harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J ={\frac{i g^{2}f_{a c d}f_{b c d}}{( 2 \pi )^{6}}}{\frac{( 1^{-}2^{-}) ( \theta_{3 1})^{2}( \theta_{3 2})^{2}}{\hat{x}_{1 2}^{2}\check{x}_{3 1 ; 2}^{2}\check{x}_{3 2 ; 1}^{2}}}\; .
\sum_{i\in K}[[x,y],t_i] \otimes abv_i + \lambda[x,y] \otimes D(ab)+ \sum_{i\in I}\Big([y,\varphi_i(x)] \otimes aD(b)u_i - [x,\varphi_i(y)] \otimes D(a)bu_i\Big)= 0
\int_{0}^{1}ln^{2}\Gamma(x)dx=\frac{\gamma^{2}}{12}+\frac{\pi^{2}}{48}+\frac{1}{3}\gamma L_{1}+\frac{4}{3}L_{1}^{2}-(\gamma+2L_{1})\frac{\zeta^{\prime}(2)}{\pi^{2}}+\frac{\zeta^{\prime\prime}(2)}{2\pi^{2}},
\sum_{\substack{q\leq X^{1/2}(\log X)^{-B}(q,M)=1}}\;\max_{(a,qM)=1}\;\Big\lvert\psi(X;qM,a) - \frac{\psi(X)}{\varphi(qM)}\Big\rvert\ll\frac{X}{\varphi(M)}(\log X)^{-A},
\dot{x}^{\mu}= 1 / 2 [ \acute{\psi}_{+}^{\mu}( \tau + \sigma ) + \acute{\psi}_{-}^{\mu}( \tau - \sigma ) ] ; \; \dot{x}^{\mu}= 1 / 2 [ \acute{\psi}_{+}^{\mu}( \tau + \sigma ) - \acute{\psi}_{-}^{\mu}( \tau - \sigma ) ] ,
\frac{\partial\Sigma^{\check{r}}(\tau,\vec{\sigma}_o)}{\partial\tau}= -[I^{-1}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_o))]^{\check{r}}_{\;i}\,\frac{\partial\alpha^i(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_o))}{\partial\tau}= + \frac{J^{\check{r}}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_o))}{J^\tau(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_o))}.
\psi^{\dagger}\sigma_{j}\frac{\partial\psi}{\partial t}=i\frac{\mu}{\hbar}\psi^{\dagger}\sigma_{j}\sigma_{i}B_{i}\psi=i\frac{\mu}{\hbar}\psi^{\dagger}(I\delta_{ij}-i\sigma_{k}\epsilon_{ijk})B_{i}\psi=\frac{\mu}{\hbar}\psi^{\dagger}(iI\delta_{ij}+\sigma_{k}\epsilon_{ijk})B_{i}\psi
\sum_{i}\gamma_{i}c_{V,i}=\sum_{i}[-\frac{V}{\omega_{i}}\frac{\partial\omega_{i}}{\partial V}][k_{B}(\frac{\hbar\omega_{i}}{k_{B}T})^{2}\frac{exp(\frac{\hbar\omega_{i}}{k_{B}T})}{[exp(\frac{\hbar\omega_{i}}{k_{B}T})-1]^{2}}]
R m_{t}(H) \sim R(y_{t}H) \rightarrow 0, \; \; \; u_{B}\sim \frac{1}{2}+ \frac{1}{2}R(y_{t}H) \rightarrow \frac{1}{2}\; \; \; u_{F}\sim \frac{1}{2}R(y_{t}H) \rightarrow 0
k_{1}^{\pm}(u)e_{1}^{\pm}(u)k_{1}^{\pm}(v)e_{1}^{\pm}(v) =k_{1}^{\pm}(v)e_{1}^{\pm}(v)k_{1}^{\pm}(u)e_{1}^{\pm}(u), f_{1}^{\pm}(v)k_{1}^{\pm}(v)f_{1}^{\pm}(u)k_{1}^{\pm}(u) =f_{1}^{\pm}(u)k_{1}^{\pm}(u)f_{1}^{\pm}(v)k_{1}^{\pm}(v),
\sigma^*_2\sigma_2=S_{\widehat X_G}P^*_{M_G}B_{M_G}P_{M_G}S_{\widehat X_G}&=S_{\widehat X_G}P^*_{M_G}\Bigr(P_{M_G}\hat S_{\widehat X_G}(P^*_{M_G}P_{M_G})^{-1}P^*_{M_G}+F)P_{M_G}S_{\widehat X_G}&=S_{\widehat X_G}P^*_{M_G}P_{M_G}\hat S_{\widehat X_G}S_{\widehat X_G}+S_{\widehat X_G}\hat FS_{\widehat X_G},
\int_{0}^{T}\int_{{\mathbb R}^d}\int_{0}^{1}\Big|{\sum_{j=1}^\infty}r_{j}(z) 2^{j(s+\alpha/p)}\Delta_{j}T^{\alpha}f(t,x)\Big|^{p}dzdxdt &\ll \int_{0}^{T}\int_{{\mathbb R}^d}\int_{0}^{1}\Big|{\sum_{j=1}^\infty}r_{j}(z) 2^{j(s+\alpha/p)}\Delta_{j}T^{\alpha}\Delta_{j}f(t,x)\Big|^{p}dzdxdt.
T_{1n}= \left(\begin{smallmatrix}1 0 0 1\end{smallmatrix}\right)\otimes\left(\begin{smallmatrix}0 1 1 0\end{smallmatrix}\right)= \left(\begin{smallmatrix}0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0\end{smallmatrix}\right),T_{kn}= \left(\begin{smallmatrix}1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0\end{smallmatrix}\right),
\overline{(P)}:= \overline{P},\overline{\overline{P}}:= P, \overline{\top}:= \bot,\overline{\bot}:= \top, \overline{(A \wedge B)}:= (\overline{A}\vee \overline{B}),\overline{(A \vee B)}:= (\overline{A}\wedge \overline{B}), \overline{\Box A}:= \Diamond \overline{A},\overline{\Diamond A}:= \Box \overline{A}.
{\frac{\partial{\cal L}}{\partial t_{k}}}={\cal L}{\cal A}_{k}- \hat{{\cal A}}_{k}{\cal L}, \, \, \,{\frac{\partial \bar{\cal L}}{\partial t_{k}}}= \bar{\cal L}{\cal A}_{k}- \hat{{\cal A}}_{k}\bar{\cal L}\, \, \, \; .
\eta \rfloor \omega ={\frac{1}{p !}}\eta_{i}\omega_{j_{1}\ldots j_{p}}g^{i j_{1}}d x^{j_{2}}\cdots d x^{j_{p}}-{\frac{1}{p !}}\eta_{i}\omega_{j_{1}\ldots j_{p}}g^{i j_{2}}d x^{j_{1}}d x^{j_{3}}\cdots d x^{j_{p}}+ \cdots .
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{(E_\Phi^r)_t(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{(E_\Phi^r)_t(\mathbb{R}^n)}.
\eta_j^{\frac{\alpha}{2}}I'_j = P_j\sum_{X'_k\in\Phi'_j\setminus X_0}H_{k0}\|\eta_j^{-\frac{1}{2}}X'_k\|^{-\alpha}&\stackrel{d}{=}P_j\sum_{X_k\in\Phi_j\setminus X_0}H_{k0}\|X_k\|^{-\alpha}=I_j &\stackrel{d}{=}\eta_j^{\frac{\alpha}{2}}P_jI_0(1),
\omega(s_1, ... , s_{M}\mid t_1, ... , t_{N}) = \sum_{\substack{m_1, ... , m_{M}, \,n_1, ... , n_{N}>0 \sum_{j=1}^{M}m_j = \sum_{k=1}^{N}n_k}}\prod_{j=1}^{M}\frac{1}{{m_j}^{s_j}}\prod_{k=1}^{N}\; \frac{1}{{n_k}^{t_k}}
n^{3n}2^{-3n^2/2+9n/2-4}\frac{2^{n(n+1)}-1}{2^n-1}= 2^{3n\log_2(n) - n^2/2+11n/2-4}\frac{2^{-n^2-n}(2^{n^2+n}-1)}{2^n-1}& = 2^{3n\log_2(n)-n^2/2 +11n/2 -4}\frac{1-2^{-n^2-n}}{2^{n}-1}.
J\phi =X^\flat\wedge (JT)^\flat +(JX)^\flat\wedge T^\flat, &={e^u}\left(dy\wedge\left(\frac{1}{2}(\textbf{H}-1)d\theta+d\varphi\right) -dx\wedge \left(\frac{1}{2}(\textbf{H}+1)d\theta+d\varphi\right) \right).
\alpha^{2}- n^{2}= 2 n \int_{0}^{\infty}{\frac{1}{1 + \left({\frac{r}{r_{0}}}\right)^{2 n}}}{\frac{d a}{d r}}d r + 2 \int_{0}^{\infty}\left( 1 - h^{2}\right) r d r .
\bigg\|\max_{0\le k\le n-1}\big|V_{k+1}-V_{k}\big|\bigg\|_{L^{\gamma p}}^{\gamma}=\bigg\|\max_{1\le k\le n}\big|\mathbf{E}\big(\frac{m^2}{n\sigma^2}|f^{-1}\mathcal{M}\big)\circ f^{n-k}\big|\bigg\|_{L^{\gamma p}}^{\gamma}\le Cn^{-\left(\gamma-\frac{2\gamma}{p}\right)},
\|\Gamma\varphi(t)-y_{0}\|=\|\int_{t_{0}}^{t}f(s,\varphi(s))ds\|\le\int_{t_{0}}^{t^{\prime}}\|f(s,\varphi(s))\|ds\le\int_{t_{0}}^{t^{\prime}}Mds=M|t^{\prime}-t_{0}|\le Ma\le b
\begin{cases}-\mathcal{L}w+z^{2-1/s}w_{zz}=0, \hbox{in}~\mathcal{C}_{\Omega}, w=0, \hbox{on}~\partial_{L}\mathcal{C}_{\Omega}, -\underset{z\rightarrow0^{+}}{\lim}w_z=d_sf, \hbox{on}~\Omega.\end{cases}
\begin{aligned}C_1+C_2+C_3=0, &C_1 e^{m_1}+C_2 e^{m_2}+C_3 e^{m_3}=0, &C_1 m_1^2\left(1-e^{m_1}\right)+C_2 m_2^2\left(1-e^{m_2}\right)+C_3 m_3^2\left(1-e^{m_3}\right)=0.\end{aligned}
\zeta_{T}( \nu | \beta ) ={\frac{\varrho^{- 2 \nu}}{\pi i}}\int_{\mu}^{\infty}d \omega \int_{C_{+}}d z{\frac{d}{d z}}\ln \left( 1 - e^{i \beta z}\right) \breve{\Phi}( \omega ; z ) ( z^{2}+ \omega^{2})^{- \nu}.
X_{ac}X_{bc}= qX_{bc}X_{ac}, \ a>b, X_{ab}X_{ac}= q^{-1}X_{ac}X_{ab}, \ b>c, X_{ac}X_{bd}= X_{bd}X_{ac}, \ a>b, c>d, X_{ac}X_{bd}= X_{bd}X_{ac}+(q-q^{-1})X_{bc}X_{ad}, \ a>b,c<d.
V(s,t) = \frac{1}{8}\{e^{-{\sqrt{2}}\, (2s+{\sqrt{3}}t )}}\ \left(1-{e^{2{\sqrt{2}}\, s}}\right)^2 = \frac{1}{2}\e^{- \sqrt{6}t}\ \sinh^2 \,(\sqrt{2}\,s)\ .
u_f'(1-t,x)(u_h'(1-t,\lambda x+\mu y)+u_f'(1-t,x)) &\qquad\quad+u_f(1-t,x)(u_h''(1-t,\lambda x+\mu y)+u_f''(1-t,x)) =\mbox{}u_g'(1-t,y)(u_h'(1-t,\lambda x+\mu y)+u_g'(1-t,y)) &\qquad\quad+u_g(1-t,y)(u_h''(1-t,\lambda x+\mu y)+u_g''(1-t,y))
(T_2)_{ij}(r^{6-n})_{,ji}= [(n-2)\sigma_1(A)g_{ij}-8A_{ij}](6-n)[(4-n)r^{2-n}x^ix^j+r^{4-n}\delta_{ij}+O(r^{6-n})] = (6-n)[4(n-4)\sigma_1(A)r^{4-n}-8(4-n)A_{ij}x^ix^jr^{2-n}]+O(r^{7-n}).
e^{- 2 U ( r )}= 1 +{\frac{( p^{1})^{2}}{8 \pi r^{2}}}\log{\frac{c ( p^{1})^{2}}{r^{2}}}\pm . . \quad , \quad z^{1}= i{\frac{\sqrt{c}\, p^{1}}{r ( 1 \mp . . )}},
\int_Xu(t,x)dP_n =\sum_{\{i\in I\mid x^i\in U(t) \}}\alpha^iu(t,x^i)+\sum_{\{i\in I\mid x^i\in X\setminus U(t) \}}\alpha^iu(t,y^i_n) & >\sum_{i\in I}\alpha^iu(t,x^i)=\int_Xu(t,x)dQ.
(dG)_{11\bar{I}\bar{J}\bar{K}\bar{L}}=-4\sqrt{2}\pi\left(\frac{\kappa}{4\pi}\right)^{2/3}(J^{(1)}\delta(x^{11}) + J^{(2)}\delta(x^{11}-\pi\rho))_{\bar{I}\bar{J}\bar{K}\bar{L}},
\eta(\tau_{n-i+j}, \tau_n^{q-1}\tau_{n-i+\ell+1})=(\tau_n \tau_1)^{i-j}\tau_n^{q-1}\tau_{n-j+\ell+1}=(\tau_n \tau_1)^{i-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}.
0 = < < \left\{J_{i}D_{i}^{\alpha}+ i g K_{i}^{*}T_{i j}^{\alpha}\psi_{j}- i g \psi_{i}^{*}T_{i j}^{\alpha}K_{j}\right\}- i \varepsilon \frac{\delta}{\delta c^{\alpha}}\delta_{B R S}O - \delta_{B R S}Y^{\alpha}> >
\sum\limits_{n = 0}^{\infty}g_{o}(n,\alpha, k,p) - g_{e}(n,\alpha, k,p)q^{n}= \frac{(q^{k};q^{k})_{\infty}}{(q;q)_{\infty}}\sum_{n =0}^{\infty}\frac{q^{pn + \alpha}}{1 + q^{pn + \alpha}}.
2 8{\cal T}_{4}( p_{1}^{\prime}, p_{1}; p_{2}^{\prime}, p_{2}) = 2 i E \int \frac{d^{2}z}{( 2 \pi )^{3}}e^{i q \cdot z}{\cal T}_{2}( p_{2}^{\prime}, p_{2}| h^{C}) ,
\phi_{f}({\cal K}) ={\frac{1}{4 \pi}}\oint_{\cal K}d x^{\mu}\oint_{{\cal K}^{f}}d y^{\nu}\epsilon_{\mu \nu \rho}{\frac{( x - y )^{\rho}}{| x - y |^{3}}}= \mathrm{l k}({\cal K},{\cal K}^{f}) .
\sum_{n = - \infty}^{\infty}\int d \tau \, d^{d - 2}x^{i}\left( \epsilon^{2}\partial_{\tau}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantom x}+ \frac{2 i n}{R}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantom x}- \partial_{i}\phi_{n}^{*}\partial_{i}\phi_{n}^{\vphantom x}- M^{2}\phi^{*}\phi \right) \ .
H_{P}= \int ( \frac{8 \pi^{2}}{g^{2}}( \pi_{i}^{a}+ \frac{1}{8 \pi}\epsilon_{i j}A^{j , a})^{2}+ \frac{g^{2}}{6 4 \pi^{2}}G_{i j}^{a}G_{i j}^{a}) d^{2}x
\mu_{n}=\left(\begin{array}{ccccccccc}0 1 1 0 1 1 -h 1 0 1 1 -h 1 0 1 \ddots 1 1 -h 0 1 0 -h 1 1 \ddots 1 1 -h\end{array}\right)\in \mathrm{M}_{2n+1,2n+1}(\mathrm{GW}(k))
M_G\left( 1+ m\frac{x^4-1}{x-1}+ ym\frac{x^4-1}{x-1}\right)=8m+1, & M_G\left( 2+ k\frac{x^4-1}{x-1}+ yk\frac{x^4-1}{x-1}\right)=2^8(4k+1), M_G\left( (x^2+1)(x-1)+ k\frac{x^4-1}{x-1}+ y\left( (x+1)+k\frac{x^4-1}{x-1}\right)\right)=-2^8(4k+1), M_G\left( (x^2+1)+ k\frac{x^4-1}{x-1}+ y\left(-(x+1)+k\frac{x^4-1}{x-1}\right)\right)=2^8(2k).
L^2(M)_r \otimes_{hM'}Y\otimes_{hM'}L^2(M)_c &=\overline{(.\#(S^{(i)}))_*[(L^2(N_{\eta_i})_c+ L^2(N_{\eta_i})_r)]}^{L^2(M)_r\otimes_{h M'}(\ell^{\infty}(I_i)\overline{\otimes}B(L^2(M)))\otimes_{h M'}L^2(M)_c}
y_{ij}=g_{i1}g_{j2}-g_{i2}g_{j1}, \eta_{kn}=g_{k1}\gamma_{n2}-g_{k2}\gamma_{n1}&x_{55}=\gamma_{51}\gamma_{52}, x_{66}=\gamma_{61}\gamma_{62}x_{56}=\gamma_{51}\gamma_{62}+\gamma_{61}\gamma_{52}
p^{\underline{a}}={\frac{1}{e ( \tau )}}( \dot{x}^{\underline{a}}- i \dot{\bar{\theta}}\Gamma^{\underline{a}}\theta ) ={\frac{1}{e ( \tau )}}( \dot{x}^{\underline{a}}- i \tilde{\psi}\chi^{\underline{a}}) .
\left[{\frac{d n_{d}( \omega , D )}{d \omega}}\right]_{\mathrm{d i v}}= r_{d}{\frac{\Gamma \left( 1 - \frac D 2 \right)}{( 4 \pi )^{D / 2}}}{\frac{m^{D - 4}}{\kappa}}\int_{\Sigma}\left[ 2 \left( m^{2}+{\frac{R}{1 2}}-{\frac{R_{i i}}{8}}\right) -{\frac{\omega^{2}}{\kappa^{2}}}P \right] .
\Phi(\xi)=0~,\qquad\eta(\Phi(X))=0~, g(\Phi(X),\Phi(Y))=g(X,Y)-\epsilon \eta(X)\eta(Y)~, d\eta(X,Y)=-2\epsilon g(\Phi(X),Y)~, \imath_{\xi}d\eta=0~,\imath_{\xi}\eta=1~.
\widetilde{X}(a)=\tau_{(23)}^{-1}\Big(X_l'(a)\otimes X_l''\Big)=X_l'(a)\otimes_1X_l'', \widetilde{X}(a)=\tau_{(12)}^{-1}\Big(X_r'\otimes X_r''(a)\Big)=X_r'\otimes_1X_r''(a).
D^{(1)}_n = \begin{cases}72(n+2)(2n+1)(2n+3)(3n+4)(3n+5)(6n+11)(6n+13)f_0f_8^5 , 10(n+2)(2n+1)(5n+7)(5n+8)(5n+9)(5n+11)f_0f_7^4 , 5(n+2)(5n+6)(5n+7)(5n+8)(5n+9)f_7^4 .\end{cases}
{}N \langle f(A)x, x \rangle^p +{N}^p\langle f(A)x, x \rangle^p\Big\{\displaystyle\sum_{n= N+1}^{\infty}\frac{1}{n^p}\Big\}< N\Big(\frac{p}{p-1}\Big)^{p}\langle f(A)x, x \rangle^p.
E_{1}\cap B_{\bar{\rho}}((\bar{z},w)) = F_{\ell}\cap \{z< \bar{z}\}\cap B_{\bar{\rho}}((\bar{z},w)) & \subset \left\{(z',w') \in \mathbb{R}\times \mathbb{R}^{n-1}: z' < \bar{z}\ \mbox{and}\r_{\ell}^{\vee}(\bar{z})+ \frac{\delta}{2}< |w'| < r_{\ell}(z') \right\}\cap B_{\bar{\rho}}((\bar{z},w)).
f_{T_n}(e_{\psi_n}g_{1,\cdots, 1}(a_1,\cdots,a_n)e_{\psi_n})(_{1\leq j\leq n}(d_j)) =\frac{1}{|U_n|}\psi\left(\sum_{i=2}^n\sum_{k=1}^{i-1}\left(\frac{y_{i,k}}{y_{i-1,k}}-\frac{y_{i-1,k}}{y_{i,k+1}}\right)\right),
\begin{aligned}\int_{\mathcal{D}_N}\left\{\sum_k W_N \left( \psi_N^{t_k}Z_N\right) \right\}f_N (0,Z_N) dZ_N = & = \int_{\mathbb{R}}\int_{\partial \mathcal{D}_N}\left[W_N \left(Z_N \right)\right]^2 f_N (t,Z_N) d\sigma_N dV_N dt\end{aligned}
\Delta u_{k+1}= u_{k+1}-Z(k+1)u_k = P_{U^{(k+1)}_{-j}}\Delta v_{k+1}, \Delta e_{k+1}= e_{k+1}-Z(k+1)e_k=P_{E^{(k+1)}_{-j}}\Delta v_{k+1}, \Delta u_0=u_0=P_{U^{(0)}_{-j}}\Delta v_{0}, \ \Delta e_0 = e_0=P_{E^{(0)}_{-j}}\Delta v_{0}.
|\phi| = \sum_{i \in \overline{\phi}}|\phi_i| + \sum_{i \notin \overline{\phi}}|\phi_i| & = |\overline{\phi}|(\dim K +1) + (m- |\overline{\phi}|)(\dim L +1) &=(\dim M +1)(\dim K +1) + (m- \dim M -1) (\dim L +1)
(\xi_1', \eta_1') + (\xi_2', \eta_2') = (\xi', \eta'), \lambda_1' = \phi_{\tilde{c_1}}(\xi_1', \eta_1') \in S_1, \ \ \lambda_2' = \phi_{\tilde{c_2}}(\xi_2', \eta_2')\in S_2, \ lambda_3' = (\psi (\xi',\eta'), \xi',\eta') \in S_3.
\mu^{\prime}= \alpha^{2}\Biggl ( N K^{2}+ \frac{1}{2}N x^{2}H^{2}+ \frac{( K^{2}- 1 )^{2}}{2 x^{2}}+ H^{2}K^{2}+ \frac{\beta^{2}}{4}x^{2}( H^{2}- 1 )^{2}\Biggr ) \ ,
m_{i,j}=\begin{cases}3 \mbox{if}\;||X_{j}-X_{i}||\leq r_{\mathsf{f}}/2 2 \mbox{if}\;r_{\mathsf{f}}/2<||X_{j}-X_{i}||\leq r_{\mathsf{f}}1 \mbox{if}\;||X_{j}-X_{i}||>r_{\mathsf{f}}\end{cases}
\delta x^{\mu}= \imath \psi^{\mu}\epsilon \; , \; \; \delta e = \imath \chi \epsilon \; \; , \delta \psi^{\mu}= \frac{z^{\mu}}{2 e}\epsilon \; \; , \delta \psi^{2 n + 1}= \frac{m}{2}\epsilon , \; \; \delta \chi = \dot{\epsilon}\; , \; \; \delta \kappa = 0 \; ,
\tilde{\nabla}_j\tilde{\nabla}_i\Phi =-u^{-1}\tilde{\nabla}_j\tilde{\nabla}_iu + u^{-2}\tilde{\nabla}_j u \tilde{\nabla}_i u=-u^{-1}\tilde{\nabla}_j\tilde{\nabla}_iu + O_3(r^{-2\delta - 2}), \tilde{\nabla}_j\tilde{\nabla}_iu =\nabla_j\nabla_iu - \Delta\Gamma^k_{ji}\nabla_ku=\nabla_j\nabla_iu + O_3(r^{-2\delta-2}),
\Gamma_r^{ss}(k): =\{H^{\sigma}(k)\gamma H^{\sigma}(k) \in \Gamma_r(k):\gamma \textrm{isrelatively semisimple}\}\Gamma_{r\tau}^{ss}(k): =\{G^{\sigma}(k)\delta G^{\theta}(k) \in \Gamma_{r\tau}(k):\delta \textrm{is relatively}\tau\textrm{-semisimple}\}.
{\rm II}= \|u(Z_\circ + r\,\cdot\,) - p_{*,Z_\circ}(r\,\cdot\,)\|_{L^2(B_{1/2}(r^{-1}(X_\circ-Z_\circ)),|y|^a)}& \leq \|u(Z_\circ + r\,\cdot\,) - p_{*,Z_\circ}(r \,\cdot\,)\|_{L^2(B_{1},|y|^a)}\leq C_hr^\lambda.
\left| \exp\left( 2i\pi \eth (x w - z y) \right) - 1 \right| = 2|\sin(\eth\pi(x w - z y))|\leq 2\pi|\eth| |x w - y z| &\leq 2\pi |\eth| \|(x,y)\|_2 \|(z,w)\|_2\leq 2\pi k |\eth| \|(x,y)\| \|(z,w)\|_2
\mathcal{L}(\bar{X}^{n},\bar{K}^{n},\bar{V}^{n},\bar{B}^{n}) =\mathcal{L}\left( X^{n},K^{n},\left\updownarrow K^{n}\right\updownarrow ,B^{n}\right) \mathcal{L}(\bar{X},\bar{K},\bar{V},\bar{B}) =\mathcal{L}(X,K,V,B)
\omega_z(a[l]a[k]^*)=\sum_{|J|=l,|K|=k}z_{J}\overline{z_{K}}\,\omega_z(s_Js_K^*)=\delta_{l,k}\sum_{|J|=l=|K|}|z_{J}|^2 |z_{K}|^2=\delta_{l,k}
\frac{f(x^2)}{f(x^1)}\leq \frac{2L(f^1)^{-1}-2\mu_p(f^1)^{2\theta-2}}{2L(f^1)^{-1}+\mu_p(f^1)^{2\theta-2}}=\frac{2L-2\mu_p(f^1)^{2\theta-1}}{2L+\mu_p(f^1)^{2\theta-1}}.
\int \left( 1 6 f_{D 3}( \tau ) + 6 4 f_{0 3^{-}}( \tau ) \right) p_{1}= \int f_{H E T}( \tau ) p_{1}= \int \frac{6}{\pi}R e \left[ \, i \, \ln ( \eta ( \tau ) ) \right] p_{1}
\int^{\pi/2}_{0}{\left(1+\frac{a^2\alpha_j\bar{\gamma}_j\lambda}{Z_1(1+\frac{Z_2}{Z_1}\lambda)\mbox{sin}^2\theta}\right)}^{-1}{d\theta}= \frac{1}{2}\left(\!1\!-\!\sum^{N_j-1}_{k=0}\frac{(^{2k}_{k})(1+\frac{Z_2}{Z_1}\lambda)^k\sqrt(a^2\alpha_j\bar{\gamma}_j\lambda/Z_1)}{4^k(1+\frac{Z_2}{Z_1}(1+a^2\alpha_j\bar{\gamma}_j/Z_2)\lambda)^{k+1/2}}\right).
\begin{gathered}x^k\cdot B_1(k)-y^{k+1}\cdot B_2(k)=x^k(\underline{{\bf y^{k+1}}}+y^k+y^{k-1}+\cdots+x^{k-1}+x^{k-2}+x^{k-3}+\cdots) -y^{k+1}(y^k+y^{k-1}+y^{k-2}+\cdots+\underline{{\bf x^k}}+x^{k-1}+x^{k-2}+\cdots),\end{gathered}
\langle x|\frac{1}{E-\hat H_0\pm i\epsilon}\hat V|\psi^{(+)}\rangle= \left( \frac{-ime^{-i\pi/4}}{\sqrt{2\pi}\hbar^2}\right)\frac{e^{ikr}}{\sqrt{kr}}\int d^2x'\,e^{-i\vec k'.\vec x'}\langle x'|\hat V| \psi^{(+)}\rangle\;,
{\cal V}[\phi_0] = V[\phi_0,m(\mu_0),\lambda(\mu_0)] +{{\hbar}\over{64 \pi^2}}\left\{{\cal M}_0^4(\mu_0) - m_0^4\right\}\ln{\mu^2 \over \mu_0^2}+\hbar X[\phi_0,m(\mu),\lambda(\mu),\mu] +O(\hbar^2).
\ddot{F}^{kl,rs}(A)B_{kl}B_{rs}=~\frac{\partial^2F(\lambda(A))}{\partial\lambda_k\partial\lambda_l}B_{kk}B_{ll}+2\sum_{k<l}\frac{\frac{\partial F}{\partial \lambda_k}-\frac{\partial F}{\partial \lambda_l}}{\lambda_k-\lambda_l}B_{kl}^2.
{4n\choose 2j}-{2n\choose j}={2n\choose j}\left(\frac{(4n-1)(4n-3)\cdots (4n-2j+1)}{1\cdot 3\cdots (2j-1)}-1\right) & ={2n\choose j}\frac{(4n-1)(4n-3)\cdots (4n-2j+1)-1\cdot 3\cdots (2j-1)}{1\cdot 3\cdots (2j-1)}.
\tilde{N}_k \sim \mathcal{N}_{\mathbb{C}}(0,1) &\tilde{Z}_{k-1}\Big| \{|X_{k-1}|=|x_{k-1}|\}\sim \mathcal{N}_{\mathbb{C}}\left(0,\frac{1}{|x_{k-1}|^2 \Delta}\right).
\int_{D\backslash\{F=0\}}|F_1-(1-b_{t_0,B}(\Psi))fF^{1+\delta}|^2e^{-\varphi+v_{t_0,B}(\Psi)-\Psi}\le \left(\frac{1}{\delta}+1-e^{-t_0-B}\right)\int_D\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|f|^2e^{-\Psi}.
R_{\overline{H}_{l_0-1}^{\perp_{\bar{f}}}}=\dim(\overline{H}_{l_0-1}^{\perp_{\bar{f}}})-\dim(\overline{H}_{\bar{f}|_{\overline{H}_{l_0-1}^{\perp_{\bar{f}}}}}^{\perp_{\bar{f}}}) =R_f-(l_0-1)-\dim(\overline{H}_{l_0-1}^{\perp_{\bar{f}}}\bigcap (\overline{H}_{l_0-1}^{\perp_{\bar{f}}})^{\perp_{\bar{f}}}) =R_f-2(l_0-1)\geq 2.
\begin{aligned}\frac{1}{(\tau^k)^2}\|A\breve{x}^{k}-b\|^2= \|A\widetilde{x}^{k}-b\|^2+\frac{1}{(\tau^{k-1})^2}\|A\breve{x}^{k-1}-b\|^2 &+2\frac{1-\tau^k}{\tau^{k}}(A\widetilde{x}^{k}-b)^T(A\breve{x}^{k-1}-b).\end{aligned}
{\Gamma}_{\mu}= - \frac{n_{\mu}}{{\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2}}B - \frac{{\partial}_{\mu}}{{\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2}}\left( C - n^{2}x^{-}B - \frac{n^{2}n_{+}}{{\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2}}{\partial}_{+}B \right)
\log \mu_t^{(n)}(y) = e^{-\frac{\sigma^2}{2}t}\log \mu_0(y) &+ \int_0^t \frac{\sigma^2}{2}e^{-\frac{\sigma^2}{2}(t-s)}\left( \frac{2}{\sigma^2}\frac{\delta F}{\delta \mu}(\nu_s^{(n-1)}, \mu_s^{(n-1)}, y) + \log \rho(y) + \operatorname{D_{KL}}(\mu_s^{(n-1)}|\rho) \right) \mathrm{d}s.
S^{\chi, C}_{C_{\lambda,r_1}, C_{\mu,r_2}}\cdot e^{2\pi i \langle \lambda, \mu \rangle}\cdot \sqrt{k/2}= \begin{cases}s_{r,r'}\lambda,\mu \in 2L'/L, (-1)^rs_{r,r'}\lambda\in 2L'/L, \mu\not\in 2L'/L \mu\in 2L'/L, \lambda\not\in 2L'/L, (-1)^{r+r'}s_{r,r'}\lambda,\mu\not\in 2L'/L.\end{cases}
D_{n,j}=D_{n,j,j}^{*}= \langle D_n \left(\phi_{n,j}\right), \phi_{n,j}\rangle_H = \frac{1}{n-1}\displaystyle \sum_{i=0}^{n-2}X_{i,j,n}X_{i+1,j,n},j \geq 1,\n \geq 2,
\frac{d}{d T}\left[ T \left( L_{\mathrm{e f f}}( \omega ) + E + \frac{\omega}{2}\right) \right] = L_{\mathrm{e f f}}( \omega ) + E + \frac{\omega}{2}+ T \frac{d \omega}{d T}\left( \frac{d L_{\mathrm{e f f}}}{d \omega}+ \frac{1}{2}\right) = 0 \, .
L_\infty(s,{\rm sym}^mf\times{\rm sym}^mg)= \Gamma_{\mathbb R}(s)^{\delta_{2\mid m}}\Gamma_{\mathbb C}(s)^{[m/2]+\delta_{2\nmid m}}\prod_{1\le \nu\le m}\Gamma_{\mathbb C}\big(s+\nu (k-1)\big)^{m-\nu+1}
\operatorname{e x p}\left\{- \sum_{i , j = 1}^{4}c_{i j}\mathrm{\bf r}_{i}\cdot \mathrm{\bf r}_{j}- \sum_{i , j = 1}^{4}d_{i j}\hat{\mathrm{\bf k}}\cdot ( \mathrm{\bf r}_{i}\times \mathrm{\bf r}_{j}) \right\},
B_{ab}^kC_{ik}^m=A_{ia}^kA_{kb}^m+A_{ib}^kA_{ak}^m,\,A_{ib}^kA_{ak}^m=B_{ab}^kC_{ik}^m+A_{ai}^kA_{kb}^m,\, B_{ab}^kC_{ki}^m=A_{ak}^mA_{bi}^k-A_{bk}^mA_{ai}^k,
\int_0^1h^n_{n-i}(t)^2 dt =d^i_{n,n-i}\int_0^1t^{n-i}h^n_{n-i}(t)dt &=\frac{(-1)^n\sqrt{2n-2i+1}d^i_{n,n-i}}{(-n)_i}\sum_{k=0}^n\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k}\frac{\pi_i(k)}{n-i+k+1},
\sqrt{1-|\xi'(\beta(q))|^2_g}=\sqrt{1-|\xi'(q)|^2_g}+\O{}(1-|\xi'(q)|_g^2) l(q,\beta(q))=\frac{2}{\kappa(0)}\sqrt{1-|\xi'|_g^2}+\O{}(1-|\xi'|_g^2).
\sum_{n=1}^{\infty}\frac{q^n}{(zq^n;q)_{n+1}(zq^{2n+2};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n+1}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}, begin{align*}6pt]\sum_{n=0}^{\infty}q^n(-zq^{n+1};q)_{n}(-zq^{2n+2};q^2)_{\infty}=\sum_{n=0}^{\infty}\frac{z^n q^{n^2+n}}{(q;q^2)_{n+1}}.
{\cal L}~ = ~ -{\scriptstyle \frac{1}{2}}\, \partial_{\mu}\, \varphi \, \partial^{\mu}\, \varphi -{\scriptstyle \frac{1}{2}}m^{2}\varphi^{2}-{\scriptstyle \frac{1}{4}}\, \lambda \, \left( \partial_{\mu}\, \varphi \, \partial^{\mu}\, \varphi \right)^{2}
\begin{aligned}W(\xi_0,k+1,x,\mathsf{L}^{k+1})= E_{\xi_0}\biggl[\varphi\biggl(x+\sum_{i=1}^{k+1}Z_i^{\mathsf{L}^{k+1}}\biggr)\biggr] = E_{1-\xi_0}\biggl[\varphi\biggl(x+\sum_{i=1}^{k+1}Z_i^{\mathsf{R}^{k+1}}\biggr)\biggr] = W(1-\xi_0,k+1,x,\mathsf{R}^{k+1}).\end{aligned}
Z = \int \prod_x \{{\rm d}v_{\mu}\,{\rm d}j^{\mu}\,{\rm d}j^{\mu}_5 \,{\rm d}|h|^2 \,{\rm d}\alpha \, K[j^{\mu},j^{\mu}_5,|h|^2] \}\, e^{{\rm i}\int{\rm d}^4 x \,{\cal L}[v_{\mu},j^{\mu},j^{\mu}_5,|h|,\alpha]},
ch(Sym^t(\nu)) =(t+1) -\sigma_1{t+1 \choose 2}+\frac{\sigma_{1,1}}{2}\left({t+1 \choose 2}+2{t+1 \choose 3}\right) - \frac{\sigma_{2}}{2}{t+1 \choose 2}+ &+\frac{\sigma_{2,1}}{6}\left({t+1 \choose 2}+3{t+1 \choose 3}\right)+\frac{1}{12}\sigma_{2,2}{t+1 \choose 3},
H\big(\mathrm{Law}(t_{n+1}X_{n+1}+ (1-t_{n+1})\widetilde{X})\big) = H\big(\widetilde{Q}\circ (t_{n+1}T + (1-t_{n+1})\mathrm{Id})^{-1}\big) &\le t_{n+1}H(Q_{n+1}) + (1-t_{n+1})H(\widetilde{Q}).
\begin{gathered}\Delta_1 \nabla_a X' (x,s,t) = \nabla_a X' (X^{-1}(x,s), s) - \nabla_a X' (X^{-1}(x,s), t), \Delta_2 \nabla_a X' (x,s,t) = \nabla_a X' (X^{-1}(x,s), t) - \nabla_a X' (X^{-1}(x,t), t),\end{gathered}
\partial_2^*(\eth_1^*(a^*)) \circ b^* = \partial_1^*(\eth_2^*(a^*)) \circ b^* , \forall a^*, b^* \in A^*, \eth_2(\partial_1(a)) \cdot b = \eth_1(\partial_2(a)) \cdot b , \forall a, b \in A.
\varrho_1 = \alpha_1^\vee/2 - p \alpha_2^\vee + p \alpha_3^\vee, \varrho_2 = \alpha_1^\vee/2 + q \alpha_2^\vee - q \alpha_3^\vee, \varrho_3 = \alpha_2^\vee, \varrho_4 = \alpha_3^\vee.
(Z_1 \cdots Z_n)(I_{N, \xi}) = \sum^N_{\substack{i_1, \ldots, i_n = 1 j_1, \ldots, j_n = 1}}((Z_{i_1, j_1; 1}\circ \cdots\circ Z_{i_n, j_n; n})(\xi)) \otimes E_\chi((i_1, \ldots, i_n), (j_1, \ldots, j_n); N)
\int_{-\infty}^{\infty}\bar{u}_y(y)\left(\psi^-_{11}-1\right) e^{-2izy}\mathrm{d}y= - z \int_{-\infty}^{\infty}\bar{u}_y(y) e^{-2izy}K\psi^-_{11}\mathrm{d}y = - \frac{1}{2}z \widehat{\bar{u}_x K \psi^-_{11}}( z).
V'(r) = - \frac{1}{f_a^{n-1}(r)}\int_0^r a_0(t) f_a^{n-1}(t) dt < 0, V''(r) = (n-1) \frac{f_a^\prime(r)}{f_a^n(r)}\int_0^r a_0(t) f_a^{n-1}(t) dt - a_0(r), \noalign{and}\left|\nabla V\big(\rho(x)\big)\right| = \left| V'\big(\rho(x)\big) \nabla \rho(x)\right| = \left|V'\big(\rho(x)\big)\right|.
|\delta_{h}(k, \eta)|^2 = \frac{2^{4 \beta -1}}{\pi^3}( 2 \beta )^{- 2 \beta}\Gamma(\beta + 1/2)^2 \biggl(\frac{H_1}{M_{\rm P}}\biggr)^2 \biggl(\frac{k}{k_1}\biggr)^{2(1 - \beta)}\biggl[ 1 +\frac{\beta}{x_{0}}\sin{(2 x_0 + \beta \pi)}\biggr],
N =\frac{(k+4)!}{(k-2)!3!3!2}+\frac{(k+4)!}{(k-3)!3!2^3}+\frac{(k+4)!}{(k-4)!4!2^3}&=\frac{(k+4)(k+3)(k+2)(k+1)k(k-1)}{24^2}\left(3k^2-3k+2\right).
\Pr\{X(\tau +t)=k+n|X(\tau)=k \}= \begin{cases}\sum_{r=1}^{\infty}\frac{(-1)^{n+r}}{r!}\int_{[\tau,\tau +t]^r}\lambda^{\beta_r(s)}\binom{\beta_r(s)}{n}ds_1...ds_r, n \geq 1 e^{-\int_{\tau}^{\tau +t}\lambda^{\alpha (s)}ds}n=0,\end{cases}
\sum_{j = 0}^k{\frac{{j^2G_j}}{{2^j}}}= 58G_0 + 94G_1 - (k^2 + 12k + 58)\frac{{G_k}}{{2^k}}&- (k^2 + 10k + 47)\frac{{G_{k + 1}}}{{2^{k - 1}}}\,,
T_{NN}(L,m,d,z)=\frac{g}{2L}\frac{1}{(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}\int d^{d-1}p\frac{1}{(\vec{p}^{\,2}+(\frac{n\pi}{L})^{2}+m^{2})}+gf_{2}(L,m,d,z).
Q_{\delta_N}L_{\delta_N}u_N= L_{\delta_N}u_N-\sum_{j=1}^m\sum_{h=1}^2C_{j,h,N}\frac{\partial}{\partial z_{j,h}}(-\delta_N^2(K(z_{N,j})\nabla V_{\delta_N, Z_N,j})).
[\bar{f}_{0,k+3}, \bar{f}_{0,l}]-(1+d+d^{-1})[\bar{f}_{0,k+2}, \bar{f}_{0,l+1}]+ (1+d+d^{-1})[\bar{f}_{0,k+1}, \bar{f}_{0,l+2}]-[\bar{f}_{0,k}, \bar{f}_{0,l+3}]=0,
\sum_{w\in B_n/B_{n-1}}\!\!\!w\cdot \frac{\prod_{1\le i\le 2n-2}\vartheta(z_n-y_i) \prod_{1\le i\le n}\vartheta(Y+z_i) \prod_{1\le i<n}\vartheta(Y-z_i)}{\vartheta(z_n)\prod_{1\le i<n}\vartheta(z_n+z_i)\vartheta(z_n-z_i)}\hphantom{\sum_{w\in B_n/B_{n-1}}\!\!\!}{}= \frac{\vartheta(2Y)}{\vartheta(Y)}\prod_{1\le i\le 2n-2}\vartheta(Y-y_i).
\||f_{1}f_{2}|^{\frac{1}{2}}\|_{L^s(\mathbb{R}^{6})}\leq N^{\frac{M \log C_{s,\epsilon}}{s\log N}+ \frac{C'}{2^M}+\frac{(-6+{\frac{3s}{2}})M}{2^{M}s}+(\gamma_{lin}+\epsilon)(1-\frac{M}{2^M})}\prod_{i=1}^{2}\|f_{i}\|_{L^{s,N}(\mathbb{R}^{6})}^{\frac{1}{2}}.
- A_{\Delta}{\bf u}_{l,N}= \boldsymbol{\delta}_{l,N}:= T_\varepsilon(\sum\limits_{\nu=1}^N \boldsymbol{\delta}_{l,\nu})= - T_\varepsilon(A_{\Delta}{\bf P}_{R_L}), \boldsymbol{\delta}_{l,\nu}= -A_{\Delta}({z_\nu}\,{\cal W}_\nu \widetilde{\bf P}_{R_l}),
\log BF_{kl}=\log \frac{p^{n}(\mathbf{y}| \mathcal{F}_{n, k})}{p^{n}(\mathbf{y}| \mathcal{F}_{n, l})}+ \log \frac{\Pi_{n, k}(\mathcal{F}_{n, k})}{\Pi_{n,l}(\mathcal{F}_{n,l})}-\log \frac{\Pi_{n,k}(\mathcal{F}_{n, k}| \mathbf{y})}{\Pi_{n, l}(\mathcal{F}_{n,l}| \mathbf{y})}.