harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d}{dt}( F(u_t),\phi )_{\Lambda}= \lim_{h\downarrow 0}\frac{1}{h}( F(u_{t+h})-F(u_{t}),\phi )_{\Lambda}&= \lim_{h\downarrow 0}\frac{1}{h}( F'(u_{t})(u_{t+h}-u_t),\phi)_{\Lambda}+ \frac{1}{h}( R(u_{t+h}-u_t),\phi )_{\Lambda}.
h^*(f \cup g)([a_1 | \dots | a_{n + p}]) = (-1)^{(n +p - 1)w(a_1) + \dots + w(a_{n + p-1})}f \cup g ([a_1 | \dots | a_{n + p}]) & = (-1)^{np + (n +p - 1)w(a_1) + \dots + w(a_{n + p-1})}f([a_1 | \dots | a_n]) g([a_{n+1}| \dots | a_{n + p}]).
p^{\prime}\left( v_{1},v_{2}\right) =\left( v_{1}^{2}-v_{2}^{2}\right)\cos\left( \frac{2\pi j\left( k_{1}-k_{2}\right)}{n}\right) +i\left(v_{1}^{2}+v_{2}^{2}\right) \sin\left( \frac{2\pi j\left( k_{1}-k_{2}\right)}{n}\right)
\epsilon_{\mu}^{( i )}( k ) \epsilon_{\nu}^{( i )}( - k ) = - ( g_{\mu \nu}- \delta_{\mu 0}\delta_{\nu 0}) + \frac{k_{\mu}k_{\nu}( 1 - \delta_{\mu 0}) ( 1 - \delta_{\nu 0})}{\vec{k}^{2}}
\mathcal{O}_{1i}=\{(t, \ell)\colon \ell \in \mathcal{U}, \s_{1\, i-1}(\ell)\leq t\leq s_{1 i}(\ell)\}i=1,\ldots,n_1+1, \mathcal{O}_2 =\{(t, \ell)\colon \ell \in \mathcal{U}, \ \tau_1(\ell)\leq t\leq \tau_2(\ell)\}, \mathcal{O}_{3i}=\{(t, \ell)\colon \ell \in \mathcal{U}, \s_{3\, i-1}(\ell)\leq t\leq s_{3 i}( \ell)\}i=1,\ldots,n_3+1,
\tilde{m}_{s b}={\frac{{\tilde{m}_{f}}^{2}}{2 \tilde{m}_{s}}}[{\frac{1}{\sqrt{1 - \tilde{m}_{s b}^{2}}}}t a n^{- 1}\sqrt{{\frac{1 + \tilde{m}_{s b}^{2}}{1 - \tilde{m}_{s b}^{2}}}}-{\frac{\pi}{4}}]
J_{\perp}={\frac{m_{f}H_{p}z^{2}\dot{\theta}_{\perp}}{\sqrt{1 - H_{p}H_{F}v_{\perp}^{2}}}}, \E ={\frac{m_{f}}{H_{F}}}\left[{\frac{1}{\sqrt{1 - H_{p}H_{F}v_{\perp}^{2}}}}- 1 \right] .
\deg_{\theta}(\phi_{\theta,i(\phi)}) + n(1+q+\dots + q^{i(\phi)-1}) \leq \frac{q-1}{r}- (n+1)+\frac{n}{q}+\frac{1}{q}+n+n(q+\dots +q^{i(\phi)-1}) &< q + n(q+\dots +q^{i(\phi)-1}) &\leq q+(q-1)(q+\dots +q^{i(\phi)-1})=q^{i(\phi)}.
\|u_n\|_{L^2}^2 =\sum_{j=1}^J\|\phi^j\|_{L^2}^2+\|R_n^J\|_{L^2}^2+o_n(1), \||\nabla|^\sigma u_n\|_{L^2}^2 =\sum_{j=1}^J\||\nabla|^\sigma\phi^j\|_{L^2}^2+\||\nabla|^\sigma R_n^J\|_{L^2}^2+o_n(1), \|u_n\|_{L^4}^4 =\sum_{j=1}^J\|\phi^j\|_{L^4}^4+\|R_n^J\|_{L^4}^4+o_n(1).
\langle 2 n{-}2 , 0 | A | 2 n , 0 \rangle = \langle 2 n , 0 | A^{\dagger}| 2 n{-}2 , 0 \rangle = \sqrt{\frac{c_{n}}{c_{n - 1}}}= \sqrt{2 n + 4 n^{2}}.
\left(\frac{|Q|}{|Q^{\prime}|}\right)^{\frac{1-s}{as}}|Q^{\prime}|^{\frac{1}{r}}=|Q|^{\frac{1-s}{as}}|Q^{\prime}|^{\frac{1}{r}-\frac{1-s}{as}}\leq|Q|^{\frac{1}{r}},
\partial_{y_j}\partial_{y_1}g_{D,\mathrm{odd}}(y) = (\partial_{y_j}\partial_{y_1}g_D)_{\mathrm{even}}(y),\partial_{y_j}\partial_{y_1}g_{N,\mathrm{even}}(y) = (\partial_{y_j}\partial_{y_1}g_N)_{\mathrm{odd}}(y)
\tilde{\cal L}=\frac{1}{2}\,\partial_\mu\phi^A\partial^\mu\phi^A+\frac{1}{2}\,\lambda\,\bigl(\phi^A\phi^A-{\rm e}^{\eta^1}\bigr)-\frac{1}{8}\,{\rm e}^{\eta^1}\partial_\mu\eta^1\partial^\mu\eta^1
g_0(\omega) := A \Big[\sum_{j=-\infty}^\infty f_0\big(\omega + \tfrac{2\pi j}{\delta}\big)\Big]^{-1}f_0^2(\omega) \omega \sin (\omega x_0)\big[\widehat{\phi}\big(\tfrac{6x_0}{\pi}(\omega-N)\big)+\widehat{\phi}\big(\tfrac{6x_0}{\pi}(\omega+N)\big)\big];
g \cdot g^{\prime}= \left( \begin{array}{r r}{a \cdot a^{\prime}+ b \cdot c^{\prime}}&{a \cdot b^{\prime}+ b \cdot d^{\prime}}\\{c \cdot a^{\prime}+ d \cdot c^{\prime}}&{c \cdot b^{\prime}+ d \cdot d^{\prime}}\\ \end{array}\right)
\bar{\Xi}_L \stackrel{\eqref{Xi}}{=}\begin{bmatrix}\bar{y}_{1,[L,L+d_1-1]}\vdots \bar{y}_{m,[L,L+d_m-1]}\end{bmatrix}\stackrel{\eqref{last_di_outputs}}{=}\begin{bmatrix}H_{d_1}(y_{1,[L,N+d_1-1]}) \vdots H_{d_m}(y_{m,[L,N+d_m-1]})\end{bmatrix}\alpha & = H_1(\Xi_{[L,N]})\alpha.
q^{-2m}(\gamma(s,\pi,\eta,\psi)-\gamma(s,\pi',\eta,\psi)) = \int_{F^\times}( W_{v_m}(t(a)w)-W_{v_m'}(t(a)w))(\omega_{\psi^{-1}}(w)\phi^m)(a)\chi(a)\eta^{-1}(a)|a|^{-s}da.
\frac{1}{1+2\eta(\min\{\varepsilon/2,2\})}= 1-2\eta(\min\{\varepsilon/2,2\})+\frac{\left(2\eta(\min\{\varepsilon/2,2\})\right)^2}{1+2\eta(\min\{\varepsilon/2,2\})}&\geq 1-2\eta(\min\{\varepsilon/2,2\})+\frac{\left(2\eta(\min\{\varepsilon/2,2\})\right)^2}{3}.
\left(\frac{b-\kappa(r)a}{b+1-\kappa(r)a}\right)^2-\left(\frac{1}{b+1-\kappa(r)a}\right)^2 = \frac{\big(b-\kappa(r)a\big)^2-1}{(b+1-\kappa(r)a)^2}= 0.
{\hat{v}}( \eta ,{\bf x}) \, = \, \frac{1}{( 2 \pi )^{3 / 2}}\int \mathrm{d}^{3}{\bf k}[{\hat{c}}_{\bf k}( \eta ) e^{i{\bf k}\cdot{\bf x}}+{\hat{c}}_{\bf k}^{\dagger}( \eta ) e^{- i{\bf k}\cdot{\bf x}}] \, ,
\frac{1}{R^{s\theta + (1-\theta)t}}= \max\big\{\vert u_\infty \vert_s, \vert v_\infty \vert_s \big\}\leq \liminf_{n\to \infty}\left( p_n^{\frac{1}{p_n}}\sqrt[p_n]{\Lambda_1(p_n)}\right) \leq \limsup_{n\to \infty}\left( p_n^{\frac{1}{p_n}}\sqrt[p_n]{\Lambda_1(p_n)}\right) \leq \frac{1}{R^{s\theta + (1-\theta)t}},
f(t)= \sum_{n=-\infty}^{\infty}f(t_n ) S_{n}(t) \textrm{with}S_{n}(t)= \frac{\int_{I}\kappa(t_n,x)\overline{\kappa(t,x)}dx}{\int_{I}\vert \kappa(t_n,x)\vert^2 dx}t\in\mathbb{R}
v_1 :\; s_1s_3s_7=1~,~~ v_4 :\; s_7=1\ , v_2 :\; s_1s_2s_9=1~,~~ v_5 :\; s_{11}=1 \ , v_3 :\; s_2s_3s_{11}=1~,~~ v_6 :\; s_{9}=1 \ .
\zeta^\wedge_{L, p}(s) =\int_{\substack{(A,\nu)\in \dot{H}v_p(A) \ge 0 v_p(A) + v_p(\nu) \geq 0}}\, \lvert \det A \rvert_p^{\, 4s-10}\, \lvert \nu \rvert_p^{\, 5s-12}\, \mathrm{d}\mu_p(A,\nu).
\Bigr(\mathrm{Tr\,}A_{B_j,+}(t,z,z)-\mathrm{Tr\,}A_{B_j,-}(t,z,z)\Bigr)dv_{U_j}(z) &=[\mathrm{Td\,}(\nabla^{T^{1,0}U_j},T^{1,0}U_j)\wedge\mathrm{ch\,}(\nabla^{L^m},L^m)]_{2n}(z)+O(t),\ \ \forall z\in U_j,
(U\circ T)(\iota_g(v)) =U\Big(\sum_{g'\in G}\iota_{g'}\big(\pi_{g'}(T(\iota_g(v)))\big)\Big) =\sum_{g'\in G}U\Big(\iota_{g'}\big(T_{g', g}(v)\big)\Big) =\sum_{g'\in G}\delta_{g'}\otimes T_{g', g}(v) =\sum_{s\in S}\delta_{gs}\otimes T_{s, e}(v)
N=q^{\lfloor n/2\rfloor}A_{n}(\frac{1}{2}\pi)=q^{\lfloor n/2\rfloor}\frac{1}{2^{n}n!}(\frac{p}{q})^{n+\frac{1}{2}}\int_{0}^{1}(1-z^{2})^{n}cos(\frac{1}{2}\pi z)dz.
e_2(t,N_h)=\sqrt{\frac{\ell}{N_h}\sum^{N_h-1}_{j=1}\Big|y(x_j,t)-y_N(x_j,t)\Big|^2}, &e_\infty(t,N_h)=\max\limits_{1\leq j\leq N_h-1}\Big|y(x_j,t)-y_N(x_j,t)\Big|,
\|p_N\mathcal S(r)\sigma_{s}\|_{L_{}(U,H)}^2=\|\sigma_{s}^*\mathcal S(r)^*p_N\|_{L_{}(H,U)}^2= \sum_{k=N}^{\infty}\|\sigma_{s}^*\mathcal S(r)^*e_k\|^2\to 0,N\to\infty,
m_{\mu \nu}{}^{i}{}_{k}m_{\rho \sigma}{}^{k}{}_{j}- m_{\rho \sigma}{}^{i}{}_{k}m_{\mu \nu}{}^{k}{}_{j}= - \eta_{\mu [ \rho}m_{\sigma ] \nu}{}^{i}{}_{j}+ \eta_{\nu [ \rho}m_{\sigma ] \mu}{}^{i}{}_{j}\ .
\theta(\mathsf{M}^n_m)^i_j = q^{-(2\rho, \lambda_i - \lambda_j)}q^{-(2\rho, \lambda_m - \lambda_n)}(\mathsf{M}^n_m)^i_j & = q^{-(2\rho, \lambda_m - \lambda_n)}\pi(K_{2\rho}^{-1})_{i}^{i}(\mathsf{M}^n_m)^i_j \pi(K_{2\rho})_{j}^{j}.
\lim_{b \to +\infty, \; b-a=c}\sqrt{\frac{4b}{\pi}}e^{2 b x}P_n(x^2)=\frac{2(b^2-a^2)^{\mu+\nu+1}}{a^\mu b^\nu \Gamma(\mu+\nu+1+n)n!}x^{\nu-\frac{1}{2}}L_n^{(\mu+\nu)}(2cx),
L = \bar{\psi}i \gamma_{\mu}\partial_{\mu}\psi - j_{\mu}( A_{\mu}+ B_{\mu}+ \epsilon_{\mu \nu}\partial_{\nu}\theta ) + \frac{1}{\pi}A_{\mu}\epsilon_{\mu \nu}\partial_{\nu}\theta + \frac{1}{2 \pi}\partial_{\nu}\theta \partial_{\nu}\theta \; ,
\sum_{k=l}^{l+n}\delta_{k,k+1}^2 \leq 4e_l^2+2^{6}\varepsilon^{-1}\Lambda_6^{2}\mu_{l}^2+(\frac{1}{2}\varepsilon C_{\rm{REL}}^{-2}-4)e_{l+n+1}^2+(\frac{\varepsilon}{4}-2^{6}\varepsilon^{-1}\Lambda_6^2)\mu_{l+n+1}^2 &+\frac{\varepsilon}{2}\sum_{k=l}^{l+n}\mu_k^2+\frac{\varepsilon}{2}C_{\rm{REL}}^{-2}\sum_{k=l}^{l+n-1}e_{k+1}^2.
\vec{V}( z ) = \Biggl ( z +{\frac{1}{4}}{\frac{({\frac{g^{2}}{2 \pi}})^{2}}{1 +{\frac{g^{2}}{2 \pi}}}}( z - \bar{z}) , -{\frac{{\frac{g^{2}}{2 \pi}}( 2 +{\frac{g^{2}}{2 \pi}})}{1 +{\frac{g^{2}}{2 \pi}}}}( z - \bar{z}) \Biggr ) .
\;\|A^{(r_k-1+s)/2}w_k(t)\|_{L^2}^2 +\int_{t_k}^{t}\|A^{(r_k-1)/2+s}w_k\|_{L^2}^2 \leq \; C\int_{t_k}^{t}\|A^{(r_k-1)/2}f(u_*,u_*)\|_{L^2}^2 \,d\tau \leq \; C\int_{t_k}^t \|u_*\|_{D(A)}^2(\|A^{s/2}u_*\|_{L^2}^2+\|A^{r_k/2}u_*\|_{L^2}^2) \,d\tau.
P(f) = a(t,x)\sqrt{F_E^0}+b(t,x)\cdot\left(1+\frac{\mathcal{I}}{mc^2}\right) p\sqrt{F_E^0}+c(t,x)\left\{cp^0\left(1+\frac{\mathcal{I}}{mc^2}\right)-\widetilde{e}(T_0)\right\}\sqrt{F_E^0}
B ={\frac{i}{2}}\kappa \ \xi \ ( \bar{\psi}^{1}\gamma_{5}\psi^{1}+ \bar{\psi}^{2}\gamma_{5}\psi^{2}) -{\frac{1}{2}}\kappa \ \theta \ ( \bar{\psi}^{1}\psi^{1}- \bar{\psi}^{2}\psi^{2}) - \kappa \ \varphi \ \bar{\psi}^{1}\psi^{2}+ . . . \ , \,
X_{z_{\pm}}=-i Q_{\pm}^{-1}\displaystyle\displaystyle\frac{\partial}{\partial \overline{z}_{\pm}}~,~~~X_{p_{\pm}}=i\displaystyle \displaystyle\frac{\partial}{\partial z_{\pm}}- i Q_{\pm}^{-1}\displaystyle\frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle\displaystyle\frac{\partial}{\partial \overline{z}_{\pm}}~,
\frac{1}{\hat{\Pi \hspace{-7pt}/}- M + i 0}= \int_0^{\infty}\! \! \! dT \! \int \! d\chi \,\exp \left [ \, - \frac{i}{2 \kappa_0}( M^2 T + M \chi) \, \right ]\, \exp \left [ \, \frac{i}{2 \kappa_0}( \hat{\Pi \hspace{-7pt}/}^2 T + \hat{\Pi \hspace{-7pt}/}\, \chi) \, \right ]
\mathcal{D}=\biggl\{\begin{pmatrix}2 1 1 4\end{pmatrix},\begin{pmatrix}2 1 4 1\end{pmatrix},\begin{pmatrix}0 2 1 4\end{pmatrix},\begin{pmatrix}0 2 4 1\end{pmatrix}\biggr\}\,.
{\displaystyle \int}{\cal D}y \, \mathrm{e x p}\Bigl [ - \int_{0}^{T}d \tau{\frac{1}{4}}{\dot{y}}^{2}\Bigr ] = \mathrm{D e t^{\prime}}_{P}^{-{\frac{1}{2}}}\bigl [ -{\partial}_{\tau}^{2}\bigr ] ={\lbrack 4 \pi T \rbrack}^{-{\frac{D}{2}}}\,
f \Bigl(\sum_{i \in S}(a_i + b_i)^*(a_i + b_i) \Bigr) \ =\ \sum_{i \in S}\| a_i + b_i\|_f^2 \ \leq\ \sum_{i \in S}\| a_i\|_f^2 +\| b_i\|_f^2 + 2\|a_i\|_f \|b_i\|_f \ \leq\A+B+ 2\sum_{i \in S}\|a_i\|_f \|b_i\|_f.
\widehat{N}(\boldsymbol{\theta}) = b(\boldsymbol{\theta})^* L(\boldsymbol{\theta}) b(\boldsymbol{\theta}) \widehat{P}, L (\boldsymbol{\theta}) := | \Omega |^{-1}\int_{\Omega}\bigl( \Lambda (\mathbf{x})^* b(\boldsymbol{\theta})^* \widetilde{g}(\mathbf{x}) + \widetilde{g}(\mathbf{x})^* b(\boldsymbol{\theta}) \Lambda (\mathbf{x}) \bigr) \, d \mathbf{x}.
\mathcal{E}(x_k'R_kx_k)= \bar{x}'_0\bar{\phi}'_{k,0}\bar{\phi}_{k,0}\bar{x}_0=\tilde{x}_0'\bar{R}_0^{\frac{1}{2}}\bar{\phi}'_{k,0}\bar{\phi}_{k,0}\bar{R}_0^{\frac{1}{2}}\tilde{x}_0 = \tilde{x}_0'\phi'_{k,0}(I_{4^{k}}\otimes\bar{R}_k)\phi_{k,0}\tilde{x}_0.
- \frac{i}{\pi}\int_{0}^{\infty}d t \int_{0}^{\pi / 2}d \theta \frac{t^{2}\left[ g ( - i t ) - g ( i t ) \right] \cos \theta}{( t^{2}+ z^{2}\cos^{2}\theta )^{3 / 2}}= - \frac{i}{\pi}\int_{0}^{\infty}d t \frac{t [ g ( - i t ) - g ( i t ) ]}{( t^{2}+ z^{2})},
\frac{d}{dt}[\begin{matrix}\omega_{1}\\ \omega_{2}\end{matrix}]=[\begin{matrix}0&-\omega_{3}(I_{3}-I_{2})/I_{1}\\ -\omega_{3}(I_{1}-I_{3})/I_{2}&0\end{matrix}][\begin{matrix}\omega_{1}\\ \omega_{2}\end{matrix}]
|\frac{-1}{r}\int_{6r}^{\infty}\frac{dw \lambda''(t+w) r^{2}w (2-2\alpha) \left((1-2\alpha)\lambda(t+w)^{-2\alpha}(\lambda'(t+w))^{2}+ \lambda(t+w)^{1-2\alpha}\lambda''(t+w)\right)}{(\lambda(t+w)^{2-2\alpha}+w^{2})^{2}}| &\leq \frac{C r}{t^{4}\log^{b+2}(t)}
{1\over \gamma_3}\int_A \omega(y) \times K_0(x, y) \, dy ={1\over \gamma_3}\int_\Omega \omega(y) \times \left[ -{y\over |x|^3}+ 3{x\cdot y\over |x|^5}x \right] \, dy + O\left({1\over |x|^4}\right).
\partial_{t}\left(\int_{0}^{\frac{t}{2}}\frac{\sin(u)b(b-1) du}{t^{2}u\log^{b+1}(\frac{t}{u})}\right) = \frac{\sin(\frac{t}{2}) b(b-1)}{t^{3}\log^{b+1}(2)}+ \int_{0}^{\frac{t}{2}}\frac{\sin(u) b(b-1)}{u}\left(\frac{-2}{t^{3}\log^{b+1}(\frac{t}{u})}-\frac{(b+1)}{\log^{b+2}(\frac{t}{u}) t^{3}}\right)du
F_{i, l}= \frac{\max\limits_{B(p_i, 2)}|v^i_l|}{\max\limits_{B(p_i, 1)}|v^i_l|}=\frac{\max\limits_{B(p_i, 2)}|\sum\limits_{t=1}^mC^i_{lt}u^i_t|}{\max\limits_{B(p_i, 1)}|\sum\limits_{t=1}^mC^i_{lt}u^i_t|}.
\begin{cases}\displaystyle\overline{A}_0 \subset B_0 \subseteq B_{R_0}\subset \mathbb{R}^3; begin{align*}10pt]\displaystyle\rho_0(x)=0,\ \forall \x\in B_0 \setminus A_0, \int_{A_0}\rho_0(x) x>0; begin{align*}10pt]\displaystyle u_0(x)|_{\partial A_0}=\overline{u}_0\end{cases}
I(s) \leq2\| |e^{iv\cdot x}\tilde{u}|^2e^{iv\cdot x}R\|_{H^k}+\|(e^{iv\cdot x}\tilde{u})^2\overline{e^{iv\cdot x}R}\|_{H^k}+2\|e^{iv\cdot x}\tilde{u}|e^{iv\cdot x}R|^2\|_{H^k}&\quad+\|\overline{e^{iv\cdot x}\tilde{u}}(e^{iv\cdot x}R)^2\|_{H^k}+\||e^{iv\cdot x}R|^2e^{iv\cdot x}R\|_{H^k}&=:I_1(s)+I_2(s)+I_3(s)+I_4(s)+I_5(s).
\frac{|I(a_1, \cdots, a_n, a_{n+1})|}{|I(a_1, \cdots, a_n)|}= \frac{q_n(q_n+q_{n-1})}{q_{n+1}(q_{n+1}+q_n)}= \frac{1}{a^2_{n+1}}\cdot \frac{1+y_n}{(1+z_n)(1+1/a_{n+1}+z_n)},
{\cal L}= \frac{i}2 \overline{{\psi}}\not \! \partial \psi - \frac{M}2 \overline{{\psi}}\psi - \frac{\sigma}2 ( \overline{{\psi}}\psi ) - \frac{N}{4 g}\sigma^{2}- \frac{N}{2 g}M \sigma .
\mathfrak{s}_{i,\mathfrak{t}}^*\mathfrak{s}_{k,\mathfrak{t}}=\delta_{i,k}\sum_{j=1}^N A_{ij}\mathfrak{s}_{j,\mathfrak{t}}\mathfrak{s}_{j,\mathfrak{t}}^*+P_{\circ_A},
\begin{array}{ll}2I=\int_{\mathbb R}[(1-\gamma)-|\omega|]\xi^2(x)-c\mu (\xi'(x))^2((1-\gamma)+\frac{b}{c}|\omega|)dx \geqq [(1-\gamma)-|\omega|]\int_{\mathbb R}\xi^2-c\mu (\xi')^2dx=[(1-\gamma)-|\omega|]\|J_c^{1/2}\xi\|^2.\end{array}
( f \star g ) ( \tilde{\bf r}) = \frac{1}{( 2 \pi )^{2}}\sum_{P^{\prime}, P^{\prime \prime}}\int_{- \pi}^{+ \pi}d \theta^{\prime}d \theta^{\prime \prime}e^{[ - 2 i A ( \tilde{\bf r}, \tilde{\bf r}^{\prime}, \tilde{\bf r}^{\prime \prime}) ]}f ( \tilde{\bf r}^{\prime}) g ( \tilde{\bf r^{\prime \prime}})
\sum_{k=0}^{n-1}(2k+1)R_k^2 \equiv 0 \pmod{n}, \sum_{k=0}^{p-1}(2k+1)R_k^2 \equiv 4p(-1)^{\frac{p-1}{2}}-p^2 \pmod{p^3}, 9\sum_{k=0}^{n-1}(2k+1)W_k^2 \equiv 0 \pmod{n}, \sum_{k=0}^{p-1}(2k+1)W_k^2 \equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad
\left({\frac{A ( q , p )}{4}}\right)_{N = 8}= \pi \left({| ( \sum_{i}| Z_{i}|^{4}- 2 \sum_{i > j}| Z_{i}|^{2}| Z_{j}|^{2}+ 8 | Z_{1}Z_{2}Z_{3}Z_{4}|}) | \right)^{1 / 2}\ ,
\frac{1}{s}\int_0^{s+M}( e^{-\lambda (u-s)}- e^{- \lambda u}) \eta (T_uA)du = \frac{1}{s}\int_0^M ( e^{-\lambda (u-s)}- e^{- \lambda u}) \eta (T_uA)du + \frac{1}{s}\int_M^{M+s}( e^{-\lambda (u-s)}- e^{- \lambda u}) \eta (T_uA)du
a_{ij}+ b_i D_t^j u_2(t_0) + c_i D_t^j v_2(t_0) + d_j D_s^i u_1(s_0) + e_j D_s^i v_1(s_0) + \nu_1 D_s^i u_1(s_0) D_t^j u_2(t_0) & + \nu_2 D_s^i u_1(s_0) D_t^j v_2(t_0) + \nu_3 D_s^i v_1(s_0) D_t^j u_2(t_0) + \nu_4 D_s^i v_1(s_0) D_t^j v_2(t_0) = D_s^i D_t^j f(s_0,t_0),
[x,[a,b]] = [b,[x^{-1},a^{-1}]]^{-xa}[a^{-1},[b^{-1},x]]^{-ba}& = [[x^{-1},a^{-1}],b]^{xa}[[b^{-1},x],a^{-1}]^{ba}& \in [\phi_{s+u},b] [\phi_{t+u},a] \leq \phi_{s+t+u}.
C^{a , b}( x_{1 2}, \partial_{2}) C^{a , b}( x_{3 4}, \partial_{4}) \frac{1}{x_{2 4}^{2 a}}= \frac{1}{x_{1 3}^{2 a_{+}}x_{2 4}^{2 a_{-}}}{\cal{E}}_{a , b}( u , v ) ,
\hat{\rho}_c(\mathbf{q}) =\int_{\mathbb{R}^D}d\mathbf{x}~\rho_c(\mathbf{x})e^{-i 2\pi\mathbf{q}\cdot\mathbf{x}}&=\left(\sum_{\mathbf{y}\in S}e^{-i 2\pi\mathbf{q}\cdot\mathbf{y}}\right)\left(\int_{\mathbb{R}^D}d\mathbf{x}'~\tilde{\rho}(\mathbf{x}')e^{-i 2\pi\mathbf{q}\cdot\mathbf{x}'}\right) &=\hat{s}(\mathbf{q})~\hat{\rho}(\mathbf{q}).
\vert \hat f(x)-f(x)\vert =\vert \hat g(\langle\hat a,x\rangle)-g(\langle a,x\rangle)\vert &\leq\vert \hat g(\langle\hat a,x\rangle)-g(\langle\hat a,x\rangle)\vert+\vert g(\langle\hat a,x\rangle)-g(\langle a,x\rangle)\vert &\leq c_0\|a-\hat a\|_{l_1^d}+c_0\vert\langle a-\hat a,x\rangle\vert\leq 2c_0\|a-\hat a\|_{l_1^d}.
[v_j,[v_0,v_1]]+[v_0,[v_1,v_j]]+[v_1,[v_j,v_0]] =0 \implies [v_j,z_b]+[v_0,0]+[v_1,0] =0 \mbox{(by Equation \ref{2bracket})}\implies [v_j,z_b] = 0 \mbox{for all $v_j$ not in the closed path of length $q$}.
x=\frac{\alpha_{1}^{\frac{1}{p-1}}}{\alpha_{1}^{\frac{1}{p-1}}+\alpha_{2}^{\frac{1}{p-1}}}P_{1}\left( x\right) +\frac{\alpha_{2}^{\frac{1}{p-1}}}{\alpha_{1}^{\frac{1}{p-1}}+\alpha_{2}^{\frac{1}{p-1}}}P_{2}\left(x\right)
\begin{array}{r@{=}l@{=}l}P \, \, \, \, p_{1}+ p_{2}\, \, \, \, -(p_{3}+ p_{4}) \, , R \, \, \, \, p_{1}+ p_{3}\, \, \, \, -(p_{2}+ p_{4}) \, , Q \, \, \, \, p_{1}+ p_{4}\, \, \, \, -(p_{2}+ p_{3}) \, .\end{array}
b_{l}^{n}=1-\frac{1}{n}\binom{l+1}{2}+\frac{1}{n^{2}}\left\{\frac{1}{2}\binom{l+1}{2}^{2}+\binom{l+1}{2}\right\}& -\frac{1}{n^{3}}\left\{\frac{1}{6}\binom{l+1}{2}^{3}+\frac{4}{3}\binom{l+1}{2}^{2}+\binom{l+1}{2}\right\}+.....
A\left( Vz,0;G_{1},\chi_{2}\right) -\overline{\chi_{1}}\left( c\right)\overline{\chi_{2}}\left( b\right) A\left( z,0;G_{2},\chi_{1}\right) & \quad=\frac{\pi i}{k}\overline{\chi_{1}}\left( -c\right) \overline{\chi_{2}}\left( b\right) s\left( d,c;G_{2},G_{1}\right) +\delta\left(z;c,d;\chi_{1},\chi_{2}\right) ,
\frac{\partial^{r+1}Q_r}{\partial x_{l_1}\cdots \partial x_{l_{r+1}}}=\displaystyle\prod_{j=1}^{r+1}(r+4-2j)\prod_{i=1}^{r+1}(-r+2-2i)\prod_{k=1}^{r+1}\dot{f}_{l_k}\,\ddot{f}_{l_k}W^{-3r-2}S_r.
\Re{( \sum_{j = 1}^{\Delta m / 2}\tilde{z}_{j})}= 0 , \quad \Im{( \sum_{j = 1}^{\Delta m / 2}\tilde{z}_{j})}= \frac{1}{2}\Im{\left( \sum_{k = 1}^{m^{+}}z_{k}- \sum_{l = m^{+}+ 1}^{m}z_{l}\right)}.
\prod_{\substack{\gcd(j_1,j_2,j_3,j_4,j_5,k)=1 j_1,j_2,j_3,j_4,j_5<k j_1,j_2,j_3,j_4,j_5\geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3+j_4+j_5}z^k}\right)^{\frac{1}{k}}= \exp\left[ \sum_{k=1}^{\infty}\left( \frac{y}{1}+ \frac{y^2}{2}+ \cdots + \frac{y^k}{k}\right)^{5}\frac{z^{k+1}}{(k+1)^4}\right].
v_* =\lim_{R\to\infty}\lim_{n\to\infty}\sum_{j=0}^\infty\dfrac{\mu^{p-1+j}}{\Gamma(p+j)}\|u^*_{n,R}\|_{L_{\alpha_0}^{p'(p-1+j)}}^{p'(p-1+j)}&\geq\lim_{R\to\infty}\lim_{n\to\infty}\dfrac{\mu^{p-1}}{\Gamma(p)}\|u^*_{n,R}\|^p_{L^p_{\alpha_0}}=\dfrac{\mu^{p-1}}{\Gamma(p)}\eta_*
S_{cl}^{I}=\frac{a^{n}}{n!}\int d^{d+1}x\int d^{d}\vec{y}_{1}...d^{d}\vec{y}_{n}[\lambda'_{n}\Psi_{1}+\lambda_{n}(\Psi_{2}+n\frac{a'}{a}+\ln\frac{(x^{d})^{n}}{\prod_{i=1}^{n}[(x^{d})^{2}+|\vec{x}-\vec{y_{i}}|^{2}]})]
\left(\mathrm{Tr}\,\mathrm{Log}\,\mathrm{A}\right)^{-(\alpha+1)}_\lambda:=\mathrm{tr}\,\widehat{\mathcal{M}}_\lambda\left[\mathrm{E}^{-\mathrm{A}};\alpha\right]=\widehat{\mathcal{M}}_\lambda\left[\mathrm{Tr}\,\mathrm{E}^{-\mathrm{A}};{\alpha}\right]=\zeta'_{A_\lambda}({\alpha})\;.
\partial_{t}^{\beta}\left(\sum_{k=1}^{\infty}\int_{0}^{r}g^{k}(s,x)dw_{s}^{k}\right) =\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{\infty}\int_{0}^{t}(t-s)^{-\beta}g^{k}(s,x)dw_{s}^{k},
\frac{\langle v_1,v_2\rangle}{|v_1||v_2|}=\frac{|v_1|^2}{|v_1||v_2|}+\frac{\langle v_1,v_2-v_1 \rangle}{|v_1||v_2|}\geq 1-\frac{|v_1||v_2-v_1|}{|v_1||v_2|}\geq 1-\frac{\epsilon\cdot c}{c-\epsilon\cdot c}=\cos\theta.
\sum_{k=0}^{n-1}\binom{n-1}{k}S_{k+1}^{(a,b)}x^{k}(x-b)^{n-1-k}(x-a-b)^{n-1-k}= \frac1b\sum_{k=0}^{n}\binom{n+k}{2k}C_k x^{2k}(ab+b^2-x^2)^{n-k}.
\begin{aligned}\frac{\partial D_{\varpi}(P\parallel Q)}{\partial \varpi}\ge \frac{e^{\varpi}\left[ \left( \sum_i p_i \frac{q_i}{p_i}\right)^{-1}-1\right]}{\sum_i{p_i e^{\left( \varpi \frac{p_i}{q_i}\right)}}}= 0, \end{aligned}
\left|d(x_{n},y_{n})-d(x'_{n},y'_{n})\right|=\left|d(x_{n},y_{n})-d(x'_{n},y_{n})+d(x'_{n},y_{n})-d(x'_{n},y'_{n})\right|\le \le \left|d(x_{n},x'_{n})+d(y'_{n},y_{n})\right|\le C_{\Omega}|x_{n}-x'_{n}|+C_{\Omega}|y_{n}-y'_{n}|\to 0.
U_E=(v_E,y_E)=\left(\frac{(2\alpha \alpha_2-\alpha^2\sqrt{y_L^2-4v_L}+\alpha^2 y_L)(\sqrt{y_L^2-4v_L}-y_L)}{4\alpha^2-2\alpha_1(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1 v_E}{\alpha^2}-\frac{\alpha_2}{\alpha}\right).
\ldots , \xi^{6}q^{- 2}w_{1}, \xi^{4}q^{- 2}w_{1}, \xi^{2}q^{- 2}w_{1}; q^{2}w_{1}, \xi^{- 2}q^{2}w_{1}, \xi^{- 4}q^{2}w_{1}, \ldots .
\begin{cases}c_{1,\infty}c_{2,\infty}= c_{3,\infty}= c_{1,\infty}c_{4,\infty}, c_{1,\infty}+ c_{3,\infty}= M_{13}c_{2,\infty}+ c_{3,\infty}+ c_{4,\infty}= M_{234}.\end{cases}
V_{0}( y ) = 4 G^{5 5}\epsilon ( y ) \bar{\Phi}^{\prime}( | y | ) \delta ( y ) + \frac{4}{\sqrt{- G}}\left( \sqrt{- G}G^{5 5}\right)^{\prime}\bar{\Phi}( | y | ) \delta ( y ) \ + 4 G^{5 5}\bar{\Phi}( | y | ) \delta^{\prime}( y )
\bar{F}^{\ast \mu}= \frac{\bar{\epsilon}^{\mu \nu \rho}}{2}\bar{F}_{\nu \rho}~ ~ ( \bar{\epsilon}^{\mu \nu \rho}= \frac{\epsilon^{\mu \nu \rho}}{\sqrt{- \bar{\eta}}}, ~ \epsilon^{0 1 2}= - 1 ) .
\mathcal{E}_T(g) \sim \Vert g \Vert_{\tilde{L}^\infty_T \tilde{L}^2_v(B^{3/2}_x)}\intertext{and}\mathcal{D}_T(g) = \Vert \nabla_x(a,b,c) \Vert_{\tilde{L}^2_T(B^{1/2}_x)}+ \Vert (\mathbf{I}-\mathbf{P})g\Vert_{\mathcal{T}^{3/2}_{T,2,2}}.
(-1)^{j+1}e_{N+1}^je_{N-j}=(-1)^{N(j+1)}y^{j+1}\left[z^{j+1}\right]\left(\left(1+\frac{z}{y}\right)(1+zx_1)\cdots (1+zx_r)\right) &= \left[t^{j+1}\right]\left((1-(-1)^{N-1}t)(1-(-1)^{N-1}x_1yt)\cdots (1-(-1)^{N-1}x_ryt)\right),
\left\langle P\left( \left\{A_{\hbar}\left( t_{i}\right) ,A_{\hbar}^{\dag}\left( t_{i}\right) \right\}_{i=1}^{n}\right) \right\rangle_{\psi_{\hbar}}\rightarrow P\left( \left\{\alpha_{i}\left( t\right),\bar{\alpha}_{i}\left( t\right) \right\}_{i=1}^{n}\right) ,
\begin{array}{cccc}q_2= q_1+p_1+p_n+p_{n-1}+ \ldots +p_3 \vdots \vdots \ddots \vdots q_j= q_1+p_1+p_n+p_{n-1}+ \ldots +p_{j+1}\vdots \vdots q_n= q_1+p_1 \end{array}
\tilde{u}(t)=\tilde{u}^{n}+ (t-t_{n})\partial_t\tilde{u}^{n}-\frac{(t-t_{n-1})(t_n-t)^2}{k_n}\partial_t^2\tilde{u}^n,\;\;\; t_{n-1}< t \leq t_n,
X(t) = X(0) + \int_{\Gamma\times[0,t]}b\bigl(s,X(s),\mathfrak{m}(s),\gamma\bigr)\rho(d\gamma,ds) &+ \int_{0}^{t}\sigma\bigl(s,X(s),\mathfrak{m}(s))\bigr)dW(s),t\in [0,T].
x \rhd y = \nabla_x y, [x,y] = - T(x,y) + R(x,y), x \rhd E = \nabla_x E, [E,x] = -Ex, E \rhd y = E y, [E_1, E_2] = - E_1 E_2 + E_2 E_1. E_1 \rhd E_2 = E_1 E_2 - E_2 E_1,
Q(z)=\begin{cases}T(z) , T(z)\left(\begin{matrix}1 0 - \dfrac{F^2}{w}\phi^{-2n}1\end{matrix}\right) , T(z)\left(\begin{matrix}1 0 \dfrac{F^2}{w}\phi^{-2n}1\end{matrix}\right) .\end{cases}
{\mathcal J}_{H,K,N}(t) :=\begin{cases}\chi_{\{t=0\}}N = 1 , K > 0 \chi_{\{H t \geq 0}\}N = 1 , K \leq 0 \left({\rm c}_\delta(t) + \frac{H}{N-1}{\rm s}_\delta(t)\right)_+^{N-1}N \in (1,\infty) \end{cases}~.
\begin{array}[c]{cccc}\mathbf{H}^{s}\times\mathbf{C}\begin{pmatrix}e^{r_{i}}b_{i}& e^{-r_{i}}\end{pmatrix}_{i=1,\ldots,s}\times(b_{s+1}) \downarrow \downarrow \mathbf{R}^{s}(r_{1},\ldots,r_{s}) \end{array}
{\mathrm{}^{*}\mathcal{L}}_{e f f}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \beta^{2}\sqrt{1 + \frac{1}{6 m^{2}\beta^{2}}H_{\mu \nu \rho}H^{\mu \nu \rho}}- \frac{\chi}{2 \theta}\varepsilon^{\mu \nu \rho \lambda}B_{\nu \rho}\partial_{\mu}A_{\lambda}.
[\begin{matrix}\hat{x}\\ \hat{y}\\ \hat{z}\end{matrix}]=[\begin{matrix}sin\theta cos\phi&cos\theta cos\phi&-sin\phi\\sin\theta sin\phi&cos\theta sin\phi&cos\phi\\cos\theta&-sin\theta&0\end{matrix}][\begin{matrix}\hat{r}\\ \hat{\theta}\\ \hat{\phi}\end{matrix}]
\begin{matrix}I_{1}\dot{\omega}_{1}+(I_{3}-I_{2})\omega_{2}\omega_{3}&=&N_{1}\\I_{2}\dot{\omega}_{2}+(I_{1}-I_{3})\omega_{3}\omega_{1}&=&N_{2}\\I_{3}\dot{\omega}_{3}+(I_{2}-I_{1})\omega_{1}\omega_{2}&=&N_{3}\end{matrix}
\psi_3(h_s)= \big( a^{ie_1-j(e_1+\dots+e_{p-(s-1)})}u_1, a^{i(e_1+e_2)- j(e_1+ \dots+ e_{p-(s-2)})}u_2 , \dots, a^{-j(e_1+\dots+e_{p-s})}u_p \big),
\begin{array}{l}Find\,\,x\, so\,as\,to\,satisfy\, \,\,\,\,\,\,\,\,\,\,{f_{ij}}\left( x \right)\left( \begin{array}{l}\le \cong \ge \end{array}\right)f_{ij}^*,\,\,i = 1,2,...m;\,j = 1,2,...,p_m^{}subject\,\,to\, \,\,\,\,\,\,\,\,\,\,\,x \in S\end{array}
\mathbf{H}_{[a:b]}^{[c:d]}= \begin{bmatrix}h_{a,c}h_{a,c+1}\cdots h_{a,d}h_{a+1,c}h_{a+1,c+1}\cdots h_{a+1,d}\vdots \vdots \ddots \vdots h_{b,c}h_{a,c+1}\cdots h_{b,d}\end{bmatrix}.
0 = \lim_{k\rightarrow \infty}E\left[h(X_{n_k}(t_i); i\leq j ) (M_g^{n_k}(t+s)-M_g^{n_k}(t))\right] &=E\left[h(X(t_i); i\leq j ) (M_g(t+s)-M_g(t))\right]
{\cal S}_{f}^{( 2 k + 1 )}= ( 2 k + 1 ) ( f^{\prime})^{k}\partial_{z}( f^{\prime})^{- 1}\partial_{z}( f^{\prime})^{- 1}\ldots \partial_{z}( f^{\prime})^{- 1}\partial_{z}( f^{\prime})^{k},
|H_{2,1}(F_{f}/2)| =\frac{1}{192}\left(|-3\tau^4_{1}+4(1-\tau^2_{1})\tau^2_{1}\tau_{2}-4(1-\tau^2_{1})(3+\tau^2_{1})\tau^2_{2}| \right. &\left.+16\tau_{1}(1-\tau^2_{1}) (1-|\tau^2_{2}|) \right) &=\frac{1}{12}\tau_{1}(1-\tau^2_{1})\left(|A+B\tau_{2}+C\tau^2_{2}| +1 -|\tau_{2}|^2\right),
G(m,t) = e^{-2t}I_m(2t) = \frac{e^{-2t}}{\pi}\int_0^{\pi}e^{2t\cos\theta}\cos m\theta\,d\theta &= \frac{e^{-2t}}{2\pi}\int_{-\pi}^{\pi}e^{2t\cos\theta}(\cos m\theta-i\sin m\theta)\,d\theta &= \frac{e^{-2t}}{2\pi}\int_{-\pi}^{\pi}e^{2t\cos\theta}e^{-im\theta}\,d\theta.
\frac{\tilde{\cal E}_{q}[ \phi ( x ) ]}{L}= - \frac{1}{4 \pi}\left( \frac{1}{2}\sum_{j}\omega_{j}^{2}\log \frac{\tilde{\omega}_{j}^{2}}{m^{2}}+ \int_{0}^{\infty}\frac{d k}{2 \pi}\tilde{\omega}( k )^{2}\log \frac{\tilde{\omega}( k )^{2}}{m^{2}}\frac{d}{d k}\left( \tilde{\delta}( k ) - \tilde{\delta}_{1}( k ) \right) \right) \, .