harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi(\tilde{T})=\left|xI_{2k}-\tilde{T}\right| =\left|\begin{array}{cc}xI_k-A -B -B xI_k-A \end{array}\right|=\left|\begin{array}{cc}xI_k-A xI_k-T -B xI_k-T \end{array}\right| &=\left|\begin{array}{cc}xI_k-A+B 0_k -B xI_k-T \end{array}\right| &=\left|xI_k-A+B\right|\left|xI_k-T\right|
\sum_{b=1}^N m^{(b)}c^{(b)}: \cos \left(2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi\!+\!gN}}U_{1 b}\Phi^{(1)}(x)+ 2\sqrt{\pi}\sum_{I=2}^N U_{I b}\Phi^{(I)}(x) - \frac{\theta}{N}\right) :
\dot\Delta(\xi_n)= -\dot \theta_N(\xi_n)2\sin \theta_N(\xi_n)+ \frac{1}{(2i\xi_n)^N}\big(\dot R_N(1,\xi_n)_1+ \dot S_N(1,\xi_n)_2\big)+ \frac{\ell^2_n}{n^{N+1}}.
E(a^3b^4h^2m,a^3b^2hn;l^2)= l_1 \cdot E(a^3b^4h^2m_1\overline{l_1}^2,a^3b^2hn_1\overline{l_1}^2;d) \times e\left(-\frac{k_2^2n_1^3}{k_3^3hdm_1^2l_1^2}\right) \cdot e\left(\frac{\overline{l_1}^2k_2^2n_1^3}{k_3^3hdm_1^2}\right).
f =\int_{\Theta}T_{w}(f)d\mu(w) &=\int_{\Theta}v(w)s(w)K\pi_{G(w)}\xi^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w) &=K\int_{\Theta}v(w)s(w)\pi_{G(w)}\xi_{w}^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w) &=KS_{\xi\chi}f.
\mathcal{S}_{q}(z)=\frac{z}{1-q}\,_{1}\phi_{1}(0;q^{3};q^{2},q^{2}z^{2}) \quad\mbox{and}\mathcal{C}_{q}(z)=\,_{1}\phi_{1}(0;q;q^{2},qz^{2})
\tilde h_{n+m}(\omega) \left(\frac{\bar s}{k^n}\right) = \tilde h_{n+m}(\omega) \left(\frac{k^m\cdot \bar s}{k^{n+m}}\right) =a_{k^m \cdot \bar s}^{n+m}(\omega) =a_{\bar s}^n(\omega) = \tilde h_n (\omega) \left(\frac{\bar s}{k^n}\right) .
\mathrm{div}_{\nabla}(\mathrm{grad}_h\rho) = b(-\nabla_{V_1}V,V_1) + b(-\nabla_{V_2}V,V_2) = b(\rho\lambda_1V_1 - V_1,V_1) + b(\rho\lambda_2V_2 - V_2, V_2) = = \rho\lambda_1 + \rho\lambda_2 - 2 = 2(H\rho - 1),
E\|Z_2-P_{(\kappa)}(Z_2)\|_{(B^\rho_{1\infty})^*}= E \sup_{\|\eta\|_{B^\rho_{1\infty}}\le 1}\left | Z_2 (\eta - P_{V^{\otimes d}_\kappa}[\eta])\right| & \lesssim \sum_{\kappa<\lambda}2^{-\lambda(\rho-d/2)}E\max_{k,j}\left | Z_2(\Phi_{\lambda,k, j}) \right| \lesssim \sum_{\kappa<\lambda}2^{-\lambda(\rho-d/2)}\sqrt \lambda =_{\kappa \to \infty}o(1).
\begin{cases}-c(x)\frac{\partial}{\partial y}w_a(y;x)-d_A\frac{\partial^2}{\partial y^2}w_a(y;x) = M_{11}w_a+M_{12}w_b, begin{align*}5pt] -c(x)\frac{\partial}{\partial y}w_b(y;x)-d_B\frac{\partial^2}{\partial y^2}w_b(y;x) = M_{21}w_a+M_{22}w_b. \end{cases}
|v_{n}(t)|^{2}=\exp\!\left\{-f(t)n^{2}-2\mu nw_{t}\right\}=\exp\biggl\{-f(t)\left(n+\frac{\mu w_{t}}{f(t)}\right)^{\!2}+\frac{\mu^{2}|w_{t}|^{2}}{f(t)}\biggr\},
\bar{A}(z) = \frac{1}{\sqrt{2}\pi^2}\int_0^{2\pi}\int_0^{2\pi}\frac{\sqrt{2}+ \sin(v)}{Q(u,v,1\,;\sqrt{z})^2}\mathrm{d}u\mathrm{d}v, \bar{V}(z) = \frac{1}{\sqrt{2}\pi^2}\int_0^{1}\int_0^{2\pi}\int_0^{2\pi}\frac{r\sqrt{2}+ r^2\sin(v)}{Q(u,v,r\,;\sqrt{z})^3}\mathrm{d}u\mathrm{d}v\mathrm{d}r,
A_{4,1}=-\frac{1}{2}M_1 V_1 + O(M_i^4), \qquad A_{4,0}=-\frac{1}{8}V_{\Phi}-\frac{13}{16}M_1 V_1+\frac{5}{6 P}e^{-\Phi_{\infty}}M_2 V_1 + O(M_i^4), begin{align*}2mm]
\frac{\partial g}{\partial u}(P)= \sum\limits_{i=i_0}^{\infty}i\alpha_iu_0^{i-1}=\pi^{{\rm ord}(\alpha_{i_0})}\sum\limits_{i=i_0}^{\infty}i\pi^{-{\rm ord}(\alpha_{i_0})}\alpha_iu_0^{i-1}= \pi^{{\rm ord}(\alpha_{i_0})}(\pi^{-{\rm ord}(\alpha_{i_0})}\alpha_{i_0}+\sum\limits_{i=i_0+1}^{\infty}i\pi^{-{\rm ord}(\alpha_{i_0})}\alpha_iu_0^{i-1}).
\int_C \frac{1}{e^{2\pi i (s+\bar{f})}-1}q^{s+\bar{f}}\frac{\Gamma (1+\frac{dH}{2\pi i}+sd+\bar{f}d)}{\Gamma (1-\frac{d^\prime H}{2\pi i}-sd^\prime+\bar{f}d^\prime)\prod_{i=1}^N\Gamma(1+\frac{w_i H}{2\pi i}+w_is+\bar{f}w_i)},
L\left(\frac{x}{1+x}\right) =-\mathrm{Li_2}(-x)-\frac{1}{2}\log x \log(1+x),(0\leq x) &=\frac{1}{2}\int_0^{x}\left\{\frac{\log (1+y)}{y}-\frac{\log y}{1+y}\right\}dy,(0\leq x).
\langle x,y\rangle:=x^{T}My=[x_{1},x_{2}][\begin{matrix}a&b\\b&d\end{matrix}][\begin{matrix}y_{1}\\y_{2}\end{matrix}]=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.
\begin{array}{l}(m,-m,1,p) \longleftrightarrow \tilde{\beta}^{\frac{2k'}{\delta}m-\frac{k}{\delta}(p-k')}, (m,-m,0,p) \longleftrightarrow \tilde{\beta}^{\frac{2k'}{\delta}m+\frac{k}{\delta}p}\tilde{\gamma}.\end{array}
\int_{S}\sum_{k=1}^{n}|f_{k}(x)g_{k}(x)|\mu(dx)\le(\int_{S}\sum_{k=1}^{n}|f_{k}(x)|^{p}\mu(dx))^{\frac{1}{p}}(\int_{S}\sum_{k=1}^{n}|g_{k}(x)|^{q}\mu(dx))^{\frac{1}{q}}.
=\frac{1}{r}\frac{d\psi}{dr}\frac{\partial x_{i}}{\partial x_{i}}+x_{i}\frac{d\psi}{dr}\frac{\partial}{\partial x_{i}}\frac{1}{r}+\frac{x_{i}}{r}\frac{\partial}{\partial x_{i}}\frac{d\psi}{dr}
\mathcal{L}= -m \sqrt{- \Delta^{1/2}(g_{\mu \nu}\dot{x^{\mu}}\dot{x^{\nu}}) - \Delta^{-1/2}\sum_i X_i^{-1}\mu_i^2 (L \dot{\phi_i}+ A^i_{\mu}\dot{x^{\mu}})^2}
\| AB-A_0B_0\|^2 &\sim \sum_{j=1}^N \sum_{i=1}^{M-1}\left[ \sum_{k=1}^{H_0-1}\{(a_{ik}-a_{iH})b_{kj}- (a^0_{ik}-a^0_{iH_0})b^0_{kj}\}+ \sum_{k=H_0}^{H-1}(a_{ik}-a_{iH})b_{kj}+ (a_{iH}-a^0_{iH_0}) \right]^2 &= \sum_{j=1}^N \sum_{i=1}^{M-1}\left\{\sum_{k=1}^{H_0-1}(a_{ik}b_{kj}- a^0_{ik}b^0_{kj}) + \sum_{k=H_0}^{H-1}a_{ik}b_{kj}+ c_i \right\}^2.
\begin{cases}u_{tt}- \Delta u + a(x)\rho(x, u_t) = 0, \, (x,t) \in \Omega \times(0, + \infty), begin{align*}0.8mm]u = 0 \,, \mbox{on}\partial \Omega \times (0,+ \infty), begin{align*}0.8mm]u(x,0) = u^0(x), \, u_t(x,0) = u^1(x) \,, \mbox{on}\Omega,\end{cases}
G_{0}^{( 2 )}( \vec{r}, z , z^{\prime}) = \frac{1}{( 2 \pi )^{d - 1}}\int d^{d - 1}p \sum_{n}e^{i \vec{p}. \vec{r}}u_{n}( z ) u_{n}^{*}( z^{\prime}) G_{0 , n}( \vec{p}) ,
F^{n-1}=-\delta u^{n-1}\cdot\nabla\varrho^{n-1}+\div((\kappa^{n-1}-\kappa^{n-2})\nabla\varrho^{n-1}), & H^{n-1}=h^{n-1}-h^{n-2}-(\delta u^{n-1}+\nabla\delta b^{n-1})\cdot\nabla u^{n-1}-(\lambda^{n-1}-\lambda^{n-2})\nabla\pi^{n-1}.
= 2\sum_{s=odd}^{\infty}e^{-\pi \tau_{2}\frac{s^{2}}{2}}(\frac{s^{2}}{4}+\frac{1}{2\pi}\frac{d}{d\tau_{2}}) \left( \frac{1}{\sqrt{2\tau_{2}}}e^{-\frac{\pi \tau_{1}^{2}s^{2}}{2\tau_{2}}}\sum_{n}e^{-\frac{\pi n^{2}}{2\tau_{2}}-\frac{n\pi \tau_{1}s}{\tau_{2}}}\right)
\widehat{S}\,\rho_{j}(\mathcal{X}; \hat{z})\,\widehat{S}^{*}= \widehat{S}\,\varrho(\mathcal{X}_{*j}; \hat{z})\,\widehat{S}^{*}= \varrho(S\mathcal{X}_{*j}; \hat{z}) = \varrho((S\mathcal{X})_{*j}; \hat{z}) = \rho_{j}(S\mathcal{X}; \hat{z}).
\varepsilon_1=\frac{p_2}{p_1}\varepsilon_2+\frac{(p_1-p_2)p_2}{2\sqrt{p_1p_2}}, \eta=\frac{p_2}{\sqrt{p_1p_2}}\varepsilon_2+\frac{q_2}{2}, 0\leqslant\varepsilon_2\leqslant\frac{\sqrt{p_1p_2}}{2}.
J_{2}V(x) = J_{2}(Ux_{1}- U(x_{2}\cdot n) \cdot n) &= U(x_{1}) \cdot m - U(x_{2}\cdot n) \cdot m \cdot n &= U(x_{1}\cdot m) + U(x_{2}\cdot n \cdot m) \cdot n &= U(J_{1}x_{1}) - U(J_{1}x_{2}\cdot n ) \cdot n &= VJ_{1}(x).
d(\varphi_i(x_0),\mathcal{Y}(x_0)g)=\min_{1\leq j\leq l}d(\varphi_i(x_0),\varphi_j(x_0)g) \leq \min_{1\leq j\leq l}(d(\varphi_i(x_0),\,\varphi_j(x_0)g_n)+d(\varphi_j(x_0)g_n,\varphi_j(x_0)g)) \leq d(\varphi_i(x_0),\mathcal{Y}(x_0)g_n)+d(g_n,g) = d(\varphi_i(x_0),\mathcal{Y}_0(x_0g_n))+d(g_n,g).
\tilde{A}^Q(h) = H_K (A^Q(h)^T)= H_K \big( Q_K^{{1}/{2}}\otimes Q_K^{{1}/{2}}\big)(A^I(h)^T) &= H_K \big( Q_K^{{1}/{2}}\otimes Q_K^{{1}/{2}}\big)H_K^T H_K (A^I(h)^T) &= H_K \big( Q_K^{{1}/{2}}\otimes Q_K^{{1}/{2}}\big) H_K^T \tilde{A}^I(h),
\left\| J_{\varphi}\left( \mathbf{x}\right) \right\|_{1}=\max_{j=1,...,d}{\{\left| \frac{{\partial \varphi}_{1}(\mathbf{x})}{{\partial x}_{j}}\right|+\mathellipsis +\left| \frac{{\partial \varphi}_{d}(\mathbf{x})}{{\partial x}_{j}}\right|\}},
\bar{g}( s ) = \frac{g s^{\frac{y}{2}}}{1 - \frac{\pi A ( y ) b ( y ) g}{y}( s^{\frac{y}{2}}- 1 )}= \frac{g_{I R}^{*}| \tilde{g}| s^{y / 2}}{1 + | \tilde{g}| s^{y / 2}}, \ \ \tilde{g}= \frac{g}{g - g_{I R}^{*}}\ \in \ \ ( - \infty , 0 ) .
AX =\sum_i \omega_i(AX)e_i =-\sum_i\omega_{i3}(X)e_i &=\frac{h(u)^2-2(h'(u))^2+h(u)h''(u)}{h(u)^2-(h'(u))^2}\mathrm{d}u(X)e_1 +\frac{\cos(u)h'(u)+\sin(u)h''(u)}{\sqrt{h(u)^2-(h'(u))^2}}\mathrm{d}v(X)e_2, Ae_1 = \frac{h(u)^2-2(h'(u))^2+h(u)h''(u)}{h(u)(h(u)^2-(h'(u))^2)}e_1, Ae_2 = \frac{\cos(u)h'(u)+\sin(u)h''(u)}{h(u)(h(u)^2-(h'(u))^2)}e_2.
\inf_{\xi\in\Gamma^{0,t}_{x,y}(\overline{\Omega})}\int^t_0L(\xi(s),\dot{\xi}(s))\ds=\inf_{\eta\in\Gamma^{-t,0}_{x,y}(\overline{\Omega})}\int^0_{-t}L(\eta(s),\dot{\eta}(s))\ds,
\|u\|_{LE}=\|u\|_E+\|\partial u\|_{l_\infty^{-1/2}(L_t^2L_x^2)}+\|r^{-1}u\|_{l_\infty^{-1/2}(L_t^2L_x^2)},\ \|u\|_{LE_m}=\sum_{|a|\leq m}\|Y^au\|_{LE},
{\cal G}_{\omega}( r , r^{\prime}, \theta ) = \sum_{n = 0}^{\infty}r^{1 - D / 2}\left[ e_{n}H_{\nu}^{( 1 )}( k r ) + f_{n}H_{\nu}^{( 2 )}( k r ) \right] C_{n}^{( - 1 +{D / 2})}( z ) \quad ( r^{\prime}> r > a ) .
\mathcal E_{\lambda,\mu}(t\bar{u}) = \frac{1}{2}\left\|\bar{u}\right\|^2 t^2 -\frac{\mu}{2^*}\int_\Omega |\bar{u}|^{2^*}dx\; t^{2^*}- \lambda\int_\Omega G(t\bar{u})dx & \leq \left( \frac{1}{2}\left\|\bar{u}\right\|^2 -\lambda M \left\|\bar{u}\right\|^2_{L^2(\Omega)}\right) t^2 -\frac{\mu}{2^*}\int_\Omega |\bar{u}|^{2^*}dx\; t^{2^*}&<0
\begin{array}{c}\Gamma^M\Gamma^N+\Gamma^N\Gamma^M=2\eta^{MN}, \qquad\eta^{MN}=(+,-,-,\dots), (\Gamma^M)^*=-\Gamma^M, \qquad (\Gamma^0)^{\rm T}=-\Gamma^0, \qquad(\Gamma^i)^{\rm T}=\Gamma^i, \quad i=1,\dots,9.\end{array}
\dot x_\text s(t) =Ax_\text s(t)+B_uu(t)-L\bigl(y(t)-C_yx_\text s(t)\bigr), begin{align*}-.75ex]\intertext{\smash{which is again an observer, and \eqref{eq:Ksd:a}--\eqref{eq:Ksd:s2a}read}}begin{align*}-4.25ex] \dot x_\text a(t) =Ax_\text a(t)+B_uu(t),x_\text a(t_i)=x_\text s(t_i) u(t) =Fx_\text a(t)+\eta(t).
I\left( a\right) =\frac{\int\limits_{{\cal F}_{a}}\left[{\cal D}\Psi{\cal D}\overline{\Psi}\right] e^{-{\cal A}_{MTM}}}{\int\limits_{{\cal F}_{0}}\left[{\cal D}\Psi{\cal D}\overline{\Psi}\right] e^{-{\cal A}_{MTM}}},
\langle (\xi,\eta),(d,w)\rangle =\langle d,-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)+\nabla\!g(\overline{x})\mu\rangle +\langle \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(d)+\nabla\!g(\overline{x})u,\eta\rangle =\langle d,\nabla\!g(\overline{x})\mu\rangle+\langle\nabla\!g(\overline{x})u,\eta\rangle.
\partial_t (\varphi^1 - \varphi^2) - \Delta(\mu^1 - \mu^2) = \mathcal P(\sigma^1 - \sigma^2) h(\varphi^1) + (\mathcal P \sigma^1 - a - \alpha u)\left( h(\varphi^1) - h(\varphi^2) \right) \mu^1 - \mu^2 = - \Delta (\varphi^1 - \varphi^2) + \psi'( \varphi^1) - \psi'(\varphi^2).
(-i\nabla - \textbf{A})^{2}= \sum_{j=1}^{d}\Bigg( \left( -i \partial_{x_j}+ \frac{b_{j}y_{j}}{2}\right)^{2}+ \left( -i\partial_{y_j}- \frac{b_{j}x_{j}}{2}\right)^{2}\Bigg) + \sum_{\ell=1}^m \partial_{z_\ell}^2.
\delta_{(n)}|A\rangle={\cal D}_n|\Lambda_n\rangle\,,\qquad{\cal D}_n\equiv\sum_{j=0}^{n-1}(-)^j\!\!\!\!\sum_{l_1,\ldots,l_{j+1}=1}^n\delta_{nl_{j+1}}\prod_{i=1}^j\frac{\varepsilon^{l_i l_{i+1}}}{\lambda_{l_i n}}{\bf a}_{l_{i+1}l_i}^{}D_{l_1}\,,
\|v \otimes u\|_{L^2(0,T;L^2)}\leq C \|v\|_{L^r(0,T;L^q)}\|u\|^{1-\theta}_{L^{\infty}(0,T;L^2)}\|u\|^{\theta}_{L^{2}(0,T;\dot H^1)},\ \ \ \theta = d/q.
\frac{\varphi^{[k:\mathbb{F}_p]}(z)}{z}= x^{[k:\mathbb{F}_p]}\Bigg( f_{\tau}\varphi(f_{\tau \circ \varphi^{-1}}) \ldots \varphi^{[k:\mathbb{F}_p]-1}(f_{\tau \circ \varphi^{-[k:\mathbb{F}_p]-1}}) \Bigg)^{-1}
A_4 (k_1,k_2,k_3,k_4) \rightarrow - i \int{d^7 k\over (2\pi)^7}{\sum_{IJ\mu}A^{I \mu}_{\J}(k_1,k_2, k) A^{J}_{\I \mu}(k_3,k_4, -k) \over k^2 - i\epsilon}\ ,
\Vert \widehat{\lambda}_h\widehat{u}_{\mathrm{nc}}-\lambda_h u_{\mathrm{nc}}\Vert_{L^2(\Omega)}\le \lambda_h \Vert u-u_{\mathrm{nc}}\Vert_{L^2(\Omega)}+\widehat{\lambda}_h \Vert u-\widehat{u}_{\mathrm{nc}}\Vert_{L^2(\Omega)}+\vert \widehat{\lambda}_h-\lambda_h\vert.
ds^2=\sum_{i,k=0}^{D-s-3}e^{2F_{i}(r)}\eta_{ik}f^{\delta_{ii_0}}(r)dX^i dX^k +e^{2B(r)}\left( r^2 d\Omega^2_{s+1}+f^{-1}(r)dr^2 \right),
A_{z}= \frac{\langle a ~ \partial_{z}g ~ g^{- 1}\rangle}{1 + \langle a ~ g ~ a ~ g^{- 1}\rangle}, A_{\bar{z}}= \frac{\langle a ~ g^{- 1}~ \partial_{\bar{z}}g \rangle}{1 + \langle a ~ g ~ a ~ g^{- 1}\rangle}.
N_{21}(x,y,a)=-\frac{1}{8ay}Re\left\{\sum_{k=0}^{\infty}\left( \frac{2\pi}{a}\right)^{k}\frac{(z-ia)^{k}}{k!}\zeta (-k)-\left( \frac{a}{2\pi}\right) (z-ia)^{-1}\right\}.
\left( (z-\lambda_j)^{\tau_j}F_\alpha(z) \Phi^{n-k}(z) \right)^{(l)}_{z=\lambda_j}=\frac{l!}{2\pi i}\int_{|z-\lambda_j|=\varepsilon}\frac{(z-\lambda_j)^{\tau_j}F_\alpha(z) \Phi^{n-k}(z)dz}{(z-\lambda_j)^{l+1}}.
= 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{\Lambda_{{\bf{k}}}^{2}({\bf{q}})}{{\tilde{\omega}}+ \omega_{{\bf{k}}}( -{\bf{q}})}- \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}\neq{\bf{k}}_{i}}\frac{\Lambda_{{\bf{k}}}^{2}( -{\bf{q}})}{{\tilde{\omega}}- \omega_{{\bf{k}}}({\bf{q}})}
\phi_1 := \phi_{2A}(\tau, z) = 2 \zeta^{\pm 1}+ 4 + (4 \zeta^{\pm 2}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}- 2 \zeta^{\pm 1}+ 16) q^2 + O(q^3)
\pi_i(\varepsilon y) = -4\pi \alpha_3 \mu_i^{\frac{1}{2}}H_\lambda(\varepsilon y,\zeta_i)+O( \mu_i^{\frac{3}{2}})=-4\pi \alpha_3 \mu_i^{\frac{1}{2}}g_\lambda(\zeta_i)+O( \varepsilon^{\frac{3}{2}}) .
\Delta_k(x)= \frac{x^{\frac{1}{2}-\frac{1}{2k}}}{\pi \sqrt{k}}\sum_{nx \le X^{1+\theta}}\frac{d_k(n)}{n^{\frac12+\frac{1}{2k}}}\cos\Big(2\pi k \sqrt[k]{nx}+\frac{k-3}{4}\pi\Big) &+I_{k}(x;\theta, \delta)+E_k(\theta, \delta, X),
\sum_{i,k=1}^d (\lambda_i-\lambda_k)^2 \langle p_i, q_k\rangle^2 \leq \left( \sum_{i=1}^r \sum_{k=1}^d+ \sum_{i=1}^d \sum_{k=1}^r+ \sum_{i,k=r+1}^{d}\right) (\lambda_i-\lambda_k)^2 \langle p_i, q_k\rangle^2.
{\cal F}^{\Sigma(T)}:=\{u\in L^2(\Sigma(T),\nu^T):\H^Tu\in \mathcal{F}^T\},{\cal E}^{\Sigma(T)}(u,v) :={\cal E}^{T}(H^{T}u,H^{T}v)\ \ \u,v\in{\cal F}^{\Sigma(T)}.
\mathcal{T}=\frac{L_{1}}{2}\dot{q}_{1}^{2}+\frac{L_{2}}{2}\dot{q}_{2}^{2},\quad\mathcal{V}=\frac{1}{2C_{1}}q_{1}^{2}+\frac{1}{2C_{12}}\left(q_{1}-q_{2}\right)^{2}+\frac{1}{2C_{2}}q_{2}^{2},\quad\mathcal{R}=\frac{R_{2}}{2}\dot{q}_{2}^{2},
\sum_{j=1}^N \left[\begin{matrix}N j \end{matrix}\right] \frac{q^{j^2}(q)_j}{(z q)_j (z^{-1}q)_j}=\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty}N_{S_1}(m, n) z^m q^n.
S_{P}^{4}= \int d^{4}x \Bigl [{\frac{1}{4}}\varepsilon^{\mu \nu \lambda \rho}B_{\mu \nu}F_{\lambda \rho}-{\frac{1}{4}}B_{\mu \nu}B^{\mu \nu}-{\frac{\mu^{2}}{2}}A_{\mu}A^{\mu}\Bigl ] ,
{\hat{\cal T}}_{J_{1}}^{R_{1}}{\hat{\cal T}}_{J_{2}}^{R_{2}}= \sum_{R , J}\left\langle \begin{array}{l}{R}\\{J}\\ \end{array}\right. \left| \begin{array}{l l}{R_{1}}&{R_{2}}\\{J_{1}}&{J_{2}}\\ \end{array}\right\rangle{\hat{\cal T}}_{J}^{R}.
\eta_{\underline{{a b}}}V_{m}^{\underline{{b}}}\eta^{i \ell_{1}}\epsilon^{m \ell_{2}\dots \ell_{d}}\epsilon_{\ell_{1}\dots \ell_{d}}+ 2 ( d ! ) a_{1}\eta_{\underline{{a b}}}\Pi_{j}^{\underline{{b}}}h^{i j}= 0
1\leq\alpha+\beta^{1/2}\biggl(\frac{1}{N}\sum_{j=1}^{N}\bigl(M_{t+h}^{j}\bigr)^{2}\biggr)^{1/2}+Ch\bigl(1+\sup_{s \in [0,t+h]}\vert Z_{s}^{i}\vert\bigr)+\sup_{s \in [t,t+h]}\bigl\vert W_{s}^{i}-W_{t}^{i}\bigr\vert.
({\rm Char}^{\cal R}(j_!{\cal F}),T^*_{X'}X')_{T^*X'}+{\rm rank}\{\cal F}\cdot \deg(X\cap A_L) &=2({\rm Char}^{\cal R}(j_!{\cal F}|_Y),T^*_YY)_{T^*Y}+({\rm Char}^{\cal R}(j_!{\cal F}),[df])_{T^*X',E}+\sum_{x\in X}({\rm Char}^{\cal R}(j_!{\cal F}),[dp_L])_{T^*X,x}.
\mu_{k}^i\left(\mathcal{B}_r(\theta^*)\right) \geq 1-C_3\sum\limits_{\theta \in \mathcal{B}_r^c(\theta^*)}\exp\left(-\frac{k}{2}\gamma(\theta)\right) & \geq 1-\sum\limits_{\theta \in \mathcal{B}_r^c(\theta^*)}\sigma{\prod_{i=1}^n \mu_0^i\left(\theta\right)^{\frac{1}{n}}}& \geq 1-\sigma
|L_r - L_u| = |\psi(r, t, X_r) - \psi(u, t, X_u)| &= |\psi(r, t, X_r) - \psi(r,t,\psi(u,r, X_u))| &\leq ||\psi||_{1,R}|X_r - \psi(u,r, X_u)|,
\qquad \! \! \! \! \! +{\frac{1}{6}}\left( \begin{array}{c}{{\sin{\phi^{\prime}}}}\\{{- \cos{\phi^{\prime}}}}\\{{0}}\\ \end{array}\right) \sin{3 v}-{\frac{1}{6}}\left( \begin{array}{c}{{\sin{\theta^{\prime}}\cos{\phi^{\prime}}}}\\{{\sin{\theta^{\prime}}\sin{\phi^{\prime}}}}\\{{\cos{\theta^{\prime}}}}\\ \end{array}\right) \cos{3 v}
S\ge\frac{B^k}{\phi(B)^k}\mathfrak{S}_B(\mathcal{L})(\log{R})^k\Bigl(\frac{J_k}{2}\log{R}\sum_{i=1}^k\frac{\phi(a_i)\phi(B)}{a_iB}\#\mathcal{P}_{L_i,\mathcal{A}}(x)-2mI_k\#\mathcal{A}(x)\Bigr).
\begin{aligned}V(u ) = V(u_{a}+u_{b}) &=V(u_{a})+V^{\prime}(u_{a})u_{b}+\frac{V^{\prime \prime}(u_{a})}{2!}u_{b}^{2}+\frac{V^{^{\prime \prime \prime}}(u_{a})}{3!}u_{b}^{3}+\frac{V^{(4)}(u_{a})}{4!}u_{b}^{4}\end{aligned}
\frac{y_j\tau_{j,m}+y_i\tau_{i,n}}{|y_j s_{j,p}- y_i s_{i,l}|}\leq \frac{y_j(\tau_{j,m}+\tau_{i,n})}{\sum_{k=i+1}^{j-1}y_k}\leq \frac{1}{3}.
\lambda_\pm(x_0,r;z;q)=\frac{1}{2}\left\{\sum^\infty_{n=0}\frac{(\frac{\rho_+}{\rho_-};q)_n}{(q;q)_n}(i\rho_-(1-q)z)^n\pm\sum^\infty_{n=0}\frac{(\frac{\rho_-}{\rho_+};q)_n}{(q;q)_n}(i\rho_+(1-q)z)^n\right\}.
\langle\phi^2(x)\rangle_{\beta,Ren.}=\langle\phi^2(x)\rangle_{\infty,Ren.}+\frac{1}{4\pi^2r^2}\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)Q_{\nu_l-1/2}\left(1+\alpha^2\frac{n^2\beta^2}{2r^2}\right).
\widehat{c}_2(t; T)={}\frac{2^T\prod_{n=1}^t \sin \left( \frac{n \pi}{6T}\right) \prod_{n=1}^{2T-t}\sin \left( \frac{n \pi}{6T}\right)}{\prod_{n=1}^{2T}\sin \left( \frac{n \pi}{6T}\right)}\prod_{n=1}^T\frac{\sin^2 \left( \frac{n \pi}{3T}\right)}{\cos \left( \frac{(2n-T-1) \pi}{6T}\right)}.
[\begin{matrix}A&B\\C&D\end{matrix}]^{-1}=[\begin{matrix}(A-BD^{-1}C)^{-1}&0\\ 0&(D-CA^{-1}B)^{-1}\end{matrix}][\begin{matrix}I&-BD^{-1}\\ -CA^{-1}&I\end{matrix}].
6 e^{2\sigma(\phi)}\left({\cal B}\sigma_i \sigma_j+ \frac{1}{2}\left({\cal B}_i \sigma_j +\sigma_i{\cal B}_j \right) \right)=A(\phi)_{i j}\;,\;\;\;{\cal B}(\phi)e^{2\sigma(\phi)}= B(\phi)\;,\;\;\;\frac{\lambda(\phi)}{{\cal B}(\phi)}=\frac{\Lambda(\phi)}{B(\phi)}
{\cal L}= \overline{\Psi}^1(i\partial\!\!\!/ - m)\Psi^1 + \overline{\Psi}^2(i\partial\!\!\!/)\Psi^2 -\overline{\Psi}^3(i\partial\!\!\!/)\Psi^3- g(\overline{\Psi}^1 \gamma^{\mu}\Psi^1 + \overline{\Psi}^2 \gamma^{\mu}\Psi^2 - \overline{\Psi}^3 \gamma^{\mu}\Psi^3)^2.
\delta = 1 - \max_{f \in F}\left(\frac{1}{\mu_f A_f}\prod_{g \in \Gamma(f)}(1+\mu_g)\right) > 0 \enspace , T_0 = \log_2 | \Omega | + |U(\sigma_1) |\cdot \log_2 \left(1 + \frac{\max_{f \in F}(\mu_f A_f)}{\min_{f \in F}A_f}\right) \enspace .
| \hat{h}'(\omega)| =|(\hat{h}_0(\omega)\hat{\phi}(\omega))'| \leq |\hat{h}_0'(\omega)\hat\phi_0(\omega/\pi)|+|\hat{h}_0(\omega)\hat{\phi}_0'(\omega/\pi)/\pi| \leq \frac{1}{2}C_{\phi_0}+\frac{\pi}{2}\cdot \frac{C_{\phi_0'}}{\pi},
\lim_{n\rightarrow+\infty}\frac{1}{n}\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes\overline A)=\lim_{n\rightarrow+\infty}\frac{1}{n}\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes\overline A^{\otimes p}).
(i+i_1+i_2)(i+i_1)[a^{i_1}q_1a^{i_2}q_2\ldots a^{i_n}q_n, a^i]=(i+i_1)i[a^{i_3}q_3\ldots a^{i_n}q_nq_1q_2, a^{i+i_1+i_2}]
\int_{1/(N+1)}^{N/(N+1)}\frac{|Z_N(t)-\sigma B_N(t)|| B_N(t)|^{p-1}}{t^{1+p/2}}dt &\leq \sup_{1/(N+1)\leq t \leq N/(N+1)}\frac{|Z_N(t)-\sigma B_N(t)|}{(t(1-t))^\zeta}\int_{1/(N+1)}^{N/(N+1)}\frac{| B_N(t)|^{p-1}}{t^{1+p/2-\zeta}}dt &=O_P\left(N^{-1/2+\zeta}\right)\int_{1/(N+1)}^{N/(N+1)}\frac{(t(1-t))^{(p-1)/2}}{t^{1+p/2-\zeta}}dt &=O_P(1).
\mathcal{B}_{n}\left( \lambda\right) =\frac{n}{\lambda-1}\sum_{k=0}^{n-1}\genfrac{\{}{\}}{0pt}{}{n-1}{k}k!\left( \frac{\lambda}{1-\lambda}\right)^{k},\,\ \ \lambda\in\mathbb{C}\backslash\{1\}.
\mathsf{H}^{1/2}(\mathbb{R}^n)= \{\psi \in \mathsf{L}^2(\mathbb{R}^n): \bigoplus\limits_{j\in\mathfrak{T}_n}(1+(\cdot)^2)^{1/4}\big(\mathcal{F}_n\psi\big)_{j}\in \bigoplus\limits_{j\in\mathfrak{T}_n}\mathsf{L}^2(\mathbb{R}_{+})\}.
{\rm SIR}_{j,0}=\frac{P_{j}Y_{j}^{-\alpha_{j}}\left|\mathbf{h}_{j,00}^{\dagger}\mathbf{f}_{j,0}\right|^{2}}{P_{1}I_{j,1C}+P_{1}I_{j,1O}+P_{2}I_{j,2}}
\vec{\sigma}\cdot \vec{p}\, \, \tilde{\xi}_{\lambda}( \vec{p}) = 2 p \lambda \, \tilde{\xi}_{\lambda}( \vec{p}) \, , \quad \vec{\sigma}\cdot \vec{p}\, \, \tilde{\eta}_{\lambda}( \vec{p}) = - 2 p \lambda \, \tilde{\eta}_{\lambda}( \vec{p}) \, .
Q_{1,k}=\int_{I_k}t^{1-2H}\left|\dot{\tilde{h}}_{k}(t)\right|^{2}dt=\int_{I_{k}}t^{1-2H}\left|\frac{1}{|I_{k}|}\dot{h}_{k}\left(\frac{t-a_{k}}{a_{k+1}-a_{k}}\right)\right|^{2}dt.
(1+x)P_{m,n}^{(\alpha,\beta)}(x)'+\beta P_{m,n}^{(\alpha,\beta)}(x){}=(-1)^{m}\frac{(\beta+n)(\alpha+n-2m+1)}{(\alpha+n-m+1)}P_{n-m}^{(\alpha+1,\beta-1)}(x)P^{(-\alpha-1,\beta-1)}_m(x).
\begin{aligned}\inf_{\tau\in\ker(b,H(\div;\mathbb{S});L^2(\mathbb{V}))}a(\tau,\tau) \geq \alpha \|\tau\|_{\div}^2, \inf_{v\in L^2(\mathbb{V})}\sup_{\tau \in H(\div;\mathbb{S})}b(\tau,v) \geq \beta \|\tau\|_{\div}\|v\|_0. \end{aligned}
z^{-w}P( \tau (\sigma ), \tau (\tilde{\sigma}))(z,t) = z^{-w}P_{(z,t)}( \tau (\sigma )(z,t), \tau (\tilde{\sigma})(-z,t)) = (-z)^{w}P_{(z,t)}( [\kappa_{H}\sigma ( \gamma (z,t) )]^{\nabla}, [\kappa_{H}\sigma ( \gamma (-z,t)) ]^{\nabla}).
\psi_K = \prod_{I<J}\prod_{i\leq j}(z_i^I - z_j^J)^{K_{I,J}}\prod_I \prod_{i<j}(z_i^I - z_j^I)^{K_{I,I}}\exp(-{1\over 2}(\sum_{i,I}\mid\! z_i^I \! \mid^2)\,,
\mathbb E_{\mu}\exp \left( t \frac{S_n}{n}\right) = \mathbb E_{\mu}\exp \left( \sum_{i=1}^k t p_i \frac{g_i}{|I_i|}\right) \leq \sum_{i=1}^k p_i \mathbb E_{\mu}\exp \left( t \frac{g_i}{|I_i|}\right).
(z_j)_{j\in S_1^-}\to (z_j)_{j=1}^{n_1}\,\,,\,\,(z_j)_{j\in S_1^+}\to (z_j)_{j=n_1+1}^{n_2}&(w_j)_{j\in S_2^-}\to (z_j)_{j=1}^{n_1}\,\,,\,\,(w_j)_{j\in S_2^+}\to (w_j)_{j=n_1+1}^{n_2}
\limsup_{h\to0}\frac{1}{h}P(N_t-N_{t-d}\ge 2, Z_t = j,d-h<U_t\le d| Z_{t-d}= j,U_{t-d}= v) \le \limsup_{h\to0}\frac{1}{h}P(N_t-N_{t-d}\ge 2| Z_{t-d}= j,U_{t-d}= v)=0.
\prod_{p | q_1q_2}p^{c_p}~=~ \prod_{1 < a < q_1/2 \atop (a, q_1) =1}{\xi}_a^{d_a}\prod_{1 < b < q_2/2 \atop (b, q_2) =1}{\xi}_b^{e_b}\prod_{1 < c < q_3/2 \atop (c, q_3) =1}{\xi}_c^{f_c}
0 = L_{11}^+(i\xi'\cdot\beta'_+) + L_{11}^-(i\xi'\cdot\beta'_-) + L_{12}^+\beta_{+N}+ L_{12}^-\beta_{-N}, -\sigma_-A^3\hat H(0) = L_{21}^-i\xi'\cdot\beta'_- + L_{22}^-\beta_{-N}, -\sigma_+A^2\hat H(0) = -L_{21}^+i\xi'\cdot\beta_+' - L_{22}^+\beta_{+N}
0=\vec 0_2=\frac{1}{p}\big(\vec 0^{(2)}-\vec 0_1^p\big)=\sum_{i=1}^n\alpha_{i2}\mu_i^p=\sum_{i=1}^n\alpha_{i2}^p\mu_i^p=\Big(\sum_{i=1}^n\alpha_{i2}\mu_i\Big)^p=0.
G_{MS}=\frac{|{{\bf{w}}_{{r}}}^T{{\bf{C}}_{{k}}}{{\bf{w}}_{{t}}}|^2}{{\left({{\bf{w}}_t^H{{\bf{w}}_t}}\right)\left({{\bf{w}}_r^H{{\bf{w}}_r}}\right)}}\bigg{|}_{{{\bf{w}}_{{t}}}={\bf{h}}_{k}^*,{{\bf{w}}_{{r}}}={\bf{g}}_{k}^*}=\lambda_k^2N_tN_r,
\sum_{a,b\in\{i,j,k,l\}}M_{ab}(h_a-h_b)^2 = M_{ij}(h_i-h_j)^2+M_{ik}(h_i-h_k)^2+M_{il}(h_i-h_l)^2 +M_{jk}(h_j-h_l)^2+M_{jl}(h_j-h_l)^2+M_{kl}(h_k-h_l)^2 \ge 0
\begin{array}{l}{{W_{q}= - \frac{1}{2}\int d x v ( x ) \int_{0}^{\infty}\frac{d \tau}{\tau}\tau^{- \frac{2}{1 + \gamma}}}}\\{{\Big ( \exp ( - \frac{\tau}{2}Q ) \frac{u_{+}\tau}{\sinh u_{+}\tau}\frac{u_{-}\tau}{\sinh u_{-}\tau}\cosh ( u_{+}\tau + u_{-}\tau ) \cosh ( u_{+}\tau - u_{-}\tau ) - 1 + \frac{\tau}{2}Q \Big )}}\\ \end{array}
a^{2}s i n h^{2}\gamma + \kappa_{5}^{2}\nu^{2}/ 3 = 1 , ~ ~ - \kappa_{5}^{2}V_{0}/ 6 = 1 , ~ ~ \Lambda_{b}^{2}\kappa_{5}^{2}/ 6 + V_{0}= 6 a^{2}/ ( \kappa_{5}^{2}- \kappa_{5}^{4}\nu^{2}/ 3 ) ,
\tau_{W_{m}}(g)=\det\left( I+\left( \widehat{g}_{o}T\left( \left( gm\right)_{e}\right) +\widehat{g}_{e}T\left( \left( gm\right)_{o}\right) -T\left(m_{o}\right) \right) T\left( m_{o}^{-1}\right) \right)
D_{pert,s\eta_s}(\psi_s)=0, D_{pert,s\eta_s^*}(\psi^*_s)=0, (\psi_s-\psi_0)\perp\ker(D|_{L^2_1}), (\psi^*_s-\psi^*_0)\perp\ker(D|_{L^2_1})
W_{\alpha \beta}( x - y ) = \left( \begin{array}{c c c c}{0}&{0}&{\delta ( x - y )}&{0}\\{0}&{0}&{0}&{\delta ( x - y )}\\{- \delta ( x - y )}&{0}&{0}&{0}\\{0}&{- \delta ( x - y )}&{0}&{0}\\ \end{array}\right) .
\gamma_{r}= \left( \begin{array}{c c}{0}&{e^{- i \theta}}\\{e^{i \theta}}&{0}\\ \end{array}\right) , \qquad \gamma_{\theta}= \left( \begin{array}{c c}{0}&{- i e^{- i \theta}}\\{i e^{i \theta}}&{0}\\ \end{array}\right) .
p^{(\varphi,id,id)}=\left[\begin{array}{ccccccc}3 2 3 2 1 2 3 2 1 2 1 1 1 3 3 \end{array}\right],p^{(id,\psi,id)}=\left[\begin{array}{ccccccc}3 2 1 3 1 1 1 2 1 3 2 3 3 2 2 \end{array}\right],
\int_{\mathbb{R}}\frac{\mathrm{d}\mu_{\omega}(x)}{z-x}=-\frac{\varphi_{a}(qz)\omega(z)+\psi_{a}(qz)}{\varphi_{a}(z)\omega(z) +a\psi_{a}(z)}\,,\ \z\in\mathbb{C}\setminus\mathbb{R}.
\|Q^{-1}B_1 x_2\|_{\hat{\mathfrak{l}}^s}^2+\|B_2 x_1+\hat{B}x_2\|_{\hat{\mathfrak{l}}^s}^2 &=\sum_{k=1}^N(1+k)^{2s}\left(\sum_{j=1}^NQ^{-1}_{k,j}\Big(\sum_{l=N+1}^\infty B_{j,l}x_l\Big)\right)^2+\sum_{k> N}(1+k)^{2s}\left(\sum_{j=1}^\infty B_{k,j}x_j\right)^2.
\mu^{n-M}\underset{=+1}{\underbrace{\left(-1\right)^{k+1+1+M}}}\det\left(C_{[n]\setminus\{i_{1},i_{1}+1,\ldots,\widehat{i_{j}+\epsilon},\dots,i_{\frac{m}{2}},i_{\frac{m}{2}}+1\}\cup\left\{M\right\}}\right)+o\left(\mu^{n-M}\right),
S_{1 0}= \int \left( \frac{1}{4}\, \hat{R}- \frac{1}{2}\, \partial_{M}\hat{\phi}\, \partial^{M}\hat{\phi}- \frac{1}{1 2}\, e^{- 2 \hat{\phi}}\, \hat{H}_{M N P}\, \hat{H}^{M N P}\right) \sqrt{- \hat{g}}\, d^{1 0}\hat{x},