harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p_{S}(x) \le \sum_{i = 1}^{d+1}\frac{dy}{\Omega_d \cdot d(y, H_i)^d}\le \frac{d + 1}{\Omega_d}\cdot \sup_{x \in \mathbb R^n}\int_{P_{1, x}}\frac{dy}{d(y, H_1)^d}.
\Psi(z,z+2\hbar,z+4\hbar,\cdot)=\Psi(z,z+2\hbar,\cdot,z+4\hbar)=\Psi(z,\cdot,z+2\hbar,z+4\hbar)=\Psi(\cdot,z,z+2\hbar,z+4\hbar)=0
A_{1/2}({\bf p};{\bf n})=B\,\left( \begin{array}{ccc}(p_0+m-p_3)e^{-i\varphi/2}\cos({\theta}/2)-p_{-}e^{i\varphi/2}\sin({\theta}/2) (p_0+m+p_3)e^{i\varphi/2}\sin({\theta}/2)-p_{+}e^{-i\varphi/2}\cos({\theta}/2)\end{array}\right),
H^{n}(S^{\otimes\bullet}\boxtimes S^{\otimes\bullet}, d_H\boxtimes d_H)=H^{n}(\Delta(S^{\otimes\bullet}),\Delta(d_H))\simeq\Delta(\wedge^n)=\bigoplus_{j=0}^n\wedge^j\boxtimes\wedge^{n-j}
\frac{2}{(1-z)^{N}(1+z)^{M}}\frac{d}{dz}A( z) =C\big( 1-z^{2}\big) \big( P_{n-1}^{(N+1,M+1)}(z) \big)^{2}-\big( P_{n}^{(N,M)}( z ) \big)^{2}.
\varepsilon . \frac{1}{\varepsilon}\ell n (_{q}< \zeta_{\ell}| \zeta_{\ell - 1}>_{q}) \cong \frac{\varepsilon}{2}( \frac{\Delta \bar{\zeta}_{\ell}}{\varepsilon}\_{q}< \zeta_{\ell}| T_{i}^{-}| \zeta_{\ell}>_{q}- \frac{\Delta \zeta_{\ell}}{\varepsilon}\_q < \zeta_{\ell}| T_{i}^{+}| \zeta_{\ell}>_{q}) \,
c(x)=1+\lambda\sum_{u:u\sim x}\rho(x)\rho(u), &p(x,\varnothing)=\frac{1}{1+\lambda\sum_{u:u\sim x}\rho(x)\rho(u)}, &p(x,\{x,y\})=\frac{\lambda\rho(x)\rho(y)}{1+\lambda\sum_{u:u\sim x}\rho(x)\rho(u)},\{\rm when}\y\sim x, &p(x,A)=0,\ \rm otherwise,
E_h(t) = \frac{1}{2\pi\rm i}\int_{\Gamma_{\theta,\delta}}e^{zt}z^{\alpha-1}(z^\alpha+A_h)^{-1}\d z \quad\mbox{and}\bar E_h(t)= \frac{1}{2\pi\rm i}\int_{\Gamma_{\theta,\delta}}e^{zt}z^{-\gamma}(z^\alpha+A_h)^{-1}\d z.
\nabla \phi_\mathrm{pd}(x^{(k_i)},z^{(k_i)})-\nabla \phi_\mathrm{pd}(x^{(k_i+1)},z^{(k_i+1)})+\begin{bmatrix}-A^Tz^{(k_i+1)}Ax^{(k_i+1)}\end{bmatrix}\in\begin{bmatrix}\partial f(x^{(k_i+1)})+\nabla h(x^{(k_i+1)}) \partial g^\ast(z^{(k_i+1)}) \end{bmatrix},
{\widehat{A}(TX,\nabla^{TX})}{\rm ch}(2^k\Theta_2(T_{C}X))=\prod_{j=1}^{k}\frac{2 x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\frac{\theta_2(x_j,\tau)}{\theta_2(0,\tau)},
H^k(\tilde{\mathbb{P}}, \mathcal{O}(\tilde{Y}+ K_{\tilde{\mathbb{P}}})) \cong H^k(\mathbb{P}, \mathcal{O}(Y+K_{\mathbb{P}})) = 0 k > 0 & H^k(\tilde{\mathbb{P}}, \mathcal{O}(\tilde{Y}+ K_{\tilde{\mathbb{P}}}- D_j)) = 0 k > 0
W_{\mathrm{tr},p}^+(\mu,\nu) = \inf_{\phi \in \mathcal{M}_{\mathrm{tr}}}\{W_{\mathrm{tr},p}(\alpha, \nu) : \mu = \phi(\alpha)\}\geq \inf_{\phi \in \mathcal{M}_{\mathrm{tr}}}\{W_{\mathrm{tr},p}(\mu, \phi(\nu))\}= W_{\mathrm{tr},p}^-(\mu,\nu).
n_{i}^{\prime\prime}=\frac{P_{i}p^{\prime}(n_{i}^{\prime}-\frac{1}{\phi}n_{i}^{\prime\prime})}{P_{i}p^{\prime}(n_{i}^{\prime}-\frac{1}{\phi}n_{i}^{\prime\prime})+P_{j}p^{\prime}((1-n_{i}^{\prime})-\frac{1}{\phi}(1-n_{i}^{\prime\prime}))}
\frac{d}{d\sigma}\frac{\partial\Lambda}{\partial u^{\alpha}}-\frac{\partial\Lambda}{\partial x^{\alpha}}=\frac{d^{2}x^{\alpha}}{d\sigma^{2}}+\Gamma_{\beta\gamma}^{\alpha}\frac{dx^{\beta}}{d\sigma}\frac{dx^{\gamma}}{d\sigma}=0,
f_1(a,b;\mathbf{t})=\int_0^{t_1}dx_1\ldots \int_0^{t_m}dx_m |x_1-x_m|^{2a}|x_2-x_1|^b |x_4-x_3|^b|x_5-x_4|^a \ldots \times |x_{m-1}-x_{m-2}|^a |x_{m}-x_{m-1}|^b,
S=S_0+ \int dt \Big\{{1\over 2}h_{MN}T^{MN}+J^{a_1 a_2 a_3; a_4}\partial_{a_4}C^{(3)}_{a_1 a_2 a_3}+30M^{+- m_1 \ldots m_4;a}\partial_aC^{(6)}_{+- m_1 \ldots m_4}\Big\}
\mathcal{H}(\mathcal{L}_+;q,t)-\mathcal{H}(\mathcal{L}_-;q,t)=z\mathcal{H}(\mathcal{L}_0;q,t), \mathcal{H}(\mathcal{L}^{+1};q,t)=t\mathcal{H}(\mathcal{L};q,t), mathcal{H}(\mathcal{L}^{-1};q,t)=t^{-1}\mathcal{H}(\mathcal{L};q,t).
D_1 = (B-n)(v_m) + \sum_{i=1}^n \Big[ \big( D(w_i) + A +1\big)(w_i) + \sum_{v_j \prec w_i}\sum_{p\in (w_i,v_j)^{\circ}}\,D(p) \,(p)\Big] + \sum_{i=1}^{m-1}\Big[D(v_i) (v_i) + \sum_{w_j \prec v_i}\sum_{p\in (v_i,w_j)^{\circ}}\,D(p) \,(p)\Big],
\underbar r_1=\frac{a_{0,\inf}-a_{2,\sup}\bar r_2-k\frac{\chi_1}{d_3}\bar r_1}{a_{1,\sup}-k\frac{\chi_1}{d_3}}, \bar r_1= \frac{a_{0,\sup}-a_{2,\inf}\underbar r_2-k\frac{\chi_1}{d_3}\underbar r_1}{a_{1,\inf}-k\frac{\chi_1}{d_3}},
\frac{1}{N^2}\sum_{i=1}^N \sum_k^{(i)}\tilde{d}_i \mathbf{e}_k^* X_i G\mathbf{e}_i \frac{\partial \omega_\iota^c}{\partial g_{ik}}=O_\prec(\Psi^2\Pi^2)\,, \iota=A,B\,.
\phi_{f}(x, y)=\frac{1}{2\pi i}\int_{b-iU}^{b+iU^{\prime}}A_z(x) \frac{y^z}{z}dz+O\left (y^{b}\sum_{n \leq x}f(n)^{-b}\left(\frac{1}{U}+\frac{1}{U^{\prime}}\right)\right).
\sum_{n=0}^{\infty}\frac{(\frac12-4k)_n (\frac12-2k)_n}{(1+2k)_n(1)_n}(3-2\sqrt 2)^{2n}= C_1 (3\sqrt 2-4)^{2k}\frac{2^{9k}}{3^{3k}}\, \frac{(1)_k \left(\frac12\right)_k}{\left(\frac{7}{12}\right)_k (\frac{11}{12})_k},
-y =(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}x =-z''(x)[[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}0 = y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}
U_{1,n}:=\left\{[s:t:x_{2}]\in \mathbb{P}^{2,3,1}:\begin{aligned}4s^{3}+27t^{2}\ne 0 Q\in E_{s,t}^{D}n x\left(Q\right)=Dx_{2}D\in \mathbb{G}_{m}\end{aligned}\right\}
\alpha_{+}= -\frac{(i-1)z^{3}+\sqrt{2}\kappa}{(i+1)z^{3}+\sqrt{2}\kappa}, \beta_{+}= -\frac{2z^{3}}{(i+1)z^{3}+\sqrt{2}\kappa}, \alpha_{-}= -\frac{(1-i)z^{3}+\sqrt{2}\kappa}{(i+1)z^{3}+\sqrt{2}\kappa}, \beta_{-}= -\frac{2iz^{3}}{(i+1)z^{3}+\sqrt{2}\kappa}.
T^{\alpha \beta \{a b \}}= \frac{1}{2}( - 1 )^{\varepsilon_{\beta}}C^{\alpha \{a}C^{\beta b \}}, \hat{T}^{\alpha \beta}\varepsilon^{a b}= - \frac{1}{2}( - 1 )^{\varepsilon_{\beta}}\varepsilon_{c d}C^{\alpha c}C^{\beta d}\varepsilon^{a b}
\| \mathcal{U}_\ast (t) \|_1 = 1 \, , \infty > \| \mathcal{U}_\ast(t) \|_\infty \ge K_0 \, t^{-\frac{N}{2+N(m_2-1)}}\textrm{and}\operatorname{supp}\, \mathcal{U}_\ast(\cdot,t) \subset B_{R_0 \, t^{\frac{1}{2+N(m_2-1)}}}^{(x_0)}\forall t > 0
\langle\langle S,\varphi\Phi\rangle\rangle = \langle\langle \Delta_y\pi,\varphi\Phi\rangle\rangle &= (-1)^2 \langle\langle \pi,\Delta_y(\varphi\Phi)\rangle\rangle &= \langle\langle \pi,\varphi\Delta_y\Phi\rangle\rangle &= \iint_{\mathbb{R}^{2n}}\varphi(x)(\Delta_y\Phi)(x,y) d\pi(x,y) &= \int_{\mathbb{R}^{n}}\varphi(x)(\Delta_y\Phi)(x,x) dx.
u(r, \omega) = v_1(r)w_1(\omega) + \sum_{k\geq 2}c_k r^{\gamma_k^+}w_k(\omega) (\u(r, \omega) = v_1(r)w_1(\omega) + \sum_{k\geq 2}c_k r^{\gamma_k^-}w_k(\omega)\ )
\begin{aligned}\hat{g}^0_{i \pm 1/2}= g_{i \pm 1/2}+ \frac{1}{12}h_i^{(4)}\Delta x^4 + O(\Delta x^5), \hat{g}^1_{i \pm 1/2}= g_{i \pm 1/2}- \frac{1}{90}h_i^{(5)}\Delta x^5 + O(\Delta x^6), \hat{g}^2_{i \pm 1/2}= g_{i \pm 1/2}- \frac{1}{12}h_i^{(4)}\Delta x^4 + O(\Delta x^5).\end{aligned}
d_{avg}(V\setminus V') =\frac{2|C_t\setminus (D_{t-1}\cup F)|-d(V\setminus V',V')}{|V\setminus V'|}=\frac{2|C_t\setminus D_{t-1}|-2|F|-d(V\setminus V',V')}{|V\setminus V'|}&=\frac{4|V\setminus V'|+4-2|F|-d(V\setminus V',V')}{|V\setminus V'|}<4,
\ddot{\chi}+ \dot{\alpha}_0 \left(1 -\frac{\ddot{\alpha}_0}{\alpha_1^2}\right) \dot{\chi}-\frac{1}{3}\left(1+\frac{\ddot{\alpha}_0}{\alpha_1^2}\right) e^{- 2 \alpha_0}\nabla^2 \chi =0 \ .
S_{b u l k}^{i}= - \frac{1}{2 \kappa_{5}^{2}}\int d^{5}x \sqrt{- g_{E}^{( 5 )}}( R + G_{I J}\partial_{\mu}X^{I}\partial^{\mu}X^{J}- \frac{1}{2{\cal V}^{2}}( \partial{\cal V})^{2}- \frac{1}{2{\cal V}^{2}}\alpha_{I}^{i}\alpha_{J}^{i}G^{I J}( X ) ) .
\times \sum_{\ell = - \infty}^{+ \infty}e^{- i \frac{\pi}{2}| \ell |}e^{i \ell ( \varphi_{f}- \varphi_{i})}e^{- \frac{i}{2}\hbar \Delta t g^{2}\ell^{2}}J_{| \ell |}\left( \frac{\omega r_{f}r_{i}}{\hbar \sin \omega \Delta t}\right) \ ,
\begin{array}{rrcl}\textrm{find}y^*(R,R') = \min y begin{align*}1mm]\textrm{subject to}G(0,1)\otimes p = (0,1)\otimes p' &G(1,e^{-\beta E})\otimes g = \frac{Z_E}{Z_{E+W}}(1,ye^{-\beta E})\otimes g' &e_{2n'}^{\rm T}G = e_{2n}^{\rm T}, &y,E,G \geq 0,\end{array}
\mathrm{\boldmath \Psi \unboldmath}= \left( \begin{array}{c}{\Psi_{+ 1}({\bf x})}\\{\Psi_{- 1}({\bf x})}\\ \end{array}\right) , \Psi_{\zeta}({\bf x}) = \left( \begin{array}{c}{\chi_{\zeta}({\bf x})}\\{\varphi_{\zeta}({\bf x})}\\ \end{array}\right) ,
P_k \geq\begin{cases}P_0 - P_0 C_\infty^H \big(C_\infty P_0 C_\infty^H\big)^{-1}C_\infty P_0+\mathcal{O}\big(\frac1k\big) m<n \frac1k \big(C_\infty^H R^{-1}C_\infty\big)^{-1}+\mathcal{O}\big(\frac1{k^2}\big) m\geq n\end{cases}
\sigma(B(e,\rho)) = \mathcal{H}^{2}(B(e,\rho) \cap \Omega), = \mathcal{H}^{2}(B(e,\rho) \cap S_i) + \mathcal{H}^{2}(B(e,\rho) \cap (\Omega \backslash S_i)), &=\pi \rho^{2}+ \mathcal{H}^{2}(B(e,\rho) \cap (\Omega \backslash S_i)) \mbox{, by}\eqref{spherelocally2unifeq}.
\begin{aligned}\log{F(s,\chi)}:= \log{\epsilon(\chi)}+ \left( \frac{1}{2}- s \right) \log{\frac{q}{2\pi}}+ \frac{1}{2}\log{\frac{2}{\pi}}+ \log{\sin{\frac{\pi}{2}(s+\kappa)}}&\quad\quad+ \log{\Gamma(1-s)}\end{aligned}
\left\langle -P_{N}\right| =\prod_{n=1}^{\infty}\exp \left[-\frac{\alpha'}{4n}\bar{\psi}_{n\mu}g^{\mu\nu}\psi_{n\nu}\right] \times{}_{B=0}\!\left\langle B_{N}\right| :\exp\left(i\int_{0}^{2\pi}d\sigma P_{\mu}(\sigma) \hat{X}^{\mu}(\sigma)\right):~.
Z = \{\overline{z}\in M^{n}: M\models \bigvee_{i=1}^{K}\exists \; \overline{b}\; \exists \; c \; (\theta^{\prime}_{i}(c, \overline{b}) \; \& \; (\varphi_{0}(\overline{z}, \theta_{i}(\cdot, \overline{b}), c, \rho(\cdot, \overline{a})))\}
(i)~{\dot R}= KR \int_{\mathbb T}\sin^2(\theta - \phi) \rho(\theta, t) ~ d\theta,{\dot \phi}= -\frac{K}{2}\int_{\mathbb T}\sin\left(2(\theta - \phi)\right) \rho(\theta, t) ~ d\theta. \cr (ii)~{\ddot R}= \frac{({\dot R})^2}{R}+ 2R(\dot\phi)^2 - 2(KR)^2 \int_{\mathbb T}\sin^2 (\theta - \phi) \cos(\theta - \phi) \rho(\theta, t) ~ d\theta.
(\tilde{\Theta}_\pm)^2\varphi =C_\pm U^\Theta K C_\pm U^\Theta K\varphi &=U^\Theta C_\pm K C_\pm K K U^\Theta K\varphi &={\rm sgn}(C_\pm K)^2 U^\Theta K U^\Theta K \varphi={\rm sgn}(C_\pm K)^2\Theta^2\varphi,
\int_\mathcal{S}\partial{_t^\varphi}fgd\mathcal{V}_t=\partial_t\int_\mathcal{S}fgd\mathcal{V}_t-\int_\mathcal{S}f\partial{_t^\varphi}gd\mathcal{V}_t-\int_{z=0}fg\partial_thdy,
h\left(x\right) =\int_{0}^{x}\varphi\left(y\right)F\left(x-y\right)dy+\int_{x}^{1}\varphi\left(y\right)F\left(x-y\right)dy; h'\left(x\right) =\int_{0}^{x}\varphi\left(y\right)F'\left(x-y\right)dy+\int_{x}^{1}\varphi\left(y\right)F'\left(x-y\right)dy.
\langle \Psi_t(\xi\oplus \eta), \Psi_t(\xi'\oplus\eta') \rangle = \exp(-t \Vert \xi - \xi' \Vert^2) \exp(-t \Vert \eta - \eta' \Vert^2) & = \exp(-t \Vert (\xi\oplus\eta) - (\xi'\oplus\eta') \Vert^2)
I{}= \biggl(T\tau+\frac{T}{2a}-\frac{\tau^2}{2}-\frac{\tau}{2a}-\frac{1}{4a^2}\biggr)e^{-2a\tau}+\frac{1}{2a^2}e^{-2aT}+\frac{T}{2a}-\frac{1}{4a^2}&{}\leq \biggl(T\tau+\frac{T}{2a}\biggr)e^{-2a\tau}+\frac{T}{2a}+\frac{1}{2a^2}e^{-2aT}.
\phi_{\lambda}(r)=\left(\lambda^{-\frac{\theta}{\theta-1}}r^{-\frac{n-p}{p}\frac{\theta}{\theta-1}}+\left(1-\lambda^{-\frac{\theta}{\theta-1}}\right)R^{-\frac{n-p}{p}\frac{\theta}{\theta-1}}\right)^{-\frac{p}{n-p}\frac{\theta-1}{\theta}}.
\int_{D_k}\partial_j\partial_{j}\left(\partial_{u}\partial_{i}g_{u i}- \partial_{u}\partial_{u}g_{ii}\right)dV_e =\int_{S_{k+1}}(\partial_{j}\partial_{u}\partial_{i}g_{u i}- \partial_{j}\partial_{u}\partial_{u}g_{ii})\nu^e_jdS^{n-1}&- \int_{S_{k}}(\partial_{j}\partial_{u}\partial_{i}g_{u i}- \partial_{j}\partial_{u}\partial_{u}g_{ii})\nu^e_jdS^{n-1},
\varphi_{2}(s,t_{0})= -\left\lbrace \left[ \frac{\partial v(s,t_{0})}{\partial s}\right]^2 + \left[ \frac{\partial w(s,t_{0})}{\partial s}\right]^2 \right\rbrace^{-1/2}\frac{\partial w(s,t_{0})}{\partial s},
(D_1)^2 =(L-2E-(mE+\sum_{j=1}^8 n_{1,j}\Gamma_j+n_{2,j}\widetilde{\Gamma}_j))^2 &=-4-2\sum_{j=1}^8 (n_{1,j}^2+ n_{2,j}^2) -2(L \cdot mE+\sum_{j=1}^8 n_{1,j}\Gamma_j+n_{2,j}\widetilde{\Gamma}_j) & \leq -4
q^{\frac{1}{2}\mathcal{E}(d)}\sum_{(d^{\bullet}; d^{\infty})}(-1)^{l(d^{\bullet})}q^{-\chi( d^{\bullet}) - \chi( d^{\infty}, d^{\bullet}) - \mathcal{E}(d^{\bullet})}\left( \prod_{i=1}^{l(d^{\bullet})}\frac{q^{-\chi( d^i , d^i )}}{(q^{-1})_{d^i}}\right) \frac{q^{-\mathcal{E}(d^{\infty})}}{(q^{-1})^{\sigma}_{d^{\infty}}},
\det\left|\begin{array}{cccc}0 \phi_{x_1}\phi_{x_2}\phi_{x_3}\phi_{x_1}\phi_{x_1x_1}\phi_{x_1x_2}\phi_{x_1x_3}\phi_{x_2}\phi_{x_2x_1}\phi_{x_2x_2}\phi_{x_2x_3}\phi_{x_3}\phi_{x_3x_1}\phi_{x_3x_2}\phi_{x_3x_3}\end{array}\right|\,=\,0.
TrD_{\mu\nu}(x-x')={\int}d^{4}x\left[3{\int}\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}+e^{2}<\rho>^{2}}+\frac{\delta^{4}(0)}{e^{2}<\rho>^{2}}\right].
\begin{cases}\tilde{\rho}\tilde{u}=\rho u, &\tilde{\rho}\tilde{u}^2+\tilde{p}=\rho u^2+p, &\tilde{\rho}\tilde{u}\tilde{v}=\rho uv, &(\tilde{\rho}\tilde{E}+\tilde{p})\tilde{u}=(\rho E+p)u+q_0\rho\phi(T)Zh, &\tilde{\rho}\tilde{u}\tilde{Z}=\rho u Z-\rho\phi(T)Zh.\end{cases}
V_{\bar{f}}^{(-)^a}\!\left(\frac{|\beta nu|}{1+u^2}\right)\cos^{(a)}\!\left(\frac{2\pi\beta n}{1+u^2}\right) &=\sum_{j+2l<2l_0}\frac{(-2\pi)^j}{j!l!}(V_{\bar{f}}^{(-)^a})^{(l)}(|\beta nu|)\cos^{(j+a)}(2\pi\beta n)u^l\left(\frac{\beta nu^2}{1+u^2}\right)^{j+l}+O_{\alpha,\sigma,l_0}\bigl(|nu|^{-\sigma}u^{2l_0}\bigr).
\Re\Big\{\sum_{n=1}^{\infty}z_n^m \Big\}\leq \frac{1}{\alpha^m}- \frac{1}{(\alpha+1-\beta_1)^{2m}}+ \Re\Big\{\frac{\delta(\chi)+\delta(\psi \chi)}{(\alpha+i\gamma')^{2m}}- \frac{\delta(\chi)+\delta(\psi\chi)}{(\alpha +1+i\gamma' -\beta_1)^{2m}}\Big\}
\frac{\partial Y}{\partial x}=\big(A_0 x^3+A_1 x^2+A_2 x+A_3 \big)Y , \frac{\partial Y}{\partial t_1}=\big( A_0 x^2+A_1 x+B_{10}\big)Y ,\frac{\partial Y}{\partial t_2}= ( -A_0 x+B_{20})Y ,
\begin{aligned}\mathcal{E}_{l_x,l_y}=[\frac{2\pi l_x}{L_{R,x}},\frac{2\pi (l_x+1)}{L_{R,x}}]\times [\frac{2\pi l_y}{L_{R,y}},\frac{2\pi (l_y+1)}{L_{R,y}}] \mathcal{E}_{m_x,m_y}=[\frac{2\pi m_x}{L_{S,x}},\frac{2\pi (m_x+1)}{L_{S,x}}]\times [\frac{2\pi m_y}{L_{S,y}},\frac{2\pi (m_y+1)}{L_{S,y}}].\end{aligned}
\lim_{N\to\infty}\|S\varphi-S_N \varphi\|_{H^p(D'_\beta)}=\lim_{N\to\infty}\|S\varphi-S[\varphi^{(N)}]\|_{H^p(D'_\beta)}\leq C_p \lim_{N\to\infty}\|\varphi-\varphi^{(N)}\|_{L^p(\partial D'_\beta)}= 0.
\beta F = - \frac{4}{3}\beta \int d^3 x \int_{\mu \sqrt{\xi^2 - \Omega_0^2 r^2}}^\infty d \omega \frac{1}{e^{\beta \omega}- 1}\frac{\xi r}{\sqrt{( \xi^2 - \Omega_0^2 r^2 )}}\left( \frac{\omega^2}{\xi^2 - \Omega_0^2 r^2}- \mu^2 \right)^{3/2}.
f(n,k-1,p_-) = f(n,k-1,p_+) \left(1 - \Theta\left( \tfrac{\omega \log^3 n}{n^2 p_+^{3/2}}\right) \right)^{n-k+1}&= f(n,k-1,p_+) \left(1 - \Theta\left( \tfrac{\omega \log^3 n}{n p_+^{3/2}}\right) \right) &\sim f(n,k-1,p_+) = (p_+ n)^{-1}\sim (p_- n)^{-1},
\eta_f[\frak{u},y](x)=(\eta[\frak{u},y+\vartheta](x+\vartheta))^\ell\cdot\prod_{1\le i\le I}(\Phi_{\mathcal{V}}(x-u_i))^{e_i}\cdot \prod_{1\le i\le I}(\Phi_{\mathcal{V}}(y-u_i))^{-e_i}.
\int \prod_{n}d X{n}^{I}d X{n}^{I I}\frac{d k_{n}}{2 \pi}\frac{d p_{n}}{2 \pi}\frac{d q_{n}}{2 \pi}\frac{d l_{n}}{2 \pi}e^{- i k_{n}X_{n}^{I}- i p_{n}X_{n}^{I}- i l_{n}X_{n}^{I I}- i q_{n}X_{n}^{I I}}e^{- \sum_{m > 0}( \frac{k_{m}^{2}}{4 \omega_{m}}+ \frac{p_{m}^{2}}{4 \omega_{m}}+ \frac{l_{m}^{2}}{4 \omega_{m}}+ \frac{q_{m}^{2}}{4 \omega_{m}})}
\overline{\wp_{\Lambda}(u)}=\overline{\frac{1}{u^2}+\sum_{\omega \in \Lambda \setminus \{0\}}\left\{\frac{1}{(u-\omega )^2}-\frac{1}{\omega^2}\right\}}=\frac{1}{\overline{u}^2}+\sum_{\omega \in \Lambda \setminus \{0\}}\left\{\frac{1}{(\overline{u}-\overline{\omega})^2}-\frac{1}{\overline{\omega}^2}\right\}.
2\dot{F}^{kl}\left(2\Lambda_k^p\nabla_lG_{1p}-\Lambda_k^p\Lambda_l^qG_{pq}\right) =2\sum_{k=1}^n\sum_{p=2}^n\dot{f}^k\left(2\Lambda_k^p\nabla_kG_{1p}-(\Lambda_k^p)^2G_{pp}\right) =2 \sum_{k=1}^n\sum_{p=2}^n\dot{f}^k\left(\frac{(\nabla_kG_{1p})^2}{G_{pp}}-\left(\Lambda_k^p-\frac{\nabla_kG_{1p}}{G_{pp}}\right)^2G_{pp}\right).
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j(M/G,L^m/G) &=(\sum^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi})\frac{1}{2\pi}\int_{X}\mathrm{Td_b\,}(T^{1,0}X)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0.
J = \frac{3\left(w^2 q_{/2}+v\,q\right)}{2\delta^2 w^3 \,q}\Big\{w^2 \!\left[ \kappa q v + 2 \delta q_{/1}+q_{/2}\left(\kappa \,w^2 + \delta ' \, v-\delta^2 \lambda \right) \right] -\delta^2 q \left( \lambda v - \delta ' \right) \Big\}.
\langle\langle \xi,\,\bar\partial^{\star}\omega\rangle\rangle_\omega= \int\limits_X \xi \wedge \bar\partial\omega_{n-1}= \int\limits_X \xi \wedge \bar\partial\omega \wedge \omega_{n-2}= -\int\limits_X \xi \wedge \partial\overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}.
\begin{aligned}1,1,1,\dots,1,1,1, 0,0,\dots,0,0,0 1,1,1,\dots,1,1,0, 0,0,\dots,0,0,1 1,1,1,\dots,1,0,0, 0,0,\dots,0,1,1 \dots \dots\dots\end{aligned}
{\cal L}= \psi^{\dagger}i \partial_{t}\psi + \psi^{\dagger}\frac{\nabla^{2}}{2 M}\psi - \frac{1}{2}C_{0}( \psi^{\dagger}\psi )^{2}\, - \frac{1}{2}C_{2}( \psi^{\dagger}\nabla^{2}\psi ) ( \psi^{\dagger}\psi ) + h . c . + \ldots .
DF(a)[v,h^{d-r+1},\ldots, h^d] = (v,Df(a)[v,h^{d-r+1},\ldots, h^d]) &=(v,Df(a)[v,0,\ldots,0]+Df(a)[0,\ldots,0,h^{d-r+1},\ldots, h^d]) &= (v,w),
j(q^{20};q^{32}) = j(q^{10};q^{16})j(-q^{10};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2}, j(q^{4};q^{32}) = j(q^{2};q^{16})j(-q^{2};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2}.
I_{1}( a , b , d ) = - \frac{1}{\left( 2 \pi \right)^{d - 2}a b}\sum_{n = 1}^{\infty}\int d^{d - 2}p \frac{1}{( \vec{p}^{\, 2}+ m^{2}+ ( \frac{n \pi}{a})^{2})}
\partial_t\Big[(P_t\ast f)_\gamma\Big]=-2\sum_{s=1}^{n-1}a^{\beta\alpha}_{rs}\Big(\big(Q^{(r)}_{\gamma\beta}\big)_t\ast\partial_s f_\alpha\Big)\qquad\mbox{in}\,\,{\mathbb{R}}^{n-1},
{\rm Tr}\left(\frac{v+U_t}{v-U_t}\cdot \frac{w+U_s}{w-U_s}\right)=\sum_{k}\frac{w+z_k(s)}{w-z_k(s)}\langle u_k(s), \frac{v+U_t}{v-U_t}u_k(s)\rangle.
\sum_{q\in [N]^r}P(x,q,A)P(y,q,B)(x_{q_1 q_2}- y_{q_1 q_2}) &=\sum_{q_3,\ldots, q_r}P(x, q, A_3) P(y,q,B_3) \biggl(\sum_{q_1, q_2}Q(x,y,q)(x_{q_1q_2}-y_{q_1q_2})\biggr),
\lambda\sigma^{-A}=\sigma^{-\frac{\theta}{p-1}}\cdot \lambda\sigma^{\frac{\theta}{p-1}-A}\le\sigma^{-\frac{\theta}{p-1}}\cdot \lambda \tilde{T}_\lambda^{\frac{1}{p-1}-\frac{A}{\theta}}=\delta^{\frac{1}{p-1}-\frac{A}{\theta}}\sigma^{-\frac{\theta}{p-1}}
[\bar{e}_{i,k+1}, \bar{e}_{i,l}]=[\bar{e}_{i,k}, \bar{e}_{i,l+1}],\ [\bar{e}_{i,k+2}, \bar{e}_{i+1,l}]-(d+d^{-1})[\bar{e}_{i,k+1}, \bar{e}_{i+1,l+1}]+[\bar{e}_{i,k}, \bar{e}_{i+1,l+2}]=0,
B((u,v),f) =\begin{dcases}\frac{B_1(u,\pi_1(f))B_2(v,\pi_2(f))}{(r-1)!}, \ (u,v)\in f, 0, ,\end{dcases}begin{align*}2mm]w(f) =\frac{w_1(\pi_1(f))w_2(\pi_2(f))}{r!}.
\mathtt{p}_{0}(u)\, =\, \frac{\left\langle \partial_{\theta}q_{0, nd}\, ,\, u\right\rangle_{-1, 1/q_{0}, d}}{\left\Vert \partial_{\theta}q_{0, nd}\right\Vert_{-1, 1/q_{0}, d}^{2}}\, .
S^{\pm}= \left( \begin{array}{l l l}{{\partial_{1}X^{0}}}&{{. . .}}&{{\partial_{1}X^{p}}}\\{{\partial_{2}X^{0}}}&{{. . .}}&{{\partial_{2}X^{p}}}\\{{. . .}}&{{. . .}}&{{. . .}}\\{{G^{0 \nu}( P_{\nu}\pm T n_{\nu})}}&{{. . .}}&{{G^{p \nu}( P_{\nu}\pm T n_{\nu})}}\\ \end{array}\right) \ ,
f(q) =q - 6q^5 + 9q^9 + 10q^{13}-30q^{17}+11q^{25}+42q^{29}-70q^{37}+ O(q^{41})\in S_3^{\textrm{new}}(\Gamma_1(16)), g(q) =q - 2q^2 - 2q^3 + 4q^4 + 4q^6 - 8q^8 - 5q^9 + 14q^{11}-8q^{12}+O(q^{14})\in S_3^{\textrm{new}}(\Gamma_1(8))
g_1 =q - 3q^4 - 3q^6 + 9q^9 + 5q^{10}+O(q^{15})\in S^{\epsilon_4}(15,3,\chi), g_2 =q^2 - 3q^3 + 5q^5 - 7q^8 + 9q^{12}+O(q^{15})\in S^{\epsilon_1}(15,3,\chi).
\mathbf{1}(t=0) N \eta \sum_{k \neq \mu, \nu, \mu_1}R_{\mu k}(\sigma_i)^{3/2}(X^G_{i \mu_1})^3 \overline{z^{3/2}R_{\nu a_1}R_{b_1 a_2}R_{b_2 a_3}R_{b_3 k}}.
a_{i}= \frac{i}{2}\Psi^{\dagger}\partial_{i}^{\hspace{- 2 . 9 m m}\raisebox{1 . 2 m m}{\leftrightarrow}}\Psi - \Psi^{\dagger}\frac{\sigma^{3}}{2}\Psi A_{i}= \frac{i}{2}\Psi^{\dagger}\nabla_{i}^{\hspace{- 3 . 8 m m}\raisebox{1 . 2 m m}{\leftrightarrow}}\Psi .
\psi((1+y)^{-1}x) = \left[1 + y + \frac{qy}{1+y}x\right]^{-1}\left[x + \frac{qy}{(1+y)(1+qy)}x^2\right] &= \left[\left(1+\frac{qy}{(1+y)(1+qy)}x\right)(1+y)\right]^{-1}\left[1 + \frac{qy}{(1+y)(1+qy)}x\right]x &= (1+y)^{-1}x.
\varphi^*_a(H)=\varphi^*_a(F_0^{\gamma_0})\cdot \varphi^*_a(F_1^{\gamma_1})\cdot\ldots\cdot\underbrace{\varphi^*_a(F_l)\cdot \varphi^*_a(F_l^{\gamma_l-1})}\cdot\ldots\cdot \varphi^*_a(F_j^{\gamma_j}).
\frac{d}{dx}\sum_{k=1}^{p+1}S_{2,\lambda}(p+1,k)\frac{x^{k}}{k}=\frac{d}{dx}\bigg(e^{-x}\sum_{n=0}^{\infty}\big((1-\lambda)_{p,\lambda}+(2-\lambda)_{p,\lambda}+\cdots+(n-\lambda)_{p,\lambda}\big)\frac{x^{n}}{n!}\bigg).
\mathbf{A}_k=\{x\in \mathbb{R}^n:x_1\in\mathbf{L}_1(\max\{\underline{M}_k, \gamma_{12}(\underline{M}_k)\}), x_2\in\mathbf{L}_2(\max\{\gamma_{21}(\underline{M}_k),\gamma_{21}\circ\gamma_{21}(\underline{M}_k)\})\},
\left|{\rm Tr}(\Gamma a_0\prod_{k=1}^{p-1}[F,a_k]F\delta(a_p)f'(s|D|))-{\rm Tr}(\mathcal{W}_p(a)Ff'(s|D|))\right| &\leq\|(\Gamma a_0\Big(\prod_{k=1}^{p-1}[F,a_k]\Big)[F,\delta(a_p)]|D|^p\|_{\infty}\cdot\||D|^{-p}f'(s|D|)\|_1=O(1),
\mathbb P\left(\sup_{0\le s\le T}\left|\int_0^s \sigma(r, X^1_r)dW^1_r\right|>a\right) \le \sum_{i=1}^d \mathbb P\left(\sup_{0\le s\le T}\left|\int_0^s \sigma^i(r, X^1_r)dW^1_r\right|>\frac{a}{d}\right)\le C_1\exp(-a^2/(C_2T)).
\psi = \left( \begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\ \end{array}\right) \ , \quad \overline{\psi}\equiv \psi^{\dag}\gamma_{3}\equiv \left( \begin{array}{c c}{\overline{\psi}_{1}}&{- \overline{\psi}_{2}}\\ \end{array}\right) \equiv \left( \begin{array}{c c}{\psi_{1}^{\dag}\sigma_{3}}&{- \psi_{2}^{\dag}\sigma_{3}}\\ \end{array}\right) \ ,
M_\varrho = \big( \varrho_{mn}\big)_{m,n\geq1}= \begin{pmatrix}\varrho_1 \varrho_2 \varrho_3 \cdots \varrho_2 \varrho_4 \varrho_6 \cdots \varrho_3 \varrho_6 \varrho_9 \cdots \vdots \vdots \vdots \ddots\end{pmatrix}.
{\cal U}(f,h,u)=\frac{(u^2-1)^2}{2e^2r^4}+\frac{\eta^2 u^2 h^2}{r^2}+\frac{\eta^2f^2}{r^2}+\frac{\lambda\eta^4}4(h^2-1)^2+\frac{\lambda\eta^4}4(f^2-1)^2 \ ,
(P_k)_i=\begin{bmatrix}0 \eta_r U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\ldots \eta_r\mu^{-\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) \overbrace{0 \ldots 0}^{(k-2)}\end{bmatrix}^{T},
T ( z ) = - \frac{1}{2}: \! \partial X \partial X \! : + i \frac{\kappa}{2}\partial^{2}X , \qquad \tilde{T}( \bar{z}) = - \frac{1}{2}: \! \bar{\partial}X \bar{\partial}X \! : - i \frac{\kappa}{2}\bar{\partial}^{2}X ,
\tau_A=\left\|\begin{array}{ccc}cosh(\tau) 0 sinh(\tau) 0 1 0 sinh(\tau) 0 cosh(\tau)\end{array}\right\|\ \quad{\bf a}_{\bf T}=\left\|\begin{array}{ccc}1+{\bf a}^2/2 -{\bf a}^T{\bf a}^2/2 -{\bf a}1 -{\bf a}-{\bf a}^2/2{\bf a}^T 1-{\bf a}^2/2\end{array}\right\|
I_{w}\left( f\right) =\int_{0}^{2\pi}\int_{0}^{R}f\left( re^{i\varphi}\right) w\left( re^{i\varphi}\right) rdrd\varphi={\displaystyle\sum_{k=0}^{\infty}}{\displaystyle\sum_{\ell=1}^{a_{k}}}\int_{0}^{R}f_{\left( k,\ell\right)}\left( r\right) w_{\left(k,\ell\right)}\left( r\right) rdr.
\Phi(A) = \left(\begin{matrix}(A_{1,1}^{(i,j)})_{i,j=1}^d \cdots (A_{1,N}^{(i,j)})_{i,j=1}^d \vdots \ddots \vdots (A_{N,1}^{(i,j)})_{i,j=1}^d \cdots (A_{N,N}^{(i,j)})_{i,j=1}^d\end{matrix}\right),
{\Bigl{\{}\ell_{\bf i}, \ell_{\bf j}, \ell_{\bf k}, u_{0},\,u_{\bf i},\,u_{\bf j},\,u_{\bf k},\,[u_{0}, u_{\bf i}],~[u_{0},\,u_{\bf j}], [u_{0}, \,u_{\bf k}]\Bigr\}}
\begin{cases}c^{k+1}=_{\frac{1}{\alpha}\|\cdot\|_1}\left( c^k - \frac{\lambda}{\alpha}A^\top(2\tau^k - \tau^{k-1})\right), \tau^{k+1}= \left(I -_{\iota_{\mathcal{F}}}\right)\left(Ac^{k+1}+ \tau^k\right). \end{cases}
|T_{w}(x)-T_{w,n}(x)|=|\sum_{k=n+1}^{\infty}w^{k}s(2^{k}x)|=|w^{n+1}\sum_{k=0}^{\infty}w^{k}s(2^{k+n+1}x)|\le\frac{|w|^{n+1}}{2}\cdot\frac{1}{1-|w|}
_{V_{2}^{*}}\langle A_{2}(u),T_{n}u\rangle_{V_{2}}=\langle-a|u|^{p-2}u,nu-n\left(I-\frac{\Delta}{n}\right)^{-1}u\rangle &=n\int_{0}^{\infty}{\rm{e}}^{-t}\left(\int_{\Lambda}-au(\xi)|u(\xi)|^{p-2}\left(u(\xi)-P_{\frac{t}{n}}u(\xi)\right)d\xi\right)\d t\leq0.
M = \sqrt{( P_{1}^{( 1 )})^{2}+ \beta^{2}}+ \sqrt{( P_{1}^{( 2 )})^{2}+ \beta^{2}}+ \sqrt{( Q_{2}^{( 1 )})^{2}+ \beta^{2}}+ \sqrt{( Q_{2}^{( 2 )})^{2}+ \beta^{2}},
\frac{1}{h^{d}}\prod_{i=1}^d \frac{\Delta_{e_i}v_h (x)}{|| e_i ||}= \bigg(h^d\prod_{i=1}^d || e_i || \bigg) \frac{1}{h^{2d}}\prod_{i=1}^d \frac{\Delta_{e_i}v_h (x)}{|| e_i ||^2}.
(\tilde{\Gamma}\circ T)^*(\bar{x}_{2i+1})=T^*\circ\tilde{\Gamma}^*(\bar{x}_{2i+1})=T^*(u^2\otimes\alpha^*(x_{2i-3}))=\Sigma u^2\otimes\Sigma^{-1}\alpha^*(x_{2i-3}),
\frac{d^2R_\ell(r)}{dr^2}+\left[\frac{(Z\alpha)^2-\ell(\ell+1)+\left(\frac{2\epsilon Z\alpha}{\hbar c}\right)r+\left(\frac{\epsilon^2-m_0^2c^4}{\hbar^2c^2}\right)r^2}{r^2}\right]R_\ell(r)=0.
1-\frac{1}{j}\leq \Vert P_2 Q_{n_j}^*H_{n_j}^iQ_{n_j}P_2\Vert &= \Vert P_2 X_{n_j}^{d+i}P_2\Vert &\leq \Vert P_{\alpha(2,2d)}\widetilde{X_{n_j}}^{d+i}P_{\alpha(2,2d)}\Vert &= \Vert P_{\alpha(2,2d)}W_{n_j}^*H_{n_j}^iW_{n_j}P_{\alpha(2,2d)}\Vert .
r_{A_1B_1A_2B_2}(a_1,b_1,a_2,b_2)=\sum_{x_1,x_2,y_1,y_2}\Big(r_{X_1Y_1X_2Y_2}(x_1,y_1,x_2,y_2) V(a_1a_2b_1b_2|x_1x_2y_1y_2)\Big) &\quad=p_{AB}(a_1,b_2)p_{AB}(a_2,b_1).