harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\sin n\theta| + p|\cos n\theta| \leq \left(p^{2+\frac{1}{\gamma}}+ \frac{C}{|\sin n\theta|^{\frac{1}{\gamma}}}\right)^{\frac{1}{2+\frac{1}{\gamma}}}= \left(p^{\frac{1}{\alpha}}+ \frac{C}{|\sin n\theta|^{\frac{1}{\gamma}}}\right)^\alpha
\sum_{\pi}\sum_{\phi \in \mathcal{B}(\pi)}\mathcal{P}_{U^{\sigma}}(\pi(\Phi^1)\phi)\overline{\mathcal{P}_{U^{\sigma}}(\phi)}=2\sum_{\Pi}\sum_{\phi_0 \in \mathcal{B}(\Pi)}\mathcal{P}_{G^{\sigma}}(\Pi(f^1)\phi_0)\overline{\mathcal{P}_{G^{\theta}}(\phi_0)}
\int_0^1 h^n_{n-j}(t) t^s dt=\sum_{k=0}^n d^j_{n,k}\frac{1}{k+s+1}&=(-1)^n\frac{\sqrt{2n-2j+1}}{(-n)_j}\sum_{k=0}^n\frac{(-n)_k(n-j+1)_k}{(k!)^2}\frac{\prod_{m=0}^{\frac{j-1}{2}}(n+k+1-2m)\prod_{m=0}^{\frac{i-3}{2}}(n-k-1-2m)}{k+s+1}.
\alpha=\{1, 3, 4, 6, 8, 9, A, C, D, E, N, O, P, Q, R, Z, a, b, c, d, j, k, l, m, n\}, &\beta= \{5, B, G, H, I, J, K, L, M, T, U, V, W, X, Y, e, f, g, h, i, o\},\mbox{and}&\gamma=\{0, 2, 7, F, S\}.
\sum_{i \in \sigma}\|\Lambda_if\|^2+\sum_{i \in (I\setminus J)\setminus\sigma}\|\Gamma_if\|^2 =\sum_{i \in \sigma\cup J}\|\Lambda_if\|^2-\sum_{i \in J}\|\Lambda_if\|^2+\sum_{i \in (I\setminus J)\setminus\sigma}\|\Gamma_if\|^2 &=\sum_{i \in \sigma\cup J}\|\Lambda_if\|^2+\sum_{i \in (\sigma\cup J)^c}\|\Gamma_if\|^2-\sum_{i \in J}\|\Lambda_if\|^2 &\ge (A-D)\|f\|^2.
F_{4}={\frac{\ell_{*}}{g_{s}r_{*}}}{\frac{Z_{4 *}^{3 / 4}}{Z_{0 *}^{5 / 4}}}\left({\frac{1}{Z_{4}}}\right)_{*}^{\prime}d x^{+}\wedge d y^{1}\wedge d y^{2}\wedge d y^{3}.
\xi_m |_{\partial B_1(0)}= \ 0 = \xi_m |_{\partial B_2(0)}\frac{\partial \xi_m}{\partial r}|_{\partial B_1(0)}= \frac{1}{\sqrt{4 \pi^2 \log m}},\frac{\partial \xi_m}{\partial r}|_{\partial B_2(0)}= 0,
M_{2,\nu}:= \max\{\bar{\sigma}_{R}(t):t\in J_{\nu}\}\varepsilon_{2,\nu}:= \min\left\{\frac{1}{2nM_{2}},\frac{\bar{b}_{\nu}-b_{\nu}}{2},\frac{a_{\nu}-\bar{a}_{\nu}}{2},\frac{1}{2n^{2}}\right\}
y^\delta \tilde{\Gamma}^\alpha{}_{\beta \delta}\tilde{B}^\beta|_1=y^\delta (\Gamma^\alpha{}_{\beta \delta}+\Gamma^\alpha{}_{\beta\delta ,\sigma}x^\sigma /2)[B^\beta +(B^\beta{}_{,\sigma}+\Gamma^\beta{}_{\sigma \tau}B^\tau )d^\sigma /2]
I_a(u) :=\int_{M\setminus\{p\}}(\eta r^{2-n}+u)P_a(\eta r^{2-n}+u)\,dv^g, J_a(u) :=\int_{M\setminus\{p\}}\eta r^{2-n}F_{\eta}\,dv^g+2\int_{M}uF_{\eta}\,dv^g+\int_{M}uP_au\,dv^g,
R_{l i j}^{( + ) k}= \partial_{i}\Gamma_{j l}^{( + ) k}- \partial_{j}\Gamma_{i l}^{( + ) k}+ \Gamma_{i m}^{( + ) k}\Gamma_{j l}^{( + ) m}- \Gamma_{j m}^{( + ) k}\Gamma_{i l}^{( + ) m}, R_{i j}^{( + )}= R_{i k j}^{( + ) k}.
\left|S^{(2)}_{\mu,\xi}(x, t)\right|\lesssim \mu_0^{-\frac{n+2s}{2}}\frac{\mu_0^{n-2s+2}}{1+|y_j|^{4s-2}}\lesssim \mu_0^{2s-(a-2s)-\sigma}(t_0)\sum_{j=1}^k\frac{\mu_j^{-2s}\mu_0^{\frac{n-2s}{2}+\sigma}}{1+|y_j|^{a}}.
a \diamond (w_1 \otimes w_2 \otimes \cdots \otimes w_n \otimes w_{n+1}) = w_1 \otimes w_2 \otimes \cdots \otimes w_n \otimes a w_{n+1}, (w_1 \otimes w_2 \otimes \cdots \otimes w_n \otimes w_{n+1}) \diamond b = w_1 b \otimes w_2 \otimes \cdots \otimes w_n \otimes w_{n+1}
\left. \frac{\partial \mathrm{E}_{\alpha ,\beta}\left( z\right)}{\partial\beta}\right\vert_{\alpha =n}=-\frac{1}{n\,\Gamma \left( \beta \right)}\sum_{m=1}^{n}Q\left( \beta ,z^{1/n}e^{i2\pi m/n}\right) .
(\begin{matrix}\tau_{xx}&\tau_{xy}\\ \tau_{yx}&\tau_{yy}\end{matrix})=(\begin{matrix}x&0\\ 0&-t\end{matrix})\cdot(\begin{matrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{matrix}),
{\displaystyle\sum\limits_{\overline{k}}}{\displaystyle\sum\limits_{i=1}^{M}}\left\langle D_{x}^{\overline{k}}u\right\rangle_{x_{i},\overline{\Omega}_{+}}^{(d_{i})}\leq C\sum_{i=1}^{M}\left\langle u\right\rangle_{x_{i},\overline{\Omega}_{+}}^{(l_{i})}.
-\int_0^s \Theta( u ,x) f'(u) \mathrm{d}u+ f(s){\Theta}(s,x)-f(0){\Theta}(0,x) =-\eta d T\big( \int_0^{s}\int_0^1 f(u)A(x,y)\Theta(u,y)\mathrm{d}y\mathrm{d}u+o(1)\big)+\sigma^2 \eta^2 T \big( \int_0^s f(u)\mathrm{d}\xi_1(u,x)+o(1)\big).
\frac{1}{\#\mathcal{G}}\sum_{v\in G_{p,m}}|Y_{p^m,w}(v,x^m)|= \frac{1}{\#\mathcal{G}}\sum_{\substack{v\in G_{p,m}x^{m-w}\mid v}}|Y_{p^m,w}(v,x^m)|+ \frac{1}{\#\mathcal{G}}\sum_{\substack{v\in G_{p,m}x^{m-w}\nmid v}}|Y_{p^m,w}(v,x^m)|.
S =\left\{(X_i^\pm,\underline{X}_i^\pm): 0\leq i \leq 1\right\}\cup \left\{(X_i^+,\underline{X}_i^+) :\ 2\leq i\leq k \right\}\cup \left\{(X_i^+,\underline{X}_i^-):\ 2 \leq i \leq k \right\},
b\Theta\wedge\omega_2+a\Theta\wedge\omega_1-c\theta_{(2k+3)(2k+4)}\wedge\omega_2=0, &c\Theta\wedge\omega_1-(-b\omega_1+a\omega_2)\wedge\theta_{(2k+3)(2k+4)}=0, &c\Theta\wedge\omega_2+(a\omega_1+b\omega_2)\wedge\theta_{(2k+3)(2k+4)}=0.
\begin{aligned}\left(\frac{\boldsymbol{u}_h^1-\widetilde{\boldsymbol{u}}_h^1}{\Delta t}+\nabla (p_h^1-p_h^0), \nabla \boldsymbol{y}_h\right) =0 \forall \boldsymbol{y_h}\in \boldsymbol{Y}\!_h (\nabla \cdot \boldsymbol{u}_h^1, q_h) =0 \forall q_h\in Q_h, \boldsymbol{u}_h^1|_{\partial \Omega}=0 \end{aligned}
\begin{aligned}V_\omega^{(k)}:= \mathbb{C}_{z_1}\times \dots \times \mathbb{C}_{z_{k-1}}\times(\mathbb{C}_{z_k}\setminus \omega_k)\times \mathbb{C}_{z_{k+1}}\times \dots \times \mathbb{C}_n, V_\omega^{(\alpha)}:= V_\omega^{(i_1)}\cap \dots \cap V_\omega^{(i_\ell)}V_\omega^{(\emptyset)}:= \mathbb{C}^n.\end{aligned}
\eta_{i a}\eta_{j b}R^{a}{}_{m}{}^{b}{}_{n}= R^{a}{}_{i}{}^{b}{}_{j}\eta_{a m}\eta_{b n}, \quad \eta_{a i}R^{a}{}_{j}{}^{l}{}_{k}= \lambda^{- 2}R^{- 1}{}^{a}{}_{i}{}^{l}{}_{k}\eta_{j a}
-\{_rT_{ik}}\{_rT_{jk}}\ +\{_rT_{il}}\{_rT_{jk}}; 0\le i<j\le n_r,k=n_r+1,\l= n_r+2,{_rT_{ij}}\{_rT_{kl}}\ -\{_rT_{ik}}\{_rT_{jk}}\ +\{_rT_{il}}\{_rT_{jk}}; 0\le i<j< k<l\le n+2
\sum\limits_{m\geq 0}a(4m+3)f(4m+3) = \sum\limits_{m\geq 0}a(2m+1)f(2m+1)- \sum\limits_{m\geq 0}a(4m+1)f(4m+1) = \sum\limits_{m\geq 0}a(2m+1)f(2m+1)- \sum\limits_{m\geq 0}a(m)f(4m+1) =\sum\limits_{m\geq 0}a(m)f(m)-\sum\limits_{m\geq 0}a(2m)f(2m)- \sum\limits_{m\geq 0}a(m)f(4m+1) =\sum\limits_{m\geq 0}a(m)(f(m)-f(2m)-f(4m+1)).
\frac{a_{n}}{a_{n - 1}}= \frac{2 n^{2}- 5 n + 3}{2 n^{2}- n - 1}= \frac{( 2 n - 3 ) ( n - 1 )}{( 2 n + 1 ) ( n - 1 )}= \frac{2 n - 3}{2 n + 1}
\left\|B\vec{t}\right\|_{\dot a^0_{p,q}(W)}\lesssim\left\|\vec{t}\right\|_{\dot a^0_{p,q}(W)}\left\{\sum_{k\in\mathbb{Z}}\sum_{l=0}^\infty\left[2^{-(E-\frac{n}{2})k_-}2^{-k_+(F+\frac{n}{2}-\frac{n}{a})}2^{-(D-\frac{n}{a})l}\right]^r \right\}^{\frac{1}{r}}\sim\left\|\vec{t}\right\|_{\dot a^0_{p,q}(W)},
[ \mathsf{L}_{1}, \mathsf{u}_{2}] = - i \hbar \mathsf{\mathcal{C}}_{1 2}\mathsf{u}_{2}, \qquad [ \mathsf{R}_{1}, \mathsf{v}_{2}] = i \hbar \mathsf{v}_{2}\mathsf{\mathcal{C}}_{1 2}
\langle ({\cal O}_{j_{1}m_{1}}^{r_{1}~ +}{\cal O}_{j_{2}m_{2}}^{r_{2}~ +}){\cal O}_{j_{3}~{- m_{3}}}^{r_{3}~ -}\rangle = \langle{\cal O}_{j_{1}m_{1}}^{r_{1}~ +}({\cal O}_{j_{2}m_{2}}^{r_{2}~ +}{\cal O}_{j_{3}~{- m_{3}}}^{r_{3}~ -}) \rangle \nonumber
\sum_{a < p \leq ca}g(p) \leq B\left(\pi(ca) - \pi(a)\right) = B\left(\frac{ca}{\log a + \log c}- \frac{a}{\log a}+ O\left(\frac{a}{\log^2 a}\right)\right) &= B(c-1)\frac{a}{\log a}\left(1 + O_c\left(\frac{1}{\log a}\right)\right) = o_{B,c}(M_g(a)).
\omega'((f_1,y_1) , (f_2,y_2)) = \underbracket{\omega'(f_1,f_2)}_{=0}+ \underbracket{\omega'(y_1,f_2)}_{=0}+ \underbracket{\omega'(f_1,y_2)}_{=0}+ \omega'(y_1,y_2) .
\sum_{\ell=0}^{N-1}\big|\mathcal{E}_{N,\ell}^{h,k,2,2}\big|\le \frac{C(1+|x|_{L^p})^{K}}{\tau^{2\kappa}}\bigl(1+t_{N-1}^{\frac{1}{2}-\frac{1}{2q}-\kappa-\beta-\gamma}\bigr)\Bigl(|(-A)^{-\beta}h|_{L^{2q}}|(-A)^{-\gamma}k|_{L^{2q}}+\frac{\Delta t}{t_{N}}|h|_{L^{2q}}|k|_{L^{2q}}\Bigr).
F(\vert\mathring{A}\vert) \le F(\sqrt{12+24H^2}) &= -12H^2 (2H^2+1) + \vert H\vert\left.\left(\frac{\sqrt{3}}{3}x^3-\frac{1}{2}\vert H\vert x^2\right)\right\vert_{x=\sqrt{12+24H^2}}\le \vert H\vert f_1(\vert H\vert),
\mathrm{ord}(D_{S_1}) = \mathrm{ord}(D) -1 \le r_q(H^1_{f,pS^{\prime}})+r_p(A(S^{\prime}))-1 &\le r_q(H^1_{f,pS^{\prime}/\ell_1})+r_p(A(S^{\prime}/\ell_1)).
[\begin{matrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\end{matrix}]=\frac{1}{\sqrt{3}}[\begin{matrix}\xi\\ \xi\\ \xi\end{matrix}]+\frac{\rho}{\sqrt{2}}[\begin{matrix}cos\theta-\frac{sin\theta}{\sqrt{3}}\\ \frac{2sin\theta}{\sqrt{3}}\\ -\frac{sin\theta}{\sqrt{3}}-cos\theta\end{matrix}].
\Delta R = \left\{q_1q_1^* H(p_1^*)+q_2^* H(p_2^*)-q_1H(p_1)-q_2q_2^*H(p_2) \right\}&-\left\{[H(p_1^*)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_2)]\times q_2^*+[H(p_2)-H(p_1^*)]\times(q_1- q_1^*)\right\}=q_1^*q_2[H(p_2)-H(p_1^*)]
H_0(t_0) =F_2(t_0)-F_1(t_0), for\: t_0\in [a,b], H_{k}(t_k) =\int_a^{t_{k-1}}H_{k-1}(t_{k-1})dt_{k-1},for\: t_{k}\in [a,b], \k=1,2,\ldots ,n.
r_{u,u'}(s,\tau,s',\tau') & =\frac{-\sigma^2(|us-u's' + u\tau -u'\tau'|)+\sigma^2(|us-u's'+u\tau|)+\sigma^2(|us - u's'-u'\tau'|)-\sigma^2(|us-u's'|)}{2\sigma(u\tau)\sigma(u'\tau')}.
\nu_{\mathbf{f}}\left( \left \{x\in \mathbb{R}^{\mathbb{N}}:x_{k}\in \left[ -m,m\right] \right \}\right) =\nu \left( \left \{\omega\in \Omega :f_{1}\left( \omega \right) \in \left[ -m,m\right] \right \}\right) =1\forall k\in \mathbb{N}.
\mathbb{P}\left[||\gamma-\gamma^{\alpha}||_{\infty}< r_{\alpha}\right] \ge \mathbb{P}\left[||\hat{f}_{t}(iy)-\hat{f}^{\alpha}_{t}(iy)||_{\infty}\le R\right] - \mathbb{P}\left[ ||\gamma (t)-\hat{f}_{t}(iy)||_{\infty}\ge R\right].
H(r^{-1}) =\int_0^Me^{-y}y\mu(ry,\infty)\,dy+\int_M^\infty e^{-y}y\mu(ry,\infty)\,dy \leq 2e\int_0^Me^{-y}yH(r^{-1}y^{-1})\,dy+\mu(Mr,\infty)\int_M^\infty e^{-y}y\,dy \leq 2eC_U(2-\delta)^{-1}M^{2-\delta}H(r^{-1})+(1+M)e^{-M}\mu(Mr,\infty)\,.
P_{n}(\lambda)= \left(\lambda-\frac{1}{2}\right)^n\left[\left((\lambda-3)^2-4\right)\left(\lambda-\frac{n+4}{2}\right)\right]-2\left(\lambda-\frac{1}{2}\right)^n(\lambda-5) &-\frac{n}{4}\left(\lambda-\frac{1}{2}\right)^{n-1}\left[(\lambda-3)^2-4\right]
\frac{d}{dt}E_*(U_1;t) =2\|U_1'\|^2+2(U_1,-AU_1+Ae^{-tA}(u_0+u_1)) &=2\|U_1'\|^2-2\|A^{1/2}U_1\|^2+2(U_1,Ae^{-tA}(u_0+u_1)) &\leq 2\|U_1'\|^2-\|A^{1/2}U_1\|^2+\|A^{1/2}e^{-tA}(u_0+u_1)\|^2
\int_{\mathbb{R}^{2n}}e^{-\pi \langle x, \mathcal{Q}x \rangle}\prod_{j=1,2}f_j(x_j)^{c_j}\, dx \ge \big(\frac{\sqrt{1-e^{-2s}}}{(2\pi)^{\frac12(c_1+c_2)-1}}\big)^n\prod_{j=1,2}\big( \int_{\mathbb{R}^n}f_j\, dx_j \big)^{c_j}
{\textstyle \sum_{K_{32}\ni j=k}^{\nu}}\,\|x^{j+1}\!-\!x^j\|\le \widehat{c}_2(\theta){\textstyle\sum_{K_{32}\ni j=k}^{\nu}}\,\Xi_{j+1}\ \{\rm with}\ \ \widehat{c}_2(\theta):=bc(1\!-\!\theta)(4/a)^{\theta}.
|\phi(x)-\phi(y)| \le|\phi(x)-(\phi)_{x,c_3^L r}|+|(\phi)_{x,c_3^L r}-(\phi)_{y,c_3^L r}|+|\phi(y)-(\phi)_{y,c_3^L r}| &\le\left(2(c_3^L)^{2n+2}+2^{2n+3}c_3^L\right)\left(\frac{c_2^L}{c_1^L}\right)^2 C_1^L (C_2^L)^{2n+1}\theta r &=C(n,L)\theta\, d_\phi(x,y)
A =\begin{pmatrix}0 -\frac{e^u}{\sqrt{2}}i\frac{e^u}{\sqrt{2}}0 0 \frac{e^u}{\sqrt{2}}0 i u_z - a_1 a_2 -i \frac{e^u}{\sqrt{2}}-i u_z 0 -i b_1 i b_2 0 a_1 i b_1 0 \sigma 0 a_2 i b_2 \sigma 0 \end{pmatrix},
\int_{\mathcal{C}_{+}\cup \mathcal{C}_{-}}z \mathrm{Tr}\left({Y}^{-1}(z){Y}^{\prime}(z)\sigma_3 \right) dz = -\int_{\mathcal{C}_\epsilon+ \mathcal{C}_R}z \mathrm{Tr}\left({Y}^{-1}(z){Y}^{\prime}(z)\sigma_3 \right)dz.
2+ b \left(y^2-1\right)+b y \left(y^2-1\right) \log \left(\frac{y+1}{3-3 y}\right) \leq 2+ b \left(y^2-1\right)+\frac{b y \left(\frac{y+1}{3-3 y}-1\right) \left(y^2-1\right)}{\frac{1}{2}\left(\frac{y+1}{3-3 y}-1\right)+1}=2-\frac{b (1-y^2) (4 y^2-3y+2)}{2-y}.
\pi'(T_{w_1}T_{w_2}) = (-1)^{\ell(w_1w_2)}\chi(T_{w_1w_2})\pi(T_{w_1w_2}) & = (-1)^{\ell(w_1) + \ell(w_2)}\chi(T_{w_1}T_{w_2})\pi(T_{w_1}T_{w_2}) & = (-1)^{\ell(w_1)}(-1)^{\ell(w_2)}\chi(T_{w_1})\chi(T_{w_2})\pi(T_{w_1})\pi(T_{w_2}).
\langle T\{{\tilde\Psi}_{L_r}(x){\tilde{\overline\Psi}}_{L_r}(y)\}\rangle= -P_L (i{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax})^{\dagger}\frac{1}{m^2_r + (i{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax})(i{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax})^{\dagger}}\delta(x-y).
\check{\tilde{R}}_{1 2}( u , v ) \check{R}_{2 3}^{\iota_{1}\iota_{2}}( u ) \check{\bar{R}}_{1 2}^{\iota_{1}}( v ) = \check{\bar{R}}_{2 3}^{\iota_{1}\iota_{2}}( v ) \check{R}_{1 2}^{\iota_{1}}( u ) \check{R}_{2 3}( u , v ) ,
T_{s_{\alpha}}\varphi_{e,\chi}(\mathfrak{s}(\mathfrak{w}_{\alpha}))=\int_{N_{\alpha}}\varphi_{e,\chi}(\mathfrak{s}(\mathfrak{w}_{\alpha})^{-1}\mathfrak{s}(v)\mathfrak{s}(\mathfrak{w}_{\alpha}))\, dv.
\sum_k\sum_{p\in\mathcal{P}_{j,k}}\frac{1}{pz_u(p)}= \sum_{k>j/2}\sum_{p\in\mathcal{P}_{j,k}}\frac{1}{pz_u(p)}+\sum_{k\le j/2}\sum_{p\in\mathcal{P}_{j,k}}\frac{1}{pz_u(p)}\ll 2^{-j/2}=y^{-a_j/2}.
c_j:= -2(N-M)b_{j-1}^{(N-M-1)}(-\frac n2-M-1)+a_{j}^{(N-M)}(-\frac n2-M-1), d_j:= 2(N-M)b_j^{(N-M-1)}(-\frac n2-M-1)-a_{j}^{(N-M)}(-\frac n2-M-1),
\mathcal{L}_{^{m,2}_{inter}}(s) ={\rm exp}\left(-\lambda_{c}\int_{\mathbb{R}^2}\left(1 -\mathcal{E}_{y_{m,2}}\left\{\frac{1}{\frac{s}{{L\left(||y_{m,2}+x_m-x||\right)}}+1}\right\}\right) dx \right),
\epsilon^{\alpha \beta \gamma \delta}\,e_\alpha^a\,e_\beta^b\,e_\gamma^c\,e_\delta^d=\det \left| \left| e_\alpha^a\right| \right|\, \epsilon^{abcd}=(\pm )\sqrt{g}\, \epsilon^{abcd}\quad{\rm and}\quad\sqrt{g}\,\epsilon^{abcd}\,e_a^\alpha \,e_b^\beta \,e_c^\gamma \,e_d^\delta=(\pm ) \epsilon^{\alpha \beta \gamma \delta}.
A_0(z)=1, A_1(z)=1+z, A_2(z)=\displaystyle 1+z+\frac{z^2}{2},A_3(z)=\displaystyle 1+z+\frac{z^3}{6}, & A_4(z)=\displaystyle 1+z+\frac{z^2}{2}-\frac{z^3}{3}+\frac{z^4}{24}, A_5(z)=\displaystyle 1+z+\frac{5 z^3}{6}-\frac{5 z^4}{24}+\frac{z^5}{120},\ldots.
F^U = -\mathcal{E}_1(U+U_s)U_{s,x}-\mathcal{E}_2(U+U_s)U_{s,y}-\ell(U_s) &=\begin{pmatrix}uu_{s,x}+vu_{s,y}uv_{s,x}+vv_{s,y}u\phi_{s,x}+v\phi_{s,y}+\phi(u_{s,x}+v_{s,y})\end{pmatrix}.
\frac{c^{i/\alpha_1-i_{0}}}{\sqrt{p+1}}\leq \frac{c^{(i_3+1)/\alpha_1-i_{0}}}{\sqrt{p+1}}\leq\frac{c^{2/\alpha_1}(\eta c^{-i_{0}})^{-1}c^{-i_{0}}}{\sqrt{p+1}}=\frac{c^{2/\alpha_1}}{\eta\sqrt{p+1}}=r
N_0^{-s_0}N_1^{-s_1}N_2^{-s_2}N_3^{-s_3}& \lesssim\begin{cases}N_1^{-s_2^{\ast}-s_3^{\ast}}N_2^{-s_1^{\ast}}N_3^{-s_0^{\ast}}, N_0 \sim N_1 \gg N_2 \ge N_3, N_1^{-s_2^{\ast}-s_3^{\ast}}\max (N_0, N_3)^{-s_1^{\ast}}\min (N_0,N_3)^{-s_0^{\ast}}, N_1 \sim N_2 \gg N_0, N_3,\end{cases}
\hat{t}_{2}( p )_{\alpha \beta \mu \nu}= \frac{i}{2 \pi}\frac{\hat{P}^{\scriptscriptstyle ( 4 )}( p )_{\alpha \beta \mu \nu}}{p^{4}}\left\{p^{4}\log \left( \frac{- ( p^{2}+ i 0 )}{M^{2}}\right) + c_{0}+ c_{2}p^{2}+ c_{4}p^{4}\right\}.
- ( 3 - \nu^{2}) \dot{a}_{1 2}^{*}+ ( 2 - \nu^{2}) \dot{a}_{1 1}^{*}a_{2 2}^{*}+ : a_{1 1}a_{1 1}^{*}a_{1 1}^{*}a_{2 2}^{*}: + \nu \varphi_{1}^{'}a_{1 1}^{*}a_{2 2}^{*};
O=\left\{\frac{2^{o_{k}}-3^{k+1}-3^{k}2^{2b_{0}+2}-\sum\nolimits_{j=1}^{k}3^{k-j}2^{2b_{0}+\sum\nolimits_{i=1}^{j}2b_{i}3^{i}+\sum\nolimits_{i=1}^{j}\upsilon_{i}+2}}{3^{k+2}}\right\},
\allowdisplaybreaks\bigvee_{j = 1}^{i-1}(-u_i + x_j) \vee (-u_{i+1}+ 1 + x_i) \leq \bigvee_{j = i+1}^{n-2}(-u_j + x_j) \vee x_{n-1}\vee (-u_{n-1}) \enspace i = 1, \dots, n-2 \bigvee_{j = 1}^{n-1}x_j \leq 0 -\infty \leq x_1 \leq x_2 \leq \dots \leq x_{n-2}\leq u_{n-1}+ x_{n-1}\, .
\begin{array}{r c l}{{\partial_{i}\partial_{j}}}&{{=}}&{{q^{- 1}\partial_{j}\partial_{i}~ , ~ ~ \mathrm{f o r}~ i < j ~ ~ \mathrm{a n d}~ ~ i \not = - j ~ ,}}\\{{\partial_{- i}\partial_{i}}}&{{=}}&{{\partial_{i}\partial_{- i}+ \lambda q^{i - 2}\Delta_{i - 1}}}\\{{}}&{{=}}&{{q^{- 2}\partial_{i}\partial_{- i}+ \lambda q^{i - 2}\Delta_{i}~ , ~ ~ \mathrm{f o r}~ i > 0 ~ .}}\\ \end{array}
\begin{array}{r c l}{Y^{k}}&{=}&{\alpha^{- \frac{k}{2}}S ( A ) w^{- k}A w^{k}}\\{}&{=}&{\alpha^{- \frac{k}{2}}< ( w^{- k})_{( 1 )}, A > ( w^{- k})_{( 2 )}w^{k},}\\ \end{array}
\min_{\mathbf{y}\in Y}\Big(\frac{1}{K}\min_{v \in \{1, \ldots v\}}R_v + \varepsilon_K \Big) y_1 + hy_2 = \min\Big\{\frac{1}{K}\min_{v \in \{1, \ldots v\}}R_v + \varepsilon_K; h\Big\}.
q_{11}'(t,k,l,\xi,\eta)=\mathbf{1}_{l\neq 0}\mathbf{1}_{|l-k|=1}\frac{\eta(l-k)}{(\eta-tl)^2+l^2}\cdot\frac{\varepsilon_0\widehat{\varphi_b}(\xi-\eta)}{2},
L = \int d^4x\varepsilon^{\mu \nu \rho \sigma}\bigg(C_1{^+}{\cal G}^{A}_{\mu \nu}{^+}{\cal G}^{B}_{\rho \sigma}+ C_2{^-}{\cal G}^{A}_{\mu \nu}{^-}{\cal G}^{B}_{\rho \sigma}+ C_3{^+}{\cal F}^{A}_{\mu \nu}{^+}{\cal G}^{B}_{\rho \sigma}+ C_4{^-}{\cal F}^{A}_{\mu \nu}{^-}{\cal G}^{B}_{\rho \sigma}\bigg)M_{AB}
\log\mathfrak{M}(q\,|\,\tau,\lambda_1,\lambda_2) =q\int\limits_0^\infty \frac{dt}{t}B(t)+\frac{1}{2}q^2\sigma^2(\mu)+\int\limits_{\mathbb{R}\setminus\{0\}}\bigl(e^{uq}-1-uq\bigr)\,d\mathcal{M}_{(\tau, \lambda_1, \lambda_2)}(u).
\,|\xi(s)-\xi^+(s)| \leqslant \,\dfrac{1}{2}\Big(|y^--\xi^+(s)|+|\xi^+(s+T^--T^*)-\xi^+(-T^*)|\Big) \leqslant \,\dfrac{1}{2}\Big(|y^--\xi^+(-T^-)|+|\xi^+(-T^-)-\xi^+(s)|+C\,\dfrac{T^+-T^-}{2}\Big) \leqslant \,C|h|
x(y-x) f(s,t-s;y-x) = \frac{x(y-x)\,\lambda e^{-\lambda (t-s)}}{2c^2s\sqrt{c^2(t-s)^2-(y-x)^2}}\,I_1\Bigl( \frac{\lambda}{c}\sqrt{c^2(t-s)^2-(y-x)^2}\Bigr)
\hat\varphi^{(1)}(x)={\rm i}\, \int d^{6}y \,\,\Big< \hat\varphi(x)\,J(y)\, \hat\varphi(y) \Big> =\frac{\kappa_{orb}\,M}{2}\, \int\frac{d^{5}k}{(2\pi)^{5}}\frac{{\rm e}^{{\rm i}k\cdot x}}{k^2}~~.
S=S(d,q_1,Q_2,E,Y)=\sum_{\substack{q_2\sim Q_2 (q_2,10)=1}}\sum_{\substack{a< d q_1q_2 (a,d q_1q_2)=1}}\sum_{\substack{|\eta|\le E/Y (\eta+a/q_1q_2d)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{d q_1q_2}+\eta\Bigr).
\mathfrak{a}^{\dagger}\left[ \mathfrak{a}\left( \mathbb{K}_{n}^{(j)}(x)\right) \right] =\mathfrak{a}^{\dagger}\left[ \Lambda_{n}^{\left( j\right)}\left( x;1,1\right) \mathbb{K}_{n-1}^{(j)}(x)\right] .
Q_{j}\phi{}_{j,k}=Q_{j}\phi{}_{j,\infty}+Q_{j}(\phi{}_{j,k}-\phi{}_{j,\infty})=\varsigma_{j-1}\phi{}_{j-1,\infty}+\varsigma_{j-1}(\phi{}_{j-1,k}-\phi{}_{j-1,\infty})=\varsigma_{j-1}\phi{}_{j-1,k},
\begin{aligned}\mathop{\max}\limits_{\left\{{{x_k}}\right\}_{k = 1}^{{N_{\rm{R}}}}}\sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({{{1 +{{\bar C}_k}\left({1 -{2^{- 1 -{x_k}}}}\right)}}}\right)}{\rm{s.t.}}\quad{x_k}\in{{\mathbb Z}^+}&\sum\limits_{k = 1}^{{N_{\rm{R}}}}{{x_k}}={B}-{B_{{\rm{min}}}}.\end{aligned}
\frac{1}{\tau}\big\|(e_n-e_{n-1})_{n=k}^N\big\|_{L^p(L^q(\varOmega))}+ \big\|(e_n )_{n=k}^N\big\|_{L^p(W^{2,q}(\varOmega))}\le C\tau^k , & \max_{k\le n\le N}\|e_n \|_{W^{1,\infty}(\varOmega)}\le C\tau^k ,
\begin{array}{l}\Delta(H_i)=H_i\otimes 1+1\otimes H_i~, \Delta(X_i^-)=X_i^-\otimes q_i^{-H_i}+q_i^{H_i}\otimes X_i^-~, \epsilon(X_i^-)=0=\epsilon(H_i)~, S(X_i^-)=-q_iX_i^-~,~~~~S(H_i)=-H_i~,\end{array}
\mathbf{a}(\theta_{0})^{T}\mathbf{v}^{0}=e^{-jA}+e^{-j2A}e^{-j\phi}+...+e^{-jNA}e^{-j(N-1)\phi}&=e^{-jA}+e^{-j(2A+\phi)}+...+e^{-j(NA+(N-1)\phi)}&=\frac{e^{-jA}(1-(e^{-jA}e^{-j\phi})^N)}{1-(e^{-jA}e^{-j\phi})}
\Gamma_{\lambda}= \begin{pmatrix}1 0 0 0 \cosh \lambda L_{2}\sinh \lambda L_{2}0 \sinh \lambda L_{2}\cosh \lambda L_{2}\end{pmatrix}\cdot \begin{pmatrix}\cosh \lambda L_{1}0 \sinh \lambda L_{1}0 1 0 \sinh \lambda L_{1}0 \cosh \lambda L_{1}\end{pmatrix}.
\left( \begin{matrix}{- \psi_{2}}\\{\psi_{1}}\\ \end{matrix}\right) = \left( \begin{matrix}{\frac{\partial \psi_{1}}{\partial t}- \frac{\partial \psi_{1}}{\partial x}}\\{\frac{\partial \psi_{2}}{\partial t}+ \frac{\partial \psi_{2}}{\partial x}}\\ \end{matrix}\right) \ .
Var_{P_n}(t_j(X_i,\theta^\prime_n)) =\int t_j(x,\theta^\prime_n)^2dP_n(x)-E_{P_n}[t_j(X_i,\theta^\prime_n)]^2 &\le M\int t_j(x,\theta^\prime_n)dP_n(x) -E_{P_n}[t_j(X_i,\theta^\prime_n)]^2\to 0.
D_i(G_k)/D_{i+1}(G_k)\cong\begin{cases}C_2 \times C_2 C_2 \times C_2 \times C_2 \text{if $i=2$, equivalently if $l = 1$,}C_2 \times C_2 \times C_2 \text{if $i = 2^l$ with $2 \le l \le k$.}\end{cases}
f^{( q )}( u ) = \sum_{n = 0}^{\infty}f^{\alpha_{1}^{\prime}\ldots \alpha_{2 n + q}^{\prime}}u_{( \alpha_{1}^{\prime}}^{+}\ldots u_{\alpha_{n + q}^{\prime}}^{+}u_{\alpha_{n + q + 1}^{\prime}}^{-}\ldots u_{\alpha_{2 n + q}^{\prime})}^{-}\ .
\begin{array}{l}{\langle : \Phi \Phi^{*}: ( x ) : \Phi \Phi^{*}: ( y ) \rangle}\\{= \int d \theta d r r r^{- d - ( 1 - \gamma ) ( D - d )}F ( \theta , r^{- \frac{1}{2}}({\bf x}_{F}-{\bf y}_{F}) , r^{- \frac{1}{2}+ \frac{\gamma}{2}}({\bf x}_{G}-{\bf y}_{G}) )}\\ \end{array}
II=\sum_{A\in\mathcal{P}}\alpha'(A)\log\frac{\beta'(A)}{\beta(A)}\le\sum_{A\in\mathcal{P}}\alpha'(A)\log\left(\max_{A\in\mathcal{P}}\frac{\beta'(A)}{\beta(A)}\right)=\log\left(\max_{A\in\mathcal{P}}\frac{\beta'(A)}{\beta(A)}\right).
\|fi(fi(M_1)x_{k_{in}}) - M_1x_{k_{in}}\| = \|fi(fi(M_1)x_{k_{in}}) - fi(M_1)x_{k_{in}}+ fi(M_1)x_{k_{in}}- M_1x_{k_{in}}\| \leq \|fi(fi(M_1)x_{k_{in}}) - fi(M_1)x_{k_{in}}\| + \|(fi(M_1) - M_1)x_{k_{in}}\|
h^{(1)}:=D^*_{S_0^c}h\cdot I_{\{i:|D^*_{S_0^c}h(i)|>\alpha/(t-1)\}}, h^{(2)}:=D^*_{S_0^c}h\cdot I_{\{i:|D^*_{S_0^c}h(i)|\leq\alpha/(t-1)\}}.
\Psi ( t ) = \sum_{\vec{n}}a_{\vec{n}}\phi_{\vec{n}}e^{- i{\cal E}_{\vec{n}}t}= e^{- i{\cal E}_{0}t}\sum_{\vec{n}}a_{\vec{n}}\phi_{\vec{n}}e^{- i \omega ( \sum_{j = 1}^{r}n_{j}f_{j}) t}.
\begin{aligned}u^{1,\dots,k}=\left(u_1(X_1,\dots,X_k),\dots,u_k(X_1,\dots,X_k),0\right),\ u^{2,\dots,k+1}=\left(0,u_1(X_2,\dots,X_{k+1}),\dots,u_k(X_2,\dots,X_{k+1})\right).\end{aligned}
\lim_{x\rightarrow q^{n}}(x-q^{n})\Big ( -\sum_{\substack{s+t=n s,t\ge 1}}\frac{q^{s(n-s)}y^{-s}z^{-(n-s)}}{1-xq^{-n}}\Big ) =\lim_{x\rightarrow q^{n}}(x-q^{n})\sum_{\substack{s+t=n s,t\ge 1}}\frac{q^{s(n-s)+n}y^{-s}z^{-(n-s)}}{x-q^{n}}&=\sum_{\substack{s+t=n s,t\ge 1}}\frac{q^{s(n-s)+n}}{y^{s}z^{n-s}},
\frac{z\mathbf{\Phi}_{{\nu}}^{\prime}(z)}{\mathbf{\Phi}_{{\nu}}(z)}=2{\nu +1}-\sum_{n\geq1}\frac{4z^{4}}{\gamma_{{\nu},n}^{4}-z^{4}},1+\frac{z\mathbf{\Phi}_{{\nu}}^{\prime \prime}(z)}{\mathbf{\Phi}_{{\nu}}^{\prime}(z)}=2{\nu +1}-\sum_{n\geq1}\frac{4z^{4}}{\gamma_{{\nu},n}^{\prime 4}-z^{4}},
\mathrm{sgn}_X(gh) = \prod_{\{u,v\}\in E(X)}\frac{u^{gh}-v^{gh}}{u - v}&= \prod_{\{u,v\}\in E(X)}\frac{u^g-v^g}{u - v}\cdot \frac{(u^g)^h - (v^g)^h}{u^g-v^g}&= \prod_{\{u,v\}\in E(X)}\frac{u^g-v^g}{u - v}\cdot \prod_{\{u,v\}\in E(X)}\frac{(u^g)^h - (v^g)^h}{u^g-v^g}&= \mathrm{sgn}_X(g) \ \mathrm{sgn}_X(h),
- e \Gamma_{\alpha \beta}( p + k )_{\beta}-{\frac{i e \lambda}{6}}\gamma_{5 , \alpha \beta}q_{\beta}+{\frac{e \kappa}{6 M^{2}}}\gamma_{6 , \alpha \beta , \mu \nu}q_{\beta}q_{\mu}( p + k )_{\nu}\quad ,
|\mathcal{F}(I-S_{L}(T))^{-1}(k)|\leq\frac{\sqrt{2}}{\varepsilon\left|\sum_{m=1}^{M}\alpha_{m}k^{r_{m}}\right|}=\left(\frac{\sqrt{2}}{c_{0}}\right)\frac{|k|^{p}}{\left|\sum_{m=1}^{M}\alpha_{m}k^{r_{m}-r_{1}}\right|}.
\frac{t}{t_1t_2}z(w+1)H_{a_1+a_2,b_1,b_2}(t,s) &= \frac{H_{a_1,b_1}(t_1,s_1)H_{a_2,b_2}(t_2,s_2)}{1 - \frac{t_1t_2}{z^2}\cdot \frac{1}{w(w+1)}\left( H_{a_1,b_1}(t_1,s_1) - \frac{z(w+1)}{t_1}\right)\left( H_{a_2,b_2}(t_2,s_2) - \frac{z(w+1)}{2}\right)}\,.
\left[\begin{array}{cc}p_{n-1}p_{n}q_{n-1}q_{n}\end{array}\right]=\left[\begin{array}{cc}p_{n-m-2}p_{n-m-1}q_{n-m-2}q_{n-m-1}\end{array}\right]\left[\begin{array}{cc}B^{(m)}_{n-m-1}B^{(m+1)}_{n-m-1}A^{(m)}_{n-m-1}A^{(m+1)}_{n-m-1}\end{array}\right]
L(\tau, I, \phi, s) =- \int_{-\infty}^{\infty}\big[ H_1(p^0( \tau+t\bar 1),q^0( \tau+ t\bar 1),I, \phi + \omega(I)t, s+t ;0 ) & \qquad-H_1(0,0,I, \phi + \omega(I)t, s+t ;0) \big] \, dt .
(1-\dfrac{\Delta t}{2}\{(1-r_1)\delta_{\alpha_1 x}^{V^-}+r_1\delta_{\alpha_1 x}^{V^+}\}) V_{i,j}^{*}=(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{V^-}+r_2\delta_{\alpha_2 y}^{I^+}\})V_{i,j}^{n}+\dfrac{\Delta t}{2}\left \{\beta S_{i,j}^{n+1/2}V_{i,j}^{n+1/2}-(\mu+\gamma) I_{i,j}^{n+1/2}\right\}
Q_{i}^{b}\frac{\partial Q_{j}^{a}}{\partial x^{b}}\:{-}\: Q_{j}^{b}\frac{\partial Q_{i}^{a}}{\partial x^{b}}- g^{lm}Q_{mij}Q_{l}^{a}= 0, g^{ij}Q_{j}^{a}Q_{i}^{b}=0.
\begin{aligned}|\kappa_n|^2= |\kappa_{-n}|^2+ 2\operatorname{Re}\bigl(\theta\overline\kappa_{-n}\overline{\widehat\mu(-n)}(1-\overline\theta)\bigr) + |1-\theta|^2|\widehat\mu(-n)|^2 &\ \ \mathbb T \m), \ \ \n\in\mathbb N. \end{aligned}
E_{\lambda_j}^{-1}(\lambda_j)\partial_{s_m}E_{\lambda_{j}}(\lambda_j)=\begin{pmatrix}\partial_{s_m}\log\Lambda_j-\frac{i}{2}|\lambda_j|^{-1/2}\partial_{s_m}d_1 \frac{1}{2}|\lambda_j|^{-1/2}\Lambda_j^{-2}\partial_{s_m}d_1 \frac{1}{2}|\lambda_j|^{-1/2}\Lambda_j^{2}\partial_{s_m}d_1 -\partial_{s_m}\log\Lambda_j+\frac{i}{2}|\lambda_j|^{-1/2}\partial_{s_m}d_1\end{pmatrix}.
a_{1}= k [ \tilde{H}]_{1}^{n - 2}( B_{0}^{2}A_{1}^{0}+ B_{1}^{1}A_{0}^{1}) + \frac{1}{\alpha}( - )^{n - 3}k{\overline{A}}_{1}^{*}B_{1}^{1}+ m_{1}A_{1}^{0},
\sup_{1\leq r\leq r_i}w_{\epsilon,\gamma}|u_i| \leq \max\left[ \sup_{r= 1}w_{\epsilon,\gamma}|u_i|, \ \frac1{4\delta(1-\delta)n}\sup_{1\leq r\leq r_i}w_{\epsilon,\gamma+2}|(\Delta-X)u_i| \right].
L_{12}\leq C\Big(\int_{1}^{\infty}\Big(\frac{\sqrt{t}}{\rho(x)}\Big)^{2(M-n)}\Big(1+\frac{\sqrt{t}}{\rho(x)}\Big)^{-2N}\Big)^{1/2}M_{V,\eta}f(x) &\leq CM_{V,\eta}f(x).