{S}^{n-1}}v(x, t, \xi) + C_{15}] = (n-1) [v(x_0, t_0, \xi_0) + C_{15}] \leq (n-1) [u_{\xi_0 \xi_0}(x_0, t_0) +2 C_{15}] \leq C_{16}(1+ | u_{\nu \nu}|).
f(x;\chi,c)=\frac{\chi^{3}}{\sqrt{2\pi}\Psi(\chi)}\cdot\frac{x}{c^{2}}\sqrt{1-\frac{x^{2}}{c^{2}}}exp\{-\frac{1}{2}\chi^{2}(1-\frac{x^{2}}{c^{2}})\},
{\left\{\ \sum_{j}A^{j}\kappa_{j}^{\alpha}\equiv \sum_{j}A^{j}\left( \eta_{j}^{\alpha}- \xi_{j}^{i}u_{i}^{\alpha}\right) \ \right\}}_{\Big \vert \displaystyle{u^{\alpha}= U^{\alpha}( z )}}= 0 \, .
\mathcal{D}_{\beta_1,\beta_2}=\left\{\psi\in \mathcal{D}_{\mathrm{max}}(\mathcal{H}_{\mathrm{c}})\Big|{\cos\beta_1c_1(\psi)+\sin\beta_1c_2(\psi)=0,\atop \cos\beta_2\psi(1)+\sin\beta_2\psi(1)=0}\right\},
B_{\theta}\left( \hbar:t\right) :=\overline{a\left( \hbar:t\right)}=W_{\hbar}^{\ast}\left( t\right) \bar{a}W_{\hbar}\left( t\right) \text{and}B_{\theta^{\ast}}\left( \hbar:t\right) :=a\left( \hbar:t\right)^{\ast}=W_{\hbar}^{\ast}\left( t\right) a^{\ast}W_{\hbar}\left( t\right) .
\psi^\tau_X(x\cdot b) =\psi^\tau_X\bigl(q^\tau_X(x')\cdot q^\tau_B(b')\bigr) &=\psi^\tau_X\circ q^\tau_X(x'\cdot b') &=q^\tau_{X/Y}\circ \psi^m_X(x'\cdot b') &\overset{*}=q^\tau_{X/Y}\circ \psi^m_X(x')\cdot q^\tau_{B/J}\circ \psi^m_B(b') &=\psi^\tau_X\circ q^\tau_X(x')\cdot \psi^\tau_B\circ q^\tau_B(b') &=\psi^\tau_X(x)\cdot \psi^\tau_B(b),
\int_{\rho_1}^1 f(\rho,R) \, d\rho = \int_{\rho_1}^1 \frac{1}{\rho}\, \psi \big( e^{\mu(\rho) - \mu(R)}\big) \leq \psi \big( e^{\mu(\rho_1) - \mu(R)}\big) \int_{\rho_1}^1 \frac{1}{\rho}\, d\rho = \log \Big( \frac{1}{\rho_1}\Big) \psi \big( e^{\mu(\rho_1) - \mu(R)}\big).
|II| \leq C \Vert f(na)\Vert_{L^1(dnda)}\left( \Vert \nabla' \Phi_2\Vert_{L^{\infty}(S')}+ \Vert \Phi_2\Vert_{L^{\infty}(S')}\right) & \leq C \lambda^{\frac{1}{m}-1}\Vert f(na)\Vert_{L^1(dnda)}\Vert \phi^1\Vert_{W^{1,m}(S')}.
\|u(t,\cdot;-n,u_n)\|_{C^{\theta}}=\sup_{x \in \Omega}{|u(t,x;-n,u_0)|}+ \sup{\frac{|u(t,x;-n,u_0)-u(t,y;-n,u_0)|}{|x-y|^{\theta}}}\le \tilde M_3^*
\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{\beta}+n^{\beta}}\int_1^{\infty}\int_1^{\infty}\frac{dy_1dy_2}{((y_1+k)^{\alpha_1}+(y_2+n)^{\alpha_2})(y_1^{\alpha_1}+y_2^{\alpha_2})},
T(m, \tau_1, \tau_2)=\sup\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\frac{\left|Z_m(\lambda)-\lambda Z_m(1)\right|}{\Bigl\{\int_0^{\lambda}\left(V_m(r; 0, \lambda)\right)^2dr+\int_{\lambda}^1 \left(V_m(r; \lambda, 1)\right)^2dr\Bigr\}^{\frac{1}{2}}}\intertext{with}V_m(r; r_1, r_2)=Z_m(r)-Z_m(r_1)-\frac{r-r_1}{r_2-r_1}\left\{Z_m(r_2)-Z_m(r_1)\right\}
(\prescript{C}{x_L}{\mathcal{D}}_{x}^{\mu}f)(x)=\prescript{}{x_L}{\mathcal{I}}_{x}^{1-\mu}\left[\frac{d}{dx}f(x)\right]=\frac1{\Gamma(1-\mu)}\int_{x_L}^x\frac{f^{\prime}(s)}{(x-s)^{\mu}}ds,x>x_L.
\pi_1(M)\geqslant\pi_1(\langle g^2,b\rangle)\geqslant\pi_1(\langle g^2,g^{-(n+1)}bg^{n+1}b\rangle)=\langle\pi_1(g^2),\pi_1(g^{-(n+1)}bg^{n+1}b)\rangle=G_1
\pi_g\Delta_g=\theta_\tau \lambda_{g}\theta_f\theta_\tau \lambda_{g^{-1}}\theta_f=\theta_\tau \theta_f\theta_\tau \lambda_g\lambda_{g^{-1}}\theta_f=I_\mathcal{X}, &\Delta_g\pi_g=\theta_\tau \lambda_{g^{-1}}\theta_f\theta_\tau \lambda_{g}\theta_f=\theta_\tau \lambda_{g^{-1}}\lambda_g \theta_f\theta_\tau \theta_f=I_\mathcal{X}.
\mathcal{D}(\mathbf{Q}, \mathbf{Q}^{'}) := \max_{\mathbf{S}_t \subseteq \overline{\mathbf{S}}, \mathbf{A}_t \in \overline{\mathbf{A}}, \overline{\pmb{\omega}}\subseteq \pmb{\Omega}}|\overline{\pmb{\omega}}^T[\mathbf{Q}(\mathbf{S}_t, \mathbf{A}_t, \overline{\pmb{\omega}}) - \mathbf{Q}^{'}(\mathbf{S}_t, \mathbf{A}_t, \overline{\pmb{\omega}})]|,
\sum_{n=m+1}^{\infty}\frac{5m^8f(\frac{n}{m})}{\alpha(m,n)}\leq 5m^8\sum_{n=m+1}^{\infty}\frac{\frac{n^6}{m^6}}{n^4(n-m)^4}\leq 5m^2\sum_{n=m+1}^{\infty}\frac{n^2}{(n-m)^4}\leq 5m^2\sum_{k=1}^{\infty}\frac{(k+m)^2}{k^4}\leq Cm^4.
|\phi_l(y)-\tilde\phi_l(y)|\lesssim \begin{cases}y^{-\gamma}y\in(0,\widetilde Ke^{-\omega_{l}s_{0}}) 0 y\in[\widetilde Ke^{-\omega_{l}s_{0}},e^{\widetilde\sigma s_{0}}) y^{2\lambda_l}y\in[e^{\widetilde\sigma s_{0}},\infty), \end{cases}
\gamma\cdot\sum_{c=0}^{q-2}t^{c}\varphi\otimes T^{-1}_{{\omega}}((T_se)_{\underline{c+\epsilon_1+a}})=\sum_{c=0}^{q-2}t^{c}\varphi\otimes T^{-1}_{{\omega}}((T_sT_{e^*(\gamma)}e)_{\underline{c+\epsilon_1+a}}).
\int_0^{\infty}K_{i\nu_{2n}}(g\,e^{x})K_{i\nu_{2m}}(g\,e^{x})\,dx =h_{2n}\delta_{n\,m}, \int_0^{\infty}K_{i\nu_{2n+1}}(g\,e^{x})K_{i\nu_{2m+1}}(g\,e^{x})dx =h_{2n+1}\delta_{n\,m},
A(t,q,y,u):=\left(\begin{matrix}q_{1,1}a_{1}(t,y,u) 0 \cdots 0 q_{2,1}q_{2,2}a_{2}(t,y,u) \ddots \vdots \vdots \vdots \vdots \ddots 0 q_{n-1,1}q_{n-1,2}q_{n-1,3}a_{n-1}(t,y,u) q_{n,1}q_{n,2}q_{n,3}\cdots q_{n,n}\end{matrix}\right).
\frac{1}{n}\sum_{i=1}^n(y_i-T\widetilde{f}_{m_0}(x_i))^2-\frac{1}{n}\sum_{i=1}^n(y_i-f(x_i))^2 + \frac{1}{n}\sum_{i=1}^n(T\widetilde{f}_{m_0}(x^{\prime}_i)-Tf(x^{\prime}_i))^2 \leq \frac{2vv_f}{m_0}+ 8B_nv\epsilon_2 + \frac{2}{n}\sum_{i=1}^n(|y_i|^2-B^2_n)\mathbb{I}\{|y_i| > B_n \}.\stepcounter{equation}
{\cal Y}_{\overline{{{\beta V}}}}^{\overline{{{\alpha V}}}}= \xi_{\overline{{{\beta V}}}}^{\overline{{{\alpha V}}}}Y_{\overline{{{\beta V}}}}^{\overline{{{\alpha V}}}}\prod_{\ell = 1}^{3}{\overline{{X}}}_{{\overline{{{\beta T}}}}^{\, \ell}}^{{\overline{{{\alpha T}}}}^{\, \ell}}\prod_{\ell = 4}^{1 4}X_{{\overline{{{\beta T}}}}^{{\, \ell}}}^{{\overline{{{\alpha T}}}}^{{\, \ell}}}
G=\zeta_{10}+\zeta_{10}^{3}+\zeta_{10}^{7}+\zeta_{10}^{9}+\cdots+(\zeta_{6}+\zeta_{6}^{5})\zeta_{17}^{j}+\cdots+(\zeta_{6}+\zeta_{6}^{5})\zeta_{17}^{k}+\cdots+(\zeta_{6}+\zeta_{6}^{5})\zeta_{17}^{17-k}+\cdots+(\zeta_{6}+\zeta_{6}^{5})\zeta_{17}^{17-j}+\cdots,
\mathbb{E}[ C_2^(a_) ] = \int_0^\infty\int_{x_1}^\infty \frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}& \cdot \log_2\left(1 + (\sqrt{1+\xi x_1}-1)\frac{x_2}{x_1}\right)dx_2dx_1.
\begin{aligned}T_2(h_i{}\leq{}1){}={}h_i+h_{i-1}-{h_i}-{h_{i-1}}\left[ (1+\kappa)\tfrac{\alpha_{i-1}}{B}-\kappa \right]{}={}h_{i-1}(1+\kappa) \left(1-\tfrac{\alpha_{i-1}}{B}\right) \end{aligned}
{\rm WF}_h'(\chi e^{-it_0h^{-1}\tilde{P}_h(z)}\tilde{R}_h(z)\chi)\cap S^*(\mathcal{M}\times\mathcal{M}) \subset \kappa\big(\{(x,\xi,y,\eta)\,:\,(e^{-t_0H_p}(x,\xi),y,\eta)\in \Delta(T^*\mathcal{U})\cup \; \Omega_+\cup E^*_+\times E^*_-\setminus\{0\}\xi=0,\eta\neq 0\}\big).
\mathrm{V}({\rho}_{\mathrm{N}};{\rho}_{\mathrm{N}}) = \frac{1}{4}( 1 + \mathrm{N})^{2}\mathrm{V}({\rho};{\rho}) + \frac{1}{4}( 1 - \mathrm{N})^{2}\mathrm{V}({\sigma};{\sigma}) + \frac{1}{2}( 1 - \mathrm{N}^{2}) \mathrm{V}({\rho};{\sigma}) .
\Vert \mathbf{y}_{k+1}\Vert =\Vert \left( \overline{\lambda}_{k}I-A^{-1}\right)^{-1}\left( \mathbf{x}_{k}+A^{-1}\mathbf{f}_{k}\right)\Vert \leq (1+\gamma_{k})\Vert (\overline{\lambda}_{k}I-A^{-1})^{-1}\Vert & =2\mathrm{sep}(\overline{\lambda}_{k},A^{-1})^{-1}\leq 2\mathrm{sep}(\alpha ,A^{-1})^{-1}<\infty .
Q_{t o t a l}= \left( \begin{matrix}{{1}}&{{- 1}}&{{- 1}}&{{- 1}}&{{- 1}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{- 1}}&{{1}}&{{0}}&{{0}}&{{\lambda_{5}+ \lambda_{6}}}\\{{0}}&{{0}}&{{0}}&{{- 1}}&{{1}}&{{0}}&{{\lambda_{3}+ \lambda_{4}}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{- 1}}&{{0}}&{{\lambda_{1}+ \lambda_{2}}}\\ \end{matrix}\right)
\Lambda_1 = x+(\zeta^4+\zeta^3)y-(\zeta^4+\zeta^3+1)z, \Lambda_2 = -(\zeta^4+\zeta^3+1)x+y+(\zeta^4+\zeta^3)z, \Lambda_3 = (\zeta^4+\zeta^3)x-(\zeta^4+\zeta^3+1)y+z,
\sum_{j = 0}^k{\sum_{s = 0}^j{( - 1)^{n(j + s) + s}\binom kj\binom js\frac{{G_0^{j - s}G_n^s}}{{G_{n + r}^j}}H_{m + nk + (r - n)j + ns}}}= ( - 1)^{nk}\left({\frac{{G_r}}{{G_{n + r}}}}\right)^k H_m,G_{n+r}\ne 0\,,
\varphi(\omega,y)=\frac{1}{16\pi}\frac{1}{y}\int_m^\infty dt \Bigg(\frac{1}{2t-y}\frac{e^{\beta t}+\omega}{e^{\beta t}-\omega}-\frac{1}{2t+y}\frac{e^{\beta (t+y)}+\omega}{e^{\beta(t+y)}-\omega}\Bigg)+\hbox{p.p.c.}
\begin{array}{r c l}{{G_{\sigma}^{( n )}( \zeta^{\prime \prime}, \zeta_{2 n - 1}, s x^{- 1}\zeta_{2 n - 1})^{\varepsilon^{\prime \prime}\, \varepsilon_{2 n - 1}\varepsilon_{2 n}}}}&{{=}}&{{\varepsilon_{2 n - 1}^{( 1 - s ) / 2}\delta_{\varepsilon_{2 n - 1}+ \varepsilon_{2 n}, 0}G_{s \sigma}^{( n - 1 )}( \zeta^{\prime \prime})^{\varepsilon^{\prime \prime}}\; ( s = \pm ) .}}\\ \end{array}
\int_{B_t(p,D)}\int_{B_0(p,D)}G(x.t;y.0) dv_{g(0)}y dv_{g(t)}x =\int_{B_t(p,D)}u_D(x,t) dv_{g(t)}x \geq \delta_1>0, ~\forall ~t\in[t_i,t_{i}+\delta_1],
I_{1}={\frac{\pi c^{2 - s}}{4 ( s - 1 ) ( s - 2 ) a \alpha^{\frac{3}{2}}}}+{\frac{( 2 c )^{2 - s}\pi}{a \alpha^{\frac{3}{2}}\Gamma ( s )}}\sum_{n = 1}^{\infty}\frac{1}{X_{n}^{2 - s}}\{K_{- s + 2}( X_{n}) - X_{n}K_{- s + 3}( X_{n}) \}\ ,
\hat{F}_{\infty}(z) - \frac{\hat{A}_{n}(z)}{R_{n}(z)}= \frac{(1+c_1^2)\gamma_{n}}{4d_1 r_{n,n}}\, \frac{1}{z^{n+1}}+ O\big((1/z)^{n+2}\big), n \geq 1.
M_{\lambda 1}^{- 2}( \beta ) = \frac{i}{2}\int [ p_{1}p_{2}p_{3}] f_{\lambda}\left( \frac{\partial}{\partial p_{3}^{2}}\tilde{\delta}\right) a_{1 \beta}^{\dagger}a_{2 \beta}^{\dagger}a_{3 \beta}+ h . c . .
V =\frac{v_{n+1}}{(-H)^{n+2}}\int_{w_-}^{w_+}\frac{w^{n+1}}{\sqrt{-U(w)}}dw, A =\frac{a_n}{(-H)^{n+1}}\int_{w_-}^{w_+}\frac{w^{n}\sqrt{1-U(w)}}{\sqrt{-U(w)}}dw,
\forall s, t \in [1/2,1), z_h^{(s)}= z_h^{(t)}= v_h \mbox{a.e. on}B_1 \setminus \left( \cup_{\mathcal{F}^{(s)}_h}B \bigcup \cup_{\mathcal{F}^{(t)}_h}B \right)
\begin{cases}(i) Y_t = \xi +\int_t^T g(s)ds + (K^1_T - K^1_t)-(K^2_T-K^2_t)-\int_t^T Z_s dB_s, t \in [0,T], (ii) \zeta_t \geq Y_{t}\geq \chi_t, \,\, t \in [0,T], (iii) \int_0^T (Y_{t}-\zeta_t)dK^1_t = 0, \int_0^T (Y_{t}-\chi_t)dK^2_t = 0, \end{cases}
\mathcal{C}_0\to\begin{pmatrix}a_0 b_0 0 1/a_0\end{pmatrix}\mathcal{C}_0,\mathcal{C}_1\to\begin{pmatrix}a_1 b_1 0 1/a_1\end{pmatrix}\mathcal{C}_1,\mathcal{C}_\alpha\to\begin{pmatrix}a_\alpha 0 b_\alpha 1/a_\alpha\end{pmatrix}\mathcal{C}_\alpha,
\lambda_1^n = \frac{\int_\Omega (-\varphi_1) (dd^c \varphi_1)^n}{\int_\Omega (-\varphi_1)^{n +1}d \nu_{f}}= \inf \left\{\frac{\int_\Omega (-\phi) (dd^c \phi)^n}{\int_\Omega (-\phi)^{n +1}d \nu_{f}}; \phi \in \mathcal E^1(\Omega), \phi \neq 0 \right\}\cdot
\inf_{\{v,v'\}}\prod_{i=1}^\infty\frac{\left(w_{\{v,v'\}}(i+\ell_0^{\{v,v'\}})\right)^2}{\big(w_{\{v,v'\}}(i+\ell_0^{\{v,v'\}})+\sum_{e\in{\cal{N}}(v)\atop e\neq{\{v,v'\}}}w_e(\ell_0^e)\big)\big({w_{\{v,v'\}}(i+\ell_0^{\{v,v'\}})+\sum_{e\in{\cal{N}}(v')\atop e\neq{\{v,v'\}}}w_{e}(\ell_0^e)}\big)}>0,
\rho^\dagger(s,\xi) = \sum_{|\alpha|<N}\frac{s^\alpha}{\alpha !}\partial_s^\alpha\left[\alpha_{-s}\left(\rho(\xi)^*\right)\right]_{s=0}+\sum_{|\alpha|=N}s^\alpha r_{N\alpha}(s,\xi) &= \sum_{|\alpha|<N}\frac{(-is)^\alpha}{\alpha !}\delta^\alpha \left[\rho(\xi)^*\right]+\sum_{|\alpha|=N}s^\alpha r_{N\alpha}(s,\xi) .
W_i(F) \begin{cases}\geq c_{\textrm{bulk}}\left(|F|^{r}+ \left(\displaystyle\frac{|F|^3}{\det F}\right)^{r-1}+(\det F)^{-s}\right) \det F>0 begin{align*}2mm] =+\infty \det F\le 0, \end{cases}
\mathcal{H}^{n}(i_0^*i^*j_*\mathcal{R}_{\ell}(^{\lambda}\mathcal{V})) \simeq R^{n}(i_0^*i^*j_*\mathcal{R}_{\ell}(^{\lambda}\mathcal{V})) = R^{n-w(\lambda)}i_0^*i^*j_*\mu_{\ell}^K(V).
\begin{pmatrix}f^+_{m_{1/2},k_{1/2}}f^-_{m_{1/2},k_{1/2}}\end{pmatrix}=\begin{pmatrix}e^{-\sqrt{m^2-a^2}r}r^{-a k_{1/2}/m}-\sqrt{\frac{m+a}{m-a}}e^{-\sqrt{m^2-a^2}r}r^{-a k_{1/2}/m}\end{pmatrix}.
\Vert K(\varepsilon ;\zeta )\Vert_{L_2\rightarrow L_2}\leqslant M_1\alpha_1^{1/2}\Vert \mathbf{D}(B^0-\zeta \overline{Q_0})^{-1}\Vert_{L_2\rightarrow L_2}&+\widetilde{M}_1\Vert (B^0-\zeta \overline{Q_0})^{-1}\Vert_{L_2\rightarrow L_2}.
Q_{1212}=\frac{3e^{3 q_3-q_2-2q_1}+2e^{2 q_3-2 q_1}-2e^{q_3+q_2-2q_1}-2 e^{2q_2-2q_1}-e^{2q_3-q_2-q_1}+e^{q_3-q_1}+e^{q_2-q_1}}{(1+e^{q_2-q_1})\Delta_3},
C^{3,3}= \left(\begin{array}{cccc}0 0 \alpha (\lambda_{k+2}+ \lambda_k) begin{align*}6pt]0 0 \beta (\lambda_{k+2}- \lambda_{k+1}) begin{align*}6pt]0 0 0\end{array}\right) \mbox{and}C(x)= C^{k+2, k+2}:= \left(\begin{array}{cccc}0_{k-1, k-1}0_{k-1, 3}begin{align*}6pt]0_{3, k - 1}C^{3,3}\end{array}\right) \mbox{for}k \ge 1.
2 \ell x = \theta - \log(y/3) = \frac{(2\gamma-1)\mu^2}{2}+ \log(3 q((1-\gamma) \mu)) + \frac{(1-\gamma)^2\mu^2}{2}= \frac{\gamma^2 \mu^2}{2}+ \log(3 q((1-\gamma) \mu)).
\frac{(x+a)^{\epsilon_A}}{(y+b)^{\epsilon_B}}= \frac{\Phi(x+a)}{\Phi(y+b)}= \Phi\left(\frac{x+a}{y+b}\right) = \left(\frac{x+a}{y+b}\right)^{\epsilon_C}= \frac{(x+a)^{\epsilon_C}}{(y+b)^{\epsilon_C}}.
\epsilon^{\alpha \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\beta \sigma \right\}}+\epsilon^{\beta \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\sigma \alpha \right\}}+\epsilon^{\sigma \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\alpha\beta \right\}}=0\,,
\kappa= g\left(\nabla_{\phi'_{\gamma(u)}(s)}\phi'_{\gamma(u)}(s),-X_1\right)=g\left(\Gamma_{22}^1\partial_u+\Gamma_{22}^2\partial_s,-\partial_u+g_{12}\partial_s\right) = -\Gamma_{22}^1\left(g_{11}-g_{12}^2\right)=-\Gamma_{22}^1,
J^{\delta,l}(\psi) =\log\left(\int e^{J^{\delta,l+1}(\psi+\psi^1)}d\mu_{C_{l+1}^{\delta}}(\psi^1)\right) &=\log\left(\int\int e^{-V^{\delta}(\psi+\psi^1+\psi^2)}d\mu_{\sum_{j=l+2}^{N_h}C_j^{\delta}}(\psi^2)d\mu_{C_{l+1}^{\delta}}(\psi^1)\right) &=\log\left(\int e^{-V^{\delta (\psi+\psi^1)}}d\mu_{\sum_{j=l+1}^{N_h}C_{j}^{\delta}}(\psi^1)\right).
W ( y ) = \frac{1}{Z^{2 / 5}}\left( - \frac{1 5}{2}+ 5 ~ Z \right) + \frac{1 5}{4 \, Z^{4 / 5}}\left( 1 + Z \right) \left( y + i \, Z^{1 / 5}\right)^{2}+ \ldots \ .
\begin{array}{rcl}\delta_{i}x^{\alpha}=0 =\displaystyle\frac{1}{i\hbar}[x^{\alpha}, L^{\mu}_{i}\circ p_{\mu}] begin{align*}3mm] \delta_{i}x^{\mu}=L^{\mu}_{i}=\displaystyle\frac{1}{i\hbar}[x^{\mu},L^{\mu}_{i}\circ p_{\mu}]\,.\end{array}
\begin{cases}F_B(x)< F_B(x_a) = s_{B}F_A(x_a)< s_{B}F_A(x), x\in [0,x_a), F_A(x)< F_A(x_b) = s_{A}F_B(x_b)< s_{A}F_B(x), x\in (x_b,1].\end{cases}
\omega^{-1}(\eta(t)) \tilde{W}[\eta|t] \omega(\eta(t)) = - 8\pi \dot{\eta}^\mu(t) \int \delta\xi ds \epsilon_{\mu\nu\rho\sigma}W^{\rho\sigma}[\xi|s] \dot{\xi}^\nu(s) \delta(\xi(s)-\eta(t)),
\theta\left (cG_-^0\right )=dG_+^0 \theta\left (cG_-^1\right )=dG_+^1 \theta\left (aF_+^0\right )= aF_-^0 \theta\left (aF_+^1\right )= aF_-^1 \zeta^2\left (cG_-^1\right )=cG_-^0 \zeta^2\left (aF_+^1\right )=aF_+^0 S\circ \zeta \left (cG_-^0\right ) = aF_+^{-2}= \zeta^4\left (aF_+^0\right )
\begin{array}{lcl}a_2 = -\frac{1}{4}u_2 a'_3 = \frac{1}{800\theta}\{9u_{2}^{2}+100\theta u'_3-280{\theta}^{2}u_2\}a_4 = \frac{49}{400}u_{2}^{2}-\frac{3}{8}\theta u'_3+\frac{41}{20}{\theta}^{2}u''_2 \vdots\end{array}
\sum_{k=1}^n \|A_{i_1}\cdots A_{i_{k-1}}D_{i_{k}}B_{i_{k+1}}\cdots B_{i_n}\| \leq K_0 \sum_{k=1}^n \|A_{i_1}\cdots A_{i_{k-1}}\| \cdot\|B_{i_{k+1}}\cdots B_{i_n}\| &\leq K_0\sum_{k=1}^n a_{k-1}b_{n-k}
A_0 = \emptyset, A_1 = \Big\{(1)\Big\}, A_2 = \Big\{(1,1), (1,2) \Big\}, A_3 = \Big\{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (1,2,3) \Big\}, A_4 = \Big\{(1,1,1,1), (1,1,1,2), (1,1,2,1), (1,1,2,2), (1,1,2,3), (1,2,1,1), (1,2,1,2), & (1,2,1,3), (1,2,2,1), (1,2,2,2), (1,2,2,3), (1,2,3,1), (1,2,3,2), (1,2,3,3), (1,2,3,4) \Big\},
\gamma_1 =\alpha_1-\beta_1=\alpha_2+\beta_2 \gamma_2 =\alpha_1-\beta_2=\alpha_2+\beta_1 \gamma_3 =\alpha_1-\beta_3=-\alpha_2+\beta_4 \gamma_4 =\alpha_1-\beta_4=-\alpha_2+\beta_3
{\cal L}= \frac{1}{2}\left\{\partial_{\mu}\varphi_{i}\partial^{\mu}\varphi_{i}+ i \bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi_{i}+ f_{i}f_{i}+ 2 f_{i}\frac{\partial{W}}{\partial \varphi_{i}}- \frac{\partial^{2}{W}}{\partial \varphi_{i}\partial \varphi_{j}}\bar{\psi}_{i}\psi_{j}\right\}.
e_{i,j}(k_n)= G\bigl(\varepsilon_{i,j},\varepsilon_{i,j\pm1}, \ldots, \varepsilon_{i,j\pm k_n},\varepsilon'_{i,j,j\pm(k_n+1)},\varepsilon '_{i,j,j\pm(k_n+2)},\ldots\bigr)
d^0(x_1,\cdots, x_k,y) =(*,x_1,\cdots,x_k,y) d^i(x_1,\cdots, x_k,y) =(x_1,\cdots,x_i,x_i,\cdots,x_k,y), 1\leq i\leq k d^{k+1}(x_1,\cdots, x_k,y) =(x_1,\cdots,x_k,y,y) s^i(x_1,\cdots, x_k,y) =(x_1,\cdots,x_{i},x_{i+2},\cdots,x_k,y) , 0\leq i\leq k.
\left({x - a}\right)_q^n = \prod\limits_{j = 0}^{n - 1}{\left({x - q^j a}\right)},\,\,\,\,\, \left({x - a}\right)_q^{(0)}=1,\,\,\,\,\, \left({x - a}\right)_q^{- n}= \frac{1}{{\left({x - q^{- n}a}\right)_q^n}}.
\begin{aligned}\epsilon^{-1}[J_1(u_1^\epsilon(\cdot),u_2(\cdot))-J_1(u_1(\cdot),u_2(\cdot))]\geq 0, \epsilon^{-1}[J_2(u_1(\cdot),u_2^\epsilon(\cdot))-J_2(u_1(\cdot),u_2(\cdot))]\geq 0.\end{aligned}
{\mathbf O}^{-1}\left((\rho^{(1)}_{xq^{\mu}}\otimes \rho^{(2)}_{xq^{-\mu}})\Delta(\mathsf{a})\right){\mathbf O}= (\mathsf{ev}^{(1)}_{x}\otimes \mathsf{ev}^{(2)}_{x}) \Delta(\mathsf{a}),
c_{3}=\lambda b_{1}\left(\left[\int_{\mathbb{R}}\frac{1}{(\alpha-t)^{2}}\frac{d\mu_{1}(t)}{\lambda}\right]^{2}-\int_{\mathbb{R}}\frac{1}{(\alpha-t)^{4}}\frac{d\mu_{1}(t)}{\lambda}\right)<0.
\Omega_{d(\mu)_i}\big(t_\mu^\Lambda\big)=\Omega_{d(\lambda)_i}\big(t_{\mu(0,d(\lambda)_i e_i)}^\Lambda\big)\otimes_{\mathcal{T}C^*(\Lambda^i)}\Omega_{d(\mu)_i-d(\lambda)_i}\big(t_{\mu(d(\lambda)_i e_i,d(\mu))}^\Lambda\big),
\bar{S_{s}}^{a b}= \frac{2^{n - 2}}{( d - 2 ) !}\varepsilon^{a b}{}_{a_{1}a_{2}\ldots a_{d - 2}}S_{s}^{a_{1}a_{2}}S_{s}^{a_{3}a_{4}}\ldots = \frac{( - i )^{n - 1}}{2}a^{a}a^{b}\, \, ( a^{0}a^{1}\ldots a^{2 n}) .
K^\perp(x) = R^X(E_1,E_2,N_2,N_1) + \langle (D_{E_1}N_1)^\top, (D_{E_2}N_2)^\top \rangle -\langle (D_{E_1}N_2)^\top, (D_{E_2}N_1)^\top \rangle &= R^X(E_1,E_2,N_2,N_1) +\left( \sum_{j=1,2}\langle A_{1j},N_1\rangle \langle A_{2j}, N_2 \rangle - \langle A_{1j},N_2\rangle \langle A_{2j}, N_1 \rangle \right).
I_{\omega,\mathbf{c}}(U(t))\le 2(S_{\omega,\mathbf{c}}(\Phi_0^{\lambda_0})-S_{\omega,\mathbf{c}}(\Phi_0))=-2(\mu_{\omega,\mathbf{c}}-S_{\omega,\mathbf{c}}(\Phi_0^{\lambda_0}))<0.
\left(\begin{array}{c}u_n^1(y) begin{align*}6pt]u_n^2(y)\end{array}\right)\propto\left(\begin{array}{c}\cos\left(\mu_n y\right)+\frac{m_-}{\mu_n}\sin\left(\mu_n y\right) begin{align*}6pt]\frac{M_n-m_+}{\mu_n}\sin\left(\mu_n y\right)\end{array}\right)
\frac 12 \frac d{ds}\delta_\Gamma^2(s) =\big(\lambda(s) - \gamma(\sigma(s))\big) \cdot \big( \lambda'(s) - \gamma'(\sigma(s))\sigma'(s) \big) &=\big(\lambda(s) - \gamma(\sigma(s))\big) \cdot \big( \lambda'(s) - \gamma'(\sigma(s))\big).
\, \equiv K_{1 2}^{T + +}( s_{1 2}, s_{1 2}, P_{0}\! - \! h_{3}) ( 1 - G_{1}^{0}G_{2}^{0}K_{1 2}^{T + +}( s_{1 2}, s_{1 2}, P_{0}\! - \! h_{3}) )^{- 1}.
\frac{m}{2J}s{\rm Tr}_N [{\hat{\bar \Psi}}{\hat \Psi}]\ +\ \frac{1}{2J}s{\rm Tr}_N [\mu ({\hat{\bar\Psi}}{\hat\Psi})^2 +\nu ({\hat{\bar\Psi}}\Gamma_0{\hat\Psi})^2 ]\ .
-\nabla \cdot (\sigma_i \nabla \partial_i P) &=(\nabla \cdot \sigma_i) \cdot \nabla \partial_i P &\stackrel{\eqref{EquationSigma}}{=}q_i \cdot \nabla \partial_i P &\stackrel{\eqref{Equationq}}{=}a(e_i+\nabla \phi_i) \cdot \nabla \partial_i P &\stackrel{\eqref{EquationCorrector}}{=}\nabla \cdot (\partial_i P\, a(e_i+\nabla \phi_i)).
\mbox{Trace}(X(\mathbf{l}_{2}) \circ b_F \circ X(\mathbf{l}_{1})\circ Y)=(-1)^{\sum_{i=1}^{d}l_i^{(2)}}\mbox{Trace}(b_F \circ Y') &=(-1)^{\sum_{i=1}^{d}l_i^{(2)}}\mbox{Trace}(X^{l_{j_{1}}}U_{j_{1}}\cdots X^{l_{j_{s}}}U_{j_{s}})\cdots\mbox{Trace}(X^{l_{j_{t+1}}}U_{j_{t+1}}\cdots X^{l_{j_{d}}}U_{j_{d}}).
\sum_{n \le N}\pi(n) R_{X}^{n}+H_{X}^{\le N}= \sum_{n \le N}\pi(n) \sum_{m=1}^{\infty}\dfrac{1}{m}R_{X}^{mn}= -\sum_{n \le N}\pi(n) \log \left( 1-R_{X}^{n}\right)= -\log \left( \prod_{n \le N}\left( 1-R_{X}^{n}\right)^{\pi(n)}\right),
D(\theta,\theta^{\prime})=\frac{1}{2}(\theta^{\prime}-\theta)^{T}(\frac{\partial^{2}}{\partial\theta_{i}^{\prime}\partial\theta_{j}^{\prime}}D(\theta,\theta^{\prime}))_{\theta^{\prime}=\theta}(\theta^{\prime}-\theta)+o((\theta^{\prime}-\theta)^{2})
\begin{aligned}q^b(t,x,y) \geq \int_{B(y,t^{1/2})}q^b(\frac{1}{4}\kappa t,x,z)q^b((1-\frac{1}{4}\kappa) t, z, y)\,dz &\geq \inf_{z\in B(y,t^{1/2})}q^b((1-\frac{1}{4}\kappa)t,z,y)\P_x(X^b_{\frac{1}{4}\kappa t}\in B(y,t^{1/2})) &\geq c_4t^{-d/2}\, t^{(d+2)/2}\frac{m_b}{|x-y|^{d+\beta}}&\geq c_4\frac{m_bt}{|x-y|^{d+\beta}},\end{aligned}
\int_{p_1}^{p_2}U(p) \, e^{i \omega \psi(p)}\, dp = \int_{p_1}^{p_1 + \delta}\dots + \int_{p_1 + \delta}^{p_2-\delta}\dots + \int_{p_2-\delta}^{p_2}\dots & =: I^{(1)}(\omega) + I^{(2)}(\omega) + I^{(3)}(\omega) \; ,
N_{R}^{0}= \sum_{n = 1}^{\infty}n ( a_{n}^{\alpha}{}^{\dagger}a_{n}^{\alpha}+ s_{n}^{A}{}^{\dagger}s_{n}^{A}) , ~ ~ N_{L}^{0}= \sum_{n = 1}^{\infty}n ( \tilde{a}_{n}^{\alpha}{}^{\dagger}\tilde{a}_{n}^{\alpha}+ \tilde{s}_{n}^{A}{}^{\dagger}\tilde{s}_{n}^{A}) .
\frac{\lambda(y^u,y^h)-\lambda(y^{u-h},y^h)}{he^{\gamma}\log y}=\int_{1}^{u/h}y^{ht-u}\omega(t)\,dt-\int_{2}^{u/h}y^{ht-u}\omega(t-1)\,dt &=\int_{1}^{2}y^{ht-u}\omega(t)\,dt+\int_{2}^{u/h}y^{ht-u}(\omega(t)-\omega(t-1))\,dt &=\int_{1}^{2}t^{-1}y^{ht-u}\,dt-\int_{2}^{u/h}y^{ht-u}t\omega'(t)\,dt.
S_{e f f}= \int d^{4}x \, \sqrt{- g}\, \left( - \frac{1}{2 \lambda_{1}h^{2}}\sigma^{2}+ \frac{1}{2 \xi_{1}h^{2}}R \sigma^{2}- \frac{1}{4 ! \lambda_{2}h^{4}}\sigma^{4}\right) - i \ln \det \left( i \gamma^{\mu}( x ) \nabla_{\mu}- \sigma \right) ,
\begin{array}{l l}{{g_{_E}( \widehat{\mathbf{f}}_{\mu}( x ) , \widehat{\mathbf{f}}_{\nu}( x ) ) = g ( \mathbf{f}_{\mu}( x ) , \mathbf{f}_{\nu}( x ) ) = g_{\mu \nu}( x )}}\\{{g_{_E}( \widehat{\mathbf{f}}_{\mu}( x ) , \mathbf{e}_{k}( x ) ) = 0}}\\{{g_{_E}( \mathbf{e}_{k}( z ) , \mathbf{e}_{l}( z ) ) = g_{0 \, k l}( x ) \, .}}\\ \end{array}