harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(-\frac{1}{2}[\begin{matrix}1&0\\ 0&1\end{matrix}]+\frac{\sqrt{3}}{2}[\begin{matrix}0&1\\ -1&0\end{matrix}])^{3}=(-\frac{1}{2}+i\frac{\sqrt{3}}{2})^{3}=(e^{\frac{2\pi i}{3}})^{3}=e^{2\pi i}=1=[\begin{matrix}1&0\\ 0&1\end{matrix}]
\int_X|Ric(\tilde\omega_i)+\tilde\omega_i|^2_{\tilde\omega_i}\tilde\omega_i^n =\int_X|3n\mu_{\alpha_i}\omega_{i}|^2_{\tilde\omega_i}\tilde\omega_i^n &\le 9n^3\int_X\tilde\omega_i^n &\to 9n^3(2\pi)^n\int_X c_1(K_X)^n &=0
\binom{k+(i+j)p}{m+ip}\equiv_{p^2}\binom{i+j}{j}\left[\binom{k}{m}+j\sum_{l=1}^{m}\binom{p}{l}\binom{k}{m-l}+i\sum_{l=1}^{k-m}\binom{p}{l}\binom{k}{k-m-l}\right] &=\binom{i+j}{j}\left[(1-j-i)\binom{k}{m}+j\binom{p+k}{m}+i\binom{p+k}{k-m}\right] &\equiv_{p^2}\binom{i+j}{i}\binom{k}{m}(1+p((i+j)H_k-jH_{k-m}-iH_m)).
\int_{-\infty}^{\infty}erf(ax+b)\frac{1}{\sqrt{2\pi\sigma^{2}}}exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}})dx=erf\frac{a\mu+b}{\sqrt{1+2a^{2}\sigma^{2}}},a,b,\mu,\sigma\in\mathbb{R}
G_1(z) =\sum_{n\ge 0}\P\{Y_n=0\}z^n=S\left(\frac{p_1 z-(p_1 z)^h}{1-p_1 z}, \ldots, \frac{p_r z-(p_r z)^h}{1-p_r z}\right) =\frac1{\displaystyle 1-\sum_{i=1}^r\frac{\frac{p_i z-(p_i z)^{h}}{1-p_i z}}{1+\frac{p_i z-(p_i z)^{h}}{1-p_i z}}}=\frac1{1-\displaystyle \sum_{i=1}^r\frac{p_i z-(p_i z)^{h}}{1-(p_i z)^{h}}}.
\Psi = ( 1 -{\frac{\rho}{4}}x^{+}x^{-})^{\frac{1}{2}}\left( \begin{array}{c}{{e^{-{\frac{\theta}{2}}}e^{\imath v}p ( x^{-})}}\\{{e^{{\frac{\theta}{2}}}e^{\imath u}f ( x^{+})}}\\ \end{array}\right)
\widetilde{\mathbf{f}}(t)=\frac{1}{L}\Big(e^{-2O_t^{(1)}}\|f^{(1)}(0)\|^2,e^{-2O_t^{(2)}}\|f^{(2)}(0)\|^2,\dots,e^{-2O_t^{(N)}}\|f^{(N)}(0)\|^2\Big)^{\top},t\in\mathbb{R}
-\frac{\partial L_{d}D_{\alpha\beta}(p\|q)}{\partial q_{j}}=\underbrace{\frac{Z_{B}}{\alpha}p^{\alpha}_{j}q^{\beta-2}_{j}}_{U_{j}}-\underbrace{\frac{Z_{A}}{\alpha}q^{\alpha+\beta-2}_{j}}_{V_{j}}
\langle E_0 | A_0^{(1)}\cdots A_0^{(n)}| E_0 \rangle =\langle \hat E_0 | \hat A_0^{(1)}\cdots \hat A_0^{(n)}| \hat E_0 \rangle\, \langle \check A_0^{(1)}\cdots \check A_0^{(n)}\rangle_G,
M_{( k , m )}= \left( \begin{array}{c c}{{0}}&{{{\cal I}_{2}}}\\{{{\cal I}_{2}}}&{{0}}\\ \end{array}\right) , \qquad M_{( 0 , 0 )}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) .
\int_{\theta_0\leq |\theta|\leq \pi}|\varphi_{t\sigma^2}(\theta)|\, d\theta=2\int_{\theta_0}^{\pi}e^{-t \sigma^2 \theta^2/2}\, d\theta = \frac{2}{\sigma \sqrt{t}}\int_{\theta_0\sigma \sqrt{t}}^{\pi \sigma \sqrt{t}}e^{-\theta^2/2}\, d\theta \leq \frac{2}{\sigma\sqrt{t}}\frac{1}{\theta_0\sigma \sqrt{t}}= \frac{2}{\theta_0\sigma^2}\frac{1}{t}.
r_{\ell}\leq e^{50\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|, 1)}\max(r_{\ell-1}, r_{\ell}, r_{\ell+1})\times\begin{cases}\max(|\ell|, e^{\delta_n q_n}), \beta_n\geq \delta_n+200\varepsilon, e^{\beta_n q_n}, \beta_n< \delta_n+200\varepsilon\end{cases}.
V^{( 0 )}= \frac{1}{4}\, \pi_{i j}\pi^{i j}+ \partial_{i}A_{0 j}\pi^{i j}- \frac{1}{4}\, A_{i j}\nabla^{2}A^{i j}+ \frac{1}{2}\, A_{i j}\, \partial^{j}\partial_{k}A^{i k}
\xi_{n,m}^{({\rm I})}\left(\tfrac{m+1}{2}\right)=\frac{\xi\left(\tfrac{n-m}{2}\right)}{\xi\left(\tfrac{n+m+2}{2}\right)}\, \frac{\prod_{\substack{1\leq j\leq [\frac{n}{2}] j\ne\frac{n-m}{2}}}\xi(n-m-2j)}{\prod_{j=1}^{[\frac{n}{2}]}\xi(n+m+2-2j)}.
L_{n_1,\ldots,n_r}^{\alpha_1,\ldots,\alpha_r}(x_1,\ldots,x_r)=L_{n_1}^{\alpha_1}(x_1)\cdot L_{n_2}^{\alpha_2}(x_2) \cdot \ldots \cdot L_{n_r}^{\alpha_r}(x_r)
\operatorname*{l i m}_{\vec{p}\rightarrow 0}\frac{C_{3}^{*}\left( \left| \vec{p}\right| \right)}{4 p_{0}V}\sim \operatorname*{l i m}_{\vec{p}\rightarrow 0}\frac{C_{3}^{*}\left( \left| \vec{p}\right| \right) \left| \vec{p}\right|^{3}}{4 p_{0}}= 0 .
\mathcal A_N(\omega_0,\omega_1,\rho_1) = \begin{pmatrix}-2N+2-2N\frac{\omega_1}{\omega_0}-\frac{2\rho_1}{\omega_0}-2 -2 -2N-4+\frac2{\omega_0}-(2N+2)\frac{\omega_1}{\omega_0}\end{pmatrix}.
k_{y}= \sum_{g\in T_y}c_+(gy,g^{-1})c_+(g,y)c_+(g,g^{-1})^{-1}u_{gyg^{-1}}= u_{(1,2,3,4)}+ u_{(1,4,3,2)}+ u_{(1,3,4,2)}+u_{(1,3,2,4)}-u_{(1,4,2,3)}-u_{(1,2,4,3)}.
A(t,k,r) =\sum_{j=0}^{\min(k,t)}(-1)^{j}\frac{k!}{j!(k-j)!}\frac{(j+t)!}{j!t!}\frac{(r-j-1)!}{(t-j)!(r-t-1)!}&=\frac{(r-1)!}{t!(r-t-1)!}\sum_{j=0}^t(-1)^{j}\binom{t}{j}\binom{t+j}{j}\frac{k(k-1)...(k-j+1)}{(r-1)(r-2)...(r-j)}.
B_1(\widehat{\phi},\widehat{w}_h)=\int_{\widehat{K}}\widehat{\phi}\frac{\partial (\widehat{w}_h-\widehat{I}_{\widehat{K}}\widehat{w}_h)}{\partial \xi_1}\;\mathrm{d}\xi_1\mathrm{d}\xi_2,
\overline{Z}_{m,m'}(1) = \overline{\sum_{y}\sqrt{W(y|X_1(m)) W(y|X_{1}(m'))}}= \sum_{y}\bigg[\overline{\sqrt{W(y|X_1(m))}}\bigg]\,\bigg[\overline{\sqrt{W(y|X_1(m'))}}\bigg] = \sum_{y}\bigg[ \sum_{x}Q(x)\sqrt{W(y|x)}\bigg]^2.
S_{x}^{\alpha}\colon \B\to B,S_{x}^{\alpha}(1)=0, S_{x}^{\alpha}(x_{k})=x_{k+\alpha}, \hphantom{S_{x}^{\alpha}\colon}\S_{x}^{\alpha}(y_{k})=y_{k},y\ne x , \alpha\in\mathbb{Z}.
V_{J}(t)=-t^{-\frac{3}{2}}+2t^{-\frac{1}{2}}-4t^{\frac{1}{2}}+6t^{\frac{3}{2}}-6t^{\frac{5}{2}}+5t^{\frac{7}{2}}-6t^{\frac{9}{2}}+3t^{\frac{11}{2}}-2t^{\frac{13}{2}}+t^{\frac{15}{2}}.
\gamma_{k}^{\langle n\rangle}= \frac{\left(\sum_{m\in \mathbb{M}_k}\sqrt{\eta_{mk}}\left|\hat{g}_{mk,n}\right|^2 \right)^2}{\sum_{m\in \mathbb{M}_k}\eta_{mk}(\beta_{mk}-\alpha_{mk}) \alpha_{mk}+\frac{1}{\gamma_t}}.
g_i^{-1}(t) = (f_i)^{-1}_{D_i \setminus S \times [0,1]}(t) \cup \left( (f_i)^{-1}_{S \times \{0\}}(t) \times [0,t] \right) \cup \left( (f_i)^{-1}_{S \times \{0\}}([0,t]) \times \{t \}\right)
\begin{aligned}\mathbb{P}\left(R_j=a\right) =\mathbb{P}\left(\varsigma_{Q_{j},m}=1\right)\cdot\prod_{i\in [a]}\mathbb{P}\left(\varsigma_{i+Q_{j-1},m}=0\right) &=\left(1-\epsilon_m\right)\epsilon^a_m.\end{aligned}
s_n(u)=\frac{(u_ne_{\bar z},e_{\bar z})}{(e_{\bar z},e_{\bar z})}=\frac{(u_ne_{\bar z})(z)}{e_{\bar z}(z)}=\frac{u_n(1+z\bar z)^{n-1}}{(1+z\bar z)^{n-1}}.
\nabla \Pi = \sqrt{- \gamma}\Pi{}^{\prime}= -{\frac{1}{\sqrt{- \gamma}}}\left[ \left({\frac{L_{2}}{\kappa_{1}}}\right)^{\prime}\eta_{2}+{\frac{L_{2}\kappa_{2}}{\kappa_{1}}}\eta_{1}-{\frac{L_{2}\kappa_{3}}{\kappa_{1}}}\eta_{3}\right] .
\begin{array}{c}{l i m}\\{\varepsilon \rightarrow 0}\\ \end{array}\frac{\partial}{\partial \theta}I ( p , m , \theta , \varepsilon ) - \frac{\partial}{\partial \theta}\begin{array}{c}{l i m}\\{\varepsilon \rightarrow 0}\\ \end{array}I ( p , m , \theta , \varepsilon ) \mid_{_{\theta_{0}^{\pm}}}\neq 0
g'(x) = \frac{\frac{1}{2}f'(x)/ \sqrt{1- f(x)}}{1 + \sqrt{1 - f(x - \frac{1}{m})}}+ \frac{(1 -\sqrt{1 - f(x)}) \frac{1}{2}f'(x-\frac{1}{m}) \big/ \sqrt{1- f(x - \frac{1}{m})}}{\left(1 + \sqrt{1 - f(x - \frac{1}{m})}\right)^2}.
S_q={\pi\over 3}T_H(L-r_+)-{1\over 12}({r_-\over r_+})^2\ln ({L-r_-\over r_+ -r_-})+{1\over 12}\ln ({L-r_+\over r_+ -r_-})+{1\over 6}\ln{r_+ \over z_0}~~,
(q^{s+m}-1) P_f(q^{-s})_{s=-m}= \lim_{N\rightarrow \infty}1 + \sum_{\alpha = 0}^N q^{-\alpha(n-m)}\left( q^{m-n}B_{f,\alpha+1}- B_{f,\alpha}\right) &= \lim_{N \rightarrow \infty}q^{-(N+1)(n-m)}B_{f,N+1}= B_f.
D_t^\alpha = D_t^{\lceil{\alpha}\rceil}{_0I_t^{\lceil{\alpha}\rceil -\alpha}}\Longrightarrow \bigl(D_t^\alpha\bigr)^\dagger ={_tI_\beta^{\lceil{\alpha}\rceil -\alpha}}\bigl(-D_t\bigr)^{\lceil{\alpha}\rceil}={_t^C\!D^\alpha},
\triangle_{[0]}=x\frac{d^2}{dx^2}+\frac{5}{2}\frac{d}{dx},\triangle_{[1]}=X\frac{d^2}{dX^2}+\frac{N}{2}\frac{d}{dX}\quad\mbox{with}X=1-x, & \dot{D}_{[0]}=\sqrt{x}\frac{d}{dx},\dot{D}_{[1]}=\sqrt{X}\frac{d}{dX}\mbox{with}X=1-x, &\|u\|_{[0]}=\Big(\int_0^1|u(x)|^2x^{3/2}dx\Big)^{1/2}, &\|u\|_{[1]}=\Big(\int_0^1|u(x)|^2X^{N/2-1}dX\Big)^{1/2}.
B^{ab,c}=\frac{3}{2}A^{abc}{}' E^{abc}=\frac{1}{12}\left(2D^{a,bc}+2D^{b,ca}+2D^{c,ab}-3A^{abc}{}'\right) F^{a,bc}=\frac{1}{4}\left(2C^{b,ca}+2C^{c,ab}+2D^{a,bc}{}'-3A^{abc}{}''\right) G^{abc}=\frac{1}{12}\left(2C^{a,bc}{}'+2C^{b,ca}{}'+2C^{c,ab}{}'-3A^{abc}{}'''\right).
{\left\{{2 ( d + 1 ) \psi{z_{1}}{z_{2}}\cdots{z_{d}}{z_{d + 1}}{z_{d + 2}}+ 2 \phi{z_{d + 1}^{d + 1}}{z_{d + 2}^{d + 1}}= 0}\right\}/ G}\, \, \, .
S_{\chi}\,=\,\int_{t^{\prime}}^{t^{\prime\prime}}\,dt\left(p_i\dot{q}^i+\mu_a\dot{\lambda}^a+i\zeta_a\dot{\eta}^a+i\zeta^{\times}{\!}_a\dot{\eta}^{\times a}\,-H\,-i\{\Omega,\phi\}\right).
\left|A+A\right| = \sum_{i=2}^{2t}i^{k}=2^{k+1}\sum_{i=1}^{t}i^k-\sum_{i=1}^{t}\left[(2i)^k-\left(2i-1\right)^k\right]-1 &\leq 2^{k+1}|A|-\frac12 (2t)^{k}=\left(2^{k+1}-(1+o(1))2^{k}\frac{k+1}{2t}\right)|A|.
w^{\varepsilon}(x,t) \ = \u(x - \frac{b}{\varepsilon}\, t,t) + \varepsilon \varkappa_1 (\frac{x}{\varepsilon})\cdot \nabla u(x - \frac{b}{\varepsilon}\, t,t) + \varepsilon^2 \varkappa_2 (\frac{x}{\varepsilon})\cdot \nabla \nabla u(x - \frac{b}{\varepsilon}\, t,t),
\tilde{u}(0) - \tilde{y}(0) = i \Phi h, \tilde{u}^{\prime}(0) - \tilde{y}^{\prime}(0) = \Phi \Phi^* \tilde{u}(0) -\Phi A_1 h = \Phi \Phi^* \tilde{y}(0) - \Phi A_1^* h.
\Phi_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\underset{v_{p}\in S_{p}}{}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p}.
\|u\|_{L_{t\in I}^{p-1}L_x^\infty}^{p-1}\leq \int_I A(t)^{p-1}dt=\int_I A(t)^\frac{m}{n}dt &\lesssim\int_I \|d_N\|_{\ell_N^2}^2\|d_N'\|_{\ell_N^2}^{p-3}dt\leq \|d_N\|_{L_{t\in I}^2\ell_N^2}^2\|d_N'\|_{L_{t\in I}^\infty\ell_N^2}^{p-3}&\leq \|u\|_{\tilde{S}_{2,\frac{2d}{d-2}}^{s_c}(I)}^{2}\|u\|_{\tilde{S}_{\infty,2}^{s_c}(I)}^{p-3}.
\sum_{i=1}^d\sum_{\ell=1}^{c-1}{\ell-1\choose i-1}{n-\ell\choose j-i}+\sum_{i=1}^{d-1}\sum_{\ell=1}^{c-1}{\ell-1\choose i-1}{n-\ell\choose (j-1)-i}+{n\choose j-1}= \sum_{i=1}^{d-1}\sum_{\ell=1}^{c-1}{\ell-1\choose i-1}\left[{n-\ell\choose j-i}+{n-\ell\choose j-i-1}\right]+\sum_{\ell=1}^{c-1}{\ell-1\choose d-1}{n-\ell\choose j-d}+{n\choose j-1},
\psi^c\left(s,\theta^{(1)}_1, \ldots, \theta^{(1)}_m, \theta^{(2)}_1, \ldots, \theta^{(2)}_n\right) = \left(c_{11}s \theta^{(1)}_1 \theta^{(2)}_1, \ldots, c_{ij}s \theta^{(1)}_i \theta^{(2)}_j, \ldots, c_{mn}s \theta^{(1)}_m \theta^{(2)}_n\right)
\sum_{x\in \xi+\ell\mathbb{Z}^4}w(h^{-1}x)e_k(a F_h(x)) H\big(h^{-1}k,h^{-2}F_h(x)\big) = \sum_{{\tiny b\,\hbox{\tiny mod}\, k}}\sum_{{\tiny \begin{tabular}{c}$z\,\hbox{\tiny mod}\, k\ell$ $z=\xi(\ell), z=b(k)$\end{tabular}}}e_k(a F_h(b)) \left(\sum_{y\in \mathbb{Z}^4}f(y) \right),
\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\smallint\limits_{T^*\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi'),
\displaystyle\mathcal F(h,h)=M^{-1}\left[\mathcal Q(Mh,Mh)+\mathcal Q(Mh,Mh)\right] begin{align*}2pt] \displaystyle\qquad=\int_{\mathbb R^d\times\mathbb S^{d-1}}\phi(|v-v_{\ast}|)b(\cos\theta)M_{\ast}(h_{\ast}^{\prime}h^{\prime}-h_{\ast}h)\, dv_{\ast}d\sigma\,.
u_k(x) = U_k(x/2) = \sum_{0 \leq j \leq \lfloor k/2 \rfloor}(-1)^j\binom{k-j}{j}x^{k-2j}, t_k(x) = 2T_k(x/2) = \sum_{0 \leq j \leq \lfloor k/2 \rfloor}(-1)^j\frac{k}{k-j}\binom{k-j}{j}x^{k-2j},
J_x^\vee = \{y \in M \,|\, (y-x)^2 \geq 0,\;(y^0-x^0) \geq 0 \}J_x^\wedge = \{y \in M \,|\, (y-x)^2 \geq 0, \;(y^0-x^0) \leq 0 \}\:.
U_tU =\frac{3(c^2-a^2)+6(ab+cd)\varepsilon_i}{8\varepsilon_i^3}t^2+\frac{ab-cd}{2\varepsilon_i}t -\frac{3(c^2-a^2)+2(ab+cd)\varepsilon_i}{8\varepsilon_i}, U_t^2+U_{tt}U =\frac{3(c^2-a^2)+6(ab+cd)\varepsilon_i}{4\varepsilon_i^3}t+\frac{ab-cd}{2\varepsilon_i},
1^{2}+2^{2}+\cdot\cdot\cdot+n^{2}=\frac{1}{3}(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1})=\frac{1}{3}(n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n).
W[{\psi}] = \int_{-\infty}^{\infty}dx^{+}\int_0^L dx^{-}{\psi}_{+}^{\star}i{\partial}^{-}{\psi}_{+}- \frac{i}{2}m \int_{-\infty}^{\infty}dx^{+}\int_0^L dx^{-}\int_0^L dy^{-}{\psi}_{+}^{\star}(x^{-},x^{+})\epsilon(x^{-}- y^{-}){\psi}_{+}(y^{-},x^{+})
\left[1-\left(u^2/2i_0\right)\right]^{-6/25}= 1+\frac{6}{25}\cdot \frac{u^2}{2i_0}+\frac{6}{25}\cdot \frac{u^4}{2(2i_0)^2}+\frac{1}{2}\cdot\frac{6}{25}\cdot \frac{u^4}{(2i_0)^2}+o(u^6).
\nabla_{P(p,q,r)}(z) =\nabla_{P(1,q,r)}(z)+\frac{(p-1)(q+r)}{4}z &= 1+ \frac{(q+1)(r+1)}{4}z + \frac{(p-1)(q+r)}{4}z =1+\frac{pq+qr+rp+1}{4}z.
/ \hspace{- 3 m m}D^{2}= \eta^{m n}D_{m}D_{n}+ i \sigma^{m n}( D_{m n}^{p}D_{p}+ R_{m n}^{p q}\sigma_{p q}+ F_{m n}^{a}) + \gamma^{m}( D_{m}\gamma^{n}) D_{n}\,
\Phi( (a,b), (c,d) ) := r_{a,b}^2+r_{c,d}^2-\|Z_{a,b}\|^2-\|Z_{c,d}\|^2 (a,b), (c,d) \in E^2, & \gamma_1(a) :=G(a) , \gamma_2(a) :=G(a)+ M (x-a) a \in E.
\begin{cases}\frac{\partial u(x,t)}{\partial t}=dD^{\alpha}_{\theta}u(x,t)-e\frac{\partial u(x,t)}{\partial x}+s(x,t), \ \ \ \t>0, \ \ 0< x <1, u(0,t)=0, \u(1,t)=0 , \ \ \ \t>0, u(x,0)=x(1-x), \ 0\leqslant x \leqslant 1,\end{cases}
v_{k}^{\prime \prime}+ \left( k^{2}- \frac{a^{\prime \prime}}{a}\right) v_{k}= 0 \; \; , \; \; \frac{a^{\prime \prime}}{a}= \frac{\gamma^{2}}{\left( 1 + \gamma^{2}\, \eta^{2}\right)^{2}}\, \, .
\nabla\Delta([0,1,2]\times[0,1,2]) = \nabla([0]\otimes[0,1,2]+[0,1]\otimes[1,2]+[0,1,2]\otimes[2]) = [0,0,0]\times[0,1,2]-[0,0,1]\times[1,2,2] \qquad+[0,1,1]\times[1,1,2]+[0,1,2]\times[2,2,2]
D_{\mu,n}(\varphi f) \leqslant \frac{C_n}{n!(n-1)!}\sum\limits_{j=0}^n \|\varphi^{(n-j)}\|^2_{\infty}\int_{\mathbb D}|f^{(j)}(z)|^2P_{\mu}(z)(1-|z|^2)^{n-1}dA(z).
J_i(\tau_i)=\mathbb{E}\Bigg\{\bar{\theta}_2\int_0^{\tau_i}e^{-\beta t}f(x_i(t))dt-e^{-\beta\tau_i}K\Bigg\}=\bar{\theta}_2\mathbb{E}\Bigg\{\int_0^{\tau_i}e^{-\beta t}f(x_i(t))dt-e^{-\beta\tau_i}\bar{K}_2\Bigg\}.
A_{3,1}: = \sup_{t > 0}\bigg( \int_t^{\infty}[\Psi_1]^{\frac{q}{p}}(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}\Psi[v;s]^{- \frac{1}{s}}(t), A_{3,2}: = \sup_{t > 0}W^{\frac{1}{q}}(t) \bigg( \int_t^{\infty}\bigg( \frac{\Psi_1(\tau)}{\Psi[v;s](\tau)}\bigg)^{\frac{s}{s-p}}\psi[v;s](\tau)\,d\tau \bigg)^{\frac{s - p}{p s}};
\varepsilon^i:= \partial_t\psi_i-u_{1,i}\vert_{x_1=0}+(\partial_t\varphi^a-u_1^a\vert_{x_1=0}+u_2^a\vert_{x_1=0}\partial_2\varphi^a)+(u^a_2+u_{2,i})\vert_{x_1=0}\partial_2\psi_i &+u_{2,i}\vert_{x_1=0}\partial_2\varphi^a=\mathcal B({\mathbf V}_i\vert_{x_1=0},\psi_i)_1\,.
\frac{\operatorname{s i n h}({\theta_{j}+ i{\frac{\eta}{2}}})}{\operatorname{s i n h}({\theta_{j}- i{\frac{\eta}{2}}})}= - \prod_{k = 1}^{m}\frac{\operatorname{s i n h}({\theta_{j}- \theta_{k}+ i \eta})}{\operatorname{s i n h}({\theta_{j}- \theta_{k}- i \eta})},
G^{{\mu}{\nu}{\sigma}}(B)=-{\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}B_{{\eta}{\delta}]}\equiv-{\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}F_{{\rho}{\eta}{\delta}}(B),
\langle P [ \{\Phi^{( I )}\}] \rangle = \frac{1}{Z}\sum_{n = 0}^{\infty}\frac{( - 1 )^{n}}{n !}\langle P [ \{\Phi^{( I )}\}] S_{i n t}[ \{\Phi^{( I )}\}]^{n}\rangle_{Q}.
{\bf H}_{1,r}{\bf V}_{1,2}^p={\bf H}_{2,r}{\bf V}_{2,1}^p\triangleq{\bf B}_1, &{\bf H}_{1,r}{\bf V}_{1,3}^p={\bf H}_{3,r}{\bf V}_{3,1}^p\triangleq{\bf B}_2, &{\bf H}_{2,r}{\bf V}_{2,3}^p={\bf H}_{3,r}{\bf V}_{3,2}^p\triangleq{\bf B}_3, &{\bf H}_{1,r}{\bf V}_{1,2}^{c}+{\bf H}_{2,r}{\bf V}_{2,3}^{c}={\bf H}_{3,r}{\bf V}_{3,1}^{c},
T_1(s,f) = \sum_{n=2}^{\infty}\frac{\lambda_f(n)c(n)}{n^s}e^{-n/X}= \sum_{M^{1-\Upsilon}<n\leq X(\log q)^2}\frac{\lambda_f(n)c(n)}{n^s}e^{-n/X}+ \mathcal{O}\Big(q^{-B}\Big).
2\binom{n}{2}\exp\biggl[- \Omega\Bigl( \frac{n^{2\gamma-2}g_r^2(0)}{\eta_{b}N_{b}}\Bigr)\biggr]+2\binom{n}{2}\exp\biggl[- \Omega\Bigl( \frac{n^{2\gamma-2}g_r^2(0)}{\eta_{\ell}N_{\ell}}\Bigr)\biggr].
\| ( A + \lambda I)^{1/2}(\hat A + \lambda I)^{-1/2}\|^2 = \| ( A + \lambda I)^{1/2}(\hat A + \lambda I)^{-1}(A+\lambda)^{1/2}\| & = \left\| \left[ I - (A+\lambda I)^{-1/2}(A -\hat{A}) (A+\lambda I)^{-1/2}\right]^{-1}\right\| & \leq \frac{1}{1-\left\| (A+\lambda I)^{-1/2}(A -\hat{A}) (A+\lambda I)^{-1/2}\right\|}.
\frac{d}{d t}\left\langle D \right\rangle_{\scriptstyle \! \Psi_{\mathrm{\!{\scriptscriptstyle ( g s )}}}}\stackrel{( a \rightarrow 0 )}{=}E_{\mathrm{( g s )}}\, \left\{1 + O \left( \left[ \kappa a \right]^{2}\ln [ \kappa a ] \right) \right\}\; .
\boldsymbol{v}_k = (\boldsymbol{v}_k^c,\boldsymbol{v}_k^s)^T, \boldsymbol{v}_k^{\perp}= (-\boldsymbol{v}_k^s,\boldsymbol{v}_k^c)^T, \textbf{curl}\, \boldsymbol{v}_k = (\textbf{curl}\, \boldsymbol{v}_k^c, \textbf{curl}\, \boldsymbol{v}_k^s)^T.
\varphi_\nu(x) = \frac{\sqrt x}{\sqrt{2\pi}}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\int_0^\infty e^{-ux/2}u^{\beta-1}du = \frac{2^{\beta}\Gamma(\alpha+\beta)}{\sqrt{2\pi}\Gamma(\alpha)}x^{1/2-\beta}& = \frac{x^{\delta/2-1}}{2^{\delta/2}\Gamma(\delta/2)}.
{\mathcal H}_{{\mbox{\boldmath $ \scriptstyle{\xi}$}}}= \mu^{-2}H_{\mbox{\boldmath $\scriptstyle{\xi}$}/\mu}=- \nabla_{\mbox{\boldmath $\scriptstyle{\xi}$}}^2- \lambda \,{\mathcal W}( \mbox{\boldmath $ \xi$})\; ,
\sum_{j,k=1}^mh(\tau(\varphi),B_{\varphi}(e_j,e_k))^2 =h\big(\tau(\varphi),\sum_{k=1}^mR^h(\tau(\varphi),d\varphi(e_k))d\varphi(e_k)\big) &=\sum_{k=1}^m h\big(\tau(\varphi),R^h(\tau(\varphi),d\varphi(e_k))d\varphi(e_k)\big) &\leq 0.
\begin{array}{l l l}{\psi}&{=}&{\exp{\phi}}\\{\varphi}&{=}&{{\displaystyle \frac{d \phi}{d z}}}\\ \end{array}\begin{array}{r r c}{\phi}&{=}&{\phi ( z , \overline{{z}})}\\{\overline{{\varphi}}}&{=}&{{\displaystyle \frac{d{\phi}}{d \overline{{z}}}}}\\ \end{array}
2^{m+1}\sum_{j \geq m+1}L_{j-m-1}^{(m+1)}(2jt)(e^{-t}y)^j = \frac{(K_{2t}(y))^2-1}{t(K_{2t}(y))^2 + (2-t)}\left[K_{2t}(y)-1\right]^{m}.
\int_{\tau}^{t}s|\sqrt{\rho}u_{tt}(s)|^2_2s+t|\nabla u_t(t)|^2_{2}\leq \tau | u_t(\tau)|^2_{D^1_0}+C\int_{\tau}^{t}s(1+|\nabla u_t|^2_2)|\nabla u_t|^2_{2}s+C
S(g)(\psi,\Psi)(z,\zeta) =([\gamma_{\psi(z)}\circ \rho_L(g^{-1})], \phi_g(z)\zeta)=(\psi(T_g(z)),\phi_g(z)\zeta) &=(\psi,\Psi)(T_g(z),\phi_g(z)\zeta)=(\psi,\Psi)\tau(g)(z,\zeta).
{\cal Z}= \int_{p e r i o d i c}D \Phi D \Phi^{\ast}\exp \left\{- \int_{0}^{\beta}d \tau \int d^{N}x \ \Phi^{\ast}[ ( \partial_{0}+ A_{0}+ i \mu )^{2}- \partial_{i}^{2}+ M^{2}] \Phi \right\}
\frac{\partial ~\rho_{n}\left( \underline{\theta}\right)}{\partial~\theta_{m}}=\rho_{n}\left( \underline{\theta}\right) ~\left\{\delta_{nm}~\sum_{\ell =1,~\ell \neq n}^{N}\left[ \cot \left( \theta_{n}-\theta_{\ell}\right) \right] -\left( 1-\delta_{nm}\right) ~\cot \left( \theta_{n}-\theta_{m}\right) \right\}~.
f(x,\phi,\theta) = \phi_1 \phi_2 \theta_3 x_1^2 + \phi_1 \phi_3 \theta_2 x_1^2 + \phi_2 \phi_3 \theta_1 x_2^2 - \phi_1 \phi_2 \theta_3 x_2^2 - \phi_1 \phi_3 \theta_2 x_3^2 - \phi_2 \phi_3 \theta_1 x_3^2,
\mathcal{R}= \sum\limits_{k = 1}^K \omega_k \mathcal{R}_{k,BD}= \sum\limits_{k = 1}^K\sum\limits_{i=1}^{d}\omega_k \log_2 (1 + \frac{\pmb{\Lambda}_k^2(i,i)}{\sigma_k^2}p_{k,i})
\left \| T(f) \right \|_{L_{\omega}^{p}}\leq C_{1}\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}+C_{2}\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}.
\ell_{\mu}\leq J_{\mu}(t_{\mu,w_{j}}w_{j}+z_{\mu,w_{j}})=J_{\mu_{j}}(t_{\mu,w_{j}}w_{j}+z_{\mu,w_{j}})+(\mu-\mu_{j})\left\vert t_{\mu,w_{j}}w_{j}+z_{\mu,w_{j}}\right\vert_{b,2}^{2}& \leq J_{\mu_{j}}(t_{\mu_{j},w_{j}}w_{j}+z_{\mu_{j},w_{j}})+o(1)=\ell_{\mu_{j}}+o(1)\leq\sup_{j\in\mathbb{N}}\ell_{\mu_{j}}+o(1)<\ell_{\mu}+o(1).
\bar{U}^{n+1}_j=\bar{U}^{n}_j-\lambda(\hat{F}(U_{j+\frac{1}{2}}^{-},U_{j+\frac{1}{2}}^{+})-\hat{F}(U_{j-\frac{1}{2}}^{-},U_{j-\frac{1}{2}}^{+}),
- \int_0^1 a^{b(\gamma)}\mathcal{D}_i \left[r \mathcal{R}_2[\tfrac{H}{r}] \right] \mathcal{D}_i H_\tau r^2 d^i \, dr = - \int_0^1 \bar{\mathcal{D}}_{i-1}\left[ D_r [ r \mathcal{R}_2[\tfrac{H}{r}] ] \right] \mathcal{D}_i H_\tau r^2 d^i \, dr.
\langle \psi^{ml}_{{\xi}}\vert \frac{{\bf{p}}^2}{2m}\vert \psi^{ml}_{{\xi}}\rangle =\frac{2 \sqrt{\beta}}{\pi}\int_{-\infty}^{+\infty}\frac{dp}{(1+\beta p^2)^2}\frac{p^2}{2m}= \frac{1}{2m\beta}\quad .
(q-1)\sum_{k=1}^{n-1}q^{k}\min\{e,\lfloor\frac{k}{d}\rfloor\}=\sum_{k=1}^{n-1}\sum_{j=q^{k}}^{q^{k+1}-1}\min\{e,\lfloor\frac{\log_{q}j}{d}\rfloor\}& =\sum_{k=1}^{q^{n}-1}\min\{e,\lfloor\frac{\log_{q}k}{d}\rfloor\}.
g^{d}(x)= |P^d_0|x^0+\,\,|P^d_1|x^1+\,\,|P^d_2|x^2+\,\,|P^d_3|x^3\cdots +2xg^{d}(x)= \enspace +2|P^d_0|x^1+2|P^d_1|x^2+2|P^d_2|x^3\cdots +2x^2g^{d}(x)= \enspace \,\,\,+2|P^d_0|x^2+2|P^d_1|x^3\cdots \vdots
\mathbb{P}[K_A=i, K_B=j|\mathbf{F}=\mathbf{f}]=\frac{\mathbb{P}[X^n\in\mathcal{A}_i,Y^n\in\mathcal{B}_j]}{\mathbb{P}[X^n\in\mathcal{A},Y^n\in\mathcal{B}]}.
S_{+}(z) = S_{-}(z)\begin{pmatrix}0 s_{j}-s_{j}^{-1}0\end{pmatrix}, z \in (-x_{j},-x_{j-1}), \, j = 2,...,m+1, & S_{+}(z) = S_{-}(z)e^{\pi i \alpha \sigma_{3}}, z \in (-x_{1},0), & S_{+}(z) = S_{-}(z)\begin{pmatrix}1 0 s_{j}^{-1}e^{\pm \pi i \alpha}e^{-2\sqrt{r}g(z)}1\end{pmatrix}, z \in \gamma_{j,\pm}, \, j = 2,...,m+1,
\int_{\rm{0}}^\infty{{x^q}\exp \left({- px}\right){I_v}\left({a\sqrt x}\right)dx}= & \frac{{\Gamma \left({q \!+\! v/2 \!+\! 1}\right)}}{{\Gamma \left({v + 1}\right)}}\frac{{{{\left({a/2}\right)}^v}}}{{{p^{q \!+\! v/2 \!+\! 1}}}}{}_1{F_1}\left({q \!+\! \frac{v}{2}\!+\! 1;v \!+\! 1;\frac{{{a^2}}}{{4p}}}\right).
\widetilde{V}^{( n^{2})}(x;N,M,\lambda_{1}) =V( x;N,M) -2\big[ \log \big( \lambda_{1}+Q_{n}^{(N,M)}(z) \big) \big]^{\prime \prime}{}= V(x;N,M) +4\big( 1-z^{2}\big)^{1/2}\frac{d}{dz}\left( \frac{(1-z)^{N+1/2}(1+z)^{M+1/2}\big( P_{n}^{(N,M)}(z) \big)^{2}}{\lambda_{1}+Q_{n}^{(N,M)}(z)}\right) ,
\begin{cases}(-\Delta)^{\frac{n}{2}}h(x,y)=0, y\in \Omega, (-\Delta)^i h(x,y) =(-\Delta)^i \left(C_n \ln\left(\frac{1}{|x-y|}\right)\right),\,\, i=0, 1, \cdots, \frac{n-1}{2}, y\in \mathbb{R}^n\backslash \Omega.\end{cases}
TP_{\left( S_{n}-te\right)^{+}}e \leq TP_{S_{n}\geq te}e\leq\dfrac{g_{S_{n}}(s)}{s^{t}}\leq\dfrac{\exp t.\exp\left( -nf\right)}{s^{t}}& =\exp t.\left( \frac{n\left\Vert f\right\Vert_{e}}{t}\right)^{t}\exp\left( -nf\right) ,
\mathrm{Fitt}_{\Lambda}\left( \mathcal{S}\right) = \mathrm{char}_{\Lambda}\left( \mathcal{S}\right) & = \mathrm{char}_{\Lambda}\left( \mathrm{Sel}_0(\mathbb{Q}_\infty, E[p^\infty])^\vee \right) & = \mathrm{char}_{\Lambda}\left( \mathbb{H}^1_{\mathrm{glob}}(T) / \Lambda \mathbf{z}_{\mathrm{Kato}}\right)
\pi_g\tau_e =\sum_{h\in G}f_h(\tau_e) \tau_{gh}=\sum_{u\in G}f_{g^{-1}u}(\tau_e) \tau_{u}=\sum_{u\in G}f_{g^{-1}u}(\tau_{g^{-1}g}) \tau_{u}=\sum_{u\in G}f_{u}(\tau_g) \tau_{u}=\tau_g, \forall g \in G
dX_t(x)= \big[-\tfrac{\partial^4}{\partial x^4}\,X_t(x)-\tfrac{\partial^2}{\partial x^2}\,X_t(x)\big]\,dt + \kappa \,X_t(x) \,dW_t(x), X'_t(0) = X'_t(1) = X^{(3)}_t(0) = X^{(3)}_t(1) = 0,X_0(x)=\xi(x)
\langle \alpha^\vee, 2\rho\rangle = 2+\sum_{\substack{\alpha \neq \beta\in \Phi^+ s_\alpha(\beta)\in \Phi^-}}\langle\alpha^\vee, \beta\rangle\geq 2 + \#\{\beta\in \Phi^+\setminus\{\alpha\}\mid s_\alpha(\beta)\in \Phi^-\}= 1+\ell(s_\alpha).
-\beta \int_0^\delta (R-s)^{N-1}\{2\frac{v'}{s}-\frac{N-1}{R}\frac{v}{s}\}\:ds=-2\beta v(0) \int_0^\delta (R-s)^{N-1}\frac{\eta'}{s}\:ds +\beta v(0)\frac{(N-1)^2}{2}R^{N-3}\delta+\beta v(0)\frac{N-1}{R}\int_0^\delta (R-s)^{N-1}\frac{\eta}{s}\:ds+\eta_2
\left( \frac{\lambda^2}{\eta}\right)^{n/2}\left(1 + \ln( \eta^{-1}+ 1 )\right)^n \eta^{-{3/2}}\leq C \lambda^{n - (2-\epsilon) \frac{n}{2}- n \frac{\epsilon}{4}- (2 - \epsilon) \frac{3}{2}}= C \lambda^{\epsilon (\frac{n}{4}+\frac{3}{2}) -3}.
\hat D_{\mu}\hat\psi \begin{array}[t]{l}=\partial_{\mu}\hat\psi+i\hat B_\mu*\hat\psi =\partial_{\mu}\hat\psi-i\hat\psi*\hat B_\mu =\partial_{\mu}\hat\psi+i(\hat\psi*\hat B_\mu -\hat B_\mu*\hat\psi )\end{array}
\begin{aligned}\max\left\{\sup_{[0,T^*]}||\rho^{k+1}(t) - \rho^{k}(t)||_\infty,\, \sup_{[0,T^*]}||\mathbf{u}^{k+1}(t) - \mathbf{u}^{k}(t)||_\infty \right\}\leq \ \frac{1}{2}\max\left\{\sup_{[0,T^*]}||\rho^{k}(t) - \rho^{k-1}(t)||_\infty,\, \sup_{[0,T^*]}||\mathbf{u}^{k}(t) - \mathbf{u}^{k-1}(t)||_\infty \right\},\end{aligned}
\omega(\tau) = A u_\ell + B v_\ell \ + \left( \frac{u_\ell(\tau)}{2\ell}\int_{-1}^{\tau}v_{\ell}(\sigma)\frac{h(\sigma)}{\sigma^2 + \ell^2}\,d\sigma - \frac{v_\ell(\tau)}{2\ell}\int_{-1}^{\tau}u_{\ell}(\sigma)\frac{h(\sigma)}{\sigma^2 + \ell^2}\,d\sigma \right).
\begin{aligned}d(x_1,x_2) \leq d(x_1, \bar x_2)+ d(\bar x_2, x_2) & \leq d(x_1,\bar x_2)+ L d(f(\bar x_2),f(x_2)) & = d(x_1,\bar x_2)+ L d(f(\bar x_2),f(x_1)) & \leq (1+L^2) d(x_1, \bar x_2),\end{aligned}
\hat{Q}_8^{(iv)}(x)= 1680(-1152\pi^2+11520) x^4+840(2688\pi^3-26880\pi) x^3 +360(-2400\pi^4+29952\pi^2-57600) x^2+120(960\pi^5-23520\pi^3 +134400\pi) x-2880\pi^6+328320\pi^4-3110400\pi^2+2073600.
\langle (E_1^{\textbf{1},\alpha}E_1^{\mathbf{1}, \overline{\alpha_N}}) | W_S^N,f|B_d\rangle=\langle E_1^{\textbf{1},\alpha}E_1^{\mathbf{1}, \overline{\alpha_N}},f|B_d | W_S^N).