harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\beta ( P_{0}; \kappa ) = - P_{0}^{2}\, \frac{e^{P_{0}/ \kappa}}{2 f ( P_{0})}\left( \frac{n \Phi ( P_{0})}{\kappa}+ \Phi^{\prime}( P_{0}) - \frac{e^{- P_{0}/ \kappa}\Phi ( P_{0})}{f ( P_{0})}\right) .
S^{(N)}\left[{\bf r}_{1},\dots{\bf r}_{N-1}\right] ({\bf r}'',{\bf r}' ; t'',t') =\sum_{j=0}^{N-1}\left[\frac{M \left({\bf r}_{j+1}-{\bf r}_{j}\right)^{2}}{2\epsilon}-\epsilon V({\bf r}_{j}, t_{j})\right]\; ,
\partial_{i}L(x) =\kappa L(x) \left\{\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\tau((i,j))-\frac{\gamma}{x_{i}}I\right\},1\leq i\leq N, \gamma :=\frac{S_{1}(\tau)}{N}=\frac{1}{2N}\sum_{i=1}^{\ell(\tau)}\tau_{i}(\tau_{i}-2i+1) .
D^{12}\phi = -D^{21}\phi=i\int_{\partial D}\frac{d\zeta}{\zeta -k}\int_{[k_0,\varsigma]}{h_{12}(\varsigma',\varsigma)}e^{{\frac{i}{2}}(\varsigma \overline{z}+\overline{\varsigma'}z)+\frac{t}{2}(\overline{\varsigma'}^2-\varsigma^2)}\overline{\phi}(\varsigma ')\overline{d\varsigma'}.
\Gamma_{P V}( A ) = ( - 1 \pm \mathrm{s i g n}( m ) ) \frac{i \pi}{2}\eta ( 0 ) + i \sum_{n = 0}^{\infty}\frac{P_{2 n + 1}}{( m^{2})^{n - ( d - 1 ) / 2}}\frac{\Gamma \left( n - \displaystyle{\frac{d}{2}}+ 1 \right)}{2 n + 1 - d}.
u^{(n)}_1=q^{2n_1+1}w_1, u^{(n)}_2=q^{(2n_2+1)}w_2, v^{(n)}_1=q^{(-2n_1+1)}w_1^{-1}, v^{(n)}_2=q^{(-2n_2+1)}w_2^{-1}.
(|(-\Delta_x)^{1/6}u_0|^p)^{1/p}_{Q_{\nu r, R\nu r}(z_0)}\le N \sum_{k = 0}^{\infty}2^{-k}F_k (R) , (|(-\Delta_x)^{1/6}u_0|^p)^{1/p}_{Q_{r, Rr}(z_0)}\le N \nu^{(4d+2)/p}\sum_{k = 0}^{\infty}2^{-k}F_k (R),
I_1 =\bigl\{\alpha : \alpha_j\leq C_j \delta L_j,\j=1,\dots,n\bigr\}I_2 =\mathbb N^n\setminus I_1=\bigl\{\alpha : \exists j\in\{1,\dots,n\}\ : \alpha_j> C_j \delta L_j\bigr\}
\P^{0,0}_{2T}(X(t_m)=x^{(m)}, m \in \{1,2, \dots, M\}) & =\prod_{m=1}^MQ(t_{m-1}, x^{(m-1)}; t_m, x^{(m)})\frac{Q(t_M, x^{(M)}; 2T, 0)}{Q(0, 0; 2T, 0)}.
\Omega\equiv\frac{d^{3}\varphi}{dz^{3}}+i\alpha Re[(c-U(z_{2}=1))\frac{d\varphi}{dz}+\varphi]-i\alpha Re(\frac{1}{Fr}+\frac{\alpha^{2}}{We})\frac{\varphi}{c-U(z_{2}=1)}=0,
\begin{aligned}-G_n(t)- 2E\left(\dfrac{Q(t)}{2(Dt)^{1/2}}\right)+1+\mu(t,D) &\ge -G_n(t)- 2E\left(\dfrac{q(\theta_n)}{2(Dt)^{1/2}}\right)+1+\mu(t,D) &=G_{n+1}(t)+1+o(1)>0\quad t\ge \tilde\theta_n.\end{aligned}
\lim_{\delta\searrow 0}\int_{\mathcal{O}_\delta^c}\frac{J_p(|x|^{-\beta}-|y|^{-\beta})}{|x-y|^{N+s\,p}}\,(\varphi(x)-\varphi(y))\,dy\,dx &=\int_{\mathbb{R}^{2N}}\frac{J_p(|x|^{-\beta}-|y|^{-\beta})}{|x-y|^{N+s\,p}}\,(\varphi(x)-\varphi(y))\,dx\,dy.
f(y) = \sum_{j=0}^{i-1}\omega^{\beta_j}\cdot p_j + 1 + f_i(y) \le \sum_{j=0}^{i-1}\omega^{\beta_j}\cdot p_j + 1 + \omega^{\beta_i}\cdot p_i = \sum_{j=0}^{i}\omega^{\beta_j}\cdot p_j \le \tau < \tau +1.
{\cal J}\overline{A}^i_n{\cal J}^{-1}= a^i_n \;\;\;;\;\;\;{\cal J}\overline{\widetilde A}^{\scriptscriptstyle L}_n{\cal J}^{-1}= L^{\scriptscriptstyle T}_n\;\;\;;\;\;\;{\cal J}\overline{C}_n{\cal J}^{-1}= c_n \;\;\;.
\widetilde{\varphi_{\ell}^{H_1}}(x_0,x_1)=(x_0^\ell+x_1^\ell)+\sum_{0<j<\ell,j\equiv 0\pmod{4}}\frac{(-1)^{\ell/4}\binom{\ell}{j}}{((-1)^{\ell/4}+2^{\ell/2-2})}x_0^{\ell-j}x_1^j
|U_2| = -U_2 := \max\Big\{\sqrt{2}s + (1+\beta)|U_\beta|(1+(L/d)^2)^{\beta/2}, \, (1+\beta)|U_\beta|(1+(L/d)^2)^{\beta/2}+ \lambda \| w \|_{L^\infty}, & \qquad\qquad\frac{2 \| w \|_{L^\infty}+ U_\beta ((L/d)^2+\epsilon^2)^{(\beta+1)/2}}{\epsilon}\Big\},
\lim_{m\rightarrow\infty}\frac{\|Tg_m\|_{q}}{\|g_m\|_{p}}= \|T\| \Rightarrow \lim_{m\rightarrow\infty}\inf_{g_\ast\in M(T)}\frac{\|g_m-g_\ast\|_p}{\|g_m\|_p}= 0
l_{n}\left( a\right) =H_{n}-\tfrac{1}{90a^{2}+2}\ln \left( n^{2}+n+\tfrac{3a+1}{3}\right) -\tfrac{45a^{2}}{90a^{2}+2}\ln \left( n^{2}+n+\tfrac{15a-1}{45a}\right)
\lim_{\nu\to\infty}S(A|A')_{(\mathcal{N}(t)\otimes\mathbb{I}_{A'})(\hat{\omega}_{AA'}(\nu))}&= \lim_{\nu\to\infty}n\left(g\left(\nu_+(\nu,t)-\frac{1}{2}\right) + g\left(\nu_-(\nu,t)-\frac{1}{2}\right) - g\left(\nu-\frac{1}{2}\right)\right) &= n\ln t + n\;,
\sup_{\begin{smallmatrix}0\le n \le m n,m\in \mathbb{N}\end{smallmatrix}}\frac{1}{(m-n+1)^p}\Bigg(\sum_{k=n}^mw(k)\Bigg)\Bigg(\sum_{k=n}^mw(k)^{-1/(p-1)}\Bigg)^{p-1}<\infty,
[\nabla'^{\beta}\nabla ,\varphi^{k+1}] = \ (k+1)[\varphi^k(\nabla \varphi)\nabla'^{\beta}+ \varphi^k(\nabla'^{\beta}\varphi)\nabla + k\varphi^{k-1}(\nabla'^{\beta}\varphi )(\nabla \varphi) &+ (k+1)\varphi^{k-1}\varphi(\nabla \nabla'^{\beta}\varphi)]
\psi_3([b,g]) =\big([a, a^{i_0}b^{i_1}a^{2i_2}], [a^2, a^{2i_0+i_1}b^{i_2}], [b, b^{i_0}a^{2i_1+i_2}] \big) &=\big([a, a^{i_0}b^{i_1}a^{2i_2}], \b_2^{-i_2}b_0^{i_2}, \b_0^{-1}b_{2i_1+i_2}\big).
(2)\,\,\,G_{i}=\left(y-\underline{\varphi}(\underline{a})\right)^{\alpha_{i}}+\sum_{\gamma\in\Delta(\underline{a})}g_{i\gamma}\cdot\left(y-\underline{\varphi}(\underline{a})\right)^{\gamma},\,\,i=1,\dots,t,
\|\theta(t)\|_\infty \lesssim \frac{\|\theta(0)\|_2}{\nu T}e^{-\nu \frac{\pi}{L}(t-T)}+\frac{L^{1-\frac2p}}{\nu}\|f\|_p \lesssim \frac{\|\theta(0)\|_2}{\nu t}+\frac{L^{1-\frac2p}}{\nu}\|f\|_p,t\geq T.
W_{r_i}=\big\langle w_{(1,r_i)},\dots,w_{(n_i,r_{i})}\big\rangle,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\leq i\leq m,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,W_{M-1}=\langle \mu\rangle.
\int_{\Omega_R}|b'(\rho)|^{\beta_0-1}\leq c\int_{\{\rho>\overline\rho-\alpha_1\}}P(\rho)\leq c\int_{\{\rho>\overline\rho-\alpha_1\}}\left(P(\rho)-P(r)-P'(r)(\rho-r)\right), \int_{\Omega_R}|b'(\rho)|^{\beta_0}\leq c\int_{\Omega_R}p(\rho).
\partial_{x_i}\Phi=\lambda y_i+a_i,\qquad\partial_{y_i}\Phi=\lambda x_i-b_i,\qquad\partial_{x_iy_i}\Phi=\lambda,\qquad\partial_{z_j}\Phi=2( z_j- c_j),\qquad\partial_{z_j z_j}\Phi=2,
\alpha A_{\varphi}^{\theta, \frac{\theta}{\alpha}}C_{\theta}- C_{\theta}A_{\bar{\varphi}}^{\frac{\theta}{\alpha}, \theta}P_{\frac\theta\alpha}\bar{\alpha}= \widetilde{H}_{\theta}H_{\varphi}C_{\alpha}P_\alpha -\widetilde{H}_{\alpha}H_{\varphi}C_{\theta}.
\bigl\langle W(\xi\otimes T\zeta'),L\xi'\otimes T\zeta\bigr\rangle &=\bigl\langle W^*(\xi\otimes\zeta),L\xi'\otimes\nabla\zeta'\bigr\rangle=\bigl\langle (L\otimes\nabla)W^*(\xi\otimes\zeta),\xi'\otimes\zeta'\bigr\rangle.
\int_{|w|<1}\frac{1}{w-z}d\mathrm{arg}(w)d\mathrm{arg}(w-1)-\int_{|w|<1}\frac{w}{1-zw}d\mathrm{arg}(w)d\mathrm{arg}(w-1)=-\pi i\left(\frac{\log(|1-z|)}{z}+\frac{\log(|z|)}{1-z}\right).
\frac{\left(-b;q^{2}\right)_{\infty}}{\left(b;q\right)_{\infty}}=\sqrt{\frac{\log q^{-1}}{\pi}}\int_{-\infty}^{\infty}{}_{1}\phi_{1}\left(\begin{array}{ccc}\begin{array}{c}a b\end{array}\vert q,-bq^{\alpha}\end{array}\right)\left(-abq^{-\alpha};q\right)_{\infty}q^{\alpha^{2}}d\alpha.
\frac{2^{1\,-\,z}- 1}{(4\,\pi)^{\frac{1-z}{2}}\,\Gamma\left(\frac{1 + (1 - z)}{2}\right)}\,G(1 - z) := \frac{2^z - 1}{(4\,\pi)^{\frac{z}{2}}\,\Gamma\left(\frac{1 + z}{2}\right)}\,G(z)\,.
|\lambda|^{1/2}\|\partial_z(\alpha v)\|_{L^\infty_H L^p_z(\Omega)}\le C_p |\lambda|^{1/p}\sup_{x_0' \in G}\biggl(|\lambda|^{1/2}\|\partial_z (\alpha v)\|_{L^p(C(x_0'; |\lambda|^{-1/2}))}+ \|\nabla_H\partial_z (\alpha v)\|_{L^p(C(x_0'; |\lambda|^{-1/2}))}\biggr),
|A_R(u(t))| = \Big| A_R(u)- A_R \left( e^{i(t+ \theta(t))}Q(x -X(t)) \right) \Big| & \lesssim e^{-CR_0}\delta(t) + \delta(t)^2+ \delta(t)^{2p}, R \ge R_0 + |X(t)|.
\left\langle \Psi_{1}{}^{\alpha \beta}\left|\right.\Psi_{2}{}^{\alpha \beta}\right\rangle=\frac{1}{N(\alpha)N(\beta)}\int{\rm Tr}\left(f_{1}^{\alpha \beta +}(q) f_{2}^{\alpha\beta}(q)\right)P_{\lambda}(q)dq^{1}\ldots dq^{n}.
p(h_{j}^{(1m)}=1|v^{m},h^{(2m)})=g(\sum_{i=1}^{D}W_{ij}^{(1m)}\frac{v_{i}^{m}}{\sigma_{i}}+\sum_{l=1}^{F_{2}^{m}}W_{jl}^{(2m)}h_{l}^{(2m)}+b_{j}^{(1m)})
\zeta_{n, \mu, p_{j}}:= \kappa_{n_j}(a_{j}+ p_{j}+ h_{p_{j}, n}^{\mu}- n_j, b_{j}+ p_{j}+ h_{p_{j}, n}^{\mu}- n_j) \textstyle{}+ \kappa_{n_j}\left(a_{j}+ p_{j}+ g_{p_{j}, n}^{\mu}- \big\lceil \frac{3n_j}{2}\big\rceil, b_{j}+ p_{j}+ g_{p_{j}, n}^{\mu}- \big\lceil \frac{3n_j}{2}\big\rceil \right).
H_{0}=\frac{1}{2}{\bf b_{0}}^{2}+\frac{e^{2}}{2\pi}{\bf a_{0}}^{2}+\frac{\pi}{4Le^{2}}(Q^{2}+Q_{5}^{2})+\frac{1}{\sqrt{2L}}{\bf a_{0}}(Q-Q_{5}) \ .
\phi_{\epsilon^v|_{E \backslash \Sigma H}}\Big(p_w - \sum_{e \in w(E \backslash \Sigma H)^1}s_e s^*_e\Big) = \phi_{\epsilon^v|_{E \backslash \Sigma H}}\circ q_{\Sigma H \backslash H}\Big(p_w - \sum_{e \in w(E \backslash H)^1}s_e s^*_e\Big) &= \phi^H_v \circ q_H\Big(p_w - \sum_{e \in wE^1}s_e s^*_e\Big) = 0.
\left| \frac1{2\pi i}\int_{\Gamma_-\cup \Gamma_+}e^{st}K(s)ds\right| \leq \frac{C_K}{\pi}\int_{r_0}^\infty r^{-\mu}e^{-rt \cos \varphi}dr &= \frac{C_K}{\pi}t^{\mu-1}(\cos \varphi)^{\mu-1}\int_{r_0 \cos \varphi}^\infty r^{-\mu}e^{-r}dr &= \frac{C_K}{\pi}t^{\mu-1}(\cos \varphi)^{\mu-1}\Gamma(1-\mu,r_0\cos \varphi),
\sum_{k=0}^n \binom{n}{k}\left| f^{(j)}(x) - f^{(j)}(\xi_k) \right| x^k (1-x)^{n-k}= \sum_{k=0 \atop \left| x - \frac{k}{n+j}\right| < \delta}^n \binom{n}{k}\left| f^{(j)}(x) - f^{(j)}(\xi_k) \right| x^k (1-x)^{n-k}& + \sum_{k=0 \atop \left| x - \frac{k}{n+j}\right| \ge \delta}^n \binom{n}{k}\left| f^{(j)}(x) - f^{(j)}(\xi_k) \right| x^k (1-x)^{n-k}
\mathrm{ad}_{R}^{\circ}(X)(\mathsf{M}_{m}^{n}) =\pi(K_{2\rho}^{-1}X_{(1)}K_{2\rho}S^{-1}(X_{(2)}))\mathsf{M}_{m}^{n}\pi(S^{-2}(X_{(3)})K_{2\rho}^{-1}S(X_{(4)})K_{2\rho}).
-\L[(\Phi^{\rm per},\psi)] = \underbrace{N\Phi^{\rm lin}\otimes \Phi^{\rm per}+ N\Phi^{\rm per}\otimes \Phi^{\rm lin}}_{=: -\L_1}+ \underbrace{\tilde{N}\Phi^{\rm lin}\otimes \Psi + \tilde{N}\Psi \otimes \Phi^{\rm lin}}_{=: -\L_2}&+ \underbrace{\bar{U}\otimes \Psi + \Psi \otimes \bar{U}}_{=: -\L_3}\, .
\ln\left(((d+1)d^{n-1})!\right) = \Big((d+1)d^{n-1}+\frac{1}{2}\Big) \ln((d+1)d^{n-1}) - (d+1)d^{n-1}+ O(1) &= n \cdot d^{n-1}\left((d+1)\ln(d)\right) & + d^{n-1}(d+1)(\ln(d+1)-\ln(d)-1) + O(n)
U_k \big( g_k+\frac{g_k^Tg_k}{p_{k-1}^T g_{k-1}}p_{k-1}\big) =\frac1{\delta_k}g_k-B_{k-1}\big( g_k+\frac{g_k^Tg_k}{p_{k-1}^T g_{k-1}}p_{k-1}\big) &=\frac1{\delta_k}g_k- g_k+\frac{g_k^Tg_k}{p_{k-1}^T g_{k-1}}g_{k-1},
\log L \geq n \log \frac{A}{\sqrt{a}}- \frac{1}{4}(1+b) \, n \log n + \frac{1}{2}n \log n - n \log e + o(n) = \left( \frac{1-b}{4}\right) \, n \log n + n ( \log \frac{A}{e\sqrt{a}}) + o(n) \;,\,
l_{\bar q}^\mu = \left(k_g + k_{\bar q}^\prime - k_{\bar q}\right)^\mu ={\vec k_{g_{\!\perp}}^{\,2}\over 2k_g^+}\ \eta^\mu \quad{\rm and}\quad l^\mu_q = \left(k_g + k_q - k_q^\prime\right)^\mu ={\vec k_{g_{\!\perp}}^{\,2}\over 2k_g^+}\ \eta^\mu\ .
T_{1}= \frac{- N_{f}/ 2 + N_{c}}{2 \pi i}, T_{0}= - \frac{1}{4 \pi i}\sum_{i = 1}^{N_{f}}m_{i}, \frac{\partial F}{\partial T_{1}}= 2 u - \frac{1}{2}\sum_{i = 1}^{N_{f}}m_{i}^{2}
\int_1^R |\nabla^{0,1}\nabla u|^2 +{\rm Ric}(\nabla u, \nabla u) = \frac{1}{2}\left(\oint_R - \oint_1\right)(\langle \nabla_{\partial_r}\nabla u, \nabla u\rangle + \langle J\nabla_{J\partial_r}\nabla u, \nabla u\rangle).
\mathcal{I}(\delta_1, 0) = \inf_{\lambda_1 \in \mathbb{R}_{+}}\left[\lambda_1 \delta_1+\sup_{\varpi \in \Pi\left(\mu_{13}, \mu_{23}\right)}\int_{\mathcal{V}}f_{\lambda, 1}(v) \, d \varpi(v)\right].
Z^{{\rm Large}N_c}=\int \prod_{i,j}[dq^{\dag i}_+][dq^i_+][ds^{ij}] \exp\left\{iS^{{\rm Large}N_c}\right\}\prod\limits_{i,j}\delta\left( s^{ij}-q^{\dag j}_+ (x)q^i_+(y)\right),
\begin{array}{l}\displaystyle{\frac{A_n}{B_n}}\sim r_0^{2\left(k+\alpha-n\right)}\qquad n-k+\frac{1}{2}-\alpha \leq 0 \displaystyle{\frac{B_n}{A_n}}\sim r_0^{2\left(n+1-k-\alpha\right)}\qquad n-k+\frac{1}{2}-\alpha> 0\end{array}
\begin{aligned}\mathbb{E}^{u_1,u_2}\bigg[\int_0^T\Big(\big(b_{v_1}(t)-\sum_{j=1}^2\sigma_j(t)h_{jv_1}(t)\big)q_1(t)-f_{v_1}(t)p_1(t)+\sigma_{v_1}(t)k_1(t)+\sum_{j=1}^2\big(\sigma_{jv_1}(t)k_{j1}(t)+h_{jv_1}(t)Q_{j1}(t)\big) + l_{1v_1}(t)\Big)\cdot v_1(t)\bigg]dt\geq 0,\end{aligned}
A(x,t)= \left(a_{ij}(x,t)\right)_{1\leq i,j \leq N}, \, \, \, b(x,t):=\left(b_1(x,t),\ldots,b_{m_0}(x,t),0,\ldots,0\right), Y:=\sum_{i,j=1}^N b_{ij}x_j\partial_{x_i}u(x,t)-\partial_t u(x,t).
\int_{\Omega}\left\langle A(x)Xu(x),Xu(x)\right\rangle^{\frac{p(x)-2}{2}}\left\langle A(x)Xu(x),Xv(x)\right\rangle \,dx & +\int_{\Omega}|u(x)|^{p(x)-2}u(x)v(x)\omega(x)^{p(x)}\,dx=\int_{\Omega}f(x)v(x)\,dx\geq 0,
[\mathbf{h}(1,1),\mathbf{h}(a,b)] =[\mathbf{h}(1,1),[\mathbf{f}(a),\mathbf{g}(b)]] &=[[\mathbf{h}(1,1),\mathbf{f}(a)],\mathbf{g}(b)]+[\mathbf{f}(a),[\mathbf{h}(1,1),\mathbf{g}(b)] &=-[\mathbf{f}(a),\mathbf{g}(b)]+[\mathbf{f}(a),\mathbf{g}(b)] &=0.
S^{\pm}_{j,k}=\mathbb{I}_{\partial_{\eta}\phi\sim \pm 2^{-j}}\mathbb{I}_{\sqrt{|\xi|^2+|\eta|^2}\sim 2^{k}}(1-\varphi^{s})\frac{\partial_{\xi}\phi}{\partial_{\eta}\phi}\frac{\partial_{\eta}^2\phi}{\partial_{\eta}\phi}.
R_Z(p,p+n) = \frac{a^2+b^2}{2}\bigl[(n+1)^{2H}-2n^{2H}+ (n-1)^{2H}\bigr] &- ab \bigl[ (2p+n+2)^{2H}- 2(2p+n+1)^{2H}+(2p+n)^{2H}\bigr] & = \bigl(a^2+b^2\bigr) R_B(0,n) -abf_p(n),
\frac{\partial F}{\partial t}- D \frac{\partial^2F}{\partial s^2}+ \frac{\partial}{\partial u}.(\mathcal{G}F)-\epsilon F\kappa : u\otimes u+\epsilon \frac{\partial}{\partial s}\Big( F\kappa:\lambda(F)\Big)=0 \textrm{on}S_2 \times ]0, 1[
{\bf D}_{\mu}( \sigma ) F ={\frac{\delta F}{\delta z^{\mu}( \sigma )}}= D_{\mu}\, f_{\rho_{1}\rho_{2}}{\cal D}z^{\prime \rho_{1}}{\cal D}z^{\prime \rho_{2}}+ 2{\cal D}^{2}( f_{\mu \rho_{2}}{\cal D}z^{\prime \rho_{2}}) - 2 f_{\rho_{1}\rho_{2}}z^{\prime \kappa}z^{\prime \lambda}R_{\mu \kappa \lambda}^{{\, \, \, \, \, \, \, \,}\rho_{1}}{\cal D}z^{\prime \rho_{2}}
\begin{aligned}P-P_{O2}^* \geq N_0\sum_{i=1}^N\frac{1}{|h_i|^2}(e^{\frac{r^*+\Delta r_i}{\alpha_i}}-1)\alpha_i &-\min_{\sum_{i=1}^K\alpha_i=1}N_0\sum_{i=1}^N\frac{1}{|h_i|^2}(e^{\frac{r^*}{\alpha_i}}-1)\alpha_i.\end{aligned}
{}\tau\left ( q_{1}, q_{2}, s, t \right ) ={}\left \{x:q_{1}+\mathcal{A}_{1}x^{m-1}\ge\theta,q_{2}+\mathcal{A}_{2}x^{m-1}\ge\theta , \left (q_{1}+\mathcal{A}_{1}x^{m-1}\right )^{T}\left (q_{2}+t\mathcal{A}_{2}x^{m-1}\right )\le s\right \}
\#\mathcal{Z}_{M,M}= \sum_{k+l=2}^n p_k(M)p_l(M) =\sum_{k=1}^{n-1}p_k(M) \sum_{l=1}^{n-k}p_l(M) \leq C \sum_{k=1}^{n-1}M^{k-1}M^{n-k}\leq C nM^{n-1}
c_n =\lim_{r\rightarrow1-}\int_{-\pi}^{\pi}f(re^{i\varphi})\overline{\phi_{n}(e^{i\varphi})}dm_{\lambda}(\varphi), \tilde{c}_n =\lim_{r\rightarrow1-}\int_{-\pi}^{\pi}f(re^{i\varphi})e^{i\varphi}\phi_{n-1}(e^{i\varphi})dm_{\lambda}(\varphi),
(a)^{\prime}\ \x \rightarrow e^{- \chi \alpha / (2a)}x, \ \Omega \rightarrow e^{- \chi \alpha / (2a)}\Omega, \e^{2 \rho}\rightarrow e^{\chi \alpha /2}e^{2 \rho}, \A \rightarrow A, \f \rightarrow f + \alpha ,
\int_{0}^{1}d t \delta_{\vec{u}}( t ) < \mathrm{T r}( H_{1 2}( \gamma ) H_{3 4}( \gamma ) ) > = -{\frac{i \pi}{k}}\mathrm{T r}( H_{1 2}( \gamma ) H_{3 4}( \gamma ) ) +{\frac{2 i \pi}{k}}\mathrm{T r}( H_{1 2}( \gamma ) ) \mathrm{T r}( H_{3 4}( \gamma ) )
M\int_{T_3}^{t-T_2}f'(x(u))\,du = \int_{T_3}^{t-T_2}\frac{f'(x(u))}{f(x(u))}\cdot \frac{Mf(x(u)))}{x'(u)}x'(u)\,du &\geq \int_{T_3}^{t-T_2}\frac{f'(x(u))}{f(x(u))}x'(u)\,du &=\log f(x(t-T_2)) - \log f(x(T_3)) &> \log\left(\frac{1}{2}\right) + \log f(x(t)) - \log f(x(T_3)).
\sup_{x\in B_k(t)}\frac{\phi_{\mu_k(t)}(x)}{\phi_{-\mu_k(t)}(x)}=\sup_{{x}\in B_k(t)}\exp\Big\{\frac{\big| \|{x}-\mu_k(t)\|_2^2-\|{x}+\mu_k(t)\|_2^2\big|}{2}\Big\}&= \sup_{{x}\in B_k(t)}\exp\{2|{x}^T\mu_k(t)| \}=\exp\{2a\},
H=\frac{|1-\Theta{\cal B}|}{2}\left\{\frac{b_1^2}{2}\tilde{p}_1^2 +\frac{b_2^2}{2}\tilde{p}_2^2 +{\vec b}_1 \cdot{\vec b}_2( \tilde{p}_1 \tilde{p}_2 + \tilde{p}_2 \tilde{p}_1 ) \right\}~,
-\dim H^2(\widehat{G},M)= -\dim H^2(\widehat{G}_1,M)-\dim H^2(\widehat{G}_2,M)+\dim H^2(\widehat{C},M) -\dim H^1(\widehat{G},M) +\dim H^1(\widehat{G}_1,M)+\dim H^1(\widehat{G}_2,M) -\dim H^1(\widehat{C},M)+\dim H^0(\widehat{G},M)-\dim H^0(\widehat{G}_1,M) -\dim H^0(\widehat{G}_2,M)+\dim H^0(\widehat{C},M).
\left(\begin{array}{c}F(t,z) G(t,z)\end{array}\right)=\left(\begin{array}{cc}1 0 0 e^{i\xi_0 t}\end{array}\right)\left(\begin{array}{cc}e^{i\nu_0 t}0 0 e^{-i\nu_0 t}\end{array}\right)\left(\begin{array}{c}F_0(z) G_0(z)\end{array}\right) + \left(\begin{array}{c}\mu(t) \nu(t)\end{array}\right)\,,
= ( \frac{3}{2}\times \frac{4}{3}\times \cdots \times \frac{2 0 1 6}{2 0 1 5}\times \frac{2 0 1 5}{2 0 1 6}) \times ( \frac{1}{2}\times \frac{2}{3}\times \cdots \times \frac{2 0 1 5}{2 0 1 6}
\Big \langle f \Big ({1 \over 1 + e^{i \theta_1}}, \dots,{1 \over 1 + e^{i \theta_N}}\Big ) \Big \rangle^{(\rm cJ)}= \langle f(x_1,\dots, x_N) \rangle^{(\rm J)}\Big |_{\lambda_1 = - \beta(N+p-1)/2 - 1 + iq\atop\lambda_2 = - \beta(N+p-1)/2 - 1 - i q}.
\langle T_{(1 - p) \rightarrow p}A h, \, g \rangle = \sum_{d=0}^m \left( -\frac{p}{1-p}\right)^d \langle h^{=d}, g \rangle \geq \mu_{p}(h) \mu_{p}(g) - \sum_{d=1}^m \left( \frac{p}{1-p}\right)^d | \langle h^{=d}, g \rangle | .
\mathcal{F}_{\boldsymbol{\eta}}(\boldsymbol{v}) = \int_{Q}\Big(\boldsymbol{u}\cdot \boldsymbol{v}- \sigma \partial_t^{1/2}\boldsymbol{\eta}\cdot \partial_t^{1/2}\boldsymbol{v}^{\perp}- \nu \, \textbf{curl}\, \boldsymbol{\eta}\cdot \, \textbf{curl}\, \boldsymbol{v}\Big) \, d\boldsymbol{x}\,dt.
a_k^3 =\mu_{(k)}(1-g_k^3), 1\leq k\leq \theta; a_{kk+1}^3 =\mu_{(kk+1)}(1-g_k^3g_{k+1}^3)-\mu_{(k+1)}\mu_{(k)}(1-\xi)^3\chi_k(g_{k+1})^3 (1-g_k^3)g_{k+1}^3 &-\lambda_{kk+1k+1}\lambda_{kkk+1}\xi^2(1-g_k^2g_{k+1})g_kg_{k+1}^2 1\leq k< \theta.
|\Phi[\varphi](x)-\Phi[\psi](x)| \leq \frac1{\rho({x})}\int_{\rho({x})}^{2\rho({x})}\left|\eta_{x,\varphi}(t) - \eta_{x,\psi}(t)\right| dt \leq \frac1{\rho({x})}\int_{\rho({x})}^{2\rho({x})}\|\varphi - \psi\| dt=\|\varphi-\psi\|.
\limsup\limits_{n \rightarrow \infty}d(\mu_n, \mathbf{0}) \leq \limsup\limits_{n \rightarrow \infty}d_{r, \emptyset,0}(\mu_n, \mathbf{0})= \limsup\limits_{n \rightarrow \infty}I_r (\mu_n) +I_r (\mathbf{0})+2^{-r}= 2^{-r}.
\sum_{k\le j\le i\le n}a_{i,j}=\sum_{i=k}^{n}\sum_{j=k}^{i}a_{i,j}=\sum_{j=k}^{n}\sum_{i=j}^{n}a_{i,j}=\sum_{j=0}^{n-k}\sum_{i=k}^{n-j}a_{i+j,i}
X_{(P_{\ell(\alpha)},\alpha)}= \sum_{S\subseteq E'}(-1)^{|S|}X_{P_{\alpha^c}+S}=\sum_{\textup{set}(\alpha)\subseteq J \subseteq[|\alpha|-1]}(-1)^{|J-\textup{set}(\alpha)|}X_{N_{|\alpha|}+\{v_iv_{i+1}\mid i\in J\}}.
\lim_{\xi\rightarrow\overline{c_{0}}^{-}}\frac{p_{\mu_{1}\boxtimes\mu_{2}}(\xi)}{\left|\xi-\overline{c_{0}}\right|^{1/2}}=\frac{2\left|a_{1}\right|}{\sqrt{\left|c_{2}\right|}}\cos\left(\frac{\theta}{2}-\frac{3\pi}{4}\right),
\gamma_0 ( t_1, \ldots, t_l )(x) = \sum_{j=1}^l t_j R w_j(x), \, \forall ( t_1, \ldots, t_l )\in [1/R^2,1]^l, \Gamma_\ast = \Big\{\gamma \in C \big( [1/R^2,1]^l, E_\lambda \setminus \{0 \}\big) \, ; \, \gamma = \gamma_0 \partial [1/R^2,1]^l \Big\}
\int\limits_{\mathbb{R}^{N}}\left\vert \nabla v\right\vert^{p}\frac{dx}{\left\vert x\right\vert^{\mu}}\geq\int\limits_{\mathbb{R}^{N}}\frac{|\nabla D_{N,d,p}v|^{p}}{|x|^{d\left( p+\mu-N\right) +N-p}}dx=\int\limits_{\mathbb{R}^{N}}|\nabla u|^{p}dx
\frac{2s_1-d}{2}|\nabla_{s_{1}}u|_{2}^{2}+\frac{2s_2-d}{2}|\nabla_{s_{2}}u|_{2}^{2}-\frac{d\lambda}{2}|u|_{2}^{2}+d\int_{\mathbb{R}^{d}}G(u)dx=0.
G_5 := qE\sup_{t_0 \leq t \leq u}\Big|\int_{t_0}^{t}|e_s^n|^{q-2}e_s^n \bar \sigma_s^n dw_s\Big| \leq K E\Big(\int_{t_0}^{u}|e_s^n|^{2q-2}|\bar \sigma_s^n|^2 ds\Big)^\frac{1}{2}& \leq K E\sup_{t_0 \leq s \leq u}|e_s^n|^{q-1}\Big(\int_{t_0}^{u}|\bar \sigma_s^n|^2 ds\Big)^\frac{1}{2}
\frac{\partial \mathbf{u}_{\varepsilon}(\mathbf{x},t)}{\partial t}=-b(\mathbf{D})^*g(\mathbf{x}/\varepsilon)b(\mathbf{D})\mathbf{u}_{\varepsilon}(\mathbf{x},t) +{\mathbf F}(\mathbf{x},t), \mathbf{x}\in \mathcal{O},\, 0< t < T,
\|u_\theta\|_{W^{1,\,\infty}(\mathbb S^{n-1})}\le \theta^2,\|\Delta_{\mathbb S^{n-1}}u_\theta\|_{L^\infty(\mathbb S^{n-1})}\leq \theta^2,\|D^2_{\tau}u_\theta\|_{L^\infty(\mathbb S^{n-1})}= \theta^{-1}.
I_1 =\sum_{0<|m|\le M}\frac{1}{2\pi i m}\sum_{n=1}^Ne^{2\pi i(n\xi-m\varphi_1(n))}\big(e^{2\pi im\psi(n)}- 1 \big), I_2 =\mathcal O\bigg( \sum_{n=1}^N\min\left\{1, \frac{1}{M\|\varphi_1(n)-\psi(n)\|}\right\}\bigg), I_3 =\mathcal O\bigg( \sum_{n=1}^N\min\left\{1, \frac{1}{M\|\varphi_1(n)\|}\right\}\bigg).
[d, \exp(-\delta)] = \sum_{n \ge 1}\frac{(-1)^n}{n!}[ d, \delta^n ] &= \sum_{n \ge 1}\frac{(-1)^n}{n!}n \delta^{n-1}d_K &= - \sum_{n \ge 0}\frac{(-1)^n}{n!}\delta^n d_K &= - \exp(-\delta) d_K
\langle 0 | O_{i}^{V}O_{j}^{V}| 0 \rangle \sim \langle 0 | \Theta_{i}^{\partial V}\Theta_{j}^{\partial V}| 0 \rangle_{d - 1}= \alpha_{i j}^{d}\int_{\partial V}\! \! d^{d - 1}z_{1}\int_{\partial V}\! \! d^{d - 1}z_{2}~ \hat{n}_{1}\cdot \hat{n}_{2}\langle 0 | \mathrm{\large ~ \v a r t h e t a ~}_{i}( \vec{z}_{1}) \mathrm{\large ~ \v a r t h e t a ~}_{j}( \vec{z}_{2}) | 0 \rangle_{d - 1}.
1- \hat{\nu}_{\Lambda}(\lambda^*) = \frac{1}{2k+1}\sum_{j=1}^k \left(2 -2\cos\left(2\pi \lambda_j \right) \right) = \frac{4\pi^2}{2k+1}\sum_{j=1}^k\left(\lambda_j^2 + O(\lambda_j^4) \right).
\int_{M}|G(p,y)||f(y)|\,dy = \int_{M\setminus B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy + \int_{B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy &\leq \int_{M\setminus B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy+ C,
\{\alpha_{j+(k-u)}= \alpha_j+\alpha_{k-u}\}_{j=1}^{u}\ \ \mbox{and}\ \ \delta_{n}^k=\alpha_{1+u}+\alpha_{k-u}\ \ \mbox{and}\ \ \{\alpha_{j+1+u}= \alpha_{j}+ \alpha_{1+u}\}_{j=1}^{k-u-1}.
\P(Z^{\ell_1}=u^{\ell_1},Z_{\ell_1+g+1}^{\ell_1+g+\ell_2}=v^{\ell_2})\leq \P(Z^{\ell_1}=u^{\ell_1})\P(Z_{\ell_1+g+1}^{\ell_1+g+\ell_2}=v^{\ell_2})\tilde{\Psi}(g).
\#(X_{p^2}\cap C_2\wr G) = \sum_{\lambda\vdash p^2}N_{\lambda}\prod_{j=1}^{p^2}2^{\lambda_j(j-1)}&=1+(p^2-1)2^{p(p-1)}+ p^2\sum_{1\neq d\mid p^2-1}\phi(d)2^{(d-1)(p^2-1)/d}.
\begin{pmatrix}X_{A}X_{B}X_{I}X_{G}\end{pmatrix}\mapsto\begin{pmatrix}G\_-\_^{\top}G 0 0 \_A-A^{\top}\_0 G\_-\_^{\top}G 0 \_B-B^{\top}\_{}[\_,B] [A,\_] (I\Omega\_^{\top}+\_\Omega I^{\top})G I\Omega I^{\top}\_\end{pmatrix}\begin{pmatrix}X_{A}X_{B}X_{I}X_{G}\end{pmatrix}
\frac{P_{k-1}(s)L_k(s)^\beta P_{k-1}(t)L_k(t)^\beta}{P_{k-1}(st)L_k(st)^\beta}\ge\frac{P_{k-1}(s)L_k(s)^\beta P_{k-1}(t)L_k(t)^\beta}{P_{k-1}(s)P_{k-1}(t)L_k(s)^\beta L_k(t)^\beta}= 1 .
~\sum_{(n,m)\in\mathbb{Z}^{2}}\frac{1}{(a+n\omega_{1}+m\omega_{2})^{k}}=\lim_{M\to\infty}\sum_{m=-M}^{m=M}\left(\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{(a+n\omega_{1}+m\omega_{2})^{k}}\right).
\left\|\sum_{k=1}^n \alpha_k \cdot b_k\right\|_X \coloneqq \sum_{k=1}^n \nu(\alpha_k), \left\|\sum_{k=1}^n \hat{\alpha}_k \cdot I(b_k)\right\|_{\hat{X}}\coloneqq \sum_{k=1}^n \hat{\nu}(\hat{\alpha}_k).
\mathfrak{M}= [\mathfrak{D}_{1}, \dots, \mathfrak{D}_{i-1}, \mathfrak{B}_{i}, \mathfrak{B}_{i+1}, \dots, \mathfrak{B}_{k}] = [\mathfrak{D}_{1}, \dots, \mathfrak{D}_{i-1}, \mathfrak{D}_{i}, \mathfrak{B}_{i+1}, \dots, \mathfrak{B}_{k}] = [\bar{\mathfrak{A}}_{i}, \mathfrak{B}_{i}] = [\bar{\mathfrak{A}}_{i}, \mathfrak{D}_{i}]
\phi_{n}(y) =\alpha_{n}y^{-\gamma}+\mathcal O(y^{-\gamma+2}), \phi_{n}(y) =\beta_{n}y^{2\lambda_{l}}+\mathcal O(y^{2\lambda_{l}-2})=\beta_{n}y^{-\gamma}y^{2l}+\mathcal O(y^{2\lambda_{l}-2}).
\frac{\partial^{3}}{\partial \rho^{3}}\left(H(1,\rho)\right){\big|_{\rho =0}}=3!\int\limits_{-1}^1f(t)G^{2}(t)dt + 3!\int\limits_{-1}^1f(t)F(t)dt=3!\int_{-1}^1f(t)r_3(t)dt=0.
\nu^{\pi^*}_{n-1}(0|0) =\sum_{x_{n-1}\in\{0,1\}}q_{n-1}(0|x_{n-1},0)\pi^*_{n-1}(x_{n-1}|0)=q_{n-1}(0|0,0)\pi^*_{n-1}(0|0)+q_{n-1}(0|1,0)\pi^*_{n-1}(1|0)
< \overline{{\psi}}_{L}^{({\cal N})}\ldots \overline{{\psi}}_{L}^{( 1 )}\psi_{R}^{( 1 )}\ldots \psi_{R}^{({\cal N})}> = ( \frac{e^{\gamma}}{4 \pi})^{\cal N}( \sqrt{\frac{\cal N}{\pi}}e_{c})^{\cal N}\quad .
U_1(z)=\sqrt{\frac{2-\sqrt{2}}{2}}e^{\frac{11\pi}{8}i}u_1 +u_2~,\hskip 0.5cmU_2(z)=-\sqrt{\frac{2+\sqrt{2}}{2}}e^{\frac{\pi}{8}i}u_1 +u_2 ~.
\frac{\mathrm{d}}{\mathrm{d}t}\left(\|\phi\|^2\right)=2\mathrm{Re}\langle\phi,\dot{\phi}\rangle=0 \quad\mbox{and}\frac{\mathrm{d}^2}{\mathrm{d}t^2}\left(\|\phi\|^2\right)= 2\left(\mathrm{Re}\,\langle\phi,\ddot{\phi}\rangle+\|\dot{\phi}\|^2\right)=0,\forall t>0.
F_N(\zeta_M) = \sum_{k=0}^{N-1}\sum_{2 < p_1,p_2 \le N-1}\zeta_M^{k(p_1+p_2)}= \sum_{2 < p_1,p_2 \le N-1}\sum_{i=0}^{N/M-1}\sum_{k=0}^{M-1}\zeta_M^{(iM+k)(p_1+p_2)}= \frac{N}{M}\sum_{2 < p_1,p_2 \le N-1}\sum_{k=0}^{M-1}\zeta_M^{k(p_1+p_2)}.
\int_{B_{5\rho/4}}u^2(x,0)dx \le n_1e^{M_1(1-\beta)+M_1\beta(1-\beta)}\left(\int_{B_{\rho/2}}u^2(x,0)dx\right)^{\beta^2}\left(N\int_{Q_{4}}u^2( x, t)\right)^{1-\beta+\beta(1-\beta)}&=n_1e^{M_1(1-\beta^2)}\left(\int_{B_{\rho/2}}u^2(x,0)dx\right)^{\beta^2}\left(N\int_{Q_{4}}u^2( x, t)\right)^{1-\beta^2}.
Y_{2}R^{-1}_{21}dK_1 R_{21}= R_{12}d K_{1}R_{21}Y_{2}\quad, \quad\tilde{Y}_{2}R_{12}dK^{\dagger}_1 R_{21}= R_{12}dK^{\dagger}_{1}R^{-1}_{12}\tilde{Y}_{1}\quad ,
\bar \rho_{\alpha}=-\rho_{\alpha+1}+2\rho_{\alpha}-\rho_{\alpha-1},\quad(\rho_{\alpha})_x+\sum_{s=1}^{min(m_1,m_2)}\sum_{k=0}^{s-1}f^s_{\alpha-k}\bar f^s_{\alpha-k}=0