harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bigl(\lambda_i(A^{(1-\theta)p_0+\theta p_1}\,\#_\alpha\,E)\bigr)_{i=1}^l\prec^{w(\log)}\bigl(\lambda_i^{1-\theta}(A^{p_0}\,\#_\alpha\,E)\lambda_i^\theta(A^{p_1}\,\#_\alpha\,E)\bigr)_{i=1}^l.
u(\varrho)=\sqrt{w}\, \varrho^\frac{1-N}{2}\le \sqrt{C}\, \varrho^\frac{1-N}{2}\,\varrho^{\frac{N-1}{2}-\frac{\gamma}{2}-\frac{\beta}{2}}\,=\sqrt{C}\, \varrho^{-\frac{\gamma+\beta}{2}},\varrho\gg 1.
\frac{1}{r^p}\int_{S^2(r)}\bigg(\frac{p(p-1)r^2}{4r^4}(-2u)^2 +\frac{2(p^2-p-1)}{r^2}u(-2u) + \frac{(p-1)(p-2)}{r^2}u^2\bigg)\,\, dS &= \frac{1}{r^{p+2}}\int_{S^2(r)}\bigg(p(p-1)-2(p^2-p-1)+(p-1)(p-2)\bigg) u^2 \, dS &= \frac{1}{r^{p+2}}\int_{S^2(r)}2u^2(2-p)\, dS < 0, p>2,
\partial_{y_i}\log Z(y) = \frac{\partial_{y_i}Z(y)}{Z(y)}={\int_\mathcal{X}\partial_{y_i}\mu(x, y) \frac{dx}{Z(y)}}={\int_\mathcal{X}\frac{\partial_{y_i}\mu(x, y)}{\mu(x, y)}\frac{\mu(x, y) dx}{Z(y)}}= \int_\mathcal{X}\partial_{y_i}\log \mu(x, y) \mu^y(dx)
\sigma(e_r[\alpha_c])=g_r[\alpha_c]\stackrel{\hbox{\scriptsize{def}}}=\sum_{1\leq i\leq j}\binom i{\frac{j-i}r}\alpha_c^{\frac{j-i}r}~e_{ij}=1+\sum_{1\leq i<j}\binom i{\frac{j-i}r}\alpha_c^{\frac{j-i}r}~e_{ij}~,
\Lambda_{F,s}[\varepsilon] = \sum_{\substack{0 \leq s_1 \leq |F_1| \ldots 0 \leq s_n \leq |F_n| s_1+\ldots +s_n =s}}\prod_{k=1}^n\Lambda_{F_k,s_k}[\varepsilon] \forall \varepsilon \in \mathcal{W}.
\begin{array}{r c l}{\partial_{j}^{}M_{k}^{\alpha}( q , u ) \!}&{=}&{\! \alpha_{j}^{}\Bigl ( \zeta ( u - v ) + \zeta ( v ) + \zeta ( \alpha ( q ) ) - \zeta ( \alpha ( q ) + u ) \Bigr )}\\{}&{}&{\times \Bigl ( M_{k}^{\alpha}( q , u ) - M_{k}^{\alpha}( q , v ) \Bigr ) ,}\\ \end{array}
_{2}F_{1}\left({\frac{1}{3}},{\frac{5}{6}}; 1 ;{- \frac{2}{2 5}}\right) = \frac{\sqrt{5}}{\sqrt{3}}{}_{2}F_{1}\left({\frac{1}{6}},{\frac{2}{3}}; 1 ;{- \frac{2}{2 5}}\right) ,
c_{i-1}^{-1}c_i^{-1}(\psi' - \psi) \sum_{{h=0}\atop{i-h{\tiny \mbox{even}}}}^{i-2}P_{h}(\psi) P_{h}(\psi')\frac{k_{i-2}b_{i-2}b_{i-1}}{k_{h}b_{h}b_{h+1}}=0.
Z = - \frac{1}{e^{2}{\phi_{0}}^{2}}\int d^{2}x \left[ \frac{e}{2}\epsilon^{i j}F_{i j}( \vert \Phi \vert^{2}-{\Phi_{0}}^{2}) + i \epsilon^{i j}( D_{i}\Phi ) ( D_{j}\Phi )^{*}\right]
\Big|\widehat{J_\varepsilon}(0) - \widehat{J_\varepsilon}(\xi) - |\xi|^2\Big| = \Big|\int_{\mathbb{R}^n}J_\varepsilon(|x|)\big(f_\xi(x) - f_\xi(0) - \nabla f_\xi(0)\cdot x - \frac{1}{2}x^TD^2f_\xi(0)x\big)\:x\Big| &\leq C(1+|\xi|^2)\int_{\mathbb{R}^n}\rho_\varepsilon(|x|)\:x \leq C(1+|\xi|^2),
a_1 =M_1+\frac{N_2(N_1-M_1)}{N_1}=N_2+\frac{M_1(N_1-N_2)}{N_1}, a_2 =M_2+\frac{N_1(N_2-M_2)}{N_2}=N_1+\frac{M_2(N_2-N_1)}{N_2}.
\Gamma_{0}( b_{i}\to b_{f}+ \gamma ) ={\frac{2}{3 \pi}}\left({\frac{M ( N_{i})^{2}- M ( N_{f})^{2}}{2 \hbar M ( N_{i})}}\right)^{3}A_{H}( N_{i}){\frac{2 J_{f}+ 1}{2 J_{i}+ 1}}{\frac{N_{f}}{N_{i}}}.
\int_0^{+\infty}| P_t f(x) -f(x)| \frac{dt}{t^{1+s}}= \int_0^{1}| P_t f(x) -f(x)| \frac{dt}{t^{1+s}}+ \int_{1}^{+\infty}| P_t f(x) -f(x)| \frac{dt}{t^{1+s}}\le C_1 \| Lf \|_\infty + C_2 \| f \|_2
\left\Vert \int_{0}^{t}\Lambda_{\theta}(t-\tau)(|u|^{\lambda}-|\tilde{u}|^{\lambda})d\tau\right\Vert_{L^{\infty}}\leq C \int_{0}^{t}(1+t-\tau)^{-\frac{n+2(\theta-1)}{2}}\left(\Vert(|u|^{\lambda}-|\tilde{u}|^{\lambda})\Vert_{L^{1}}+\Vert(|u|^{\lambda}-|\tilde{u}|^{\lambda})\Vert_{H^{s}}\right)d\tau &:=J_1+J_2.
\Theta_{j_1,m_1}(z)\tilde \Phi_{j_2,m_2}(w)\sim \left| z-w\right|^{-\frac4{k-2}j_1j_2}(\tilde \Phi_{j_1+j_2,m_1+m_2}(w)-\frac{2j_1}{k-2}\ln \left|z-w\right| \Theta_{j_1+j_2,m_1+m_2}(w))+...
\gamma_{m}^{b}=\frac{\rho_{1,0}((\pi_{R,m}^{in})^{-1}-1)((\gamma_{m+1}^{b})^{-1}-1)+(1-\rho_{1,0})}{(\rho_{0,0}+\rho_{1,0})((\pi_{m+1}^{in})^{-1}-1)((\gamma_{m+1}^{b})^{-1}-1)+\rho_{1,1}+\rho_{01}},
\left\|U^{m,n}_{\psi,\overrightarrow{s}}\right\|_{L^{p_1}_{\omega_1}\left(\mathbb R^d\right)\times\cdots\times L^{p_m}_{\omega_m}\left(\mathbb R^d\right)\to L^{p}_{\omega}\left(\mathbb R^d\right)}\geq \varepsilon^{\beta\varepsilon}\cdot\left|\,\int_{[\varepsilon,1]^n}\prod_{k=1}^m|s_k(t)|^{-\frac{d+\alpha_k}{p_k}-\varepsilon_k}\psi(t)dt\right|.
4g_{\mu \upsilon}= 2{\partial \psi^\beta \over \partial \omega_1}{\partial \psi^\beta \over \partial \omega_1}{M_\mu M_\upsilon\over (\omega_1)^2}+{4(S_1)_{\mu \upsilon}\over (\omega_1)^2}\sin^2 \left({\theta_1 + 2\theta_2 \over 2}\right)
\mathbb P[E_3] \geq \prod_{0<m\leq CL}\left[\frac{1}{32CL}\frac{m!\Gamma(n)}{\Gamma(m+n)}(1-r^2)^{L}\right]^{\frac{\Gamma(n+m)}{m!\Gamma(n)}}& =\left[\frac{\Gamma(n)}{32CL}(1-r^2)^{L}\right]^{\sum\limits_{0<m\leq CL}\frac{\Gamma(n+m)}{m!\Gamma(n)}}\prod_{0<m\leq CL}\left(\frac{m!}{\Gamma(m+n)}\right)^{\frac{\Gamma(n+m)}{m!\Gamma(n)}}.
\frac{d\delta_0}{dv}(v) =\frac{\lambda}{6\kappa}(1+y_m(v))v^2, \frac{d\delta_1}{dv}(v) =\left(V_m(v)-c_{+0}-\frac{\kappa}{2}(1+y_m(v))v\right)\frac{\lambda}{3\kappa^2}v+(V_m(v)-c_{+0})\phi_{m+1}(v).
\pi(p)=k(p)\left[\frac{\partial \psi}{\partial y}\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial z}+\frac{\partial \psi}{\partial z}\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}-\frac{\partial \psi}{\partial x}\frac{\partial}{\partial y}\wedge \frac{\partial}{\partial z}\right].
\frac{i}{\sqrt{2 p^{+}}}\left[ \gamma_{0}( \tau ) -{\gamma_{0}^{\dagger}}( \tau ) \right] = \frac{i}{2 \sqrt{\mu p^{+}}}\left( e^{- 2 i \mu \tau}a_{0}+ e^{2 i \mu \tau}\bar{a}_{0}\right) - \frac{m R_{1}}{2 \mu l_{s}^{2}p^{+}}.
Q = +{\frac{\pi}{2}}+ \rho \left[ \int_{1}^{n^{2}}{\frac{d u^{\prime}}{1 - u^{\prime}}}\Im \left\{{\frac{i \sqrt{u^{\prime}- 1}}{\sqrt{n^{2}- u^{\prime}}+ i \rho \sqrt{u^{\prime}- 1}}}\right\}\right] .
(x) = \frac{(1-\epsilon_+ x)(1+\epsilon_- x)}{2\pi \sqrt{1-x^2}\prod_{1\leq r\leq d}(1+2a_r x+a_r^2)},^{(m+1)}_l = \Delta_l^{(m+1)}\frac{(1-\epsilon_+ x^{(m+1)}_l)(1+\epsilon_- x^{(m+1)}_l)}{\prod_{1\leq r\leq d}(1+2a_r x^{(m+1)}_l+a_r^2)}.
(1-\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{S^-}+r_2\delta_{\alpha_2 y}^{S^+}\}) S_{i,j}^{n+1}=(1+\dfrac{\Delta t}{2}\{(1-r_1)\delta_{\alpha_1 x}^{S^-}+r_1\delta_{\alpha_1 x}^{S^+}\}) S_{i,j}^{*}-\dfrac{\Delta t}{2}\left \{\mu-\mu S_{i,j}^{n+1/2}-\beta S_{i,j}^{n+1/2}V_{i,j}^{n+1/2}\right\}
\mathcal{I}(0)=\int_{\mathcal{X}}\left[\sup_{\gamma(\cdot|x) \in \Pi(\mu_{1|3}, \mu_{2|3})}\int_{\mathcal{Y}_1\times\mathcal{Y}_2}f(y_1,y_2,x) \, d \gamma(y_1,y_2|x)\right]d\gamma_X(x),
\begin{array}{rl}\displaystyle \min_{z_0,\ldots,z_T \in \real^n}\frac{1}{2}\left(\|z_0 - b\|^2_{(B^{\infty})^{-1}}+\sum_{t=0}^{T}\|y_t - \mathcal{H}_t(z_{t})\|^2_{R_t^{-1}}\right), \mbox{s. t.}z_t = \mathcal{M}_t(z_{t-1}), ~t=1,\ldots,T.\end{array}
\begin{aligned}E_n =\int_{x^*}^{x_n}(f(x)-p_n(x))\dx=-\frac{f^{(n+1)}(\xi_1)}{(n+1)!}h^{n+2}\int_{\frac{x^*-x_0}{h}}^{\frac{x_n-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)\d\mu &+\frac{f^{(n+2)}(\xi_2)}{(n+2)!}h^{n+3}\int_{\frac{x^*-x_0}{h}}^{\frac{x_n-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)(\mu-\frac{x_n-x_0}{h})\d\mu.\end{aligned}
\Sigma := \Bigl(T^2\setminus D_1^+\Bigr) \cup_{\bar\gamma_1}\Bigl(T^2\setminus(D_1^-\cup D_2^+)\Bigr) \cup_{\bar\gamma_2}\Bigl(T^2\setminus(D_2^-\cup D_3^+)\Bigr) \cr \cup_{\bar\gamma_3}\Bigl(T^2\setminus(D_3^-\cup D_4^+)\Bigr) \cup_{\bar\gamma_4}\Bigl(T^2\setminus D_4^-\Bigr).
\Gamma_{+}^{- 1}A^{( r )}C_{r}- \alpha_{r}C_{3}\Gamma_{+}^{- 1}A^{( r )}= C \delta_{r 3}- \frac{1}{2}\frac{\alpha}{1 + \mu \alpha k}C U_{3}^{- 1}Y Y^{\mathrm{T}}A^{( r )}C U_{r}^{- 1}.
U(\prod_{i=1}^{g}\left(g_{a_i}^{~}g_{b_i}^{~}g_{a_i}^{-1}g_{b_i}^{-1}\right)) =V(\prod_{i=1}^{g}\left(g_{a_i}^{~}g_{b_i}^{~}g_{a_i}^{-1}g_{b_i}^{-1}\right)) =1~,
S_{1}={\displaystyle\sum\limits_{1<i_{1}<\cdots<\text{\}i_{r-1}<k}}n_{i_{1}}a_{i_{1}}\cdots n_{i_{r-1}}a_{i_{r-1}}, S_{2}={\displaystyle\sum\limits_{1<i_{1}<\cdots<\text{\}i_{r-2}<k}}n_{i_{1}}a_{i_{1}}\cdots n_{i_{r-2}}a_{i_{r-2}}.
L^{2}=-\left(\frac{d\tau}{d\lambda}\right)^2+l^2\cosh^2 \left( \frac{\tau}{l}\right)\sum_{j=1}^{d-1}\left( \prod_{i=1}^{j-1}\sin^2 \theta_{i}\right)\left( \frac{d\theta_j}{d\lambda}\right)^2,
(1-2xt)\frac{d}{dt}M(x;t) =2x(1-2x)\frac{d}{dx}M(x;t) &+\left(a(x)+xb(x)-2x(1-2x)\frac{d}{dx}a(x)\right)R^+(x;t) &+\left(2xb(x)+a(x)-2x(1-2x)\frac{d}{dx}b(x)\right)R^-(x;t).
\sum_{\substack{(t_1)^1\in \left(Y^p\right)^1, t_2\in Y^q p+q+1=n}}(t_1)^1 \vee t_2 +\sum_{\substack{t_1\in Y^p, (t_2)^1\in \left(Y^q\right)^1 p+q+1=n}}t_1\vee (t_2)^1
{\bf E}\bigl[\beta_{2,2}^q(\delta)\bigr] = \frac{G(\delta)}{G(q+\delta)}\frac{G^2(q+\delta+1/2)}{G^2(\delta+1/2)}\frac{G(\delta+1)}{G(q+\delta+1)}, & = \prod\limits_{k=0}^\infty \Bigl[\frac{\delta+k}{q+\delta+k}\frac{(q+\delta+1/2+k)^2}{(\delta+1/2+k)^2}\frac{\delta+1+k}{q+\delta+1+k}\Bigr]^{k+1}
\left[{\frac{1}{2}}( 1 + \sigma_{3}) ( i{\frac{\partial}{\partial t}}- g \phi ) + i \mathrm{\boldmath ~ \s i g m a ~}\cdot ( \mathrm{\boldmath ~ \nabla ~}- i g \mathrm{\boldmath ~ \p h i ~}) + M ( 1 - \sigma_{3}) \right] \psi = 0 .
[L_n,G_m]_t=[L_n,G_m]+\sum_{i\geq p}a_i\varphi_1(L_n,G_m)t^i; &[L_n,L_m]_t=[L_n,L_m]+\varphi_0(L_n,L_m); &[L_n,G_m]_t'=[L_n,G_m]+\lambda\varphi_1(L_n,G_m)t; &[L_n,L_m]_t'=[L_n,L_m]+\varphi_0(L_n,L_m).
=\ - \sum_{i=1}^N \left[ d_i^2 + \nu \sum_{j\neq i}\frac 2{x_i - x_j}d_i- \nu\sum_{j\neq i}\frac 1{(x_i-x_j)^2}(1 - K_{ij}) \right] +\sum_{i=1}^N x_i^2 \ =
J(v)\lesssim \iint_\Omega |\nabla (e^{-|t|}\Delta_i^h g)|^2\frac{dtdx}{|t|^{n-d-1}}+ \iint_{\Omega}|\nabla u(x+he_i, t)|^2\frac{dtdx}{|t|^{n-d-1}}:= J_3 + J_4.
(U_l^*-V_l^*)^\top\nabla_{U_l}F(U^*,V^*;\gamma) + (V_l^*-U_l^*)^\top\nabla_{V_l}F(U^*,V^*;\gamma) &=(U_l^*-V_l^*)^\top\left\{2\gamma I_m-\left(\sum_{ij\in E_{\mathrm{ss}}}\alpha_{ij}^*\bar{A}_{ij}+\sum_{ik\in E_{\mathrm{sa}}}\alpha_{ik}^*\bar{A}_{ik}\right)\right\}(U_l^*-V_l^*) = 0.
r_n^{[2]}(h):=\frac{2^{2p+1}\cdot y_{4n}\left(\frac{h}{4}\right)-3\cdot 2^p\cdot y_{2n}\left(\frac{h}{2}\right)+ y_n(h)}{\left(2^p-1\right) \left(2^{p+1}-1\right)}
\pi^{(i)}_*\left[ (1-\beta T_i)^{-q_i}\cdot \tilde{c}_{q_i}(i) \right] = (1-\beta T_i)^{-q_i+p_i-i}\cdot c_{q_i-p_i+i}(i) = (1-\beta T_i)^{-\lambda_i}\cdot c_{\lambda_i}(i).
\begin{aligned}\varphi(a, b) \leq d(a, a') + \varphi(a', b), d(a, a') \leq \varphi(a, b) + \varphi(a', b), \varphi(a, b) \leq d(b, b') + \varphi(a, b'), d(b, b') \leq \varphi(a, b) + (a, b')\end{aligned}
E(t )\leq E(0) &+\int_0^t C(\|\partial_{\alpha}\xi_0\|_{H^{s-1}},\|v_0\|_{H^s}, \|w_0\|_{H^s}, \mathcal{E}(\tau), d_I(\tau)^{-1}, d_P(\tau)^{-1}, N\lambda_{max}, C_1, C_2, \alpha_0) ds.
\sigma(H_0^{-1})(X,\Xi,\hat{x}_{n},X,\xi) = \sigma(H_0^{-1})(X,\Xi,\hat{x}_{j,n},Y,X,\xi) &\ \ \ + (x_j-Y) \cdot R_{\mathrm{od},1}(X,\Xi,\hat{x}_{j,n},Y,X,\xi) \ \ \ + (x_j-Y)^T R_{\mathrm{od},2}(X,\Xi,\hat{x}_{n},X,\xi) (x_j-Y)
\int I(\xi (t) ,v(t),\theta(t)|\bar{\xi}(t),\bar{v}(t),\bar{\theta}(t))\:dx\leq \, e^{Ct}\int I(\xi^0 ,v^0,\theta^0|\bar{\xi}^0,\bar{v}^0,\bar{\theta}^0)\,dx &+\frac{1}{2}\int_{0}^{t}e^{C(t-s)}\int\left(\mu\,\theta(s)\frac{|\nabla\bar{v}(s)|^2}{\bar{\theta}(s)}+k\frac{|\nabla\bar{\theta}(s)|^2}{\bar{\theta}(s)}\right)\:dsdx
F(x^+,x^-,0)=\frac{1}{2 \pi}\sum_{K,N_\perp,\alpha}\frac{1}{2L}\frac{1}{l}\left(\frac{\pi K}{L}\right)^3 e^{-iP^-_\alpha x^+ -iP^+ x^-}\frac{|\langle u|\alpha\rangle|^2}{l K^3 |N_u|^2}\,.
f_{m,n}\leq\frac{2n+1}{n\mu_{0}/2\sqrt{n(1-\mu_{0})/2}}\overset{n\geq1}{\leq}\frac{6\sqrt{2}}{\mu_{0}\sqrt{1-\mu_{0}}}n^{-1/2}=:c_{7}n^{-1/2}, f_{m,n}\geq\frac{\sqrt{n+\mu_{0}n}\sqrt{\mu_{0}n}}{\sqrt{n-\mu_{0}n}(1+\mu_{0})n}=:c_{8}n^{-1/2}.
\Phi^{I}= P^{I}+ \frac{2 i}{\sqrt{2 p^{+}}}\left( \, ( p^{+}- h P^{-}) \delta^{I J}+ \frac{1}{2}h X^{-}s^{I J}\, \right) \psi^{J}~ \approx ~ 0 +{\cal O}( \psi^{3}) ~ ,
g_0 = h_0 a^{k_0}\kappa_1(h_1 a^{k_1}\kappa_1(g_2) d_1) d_0 &= h_0 a^{k_0}\kappa_1(h_1 a^{k_1}) \kappa_2(g_2) \kappa_1(d_1) d_0 &= h_0 a^{k_0}\kappa_1(h_1 a^{k_1}) \kappa_2(g_2) d_0 \kappa_1(d_1).
\kappa ( h ) = \prod_{\alpha > 0}\left[ \frac{\pi ( h , \alpha )}{a_{0}}\prod_{n = 1}^{\infty}\left( 1 - \frac{( h , \alpha )^{2}}{a_{0}^{2}n^{2}}\right) \right] = \prod_{\alpha > 0}\sin \frac{\pi ( h , \alpha )}{a_{0}}\ .
\Gamma_{e . f .}= \int \{Y_{1}( -{e}z_{e}\sqrt{\frac{z_{2}}{z_{1}}}\sqrt{z_{A}}\bar{\varphi}_{2}c + x_{1}q_{1}) \; + \; Y_{2}({e}z_{e}\sqrt{\frac{z_{1}}{z_{2}}}\sqrt{z_{A}}( \bar{\varphi}_{1}+ \frac{\sqrt{z_{m}}m}{\sqrt{z_{1}}{z_{e}}e}) c + x_{2}q_{2}) \}
\eqref{0-1term1}= \frac{1}{\sqrt{N}}\int dx_1dx_2 dy_1\Big\{\quad\u(x_1, x_2)v_N(x_1-x_2)\phi(x_2)\delta(y_1- x_1) + u(x_1, x_2)v_N(x_1-x_2)\phi(x_2)p(y_1, x_1) &+ (\bar p\circ u)(x_1, x_2)v_N(x_1-x_2)\phi(x_2)\delta(y_1-x_1) + (\bar p\circ u)(x_1, x_2)v_N(x_1-x_2)\phi(x_2)p(y_1, x_1) \Big\}a_{y_1}.
\begin{aligned}A(t) =p_2(t)+e^{q_3(t)-q_2(t)}p_3(t)= &\frac{1}{2}\left(H_0\left(1+e^{q_3-q_2}\right)+\sqrt{(1-e^{q_3(t)-q_2(t)})(4H_1-(1+e^{q_3(t)-q_2(t)})H_0^2)}\right),\end{aligned}
\left( P^{0}- 2 \pi B R_{1}n \right)^{2}-{\vec{p}}^{2}= \left( \frac{m}{R_{1}}\right)^{2}+ \left( \frac{n R_{1}}{\alpha^{\prime}}\right)^{2}+ \frac{2}{\alpha^{\prime}}\left( N + \tilde{N}- 2 \right)
L_1(t_1,t_2):= \int_{t_1}^{t_2}\frac{dt}{1-t}=\log \frac{1-t_1}{1-t_2}L_{-1}(t_1,t_2):= \int_{t_1}^{t_2}\frac{-dt}{1+t}=\log \frac{1+t_1}{1+t_2}L_0(t_1,t_2):= \int_{t_1}^{t_2}\frac{dt}{t}=\log \frac{t_2}{t_1}.
\mathcal{N}_{\lambda}= \left\{(y_1, y_2)\in W_{0}^{1, \mathcal{H}}(\Omega) \times W_{0}^{1, \mathcal{H}}(\Omega)\backslash\{(0,0)\}: \psi_{(y_1, y_2)}^{\prime}(t) \bigg\vert_{t = 1}= 0 \right\}.
C_{T}=c_{0}^{-1}(1+\left\|a\right\|_{L^{\infty}})e^{c_{0}^{-1}\left(1+\|a^{\prime}\|_{L^{\infty}}+\|q\|_{L^{\infty}}+2\left\|a\right\|_{L^{\infty}}\right)T}.
\sup_{D_r}\frac{u}{V(d_D)}- \inf_{D_r}\frac{u}{V(d_D)}\leq \sup_{D_{r_k}}\frac{u}{V(d_D)}- \inf_{D_{r_k}}\frac{u}{V(d_D)}\leq c_2 (M_k-m_k) = c_2 V(r_{k+1}/ 2)^\gamma \leq c_2 V(r)^\gamma.
\tilde{f}^{\alpha a}_2=\sum_{\tiny\begin{matrix}\beta+\gamma=\alpha, b + c = a |\beta|+|b|,|\gamma|+|c|<|\alpha|+|a| \end{matrix}}C_\alpha^\beta C_a^b\nabla(\nabla^\perp H^{(\beta,b)}\nabla V^{(\gamma,c)}).
(-b)_k\prod_{\substack{1\leq j\leq a-k j\neq a-b}}(j-a+b)=\lim_{z\to 0}\frac{(-b-z)_k(1-a+b+z)_{a-k}}{z}= (-1)^a\lim_{z\to 0}\frac{(-b-z)_{a-k}(1-a+b+z)_{k}}{z}=(-1)^a(-b)_{a-k}\prod_{\substack{1\leq j\leq k j\neq a-b}}(j-a+b).
\begin{aligned}\|(1-P_R)Q_\epsilon u_\epsilon\|_{X}\leq 1/R\sum_{1\leq l\leq s}\|\mathcal{K}_{\mathcal{I}_l}u_\epsilon\|_X\lesssim_{p,q}1/R\sum_{1\leq l\leq s}\|\mathcal{K}_{\mathcal{I}_l}u_0\|_{L^2}.\end{aligned}
< \phi_{\omega}, \phi_{\sigma}> = \int_{\Sigma}\sqrt{h}d^{D - 1}x \left[ ( \omega + \sigma ) \phi_{\omega}^{*}\phi_{\sigma}+ i \phi_{\omega}^{*}a^{i}( \nabla_{i}+ i \sigma a_{i}) \phi_{\sigma}- i ( \nabla_{i}- i \omega a_{i}) \phi_{\omega}^{*}a^{i}\phi_{\sigma}\right] ,
\langle e_i, g\!\cdot\! v \rangle = \frac{1}{\|gv\|}\sum_{j=1}^d v_j \langle e_i, g e_j \rangle = \frac{\sum_{j=1}^d v_j g^{i,j}}{\sum_{i=1}^d \sum_{j=1}^d v_j g^{i,j}}.
\Psi_{n , m}( x , y ) = x^{p_{1}}y^{p_{2}}L_{n}^{( - 1 / 2 + p_{1})}( \omega x^{2}) L_{m}^{( - 1 / 2 + p_{2})}( \omega y^{2}) e^{- \frac{\omega x^{2}}{2}- \frac{\omega y^{2}}{2}}\,
\left\|\psi\left(\frac{\mathbf y}{\tau}\right)\right\|_{L^8(\mathcal K)}=\tau^{\frac{1}{4}}\|\psi\left(\mathbf x\right)\|_{L^8(\mathcal K)}=\mathcal O(\tau^{-\frac{5}{12}}),\mbox{as $\tau \rightarrow \infty$.}
P_{1,n+1}(a)Q_{1,n+1}^*(a)-a b Q_{1,n+1}(0)P_{2,n}^*(0)-P_{1,n+1}(0)Q_{2,n}^*(0)=0, (b-a)Q_{4,n}(a)P_{3,n}^*(a) - b Q_{4,n}(0)P_{3,n}^*(0) - a P_{4,n}(0)Q_{3,n}^*(0) = 0.
\Psi^l=\Psi^l\left(x_{i},u_j,(u_j)_{x_{i_1}},(u_j)_{x_{i_1}^{j_1},x_{i_2}^{j_2}},\dots,(u_j)_{x_{i_1}^{j_1},x_{i_2}^{j_2},x_{i_3}^{j_3},\dots,x_{i_n}^{j_n}}\right),
f ( r , Z_{0}) = - 2 i \int_{0}^{\omega - \alpha}d p \, \frac{p e^{- i p r}}{( \omega^{2}- p^{2})^{3 / 2}}\cosh ( \sqrt{\omega^{2}- p^{2}}\, Z_{0}) + i \sqrt{\frac{2}{\omega \alpha}}\, e^{- i \omega r}- \pi Z_{0}e^{- i \omega r}.
\theta r =[\alpha_0|\nabla\theta|_2^2+ \alpha_1(d|\nabla \theta)^2]/\theta + 2\mu_s|D|_2^2 + \mu_b|{\rm div}\, u|^2 &+\frac{1}{\gamma}|P_d({\sf a}-\mu_PDd)|_2^2+\mu_L |P_d Dd|^2 +\mu_0 (Dd|d)^2.
S_{\mathrm{P 5 h}}= - T_{\mathrm{P 5 h}}\int d^{6}\xi \e^{- 2{\phi}}k^{- 1}\sqrt{| \mathrm{d e t}\ ( \partial_{i}X^{\mu}\partial_{j}{X}^{\nu}g_{\mu \nu}- k^{2}F_{i}F_{j}) |}+ \mathrm{W Z},
g(0)g(\infty)= \frac{(x_0+\sqrt{a}x_1)(\sqrt{b}x_2+\sqrt{ab}x_3)}{(\sqrt{b}x_2-\sqrt{ab}x_3)(x_0-\sqrt{a}x_1)}>0 \Leftrightarrow\ (x_0^2-ax_1^2)(bx_2^2-abx_3^2)>0 \Leftrightarrow\ (x_0^2-ax_1^2)(x_0^2-ax_1^2-1)>0
A(p) := \textstyle{\frac{1}{100}}{\footnotesize \begin{pmatrix}128 p^2 - 72 p - 32 295 p^2 - 199 p + 4 165 p^2 - 234 p + 46 -82 p^2 - 59 p + 270 -266 p^2 + 144 p - 73 -147 p^2 - 210 p + 286 70 p^2 + 296 p - 80 43 p^2 + 96 p + 8 15 p^2 + 146 p - 251 \end{pmatrix}}
g_{ij}(x,t)=h_{ij}(x)e^{2\lambda(x,t)}=\frac{\cosh^{2}t}{2\Lambda}\frac{\partial\phi_{\xi}^{k}(x)}{\partial x^{i}}\frac{\partial\phi_{\xi}^{l}(x)}{\partial x^{j}}h_{kl}\left(\phi_{\xi}(x)\right),
I_{\varepsilon}\left(t_{\varepsilon}(Z_{\varepsilon,\xi})Z_{\varepsilon,\xi}\right)=\frac{1}{2}t_{\varepsilon}^{2}\|Z_{\varepsilon,\xi}\|_{\varepsilon}^{2}-\frac{1}{p}t_{\varepsilon}^{p}|Z_{\varepsilon,\xi}|_{\varepsilon,p}^{p}+\frac{1}{\varepsilon^{n}}qt_{\varepsilon}^{2}\int_{M}\psi(t_{\varepsilon}Z_{\varepsilon,\xi})Z_{\varepsilon,\xi}d\mu_{g}
\lim_{\beta \to \infty}\frac{J_{\{m \}}}{J_{0}}= \lim_{\beta \to \infty}\frac{\int_{G}d \mu_{H}\, \exp ( - \beta ( 1 - 1 / N \Re \mathrm{t r}( g ) ) ) \, \, \frac{\mathrm{t r}( \pi ( g ) )}{\mathrm{d i m}( \pi )}}{\int_{G}d \mu_{H}\, \exp ( - \beta ( 1 - 1 / N \Re \mathrm{t r}( g ) ) )}= 1
\widetilde{j}_{\mu}^{a , \mathrm{r e g}}= \widetilde{\psi}_{\varepsilon}^{+}T^{a}\gamma_{\mu}\widetilde{\psi}_{\varepsilon}= \psi_{\varepsilon}^{+}\Omega_{+}T^{a}\gamma_{\mu}\Omega_{-}\psi_{\varepsilon}.
\lim_{x\to+\infty}\frac{\ell(t x)}{\ell(x)}=\lim_{x\to+\infty}\frac{(tx)^{L-1}\, (Ltx+\gamma)^{\alpha-L}\, (tx)^{1-\alpha}}{x^{L-1}\, (Lx+\gamma)^{\alpha-L}\, x^{1-\alpha}}&=\lim_{x\to+\infty}t^{L-\alpha}\left(\frac{Lx+\gamma}{L tx+\gamma}\right)^{L-\alpha}=1.
M=F_{-1}, &C^\bullet\colon 0=C^2\subseteq (C^1_F)_{-1}\subseteq F_{-1}=C^0, &D_\bullet\colon 0=D_{-1}\subseteq (D_0^F)_{-1}\subseteq F_{-1}=D_1,\textup{and}&\varphi_0=\pi_{-1}\varphi^F_{0},\ \varphi_{1}=\pi_{-1}\varphi^F_{1}.
\Phi\,(\log\frac{p}{\Lambda}) = 1 + \frac{a_1}{\log\frac{p}{\Lambda}}+ \frac{a_2 + a_3\log(\log\frac{p}{\Lambda})}{\log^2\frac{p}{\Lambda}}+ \mathcal{O}\,(\frac{1}{\log^3\frac{p}{\Lambda}})\,.
T_{2}\ll\frac{1}{n^{2}}\sum_{j=1}^{i}\Vert{\mathbb{E}}({\widetilde{X}}_{ij}|{\mathcal{G}}_{i,j-q})\Vert_{2}^{2}\ll\frac{1}{n}\eta_{q}^{2}+\frac{1}{n^{2}}\sum_{j=1}^{i}\Vert X_{ij}^{2}I(|X_{ij}|>A)\Vert_{1}\,.
\mathrm{tr}\left[ (A_\eta - \lambda)^{-3}- (A_0 - \lambda)^{-3}\right]= -\frac{1}{2}\mathrm{tr}\left[ \frac{\mathrm{d}^{2}}{\mathrm{d}\lambda^{2}}\left( (I_4 + \eta M(\lambda))^{-1}\eta \frac{\mathrm{d}}{\mathrm{d}\lambda}M(\lambda) \right) \right]
=\sum_{r_1=1}^{n}\frac{a_{r_1}b_{r_1}}{(r_1+k+1)^2}+\sum_{\substack{{r_1,r_2=1}r_1\ne r_2}}^{n}\frac{a_{r_1}b_{r_2}}{r_2-r_1}\left(\frac{1}{r_1+k+1}-\frac{1}{r_2+k+1}\right).
{}_{1}F_{1}\left(1,s,z\right)= \sum_{k=0}^{\infty}\frac{(1)_{k}}{(s)_{k}}\frac{z^{k}}{k!}= 1+\sum_{k=1}^{\infty}\frac{k!}{s(s+1)\hdots(s+k-1)}\frac{z^{k}}{k!}= 1 +\frac{z}s \sum_{k=1}^{\infty}\frac{(k-1)!}{(s+1)\hdots(s+k-1)}\frac{z^{(k-1)}}{(k-1)!}=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\lbrack (p_{\mu}\gamma \cdot p^{^{\prime}}+p_{\mu}^{^{\prime}}\gamma\cdot p-p\cdot p^{^{\prime}}\gamma_{\mu})+((p+p^{^{\prime}})_{\mu}+i(p^{^{\prime}}-p)_{\alpha}\epsilon_{\alpha \mu \nu}\gamma_{\nu})w+w^{2}\gamma_{\mu}]
\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}g^{j,t}\|_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}\leq \sum_{k\geq 0}\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}g^{j,t}_k\|_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}.
[D(2,2n):D(3,s)]_q =q^{2(2n-s)}[D(2,2n-3):D(3,s-3)]_q+q^{2(2n-1)}[D(2,2n-2):D(3,s)]_q &+q^{6n-2s-3}[D(2,2n-4):D(3,s-3)]_q,
\Gamma =\Big(\frac{\gamma_1\cdots\gamma_m}{\min_{i\in I}\gamma_i}\Big)^{1/2}\quad\Big(\sum_{i\in I}\frac{\lambda_i}{2-\lambda_i}\Big) -\min_{i\in I}\frac{\lambda_i}{2-\lambda_i}=\Big(\sum_{i\in I}\frac{1}{\beta_i}\Big) -\frac{1}{\max_{i\in I}\beta_i}.
\tilde{f}_\mu(p) = \sum \limits_{i}^{N}\, \alpha_i \frac{u_{i\mu}}{u_i \cdot p}, \,\,\,\,\,\,\,\sum \limits_{i}^{N}\alpha_i = 0, \,\, u_i \cdot u_i =1, \, i=1, \ldots, N.
P^{(C_n)}_{(r)}(x)={(q;q)_r\over (t;q)_r}\sum_{j=0}^{\lfloor r/2\rfloor}G_{r-2j}(x;q,t){(b/t;q)_j(t^{-n}q^{-r};q)_j\over(q;q)_j(t^{-n+1}q^{-r+1}/b;q)_j}{1-t^{-n}q^{-r+2j}\over 1-t^{-n}q^{-r}}(t^2/qb)^j,
\int_0^{a^{p/r}}v(b)^{1+\lambda/n}db = \int_0^{a^{p/r}}v(b)^{1+\lambda/n}b^{(\lambda +n)(r-1)/n}b^{-(\lambda +n)(r-1)/n}db &\geqslant \bigg(\int_0^{a^{p/r}}v(b) b^{r-1}db \bigg)^{1+\lambda/n}\bigg(\int_0^{a^{p/r}}b^{(\lambda +n)(r-1)/\lambda}db \bigg)^{-\lambda/n}.
\left\{\prod\limits_{j=1}^{N-1}(1-R\kappa_j(x))\right\}^{\frac 12}- \left\{\prod\limits_{j=1}^{N-1}(1+R\kappa_j(x))\right\}^{\frac 12}= c\ \mbox{for every}x \in \partial\Omega,
\mathcal M_{\alpha, \beta, j+1}=(\nabla_x \cdot \nabla_y) \mathcal M_{\alpha, \beta, j}-\frac{j( n-1-3j-2 \alpha -2\beta )}{(\alpha +1-\rho) (\beta +1-\rho)}\mathcal M_{\alpha+1, \beta+1, j-1}\mathcal L_x\mathcal L_y.
L = -{\frac{1}{4}}G_{\mu \nu}^{2}+{\sum_{i = 1}^{N_{f}}}\bar{\psi}_{i}i \gamma^{\mu}D_{\mu}\psi_{i}+ G ~{\sum_{i = 1}^{n_{f}}}( \bar{\psi}_{i}\psi_{i})^{2}~ .
[\begin{matrix}V_{1}^{*}\\V_{2}^{*}\end{matrix}]M^{*}M[\begin{matrix}V_{1}&V_{2}\end{matrix}]=[\begin{matrix}V_{1}^{*}M^{*}MV_{1}&V_{1}^{*}M^{*}MV_{2}\\V_{2}^{*}M^{*}MV_{1}&V_{2}^{*}M^{*}MV_{2}\end{matrix}]=[\begin{matrix}D&0\\ 0&0\end{matrix}].
\begin{array}{lll}\|\Phi(t,s)P(s)\| = \|\Phi(t,\tau)\Phi(\tau,s)P(s)\| &= \|\Phi(t,\tau)P(\tau)\Phi(\tau,s)\| &= \|\Phi(t,\tau)P(\tau)\Phi^{-1}(s,\tau)\|.\end{array}
\varepsilon_{mod}= \left|\left| \frac{F_{approx}(t)}{F_{ex}(t)}- 1 \right|\right| = \left( \frac{1}{t_{max}-t_{min}}\int_{t_{min}}^{t_{max}}\left| \frac{F_{approx}(t)}{F_{ex}(t)}- 1 \right|^2dt\right)^{1/2}.
S_{3 / 2}= \int_{M}d^{4}x \sqrt{g}\biggr [{\frac{1}{2}}\tilde{\kappa}_{\mu}{\widehat \nabla}\kappa^{\mu}+ \tilde{\epsilon}{\widehat \nabla}\epsilon +{\frac{1}{2}}\tilde{\eta}{\widehat \nabla}\eta \biggr ] \; ,
\tilde{R}= \frac{\tilde{a}M^2 (\gamma - 1)^2 e^{\phi_0}}{\rho^4}\;\left(1 - \frac{\rho_1}{\rho}\right) \;\left(1 - \frac{\rho_0}{\rho}\right)^{- \frac{1 + 3 k^2 - 2 l}{1 + k^2}}\; .
\lim_{s\uparrow S}(S-s)^{2\alpha}((S-s)^{1-2\alpha}- S^{1-2\alpha})^{\frac{2\alpha}{2\alpha-1}}= \lim_{s\uparrow S}\left(1- S^{1-2\alpha}(S-s)^{2\alpha - 1}\right)^{\frac{2\alpha}{2\alpha-1}}= 1, \lim_{t\uparrow T}(T-t)^{2\beta}((T-t)^{1-2\beta}- T^{1-2\beta})^{\frac{2\beta}{2\beta-1}}= 1,
J_{\varepsilon}^{\prime}(W_{\varepsilon,\xi}+\phi_{\varepsilon,\xi})\left[\left. \frac{\partial}{\partial y_{h}}\phi_{\varepsilon,\xi(y)}\right\vert_{y=0}\right] =\sum_{l=i}^{n}C_{\varepsilon}^{l}\left\langle Z_{\varepsilon,\xi}^{l},\left. \frac{\partial}{\partial y_{h}}\phi_{\varepsilon,\xi(y)}\right\vert_{y=0}\right\rangle_{\varepsilon}.
i\Delta I_h z(t)= i\Delta S(t) I_h z_0+\int_0^t \Delta S(t-s) I_h(\nu z(s) -i \epsilon (\gamma z(s) -\mu \overline{z}(s)))\d s -\tfrac{i}{2}\int_0^t \Delta S(t-s)I_h(z(s)F_\Phi )\d s - \kappa \int_0^t \Delta S(t-s)I_h(\theta_R(|z|_{X^{\mathfrak{s}}_s})|z(s)|^2z(s))\d s +\int_0^t \Delta S(t-s)I_h(z(s)\d W(s))
\frac{\partial}{\partial t}[\rho(h+\overline{\eta})\overline{u}]+\nabla\cdot[\rho(h+\overline{\eta})\overline{u}\otimes\overline{u}+S+\frac{1}{2}\rho g(h+\overline{\eta})^{2}I]=\rho g(h+\overline{\eta})\nabla h+\tau_{w}-\tau_{b},
\begin{cases}\hat{\lambda}^k=\lambda^k-(1-\tau^k)\beta^k(A{x}^{k}-b), \hat{x}^k=x^k+\frac{\tau^k(1-\tau^{k-1})}{\tau^{k-1}}(x^k-x^{k-1}), x^{k+1}\in\arg\min\limits_x\{f(x)+x^T\nabla_x\varphi^k(\hat{x}^k,\hat{\lambda}^k)+\frac{\beta^k}{2}\|x-\hat{x}^k\|_D^2+\frac{\sigma(1-\tau^k)}{2\tau^k}\|x-x^k\|^2\}.\end{cases}
p_{\widehat{K}}(\hat{k}) =\mathbb{P}\big(\hat{k}-1/2\le \widehat{K}^\mathbb{R}\le \hat{k}+1/2\big)=F_{\widehat{K}^\mathbb{R}}(\hat{k}+1/2)-F_{\widehat{K}^\mathbb{R}}(\hat{k}-1/2).
\tau_n:=\inf\{t\in [0,T]: \|S^n\|_{\mathcal C([0,t];\mathbb R^N\times \mathbb R^N)}\ge n\}\wedge \inf\{t\in [0,T]: \min_{i=1}^N \min_{s\in [0,t]}\rho_i^n(s)\le \frac 1{n}\},
V^{+,+}= \left\{(A^+,\underline{A}^+) :\A\in \binom{[n]}{k}\right\}, V^{+,-}= \left\{(A^+,\underline{A}^-) :\A\in \binom{[n]}{k}\right\}, V^{-,+}= \left\{(A^-,\underline{A}^+) :\A\in \binom{[n]}{k}\right\}, V^{-,-}= \left\{(A^-,\underline{A}^-) :\A\in \binom{[n]}{k}\right\}.
\xi^{(0)}=\sup\bigg\{a\ge 0\colon \lim_{n\to\infty}\sup_{v\in\mathbb{Z}^d,\,\|v\|\ge n}\mathbb{P}\big( \gamma(0,v)\subset\mathrm{cyl}(0,v,\|v\|^a) \big) <1 \bigg\},