harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\aligned t^{-\delta}\int_0^t |t-s|^{-h/\gamma_1}s^{\delta-1}\|u(s)\|_{B^{k}_{p,q}}= t^{-\delta}\|u\|_{(k-s_1)/\gamma_1;k,p,q}\int_0^t |t-s|^{-h/\gamma_1}s^{\delta-1-(k-s_1)/\gamma_1}ds \leq C\|u\|_{(k-s_1)/\gamma_1;k,p,q}t^{-\delta}t^{-h/\gamma_1}t^{\delta-1-(k-s_1)/\gamma_1+1}\leq Ct^{-(k+h-s_1)/\gamma_1}\|u\|_{(k-s_1)/\gamma_1;k,p,q}\endaligned
|Y_2||Y_1||S_a| =|Y_2||Y_1+rS_a| &\leq{\left|Y_1+\frac{c-d}{e-f}Y_2\right||aS_a+Y_2|}&\leq{\left|eY_1-fY_1+cY_2-dY_2\right||aS_a+Y_2|}.
\Delta_{l}^{( 3 )}\Delta_{l}^{( 4 )}( \omega ) = \frac{1}{2 \omega}\sin ( 2 \omega R - l \pi ){h_{l}}^{( 1 )}( \omega a ){\frac{d}{d \omega}}[ \omega{h_{l}}^{( 1 )}( \omega a ) ] ,
g(0^{+})= \|v_{k}(\cdot,0)\|_{L^{3}}\geq g_{+}(0^{+})=\frac{1+\sqrt{1-4ab(0)^{2}C_{1}^{2}}}{2C_{1}b(0)}\geq \frac{2-4 C_{1}^{2}\|v_{k}(\cdot,0)\|_{L^{3}}}{2C_{1}}=\frac{1}{C_{1}}-2C_{1}\|v_{k}(\cdot,0)\|_{L^{3}},
\frac{d^{2}y}{dx^{2}}+\left( \frac{\gamma_{1}}{x-t_{1}}+\frac{\gamma_{2}}{x-t_{2}}+\frac{\gamma_{3}}{x-t_{3}}\right) \frac{dy}{dx}+\frac{\alpha\beta(x-t_{3})-q}{\prod_{j=1}^{3}(x-t_{j})}y=0.
G'_{\alpha}(\vec{\beta},{\bf w}) \, = \, -\alpha ( U \otimes (w_1 - w_2) + (w_1 - w_2) \otimes U ) - (\beta_1 - \beta_2) ( V \otimes w_1 + w_1 \otimes V ) ~~~- \beta_2 ( V \otimes (w_1 - w_2) + (w_1 - w_2) \otimes V ) - w_1 \otimes (w_1 - w_2) - (w_1 - w_2) \otimes w_2 \,.
V(\overline{\rho})=\frac{3}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\ln(p^{2}+e^{2}\overline{\rho}^{2})+\frac{1}{2}m^{2}\overline{\rho}^{2}+\frac{\lambda}{24}\overline{\rho}^{4}.
\frac{2}{\pi}\int_0^\infty dr \left(\psi_\ell^*(k,r) \psi_\ell(k,r) - \psi_\ell^{(0)}{}^*(k,r) \psi_\ell^{(0)}(k,r)\right) =\frac{1}{\pi}\frac{d\delta_\ell}{dk}
\Lambda = v^{\mu}\langle \partial_{\mu}\gamma_{2 1}\psi^{- 1}\rangle_{0}- v^{\mu}( \partial_{\mu}\ln{\rho}) \langle \psi \gamma_{2 1}\psi^{- 1}\rangle_{0}- v^{\mu}\partial_{\mu}\beta \langle \gamma_{5}\psi \gamma_{2 1}\psi^{- 1}\rangle_{0}.
I_{2}( k = 1 ) = - \frac{9 \pi}{8 1 9 2}K_{c}\approx - 0 . 0 9 6 8 3 , K_{c}\equiv \int_{0}^{\infty}\frac{( k_{c}^{2}+ 1 )^{3}( k_{c}^{2}+ 9 )^{2}}{( k_{c}^{2}+ 1 3 ) \cosh^{2}\frac{k_{c}\pi}{2}}\approx 2 8 . 0 5 4 9 .
\mathcal{F}(D(\lambda)f)(\xi) =-i \tilde{c}_\lambda r^{2\lambda+2-n}(\xi)\big[(n-2\lambda)\xi_nr^{n-2\lambda-2}(\xi)+r^{n-2\lambda}(\xi)\partial_n\big]\mathcal{F}(f)(\xi) =\tilde{c}_\lambda\mathcal{F}\big((n-2\lambda)\partial_n f-\Delta(x_n\cdot f) \big)(\xi),
V^{(1)}_3(r) = \lambda A - \frac{\lambda A}{1+\lambda r^2}+ \lambda B_1 (1+\lambda r^2) + \lambda B_2 (1+\lambda r^2)^2 + \lambda B_3 (1+\lambda r^2)^3{}+ \lambda B_4 (1+\lambda r^2)^4 + \lambda B_5 (1+\lambda r^2)^5 + \lambda B_6 (1+\lambda r^2)^6
\Pi^{i n}= - \frac{i}{R}\left\{\partial_{x}+ \frac{\sigma^{3}}{x}\left[ l - l_{0}- \frac{1}{2}\left( 1 - \sigma^{3}\right) + \vartheta \left( x \right) + \xi \rho_{R}x^{2}\right] \right\}\sigma^{1}, \; \rho_{R}= \gamma R^{2}/ 2 \; .
\Delta_J(rJ) + V \sum_{m=rJ+1}^{(r+1)J}D(m,a_m^*) \leq UJ + V \sum_{m=rJ+1}^{(r+1)J}D(m,a_m^*)+ \sum_{m=rJ+1}^{(r+1)J}q(m)y(m) = UJ + V \sum_{m=rJ+1}^{(r+1)J}D(m,a_m^*) + \sum_{m=rJ+1}^{(r+1)J}q(rJ+1)y(m) & + \sum_{m=rJ+1}^{(r+1)J}(q(m) - q(rJ+1)) y(m).
C\|G\mathbf{h}_i\|\|G\mathbf{e}_i\|=\frac{C}{\eta}\sqrt{\Im \mathbf{h}_i^* G\mathbf{h}_i}\sqrt{\Im G_{ii}}= \frac{C}{\eta}\sqrt{\Im \mathcal{G}_{ii}}\sqrt{\Im G_{ii}}\leq C'\frac{\Im(G_{ii}+\mathcal{G}_{ii})}{\eta},
\chi(k,\xi) = (k+N)^{\i\nu}\cdot\exp[\frac{1}{2\pi\i}\int\limits_{-N}^{k_0}\frac{\ln\frac{1+|r(s)|^2}{1+|r(k_0)|^2}\ \d s}{s-k}+\frac{1}{2\pi\i}\int\limits_{-\infty}^{-N}\frac{\ln(1+|r(s)|^2)\ \d s}{s-k}].
\lim_{m\rightarrow\infty}\sum_{0\leq n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i+1)-1-\delta-\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\frac{1}{2^{m+1}}=\frac{1}{2}\int_{0}^{1}(1-x)^{\frac{\gamma(i+1)-\delta-1-\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\mathrm{d}x,
{\cal S}={\frac{1}{2}}\mathrm{t r}[ \alpha \alpha^{t}] ={\frac{1}{2}}\mathrm{t r}[ a^{- 1}m b b^{t}m^{t}( a^{- 1})^{t}] ={\frac{1}{2}}m_{n}m_{k}\mathrm{t r}[ \alpha_{n}\alpha_{k}^{t}] = M^{t}G M ,
\sum_{J}\frac{(-i)^{|J|}}{J !}\partial_{\xi}^{J}q \, \partial_{x^\prime}^{J}q - \sum_{J}\frac{(-i)^{|J|}}{J !}\partial_{\xi}^{J}b \, \partial_{x^\prime}^{J}q - \frac{\partial q}{\partial x_n}+ c = 0,
\begin{aligned}\xi^{\prime}(x)+\eta^{\prime}(x) =\lambda\xi(x), x\in (0,1), \eta^{\prime\prime}(x)+\eta^{\prime}(x)+\xi^{\prime}(x) =\lambda\eta(x), x\in (0,1), \xi(0) =\xi(1), \eta(0)=0,\ \ \eta(1) =0.\end{aligned}
A_{\cal F}= - \tilde{A}^{T}K \tilde{A}- \frac{1}{4}\tilde{A}^{T}Z \tilde{A}- \tilde{A}^{T}J \tilde{A}+ 4{\cal M}_{\mu \nu}^{- 1 \A B}\partial_{\rho}{\cal H}_{B}^{\rho \nu}\partial_{\sigma}{\cal H}_{A}^{\sigma \mu}
P:=\frac{F_s}{|x-y|^2}\frac{\rho^{s-1}}{(\rho+1)^{\frac{N}{2}-1}}+\frac{2 V_s(\rho)\nabla F_s\cdot\nabla \rho +F_sV_s'(\rho) |\nabla \rho|^2 +F_sV_s(\rho)\Delta \rho}{4(s-1)}.
a_0 =p_1q_1(q_1-1)+\theta^\infty_2q_1+\theta^0, a_1 =\frac{1}{t_2}\big(\big(p_1q_1+\theta^\infty_2\big)(1-q_1)-p_2q_2+\theta^1\big), (B_1)_{32}=(q_1-1)(p_1q_1-p_2q_2)+\big(\theta^\infty_2-\theta^\infty_1\big)q_1+\frac{t_2}{t_1}(q_1-p_2)-\theta^1-\theta^\infty_2, (B_1)_{33}=(p_1+t_1)(q_1-1)+(p_1+q_2)q_1-p_2q_2.
\partial_a \partial_\nu H_{a, [\nu]}(x,y) = \frac{a}{16 \pi}\:\sum_{j=1}^\infty \frac{(-1)^j}{j!\:(j+1)!}\:(j+1)(-j)\:\frac{a^{j-1}}{4^j}\: \big( \xi^2 - \nu \big)^{j-1}&= \frac{a}{16 \pi}\:\sum_{j=0}^\infty \frac{(-1)^j}{j!\:(j+1)!}\:\frac{a^{j-1}}{4\cdot4^j}\: \big( \xi^2 - \nu \big)^{j}= \frac{1}{4}\: H_{a, [\nu]}(x,y)
q(3,3)^4 =1+r_8+r_4\pmod{2}, q(2,3)^2 =1+r_8+r_4+(r_6+r_{10}+r_{14})\pmod{2}, q(2,5)^2 =1+r_8+r_4+(r_2+r_6+r_{10})\pmod{2}, q(2,7)^2 =1+r_8+r_{12}+(r_2+r_{10}+r_{14})\pmod{2}.
\begin{array}{l}{L = N \sum_{k = 1}^{N}\left( ( \partial_{\mu}\phi_{k})^{2}+ V ( \phi_{k}) \right)}{+ \sum_{i , j = 1}^{N}( \phi_{i}- \phi_{j})^{2}| B_{\mu}^{i j}|^{2}}\end{array}
\sum_{i=t_0}^{t}n_i \leq 2n+ \sum_{i=t_0+2}^{t}n_i\leq 2n+(\lambda'+1)\binom{n}{2}- \sum_{i=1}^{t_0+1}n_i \leq 2n+(\lambda'+1)\binom{n}{2}-f(\lambda',n)= 2n+\binom{n}{2}<2\binom{n}{2}-2n,
0 = c_0 + 2\sum_{i=0}^{k-1}e(a_i) w_i\quad\mbox{and}0 = c_j + \sum_{i=0}^{j-1}(e(a_i) w_{k-j+i}+ \alpha e(a_{k-j+i}) w_i) + \sum_{i=j}^{k-1}(e(a_i) w_{i-j}+ e(a_{i-j}) w_i)
\delta D_{pk}({\bf r}^{\prime\prime},{\bf r}^\prime , i \omega) =D_{pj}(\varepsilon,{\bf r}^{\prime\prime},{\bf r}_2 , i \omega) \delta\varepsilon (i |\omega|) D_{jk}(\varepsilon,{\bf r}_2 ,{\bf r}^\prime , i \omega).
\|h\nabla u\|^2_{L^2(M)}=(-h^2\Delta_g u|u)_{L^2(M)}&=(A_\varepsilon u|u)_{L^2(M)}+\||\nabla\varphi_\varepsilon|u\|_{L^2(M)}^2 &\le C\|A_\varepsilon u\|_{L^2(M)}^2+C\|u\|_{L^2(M)}^2.
\psi_{\kappa}(\theta, x) \ = \E^{\hat v_1, \hat v^*_2}_x \Big[ e^{\frac{\kappa \|r_2\|_{\infty}}{\alpha}}e^{\theta \int^{T_{\kappa}}_0 e^{-\alpha s}r_2(X(t), \hat v_1(\theta(t),X(t)), \hat v^*_2(\theta(t), X(t))) dt}\Big] .
(d\Psi)_{x_0}= \left (\begin{matrix}\left(\frac{\partial \psi^i}{\partial x^j}(x_0)\right) 0 0 0 \left(f^\mu_\nu(x_0)\right)^t 0 0 0 \left(h^I_J(x_0)\right)^t\end{matrix}\right ).
{\mathbf O}\,\,{\mathbf R}\,\,{\mathbf O}^{-1}\:=\: -8m +{\cal M}_{\nu}^{\mu}{\mathbf a}^{\dagger}_{\nu}{\mathbf a}^{\mu}+{\cal N}_{\nu}^{\mu}{\mathbf b}^{\dagger\nu}{\mathbf b}_{\mu}.
\mathcal{N}=(\log \log T)^{-1/(2^{2d+3}\dim G)}+ \frac{M^{10^d \dim G}}{(\log \log T)^{1/2^{d+2}}}\ll \frac{M^{10^d \dim G}}{(\log \log T)^{1/(2^{2d+3}\dim G)}}.
||\dot{{\mathbf U}}||_{s,\ast,T}+||\varphi||_{H^s(\Gamma_T)}\leq C(K)T^{\frac{1}{2}}e^{C(K)T}\Big(||\mathbf{F}||_{s,\ast,T}+(||\dot{{\mathbf U}}||_{6,\ast,T}& \quad+||\varphi||_{H^6(\Gamma_T)}+||\mathbf{F}||_{4,\ast,T})||\hat{W}||_{s+4,\ast,T}\Big).
\frac{1}{(2\pi )^{n-1}n^{1/2}}\int_{\mathbb{A}^{(n)}_{\texttt{a}}}f(\boldsymbol{\xi}) \boldsymbol{\xi}=\frac{1}{n \, m^{n-1}}\sum_{\lambda\in\Lambda^{(m,n)}_{\texttt{a}}}f \left(\frac{2\pi \lambda}{m}\right) \delta^{(m,n)}_{\texttt{a};\lambda}(1)
V_k \sim V_\star + \frac{1}{\eta k + (V_0-V_\star)^{-1}}\quad\eta = \frac{(m^2-\tilde M \delta^2)^3}{4m^2\bigl( c^2(\tilde M-m^2) + 2G^2m^2(1-\delta^2)\bigr)}
\left|(u_k+\delta)\right|_{\tau_{n+1}}\leq\left(C|\Omega|^{1-\frac{1}{q}-\frac{2s}{N}}\right)^{\left(\sum\limits_{i=0}^{n}\frac{q}{\tau_i}\right)}\prod\limits_{i=0}^{n}\left(\frac{\tau_i}{q}\right)^{\frac{q}{\tau_i}}|(u_k+\delta)|_{q}
\int d x \left({\frac{1}{2}}\tilde{g}^{i j}{\frac{\delta^{2}\Psi}{\delta \phi^{i}\delta \phi^{j}}}+{\frac{1}{2}}g_{i j}\partial_{x}\phi^{i}\partial_{x}\phi^{j}\Psi \right) = i{\frac{\partial}{\partial t}}\Psi .
-\int_{\frac{a_n}{n-1}}^1 \log(1-s)\mathrm{B}_{k,n-k}(s) \, ds \leq \frac{n-1}{n-k-1}\int_{\frac{a_n}{n-1}}^1 \mathrm{B}_{k,n-k-1}(s) \, ds = o(n^{-(3-\epsilon)}),
- \frac{2 \pi}{2 \pi}{L}\hbar p \sum_{k \in \cal Z}\frac{1}{2}[ a_{k}^{\dagger}, a_{k + p}]_{-}\cdot | \lambda \varepsilon_{k , \mathrm{R}}|^{- s / 2}| \lambda \varepsilon_{k + p , \mathrm{R}}|^{-{3 s}/ 2}.
\begin{array}{l}\delta_1^{(1)}h_{\alpha \beta}= g(\partial_{\pi \kappa}\partial_{\kappa \nu}\xi_{\alpha \pi}h_{\nu \beta}+\partial_{\kappa \nu}\partial_{\nu \beta}\xi_{\kappa \pi}h_{\alpha \pi}- s\partial_{\alpha \pi}\partial_{\kappa \nu}\xi_{\pi \kappa}h_{\nu \beta}) + (\alpha \Leftrightarrow\beta )\end{array}
\begin{cases}\Delta^2u_k=g(|\lambda_k x|)u_k^p\quad \mbox{in}B_{\frac{1}{\lambda_k}}(0) u_k=\Delta u_k-(1-\sigma)\lambda_k(u_k)_n=0\quad \mbox{on}\partial B_{\frac{1}{\lambda_k}}(0),\end{cases}
\inf\left\{R:D_{1+s}(P_{X^{n}Y^{n}|U_{n}}\|\pi_{X^{n}Y^{n}}|P_{U_{n}})\rightarrow0\right\}\leq \inf_{\substack{\{P_{W^{n}},P_{X^{n}|W^{n}},P_{Y^{n}|W^{n}}\}_{n=1}^{\infty}: D_{1+s}(P_{X^{n}Y^{n}}\|\pi_{X^{n}Y^{n}})\rightarrow0}}\qquad\limsup_{n\to\infty}\frac{1}{n}D_{1+s}(P_{X^{n}Y^{n}|W^{n}}\|\pi_{X^{n}Y^{n}}|P_{W^{n}}).
(ds)^2 = -\frac{1}{\sqrt{\Lambda}}\left(1+\frac{2}{3}\delta{\rm cos}R\right){\rm sin}^2R d^2T + \frac{1}{\Lambda}\left(1-\frac{2}{3}\delta{\rm cos}R\right)d^2R + \frac{1}{\Lambda}\left(1-2\delta{\rm cos}\right)d^2\Omega_{(2)}
\bar{\cal X}_{, i}^{\mu}\bar{\cal X}_{, j}^{\nu}{\cal G}_{\mu \nu}= \bar{g}_{i j}, \bar{\cal X}_{, i}^{\mu}\bar{\eta}^{\nu}{\cal G}_{\mu \nu}= 0 , \bar{\eta}^{\mu}\bar{\eta}^{\nu}{\cal G}_{\mu \nu}= \varepsilon
H = \left[ \begin{array}{c c}{{1 \}}&{{\ 0}}\\{{0 \}}&{{\ - 1}}\\ \end{array}\right] \ , \ \E_{+}= \left[ \begin{array}{c c}{{0 \}}&{{\ 1}}\\{{0 \}}&{{\ 0}}\\ \end{array}\right] \ , \ \E_{-}= \left[ \begin{array}{c c}{{0 \}}&{{\ 0}}\\{{1 \}}&{{\ 0}}\\ \end{array}\right] \ .
[ a_{i j}( k^{+},{\bf k}^{\perp}) , a_{k l}^{\dagger}({\tilde{k}}^{+},{\tilde{\bf k}}^{\perp}) ] = \delta_{i k}\delta_{j l}\delta ( k^{+}-{\tilde{k}}^{+}) \delta ({\bf k}^{\perp}-{\tilde{\bf k}}^{\perp}) .
\Pi^{\mu \nu}=-i\frac{e^2}{12\pi^2}(p^\mu p^\nu -g^{\mu \nu}p^2)\{c_A(m;\{{\bar{\mu}},{\bar{c}}_i^0\})+6\int_0^1dzz(1-z)\ln[1-z(1-z)p^2/m^2]\};
\left( u_i^*, w_i^*, a_i^* \right) \mapsto \left( q_i^*, a_i:= a(u_i^* + w_i^* + \tilde{q}_i^*),\left\{a_i \diamond \partial_1^2 v_i (\cdot, a_0) \right\}\right) \stackrel{Thm. \ref{linear_theorem}}{\longmapsto}\left(u_i, w_i, a_i \right),
\begin{aligned}b \mathcal{L}\eta \leq \, \Big(C \delta^2 - \frac{\delta}{2}+ \frac{C}{U_{11}}\Big) F^{ii}U_{ii}^2 + C b^2 \sum_{i \in J}F^{ii}+ C \sum F^{ii}+ C (\delta^2 U_{11}^2 + b^2) F^{11}+ C.\end{aligned}
l(\gamma) = \int_0^{l(S)}||\gamma'(u)|| \du = \int_{0}^{l(S)}\omega(\varphi(u),\gamma'(u))\cdot||\varphi'(u)|| \du = = \int_0^{l(S)}\omega(\varphi(u),\gamma'(u)) \du = \int_0^{l(S)}\omega(\varphi(u),\gamma(u))' - \omega(\varphi'(u),\gamma(u)) \du = = \int_0^{l(S)}\omega(\gamma(u),\varphi'(u)) \du = \int_0^{l(S)}h_{\gamma}(u) \du.
( S_{n}^{a}+ i \eta M_{a b}\tilde{S}_{- n}^{b}) | B , \eta \rangle = 0 , \qquad ( S_{n}^{\dot{a}}+ i \eta M_{\dot{a}\dot{b}}\tilde{S}_{- n}^{\dot{b}}) | B , \eta \rangle = 0 .
\sum_{s=0}^ns\frac{x_s}{s!}\frac{B_{n-s, k-1}(x_1, x_2, \ldots)}{(n-s)!}= \frac{n}{k}\sum_{s=0}^n\frac{x_s}{s!}\frac{B_{n-s, k-1}(x_1, x_2, \ldots)}{(n-s)!}= \frac{1}{(n-1)!}B_{n,k}(x_1, x_2, \ldots)
2\delta(A^2)~= ~\lim\limits_{i\in \mathcal{I}}~2\delta(A\rho_i A) ~= ~\lim\limits_{i\in \mathcal{I}}~(2A^2\delta(\rho_i)+(6A\rho_i+2\rho_i A)\delta(A)-2\rho_i\delta(A^2)-(2A^2\rho_i+4A\rho_i A)\xi) ~= ~2\lim\limits_{i\in \mathcal{I}}A^2\delta(\rho_i)+8A\delta(A)-2\delta(A^2)-6A^2\xi.
\|\nabla v\|_{L^2(\Omega\setminus\Sigma_{p_1\varepsilon})}=\bigg(\int_0^{p_1\varepsilon}\int_{\partial S_t}|\nabla v|^2 dS\bigg)^{\frac{1}{2}}\leq C\varepsilon^{\frac{1}{2}}\Big\{\|F\|_{L^q(\Omega)}+\|h\|_{W^{1,q}(\Omega)}\Big\},
\mathbf{I}-\mathbf{\Delta}=\left(\mathbf{S}_{f}\right)^{-1}\mathbf{S}_{fg}\left(\mathbf{S}_{g}\right)^{-1}\Leftrightarrow\mathbf{S}_{f}\mathbf{\Delta}\mathbf{S}_{g}+\mathbf{S}_{fg}=\mathbf{S}_{f}\mathbf{S}_{g}.
\widetilde{V}^{( \nu^{2k+1})}(x;\Lambda_{2k})=V(x)-2\left\{\log \left[\psi_{\nu}(x)\prod\limits_{l=1}^{k}\left( \lambda_{2l}+\int_{x_{0}}^{x}dt\big( \Psi_{\nu}^{( \nu^{2l-1})}(t;\Lambda_{2l-1})\big)^{2}\right) \right]\right\}^{\prime \prime}.
( X )_{l l^{\prime}}= \int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2 \pi}\Psi_{n l m}^{*}( r , \theta , \varphi ; \delta_{1}, \delta_{2}) X \Psi_{n l^{\prime}m}( r , \theta , \varphi ; \delta_{1}, \delta_{2}) \, r^{2}\sin \theta d r d \theta d \varphi .
\nabla_{\theta}(P_nH(\theta)P_n) = (\nabla_{\theta}P_n)H(\theta)P_n + P_n(\nabla_{\theta}H(\theta))P_n + P_nH(\theta)\nabla_{\theta}P_n &= P_n(\nabla_{\theta}H(\theta))P_n + E_n\nabla_{\theta}(P_nP_n)=P_n(\nabla_{\theta}H(\theta))P_n + E_n\nabla_{\theta}P_n.
\nabla^L_{X_j}X_j=0,~~~1\leq j\leq 2,~~~\nabla^L_{X_1}X_2=\frac{1}{2}X_3,~~~ \nabla^L_{X_2}X_1=-\frac{1}{2}X_3, &\nabla^L_{X_1}X_3=-\frac{L}{2}X_2,~~\nabla^L_{X_3}X_1=-\frac{L}{2}X_2-X_3, & \nabla^L_{X_2}X_3=\nabla^L_{X_3}X_2=\frac{L}{2}X_1,~~\nabla^L_{X_3}X_3=LX_1.
\int_{0}^{T}\int_{\mathbb{T}}s|v|^2dxdt+\int_{\mathbb{T}}s(|v|_{t=0}^{2}+|v_{t=T}^{2})dx\leq C\left(\int_{0}^{T}\int_{\mathbb{T}}|e^{s\varphi}Pw|^2dxdt+\int_{0}^{T}\int_{\omega}s|v|^2dxdt\right),
\begin{aligned}g_1 =\sum_{\mu=0,1}\exp\left(\sum_{i=1}^N\mu_i(\xi_i-\phi_i)+\sum_{i<j}^N \mu_i\mu_j\gamma_{ij}\right), g_2 = \sum_{\mu=0,1}\exp\left(\sum_{i=1}^N\mu_i(\xi_i+\phi_i)+\sum_{i<j}^N \mu_i\mu_j\gamma_{ij}\right),\end{aligned}
\begin{aligned}|K_{\alpha}(x,y_{1},\dots,y_{j}, \dots , y_{m}) -K_{\alpha}(x,y_{1},\dots,y_{j}', \dots , y_{m}) | \le \dfrac{A}{\Big (\sum\limits_{j=1}^{m}|x-y_{j}| \Big)^{mn-\alpha}}\omega \Big( \frac{|y_{j}-y_{j}'|}{|x-y_{1}|+\cdots+|x-y_{m}|}\Big)\end{aligned}
\left( \frac{1}{c^{2}}\left( \frac{\partial}{\partial t}\right)^{2}-\sum_{j = 1}^{3}\left( \frac{\partial}{\partial x^{j}}\right)^{2}\right)A_{0}(x;y) = 4\pi m_{y}G \delta ({\bf x}-{\bf y}(t)),
{P(x,T,T^*) ={{{\mid H(T) \mid}^2}\over{{T_r}^3{\mid \eta (T) \mid}^6}}2 x A^2 e^{-2 \alpha x}{\left({{\alpha - \beta}\over{1 + 2 x \beta}}\right)}^2 \left( 1 + \alpha + \beta + 2 x \alpha \beta -{1/{4 x^2}}\right)}
|\epsilon \int_U u u_t(x,t) \, dx| \leq \frac{\epsilon}{2}(\int_U u(x,t)^2 \, dx) + \frac{\epsilon}{2}\int_U (u_t(x,t))^2 \, dx \leq \frac{\epsilon B^2}{2}\int_U (u_{xx}(x,t))^2 \, dx + \frac{\epsilon B^2}{2}\int_U (u_t(x,t))^2 \,dx \leq \epsilon B^2 E(t).
\frac{1}{2}\int_0^\tau \!\!\!\int_0^L |u|^2 dxdt = \frac{1}{2}\int_0^\tau\!\!\!\int_0^{\sqrt{\varepsilon}}|u|^2 dxdt + \frac{1}{2}\int_0^\tau \!\!\! \int_{\sqrt{\varepsilon}}^L |u|^2 dxdt =:I_1 + I_2.
\int_{\mathbb R}\vert\varphi(t)\vert^2dx_2-\int_0^t\!\!\!\int_{\mathbb R}(\partial_2\hat u^+_2)\varphi^2\,dx_2\,ds-2\int_0^t\!\!\!\int_{\mathbb R}\dot u^+_N\varphi\,dx_2\,ds-2\int_0^t\!\!\!\int_{\mathbb R}(\partial_1\hat u^+_N)\varphi^2\,dx_2\,ds=0\,,
f(z) = (z - \xi_k) a_{k}^{\#}(z) \widetilde{f}(z) + (p(z) - p(\xi_k)) + p(\xi_k) & = (z - \xi_k) \Big(a_{k}^{\#}(z) \widetilde{f}(z) + \frac{p(z) - p(\xi_k)}{z - \xi_k}\Big) + p(\xi_k) & = (z - \xi_k) f_{k}(z) + f(\xi_k),
\frac{\sqrt{h}}{2}h^{ab}\left[ 2g_{\mu\nu}(X^\alpha)\partial_a\partial_bX^\mu+2\partial_aX^\mu\partial_bX^\rho\partial_\rho g_{\mu\nu}(X^\alpha)+\partial_aX^\mu\partial_bX^\eta\partial_\nu g_{\mu\eta}(X^\alpha)\right]=0
\sigma_i = 2^{1/2}\left( \begin{array}{cc}e^3_i e^2_i e^1_i -e^4_i\end{array}\right);\qquad\bar\sigma_i = 2^{1/2}\left( \begin{array}{cc}- e^4_i - e^2_i - e^1_i e^3_i\end{array}\right).
c_\alpha^\top \alpha + c_\beta^\top \beta \to \max &\begin{cases}\alpha \leq b_{1:k}A \alpha + B \beta \leq b_{k+1:k+s}\bar A \alpha + \bar B \beta \leq b_{d}\alpha \in \mathbb{Z}^k,\, \beta \in \mathbb{Z}^s. \end{cases}
[M] = \frac{\prod\limits_{i=0}^N v_i^{x_i}}{\prod\limits_{n=1}^\infty (1- z^n)^{N}\prod\limits_{n=1}^\infty \prod\limits_{1 \leq i < j \leq N}(1-v_j v_i^{-1}z^{n-1}) (1- v_i v_j^{-1}z^n)}.
K=\tilde{p_1}^2-\tilde{p_1}\tilde{p_2}+\tilde{p_2}^2+\frac{4}{3}\tilde{q_1}^2+\frac{4}{3}\tilde{q_1}\tilde{q_2}+\frac{4}{3}\tilde{q_2}^2+\frac{4}{9}\sqrt{3}\tilde{q_1}^2\tilde{q_2}+\frac{4}{9}\sqrt{3}\tilde{q_1}\tilde{q_2}^2.
\Phi ( \xi \eta ; P ) = \sum_{1 2 3}D ({\hat{q}}_{1 2}){\Delta_{1}}^{- 1}{\Delta_{2}}^{- 1}\phi ({\hat{\xi}}{\hat{\eta}}; P ) \times \sqrt{\frac{\delta ( \nu_{3}-{\hat{\omega}}_{3}/ M )}{M{\hat{\omega}}_{3}{\Delta}_{3}}}
R = \left[ \begin{array}{l l l l}{{q}}&{{\,}}&{{\,}}&{{\,}}\\{{\,}}&{{1}}&{{0}}&{{\,}}\\{{\,}}&{{\lambda}}&{{1}}&{{\,}}\\{{\,}}&{{\,}}&{{\,}}&{{q}}\\ \end{array}\right] \; , \quad \hat{R}= \left[ \begin{array}{l l l l}{{q}}&{{\,}}&{{\,}}&{{\,}}\\{{\,}}&{{\lambda}}&{{1}}&{{\,}}\\{{\,}}&{{1}}&{{0}}&{{\,}}\\{{\,}}&{{\,}}&{{\,}}&{{q}}\\ \end{array}\right] \; , \quad{\cal P}R{\cal P}= R^{t}\; ,
D=\det \left(\begin{array}{cc}g_{tt}\lambda D_{t}\Phi^J -\lambda D_t\Phi^I g^{IJ}+i\lambda [\Phi^I,\Phi^J] \end{array}\right)=\det \left(\begin{array}{ccc}D_{tt}D_{tj}D_{tb}D_{it}D_{ij}D_{ib}D_{at}D_{aj}D_{ab}\end{array}\right) \ ,
p^{22}=\left[\begin{array}{ccccc}1 1 2 3 3 2 2 1 2 1 3 3 3 1 2 \end{array}\right],p^{23}=\left[\begin{array}{ccccc}1 1 2 1 2 2 2 1 3 3 3 3 3 2 1 \end{array}\right],p^{24}=\left[\begin{array}{ccccc}1 1 2 2 3 2 2 1 3 1 3 3 3 1 2 \end{array}\right],
e_{+}= \frac{1}{\sqrt{1+ |a|^2 \l^{2}}}\begin{pmatrix}e^{i \theta}a \l -\bar a\l e^{-i \theta}\end{pmatrix}, \;e_{-}= \frac{1}{\sqrt{1+ |b|^2 \l^{-2}}}\begin{pmatrix}e^{i \kappa}b \l^{-1}-\bar b\l^{-1}e^{-i \kappa}\end{pmatrix},
\mu_N(P) \leq \exp \left\{M_k \left(- \log Z_N + \frac{N \delta \log b}{2}\right)\right\}= \exp\left\{M_k \log b \left(-\log_b Z_N + \frac{\delta N}{2}\right)\right\}= b^{-M_k \left(\log_b Z_N - \frac{\delta N}{2}\right)}.
({\gamma}^{\mu}i{\partial}_{\mu}- m_{2}){\psi}_{2}( x , t ) - \frac{1}{2}e_{2}b ( t ){\gamma}^{1}{\psi}_{2}( x , t ) + e_{2}\int_{0}^{\mathrm{L}}d y D ( x , y | \mathrm{L}) J^{0}( y , t ){\gamma}^{0}{\psi}_{2}( x , t ) = 0 .
\lambda_{h}^{T, (n,k)}= i R_0 - h^{\frac 23}a_n^+(\kappa) e^{-i\pi/3}+ h (2k-1)\frac{e^{-i\pi/4}}{\sqrt{2 R_0}}+ h^{\frac 43}\frac{e^{-i\pi/6}}{2R_0\, a_n^+(\kappa)}+ \mathcal O(h^{\frac 53})\, .
{\bf V}= \left( \begin{matrix}{{\cos \varphi_{p}}}&{{- \epsilon ( p_{0}) e^{\beta \mu / 2}\sin \varphi_{p}}}\\{{\epsilon ( p_{0}) e^{- \beta \mu / 2}\sin \varphi_{p}}}&{{\cos \varphi_{p}}}\\ \end{matrix}\right) , \qquad \cos \varphi_{p}={\frac{\theta ( p_{0}) e^{x / 4}+ \theta ( - p_{0}) e^{- x / 4}}{\sqrt{e^{x / 2}+ e^{- x / 2}}}},
F_2(x,y)=\begin{cases}\min{(x, y)}y< g(x) y=g(x) x<g^2(x), \max{(x,y)}y>g(x) y=g(x) x>g^2(x), \min{(x, y)}\max{(x,y)}y=g(x) x=g^2(x). \end{cases}
C^0_{com}(\mathfrak{g}, M) :=~ \{m \in M ~|~ x \cdot_1 m - m \cdot_1 x = x \cdot_2 m - m \cdot_2 x, ~\forall x \in \mathfrak{g}\}, C^n_{com}(\mathfrak{g}, M) :=~ \underbrace{C^n (\mathfrak{g}, M) \oplus \cdots \oplus C^n (\mathfrak{g}, M)}_{n}, ~ n \geq 1.
\delta A_d[q_d] =\delta\sum_{m=0}^{N-1}L_d (q_m,q_{m+1}), =\sum_{m=0}^{N-1}[D_1 L_d (q_m,q_{m+1}) \cdot \delta{q_m}+D_2 L_d (q_m,q_{m+1}) \cdot \delta{q_{m+1}}].
F_{0 1}= \frac{F_{0 1}^{c}}{\sqrt{1 + \left( \frac{\alpha^{\prime}s}{R^{2}G_{o}^{2}}\right)^{2}}}, \quad F_{0 1}^{c}= \frac{2 \pi R^{2}}{s}\frac{1}{G_{o}^{2}( 2 \pi \alpha^{\prime})^{2}}
\hom(G,H) \leq s(\delta,H)(k^2-1)k^{n-\delta-2}+|V(H)|^\delta(k-1)^{cn}k^{n-cn-\delta}&=s(\delta,H)k^{n-\delta}-s(\delta,H)k^{n-\delta-2}+|V(H)|^\delta(k-1)^{cn}k^{n-cn-\delta}&<s(\delta,H)k^{n-\delta}
\nu = \begin{cases}\frac{3n-2 - p(2n-1)}{2n(p-1)}, \mbox{if}\ 1< p \le \frac{3n-2}{2n-1}, 0, \mbox{if}\ \frac{3n-2}{2n-1}< p \le 2-\frac{1}{n}, \end{cases}
\begin{array}{r l r l r l}{\hat{O}}&{}&{\stackrel{\cal W}{\longleftrightarrow}}&{}&{\tilde{o}( z ) \nonumber}\\{}&{\searrow \nwarrow}&{}&{W \swarrow \nearrow}&{\nonumber}\\{}&{}&{o ( y , x )}&{}&{}\\ \end{array}
\delta \bar{\theta}^{A}( \sigma ) = - \bar{\epsilon}^{A}+ \bar{\kappa}^{B}( \sigma ) ({\bf 1}\, \delta^{B A}+ \Gamma^{B A}( \sigma ) ) + f^{A}{}_{B C}\Lambda^{B}( \sigma ) \bar{\theta}^{C}( \sigma ) \, .
H_{c}^{r}={\frac{1}{2}}\int \left[ \left( \pi_{1}\phi_{2}- \pi_{2}\phi_{1}\right)^{2}-{\frac{1}{\phi_{1}^{2}+ \phi_{2}^{2}}}\left( \phi_{1}\partial_{i}\partial_{i}\phi_{1}+ \phi_{2}\partial_{i}\partial_{i}\phi_{2}\right) \right] d x .
\sum_{\substack{|K| \leq N' -O(\gamma) \leq |J| \leq N|E(\gamma)|}}\mathbf{t}^{J - n/2}\int_{D_{U_{\delta}, \gamma}}e^{-\sum_{e \in E(\gamma)}Q^{}_e(w, y)/4 t_e}c_{J, K}(w) y^K\, dy dw = \sum_{\substack{|K| \leq N', -O(\gamma) \leq |J| \leq N|E(\gamma)|}}\mathbf{t}^{J - n/2}\int_B c_{J, K}(w) \mathcal{I}^K(w, \mathbf{t})\, dw
u_i =\frac{\log{e_i}}{\log{R}}, n_j(p) =\#\{i\in\{1,\dots,j\}:p\nmid W_i\}, g_j(d) =\prod_{p|d}(g(p)+n_j(p)), c_j =\prod_p\Bigl(1+\frac{n_j(p)}{g(p)}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^j.
\widehat{S}_{t}^{H}\psi(x)=\left( \frac{1}{2\pi i\hbar}\right)^{1/2}\frac{i^{m}}{\sqrt{|\sin t|}}\int_{-\infty}^{\infty}\exp\left[ \frac{i}{\hbar}\frac{(x^{2}+x^{\prime2})\cos t-2xx^{\prime}}{2\sin t}\right]\psi(x^{\prime})dx^{\prime}.
\cdots \stackrel{a ( l - 1 )}{\longleftarrow}{\cal E}( l - 1 ) \stackrel{a ( l )}{\longleftarrow}{\cal E}( l ) \stackrel{a ( l + 1 )}{\longleftarrow}{\cal E}( l + 1 ) \stackrel{a ( l + 2 )}{\longleftarrow}\cdots
D^{+}(n,k) =(k+1)D^+(n-1,k)+(n-2k+1)D^+(n-1,k-1)+D^-(n-1,k)+ &kT^+(n-1,k)+(n-2k+1)T^+(n-1,k-1) &=k(D^+(n-1,k)+T^+(n-1,k))+(n-2k+1)(D^+(n-1,k-1)+T^+(n-1,k-1)) &+D^+(n-1,k)+D^-(n-1,k) &=kR^+(n-1,k)+(n-2k+1)R^+(n-1,k-1)+D(n-1,k),
\frac{\Phi(1)^*}{|\Phi(1)|^2}= E(z)^\top + E(z)^\top \frac{[\Phi(z) - \Phi(1)] \Phi(1)^*}{|\Phi(1)|^2}+ Q_{\Phi(z)}Q_{E(z)}^\top \frac{\Phi(1)^*}{|\Phi(1)|^2},
\|\mathcal{M}_{p_0}(\vec{f})\|_{L^{p(\cdot)}(w_1^{p(\cdot)}\cdots w_N^{p(\cdot)})}= \|\mathcal{M}_{p_0}(\vec{f})w_1\cdots w_N\|_{L^{p(\cdot)}}\leq \|\prod_{i=1}^N\{\mathcal{M}_{p_0}(f_i)w_i\}\|_{L^{p(\cdot)}}\leq C\|\mathcal{M}_{p_0}(f_1)w_1\|_{L^{p_1(\cdot)}}\cdots\|\mathcal{M}_{p_0}(f_N)w_N\|_{L^{p_N(\cdot)}},
\eta_1=\frac{2d_2^2}{d_1\gamma_{22}^1}\left(\zeta_1+\frac{\gamma_{11}^1}{4d_1}\right),\eta_2=\frac{2d_2^2}{d_1\gamma_{22}^1}\left(\zeta_2+\frac{\gamma_{22}^2}{4d_2}+\frac{\gamma_{22}^1}{d_2}\right).
K_{ij}(x_0 ,x;y_0 ,y;t) = \frac{1}{\beta (2\pi)^3}\sum_{r=-\infty}^{\infty}\int d^3 k \, e^{i(k(x-y) + 2\pi r (x_0 -y_0 )/\beta )}e^{-tk^2}e^{-t(2\pi r/\beta -a_0 (y) )^2}\left( e^{-(A+B)}\right)_{ij}
A^{n,q}_{F\otimes L,h_F\otimes h_\nu,\omega}=[i\Theta_{F,h_F}\otimes\mathrm{id}_L,\Lambda_\omega]+[i\Theta_{L,h_\nu}\otimes\mathrm{id}_F,\Lambda_\omega] \geq B_{h_F,\omega}-\beta_\nu[\omega\otimes\mathrm{id}_F,\Lambda_\omega]=B_{h_F,\omega}-q\beta_\nu.
\mathcal{G}_{\alpha, \beta, z_{21}, z_{12}, \tilde{z}_{21}, \tilde{z}_{12}}= \big\{(X, Z): \|Z_{21}\|_F\leq \tilde{z}_{21}, \|Z_{12}\|_F\leq \tilde{z}_{12}, & (X, Z)\in \mathcal{F}_{r, \alpha, \beta, z_{21}, z_{12}}\big\}.
\mathbb{E}[\delta_Z(z)|\mathcal{F}_t] = \frac{1}{2\pi}\int_{\mathbb{R}}\exp\big[\int_0^t\int_{\mathbb{R}}ix\psi(s,\zeta)\tilde{N}(ds,d\zeta) +\int_0^t ix\beta(s)dB(s) &+\int_t^{T_0}\int_{\mathbb{R}}(e^{ix\psi(s,\zeta)}-1-ix\psi(s,\zeta))\nu(d\zeta)ds-\int_t^{T_0}\frac{1}{2}x^2\beta^2(s)ds-ixz\big]dx.
I = \int d^{4}x [ - \frac{1}{4}a_{m n}a^{m n}- \frac{1}{1 2}H_{m n p}H^{m n p}- \frac{1}{4}\mu \epsilon^{m n p q}B_{m n}a_{p q}+ \frac{1}{2}\epsilon^{m n p q}a_{m n}\partial_{p}a_{q}] .
g=\frac{1}{2}{\textstyle\sum_{k,l=1}^{s+1}}g_{kl}\left( \mathrm{d}z_{k}\otimes\mathrm{d}\overline{z_{l}}+\mathrm{d}\overline{z_{l}}\otimes\mathrm{d}z_{k}\right) \qquad\omega={\textstyle\sum_{k,l=1}^{s+1}}\frac{i}{2}g_{kl}\mathrm{d}z_{k}\wedge\mathrm{d}\overline{z_{l}}
\left[ \begin{array}{c}{A_{j m}^{( 1 )}}\\{A_{j m}^{( 2 )}}\\{A_{j m}^{( 3 )}}\\ \end{array}\right] = R \cdot \left[ \begin{array}{c}{a_{j m}^{( 1 )}}\\{a_{j m}^{( 2 )}}\\{a_{j m}^{( 3 )}}\\ \end{array}\right] ,
[M_{0j},P_0] = \frac{1}{2}[\big( d\Gamma(X_jP_0)+d\Gamma(P_0X_j)\big),d\Gamma (P_0)] &=\frac{1}{2}\big( d\Gamma([X_j,P_0]P_0)+d\Gamma(P_0[X_j,P_0])\big) &=i \, d\Gamma(V_jP_0)=i \, d\Gamma(P_j)=iP_j,