harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{E}J_1^1(h+y_0^1,h+y_0^2,\lambda) =\frac{2h^2}{2(b-\lambda-(d_{\tau}+(1+\eta t)d_0'))}&+\frac{1}{2}\sum_{p=0}^{\tau-1}\frac{1}{n_p}\log\frac{b-\lambda-(d_{\tau}+(1+\eta t)d_{p+1}')}{b-\lambda-(d_{\tau}+(1+\eta t)d_p')}&+\frac{1}{2}\sum_{p=\tau}^{k}\frac{1}{n_p}\log\frac{b-\lambda-d_{p+1}}{b-\lambda-d_p}
{\frac{1}{p}}\log\Biggl({\frac{1}{k}}\sum_{i=1}^k\lambda_i^p(A)\Biggr) \searrow{\frac{1}{k}}\sum_{i=1}^k\log\lambda_i(A),{\frac{1}{p}}\log\Biggl({\frac{1}{k}}\sum_{i=1}^k\lambda_i^p(B_\xi)\Biggr) \searrow{\frac{1}{k}}\sum_{i=1}^k\log\lambda_i(B_\xi)
X^{i_{1}}Y^{j_{1}}\cdot\cdot\cdot X^{i_{k}}Y^{j_{k}}=\frac{[X^{(i_{1})}Y^{(j_{1})}\cdot\cdot\cdot X^{(i_{k})}Y^{(j_{k})}]}{i_{1}+j_{1}+\cdot\cdot\cdot+i_{k}+j_{k}}.
\Phi(\bar u) = \frac{2d^2}{(b-a)^2}\int_a^\frac{a+b}{2}p(x)\,dx+\frac{2d^2}{(b-a)^2}\int_a^\frac{a+b}{2}q(x)(x-a)^2\,dx+\frac{d^2}{2}\int_\frac{a+b}{2}^b q(x)\,dx &\ge \frac{d^2 p_0}{b-a}&> r.
\lim_{x_+=0,x_2\rightarrow 0}v=-m^2{\pi |x_2|^2\over 2 (\alpha')^2 C^2}\log(|x_2|)= -m^2\Lambda^4{\pi |\tilde u|^2\over 2 C^2}(\log(|\tilde u|)+{\rm const.}) \, .
E_w^0 = \{z \in BS(2,1)^+ : z \le w\}, E_w^1 = \{(z, z a): z, za \in E_w^0\}\bigcup \{(z, z b): z, zb \in E_w^0\}& = \{(z, z l): z, z l \in E_w^0 l = a,b\}
\|\phi_{m}\circ \ldots \circ \phi_1-\phi_n \circ \ldots \circ \phi_1\|_{L^{\infty}}\leq \sum_{k=n}^{m-1}\|\phi_{k+1}-\operatorname{id}\|_{L^{\infty}}\leq \sum_{k=n}^{m-1}C\tau_0|M_{k+1}|^{1/2}.
\big|B_{|\mathcal{S}|}(x)\triangle \mathcal{S}\big|\geq \big|B_{\nu_m}(x_0) \triangle \mathcal{S}\big|-\big||\mathcal{S}|-|B_{\nu_m}(x_0)|\big|=\frac{\tilde\gamma^2}{4}-\big||\mathcal{S}|-\nu_m\big| \geq \frac{\tilde\gamma^2}{8}\geq \frac{\gamma^2}{8},
e_{s+1}\tau^{r-s-1}nd=|P_{s+1}| =\sum_{v\in C'}f_{s+1}(v)+\sum_{v\in D}f_{s+1}(v) &\le \sum_{v\in C'}\zeta\tau^{r-s-1}d(v) + \sum_{\sigma\in \Gamma_s}d_{s+1}(\sigma)+ \sum_{v\in D}d_{s+1}(v) &\le \zeta\tau^{r-s-1}nd + \sum_{\sigma\in \Gamma_s}d_{s+1}(\sigma)+ \tau^{r-s-1}nd(\mu(D)+r\delta)\,,
(\psi_\varepsilon)^*(x)= \sup_{z \in \mathbb{R}^n}\left[\langle z, x \rangle + \log \int_{\mathbb{R}^n}e^{-\psi(y)}\gamma_{2\varepsilon}(z-y)\, dy\right] \le \sup_{z \in \mathbb{R}^n}\left[ \langle z, x \rangle + \log \int_{\mathbb{R}^n}e^{\psi^*(x) - \langle x, y \rangle}\gamma_{2\varepsilon}(z-y)\, dy \right] = \psi^*(x)+\varepsilon |x|^2.
\frac{1}{\sqrt{\lambda_t}}-1\geq\frac{1}{2}\alpha_0\sum_{i=1}^t\sqrt{\gamma_i/\gamma_0}\quad\Rightarrow\lambda_t\leq4\left(2+\alpha_0\sum_{i=1}^t\sqrt{\gamma_i/\gamma_0}\right)^{-2}.
|f_n(z)| \leq \frac{2|z| \, |\kappa(z)|^{2^{n}}}{1 - |\kappa(z)|^{2^{n}}}=\frac{2 \sqrt{2}t \kappa(s)^{2^{n-1}}}{1 - \kappa(s)^{2^{n-1}}}= \frac{\sqrt{2}s \kappa(s)^{2^{n-1}}}{1 - \kappa(s)^{2^{n-1}}}(1+2t^2),
\det \left( \det X \begin{pmatrix}(J \setminus (j_\alpha)) \sqcup K (J \setminus (j'_\beta)) \sqcup K' \end{pmatrix}\right)_{1 \le \alpha, \, \beta \le q}=\left( \det X \begin{pmatrix}J \sqcup K J' \sqcup K' \end{pmatrix}\right)^{q-1}\cdot\det X \begin{pmatrix}K K' \end{pmatrix}.
\gamma^{\mu\rho\nu}\nabla_{\rho}\Psi_{\nu}^{A}-\gamma^{\mu\nu}\gamma^{5}\partial_{5}\Psi_{\nu}^{A}+k\epsilon\gamma^{\mu\nu}\gamma^{\hat{5}}\Psi_{\nu}^{A}-\sqrt{2}\left(g_{R}R\epsilon(\sigma_3)^{A}_{B}+g_{S}S(\sigma_1)^{A}_{B}\right)\gamma^{\mu\nu}\Psi_{\nu}^{B}=0\ ,
\int_0^T\int_{\mathbb{R}^{N}}\omega\varphi dx dt+\int_{\mathbb{R}^{N}}u_0(\varphi(0,x)-k\Delta\varphi(0,x))dx\leq C(p)\biggr(\mathcal{I}_1+k\mathcal{I}_2+\mathcal{I}_3\biggr),
l_i(u^a) \nabla r_j(u^a) \big[\lambda_i(u^k)\mathbb{I}- f'(u^k)\big] u^k_x= l_i(u^a) \nabla r_j(u^a)\Big\{\big[\lambda_i(u^k) - s_k\big] u^k_x - (a^2 - s_k^2) u^k_{xx}\Big\}\leqslant O(1) \big( |\lambda_i(u^k) - s_k||u^k_x|+ |u^k_{xx}|\big)\leqslant O(1)\delta |u^k_x|.
\begin{aligned}A_+ := \{a+1 \;:\; a \in A \}A_- := \{a-1 \;:\; a \in A\setminus[0,1) \}M_1 := \cup_{i=0}^\infty [2i,2i+1), M_2 := M_1^c, X := (A \cap M_1) \cup (A_+^c \cap M_2), Y := (A \cap M_2) \cup (A_-^c \cap M_1).\end{aligned}
\partial_\delta U_{\delta,\xi}(x)= \alpha_n\frac{n-2}{2}\delta^\frac{n-4}{2}\frac{|x-\xi|^2-\delta^2}{(\delta^2+|x-\xi|^2)^\frac{n}{2}},\partial_{\xi_i}U_{\delta,\xi}(x)=-\frac{\alpha_n\delta^\frac{n-2}{2}(x_i-\xi_i)}{(\delta^2+|x-\xi|^2)^\frac{n}{2}},\i=1,\ldots, n,
2(\begin{matrix}n\\r\end{matrix})2^{(\begin{matrix}n\\ 2\end{matrix})-(\begin{matrix}r\\ 2\end{matrix})}<2^{(\begin{matrix}n\\ 2\end{matrix})}\Leftrightarrow(\begin{matrix}n\\r\end{matrix})2^{1-(\begin{matrix}r\\ 2\end{matrix})}<1
J_1(t) =\langle E(t),F_n(U(t))-F_n(\tilde U(t))\rangle+\langle E(t),F_n(\tilde U(t))-F_n(\mathbb U(t))\rangle &\le \frac{3}{2}\|E(t)\|^2+\frac{1}{2}\|F_n(\mathbb U(t))-F_n(\tilde U(t))\|^2,
\sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}\tau_k(n)\tau(n+h)\tau(n-h) =\widehat{W}(1)XHQ_{k+1}(\log X) &+O\left(X^{\varepsilon}(H^2+HX^{1-\frac{1}{k+2}}+XH^{1/2}+X^{3/2}H^{-1/2})\right),
W^{\mathbf{k}}(G) =\sum_{\mathbf{v}\in[n]^m}\Bigg(\prod_{e\in E(G_{[m-1]})}b_{v(e)}^{k_e}\Bigg)\Bigg(\prod_{\{i,m\}\in E(G)}b_{v_iv_m}^{k(i)}\Bigg) &=\sum_{\mathbf{v}\in[n]^m}\prod_{\{i,m\}\in E(G)}\Bigg(b_{v_iv_m}^{k_m'}\prod_{e\in E(G_{[m-1]})}b_{v(e)}^{k_e}\Bigg)^{k(i)/k_m'},
\sigma^{1\dot ab}=\begin{pmatrix}0 -1 -1 0\end{pmatrix},\sigma^{2\dot ab}=\begin{pmatrix}0 -i i 0\end{pmatrix},\sigma^{3\dot ab}=\begin{pmatrix}-1 0 0 1\end{pmatrix},\sigma^{4\dot ab}=\begin{pmatrix}1 0 0 1\end{pmatrix}.
\sigma_{n}(C_{n})\,\mbox{{\rm cov}}(\delta_{i,n},u_{i,n})=q_{i,n}^{1/2}\int_{\,I_{i,n}}\left({\bf E\,}\left| T_{\eta}(x)\right| \,u_{i,n}\,q_{i,n}^{-1/2}\right) \sqrt{k_{n}(x)}\,dx.
\sum_{k_j=1}^\infty\frac{1}{\Gamma(k_j)}\biggl(\log\frac{1-x_{j+1}}{1-x_j}\biggr)^{k_j-1}X_j^{k_j-1}=\exp\biggl(X_j\log\frac{1-x_{j+1}}{1-x_j}\biggr) &=\biggl(\frac{1-x_{j+1}}{1-x_j}\biggr)^{X_j}
\mathrm{CWE}(C) =\sum_{\mathsf{c}\in C}z_0^{k_0}z_1^{k_1}\cdots z_{p-1}^{k_{p-1}}&=\sum_{(k_0,k_1,\cdots,k_{p-1})\in B_n}A(k_0,k_1,\cdots,k_{p-1})z_0^{k_0}z_1^{k_1}\cdots z_{p-1}^{k_{p-1}},
\int\limits_{0}^{|l|}\prod\limits_{i=1}^{n}1_{\{l(t_i)=x_{i}\}}\,dt_i= \sum\limits_{\sigma\in\mathfrak{S}_n}\int\limits_{0<t_{\sigma(1)}<\cdots<t_{\sigma(n)}<|l|}\prod\limits_{i=1}^{n}1_{\{l(t_{\sigma(i)})=x_{\sigma(i)}\}}\,dt_{\sigma(i)}= \sum\limits_{\sigma\in\mathfrak{S}_n}\int\limits_{0<s_1<\cdots<s_n<|l|}\prod\limits_{i=1}^{n}1_{\{l(s_i)=x_{\sigma(i)}\}}\,ds_i.
T_{\mathrm{Br}}H_{\mathrm{BrLie}}\mathbb{I}_{\mathrm{BrLie}}^{s}\overset{(\ref{diag:HBrLies})}{=}T_{\mathrm{Br}}\mathbb{I}_{\mathrm{Br}}^{s}H_{\mathrm{BrLie}}^{s}\overset{(\ref{diag:Tbrs})}{=}\mathbb{I}_{\mathrm{BrAlg}}^{s}T_{\mathrm{Br}}^{s}H_{\mathrm{BrLie}}^{s}.
\delta(n)=\frac{1}{Q}\sum_{1\leq q\leq Q}\;\frac{1}{q}\; \sideset{}{^\star}\sum_{a\bmod{q}}e\left(\frac{na}{q}\right) \int_\mathbb{R}g(q,x) e\left(\frac{nx}{qQ}\right)\mathrm{d}x
\mathbf{\Phi}_{0}\underline{\mathbf{Z}}_{\tau}-\underset{i=1}{\overset{\mathbf{P}}{\sum}}\mathbf{\Phi}_{i}\underline{\mathbf{Z}}_{\tau -i}=\nabla_{S}^{-\underline{\mathbf{D}}}(L)\underline{\mathbf{u}}_{\tau},
\frac{\partial}{\partial R_{{\lambda}{\tau}}}\int d^{D}p e^{- \frac{1}{2}p \cdot R \cdot p}= - \frac{1}{2}( 2 \pi )^{D / 2}\left[ \frac{\partial}{\partial R_{{\lambda}{\tau}}}( \det R ) \right] ( \det R )^{- 3 / 2}
\begin{matrix}{\quad}\\{\vec{\varphi}( x )}\\{\quad}\\ \end{matrix}= \left( \begin{matrix}{c_{1}\: \cos ( r x ) + c_{2}\: \sin ( r x )}\\{- c_{1}\: \sin ( r x ) + c_{2}\: \cos ( r x )}\\{c_{3}\: x}\\ \end{matrix}\right) ,
\left[ \begin{array}{cccc}_rT_{0,n_r+1}_rT_{1,n_r+1}\cdots_rT_{n_r,n_r+1}_rT_{0,n_r+2}_rT_{1,n_r+2}\cdots_rT_{n_r,n_r+2}\end{array}\right], r=1,\ldots,\mathbf r.
\|y\|_p = \int_{\Omega}\|m_{\mathfrak e \mathfrak e}^Ry\|_p dP\leq K \int_{\Omega}\|m_{\mathfrak e}^Ry\|_p dP= K \int_{\Omega}\|\sum_{j=1}^{n_\pi}(Q_\pi^{-\frac{1}{2}})_{jj}\varepsilon_j^{(\pi)}u_{ij}^{(\pi)}\|_p dP & \leq CK\sqrt p \|((Q_\pi^{-\frac{1}{2}})_{jj}u_{ij}^{(\pi)}D^{\frac{1}{p}})\|_{CR_p[L^p(\mathbb G)]},
\partial_{\underline i}\epsilon_I +{\textstyle\frac{1}{4}}\biggl ({\partial_{\underline i}{\overline{\cal H}_2}\over{\overline{\cal H}_2}}-{\partial_{\underline i}{\cal H}_2\over{\cal H}_2}\biggr )\epsilon_I -{\textstyle\frac{1}{2}}\left( \partial_{\underline i}U\right)\epsilon_I=0\, .
\tilde{S}_{rs}=\oint\frac{d\xi}{2\pi\imath}\frac{d\xi'}{2\pi\imath}\xi^{-r-\frac{1}{2}}\xi'{}^{-s-\frac{1}{2}}\frac{2\imath}{\sqrt{1+\xi^2}\,\sqrt{1+\xi'{}^2}}\ln\left(\frac{(1+i\xi)(1-i\xi')}{(1-i\xi)(1+i\xi')}\right).
+\sin\psi\cot\eta\cdot\left(\frac{F[-q,p](i\mu_{0})w_{1}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{2}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
0=\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\psi\rangle+\langle\mathbf{1}|\tilde{a}(\mathrm{m})\varphi\rangle\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\mathbf{1}\rangle,
\varphi_{\boldsymbol{0}}((baaxa(ba)^{2l}x)^2) = \varphi_{\boldsymbol{0}}(x(ba)^{4l+2}x) &= \omega(x)(ab)^{2l+1}\omega(x)=\begin{cases}(ba)^{2l+1}\omega(x)=a (ab)^{2l+1}\omega(x) =1.\end{cases}
\left( \begin{array}{c}v(t) v'(t)\end{array}\right)=e^{t\mathcal{A}_0}\left( \begin{array}{c}u^1 v^1\end{array}\right)+\int_0^t e^{(t-s)\mathcal{A}_0}\left( \begin{array}{c}0 F(s)\end{array}\right)ds.
F_{A}= \left( \begin{matrix}{{{\frac{1}{2}}F^{0}}}&{{0}}\\{{0}}&{{-{\frac{1}{2}}F^{0}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) h^{2}d z \wedge d{\bar{z}}\quad .
g =(g_1, \; e, \;g_3,\; e,\; g_5,\; \ldots, \; \;e, \;g_{2m-1},\; g_{2m}, \; e) \in (N_G(\c))^{n-1}\times G \times G, h = (e, \;h_2,\; e, \;h_4,\; e, \;\ldots,\;h_{2m-2}, \; e, \;h_{2m}, \; e)\in (N_G(\c))^{n+1},
[v_1, v_2]=-4\kappa v_3, [v_1, v_3]=4\kappa v_2, [v_1, v_4]=-2\kappa v_5, &[v_1, v_5]=2\kappa v_4, [v_2, v_3]=4\kappa v_1+r v_6, [v_2, v_4]=2\kappa v_5, [v_2, v_5]=2\kappa v_4, &[v_3, v_4]=-2\kappa v_4, [v_3, v_5]=2\kappa v_5, [v_4, v_5]=- \kappa v_6,
f_{a,b,c}^{-1}\bigl([0,a,c]\bigr) = \{Y=0\}\setminus\bigl\{[1,0,0]\bigr\}, f_{a,b,c}^{-1}\bigl([1,1,0]\bigr) =\{Z=0\}\setminus\bigl\{[0,1,0], [1,0,0] \bigr\}, f_{a,b,c}^{-1}\bigl([0,0,1]\bigr) =\emptyset, f_{a,b,c}^{-1}\bigl([a,at,ct-c]\bigr) =\emptyset~
\langle \hat{\phi}( x ) \hat{\phi}( x^{\prime}) \rangle_{\beta}= \int_{0}^{\infty}d \omega \int d^{2}k \left[ n_{\omega}( \beta ) ~ U_{\omega , k}^{*}( x ) U_{\omega , k}( x^{\prime}) + ( n_{\omega}( \beta ) + 1 ) U_{\omega , k}( x ) U_{\omega , k}^{*}( x^{\prime}) \right] ~ ,
S^Y(\omega) =\int_Y|V_{\omega}|^2\,d\mu_Y+2\sum_{n=1}^\infty\Re\int_Y P_\omega^nV_{\omega}\,\bar V_{\omega}\,d\mu_Y & =\int_Y|V_{\omega}|^2\,d\mu_Y+2\sum_{n=1}^\infty\Re\int_Y e^{-i\omega r_n}V_{\omega}\,\bar V_{\omega}\circ F^n\,d\mu_Y.
<\phi_\omega,\phi_\sigma>={1 \over (\omega-\sigma)}\lim_{r\rightarrow \infty}\int_{C_r}d\sigma^i\left(\nabla_i\phi_\omega^{*}\phi_\sigma-\phi_\omega^{*}\nabla_i\phi_\sigma-i(\omega+\sigma)a_i\phi_\omega^{*}\phi_\sigma\right)~~.
0\leq I(F;Y)-I(F;\hat{f}) = \int\int p(f|y)\log p(f|y)df\,dP_{Y}(dy)-\int p(f)\log p(f)df &\qquad- \Big(\int\int p(f,\hat{f})\log p(f|\hat{f})dfd\hat{f}-\int p(f)\log p(f)df\Big) &= \int\int p(f|y)\log p(f|y)df\,dP_{Y}(dy){+}H(F|\hat{f}).
\begin{array}{l}{T ( z ) = \hat{T}( z ) + T_{Q}( z )}\\{\}\\{G^{+}( z ) = \hat{G}^{+}( z ) + G_{Q}^{+}( z )}\\{\}\\{G^{-}( z ) = \hat{G}^{-}( z ) + G_{Q}^{-}( z )}\\{\}\\{J ( z ) = \hat{J}( z ) + J_{Q}( z )}\\ \end{array}
\langle x_{m}\frac{\partial H}{\partial x_{n}}\rangle=\delta_{mn}\frac{1}{\rho}\frac{\partial}{\partial E}\int_{H<E}(E-H)d\Gamma=\delta_{mn}\frac{1}{\rho}\int_{H<E}d\Gamma=\delta_{mn}\frac{\Omega}{\rho}.
\left. \prod_{i\ge 0}\left( \prod_{s=-2k+4i+1}^{-k+2i}(2n+s+1)\prod_{s=k-2i}^{2k-4i-2}(2n+s+1) \right) \middle/ \prod_{i=1}^{k-1}(2i+1)^{k-i}\right..
h ({\bf u}) = \frac{1}{V ( S_{3})}\ \frac{2 \pi J^{\prime}}{V ( S_{1 1}) V ( B_{1 2})}\int_{S_{3}}\frac{h ({\bf u}^{\prime})}{({\bf u}-{\bf u}^{\prime})^{2}}\d^{3}{\bf u}^{\prime}
\begin{aligned}\frac{2\lambda^k}{m}\sum_{i=1}^m\left\langle g_i^k, \bar x^k -\theta^* \right\rangle = \frac{2\lambda^k}{m}\sum_{i=1}^m\left\langle g_i^k-\nabla f_i(\bar{x}^k), \bar x^k -\theta^* \right\rangle \qquad+ \frac{2\lambda^k}{m}\sum_{i=1}^m\left\langle\nabla f_i(\bar{x}^k), \bar x^k -\theta^* \right\rangle \end{aligned}
\mathrm{Spec}(\Delta_0) =\{c_{\gamma}\;;\;\gamma\in\widehat{G}\;\mathrm{s.t.}\;\mathrm{mult}(\mathrm{triv.repr.},\mathrm{Res}^{G}_{K}(\rho_{\gamma}))\neq0\}\,, \intertext{and}\mathrm{Spec}(\Delta_1) =\{c_{\gamma}\;;\;\gamma\in\widehat{G}\;\mathrm{s.t.}\;\mathrm{mult}(\rho,\mathrm{Res}^{G}_{K}(\rho_{\gamma}))\neq0\}\,.
\mu^{n+1}(t,\cdot)=\mathbb{P}_{\delta}\mu_{0}+\int_{0}^{t}\left[\Delta\mu^{n+1}(\tau,\cdot)-\varepsilon\mathrm{div}\left(\left(\bar{m}+\mu^{n}(\tau,\cdot)\right)\Theta_{p}(\tau,\cdot,\mu^{n},Dw^{n})\right)\right]\d\tau,
g_k = (\sum_{s=0}^{d-a_i}(-1)^s\binom{d-a_i}{s}y^s_\alpha y^{d-a_i-s}_\beta)(\sum_{j=0}^{a_i-k}\mu_jy^j_\alpha y^{a_i-k-j}_\beta) = \sum_{s=0}^{d-a_i}\sum_{j=0}^{a_i-k}(-1)^s\mu_j\binom{d-a_i}{s}y^{s+j}_{\alpha}y^{d-k-s-j}_{\beta}= \sum_{j=0}^{a_i-k}\sum_{l=j}^{j+d-a_i}(-1)^{l-j}\mu_j\binom{d-a_i}{l-j}y^l_\alpha y^{d-k-l}_\beta.
A = (2a+2)(2a+3), E_{1,l}= \lambda[2(4a+3)z_1 + 8(a+1)b_1 + 2(a+1)(2l+d) - 2b_1 - l(l+d-1)], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1 - z_1(1+\lambda r^2)] \exp(-\lambda b_1 r^2),
(\alpha,\pi_n)_{L,n}= (\alpha_m,v_m)_{E_{\rho}^m,m}= -(v_m, 1+b_m)_{E_{\rho}^m,m}= - (\Phi_{E_{\rho}^m}(1+b_m)(v_{2m})-v_{2m}) = - (\Phi_K(\operatorname{N}_m(1+b_m))(v_{2m})- v_{2m}).
R(V):=\log \prod_{j\in V}\frac{p_j}{q_j}\prod_{j\in V^c}\frac{1-p_j}{1-q_j}=\sum_{j\in V}\log\big(1+(1-q_j)\omega \beta_j \big)+ \sum_{j\in V^c}\log\big( 1- q_j\omega \beta_j\big).
\prod_{n = R}^{\infty}\left( 1 - \frac{R^2 t^2}{n^2}\right)^2 \leq e^{-2RV(t)}\leq \prod_{n=R+1}^{\infty}\left( 1 - \frac{R^2 t^2}{n^2}\right)^2 , t \in [-1,1].
[\begin{matrix}x(t)\\y(t)\end{matrix}]=[\begin{matrix}4e^{t}/3-e^{-5t}/3&2e^{-5t}/3-2e^{t}/3\\ 2e^{t}/3-2e^{-5t}/3&4e^{-5t}/3-e^{t}/3\end{matrix}][\begin{matrix}1\\ 1\end{matrix}]
= \; C_{m_d}(w\!-\!z) \; - \; \frac{mc}{2\pi}\frac{4\pi^2}{\pi\!+\!g}\;\int_\Lambda d^2 x \;C_{m_d}(x\!-\!w)C_{m_d}(x\!-\!z) \; 2 \cos(\theta)
k_\lambda(t_0)=2\lambda \frac{(r-p)(\lambda+1)^{\frac{p-1}{r-1}}}{(2\lambda(p-1))^{\frac{p-1}{r-1}}}+(\lambda+1) \frac{(2\lambda(p-1))^{\frac{r-p}{r-1}}}{((r-p)(\lambda+1))^{\frac{r-p}{r-1}}}.
4(3f(x)e^{3F(x)})=3(2p-1)xf(x)^2f^{(1)}(x)+12pKf(x)f^{(1)}(x)-3f(x)^3, &4Z(x)=4e^{3F(x)}=3(2p-1)\int_0^x tf(t)^2f^{(1)}(t) \;dt + 6pKf(x)^2-3\int_0^x f(t)^3 \;dt +4e^{3F(0)}.
\|D_Hf(x)\|^Q =\|g^{-1}f^*h(x)\|^{Q/2}\leq K^{(2n+2)/2n}\det(g^{-1}f^*h(x))^{(2n+2)/4n}&=K^{(n+1)/n}HJ(x,f)^{(n+1)/n}=K^{(n+1)/n}J(x,f).
\left[ a_{i j}( p^{+}) , a_{l k}^{\dagger}( q^{+}) \right] = \left\{b_{i j}( p^{+}) , b_{l k}^{\dagger}( q^{+}) \right\}= \delta ( p^{+}- q^{+}) \left( \delta_{i l}\delta_{j k}- \frac{1}{N}\delta_{i j}\delta_{k l}\right)
s(p,q):=\inf\left\{\left.\int_0^{r_p(q)}e^{-2\frac{\phi(\gamma_t)}{n-m}}\d t\;\;\right|\left.\begin{array}{ll}\!\!\!\!\gamma: &\!\!\!\!{\gamma}(0)=p,{\gamma}(r_p(q))=q\end{array}\right.\right\}.
\mathcal{L}(\lambda) = \left[\begin{array}{c|c|c}-P_d M_{12}(\lambda) 0 \hline\sigma M_{12}(\lambda)^T \phantom{\Big{(}}M_{22}(\lambda) \phantom{\Big{(}}L_{t-1}(\lambda)^T \otimes I_n \hline0 \sigma L_{t-1}(\lambda)\otimes I_n 0\end{array}\right], \mbox{with}t=\frac{d}{2},
\frac{\partial U_{\Lambda}( \varphi , T )}{\partial \Lambda}= - \frac{K_{3}}{2}2 \pi T^{2}\Lambda g \left( \frac{\Lambda}{T}\right) \ln \det \left[ \Lambda^{2}+ \frac{\partial^{2}U_{\Lambda}( \varphi )}{\partial \varphi_{i}\partial \varphi_{j}}\right]
\varepsilon(F_{a}\triangleright\mathsf{Q}_{k}^{j})\varepsilon(E_{a}\triangleright\mathsf{Q}_{i}^{k})=\sum_{m,n,o,p}c_{n}^{m}c_{p}^{o}\varepsilon(F_{a}\triangleright(\mathsf{N}_{m}^{n})_{k}^{j})\varepsilon(E_{a}\triangleright(\mathsf{N}_{o}^{p})_{i}^{k}).
\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k\equiv0\pmod{\Phi_n(q)}.
v(\mathbf{g}') - v(\mathbf{g}) = 2 + 2(g_{n-1}+ p) - s(g_{n+1}+ p) - 3 g_{n-2}+ 1 - 2g_{n-1}+ s(g_{n-1}) \geq 2p - 3g_{n-2}- \lceil \log_2(p) \rceil + 3 \geq \frac{11p}{7}- \lceil \log_2(p) \rceil + \frac{12}{7}> 0.
[A,K] = \sum_{i=1}^{L+1}\alpha_i\{A^i,C\}+\delta\{B,C\}+2\epsilon C + \beta\{\{A,B\},C\}&\ \ + \sum_{i=1}^{L+1}\frac12 m_i\{A^i,C\}+\frac{n}{2}\{A,\{B,C\}\}+ \ell_1 C + \ell_2\{B,C\}.
\begin{array}{l}\vec{a}\,' = sin(2\pi p(\sigma-t)/L + \alpha)\hat{x}+ cos(2\pi p(\sigma-t)/L + \alpha)\hat{z}, \vec{b}\,' = sin(2\pi q(\sigma+t)/L + \beta)[cos\Psi \hat{x}+sin\Psi \hat{y}] + cos(2\pi q(\sigma+t)/L + \beta)\hat{z}.\end{array}
\displaystyle{\langle \Psi^{\rho r}(z_{1})\Psi^{\sigma s}(z_{2})\rangle = C_{\Psi}\frac{I^{\rho\sigma}(\hat{{\cal X}}_{12})I^{rs}(V(z_{12}))}{(\det{\cal X}_{12})^{\frac{1}{2}\eta}}}
\frac{1}{N}\sum_{c=1}^{m}\sum_{i\in I_{c}^{N}}g^{(c)}(T,\,z_{i}^{N}(T))-\frac{1}{N}\sum_{c=1}^{m}\sum_{i\in I_{c}^{N}}g^{(c)}(0,\,z_{i}^{N}(0))=\Theta_{1}+\Theta_{2}+\Theta_{3}+\Theta_{4}
P_{1,n+1}(0) - P_{1,n+1}(a) = -a \Sigma_{1, n+1}^* L_{1, n+1}R_{n}(a) v_{n}= -a \Sigma_{1, n+1}^* v_{n+1}u_n^* \widetilde H_{1,n}^{-1}R_{n}(a) v_{n}= -a P_{1,n+1}(0) u_n^* \widetilde H_{1,n}^{-1}R_{n}(a) v_{n}.
S_{\mathbf{n}}(t)=\Gamma(t)-\tfrac{1}{2}\tilde{\mathbf{n}}^{1/2}h_{\mathbf{n}}^{(k+4)/2}f(u_{0})t^{T}\Lambda\otimes\bigl(\Phi(u_{0})\zeta(u_{0})\bigr)-\Upsilon_{\mathbf{n}}(t)+\tilde{R}_{\mathbf{n}}(t),
\displaystyle \frac{1}{\lambda - x}= \displaystyle \frac{1}{\epsilon}= \displaystyle \frac{( - 1 )^{\delta}e^{i Z_{N}( x + \epsilon )}i Z_{_{N}}^{\prime}( x + \epsilon )}{1 + ( - 1 )^{\delta}e^{i Z_{N}( x + \epsilon )}}+ . . . ={\cal R}_{\delta}( x + \epsilon ) + . . .
A_H= \begin{bmatrix}a_1 a_2 \dots a_{n-1}a_n a_2 a_3 \dots a_{n}a_{n+1}\vdots \vdots \reflectbox{$\ddots$}\vdots \vdots a_{n-1}a_{n}\dots a_{2n-3}a_{2n-2}a_n a_{n+1}\dots a_{2n-2}a_{2n-1}\end{bmatrix}.
x^r \left(\frac{x}{x-a}\right)^k = x^r \left(1 - \frac{a}{x}\right)^{-k}&= x^r \sum\limits_{i=0}^\infty{-k \choose i}\left(- \frac{a}{x}\right)^i &= \sum\limits_{i=0}^\infty{-k \choose i}(-a)^i x^{r-i}.
\|\partial_xw\|^2_{L^2\left(\Omega_1\right)}+\|\partial_zw\|^2_{L^2\left(\Omega_1\right)}+\|\partial_xw\left(\cdot,z_\iota\right)\|^2_{L^2\left(\Omega_x\right)}\leq c \|\partial_x\xi_2^\prime\|_{L^2\left(\Omega_2\right)}\|\partial_xw\left(\cdot,z_\iota\right)\|_{L^2\left(\Omega_x\right)},
F^{}(x_{i+1/2}) = \underbrace{\sum_{k=1}^{i}x_k n_k \Delta x_k \Bigg( \sum_{j=\alpha_{i,k}}^{I}n_j\beta_{j,k}\Delta x_j +n_{\alpha_{i,k}-1}\beta_{\alpha_{i,k}-1,k}(x_{\alpha_{i,k}-1/2}- (x_{i+1/2}-x_k))\Bigg)}_{=:J^{}_{i+1/2}(n)}&+{\cal O}(\Delta x^2)
\left[ \left( \frac{r}{g}\right)^{2}-\left( \frac{p}{f}\right)^{2}\right]^{\frac{1}{2}}\left[ 2+\left( \frac{p}{f}\right)^{2}\left( \frac{r}{g}\right)^{-2}\right] =\frac{\frac{d}{df}\left( \dfrac{f\dot{p}}{p}\right)\left( \dfrac{p}{f}\right)}{2H_{0}\frac{d}{df}\left( \dfrac{f}{p}\right)}.
( 1 + \frac{K}{2 f_{5}}) d x^{+}d x^{-}+ \frac{R_{s}^{2}}{R^{2}}( h + \frac{K}{4 f_{5}}) d x^{-}d x^{-}+ \frac{R^{2}}{R_{s}^{2}}\frac{K}{4 f_{5}}d x^{+}d x^{+}
G_0^0(x) = ~-\frac1{2\pi}\int_1^x \frac1{\xi}~d\xi+ \frac1{2\pi}\int_0^1\frac{e^{i\xi}-1}{\xi}~d\xi+\frac{1}{2\pi}\int_1^{\infty}\frac{e^{i\xi}}{\xi}~d\xi &-\frac1{2\pi}\int_1^{2}\frac{\chi(\xi)}{\xi}~d\xi = ~-\frac1{2\pi}\log|x|+\frac{c_1}{2\pi}.
{\frac{1}{\omega}}\left( -{\frac{9}{2 0}}-{\frac{z_{0}^{2}}{l^{2}}}\right) <{\frac{1}{3 2 \omega^{2}l^{2}}}\left( - 3 2 \omega z_{0}^{2}+ 9 l^{2}+ 3 \sqrt{- 6 4 \omega z_{0}^{2}l^{2}+ 9 l^{4}- 6 4 l^{2}z_{0}^{2}\omega^{2}}\right) \, ,
\mathfrak{g}=\bigoplus_{k\in \mathbb{Z}}\mathfrak{g}^{k,-k}\quad\quad\mathfrak{g}^{k, -k}= \{X\in\mathfrak{g}|~XH^{r, n-r}_p\subseteq H^{r+k, n-r-k}_p,\ \forall r\}.
K_{u v}^{x}k_{\Lambda}^{u}k_{\Sigma}^{v}-{\o{\lambda}{2}}\varepsilon^{x y z}{\cal P}_{\Lambda}^{y}{\cal P}_{\Sigma}^{z}={\o{1}{2}}f_{\phantom{\Delta}\Lambda \Sigma}^{\Delta}{\cal P}_{\Delta}^{x}
\left( \begin{array}{c c c}{1}&{0}&{0}\\{0}&{- 1}&{0}\\{0}&{0}&{0}\\ \end{array}\right) \quad \mathrm{a n d}\quad \left( \begin{array}{c c c}{0}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{- 1}\\ \end{array}\right) .
\mathfrak{d}_2^{m+1}\dot{\phi}(n) &=\frac{\Gamma(N+\frac{1}{2}+m)}{\Gamma(N+\frac{1}{2})}\int_0^2\cdots\int_0^2 (1+n+t_1+\cdots +t_m)^{-N-\frac{1}{2}-m}\,dt_1\cdots dt_m.
\begin{aligned}(u^e_1)_t-\mu_1\Delta u_1^e -c^e_1 u_1^e = \alpha_1 u_1 u_2^e (u^e_2)_t-\mu_2\Delta u_2^e -c^e_2 u_2^e = \alpha_2 u_2 u_1^e\end{aligned}
\begin{aligned}U^{\ast}_{\psi}\omega U^{}_{\psi}= \widetilde{\psi}^{\ast}\omega = \omega \circ \widetilde{\psi}\forall \omega \in A^\ast(G_1) , U^{\ast}_{\psi}\delta_{\psi_2}U^{}_{\psi}=\delta_{\psi_2 \psi}-\delta_{\psi}\forall \psi_2 \in G_2.\end{aligned}
< x | e^{- \beta ( \frac{1}{2}K^{i j}v_{i}v_{j}- J^{i}v_{i})}| y > = \sum_{n = 0}^{\infty}< x | e^{\varphi^{i}v_{i}}( - \frac{1}{2}\beta K^{i j}v_{i}v_{j})^{n}| y > / n !
\widehat{h}(\widehat{x}):= \frac{\widehat{h}(\widehat b_k)-\widehat{h}(\widehat a_k)}{\widehat{b}_k-\widehat{a}_k}\cdot (\widehat{x}-\widehat{a}_k)+\widehat{h}(\widehat{a}_k).
\Gamma\in \mathcal{M}(D_6, D_{10}, D_{14}, D_{15}, D_{17}, D_{18}, D_{21}, D_{22}, D_{24}) &= \mathcal{M}(D_{10}, D_{21}, D_{22}, D_{24})\cup\mathcal{M}(D_6, D_{14}, D_{15}, D_{17}, D_{18}).
\int \dfrac{x}{(- x^2 (1-q)^2;q^2)_\infty}\left(\frac{q^{-1}-q^{-\nu}[\nu]_q^2}{x}+qx \right) J_\nu^{(2)}( x |q^2) d_qx = & \frac{x}{q(- x^2 (1-q)^2;q^2)_\infty}\left( J_\nu^{(2)}( x |q^2)-x D_{q^{-1}}J_\nu^{(2)}( x |q^2) \right),
T(h\otimes \chi_{\{y_0\}})(\xi,y_0) =\int (h\otimes \chi_{\{y_0\}})d T^*(\delta_{(\xi,y_0)}) = \sum_{\eta \in \alpha}h(\eta)\cdot T^*(\delta_{(\xi,y_0)})(\{(\eta,y_0)\})= \sum_{\eta \in \alpha}r(\eta,\xi)\cdot h(\eta).
\sigma = 12\kappa_2+6(\kappa_1-\kappa_2)\vert T\vert^2, a_{ii}=3\kappa_2+(\kappa_1-\kappa_2)(2T_i^2+\vert T\vert^2), \vert\mathring{a}\vert^2 = 3(\kappa_1 - \kappa_2)^2\vert T\vert^4.
{\cal E}=\frac{1}{2}\,\mu^{1-2\nu}\sum_{n=1}^\infty\int\frac{d^2p}{(2\pi)^2}\left\{{\bf p}^2+\frac{\pi^2}{\ell^2}\left[(n+s)^2+\frac{g}{3}\right]\right\}^{\nu}\Bigg|_{\nu=1/2}.
\sum_{k}\frac{\partial^2{q^{0}}}{\partial \xi_k\partial\gamma_k}(h'(\xi),\xi)= \sum_{k}\frac{\partial q_k}{\partial \xi_k}-\sum_{j,k}\frac{\partial q_j}{\partial\gamma_k}\frac{\partial^2 h}{\partial\xi_k\partial\xi_j}= 2\sum_{k}\frac{\partial q_k}{\partial \xi_k}+ \mathrm{div}(H_{q^{0}}).
\deg e_{\ell_1,\ell_2,\ell_3}= \deg \left( X_1^{s-\ell_3}X_2^{s- \ell_3}X_3^{s- \ell_1}\right) = (s-\ell_3)m_1 +(s-\ell_2)m_2+ (s-\ell_1)m_3.
J_s = \left[\begin{array}{cccccc}0 1 0 \cdots 0 0 0 0 1 \cdots 0 0 0 0 0 \cdots 0 0 \vdots \vdots \vdots \vdots \vdots 0 0 0 \cdots 0 1 0 0 0 \cdots 0 0 \end{array}\right].
P ( u , x ) = \lim_{N \rightarrow \infty}\left. \, P_{2 n}( \frac{x}{N}) \right|_{n = N t / 2}= \cos u x = \Gamma ( \frac 1 2 ) \frac{J_{- \frac 1 2}\left( u x \right)}{\left( u x / 2 \right)^{- \frac 1 2}}.
\operatorname*{rot}\left(^{\left( \alpha \right)}\bar{T}\left(u,v\right) \right) =\frac{1}{w}\left[ \frac{\partial \left( T^{1}g_{12}+T^{2}g_{22}\right)}{\partial u}-\frac{\partial \left( T^{1}g_{11}+T^{2}g_{12}\right)}{\partial v}\right] .
a^{(\rho)}=(1-\widetilde{F}_{0}^{(\rho)}(z)) \prod_{i=\rho-p}^{\rho-1}\widetilde{F}_{0}^{(i)}(z)=(-C_{1}^{(\rho)}z^{-1}+O(z^{-2}))(z+O(1))=-C_{1}^{(\rho)},
g_{y\bar{y}}=\partial_y\partial_{\bar{y}}K=\frac n{n+1}+{|y|^{2(n-1)}\over(n-1)!}{e^{-|y|^2}\over Q(|y|^2)}\left(n-|y|^2-{|y|^{2n}\over(n-1)!}{e^{-|y|^2}\over Q(|y|^2)}\right),
\left[ \frac{\mathrm{d}^{2}}{{\mathrm{d}{\rho}^{*}}^{2}}+ \left( \omega - e \frac{Q}{\rho}\right)^{2}- \mu^{2}\frac{\Delta ( \rho )}{R^{2}( \rho )}\right] \chi ({\rho}^{*}) = 0 ,
\min\, \tau_2=3a_{0,1}+a_{0,2}+2a_{1,1}+a_{1,2}\textrm{s.t.}2a_{0,1}+a_{0,2}+4a_{1,1}+2a_{1,2}+a_{2,0}+2a_{2,1}+a_{2,2}=1, &2a_{1,1}+a_{1,2}+a_{2,0}+2a_{2,1}+a_{2,2}\le\mu_R, &a_{0,1}+a_{0,2}+2a_{1,1}+2a_{1,2}+a_{2,1}+a_{2,2}\le\mu_T.
\cos \phi^{( 1 )}= \frac{a}{L^{( 1 )}}( \frac{l}{4 d}( a_{l}+ a_{r}) + \frac{w_{1}}{2}) , \sin \phi^{( 1 )}= \frac{a}{L^{( 1 )}}\frac{\sqrt{3}l}{4 d}( a_{l}- a_{r}) ,
R(x_2, x_1) x_1 = - (3/4) \lambda^2 x_2 + \lambda x_n, R(x_2, x_i) x_i = (\lambda /2) x_n (\mbox{for $i=3, \ldots, n-1$}) , R(x_2, x_n) x_n = (1/4) \lambda^2 x_2 .
\lim_{t\to\infty}\hat{\psi}(a_{\Lambda}J_t(b_{\Lambda^{'}})) = \sum_{i,j}\left(\prod_{x\in \Lambda}\mathrm{Tr}(h_{x,i}h^\ast_{x,j}a_x)\right)\beta_{\Lambda^c, i, j}\alpha^{(|\Lambda^{'}|)}(J^{-1}_{\Lambda^{'}}(b_{\Lambda^{'}}))