harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{deg}_a Z_{f_{t_0}}=\mathrm{deg}(L\cdot Z_{f_{t_0}})= &=\mathrm{deg}(L \cdot Z_{f_t})= &=\sum_{b\in L\cap f_t^{-1}(0)}i(L\cdot Z_{f_t},b)\{b\}\geq &\geq i(L\cdot Z_{f_t},a)\{a\}\geq \mathrm{deg}_a Z_{f_t},
(U \otimes I_\varrho)\sigma_{\varrho, \rho}=UV^*\varrho(V)=(V')^*\varrho(V)=(V')^*\varrho(V')\varrho((V')^*V)=\sigma_{\varrho, \rho'}\varrho(U)=\sigma_{\varrho, \rho'}(I_\varrho \otimes U),
\cal{F}_{F(\lambda)}(z)= F_{11}(\lambda)+\frac{F_{12}(\lambda)F_{21}(\lambda)z}{1-F_{22}(\lambda)z}=x_1(\lambda)+\frac{(x_1(\lambda)x_2(\lambda)-x_3(\lambda))z}{1-x_2(\lambda)z}= \frac{x_1(\lambda)-x_3(\lambda)z}{1-x_2(\lambda)z}=\frac{x_3(\lambda)z-x_1(\lambda)}{x_2(\lambda)z-1}=\Psi(z,x(\lambda))
\int_{0}^{\infty}e^{-t}\exp\left( itsH\right) dt=\frac{1}{I-isH}=\frac{1}{\det\left( I-isH\right)}~\sum_{n=0}^{\operatorname*{rank}\left(H\right) -1}\left( isH\right)^{n}\operatorname*{Trunc}_{\operatorname*{rank}\left( H\right) -1-n}\left[ \det\left( I-isH\right)\right] \ ,
{\mathrm{Res}}_{z=0}\frac{H^{(1)}_{\mu}(\lambda z)}{H^{(1)}_{\nu}(z)}F(z)={\mathrm{Res}}_{z=0}\frac{J_\nu (z)J_\mu (\lambda z)+Y_\nu (z)Y_\mu (\lambda z)}{J_\nu^2(z)+Y_\nu^2(z)}F(z)
\begin{aligned}x^{k+1}=\left((I+\gamma_1^kR)\otimes I_d\right) x^k+\gamma_1^k \zeta_w^k-\lambda^k y^k y^{k+1}=\left(((1-\alpha^k)I+\gamma_2^kC)\otimes I_d\right) y^k +\gamma_2^k \xi_w^k +g^{k+1}&\qquad-(1-\alpha^k)g^{k}\end{aligned}
\overline{{\Theta}}_{\mu \nu}^{a b}= - g_{\mu \nu}\left[ \left( \nabla_{\lambda}- i g ( n^{c}T^{c}) \overline{{A}}_{\lambda}\right)^{2}\right]^{a b}+ i g ( n^{c}T^{c})^{a b}\overline{{F}}_{\mu \nu}- \delta^{a b}R_{\mu \nu}.
S(\omega, \theta) = -\int_{B_R}V e^{{\rm i}\sqrt[4]{\lambda}(\omega - \theta)\cdot x}{\rm d}x - \int_{B_R}R_V(\lambda) (-V \varphi_{\omega}) V\varphi_{-\theta}{\rm d}x- 2\sqrt{\lambda}\int_{\partial B_R}\varphi_{\omega}\partial_\nu (\varphi_{-\theta}){\rm d}s(x).
\hat{{\cal H}}( s , \chi ) = s \left( m^{2}- \Pi^{2}+ \frac{i g}{2}F_{\alpha \beta}\tilde{\gamma}^{\alpha}\tilde{\gamma}^{\beta}\right) + \left( \Pi_{\nu}\tilde{\gamma}^{\nu}- m \tilde{\gamma}^{5}\right) \chi \; .
Z_{\rm gr}[\varphi,\mu,\bar{\mu}]=e^{K[\varphi,\mu,\bar{\mu}]}\int{\cal D}T{\cal D}\overline{T}\,\,|\chi[\mu,T]|^2 e^{-{1\over 2}\int d^2w\,\, e^{-\varphi}T\overline{T}},
-\frac12[f'(s|D|),\delta(a)] =\delta^2(a)\cdot sf(s|D|)+s|D|f^{1/2}(s|D|)\cdot[f^{1/2}(s|D|),\delta(a)] &+[f^{1/2}(s|D|),\delta(a)]\cdot s|D|f^{1/2}(s|D|)+[f^{1/2}(s|D|),\delta^2(a)]\cdot sf^{1/2}(s|D|),
\left[C_{\gamma}(\mathcal{Z}_{m,n}^{\gamma}) \right](z) = \dfrac{(-1)^{n}(\gamma+1)_{m+n}(\gamma+m+2)_{n-1}(-m-1)_{n}}{(\gamma +1)_{n}(m-n+2)_{n-1}(m+1)}\left(1-|z|^2\right)^{\gamma+1}& \frac{(\gamma+2)_{m}(\gamma+2)_{n-1}}{\left((\gamma+2)_{m+n-1}\right)^2}\mathcal{Z}_{m,n-1}^{\gamma+1}(z,\bar{z}),
D_{s\eta^i}=(\phi_s^i)^{-1}_*\circ D_s= e_1\cdot(\phi_s^i)^{-1}_*(\partial_{\tau})+e_2\cdot(\phi_s^i)^{-1}_*(\partial_u)+e_3\cdot(\phi_s^i)^{-1}_*(\partial_{\bar{u}}) &+\frac{1}{2}\sum_{i=1}^3e_i\sum_{k<l}(\phi_s^i)^{-1}_*(\omega_{kl}(e_i))e_ke_l.
\begin{array}{l}{\lambda\in N_{K_\varphi}(\varphi(\bar x,\bar u,\bar v)),}(\alpha,0,0)\in \partial\langle \lambda, \varphi \rangle(\bar x,\bar u,\bar v)+\{0\}\times N_{U}(\bar u)\times \{0\}\end{array}\Longrightarrow \alpha=0
\Lambda_{k}\left( P\right) \equiv \left\{q_{k}\left( \overline{\mathbf{A}}_{k},\overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k}\right) :E_{P}\left[q_{k}\left( \overline{\mathbf{A}}_{k},\overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k}\right) |\overline{\mathbf{A}}_{k-1},\overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k}\right] =0\right\}.
\int_M df\wedge d^c_B f\wedge \omega^{n-1}_h\wedge \eta\leq \sum \int_M df\wedge d^c_B f\wedge \omega_u^k\wedge \omega^{n-1-k}_v\wedge \eta=-\int_M \frac{f}{2}(\omega_u^n-\omega_v^n)\wedge \eta,
M_{U\ \scriptstyle\rm min}^{E_8 - E_6}\approx 5.49\times 10^{17}\times g^{E_8 - E_6}_U\times\frac{1}{\sqrt{1 + 0.08 \times \left(g^{E_8 - E_6}_U\right)^2}}\,{\rm GeV}\, .
- \frac{d M ( u )}{d u}= \frac{1}{4 \pi}\int d^{2}{\cal S}\Bigl [ \Bigl ( \frac{\partial}{\partial u}C_{1}\Bigr )^{2}+ \Bigl ( \frac{\partial}{\partial u}C_{2}\Bigr )^{2}\Bigr ] + \frac{1}{8 \pi}\int d^{2}{\cal S}\: \frac{1}{4}\: r^{2}R^{\mu \nu}\nabla_{\mu}v \nabla_{\nu}v \biggl |_{{\cal I}^{+}}
\tilde{\theta}^{0}= \theta^{s - 1}\in H^{1}_{0}(\Omega) \tilde{g}= \partial_{t}^{s - 1}\bar{g}\in C^{0}\big([0, T], L^{2}(\Omega)\big) \hookrightarrow L^{2}\big(0, T; L^{2}(\Omega)\big)
\mathcal{A}(\{n, n\}, \{n, n\}) = 2\left[\binom{x + n - 1}{n}^2 \bigcirc \binom{x + n}{n}^2\right] = 2\sum_i (-1)^i\binom{n}{i - n}\binom{i}{n}\binom{n + i - 1}{n}^2.
\varphi :=z\phi_{x^0}+\frac{s}{r}\phi_r+\phi_s, W :=\frac{1}{\phi}\left\{\frac{\varphi}{2}-s\phi U - \frac{\phi_z\Omega}{2\Lambda}(\varphi_z-2\phi_{x^0})-(r^2-s^2)\left[{\phi_s}U-{\phi_z}V\right]\right\}, U :=\frac{1}{2\Lambda}\left\{\left(\varphi_s-\frac{2}{r}\phi_r\right)\phi_{zz}-\left(\varphi_z-2\phi_{x^0}\right)\phi_{sz}\right\},
\Big[ \frac{1}{lk}\Big( \underset{i=1}{\overset{k}{\sum}}\underset{j=1}{\overset{l}{\sum}}(B^j_i - p_i) \Big)^2 \Big] = \frac{1}{l^2k^2}\Big[ \Big(\sum_{j=1}^l \sum_{i=1}^k (B_i^j - p_i) \Big)^2\Big] \leq 1/8.
\mathbb{E}[ C_2^(a_) ] = \frac{e^{\frac{2}{\beta\xi}}}{\ln(4)}E_1\left(\frac{2}{\beta\xi}\right) + \int_0^\infty \frac{2}{\beta\ln(2)}\cdot \exp\left(-\frac{x}{\beta}\left(\frac{\sqrt{1+\xi x}-2}{\sqrt{1+\xi x}-1}\right)\right) E_1\left( \frac{x\sqrt{1+\xi x}}{\beta(\sqrt{1+\xi x}- 1)}\right)dx.
H_{\Delta_k}(y_1 , y_2 ;m) = (\frac{4}{m}+2) H_{2,1,1}\left(y_1,y_2;m\right) -\frac{4 y_1 H_{2,2,1}\left(y_1,y_2 ;m\right)}{m}-\frac{8 H_{3,1,1}\left(y_1,y_2;m\right)}{m}.
F_{B(F)}\left(\beta,V,\frac{1}{m}\right)= -\frac{V}{2^{\frac{d}{2}+1}\pi^{\frac{d}{2}}}\int_{0}^{\infty}ds\;s^{-1-\frac{d}{2}}\left[ \theta_{3(4)}\left(0\left|\frac{i\beta^{2}}{2\pi s}\right.\right)-1\right] e^{-\frac{m^{2}s}{2}}\;.
\begin{aligned}C_{m_0,n}:= \{x \in [0,1] \mid t_i^{m_0,n}(x) \notin U i \}, C_{m,n}:= \{t_i^{m,n}(x) \mid x \in C_{m,n}\}, C_m := \bigcap_{n \geq m}C_{m,n}. \end{aligned}
\left(c;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{\left(-\sqrt{q};\sqrt{q}\right)_{n}q^{\binom{n}{2}}c^{n}}{\left(\sqrt{q},\sqrt{c},-\sqrt{c};\sqrt{q}\right)_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}c^{n}A_{q}^{2}\left(-cq^{n-1/2}\right)}{\left(q;q\right)_{n}}
\int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \partial\omega_{n-2}= \int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \partial\omega\wedge\omega_{n-3}= - \int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \bar\partial \rho_\omega^{2,\,0}\wedge\omega_{n-3}.
\sum_{n=1}^{\infty}n\Vert f_n^d \Vert_{L^{1}(S_2)}\leq \Big[\sum_{n=1}^{\infty}\dfrac{1}{n^2}\Big]^{1/2}\Big[\sum_{n=1}^{\infty}n^4\Vert f_n^d \Vert_{L^{1}(S_2)}^2\Big]^{1/2}
x_+(t)+\widetilde{x}_+(-t) =x_+(t_0)+\widetilde{x}_+(-t_0)+\int_{t_0}^t q_t(\tau-t_0,x(t_0)+x_+(t_0))d\tau -\int_{t_0}^t q_t(\tau-t_0,x(t_0)-\widetilde{x}_+(-t_0))d\tau \geq \varepsilon(t-t_0)+x_+(t_0)+\widetilde{x}_+(-t_0).
v^a \mapsto \big(\Sigma v^a\big)^{A}{}_{B}= \,\,{\mathbb Y}_a{}^{A}{}_{B}v^a+{\mathbb Z}_a{}^b{}^A{}_B\big((\nabla_bv^a)_o+\tfrac{1}{6}\delta_b{}^{a}\nabla_cv^c\big) & -\tfrac{1}{3}{\mathbb W}^{A}{}_{B}\nabla_cv^c-{\mathbb X}^a{}^A{}_B(\tfrac{1}{3}\nabla_a\nabla_b+P_{ab})v^b .
Z_{i}= \frac{2 \pi M_{1 1}^{3}V_{7}}{V_{3 , i}^{2}}( i \leq N_{3}) \ , \quad \frac{1}{2 \kappa_{4}^{2}}= Z_{N_{3}+ 1}={2 \pi M_{1 1}^{9}V_{7}}\ .
f(x,\sigma^{2}|\mu,\lambda,\alpha,\beta)=\frac{\sqrt{\lambda}}{\sigma\sqrt{2\pi}}\frac{\beta^{\alpha}}{\Gamma(\alpha)}(\frac{1}{\sigma^{2}})^{\alpha+1}exp(-\frac{2\beta+\lambda(x-\mu)^{2}}{2\sigma^{2}})
\displaystyle \int_{-\infty}^{\infty}\frac{1- \displaystyle e^{-\frac{1}{2}hu^2}}{u^2}\,du = \sqrt{2\pi h}\textup{P.V.}\int_{-\infty}^{\infty}\frac{tu}{u^2}\, e^{-\frac{1}{2}hu^2}\,du=0,
\langle x,\tau\vert x'',0\rangle=\delta^2(x-x')+e\int_0^\tau d\tau'd^2x''{d^2p\over (2\pi)^2}\langle x,\tau'\vert x'',0\rangle{p\!\!\!\slash A\!\!\!\!\slash(x')\over p^2}e^{-ip\cdot (x''-x')}\;\;
\begin{cases}R_1 || |\nabla_{H}u| ||_{R_1,2R_1}\leq C(1+K^{1/2}) ||u\psi^{1/2}||_{4R_1}, R_2|| |\nabla_{H}u| ||_{R_2,2R_2}\leq C( 1+ K^{1/2}) || u \psi^{1/2}||_{R_0}.\end{cases}
\left\|\dfrac{R-q}{\bar{z}}\right\|_{L^2(\Omega)}^2= \int_r^1\int_{0}^{2\pi}\dfrac{|R(te^{i\theta})-q(t)|^2}{t^2}tdtd\theta \leq \int_r^1\int_{0}^{2\pi}\dfrac{|\partial_{\theta}R(te^{i\theta})|^2}{t^2}tdtd\theta \leq \left\|\nabla{R}\right\|_{L^2(\Omega)}^2\leq C\epsilon_0.
M_2 := \left(\int_{1}^{t}\frac{M_{|g|}(u)}{u^{2}}du\right)^{-1}\int_{t^{\kappa}}^t \frac{|\Delta_h(u)|}{u^2}du, M_3 := R_h(\lambda)\left(\int_1^t \frac{M_{|g|}(u)}{u^2}du\right)^{-1}\left(\int_1^{t^{\kappa}}\frac{M_{|g|}(u)}{u^2\log^{\lambda}(3u)}du\right).
2 Im(F(h)) = \frac{W_{\delta,2\delta_s}(\omega',\omega_0')}{|I'_0(h)|^2}+ \frac{W_{\delta,\delta_c^1+\delta_c^2}(\omega',\omega_0')}{|I'_0(h)|^2}= \frac{W_{\delta,\delta_s}(\omega',\omega_0')}{|I'_0(h)|^2}, \;\; h \in (-\infty,h_c^1) .
X := H^{1}_{0}(\Omega)^{d}= \{v \in H^{1}(\Omega)^d : v = 0 \; on \; \partial \Omega \}, \;Q := L^{2}_{0}(\Omega) = \{q \in L^{2}(\Omega) : (1,q) = 0 \}, V := \{v \in X : (q,\nabla \cdot v) = 0 \; \forall \; q \in Q \}.
\frac{d_{+}}{d_{-}}= \frac{\Gamma ( - \frac{2 j + 1}{k + 2})}{\Gamma ( \frac{2 j + 1}{k + 2})}\; \sqrt{\frac{\Gamma ( \frac{2 j}{k + 2}) \; \Gamma ( \frac{2 j + 2}{k + 2})}{\Gamma ( - \frac{2 j}{k + 2}) \; \Gamma ( - \frac{2 j + 2}{k + 2})}}\; = \; \frac{\gamma_{-}}{\gamma_{+}},
P_{\epsilon}((x_{\epsilon,\eta})_{*}) =u_{\epsilon}(x_{\epsilon,\eta}) -\frac{1}{\epsilon}|(x_{\epsilon,\eta})_{*}-x_{\epsilon,\eta}|^{2}= u(x_{\epsilon,\eta})_{*}), \nabla P_{\epsilon}((x_{\epsilon,\eta})_{*}) =\nabla u_{\epsilon}(x_{\epsilon,\eta}), \nabla^{2}P_{\epsilon}((x_{\epsilon,\eta})_{*}) =\nabla^{2}u_{\epsilon}(x_{\epsilon,\eta}).
f:= \sum_{j\in H}\left(100(x_{j+1}-x_{j}^2)^2 + (1-x_{j})^2 + 90(x_{j+3}-x_{j+2}^2)^2 \right. \left.+ (1-x_{j+2})^2 + 10(x_{j+1}+x_{j+3}-2)^2 + 0.1(x_{j+1}-x_{j+3})^2\right)
\frac{L^\alpha[e^a]}{e^a}= \int \left(1-\frac{\exp(a(\eta^\epsilon(x,y),t)/\epsilon)}{\exp(a/\epsilon)}\right) K dy &= \frac{L^\alpha[e^g]}{e^g}+ \int \left( \frac{\exp(g(\hat{x}|x|^{1/\epsilon}-y))}{\exp(g(\hat{x}|x|^{1/\epsilon}))}- \frac{\exp(a(\eta^\epsilon(x,y),t)/\epsilon)}{\exp(a/\epsilon)}\right) Kdy,
\delta Z_{A \phi}= - \frac{g^{2}}{M_{W}( h ) + M_{A \phi}( h )}\frac{1}{\pi}+ \left( \frac{g^{4}h^{2}}{M_{W}( h )^{3}}+ \frac{( \frac{\lambda}{6}+ g^{2})^{2}h^{2}}{M_{A \phi}( h )^{3}}\right) \frac{1}{4 8 \pi}.
W(\tilde u, u_1) = W(\tilde u, u_2) + \langle W'(\tilde u, u_2), u_1-u_2 \rangle + W(u_2, u_1), w(u_1) = w(u_2) + \langle w'(u_2), u_1 - u_2 \rangle + W(u_2,u_1).
u_{x^{i}}=\frac{\partial u}{\partial x^{i}}=D_{^{i}}u,D_{x}^{\mathfrak{a}}u=D_{1}^{\mathfrak{a}_{1}}\cdots D_{d}^{\mathfrak{a}_{d}}u,\quad|\mathfrak{a}|=\mathfrak{a}_{1}+\cdots+\mathfrak{a}_{d}.
\lim_{n \rightarrow \infty}T^{-1}\left( \sum_{k=0}^{n}\bar{\alpha}_{k}e_{k}\otimes \bar{e}_{n}\right) T \eta = T^{-1}\lim_{n \rightarrow \infty}\sum_{k=0}^{n}(T \zeta | e_{k}) e_{k}= \zeta,
\frac{1}{l}\sum_{i=1}^{k}\Biggl(\int^{t+l}_{t}\bigl\| f(s+\tau,x_{i})-f(s,x_{i})\bigr\|^{q}\, ds \Biggr)^{1/q}\leq \frac{1}{l}l^{(1/q)-(1/p)}\sum_{i=1}^{k}\Biggl(\int^{t+l}_{t}\bigl\| f(s+\tau,x_{i})-f(s,x_{i})\bigr\|^{p}\, ds \Biggr)^{1/p}\leq \epsilon l^{(1/q)-1},t\in I.
\prod^{M}_{m=1}\left(\frac{m!}{\Gamma(m+n)}\right)^{\frac{\Gamma(n+m)}{m!\Gamma(n)}}\geq \left(\frac{M!}{\Gamma(M+n)}\right)^{M\frac{\Gamma(n+M)}{M!\Gamma(n)}}\geq \left(\frac{M!}{\Gamma(M+n)}\right)^{\frac{\Gamma(n+M)}{\Gamma(M)}}&\geq \left(\frac{\Gamma(CL+1)}{\Gamma(CL+n)}\right)^{\frac{\Gamma(n+CL)}{\Gamma(CL)}}.
{\cal I}_{\mathrm{1 - l o o p}}^{I}= - \frac{i V^{( 8 )}B^{4}}{2 \pi}\sum_{w}\frac{T_{2}}{w^{I}G_{I J}w^{J}}\bigg [ \sum_{i j}e^{2 \pi i ( a_{i}+ a_{j})_{I}w^{I}}( q_{i}+ q_{j})^{4}- \sum_{i}e^{4 \pi i a_{i I}w^{I}}( 2 q_{i})^{4}\bigg ]
\tilde{w}_s^{-1}\Psi([\tilde{u},\tilde{v}])\tilde{w}_s =\Psi([\tilde{u_1},\tilde{v_1}])\qquad\qquad \tilde{w}_s^{-1}\Psi\big(\tilde{R}(u,v)\big)\tilde{w}_s =\Psi\big(\tilde{R}(u_1,v_1))\big).
\frac{\left\|V^{N}\right\|^2- \left\|V^{0}\right\|^2}{2}\leq k_1 \left( \kappa\left\|V^{1}\right\| + \left\|g^{1/2}\right\| \right) \left\|V^{1}\right\| + \sum\limits_{n=2}^{N}k_n \left( \kappa \left\|V^{n-\frac{1}{2}}\right\| + \left\|g^{n-\frac{1}{2}}\right\| \right) \left\|V^{n-\frac{1}{2}}\right\|.
Riem(g)^{\alpha \beta \gamma \delta}Riem(g)_{\alpha \beta \gamma \delta}= Riem(g)_{ab}^{\ \cd}Riem(g)_{cd}^{\ \ab}+ 4 Riem(g)_{a0}^{\ \c0}Riem(g)_{c0}^{\ \a0}& \ \- 4 g_{cc'}g_{dd'}g^{bb'}Riem(g)_{0b}^{\ \cd}Riem(g)_{0b'}^{\ \c'd'}.
M ={\frac 1 2}\int d \sigma \sigma g^{\mu_{1}\mu_{2}}{\bf D}_{\mu_{1}}( \sigma ){\bf D}_{\mu_{2}}( \sigma ) , \quad N ={\frac 1 2}\int d \sigma \sigma g_{\mu_{1}\mu_{2}}z^{\prime \mu_{1}}z^{\prime \mu_{2}}
{\Lambda_n}= \frac{1}{{{{\left({2\beta}\right)}^U}}{{{\ln}^n}2}}\sum\limits_{k = 0}^\infty{{c_k}\sum\limits_{q = 0}^k{\frac{{{{\left({- k}\right)}_q}}}{{q!\Gamma \left({U + q}\right)}}{{\left({\frac{U}{{2\beta \xi}}}\right)}^q}}}&\times J\left({q + U,\frac{1}{{2\beta}},n}\right).
D_j = \left(\frac{4}{3}\right)^{j/3}\frac{e^{-2\pi ij/3}}{2^{3/2}\pi i}\oint_{(1/2+)}\frac{z^{-1/2}}{w(z)^j}dz = \left(\frac{4}{3}\right)^{j/3}\frac{1}{2\pi i}\oint_{(1/2+)}\frac{(2z)^{-1/2}}{(w^3(z))^{j/3}}dz.
\int \frac{x^2}{(x^2 q^{-1}(1-q);q^2)_\infty}\cos(x;q)d_qx = \frac{q}{(\frac{x^2}{q}(1-q);q^2)_\infty}\Big(x \cos(\frac{x}{q};q) +\sqrt{q}\sin( q^{\frac{-1}{2}}x;q)\Big),
\phi_{m}= N_{m}e^{{\frac{5}{4}}k \rho}\left[ J_{3 / 2}\left({\frac{2 m}{k}}\right) Y_{5 / 2}\left({\frac{2 m}{k}}e^{{\frac{k}{2}}\rho}\right) - Y_{3 / 2}\left({\frac{2 m}{k}}\right) J_{5 / 2}\left({\frac{2 m}{k}}e^{{\frac{k}{2}}\rho}\right) \right]
\Theta_{2}\leq C 2^{-2(1+s)q}b_q\|\partial_1\theta\|_{L^\infty}\|\omega\|_{H^s}\|\theta\|_{H^{1+s}}&\quad+ C 2^{-2(1+s)q}b_q(\|\omega\|_{H^s}+ \|\theta\|_{H^{1+s}})\|\partial_1\theta\|_{H^{1+s}}.
R=A_{1}{\rho^{z_{1}}\left[-\mu+\lambda+{2 c_{2}\lambda \rho^{2\lambda}\over1-c_{2}\rho^{2\lambda}}\right]^{2}\over [(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})]^{1+{1\over d}}(1-c_{2}\rho^{2\lambda})^{z_{2}}}+A_{2}{\rho^{z_{1}+2\lambda}\left({1-c_{2}\over 1-c_{2}\rho^{2\lambda}}\right)^{z_{2}+2}\over [(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})]^{1+{1\over d}}}
f_{TTT}=-{2i\over\pi}\frac{j_T(T)}{j(T)-j(U)}\left\{\frac{j(U)}{j(T)}\right\}\left\{\frac{j_T(T)}{j_U(U)}\right\}\left\{\frac{j(U)-j(i)}{j(T)-j(i)}\right\}\equiv 2 W(T,U).
X^{s,\gamma,p}_{(0,t)}=\begin{cases}L^\gamma(0,t;W^{s,p}_x),\ \ frac{2\alpha}{\gamma}+ \frac 3p = 2\alpha-1+s, 1\leq \gamma <\infty, 1\leq p\leq\infty, s\geq 0, C([0,t];W^{s,p}_x),\ \,\ \ \gamma = \infty, 1\leq p\leq \infty, s\geq 0. \end{cases}
\Sigma ( z ) = \frac{\int_{- \infty}^{z}\sqrt{- g^{( 4 )}}d z^{\prime}\left( T_{\; \; t}^{t}- T_{\; \; z}^{z}- T_{\; \; x}^{x}- T_{\; \; y}^{y}\right) \int d x d y}{\sqrt{g^{( 2 )}}\int d x d y}.
\begin{cases}x = x_0 =\dots = x_k y = y_0 = r_0x_0^{m_0}+r_1x_0^{m_0+m_1}+\dots +r_{k-1}x_0^{m_0+\dots+m_{k-1}}+y_kx^{m_0+\dots +m_{k-1}}.\end{cases}
{\frac{2 Q_{\mathrm{e}}}{g_{\mathrm{Y M , e f f}}}}= q -{\frac{p}{2 \pi}}\left[ \arctan \Bigl ({\frac{m_{1}- a_{1}}{m_{2}}}\Bigr ) + \arctan \Bigl ({\frac{m_{1}+ a_{1}}{m_{2}}}\Bigr ) \right] .
\begin{aligned}\overline{H}(t_{1},x_{1},p_{1},P_{1}) - \overline{H}(t_{2},x_{2},p_{2},P_{2}) = \overline{H}(t_{1},x_{1},p_{1},P_{1}) - \overline{H}(t_{1},x_{2},p_{1},P_{1}) + \overline{H}(t_{1},x_{2},p_{1},P_{1})- \overline{H}(t_{2},x_{2},p_{2},P_{2}). \end{aligned}
d s^{2}= H ( x_{\perp})^{- 4 \frac{\tilde{d}}{\Delta ( D - 2 )}}d x_{\parallel}^{\mu}\otimes d x_{\parallel}^{\nu}\eta_{\mu \nu}+ H ( x_{\perp})^{4 \frac{d}{\Delta ( D - 2 )}}d x_{\perp}^{I}\otimes d x_{\perp}^{J}\delta_{I J}
p_{d+i}-p_{d+i}' = h_i(p_1,\ldots,p_d)-h_i(p_1',\ldots,p_d')+a_{i1}(p_1^{b_{i1}}-(p_1')^{b_{i1}})+\cdots+a_{id}(p_d^{b_{id}}-(p_d')^{b_{id}}).
\sum_{m=1}^{M}(\kappa_m-\alpha_m)=\sum_{n=1}^{N-1}\mu_n,\sum_{m=j}^{M}(\kappa_m-\alpha_{m-j+1})^+\leq\sum_{n=j}^{N-1}\mu_n,\quad\forall j=1,\dotsc,M.
\kappa(a,b,c,d)=\begin{vmatrix}b_1 \boxed{d_1}b_2 d_2\end{vmatrix}^{-1}\begin{vmatrix}b_1 \boxed{c_1}b_2 c_2\end{vmatrix}\begin{vmatrix}a_1 \boxed{c_1}a_2 c_2\end{vmatrix}^{-1}\begin{vmatrix}a_1 \boxed{d_1}a_2 d_2\end{vmatrix},
\lim_{n\rightarrow\infty}G_{\ell,n}(p) =\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right) \left\{\frac{x_{i}}{x_{\ell}}\tau (\ell,i) \mathrm{d}\mu_{S}(x) +\mathrm{d}\mu_{S}(x) \tau( \ell,i) \right\},
I_\zeta^{(\alpha)}(r,s) := \int_0^\infty \frac{\Gamma(\tfrac\alpha2+1)\sin(\tfrac{\pi\alpha}{2})}{\pi\,\tau^{1+\alpha/2}}\cdot \frac{(rs)^{\frac12-\zeta}}{2\tau}\exp\left(-\frac{r^2+s^2}{4\tau}\right)I_{\zeta-\frac12}\left(\frac{rs}{2\tau}\right)\,d\tau
m_{d+1,1^{k-1}}(x_1,x_2,\ldots,x_i) = \begin{cases}m_{d,1^{k-1}}(x_1,x_2,\ldots,x_i) x_{i+1}(d\ge2), k m_{1^{k}}(x_1,x_2,\ldots,x_i) x_{i+1}(d=1). \end{cases}
\| \sum_{i=1}^q\epsilon_{Y_j}(A) m_f(r,Y_j)\le(\delta(n+1)+\epsilon) T_{f,A}(r,s) &+c\left(m_0(\mathfrak{L};r,s)+{\rm Ric}_{p}(r,s)+\kappa \log^+ \mathcal{Y}\left(r^{2}\right)+k \log^+ r\right).
\partial_{\bar{a}}C_{i_{1}. . . i_{s}}^{g}= f a c t_{g}({\bar{C}}_{\bar{a}}) - \sum_{i = 1}^{s}{\bar{C}}_{{\bar{a}}i}^{l}C_{\sigma_{1}( l ) i_{1}. .{\hat{i}}. . i_{s}}^{g}+{\bar{C}}_{{\bar{a}}0}^{l}C_{\sigma_{2}( l ) i_{1}. . . i_{s}}^{g}
\phi(\frac 1 z,\frac 1 w) = \phi_0(z,w)-\phi_0(z,0)-\phi_0(w,0) - \int_u (\phi_0(z,u)-\phi_0(z,0)-\phi_0(u,0)) \omega(u) & - \int_u (\phi_0(w,u)-\phi_0(w,0)-\phi_0(w,0))) \omega(u) =\phi(z,w)-\phi_0(z,0)-\phi_0(w,0) + \phi_0(z,0) \int_u \omega(u) + \phi_0(w,0)\int_u \omega(u) &=\phi(z,w).
\begin{aligned}\frac{d}{dt}\int_{-\infty}^t a(t-s)u(s) \,{\rm d}s = A u(t) + f(\tilde{u}(t)) + g \,\left[r(\gamma (t), \tilde u(t)) + \dot W(t) \right], t \in [0,T] u(t) = u_0(t), t \le 0. \end{aligned}
\mathbb{E}[\int_0^T \mathbb{E}[D_tF|\mathcal{F}_t]^2 dt ] = \mathbb{E}[(\int_0^T\mathbb{E}[D_tF|\mathcal{F}_t] dB(t))^2] = \mathbb{E}[(F^2-\mathbb{E}[F]^2)]
b_i\ge\sum_{r=i+1}^{k-1}\binom{\left\lfloor\frac{\alpha m}k\right\rfloor}{r-i}b_r\ge \sum_{r=i+1}^{k-1}\binom{\left\lceil\frac{\alpha m}k\right\rceil-k+r-i-1}{r-i}b_r\ge \sum_{r=i+1}^{k-1}(-1)^{r-i+1}\binom{\left\lceil\frac{\alpha m}k\right\rceil-k+r-i-1}{r-i}b_r,
N^{\dagger}X^{-1}=V\begin{pmatrix}I 0\end{pmatrix}\Sigma^{-1}X_{1}^{-1}\begin{pmatrix}I 0\end{pmatrix}U^{\ast}, NN^{\dagger}(I+R^{\ast}R)^{-1}(I+R^{\ast}) =U\begin{pmatrix}I 0\end{pmatrix}(I+G^{\ast}G)^{-1}\begin{pmatrix}I G^{\ast}\end{pmatrix}U^{\ast}.
\zeta_{p,0}(s,c):=\sum_{m,n\neq0}\frac{p^{2}+\lambda_{m}^{2}}{p(p+\frac{1}{\pi})+\lambda_{m}^{2}}\,\frac{1}{\left(\lambda_{m}^{2}+c\,\left(n-\frac{1}{2}\right)^{2}\right)^{s}},\,\,\,\,(s)>1.
\int_{t_o}^\infty e^{\int_{t_o}^x \left[ G^*(e^{t^2/2}) e^{-t^2/2}- t \right]dt}dx \leq \int_{t_o}^\infty \left( \frac{x}{t_o}\right)^{-1.228}dx \, = \, \frac{1}{0.228 t_o}.
w(\rho)=c_0 \rho - \tilde \alpha_p \rho \int_{\rho_0}^\rho s h(s) ds + \tilde \alpha_p (1-\rho^2)^{-\frac{2}{p-1}}h(\rho)\int_{0}^\rho s^2 (1-s^2)^{\frac{2}{p-1}}ds
V^{(2)}_3(r) = \lambda A - \frac{\lambda A}{1+\lambda r^2}+ \frac{\lambda B_1}{(1+\lambda r^2)^2}+ \frac{\lambda B_2}{(1+\lambda r^2)^3}+ \frac{\lambda B_3}{(1+\lambda r^2)^4}+ \frac{\lambda B_4}{(1+\lambda r^2)^5}{}+ \frac{\lambda B_5}{(1+\lambda r^2)^6}+ \frac{\lambda B_6}{(1+\lambda r^2)^7}
P_{m+d+1}^{c}\left( x\right) =\left( x-\beta_{m+d}\right) P_{m+d}^{c}\left(x\right) -\sum\nolimits_{\nu=0}^{d-1}\gamma_{m+d-\nu}^{d-1-\nu}P_{m+d-1-\nu}^{c}\left( x\right) ,\ \m\geq k.
\delta(\rho X\sigma) =\delta(\rho X)\sigma +\rho \delta(X\sigma)-X\delta(\rho\sigma) -\rho \delta(X)\sigma +\rho X\delta(\sigma) &=\delta(\rho X)\star X +\rho \delta(X\wedge \star X)-X\rho V -\rho \delta(X)\star X -\rho X\wedge \star d X &=\rho(\pi-XV) + \delta(\rho X)\star X +\frac{\rho}{2}\delta(X\wedge \star X)-\rho X\wedge \star d X.
C_t(y^{t-1}_{t-M})=\sup_{\big\{{\pi}_i(dx_i|y_{i-M}^{i-1}):~i=t,t+1,\ldots,n\big\}}{\bf E}^{\pi}\bigg\{\sum_{i=t}^n\log\Big(\frac{d{q}_i(\cdot|y^{i-1}_{i-M}, x_i)}{d\nu^\pi_t(\cdot|y^{i-1}_{i-M})}(Y_i)\Big)\Big{|}Y^{t-1}_{t-M}=y^{t-1}_{t-M}\bigg\}.
W(u_1,u_2,u_3)= \left[D_2-C_1D_1+D_1\, W(u,\partial_\epsilon v) +\frac{1}{2}W(u,\partial_\epsilon^2 u)\right]u-D_1^2v-D_1\partial_\epsilon u &-(D_1v+\partial_\epsilon u)W(u,\partial_\epsilon u)
n^{-1}\int_{t_l}^{t_{m+1}}\psi_e(s) \left( \bar \Lambda^n (e_x \mid s) - \bar \xi^n_{(2)}(e_x \mid s) \right) ds = n^{-1}\sum\limits_{k=l}^{m}\left( \delta_{\bar \nu^{n,k+1}}(e_x) - \bar \mu^{n, k+1}(e_x) \right)
\begin{aligned}{^C_0}D^\alpha_tf(x,t_n) \cong \frac{1}{\tau^\alpha}\sum_{k=0}^{n}\omega^\alpha_kf(x,t_{n-k}) &\ -\frac{1}{\tau^\alpha}\sum^{m-1}_{l=0}\sum^{n}_{k=0}\frac{\omega^\alpha_k f^{(l)}(x,0)t_{n-k}^{l}}{l!}.\end{aligned}
\inf_{y \in Y}|\beta_{k}(y)|^2 \int_{S_{\sqrt{\rho}}(r)}d^c \sqrt{\rho}\wedge (dd^c \sqrt{\rho})^{n-1}= \inf_{y \in Y}|\beta_{k}(y)|^2 \int_{Y}(dd^c \sqrt{\rho})^n \leq \int_{Y}|\beta_{k}|^2 (dd^c \sqrt{\rho})^n \leq \int_{S_{\sqrt{\rho}}(r)}|\beta_{k}|^2 d^c \sqrt{\rho}\wedge (dd^c \sqrt{\rho})^{n-1}
\begin{aligned}S_{\lambda}(T_{\alpha,q}v)(x) = T_{\alpha}(D_{\lambda^{-\alpha}}v)(x)x\in B_R, T_{\alpha}^{-1}(S_{\lambda}u)(y) = D_{\lambda^{-\alpha}}(T^{-1}_{\alpha,q}u)(y)y\in \mathbb{R}^n.\end{aligned}
{\cal V}( \phi_{0}, \mu ) = V ( \phi_{0}, \mu_{0}) +{\frac{\hbar}{4 \pi}}\left\{V^{\prime \prime}( \phi_{0}, \mu ) - m^{2}( \mu ) \right\}\ln{\frac{\mu}{m ( \mu )}}+ \hbar m^{2}( \mu ) \; X [ V ( \phi_{0}, \mu ) / m^{2}( \mu ) , \mu ] + O ( \hbar^{2}) .
i\partial_t v_2 = \lambda_6 t^{-1}\mathcal{F}M(t)^{-1}\mathcal{F}^{-1}\mathcal{N}_2(\mathcal{F}M(t){{\mathcal F}}^{-1}v_1, \mathcal{F}M(t){{\mathcal F}}^{-1}v_2) &= \lambda_6 t^{-1}\mathcal{N}_2(v_1,v_2) + iR_2(s).
\overline{\{r_{2n-1}:n\in\mathbb N\}}=\overline{\{r_{2n}:n\in\mathbb N\}}=\mathbb Q, A=\bigcup_{n=1}^\infty \{r_{2n-1}\}\times [0,1],B=\bigcup_{n=1}^\infty \{r_{2n}\}\times [0,1].
\varepsilon_F^\ell(L^r;x) = r \cdot \sup_{m \ge 1}\frac{p^{s^\ell_F(L^{rm};x)-1}}{rm/(\ell+1)}\le r \cdot \sup_{m' \ge 1}\frac{p^{s^\ell_F(L^{m'};x)-1}}{m'/(\ell+1)}= r \cdot \varepsilon_F^\ell(L;x)
G (\Theta(x),y)=\frac{2(1+\alpha)}{1+b_0}\sum_{k=0}^{\infty}(\alpha\beta)^k\log|x-{\bf 2k}-\Theta(y)| &\quad-\frac{2(1+\alpha)}{1+b_0}\sum_{k=0}^{\infty}(\alpha\beta)^k\log|x-{\bf 2k}|+\frac{(1-\beta)(1+\alpha)}{1-\alpha\beta}\log|y|.
g_{\alpha \beta}^j(f_{jk}^1(\xi_k,t),...,f_{jk}^n(\xi_k,t))=\sum_{r,s=1}^n g_{rs}^k(\xi_k,t)\frac{\partial f_{jk}^{\alpha}}{\partial \xi_k^r}\frac{\partial f_{jk}^{\beta}}{\partial \xi_k^s}
\Theta^4 \Big[ \begin{array}{c}0 \cr0 \end{array}\Big] (\tau) -\Theta^4 \Big[ \begin{array}{c}0 \cr1 \end{array}\Big] (\tau) -\Theta^4 \Big[ \begin{array}{c}1 \cr0 \end{array}\Big] (\tau) = 0,
lim_{n\rightarrow\infty}(\sum_{k=1}^{n}\frac{1}{0.1k}-\int_{1}^{n}\frac{dk}{0.1k})=10lim_{n\rightarrow\infty}(\sum_{k=1}^{n}\frac{1}{k}-\int_{1}^{n}\frac{dk}{k})=10\gamma
\langle\prod_{i=1}^{2n}\sigma(z_{i})\rangle^{2}=\frac{1}{2^{n}}\sum_{\begin{matrix}\epsilon_{i}=\pm1\\ \sum_{i=1}^{2n}\epsilon_{i}=0\end{matrix}}\prod_{1\le i<j\le2n}|z_{ij}|^{\frac{\epsilon_{i}\epsilon_{j}}{2}}
g_j^* - \mu_{s}\le \mu_j - \mu_{s}\le \sqrt 2 (1+\delta_{s})(\sqrt{\log(9ep/j)}- \sqrt{\log(9ep/(j+s))}), = \sqrt 2 (1+\delta_{s})\frac{\log(1+s/j)}{\sqrt{\log(9ep/j)}+\sqrt{\log(9ep/(j+s))}}, \le \frac{\sqrt 2 (1+\delta_{s})}{\sqrt{\log(9ep/s)}}\log(1+s/j).
H=\sum_{\vec{x}}a^3 \left\{\psi_{\gamma \vec{x}}^{\dagger}m\beta_{\gamma \delta}\psi_{\delta \vec{x}}+\frac{i}{2a}\sum_{l=1}^{3}[\psi_{\gamma \vec{x}+\vec{l}}^{\dagger}(\alpha_{l})_{\gamma \delta}\psi_{\delta \vec{x}}- \psi_{\gamma \vec{x}}^{\dagger}(\alpha_{l})_{\gamma \delta}\psi_{\delta \vec{x}+\vec{l}}] \right\}\quad .
(\mathcal{W}_X^{-1})_{ij}= ~\left(\bar{g}^{jq}-\frac{\langle \bar{g}^{ja}\bar{\nabla}_as,\bar{g}^{qb}\bar{\nabla}_bs\rangle}{1+s^2+|\bar{\nabla}s|^2}\right)\tau_{qi}\sqrt{\frac{1+s^2}{1+s^2+|\bar{\nabla}s|^2}}
\displaystyle J(f) =\int_{H(F)\backslash G(F)}\int_{H(F)}{}^gf(h)\xi(h) dhdg =\int_{H(F)\backslash G(F)}\int_{H_x(F)\backslash H(F)}\int_{H_x(F)}\eta_x^H(h_x)\;{}^{hg}f(h_x)\xi_x(h_x)dh_xdhdg =\int_{H(F)\backslash G(F)}\int_{H_x(F)\backslash H(F)}\int_{H_x(F)}\eta_{x,G}^H(h_x)^{1/2}({}^{hg}f)_{x,\Omega_x}(h_x)\xi_x(h_x)dh_xdhdg
\max_{i\in \frac{1}{4}I}\sum_{j\in I}\mathcal{U}_{ij}(t,s)F_{j}^{(2,\mathrm{out})}=\max_{i\in \frac{1}{4}I}\frac{1}{N}\sum_{j\in I\backslash \frac{1}{2}I}\sum_{k\in I^c}\mathcal{U}_{ij}(t,s)F^{(2)}_{jk}(s)\,,