harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}|\ss_2(G_1)\cap \left(\ss_2(c),\ss_2(d)\right)| =|\ss_2\left(G_1\cap (c,d)\right)|=d\ss\left(G_1\cap (c,d)\right) &<d\ss(G_2\cap (c,d))=d\ss_2((c,d)) &=|\left(\ss_2(c),\ss_2(d)\right)|.\end{aligned}
= \frac{px}{(1-p^2x)(1-px)}\bigg(\frac{px^2}{1-px^2}- \frac{p^2x^3}{1-p^2x^3}\bigg) - \frac{px}{(1-p^2x)(1-p^{-1})}\bigg( \frac{p^3x^3}{1-p^3x^3}- \frac{p^2x^3}{1-p^2x^3}\bigg)
\begin{aligned}\mathbb{P}(\frac{1}{n}\kappa(Y_{n}) \in O_{2}) \geq \mathbb{P}(\kappa(Y_{k_{n}n_{0}+(n-k_{n}n_{0})}) \in k_{n}n_{0}O_{1}+ M_{n_{0}}) \geq \mathbb{P}(\kappa(Y_{k_{n}n_{0}}) \in k_{n}n_{0}O_{1}). \mathbb{P}(Y_{n-k_{n}n_{0}}\in L_{n_{0}}) \geq e^{-n_{0}k_{n}\alpha}\frac{1}{2}\end{aligned}
v_{ea}^1(f,g_{avg}^*) =\lim_{\beta\uparrow 1}(1-\beta) v_\beta^1(f,g_\beta^*) &\leq \lim_{\beta\uparrow 1}(1-\beta) v_\beta^1(f_\beta^*,g_\beta^*) &=v_{ea}^1(f^*_{avg},g^*_{avg}).
u^{1}_{s}+ 2j \cdot s^{2j-1}w^{1}_{s}= 0, u^{1}_{t}+ v^{1}_{s}+ 2j \cdot s^{2j-1}w^{1}_{t}+ 2k \cdot t^{2k-1}w^{1}_{s}= 0, v^{1}_{t}+ 2k \cdot t^{2k-1}w^{1}_{t}= 0,
\delta \psi_{\mu}= \partial_{\mu}\epsilon -{\frac{1}{4}}\omega_{\mu}^{a b}\gamma_{a b}\epsilon +{\frac{( - 1 )^{p}}{8 ( p + 2 ) !}}e^{\phi}F_{\mu_{1}. . . \mu_{p + 2}}\gamma^{\mu_{1}. . . \mu_{p + 2}}\gamma_{\mu}\epsilon_{( p )}^{'}
P_{\mathrm{r e d}}( i \rightarrow f ) = \sum_{m = 0}^{+ \infty}\sum_{\ell = - \infty}^{+ \infty}< r_{f}, \theta_{f}| m , \ell > \, e^{- \frac{i}{\hbar}( t_{f}- t_{i}) E_{m , \ell}}\, < m , \ell | r_{i}, \theta_{i}> \ \ \ .
e^{-\frac{|O(\alpha t)x-y|^2}{4t}}\,=\, e^{-\frac{|x|^2}{4t}}+ \frac{\langle O (\alpha t) x, y\rangle}{2 t}e^{-\frac{|x|^2}{4t}}+ \frac{\langle y, Q y\rangle}{8 t^2}e^{-\frac{|O(\alpha t)x-\theta y|^2}{4t}}\,,
\bar J_K^{2,\pm}u(x) := \{(p,X)\in \mathbb{R}^n \times \mathbb{S}^n| \ \exists x_k\in K \{\rm and}\ (p_k, X_k)\in J_K^{2,\pm}u(x) \{\rm such \that}(x_k, u(x_k), p_k, X_k)\rightarrow (x, u(x), p, X) \{\rm as}\k\rightarrow +\infty \},
\eta_t \leq c\log\Big(1+\frac{1}{m^2t}\Big){\bf 1}_{d=2}+ c m\Big(\sqrt{1 + \frac{1}{tm^2}}- 1\Big){\bf 1}_{d=3}, &\gamma_t \leq \frac{c}{m^2(m^2t+1)}{\bf 1}_{d=2}+ c\log\Big(1+\frac{1}{m^2t}\Big){\bf 1}_{d=3}.
\int_{0}^{\infty}\frac{\cosh t \sinh^{4}\frac{\alpha}{2}t}{\sinh^{3}t}d t = \frac{\alpha^{2}}{8}\left( \psi \left( \frac{\alpha}{2}\right) + \psi \left( - \frac{\alpha}{2}\right) - \psi \left( \alpha \right) - \psi \left( - \alpha \right) \right) .
z_i = \begin{cases}q^d (q^i-1)(1 - x_1^{-2}q^{i-2}) , - q^d (q^i+1) (1 + x_1^{-2}q^{i-2}) + \tau x_1^{-1}q^{i-1}(q^{d+1}+1) . \end{cases}
\widetilde{B(u,u)}=\begin{pmatrix}G-\frac{P}{G}-\frac{E-EP/G^2}{(e-E^2/G^2)^{1/2}}-|\dot{u}| -\left(P-P^2/G^2-\frac{(E-EP/G^2)^2}{e-E^2/G^2}\right)^{1/2}\end{pmatrix}.
\tilde{t}_{1}= \tilde{\gamma}+ \tilde M_{1}, \tilde{t}_{2}= \tilde\gamma (\tilde\gamma -\tfrac{1}{3}) + \tilde M_{1}(\tilde M_{1}+\tilde\gamma-\tfrac{2}{3}) - \tilde{M}_{2},\tilde{t}_{3}= 2 \tilde{\gamma}- \tfrac{1}{3}+ \tilde M_{1},
\frac{\eta}{2}> - b (x) \cdot D\varphi (x) + f (x) > - b (x) \cdot D\psi (x) + g (x) - \frac{\eta}{4}= - b (x) \cdot D\varphi (x) + \tilde G(x) - \tilde G(0) -\frac{\eta}{4}> - b (x) \cdot D\varphi (x) + G(x, 0) - G(0, 0) - \frac{\eta}{2}.
\dot{Y}(t)=\frac{i}{\hbar}[H_{sys},Y(t)]-\frac{i}{2\hbar}[X,\{Y(t),\xi(t)-\int_{t_{0}}^{t}f(t-t_{0})\dot{X}(t^{\prime})dt^{\prime}-f(t-t_{0})X(t_{0})\}],
\int_{\mathcal A_1^-(n)}\mathbf h_p(h) dh = \sum_{j\geq 0}\int_{\mathcal L_{2j+1}}\mathbf h_p(h) dh = -p \sum_{j\geq 0}\mathrm{vol}(\mathcal L_{2j+1}) = -p \sum_{j\geq 0}p^{-2j-1}(1-p^{-1})^2 = = -(1-p^{-1})^2 \sum_{j\geq 0}p^{-2j}= \frac{-(1-p^{-1})^2}{1-p^{-2}}= \frac{-(1-p^{-1})}{1+p^{-1}}= \frac{1-p}{p+1}.
\begin{aligned}\iota'\bigl(f\bigr)(s) = \operatorname{Ad}(u(s)) \Bigl( \operatorname{diag}\bigl[\overbrace{c(f(0)), \dots, c(f(0))}^a, f(t_1(s)), \dots, f(t_k(s)), \underbrace{c(f(1)), \dots, c(f(1))}_b \bigr] \Bigr) \end{aligned}
\delta g_{\mu \nu}( x , \tau , \sigma , f ) = ( F \xi )_{\mu \nu}( x ) + 2 [ f^{\star}\delta \sigma \bar{g}_{\mu \nu}] ( x ) + [ f^{\star}\frac{\partial \bar{g}_{\mu \nu}}{\partial \tau_{i}}\delta \tau_{i}] ( x ) \; .
\mathbf{Var}_{i\sim \mathbf{u}}[p_{i}^{t+1}] =\frac{1}{q}\sum_{i=1}^{q}\left(p_{i}^{t+1}-\frac{1}{q}\sum_{s=1}^{q}p_{s}^{t+1}\right)^{2}=\frac{1}{q}\sum_{i=1}^{q}\left(p_{i}^{t+1}-\frac{1}{q}\right)^{2}\geq\frac{1}{q}\left(p_{i^{t}}^{t+1}-\frac{1}{q}\right)^{2}=\frac{1}{q^{3}}.
{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y) =\int_{0}^{1}t^{x-1}(1-t)^{y-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right) dt &=2\int_{0}^{\frac{\pi}{2}}\cos^{2x-1}(\theta) \sin^{2y-1}(\theta){}_{0}\Psi_{1}\left( \alpha,\beta;-p \sec^{2}(\theta) \csc^{2}(\theta)\right)d\theta.
\langle T^\mu_\mu \rangle_R - (T_\mu^\mu)_0 = 2\pi m^2\int_{-\infty}^{+\infty}\frac{d\theta}{2\pi}\left[\frac{e^{-m R \cosh\theta}}{1+ e^{-mR \cosh\theta}}+2 \frac{e^{-2 m R \cosh\theta}}{(1+e^{-m R \cosh\theta})^2}+\cdots \right]\,\,\,.
ge_{\overline{\kappa(i)}}=e_{\overline{\kappa(i)}}-\mu \varphi^*(e_{\overline{\kappa(i)}})-\frac{\mu^2}{2}\varphi\varphi^*(e_{\overline{\kappa(i)}})=e_{\overline{\kappa(i)}}-\mu f_k-\frac{\mu^2}{2}\varphi(f_k) & =e_{\overline{\kappa(i)}}-\mu f_k-\frac{\mu^2}{4}\varepsilon\delta_{k,c}e_i.
\Gamma_{reg}[V]=-i\sum_i c_i\sum_{n=2}^{\infty}\frac{1}{n}\int\,dz_1\dots dz_n v(z_1)\star \bar{D}^2 D^2 G(z_1,z_2) \star\dots\star v(z_n)\star \bar{D}^2 D^2 G(z_n,z_1),
\mathbf{e}^{j}(l,l)=\frac{(-1)^{l}}{\mu^{n}_{l}}(J_{+})^{l},\ \mathbf{e}^{j}(l,m-1)=\frac{1}{\beta_{l,m}}\left[ J_{-},\mathbf{e}^{j}(l,m)\right] ,
\begin{array}{lll}\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\exp (t+s)X \exp (tY) \exp(tZ)}\ & =\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\exp(sX)\alpha(t)}=\displaystyle{\left.\frac{d}{ds}\right|_{s=0}R_{\alpha(t)}(\exp (sX))}=(dR_{\alpha(t)})_eX=X^R_{\alpha(t)},\end{array}
\left( \begin{array}{l}{- \psi_{2}}{\psi_{1}}\end{array}\right) = \left( \begin{array}{l}{\, \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{array}\right) \ .
\frac{j-1}{n-j}\sum_{i=j+1}^{n}\frac{1}{i-2}-\frac{j}{n-j-1}\sum_{i=j+2}^{n}\frac{1}{i-2}=\frac{1}{n-j}-\frac{n-1}{(n-j)(n-j-1)}\sum_{i=j+2}^{n}\frac{1}{i-2}&=\frac{n-1}{n-j}\left(\frac{1}{n-1}-\frac{1}{n-j-1}\sum_{i=j+2}^{n}\frac{1}{i-2}\right)<0,
{\mathcal I}_*(t)= -\frac{t^3}{\pi}\int_{0}^{\infty}\zeta^{-1/2}\left( \Xi (t,\zeta) N_* + N_* \Xi (t,\zeta) - \zeta \,\Xi (t,\zeta) N_* \Xi (t,\zeta)\right) \, d\zeta.
A(x,v) := A_+(x)\chi_{\{v>0\}}+A_-(x) \chi_{\{v<0\}}, f(x,v) := f_+(x)\chi_{\{v>0\}}+f_-(x) \chi_{\{v<0\}}, \lambda(v) := \lambda_+ \chi_{\{v>0\}}+ \lambda_- \chi_{\{v\le 0\}}.
\begin{aligned}J^4(v_\lambda,v_\mu^{x_0}) = \g_{[<\lambda]}J^4_{main}(v_\lambda,v_\mu^{x_0})+ (g_{[<\mu]}^{x_0}- g_{[<\lambda]}) \left(M(v_\mu^{x_0}) E(v_\lambda)- P(v_\lambda) P(v_\mu^{x_0})\right) : = \J^{4,a}(v_\lambda,v_\mu^{x_0})+J^{4,b}(v_\lambda,v_\mu^{x_0}),\end{aligned}
M\models \forall \; \overline{z}\; \bigvee_{i=1}^{K}\exists \; \overline{b}\; \exists \; c \; (\theta^{\prime}_{i}(c, \overline{b}) \; \& \; (\varphi_{0}(\overline{z}, \theta_{i}(\cdot, \overline{b}), c, \rho(\cdot, \overline{a})) \vee \neg \psi_{0}(\overline{z},\theta_{i}(\cdot, \overline{b}), c, \rho(\cdot, \overline{a}))))
(L_u+ (\lambda +1)Id)^{-1}- S^*(L_u+\lambda Id)^{-1}S & = - \frac{\langle \, \cdot \, | S^*(L_u+\lambda Id )^{-1}1\rangle}{\langle (L_u+\lambda Id)^{-1}1 | 1 \rangle}S^*(L_u+\lambda Id)^{-1}1.
L_{\varphi}=e^{\frac{1}{2}(3\varphi+\chi)}(-\frac{1}{2}e^{-\varphi}\partial_{\alpha}\varphi\partial_{\alpha}\varphi-e^{-\varphi}\partial_{\alpha}\varphi\partial_{\chi}\varphi+\frac{3}{2}e^{-\chi}\partial_{0}\varphi\partial_{0}\varphi)
Z_\alpha^+(x, y) = \frac{\chi_{(\alpha+1, \infty)}(x)\chi_{(-\alpha,\alpha)}(y)}{2\pi(x-y)},\textup{and}\ \Z_\alpha^-(x, y)= \frac{\chi_{(-\infty, -\alpha-1)}(x)\chi_{(-\alpha,\alpha)}(y)}{2\pi(x-y)}
\int_{E}\int_{E}g(x,y)~dxdy & =\frac{1}{2}\int_{\mathbb{S}^{n-1}}\int_{\Pi^{\xi}}\int_{E_{z}^{\xi}}\int_{E_{z}^{\xi}}g(z+s\xi,z+t\xi)|t-s|^{n-1}dsdtd\mathcal{H}^{n-1}(z)d\mathcal{H}^{n-1}(\xi)~.
\mathcal{I}(\phi)= \frac{1}{8h}\epsilon_1(\phi_2-c^a-c^a_1\phi_1)^2+\frac{1}{8h}\sum_{j=2}^{N-1}\epsilon_j(\phi_{j+1}-\phi_{j-1})^2+\frac{1}{8h}\epsilon_N(c^b+c^b_N\phi_N-\phi_{N-1})^2 +\frac{1}{8\beta_a}(c^a+(1+c^a_1)\phi_1)^2+\frac{1}{8\beta_b}(c^b+(1+c^b_1)\phi_N)^2.
X \to q^{m{\rho}^{\vee}}X{\bar{q}}^{m{\rho}^{\vee}}\, , \, \, \,{\rho}^{\vee}= \sum_{i}{\Lambda}_{i}^{\vee}\, \, \, ( \, \Rightarrow \, \, E_{i}\to q^{m}E_{i}\, , \, \, F_{i}\to{\bar{q}}^{m}F_{i}\, ) \, .
T_{2}\left(W',Z'\right)=\left(\frac{1}{I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)}\otimes W',V\otimes\left( Z'-A\otimes W \right)\right),\quad\mbox{for a suitable matrix $V=V\left(W',Z'\right)\in\mathcal{M}_{q'N'\times q'N'},$}
\langle U , U^* \rangle_{\mathcal{H}_\kappa}= \big\langle(u,v,\tau,\theta)^T , (u^*,v^*,\tau^*,\theta^*)^T\big\rangle_{\mathcal{H}}+ \int_{\Omega}\kappa m_{ij}q_i^* q_j \, \mathrm{d}x
R=LD=[\begin{matrix}1&0&\cdot\cdot\cdot&0\\r_{12}/r_{11}&1&\cdot\cdot\cdot&0\\ \vdots&\vdots&&\vdots\\r_{1n}/r_{11}&r_{2n}/r_{22}&\cdot\cdot\cdot&1\end{matrix}][\begin{matrix}r_{11}&0&\cdot\cdot\cdot&0\\ 0&r_{22}&\cdot\cdot\cdot&0\\ \vdots&\vdots&&\vdots\\ 0&0&\cdot\cdot\cdot&r_{nn}\end{matrix}].
\begin{cases}{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{2\alpha}u_{2\alpha}(x,t)= H^{2\alpha}\frac{\partial^2}{\partial x^2}u_{2\alpha}(x,t),\alpha \in (\frac{1}{2},1), u_{2\alpha}(x,0)=\delta(x), \frac{\partial}{\partial t}u_{2\alpha}(x,t)|_{t=0}= 0.\end{cases}
\widehat Lu = \sum_{i=1}^n \left(x_i\alpha_{ii}(z)u_{x_ix_i}+ b_i(z)u_{x_i}\right) + \sum_{i,j=1}^n x_ix_j \tilde \alpha_{ij}(z)u_{x_ix_j}&\quad+\sum_{i=1}^n\sum_{l=1}^m x_i c_{il}(z)u_{x_iy_l}+\sum_{k,l=1}^m a_{n+k,n+l}(z)u_{y_ky_l}+ \sum_{l=1}^m e_l(z)u_{y_l},
\delta \left[ \epsilon_{A B C}\epsilon^{\alpha \beta \gamma \delta}\partial_{A}v_{\alpha}\partial_{B}v_{\beta}\partial_{C}v_{\gamma}v_{\delta}\right] = 3 \epsilon_{A B C}\epsilon^{\alpha \beta \gamma \delta}\partial_{A}\delta v_{\alpha}\partial_{B}v_{\beta}\partial_{C}v_{\gamma}v_{\delta}
\zeta_{\bar{\cal V}}( s ) = \frac{m L}{\sqrt{8 \pi}}\frac{1}{\Gamma ( s )}\int_{0}^{\infty}d \beta \, \beta^{s - \frac{3}{2}}e^{- 4 \beta}= \frac{m L}{\sqrt{8 \pi}}\frac{1}{4^{s - \frac{1}{2}}}\frac{\Gamma ( s - \frac{1}{2})}{\Gamma ( s )}
I_p(x):=\left( \frac{x}{2}\right)^p \sum_{k=0}^{\infty}\frac{(x/2)^{2k}}{k! \Gamma(k+p+1)}=\frac{(x/2)^p}{\Gamma(1+p)}{}_0F_1\left(-;1+p;\frac{x^2}{4}\right), (-p \notin \mathbb{N}),
d(\mathcal{C}_{L_1},\mathcal{C}_{L_2}):=\max\left\{\sup_{\mathcal{C}_{L_1}}\inf_{\mathcal{C}_{L_2}}d((f_1,g_1),(f_2,g_2)),\sup_{\mathcal{C}_{L_2}}\inf_{\mathcal{C}_{L_1}}d((f_2,g_2),(f_1,g_1))\right\}.
((1-q^{2d-2}) \chi_{\{P\}}+ \chi_{P^\sim}) ((1-q^{2d-2-s}) \chi_{H_s}+ (1-q^{s}) \chi_{H_s^\perp}+ \chi_{C(H_s)})^\top = (1-q^{2d-2}) (1-q^{2d-2-s}) + (1-q^{2d-2-s}) |P^\sim \cap H_s| + (1-q^{s}) |P^\sim \cap H_s^\perp| + |P^\sim \cap C(H_s)| = (q^s-1)(q^{2d-2-s}-1)\frac{q^{2d-2}-1}{q^2-1}.
\hat{p}(y_{k}|y_{0},\cdot\cdot\cdot,y_{k-1})=\int p(y_{k}|x_{k})\hat{p}(dx_{k}|y_{0},\cdot\cdot\cdot,y_{k-1})=\frac{1}{N}\sum_{i=1}^{N}p(y_{k}|\xi_{k}^{i})
|D_4(S)| \geq |D_2(S)| +2 = |D_{\{1 \}}(S) \cup [D_{\{2\}}(S) \setminus D_{\{1\}}(S)]| +2 \geq (n-1) +\left(\frac{n}{3}- \frac{7}{3}\right)+2=\frac{4}{3}n-\frac{4}{3}.
\hat{H}: \ \ \ - \frac{\hbar^{2}}{2}\left[ \partial_{r}^{2}+ \frac{1}{r}\partial_{r}+ \frac{1}{r^{2}}\partial_{\theta}^{2}+ \partial_{z}^{2}\right] \, + \, \frac{1}{2}\omega^{2}r^{2}\ \ \ .
\eta_{t}= \frac{1}{k_{1}\epsilon}\theta_\xi - \epsilon\frac{k_{2}}{k_{1}}\eta\eta_{t}+ \delta^{2}\frac{k_{3}}{k_{1}}u_{0,xxt}, \eta_{x}= \frac{1}{k_{1}\epsilon}\theta_{x}- \epsilon\frac{k_{2}}{k_{1}}(\eta^{2})_{x}+ \delta^{2}\frac{k_{3}}{k_{1}}u_{0,xxx},
A_{u u}={\frac{1}{2}}( A_{0 0}- A_{y y}+ 2 i A_{0 y}) ; ~ A_{u v}={\frac{1}{2}}( A_{0 0}+ A_{y y}) ; ~ A_{v v}={\frac{1}{2}}( A_{0 0}- A_{y y}- 2 i A_{0 y}) ~ ,
0 =\int_{B_t}|\nabla v_0|^{p-2}\Big(|\nabla v_0|^2-\Gamma_{ij}^l\nabla v_0^i\cdot \nabla v_0^j v_0^l\Big)\eta dx &\geq a_{\kappa}(2R)\int_{B_t}|\nabla v_0|^{p-2}h_{ij}\nabla v_0^i\cdot \nabla v_0^j dx &\geq ca_{\kappa}(2R)\int_{B_t}|\nabla v_0|^{p}dx,
\begin{aligned}Q_{j}(\vec{1}) = (2m)^{j}P_{j}(\vec{1}) \frac{\partial Q_j}{\partial w_1}(\vec{1}) = (2m)^{j-1}\sum_{k=1}^{n}\frac{\partial P_j}{\partial z_k}(\vec{1}) (m+x_k) \frac{\partial Q_j}{\partial w_2}(\vec{1}) = (2m)^{j-1}\sum_{k=1}^{n}\frac{\partial P_j}{\partial z_k}(\vec{1}) (m-x_k) \end{aligned}
A_{k}^{i}\left. \left| 0\right\rangle \!\right\rangle =B_{m}^{j}\left.\left| 0\right\rangle \!\right\rangle =\tilde{A}_{k}^{i}\left. \left|0\right\rangle \!\right\rangle =\tilde{B}_{m}^{j}\left. \left|0\right\rangle \!\right\rangle =0.
( z - w )^{- 4}-{\frac{1}{2}}\sum_{n = 0}^{\infty}\left({-{\frac{1}{2}}\atop n + 1}\right){\frac{n + 1}{n + 2}}w^{- n - 2}( z - w )^{n - 2}={\frac{1}{2}}{\frac{z + w}{\sqrt{z w}}}( z - w )^{- 4}\, ,
\widetilde{L}_3K_0(\sqrt{u}t)={}\frac{(2u^{2}-25u+32)t^2+2 (u-4)}{2}K_0( \sqrt{u}t){}-\frac{[ (u-16) (u-4)t^2+12 (u-6)]\sqrt{u}t}{8}K_1( \sqrt{u}t),
\begin{array}{ll}<{\O1_{i}}_{\lambda},{\O1_{j}}_{\tau}>=\varepsilon_{i}\delta_{ij}{}^{*}\delta_{\lambda\tau}\qquad (<{\O1_{i}}^{\lambda},{\O1_{j}}^{\tau}>=\varepsilon_{i}\delta_{ij}{}^{*}\delta^{\lambda\tau}), <{\O1_{i}}^{\lambda},{\O1_{j}}_{\tau}>=\varepsilon_{i}\delta_{ij}\delta^{\lambda}_{\tau},\end{array}
\alpha_{\pm}= - \left( \alpha_{1}A \right)^{- 1}\left( \cosh \frac{\triangle \theta_{n}}{4}- i \sinh \frac{\triangle \theta_{n}}{4}\right)^{- 1}\left( \pm 1 - A^{2}\sinh^{2}\frac{\triangle \theta_{n}}{4}\right) .
\begin{aligned}\|\mathbf{w}_{xs}-\nu\mu\mathbf{e}\|^2 =\|\mathbf{w}_{xs}-\mu\mathbf{e}\|^2 + 2\|\mu\mathbf{e}-\nu\mu\mathbf{e}\|^2 +2(1-\nu)\mu(\mathbf{w}_{xs}-\mu\mathbf{e})^T\mathbf{e}&\le \left(\gamma^2/2+(1-\nu)^2k\right)\mu^2.\end{aligned}
\left. - \frac{1}{4}e^{\varphi}\,{\widetilde F}_{\mu \nu}^{J}{\widetilde F}^{J \mu \nu}- \frac{1}{\sqrt 2}e^{\varphi}{\widetilde F}_{\mu \nu}^{J}{\widetilde C}^{[ \mu}\partial^{\nu ]}{\tilde{D}}^{J}- \frac{1}{2}e^{\varphi}{\widetilde C}_{[ \mu}\partial_{\nu ]}{\tilde{D}}^{J}{\widetilde C}^{[ \mu}\partial^{\nu ]}{\tilde{D}}^{J}\right.
|| \nabla E(\boldsymbol{h}) ||^2_{l^2(\mathbb{Z})}= (V_2(z_{-N-1},z_{-N})+V_1(z_{-N},x_{-N+1}))^2 + (V_2(z_{-N},x_{-N+1})+V_1(x_{-N+1},x_{-N+2}))^2 & + \sum_{j =-N+2}^{N-1}(V_2(x_{j-1},x_j)+V_1(x_j,x_{j+1}))^2 & + (V_2(x_{N-1},x_N)+V_1(x_N,z_{N+1}))^2 + (V_2((x_N,z_{N+1}))+V_1(z_{N+1},z_{N+2}))^2.
\frac{1}{\left(\sqrt{2\pi}\right)^n}\int_{\mathbb{R}^n}e^{-\mathbf{x^TAx}-\mathbf{b^Tx}}\mathrm{d}\mathbf{x}= \frac{1}{\sqrt{2^n\det \mathbf{A}}}e^{\frac{\mathbf{b^TA^{-1}b}}{4}}.
Y^{(n)}(t)=Y^{(n)}(t_{n, i})+\frac{t-t_{n, i}}{t_{n, i+1}-t_{n, i}}(Y^{(n)}(t_{n, i+1})-Y^{(n)}(t_{n, i})), t_{n, i}\leq t \leq t_{n, i+1}.
RR(k)=\dfrac{\|{\bf r}^k\|_2^2/\sigma^2}{\|{\bf r}^{k-1}\|_2^2/\sigma^2}\sim \dfrac{\chi^2_{n-k}}{\chi^2_{n-k}+\chi^2_1}\sim \mathbb{B}(\dfrac{n-k}{2},\dfrac{1}{2})
{\tilde{\rho_{1}}}^{1 2}={\tilde{\rho_{2}}}^{1 2}={\tilde{\rho_{3}}}^{1 2}={\tilde{\rho_{4}}}^{2}={\tilde{\rho_{4}}}{\tilde{\rho_{5}}}^{2}= \mathrm{i d e n t i t y}.
f(x) =\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}M_1(u)u^{k-1}e^{-xu}u^{\lambda-1}\, du =\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}N_1(s)s^{-k-\lambda}e^{-x/s}\, ds.
s-t \leqslant v_{i+1}-v_i = \left(\dfrac{n+2}{n+1}\right)^i t \geqslant v_i = \sum_{k=0}^{i-1}\left(\dfrac{n+2}{n+1}\right)^k = (n+1)\left( \left(\dfrac{n+2}{n+1}\right)^i -1\right).
r\frac{d}{dr}\left[\frac{1}{1-z\bar{w}}\right] =\frac{z\bar{w}}{(1-z\bar{w})^2},z=re^{i\theta}, r\frac{d}{dr}\left[\frac{1}{|1-zw|^{2\lambda}}\right] =2\lambda\frac{{\mathrm Re}\,(zw)-|zw|^2}{|1-zw|^{2\lambda+2}},z=re^{i\theta}, r\frac{d}{dr}A =A\left(1+2\frac{{\mathrm Re}\,(zw)-|zw|^2}{|1-zw|^{2}}\right)=A\frac{1-|zw|^2}{|1-zw|^{2}},z=re^{i\theta}.
\mathbb P\left(\left\|\frac{R}{ks}-\mathrm{I}_{nk}\right\|>\varepsilon\right) =\mathbb P\left(\left\|\frac{R}{ks}-\mathrm{I}_{nk}\right\|^2>\varepsilon^2\right) &\leq \mathbb P\left(\mathrm{Tr}\left(\frac{R}{ks}-\mathrm{I}_{nk}\right)^2>\varepsilon^2\right) &\leq\frac{\mathbb{E}\mathrm{Tr}\left(\frac{R}{ks}-\mathrm{I}_{nk}\right)^2}{\varepsilon^2}.
\mathrm{RW}(d) =\sup_{\lambda \ge 1}\left[\inf_{\pi \in \Pi(\mu_{13}, \mu_{23})}\int_{\mathcal{V}}\min\{y_2 + \varphi_{\lambda, 1}(x_1, x_2), y_1 + \varphi_{\lambda, 0}(x_1, x_2) \}d \pi(v) - \langle \lambda, \delta \rangle\right],
{\cal L}_{\mathrm{e f f}}= \frac{1}{8 \pi}\mathrm{I m}\! \int \! \! \mathrm{d}^{2}\theta \tau_{D}( A_{D}) W_{D}^{2}- \frac{1}{4 \pi}\mathrm{I m}\! \int \! \! \mathrm{d}^{2}\theta \mathrm{d}^{2}\bar{\theta}\bar{A}_{D}A ( A_{D}) .
\mathcal{Q}= Z_{3}\mathcal{Q}^{\mathrm{R}}, \quad \mathcal{Q}^{\mathrm{R}}: ={\frac{1}{2}}a_{\mu}^{\mathrm{R}}( x ) a^{\mu}{}^{\mathrm{R}}( x ) + \alpha^{\mathrm{R}}i \bar{C}^{\mathrm{R}}( x ) C^{\mathrm{R}}( x ) .
\sum_{j=1}^{n}\binom{n}j (-1)^{j-1}\frac{a_j}{j^k}= \sum_{j=1}^{n}\sum_{i=j}^{n}\binom{i-1}{j-1}(-1)^{j-1}\frac{a_j}{j^k}=\sum_{i=1}^{n}\frac1i \sum_{j=1}^{i}\binom{i}{j}(-1)^{j-1}\frac{a_j}{j^{k-1}}.
a_{\tau}={\frac{\sqrt{\pi}\Gamma ( \tau + 1 )}{2^{\frac{\tau + 1}{2}}\Gamma \! \left({\frac{\tau + 2}{2}}\right)}}, \qquad b_{\tau}={\frac{\sqrt{2}\Gamma \! \left({\frac{\tau + 2}{2}}\right)}{\Gamma \! \left({\frac{\tau + 1}{2}}\right)}}.
|||(\tilde{\mathcal{A}}\mathcal{F})(t)|||^2_{s-1,\ast}\lesssim||\tilde{\mathcal{A}}J^T\mathbf{F}||^2_{s,\ast,t}&\leq C(K)\Big(||\mathbf{F}||^2_{s,\ast,t}+||\mathbf{F}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s,\ast,t}\Big) &\leq C(K)\mathcal{M}(t).
\begin{array}{c c}{{G_{\mu \nu}( X^{i}) =}}&{{\left( \begin{array}{c c}{{\left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right)}}&{{0}}\\{{0}}&{{G_{i j}}}\\ \end{array}\right) ,}}\\ \end{array}
J_1:=\left(\begin{smallmatrix}\;1\; \;0\; & \;0\; \;1\;\end{smallmatrix}\right),J_2:=\left(\begin{smallmatrix}\;0\; \;1\; & -1 \;0\;\end{smallmatrix}\right),J_3:=\left(\begin{smallmatrix}0 \sqrt{-1}\sqrt{-1}0\end{smallmatrix}\right),J_4:=\left(\begin{smallmatrix}\sqrt{-1}0 0 -\sqrt{-1}\end{smallmatrix}\right).
\Phi((a,b),(c,d)) \geq - \langle \gamma_1(a), \gamma_2(d) \rangle-\langle \gamma_1(c), \gamma_2(b) \rangle = - \langle \gamma_1(a) ,\gamma_1(c)+\gamma_2(d) \rangle & = -2 \langle \gamma_1(a), Z_{c,d}\rangle = -2 \langle Z_{a,b}, Z_{c,d}\rangle.
q_1 = \frac{1}{a_1}q_i = \sum_{j=i}^N\frac{(-1)^{j-i}}{a_1\cdots a_j}q_m = (-1)^{N-1}\frac{1 - a_N + \sum_{k=2}^{N-1}(-1)^k a_N \prod_{l=2}^k a_{N-l+1}}{\prod_{k=1}^N a_k - (-1)^N},
\begin{array}{c | c c c c}{{}}&{{m}}&{{n}}&{{p}}&{{p^{\prime}}}\\ \hline{{( a )}}&{{1}}&{{2}}&{{}}&{{4}}\\{{( b )}}&{{1}}&{{4}}&{{}}&{{2}}\\{{( c )}}&{{2}}&{{4}}&{{4}}&{{2}}\\{{( d )}}&{{2}}&{{2}}&{{4}}&{{2}}\\{{( e )}}&{{2}}&{{3}}&{{4}}&{{2}}\\{{( f )}}&{{2}}&{{2}}&{{3}}&{{3}}\\ \end{array}
\phi_{k,\lambda}=\frac{1}{e}\sum_{n=0}^{\infty}\frac{1}{n!}(n)_{k,\lambda}=\frac{1}{e}\sum_{n=1}^{\infty}\frac{1}{(n-1)!}(n-\lambda)_{k-1,\lambda},(k\in\mathbb{N}).
\int_{I^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k \geq S(I)^k(\ln n)^{-k}\int_{(0,l_n)^k}\phi_{k,n}\big(\gamma_n(u_1),\cdots,\gamma_n(u_k)\big)\prod_{j=1}^k\beta_n(u_j)du_1\cdots du_k.
B ( x ) = \frac{( 2 \rho^{2}+ 4 \rho - 2 ) - 8 \rho x + 4 \rho^{2}e^{- 2 x}}{[ e^{2 x}- \rho^{2}e^{- 2 x}+ ( 1 - \rho^{2}) + 4 \rho x ]}.
\sum \limits_{k=1}^{n}\alpha_k =0, \alpha_{n+1}=0, \alpha_{n+2}=\frac{1}{2n \sin \frac{\pi}{n}}\sum \limits_{k=1}^{n}\alpha_k \cos \frac{\pi(2k-1)}{n}, \alpha_{n+3}=\frac{1}{2n \sin \frac{\pi}{n}}\sum \limits_{k=1}^{n}\alpha_k \sin \frac{\pi(2k-1)}{n},
k_ik_i^{-1}= k_i^{-1}k_i = 1, k_0k_1 = k_1k_0, &k_ie_i^{\pm}= q^{\pm2}e_i^{\pm}k_i, k_ie_j^{\pm}= q^{\mp2}e_j^{\pm}k_i, i \neq j, &e_i^+e_i^- - e_i^-e_i^+ = \frac{k_i - k_i^{-1}}{q-q^{-1}}, e_0^{\pm}e_1^{\mp}- e_1^{\mp}e_0^{\pm}= 0,
xq\big(1-q^{2}\big)\big(xqq^{\prime}\big)^{\prime}+ x\left( (xq^{\prime})^{\prime}+ \frac{q}{4}\right)\big(1-q^{2}\big)^{2}+ x^{2}q \left( qq^{\prime}\right)^{2}= \alpha^{2}\frac{q}{4},
\begin{array}{r c l}{{\Delta ( x - 1 \epsilon ( x ) )}}&{{=}}&{{x_{( 1 )}\otimes x_{( 2 )}- 1 \epsilon ( x ) \otimes 1}}\\{{}}&{{=}}&{{( x - 1 \epsilon ( x ) ) \otimes 1 + x_{( 1 )}\otimes ( x_{( 2 )}- 1 \epsilon ( x_{( 2 )}) )}}\\ \end{array}
\partial_{t}\widetilde{E}=b^{11}\partial_{t}b_{11}-\tilde{d}\frac{h_{t}}{h}+2\tilde{l}\sum_{j}h_{j}h_{jt}=b^{11}(h_{11t}+h_{t})-\tilde{d}\frac{h_{t}}{h}+2\tilde{l}\sum_{j}h_{j}h_{jt}.
\left| \int_{\mathbb S^1}B_xJ_{x}B_x\, dx\right|\leq \ \|J_x\|_{L^\infty}\|B_x\|^2_{L^2}, \left| \int_{\mathbb S^1}BJ_xB_{xx}\, dx\right|\lesssim \ \|J_x\|_{L^\infty}\|B_{xx}\|_{L^\infty}\|B_x\|_{L^2}.
\left[ - x \sum_{k = 0}^{n_{c}- 1}( - 1 )^{k}s_{k}( \phi^{2}) x^{n_{c}- 1 - k}+ m_{1}\cdots m_{n_{f}}\right]^{2}- \prod_{i = 1}^{n_{f}}( x + m_{i}^{2}) = - \frac{\alpha_{n_{c}}^{2}}{\Lambda^{2}}x \prod_{a = 1}^{n_{c}- 1}( x - \alpha_{a})^{2}.
x^{( 1 / 3 )}, \quad y^{( 1 / 3 )}={x^{( 1 / 3 )}}^{\prime}, \quad z^{( 1 / 3 )}, \qquad x_{d}^{( 1 / 3 )}, \quad y_{d}^{( 1 / 3 )}={x_{d}^{( 1 / 3 )}}^{\prime}, \quad z_{d}^{( 1 / 3 )},
u=\frac1{\sqrt{t^2+1}}(w^1\cos\tau-w^2\sin\tau)+\frac{tx}{t^2+1}-\frac{\kappa y}{t^2+1}, v=\frac1{\sqrt{t^2+1}}(w^1\sin\tau+w^2\cos\tau)+\frac{ty}{t^2+1}+\frac{\kappa x}{t^2+1},
{\frac{1}{\beta_{c}}}= b ( d - 2 ) ( 1 - \kappa ){\int}_{- 1}^{1}d \xi H_{c}( \xi ){\frac{\sqrt{1 - \xi^{2}}}{\sqrt{( \xi + \frac{3 + \kappa}{1 - \kappa}) ( \xi + \frac{1 + 3 \kappa}{1 - \kappa})}}}.
\quad\,x_{(r-1)}\sum^{r-1}_{l=2}\bigg(\sum^{l-1}_{k=1}(q-1)q^{r-k-1}(c_k-c_r)\bigg)T_{r-1}\cdots T_l &=x_{(r-1)}\sum^{r-1}_{l=2}\big((q^{r-l+1}-q^{r-l})c_{q,1,l-1}-(q^{r-1}-q^{r-l})c_r\big)T_{r-1}\cdots T_l.
T_{5,1}=e^{i \pi /4}\sqrt{\frac{4\pi}{3}}A |A|^2 \frac{e^{ia \ln|\xi|}}{|\xi|}\int_{\nu>0}e^{-3i\nu}\frac{1}{\sqrt{|\nu|}}d\nu + O(|\xi|^{-2\gamma}) + O(|\xi|^{1-3\gamma}) + O(|\xi|^{-2+\gamma/2}) = \frac{2\pi}{3}A |A|^2 \frac{e^{ia \ln|\xi|}}{|\xi|}+ O(|\xi|^{-2+\gamma/2})
\varphi^* \left(\begin{array}{c c}a b 0 d \end{array}\right) :=\left(\begin{array}{c c c}a^{\ell_1}0 a^{p^{e_2}}\, b^{p^{e_2}}\, d^{- \ell_3}0 d^{- \ell_2}a^{p^{e_1}}\, b^{p^{e_1}}\, d^{- \ell_3}0 0 d^{- \ell_3}\end{array}\right) .
\| w \|_{L^\infty_1 (\Omega)}\leq C_3 \bigg ( |\alpha|^{-\frac{1}{2}}\big ( |\alpha | + |b_\Omega[f]| + \| F\|_{L^2 (\Omega)}+ \| f \|_{L^2 (\Omega_{6 R_0})}\big ) & \qquad+ \big | \log |\alpha|\big |\, \| F \|_{L^\infty_{2}(\Omega)}+ \| F_{12}- F_{21}\|_{L^1 (\Omega)}\bigg ) \,,
\Big( \sum_{i=-N}^N \big\| \|A_t^\phi (F,G)(x,y)\|_{\textup{V}_t^2([2^i,2^{i+1}],\mathbb{C})}\big\|^2_{\textup{L}^2_{(x,y)}(\mathbb{R}^2)}\Big)^{1/2}\lesssim_{\lambda}C_3^{1/2},
0\leq \chi^{\pm}_{\varepsilon}\chi^{\pm}\leq 1\quad\chi^{\pm}_{\varepsilon}\chi^{\pm 2}=\chi^{\pm}_{\varepsilon}, &\forall (f,w)\in\L^1(U)\times\L^1(U):\;\chi^{\pm}_{\varepsilon}(w)f\underset{\varepsilon\rightarrow 0}{\longrightarrow}\chi^{\pm}(w)f\quad\quad\L^1(U).
F=\frac{1}{2}\begin{pmatrix}(\partial_{\xi_j}\partial_{x_k}q(x,\xi))_{j,k=1}^d (\partial_{\xi_j}\partial_{\xi_k}q(x,\xi))_{j,k=1}^d -(\partial_{x_j}\partial_{x_k}q(x,\xi))_{j,k=1}^d -(\partial_{x_j}\partial_{\xi_k}q(x,\xi))_{j,k=1}^d\end{pmatrix}.
2 X_{[ij}\cdot P_{e_1}X_{kl]}\cdot P_{e_2}= X_{[ij}\cdot X_{kl]}\cdot( P_{e_1}P_{e_2}) - (X_{[ij}X_{kl]}\cdot P_{e_1}) P_{e_2}- P_{e_1}(X_{[ij}X_{kl]}\cdot P_{e_2}),
{\frac{1}{a}}\cdot \left( \begin{array}{c c c c}{{1}}&{{-{\frac{b}{a}}}}&{{\left({\frac{b}{a}}\right)^{2}}}&{{\dots}}\\{{{\frac{c}{a}}}}&{{\left({\frac{d}{a}}-{\frac{2 b c}{a^{2}}}\right)}}&{{\dots}}&{{}}\\{{\left({\frac{c}{a}}\right)^{2}}}&{{\dots}}&{{}}&{{}}\\{{\vdots}}&{{}}&{{}}&{{}}\\ \end{array}\right) \left( \begin{array}{l}{{1}}\\{{z}}\\{{{z}^{2}}}\\{{\vdots}}\\ \end{array}\right) .
V_{5 ( 0 )}^{b}\sim \frac{i}{2 \sqrt{2}}\biggl [ ( 1 - i ) e^{i H_{5}}e^{i H_{8}}+ ( 1 + i ) e^{i H_{5}}e^{- i H_{8}}+ ( 1 + i ) e^{- i H_{5}}e^{i H_{8}}+ ( 1 - i ) e^{- i H_{5}}e^{- i H_{8}}\biggr ]
\frac{1}{4 \pi}L_{C M}= \frac{v g}{2 k}\left( \sum_{j = 1}^{k}\vec{V}_{j}\right)^{2}- \frac{v}{2 g k}\left( \sum_{j = 1}^{k}q_{j}\right)^{2}+ \frac{b}{k}\left( \sum_{j = 1}^{k}q_{j}\right) \left( \sum_{l = 1}^{k}V_{l \chi}\right) ,
\sup_{r_{1,1},...,r_{1,d_1},r_{2,1},...,r_{2,d_2}\in [0,2]}\left|\mathbb E\left[\prod_{k=1}^{d_1}B_1(r_{i,1})\prod_{k=1}^{d_2}B_2(r_{2,k})\right]\right|<\infty,