harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2\partial_{\bar z}\mbox{\bf G}+\mbox{\bf G}A_2^*-A_1^*\mbox{\bf G}=0\mbox{in}\,\,\Omega\setminus \{x\in \bar\Omega\vert det\, \mathcal Y=0\},\mbox{\bf G}\vert_{\tilde \Gamma}=I, \partial_{\vec \nu}\,\mbox{\bf G}\vert_{\tilde \Gamma}=0.
\Omega_{crit}^{\pm}(\rho)=b(\rho)+\sum_{m=1}^M m\mbox{Im}(\beta_m)\rho^{2m}\pm\sqrt{\left[ \sum_{m=1}^M m\mbox{Im}(\beta_m)\rho^{2m}\right]^2-\frac{a(\rho)}{\rho}\frac{\partial a(\rho)}{\partial \rho}}.
A(k,\xi)-A(k,\xi-\eta) = M(k,\xi)\left( (1+k^2+\xi^2)^{N/2}-(1+k^2+(\xi-\eta)^2)^{N/2}\right) & + (M(k,\xi)-M(k,\xi-\eta))(1+k^2+(\xi-\eta)^2)^{N/2}& ={\rm com_1+com_2}.
g_{\sigma(i)}= \begin{cases}\beta_{i}\mbox{if $1 \leq i \leq p$}, \beta_{\sigma(i) + p - i}^{-1}\cdots \beta_{1}^{-1}\alpha_{i-p}\beta_{1}\cdots \beta_{\sigma(i) + p - i}, \mbox{if $p+1 \leq i \leq p+q$}\end{cases}
\left(sM_s\right)_{v\sigma}=\sum_{\rho\in\Sigma}s(\rho v)\cdot (M_s)_{\rho v,v\sigma}&= \sum_{\rho\in \Sigma}\frac{s(\rho v)\cdot s(v\sigma)}{\sum_{\tau\in\Sigma}s(v\tau)}&= s(v\sigma)\cdot\frac{\Sigma_{\rho\in\Sigma}s(\rho v)}{\Sigma_{\tau\in\Sigma}s(v\tau )}=s(v\sigma),
H_\lambda D^{r}\Bigl(\frac{C(\lambda,r)}{H_\lambda}\Bigr) = \frac{(2r)!}{(r+1)!}|\lambda|, H_\lambda D^{r+1}\Bigl(\frac{C(\lambda,r)}{H_\lambda}\Bigr) = \frac{(2r)!}{(r+1)!}, H_\lambda D^{r+2}\Bigl(\frac{C(\lambda,r)}{H_\lambda}\Bigr) = 0.
\delta x : = \overline{{{\varepsilon}}}\psi + \overline{{{\psi}}}\varepsilon , \qquad \delta \psi : = - ( \mathrm{i}\dot{x}+ \Phi ( x ) ) \varepsilon , \qquad \delta \overline{{{\psi}}}= ( \mathrm{i}\dot{x}- \Phi ( x ) ) \overline{{{\varepsilon}}},
\widetilde{G}_{np}= \begin{cases}\sqrt{\dfrac{2}{\pi}}\sum_{l=1}^\infty (-1)^l \langle u_T, \phi_{2ln}\rangle, p=0, \sum_{l=1}^\infty (-1)^l [ \langle u_T, \phi_{p+2ln}\rangle + \langle u_T, \phi_{-p+2ln}\rangle], 1 \le p \le n-1.\end{cases}
\mathcal{E}_1(\gamma) - \mathcal{E}_2(\gamma) = \frac{1}{\ell}\int_0^\ell \{\cos^2(v_0 t + \theta_0) - \sin^2(v_0 t + \theta_0) \}\: dt = \frac{1}{\ell}\int_0^\ell \cos 2(v_0 t+\theta_0) \: dt = 0,
\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/d q_1q_2)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{q_1q_2d}+\eta\Bigr)\le \Sigma_1(\Sigma_2\Sigma_3'+\Sigma_4\Sigma_5'),
0 =\nabla f(\bar x)+\sum\limits_{i\in I^g(\bar x)}\bar \lambda_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\bar \rho_j\nabla h_j(\bar x) +\sum\limits_{l\in I^{\varphi_\textup{KS}}(\bar x)}\bar \xi_l\bigl(H_l(\bar x)\nabla G_l(\bar x)+G_l(\bar x)\nabla H_l(\bar x)\bigr)
- \frac{\hat{y}_{2 3}}{\hat{y}_{1 2}\, \hat{y}_{3 4}}\, ( \sum_{i = 0}^{4}c_{n}( s , t ) v^{n}= ( 1 + v ) ( s + v ( 1 + s - t ) + v^{2}) O ( \frac{1}{( \hat{y}_{1 2}\, \hat{y}_{3 4})^{2}})
\hat{R}_{12}(\sigma^{1}K' \sigma^{1})_{1}\hat{R}_{12}^{-1}(\sigma^{1}K'\sigma^{1})_{1}= (\sigma^{1}K' \sigma^{1})_{1}\hat{R}_{12}^{-1}(\sigma^{1}K' \sigma^{1})_{1}\hat{R}_{12}\;.
\| f \|_{W^{\gamma, b}}:= \sum_{k\in\mathbb{Z}}\big(2^{\gamma k_+}+ 2^{b k_{-}}\big) \| P_{k}f\|_{L^\infty},\| f\|_{W^{\gamma}}: = \sum_{k \in \mathbb{Z}}(2^{N_2 k}+ 1) \| P_k f \|_{L^\infty},
\sum_{n=1}^{\infty}\frac{\mu(n)}{n^k}\exp\left({-\frac{x}{n^2}}\right) =\frac{\Gamma(\frac{k}{2})}{x^\frac{k}{2}}\sum_{n=1}^{\infty}\frac{\mu(n)}{n}{}_1F_1 \left(\frac{k}{2}; \frac{1}{2}; - \frac{\pi^2}{n^2x}\right) + \frac{1}{2}\sum_{\rho}\frac{\Gamma(\frac{k-\rho}{2})}{\zeta'(\rho)}x^{-\frac{(k-\rho)}{2}},
\begin{aligned}\left\Vert\frac{1}{T}\int_0^T B(s)Ke^{-isH}ds\right\Vert^2=\left\Vert\frac{1}{T}\int_0^T e^{isH}K^*B(s)^*ds\right\Vert^2 &=\sup_{\Vert f\Vert=1}\left\Vert\frac{1}{T}\int_0^T e^{isH}K^*B(s)^*fds\right\Vert^2 &=\sup_{\Vert f\Vert=1}\frac{1}{T^2}\int_0^T\int_0^T (B(s)^* f,\psi)(\psi,B(t)^* f)(e^{-i(t-s)H}\phi,\phi)dt ds.\end{aligned}
[\begin{matrix}t_{i}\\A_{i}\end{matrix}]=[\begin{matrix}0,&\pi/\omega_{d},&2\pi/\omega_{d},&3\pi/\omega_{d}\\K^{3}/(K+1)^{3},&3K^{2}/(K+1)^{3},&3K/(K+1)^{3},&1/(K+1)^{3}\end{matrix}]
J_{F}(r,\varphi)=[\begin{matrix}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial\varphi}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial\varphi}\end{matrix}]=[\begin{matrix}cos\varphi&-rsin\varphi\\sin\varphi&rcos\varphi\end{matrix}]
Q(X,\tau)=\left[\widehat{A}(TX){\rm ch}(\triangle(X))+32\widehat{A}(TX)\right]^{(8)}&+\left[2\widehat{A}(TX){\rm ch}(\triangle(X)){\rm ch}(\widetilde{TX})+32\widehat{A}(TX){\rm ch}(\widetilde{TX}+\wedge^2\widetilde{TX})\right]^{(8)}q+O(q^{\frac{3}{2}}).
0=\sum_{\substack{0\leq d_i\leq a_i 1\leq e_j\leq b_{j}-1}}\Big( \prod_{i=1}^n Q_{d_i}(a_i)\psi_{i}^{d_i}\Big) \Big( \prod_{j=1}^m Q_{e_j+1}(b_j) \Big) \kappa_{e_1,\dots,e_m}
L(t) =e^{(n-\nu-2)t}\int_XS(t)\omega(t)^n &=ne^{(n-\nu-2)t}(2\pi c_1)\cdot(-2\pi(1-e^{-t})c_1(X)+e^{-t}\alpha)^{n-1}&=\sum_{k=0}^{\nu-1}A_k(1-e^{-t})^ke^{(k-\nu-1)t},
\nabla_{G\xi}Y_{\tau}^{n,t,x}=\int_\tau^Te^{-(s-\tau){A}}G \nabla_{}B^n(s,\Xi^{t,x}_s)e^{(s-t)A}G\xi\,ds+\int_\tau^Te^{-(s-\tau){A}}\nabla_{G\xi}Z_s^{n,t,x}\, B^n(s,\Xi^{t,x}_s)\,ds &+\int_\tau^Te^{-(s-\tau){A}}Z_s^{n,t,x}\,\nabla_{}B^n(s,\Xi^{t,x}_s)e^{(s-t)A}G\xi\,ds - \int_\tau^Te^{-(s-\tau){A}}\nabla_{G\xi}Z^{n,t,x}_{s}\;d W_s.
U' (x(T,y)) &=\frac{1}{x(T,y)}= \frac{1}{x}\exp\{-\int_0^T[\pi(s,y)b_0(s,y)-\frac{1}{2}\pi^2(s,y)\sigma_0^2(s,y)]ds - \int_0^T\pi(s,y)\sigma_0(s,y)dB(s)\}
\mathrm{R e}\, \Gamma ( \varphi , \Lambda ) \; = \; \frac{1}{2}\, \sum_{n}\, \log \left( 1 \, + \, \frac{\Lambda^{2}}{( 2 \pi )^{2}\, ( n + \varphi )^{2}}\right) \; - \; \frac{1}{2}\, \sum_{n}\, \log \left( 1 \, + \, \frac{\Lambda^{2}}{( 2 \pi )^{2}n^{2}}\right) \; .
g_1 =\begin{pmatrix}1 &1 & 1\end{pmatrix}g_4 =\begin{pmatrix}-1 -1 & -1\end{pmatrix}g_2 =\begin{pmatrix}1 1 &1 \end{pmatrix}g_5 =\begin{pmatrix}-1 &-1 -1 \end{pmatrix}g_3 =\begin{pmatrix}1 & 1 1 \end{pmatrix}g_6 =\begin{pmatrix}-1 & -1 &-1 \end{pmatrix}
\int_{S^{d-1}}|\mathcal{F}(f\circ \theta_a^X)(Ru)|^2 du{}= \int_{S^{d-1}}\bigg|\int g_a(x)e^{-2\pi i Rx \cdot u}dx\bigg|^2 du &=\frac{1}{(2\pi R)^2}\int_{S^{d-1}}\bigg|\int \nabla_u g_a(x) e^{2\pi i R x \cdot u}dx\bigg|^2 du.
\nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\nabla^\frac{H}{2}\psi_u(u)\right)(u) = \frac{\partial^2f}{\partial x}(s,x_s)\nabla^\frac{H}{2}x_s(u)\nabla^\frac{H}{2}\psi_u(u) + \frac{\partial f}{\partial x}(s,x_s)\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u(u,u)
z^{m}\in H^{s - m}(\Omega) \cap H^{1}_{0}(\Omega) m = 2, \dots, s - 1 z^{s}\in L^{2}(\Omega), \theta^{k}\in H^{s + 1 - k}(\Omega) \cap H^{1}_{0}(\Omega) k = 1, \dots, s - 2 \theta^{s - 1}\in H^{1}_{0}(\Omega).
\widetilde T_n^\alpha =\frac{1}{N}\sum_{i=1}^n \widetilde T_{n,i}^\alpha =\frac{1}{N}\sum_{i=1}^n R_{n,i}T_{n,i}^\alpha, \shortintertext{and}\Psi_n^\alpha =\frac{1}{n}\sum_{i=1}^nE[W_{n,i}^\alpha].
{\cal T}_{s t a n}\, = \, \left[ \begin{array}{c c}{{1}}&{{\qquad 0}}\\{{}}&{{}}\\{{1}}&{{\qquad 1}}\\ \end{array}\right] \qquad \qquad{\cal S}_{s t a n}\, = \, \left[ \begin{array}{c c}{{0}}&{{\qquad - 1}}\\{{}}&{{}}\\{{1}}&{{\qquad 0}}\\ \end{array}\right]
F_2(d;a_1,a_2;b_1;b_2;x,y)\displaystyle=\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{(d)_{m+n}(a_1)_m(a_2)_n}{(b_1)_m(b_2)_nm!n!}x^my^n.
\sum_{j}B^{1}_{1a,j}\omega_j=\mu\omega_{2a}, ~~~\sum_{j}B^{2}_{1a,j}\omega_j=\mu\omega_{1a}, &\sum_{j}B^{1}_{2a,j}\omega_j=\mu\omega_{1a}, ~~~\sum_{j}B^{2}_{2a,j}\omega_j=-\mu\omega_{2a}, &~~~~~~~~~~B^{1}_{ab,j}=0, ~~~B^{2}_{ab,j}=0.
\int_{B_{d_x}(x)}|\nabla (\psi f)|^p \le \left( \int_{B_{d_x}(x)}1\right)^{1-\frac{p}{k}}\left(\int_{B_{d_x}(x)}(|\nabla (\psi f)|^p)^{\frac{k}{p}}\right)^{\frac{p}{k}}& \leq C M^p d^{n(1-\frac{p}{k})+ [{k\alpha -k+ n}]\frac{p}{k}}=C M^p d^{p\alpha - p +n}.
~~~~(\beta M_1^H+\alpha M_2^H)^H(\beta M_1^H-\alpha M_2^H)+(\beta M_1^H-\alpha M_2^H)^H(\beta M_1^H+\alpha M_2^H) =2\vert\beta\vert^2(M_1M_1^H-\tfrac{\vert\alpha\vert^2}{\vert\beta\vert^2}M_2M_2^H) \geq 2\vert\beta\vert^2(1-\tfrac{\vert\alpha\vert^2}{\vert\beta\vert^2})M_2M_2^H \geq 0.
P_{eg}(\tau)=\lambda^{2}|\left<e\right|m(0)\left|g\right>|^{2}\int_{0}^{\tau}{\rm d}\tau^{'}\int_{0}^{\tau}{\rm d}\tau^{''}e^{iE(\tau^{''}-\tau^{'})}\left<\Phi_{i}\right|\varphi(x(\tau^{'}))\varphi(x(\tau^{''}))\left|\Phi_{i}\right>,
\Delta w_1 = \frac{2k_{11}}{\sqrt{|K|}}\bigg(e^{u_0^{'}+\sqrt{|K|}w_1}-1\bigg) &+\frac{2k_{12}}{\sqrt{|K|}}\bigg(e^{u_0^{''}+(|K|w_2+k_{21}\sqrt{|K|}w_1)/k_{11}}-1\bigg)+h_1 \Delta w_2 =2\bigg(e^{u_0^{''}+(|K|w_2+k_{21}\sqrt{|K|}w_1)/k_{11}}-1\bigg)+h_2,
(R^{-1}R)_{X,Y}=c_{X,Y}^{-1}F(\Psi_{X,Y}^{-1})c_{Y,X}\Psi^{{\rm Vec}}_{F(X),F(Y)}\Psi^{{\rm Vec}-1}_{F(X),F(Y)}c^{-1}_{Y,X}F(\Psi_{X,Y})c_{X,Y}=c_{X,Y}^{-1}c_{X,Y}=\Delta(1)_{X,Y}
\sum_{j=1}^\infty \bigl[2^{a(j)}\cdot (2^{s(j)})^x\bigl]\ \leq\ \ \sum_{j=1}^\infty 2^{g(x) + 1 - |n-j|}\ =\ 2^{g(x)+1}\sum_{j=1}^\infty 2^{-|n-j|}&\leq\ 2 \cdot r^x f(x) \cdot \sum_{j=-\infty}^\infty 2^{-|n-j|}\ =\ 2 \cdot r^x f(x) \cdot 3.
\frac{q^{k_m^2}}{1-q^{n-k_m+1}}=q^{k_m}(1+q^{n-k_m+1}+q^{2(n-k_m+1)}+\dots) &=q^{k_m^2}+q^{n+1+k_m^2-k_m}+(\,\,) &=q^{k_m^2}
\mathbb{E}\left[|\widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}|\right] \leq c_6\sum_{i \in [M]}\sum_{t=0}^{T-1}\exp\left(-c_7 N_in_x \left(\frac{\alpha \rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right),
\mu_{k}^i\left(B\right) \geq 1\textendash C_3\sum\limits_{\theta \in B^c}\prod\limits_{t=1}^{k}\prod\limits_{j=1}^{n}\left( \frac{\ell^j(s_{t}^j|\theta)}{\ell^j(s_{t}^j|\theta^*)}\right)^{\frac{1}{n}}
\int_0^{\pi}e^{2ij\theta}\left(e^{2i\theta},e^{-2i\theta};q\right)_{\infty}\frac{d\theta}{2\pi}=\frac{(-1)^{j}}{(q;q)_{\infty}}\left(q^{\binom{j}{2}}+q^{\binom{-j}{2}}\right).
{\mathbb K}(Z,W)(P) = &{\mathbb H}_0(Z) \begin{bmatrix}({\rm id}_{n \times m}\otimes \pi)(P) ({\rm id}_{n \times m}\otimes \pi)(P) \cdot{\rm id}_{n \times m}\otimes J ({\rm id}_{n \times m}\otimes \pi)(P) \cdot{\rm id}_{n \times m}\otimes J ({\rm id}_{n \times m}\otimes \pi)(P) \end{bmatrix}{\mathbb H}_0(W)^*
w_1d-d =\sum_{n=0}^\infty u_n (v_{n+1}v_n^*v_\infty a v_\infty^*-v_\infty av_\infty^*)u_n^* &=\sum_{n=0}^\infty u_n(v_{n+1}v_n^*-v_\infty v_\infty^*)v_\infty av_\infty^*u_n^*,
\int_{- \infty}^{\infty}d \kappa \frac{e^{c \kappa}}{\frac{2}{\kappa}\operatorname{s i n h}\left( \frac{\kappa \pi}{2}\right) 2 \operatorname{c o s h}( \frac{\kappa \pi}2 )}= \frac{1}{4 \operatorname{c o s}({\frac{c}{2}})^{2}}\, .
(\chi' \ast \kappa)(b) = \,\,\ast &=\chi'(b) \kappa(b) (\ref{action.eq.}) &=a^{-1}\chi(b)\kappa(b) a \kappa(b)^{-1}\kappa(b) &=a^{-1}\chi(b)\kappa(b) a \,\,\ast &= a^{-1}(\chi \ast \kappa)(b) \,a.
Q_\alpha = \left[\frac{\sin\left(\alpha u_2\right)}{\sin\left(\alpha u_1\right)}\right]^{\alpha/(1-\alpha)}\left[\frac{\sin u_1}{\sin u_2}\right]^{\frac{1}{1-\alpha}}\frac{\sin\left( u_2 - \alpha u_2\right)}{\sin\left( u_1 - \alpha u_1\right)}
\sum_{A_1', \ldots, A_k'}\left( g_{A_1' \ldots A_k'}, G_{A_1' \ldots A_k'}\right)_{\varphi}= \sum_{A_1', \ldots, A_k'}\left( g_{A_1' \ldots A_k'},G_{(A_1' \ldots A_k')}\right)_{\varphi}
\int_{\Omega}\left( a(x)\left\lvert du \right\rvert^{p}+ \langle f ; u \rangle \right) = \int_{\Omega}\left( a(x)\left\lvert du \right\rvert^{p}- \langle F ; du \rangle \right) + \int_{\partial\Omega}\langle F ; \nu \wedge u_{0}\rangle .
\sum_{n=1}^{\infty}\frac{(q)_{n-1}z^nq^n}{(1-d q^n) (zq)_n}=z\sum_{n=1}^{\infty}\frac{(zq/d)_{n-1}}{(zq)_n}\frac{d^{n-1}q^n}{1-zq^{n}}
\|1_A\| \ = \ \|z-1_{B\backslash E}- P_E(z)\| \ \leqslant\ \mathbf C_\ell \|z-P_E(z)\|\ \leqslant\ \mathbf C_\ell \mathbf C_{\lambda, dkp}\sigma^{\mathcal{U}_X, L}_m(z) \ \leqslant\ \mathbf C_\ell \mathbf C_{\lambda, dkp}\|1_B\|.
N_1(m_{n-1})+\sum_{i=2}^{C_1(m_{n-2})}\,N_1(v_i(m_{n-2}))+\sum_{i=2}^{C_1(m_{n-3})}\,\sum_{j=1}^{C_1(v_i(m_{n-3}))}\,N_1(v_{i,j}(m_{n-3})),
\sin \phi_3 = \frac{4A}{2u_1u_2}\cos \phi_3 = \frac{u_1^2+u_2^2-u_3^2}{2u_1u_2}1 + \cos \phi_3 = \frac{(u_1+u_2)^2 - u_3^2}{2u_1u_2}
F_1(t) = \int_{t_0}^{t_{mid}}S(t-s) \mathbb P{\rm div}[ \mathbf{1}_{\mathcal F_0}(w_a - \ell_a) \otimes w_b ]{\rm d}s F_2(t) = \int_{t_{mid}}^{t}S(t-s) \mathbb P[w_a \cdot \nabla w_b]{\rm d}s F_3(t) = \int_{t_{mid}}^{t}S(t-s) \mathbb P[\ell_{a}\cdot \nabla w_b]{\rm d}s.
c = 3 (q_1^2 + q_2), - 5 q_4 = q_2^2 + 4 q_1 q_3, q_5 = q_2 q_3, 35 q_6 = 2 q_2^3 - 12 q_1 q_2 q_3 - 5 q_3^2, &5 q_7 = 2 q_3 (q_2^2 - q_1 q_3), - 525 q_8 = 9 q_2^4 + 72 q_1 q_2^2 q_3 - 56 q_1^2 q_3^2 + 100 q_2 q_3^2,
k(2s) =2^{-d+2}\mu(4s^2)s^{-d+2}\ge c_0 2^{-d}\phi'(4^{-1}s^{-2}) s^{-d-2}&\ge c_0 2^{-d}\phi'(s^{-2}) s^{-d-2}\ge c_0 2^{-d}(1-2e^{-1}) k(s).
\gamma_i^{\pm}(u) = \sum_{r=0}^{\infty}\gamma_{i, \pm r}^{\pm}u^{\pm r}= q_i^{\deg Q_i - \deg R_i}\frac{Q_i(uq_i^{-1})R_i(uq_i)}{Q_{i}(uq_i) R_{i}(uq_i^{-1})},
\ \sum_{\substack{a=1 (a,q)=1}}^qf(\frac{a}{q}+\beta)^3h(-\frac{a}{q}-\beta)e(-\frac{\overline{a}\,v}{q}) = \ \frac{1}{q^4}\mathcal{I}(\beta,q)\sum_{\substack{a=1 (a,q)=1}}^qS(q,a)^3e(-\frac{\overline{a}\,v}{q}) &\ \ \ \ \ +D_1+3D_2+3D_3+D_4+D_5+3D_6+3D_7.
T_{1}\ll\frac{1}{n^{3/2}}(Aap^{2})\sigma^{2}+\frac{p}{n^{2}}\sum_{j=1}^{i}\Vert X_{ij}^{2}I(|X_{ij}|>A)\Vert_{1}+\frac{1}{n^{2}}k_{i}p^{2}\gamma_{aq}\,.
\left( \begin{array}{c}{\hat{a}_{i}( t )}\\{\hat{a}_{i}^{\dag}( t )}\\ \end{array}\right) = \sum_{j = 1}^{2}\left( \begin{array}{c c}{\alpha_{i j}( t )}&{\beta_{i j}( t )}\\{\beta_{i j}^{\ast}( t )}&{\alpha_{i j}^{\ast}( t )}\\ \end{array}\right) \left( \begin{array}{c}{\hat{a}_{j}}\\{\hat{a}_{j}^{\dag}}\\ \end{array}\right) ,
\tau_{{}_{\mathbb{HP}^n_q}}:=\mathcal{P}_{SU_q(2)}(S^{4n+3}_q)\Box^{\mathcal{O}(SU_q(2))}V^{\vee}, &\tau^*_{{}_{\mathbb{HP}^n_q}}:=\mathcal{P}_{SU_q(2)}(S^{4n+3}_q)\Box^{\mathcal{O}(SU_q(2))}V.
Z_r\left(gg^\top ,\frac{rs}{2}\right) = \zeta(rs) \sum_{v\in \Lambda}{|v\cdot g|^{-rs}}= 2\zeta(rs) \sum_{v\in \{\pm 1\}\backslash\Lambda}{|v\cdot g|^{-rs}}&= 2\zeta(rs)\sum_{\Psi(v)\in P\cap \Gamma\backslash \Gamma}\varphi_s^P(\Psi(v) g) = 2\zeta(rs)E^P_{s,\varphi}(g),
\xi_1^{(i)}(x) \equiv \langle \lambda_{max}^{(i)}|{\rm exp(}\frac{1}{2}\Phi) T(x),{}~~~~\xi_2^{(i)}(x) \equiv \langle \lambda_{max}^{(i)}|E_i{\rm exp(}\Psi_+){\rm exp(}\frac{1}{2}\Phi) T(x)
E(\hat{\mathcal{M}}(t))=\hat{r}_{1}t, \ \operatorname{Var}(\hat{\mathcal{M}}(t))=\hat{r}_{2}t,\ \operatorname{Cov}(\hat{\mathcal{M}}(s),\hat{\mathcal{M}}(t))=\hat{r}_{2}\min\{s,t\}.
\left(\omega_{\psi}\begin{pmatrix}1 -1 \end{pmatrix}\right)f(x) =\gamma(\psi)\hat f(x), \left(\omega_{\psi}\begin{pmatrix}1 b 1 \end{pmatrix}\right) f(x) =\psi(bx^2)f(x), b\in F \left(\omega_{\psi}\begin{pmatrix}a a^{-1}\end{pmatrix}\right) f(x) =|a|^{1/2}\frac{\gamma(\psi)}{\gamma(\psi_a)}f(ax), a\in F^\times,
J^{( N )}( n ; 1 , \ldots , 1 ) = \frac{1}{\Lambda^{( N )}}\left( \prod m_{i}^{2}\right) \sum_{i = 1}^{N}\frac{1}{m_{i}^{2}}F_{i}^{( N )}J_{i}^{( N )}( n ; 1 , \ldots , 1 ) .
\int_{E\setminus B_R}\int_{B_{R/2}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dy\,dx \ge c\,R^N\,\int_{E\setminus B_R}\frac{|u|^p}{|x|^{N+s\,p}}\,dx &-c\, \left(\int_{E\setminus B_R}\frac{1}{|x|^{N+s\,p}}\,dx\right)\, \int_{B_{R/2}}|u|^p\,dy.
\lim_{h\to 0}\int_0^t \left\langle f(\tau, \cdot), \frac{G_{\Lambda}^2(\tau+h, D_v) - G_{\Lambda}^2(\tau, D_v)}{2h}f(\tau+h,\cdot) \right\rangle \,\mathrm{d}\tau &= \frac{1}{2}\int_0^t \left\langle f(\tau, \cdot), \left(\partial_{\tau}G_{\Lambda}^2\right)(\tau, D_v) f(\tau,\cdot)\right\rangle \,\mathrm{d}\tau.
HP^\bullet(\mathbb{C}\Gamma)= \prod_{\hat{\gamma}\in \langle \Gamma \rangle'}\big( H^\bullet (N_\gamma, \mathbb{C}) \otimes HP^\bullet (\mathbb{C})\big) \times \prod_{\hat{\gamma}\in \langle \Gamma \rangle''}H^\bullet(N_\gamma, \mathbb{C}),
J_{\rm{ACKS}}[u]:=\int_\Omega|\nabla u|^2+{Area(\partial\Omega^+)}\ge \int_\Omega|\nabla u|^2+c_n \left[\mathcal L^n(\Omega^+)\right]^{1-\frac1n}\qquad\quad\ge \int_\Omega|\nabla u|^2+c_o\,\Phi_0\left(\mathcal L^n(\Omega^+)\right)=:J[u],
P\left(\sum^{n}_{i=1}x_{ij}>xV_{nj}\right)\leq P\left(\sum^{r}_{i=1}(A_{ij}+B_{ij})>xV_{nj}\right) \leq P\left(\sum^{r}_{i=1}A_{ij}>xV_{nj}/2\right)+ P\left(\sum^{r}_{i=1}B_{ij}>xV_{nj}/2\right) \leq P\left(\sum^{r}_{i=1}A_{ij}>xV_{1nj}/2\right)+ P\left(\sum^{r}_{i=1}B_{ij}>xV_{2nj}/2\right) \leq 2\exp(-x^2/8),
\mathbb{E}|V_N(z)|^2 = \mathbb{E}|(N/d_{N})(s_{N}(E+z/d_N)-\mathbb{E}s_{N}(E+z/d_N))|^{2}&\leq 2\mathbb{E}|(N/d_{N})(s_{N}(E+z/d_N)-s(E+z/d_N))|^{2}+ 2|(N/d_{N})\mathbb{E}(s_{N}(E+z/d_N)-s(E+z/d_N))|^{2}&\leq4\mathbb{E}|(N/d_{N})(s_{N}(E+z/d_N)-s(E+z/d_N))|^{2}
\varphi ( x ) = \frac{1}{\left( 2 \pi \right)^{3 / 2}}\int \frac{d^{3}k}{\sqrt{2 \omega}}\left[ a ( \vec{k}) \; e^{i k \cdot x}+ a^{+}( \vec{k}) \; e^{- i k \cdot x}\right]
\mathbb{E}\big[ \mathfrak{l}^{(p,p)}(z)\big]=\mathbb{E}\big[O_\prec(\hat{\Pi}^2)\mathfrak{l}^{(p-1,p)}(z)\big]+\mathbb{E}\big[O_\prec(\hat{\Pi}^4) \mathfrak{l}^{(p-2,p)}(z)\big]+\mathbb{E}\big[O_\prec(\hat{\Pi}^4) \mathfrak{l}^{(p-1,p-1)}(z)\big]\,.
\frac{\tilde{A}\left(W',Z',\overline{W}',\overline{Z}'\right) W'-\left(\overline{\tilde{A}\left(W',Z',\overline{W}',\overline{Z}'\right)\otimes W'}\right)^{t}}{2\sqrt{-1}}, \quad\quad\mbox{for all $i,j=1,2$,}
e T_{\mu \nu}={\tilde{\overline{{\Psi}}}_{L}}{\gamma_{\mu}}i \left( D_{\nu}- \frac{1}{2}\Gamma_{\alpha \nu}^{\alpha}\right){\tilde{\Psi}}_{L}- \frac{i}{2}g_{\mu \nu}\partial_{\alpha}({\tilde{\overline{{\Psi}}}}_{L}\gamma^{\alpha}{\tilde{\Psi}_{L}}) . \nonumber \,
\binom{t}{j}\binom{t+j}{j}=\frac{(t+j)!}{(t-j)!(j!)^2}<\sqrt{t+j}\frac{(t+j)^{t+j}}{(t-j)^{t-j}j^{2j}}\leq \sqrt{t+\frac{t}{\sqrt{2}}}\left(\frac{\big(1+\frac{1}{\sqrt{2}}\big)^{1+\frac{1}{\sqrt{2}}}}{\big(1-\frac{1}{\sqrt{2}}\big)^{1-\frac{1}{\sqrt{2}}}2^{-\frac{1}{2\sqrt{2}}}}\right)^t &=\sqrt{t+\frac{t}{\sqrt{2}}}L^t.
\sigma_{\Sigma,h}:\Psi_{\Sigma,\delta}(X;m,\mathcal{L}(\mathcal{B},\mathcal{H})) &\to S_{\Sigma,\delta}(T^\ast X;m,\mathcal{L}(\mathcal{B},\mathcal{H}))/S_{\Sigma,\delta}(T^\ast X;m\langle h^{-\delta}d(\Sigma,\cdot)\rangle^{-1},\mathcal{L}(\mathcal{B},\mathcal{H})).
\begin{array}{rcl}\hat{J}_\pm^\prime D^{J}_{MK}= \sqrt{[J\mp M][J\pm M-1]}D^{J}_{M\pm 1,K}~, begin{align*}3mm]\hat{J}_3^\prime D^{J}_{MK}= K D^{J}_{MK}~, ~~~~\hat{J}_z^\prime D^{J}_{MK}=M D^{J}_{MK}~.\end{array}
Z_g\big(s, \chi, S(\Delta_{\gamma_3}\bigcap(\mathbb{N}^2\setminus \{0\}))\big)= \dfrac{\tilde{G}_3(q^{-s})}{1-q^{-1-s}}\sum_{a=1}^{\infty}q^{-a(i_0+j_0+i_0j_0s)}= \dfrac{G_3(q^{-s})}{(1-q^{-1-s})(1-q^{-i_0-j_0-i_0j_0s})},
\int_{\vert y\vert \geq m\tau^{1/\beta}}\psi(T,x-y)dy\leq \int_{\vert z\vert \geq \frac m 2 \tau^{1/\beta}}\psi (T,z)dz=e^{-T}\sum_{k=1}^\infty \frac{T^{k}}{k!}\int_{\vert z\vert \geq \frac m 2 \tau^{1/\beta}}J^{*(k)}(z)dz.
I [ A , \lambda ] = \int_{{\cal M}^{2 l - 1}}B ( A , \lambda ) = \int_{{\cal M}^{2 l - 1}}\sum_{\alpha = 0}^{\infty}\lambda^{N}B_{N}( A ) = \sum_{\alpha = 0}^{\infty}\lambda^{N}I_{N}[ A ] \ .
V^{a}\, = \,{\frac{1}{2}}\epsilon^{a b}\, \Big ( \phi_{A}^{\ast \, b}{\frac{\partial^{r}}{\partial{\bar{\phi}}^{A}}}- ( - 1 )^{\epsilon_{A}}\pi_{A}^{b}{\frac{\partial^{r}}{\partial \phi^{A}}}\Big ) \, \, .
\sum_{n=-\infty}^{\infty}\!q^{n(n-1)/2}\left[z^{n}A_{q}\!\left(zq^{n+m}\right)\!A_{q}\!\left(zq^{n-m-1}\right)-z^{1-n}A_{q}\!\left(z^{-1}q^{n+m}\right)\!A_{q}\!\left(z^{-1}q^{n-m-1}\right)\right]=0,
\frac{1}{g_{h}^{2}}= \frac{1}{g_{h}^{2}}+ \frac{b_{0}}{8 \pi^{2}}\mathrm{l n}\frac{M}{M^{\prime}}- \sum_{i}\frac{t_{2}( i )}{8 \pi^{2}}\mathrm{l n}Z_{i}( M , M^{\prime}) .
\left( U ( g ) f \right) ( z ) = \epsilon ( g ; z ) ( \alpha + \overline{{\beta}}\overline{{z}})^{- \frac{1}{2}}( \overline{{\alpha}}+ \beta z )^{- \frac{1}{2}}f \left( \frac{\alpha z + \overline{{\beta}}}{\beta z + \overline{{\alpha}}}\right)
\| T_{\mu}x -f\|^{2}= \langle T_{\mu}x -f , x_{2}^{*}\rangle= \mu_{t}\langle T_{t}x -f , x_{2}^{*}\rangle \leq \sup_{t}\|T_{t}x -f\|\|T_{\mu}x -f\| < \|x -f\| \|T_{\mu}x -f\|,
2\lambda_\varepsilon ||\xi^\perp||(f_4||\xi^\perp||-\varepsilon f_3)g(\phi X,Y)= g((\nabla_XA)\xi^\perp,Y)-g((\nabla_YA)\xi^\perp,X) = -2c||\xi^\perp||(||\xi^\perp||-\varepsilon)g(\phi X,Y)
\psi' \left( \prod_{i=1}^{n_1}((c_{1i}, \ldots,c_{ki}),2) \prod_{i=0}^{n_2}((c_{1i}',\ldots,c_{ki}'),-1) \prod_{i=1}^{n_3}((c_{1i}'',\ldots,c_{ki}''),2)\right) = \sum_{i=1}^{n_1}(c_{1i}z_1^{i-n_1-n_2-1},\ldots,c_{ki}z_1^{i-n_1-n_2-1})+\sum_{i=0}^{n_2}(c_{1i}'z_1^{i-n_2},\ldots,c_{ki}'z_k^{i-n_2})+\sum_{i=1}^{n_3}(c_{1i}''z_1^{i},\ldots,c_{ki}''z_k^{i}).
p(t,x,y)=\int_{0}^{y}f(t,x, y-z) \beta_3 e^{-\alpha_3(t)-\beta_3 z}\sqrt{\frac{\alpha_3(t)}{\beta_3 z}}I_1(2\sqrt{\alpha_3(t)\beta_3 z})dz + e^{-\alpha_3(t)}f(t,x,y)
S_{eff}=NS_{0}(\sigma^{0})+S_{1}(\sigma^{0})-{i\over 2}C_{2}+{1\over 2}D_{2}+\sum_{n=1}^{\infty}{1\over\sqrt{N}^{n}}\Big[i(-1)^{n+1}[{C_{n+2}\over (n+2)}+mL^{d}\delta^{d}(0){C_{n}\over n}]+{1\over (n+2)!}D_{n+2}\Big]
(u+1)z^2\sum_{k\in \mathbb{Z}/d\mathbb{Z}}x_k + (u+2)z\sum_{k\in \mathbb{Z}/d\mathbb{Z}}E^{(k)}+ \sum_{k \in \mathbb{Z}/d\mathbb{Z}}{\rm tr}(e_1^{(k)}e_2)=0.
\tilde{p}^{\prime 1}= \sum_{a = 2}^{n - 1}\tilde{p}^{\prime a}= \tilde{p}^{1}+{\frac{2}{( D - 2 )}}\tilde{p}^{n - 1}+{\frac{l}{2 \, ( D - 2 )}}\tilde{p}\, ,
\frac{m_{n}}{c_{n+1}}\leq \frac{3h_{n}}{h_{n+1}/(n+1)^{1 + \epsilon}}\leq \frac{3h_{n}(n+1)^{1 + \epsilon}}{r_{n}h_{n}}= \frac{3(n+1)^{1+\epsilon}}{r_{n}}\to 0,
Hom^{G}(\mathcal{H}(n+1),Poly_{\mathbb{R}}(S^{2})_{\leq n})\simeq{\displaystyle\prod\limits_{l=0}^{n}}Hom^{G}(\mathcal{H}(\psi_{l}),Poly(\psi_{l}))\simeq\mathbb{R}^{n+1}
\nabla f = \dfrac{\partial f}{\partial z}\begin{pmatrix}1 i\end{pmatrix}+ \dfrac{\partial f}{\partial \bar z}\begin{pmatrix}1 -i\end{pmatrix},\div\begin{pmatrix}f i f\end{pmatrix}=2\dfrac{\partial f}{\partial \bar z},\div\begin{pmatrix}f -i f\end{pmatrix}=2\dfrac{\partial f}{\partial z}\,,
f\ast g:= f\cdot g+\sum^{\infty}_{n=1}{(\lambda/2)^n\over n!}\epsilon^{r_{1}s_{1}}\dots\epsilon^{r_{n}s_{n}}{\partial^n f\over\partial\phi^{r_{1}}\dots\partial\phi^{r_{n}}}{\partial^n g\over\partial\phi^{s_{1}}\dots\partial\phi^{s_{n}}},
{\cal L}^{\beta,\mu}_{\rm eff}(E) ={\cal L}^{\beta,\mu}_{\rm eff}(0)- \frac{(eE)^2}{24\pi^2}\frac{|\mu - eA_{0}|}{\sqrt{(\mu - eA_{0})^2 - m^2}}\,\theta \left( |\mu - eA_{0}| - m \right) \ ,
\tilde{\sigma}_\varepsilon := \inf \{t>0 | dist (v_t ,M)> \varepsilon^{\kappa '}\or \ \|v_t\|_{L^\infty}>2C_0 \or \v_t \notin H^\alpha +m \}&\bar{\sigma}_\varepsilon := \inf \{t>0 | dist (w_t ,M)> \varepsilon^{\kappa '}\or \ \|w_t\|_{L^\infty}>2C_0 \or \w_t \notin H^\alpha +m \}
\int f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m})Y_{n_{k_l}}\mathrm{d}\mathbb{P}_{n_{k_l}}= \int f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m})\mathrm{d}\mathbb{Q}_{n_{k_l}}\to \int f(Z_1',\ldots,Z_{m}') \mathrm{d}Q.
\int k\psi^{k-1}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\nabla f\nabla\psi +n(k-2)\int\frac{1}{f}\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}&\qquad+(k-1)\int\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\Delta f=0.
-(3 - \nu^{2})\dot{a}_{12}^{*}+ (2 - \nu^{2})\dot{a}_{11}^{*}a_{22}^{*}+:a_{11}a_{11}^{*}a_{11}^{*}a_{22}^{*}: + \nu\varphi_{1}^{'}a_{11}^{*}a_{22}^{*};
\tilde{e}'''_k:=\left[\begin{array}{c}0 0 \delta H^+_k\cdot (S_{\theta_k}H^+_k-H^+_{k+\frac{1}{2}})-\delta H^-_k\cdot (S_{\theta_k}H^-_k-H^-_{k+\frac{1}{2}})\end{array}\right]\,.
\begin{array}{l}e^{-i\chi_s}[-\partial_r+s(n-\Phi^{(0)}+\Upsilon)r^{-1}]f_n(r)=(E+m)g_n(r), e^{i\chi_s}[\partial_r+s(n-\Phi^{(0)}+\Upsilon+s)r^{-1}]g_n(r)=(E-m)f_n(r).\end{array}
\frac{q}{S_*}\|\nabla f(x_N)\|^2 + \sum_{i=0}^{N-1}\frac{p_i}{S_*}\min\{\|\nabla f(x_i)\|^2 \,,\, \|\nabla f(x_{i+1})\|^2\}{}\leq{}\frac{f(x_0)-f_*}{S_*}.
\begin{array}{l c c c c}{\mathrm{f o r}1}&{:}&{1 ,}&{1 ,}&{1}\\{\mathrm{f o r}\epsilon}&{:}&{1 ,}&{1 ,}&{- 1}\\{\mathrm{f o r}\rho}&{:}&{2 ,}&{- 1 ,}&{0 .}\\ \end{array}
\Phi(t,\omega)=\left[\begin{array}{cccc}\Phi_{11}(t,\omega) 0 \cdots 0 \Phi_{21}(t,\omega) \Phi_{22}(t,\omega) \cdots 0 \vdots \vdots \ddots \vdots \Phi_{n1}(t,\omega) \Phi_{n2}(t,\omega) \cdots \Phi_{nn}(t,\omega)\end{array}\right]
2\,\sum_{i<j}^n \bigg(\frac{m_i+m_j}{m_i m_j}\bigg) \rho_{ij}p^2_{\rho_{ij}}+ \,\sum_{i \neq j,i\neq k, j< k}^n\,\frac{2}{m_i}(\rho_{ij}+ \rho_{ik}- \rho_{jk})p_{\rho_{ij}}p_{\rho_{ik}}+V(\rho_{ij})\ =\E\ ,
S = - \frac{1}{2}\int d^{5}x \frac{g_{5}^{2}}{4 \pi^{2}R^{2}}( \phi^{\dagger}\phi )^{2}= - \frac{1}{4}\int d^{4}x \, g_{4}^{2}( \phi^{\dagger}\phi )^{2}~ ,
{\cal L}={\cal L}_{1}+{\cal L}_{2}+{\cal L}_{3}+ \frac{g^{' 2}}{\kappa^{4}}- \sqrt{3}\frac{g^{\prime}}{\kappa}\bar{\chi}\tau^{2}\chi - \frac{\sqrt{3}g^{\prime}}{4 \kappa}\bar{\psi}_{a}\tau^{2}\gamma^{a b}\psi_{b}