harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\varphi'}_{\sigma'}^{\sigma''}[x'] \circ \varphi_{\sigma}^{\sigma'}[x] = \sum_{x'' \in S''_{\sigma''}\backslash G / S_{\sigma}}\bigl({\varphi'}_{\sigma'}^{\sigma''}[x'] \circ \varphi_{\sigma}^{\sigma'}[x]\bigr)[x'']
\|H_\ell (r_1,r_2,\cdot)*f\|_{L^6(d\theta_2)}\lesssim\begin{cases}(\lambda r_1 r_2)^{-\frac 12}2^{\frac{\ell}{3}}\|\beta_\ell f\|_{L^2(d\theta_1)}2^\ell \leq (\lambda r_1 r_2)^{\frac 12}2^{-\frac{2 \ell}3}\|\beta_\ell f\|_{L^2(d\theta_1)}2^\ell > (\lambda r_1 r_2)^{\frac 12}\end{cases}.
\mathcal{L}_M(\varepsilon) = \sum\limits_{k=0}^{- \lfloor \alpha\rfloor}\varepsilon^{\alpha +k -1}\, \mathfrak{b}_{\alpha +k -1}+ \sum\limits_{k= 1}^{M +\lfloor \alpha\rfloor}\varepsilon^{k -1}\, \mathfrak{b}_{k -1}+ \sum\limits_{k=- \lfloor \alpha\rfloor + 1}^{M}\varepsilon^{\alpha +k -1}\, \mathfrak{b}_{\alpha +k -1},
\underset{\mathbf{x}\in \mathbb{R}^{np}}{}\,\, f(\mathbf{x}) + \frac{\rho (1-\eta)}{4}\|\mathbf{x}\|^2_{\mathbf{V}\otimes \mathbf{I}_p}\,\,\,\, \, \sqrt{\eta}\sqrt{\mathbf{V}}\mathbf{x}= \mathbf{0}.
I_2(k=1)=-\frac{9\pi}{8192}K_c\approx -0.09683, \;\;\; K_c\equiv \int^{\infty}_0\frac{(k^2_c+1)^3(k^2_c+9)^2}{(k^2_c+13)\cosh^2\frac{k_c\pi}{2}}\approx 28.0549.
(z-z_1) \cdots (z-z_{r+1}) = 1 \cdot z^{r+1}+Z_1(z_1,\ldots,z_{r+1}) z^r+\cdots +Z_{r+1}(z_1,\ldots,z_{r+1}), \phi_{r+1}(z)+b_1 \phi_r(z) + \cdots +b_r \phi_1(z) = 1 \cdot z^{r+1}+B_1(b_1,\ldots,b_r) z^r+\cdots +B_{r+1}(b_1,\ldots, b_r).
{\cal L}={1\over 2 g \sqrt{\alpha'}}~{\rm Tr}~\left(\left({\dot \Phi}_i -{i \lambda\over 2(2\pi\alpha^\prime)}\epsilon_{ijk}\left[\Phi_i,\Phi_j\right]\right)^2 +{1+\lambda^2\over 2 (2\pi\alpha')^2}\left[ \Phi_i,\Phi_j \right]^2 \right)~.
\begin{aligned}df^i =0, 1\leq i\leq 6, df^7 =\frac{\sqrt{6}}{6}\,y(t)^{-1}z(t)^{-2}\Big(y(t)^{-2}f^{12}+z(t)^{-2}f^{34}+z(t)^{-2}f^{56}\Big).\end{aligned}
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(36n+18)q^n \equiv 2qf_{12}^3+3qf_{12}^3\cdot\dfrac{f_2^2}{f_4}+\dfrac{f_6^2}{f_{12}}\cdot f_4^3 &\equiv 2qf_{12}^3+3qf_{12}^3\left(\dfrac{f_{18}^2}{f_{36}}-2q^2\dfrac{f_6f_{36}^2}{f_{12}f_{18}}\right) &\quad+\dfrac{f_6^2}{f_{12}}\left(f_{12}a(q^{12})-3q^4f_{36}^3\right)~(\textup{mod}~4).
\frac{1}{t^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}- G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) +G(\mbox{Hor}(P_VU),\mbox{Hor}(P_VU)) = 0.
\ell(\xi) \, J W(\eta) J \, \Omega = \ell(\xi) \, W(A^{1/2}\overline \eta) \Omega &= \ell(\xi) \, A^{1/2}\overline \eta &= \xi \otimes A^{1/2}\overline\eta J W(\eta) J \, \ell(\xi) \, \Omega = J W(\eta) J \, \xi &= \xi \otimes A^{1/2}\overline\eta + \overline{\langle \overline \xi, A^{1/2}\overline\eta\rangle}\, \Omega.
{\cal H}_{c}= \int d^{d}x \, \left\{{\frac{1}{2}}\sum_{i = 1}^{N}\Bigl [ ( \partial_{\mu}\Phi_{i})^{2}+ r \Phi_{i}^{2}\Bigr ] +{\frac{1}{4 !}}\Bigl [ u ( \sum_{i}^{N}\Phi_{i}^{2})^{2}+ v \sum_{i}^{N}\Phi_{i}^{4}\Bigr ] \right\}.
\textbf{z}_k=A_k\{(\textit{A}^T)^2\textbf{z}_{k-2}+B_k \textit{A}^T\textbf{z}_{k-2}+C_k \textbf{z}_{k-2}+F_k \textit{A}^T\textbf{z}_{k-3}+G_k \textbf{z}_{k-3}\}.
P_d=-M_{11}(\lambda), &\lambda^t P_d = M_{12}(\lambda)(\Lambda_{t-1}(\lambda)\otimes I_n) = (\Lambda_{t-1}(\lambda)^T\otimes I_n)M_{21}(\lambda), &Q(\lambda) = (\Lambda_{t-1}(\lambda)^T\otimes I_n)M_{22}(\lambda)(\Lambda_{t-1}(\lambda)\otimes I_n).
\left[\begin{array}{ll}\partial_t \widetilde{v}(t,x) = \partial_{xx}\widetilde{v}(t,x) + f(\widetilde{v}(t,x)) \dot{\widetilde{W}}(t,x), (t,x) \in [0, T] \times [0, \pi], \widetilde{v}(t,0) = \widetilde{v}(t, \pi) = 0, \textrm{for all}\,\, t \in [0,T], \widetilde{v}(0,x) = 0, \textrm{for all}\,\, x \in [0, \pi],\end{array}\right.
\begin{cases}Z^j_t=z_j+\sum_{i=1}^m X^{i,j}\circ C^i_t+Y^j_t 1\leq j\leq m Z^{m+j}_t=z_{m+j}+\sum_{i=1}^m X^{i,m+j}\circ C^i_t+Y^{m+j}_t+\sum_{i=1}^{n}C^{m+i}_t\beta_{i,j}1\leq j\leq n\end{cases}
A_{trans}^{\prime}=\frac{I_{trans}}{I_{inc}}=(1-R_{1})(1-R_{2})A_{circ}=\frac{(1-R_{1})(1-R_{2})}{(1-\sqrt{R_{1}R_{2}})^{2}+4\sqrt{R_{1}R_{2}}sin^{2}(\phi)}.
\|\langle x \rangle^{s}(P(h)-E \pm i \varepsilon)v\|_{L^2}\le \|(P(h)-E \pm i \varepsilon)\langle x \rangle^{s}v \|_{L^2}+ \|[P(h),\langle x \rangle^{s}]\langle x \rangle^{-s}\langle x \rangle^{s}v \|_{L^2}& \le C_{\varepsilon, h}\| \langle x \rangle^{s}v \|_{H^2},
\sum_{k=0}^n \binom{n}{k}p(k) (-1)^{n-k}= \sum_{k=0}^n \binom{n}{k}\left(c_0+c_1k+\cdots+c_nk^n\right) (-1)^{n-k}= \left(\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}\right)c_0 +\cdots+ \left(\sum_{k=0}^n \binom{n}{k}k^n (-1)^{n-k}\right)c_n=n!c_n.
u = \frac{\frac{\sqrt 3}{2 \pi}\, \theta(t-t_0) + C_0}{(t-t_0)^2}\, \exp\left[-\frac{(x-x_0)^2}{4(t-t_0)}-\frac{3}{(t-t_0)^3}\left(y-y_0 -(t-t_0) \frac{x+x_0}{2}\right)^2 \right]
\tilde{S}_{Q 1}^{k i n}( B_{i}) = \frac{2}{N}\mathrm{T r}\sum_{j = - L / 2 + 1}^{L / 2}( 1 - \cos k_{j}) B_{j}^{2}= \tilde{S}_{Q 1 f}^{k i n}( B_{i})
- \frac{1}{e}\frac{\lambda^{2}}{( x_{1}^{2}+ \lambda^{2})^{\frac{3}{2}}}\left[ \mathrm{a r c t a n}\left( \frac{x_{2}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\right) - \pi \Big ( \frac{1}{2}+ m \Big ) \right] .
\nabla \cdot A = u \cdot \nabla \varphi + c \cdot V -{u - c \over \gamma_n}\cdot \nabla\left({m \cdot x \over |x|^n}\right) = |u|^2 - (u-c) \cdot \left( V +{1 \over \gamma_n}\nabla\left({m \cdot x \over |x|^n}\right) \right).
\big((\hat{v}+\omega) \times B \big)_a= (-1)^{a-1}\big[(\hat{v}+\omega)_{3-a}(\omega\cdot \mathbf{e}_3) (B\cdot \omega ) -\big(\sum_{i=1,2,3}(\hat{v}+\omega)_{3-a}( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i))+ (\hat{v}+\omega)_3 B_2\big)\big].
2\rho(1-\psi)(\theta)(r_1+r_2)^{\frac 12}a_\lambda(r_1 + r_2)\int_0^\infty e^{i\lambda r_1 r_2s^2}\frac{\theta}{s^2+\theta^2}\,ds = O\left((\lambda r_1 r_2)^{-\frac 12}\right).
( e^{- 2 \phi})_{\mathrm{f i x}}={\frac{| n_{2}m_{1}- n_{1}m_{2}|}{m_{1}^{2}+ m_{2}^{2}}}\ , \qquad ( a )_{\mathrm{f i x}}={\frac{n_{2}m_{2}+ n_{1}m_{1}}{m_{1}^{2}+ m_{2}^{2}}}\ .
(L_\omega f)(\lambda){=}(\mathcal{J}L_{\omega;1}\mathcal{J}^{-1}f) (\lambda){=}( L_{\omega;1}\mathcal{J}^{-1}f)(\lambda){=}\sum_{\nu\in W_0\omega}( \mathcal{J}^{-1}f)(\lambda + \nu){=}\sum_{\nu\in W_0\omega}t[\lambda + \nu] f(\lambda + \nu)+d_{\lambda,\nu}(1-t^{-1}_{\vartheta}) f(\lambda) .
\sum\limits_{\{h,f\}\in C_q}x_{hf}+ x_{i_1j_1}= \sum\limits_{\{h, f \}\in C_q: f < i_1}x_{hf}+\sum\limits_{\{h, i_1\}\in C_q}x_{hi_1}+\sum\limits_{\{h, j_1\}\in C_q}x_{hj_1}
\phi^{\alpha,n}(g)^\ast \left(S, f_\beta \otimes \rho_{n}(E^\beta) \right) = - \left(S, \overline{g^{S^\flat}}\cdot f_\beta \otimes \rho_n([E^\alpha,E^\beta]) \right) + \left\langle (S, f_\beta \otimes \rho_{n}(E^\beta)), \phi^{\alpha,n}(g) 1 \right\rangle 1.
\pi_{5}(a(2j+1,k)) =\pi_{5}\left(\sum_{i=1}^{\infty}a(2j,i)m(6i+2,k+i)\right) \geq \min_{i\geq1}\left(j+\left\lfloor\dfrac{5i-3}{2}\right\rfloor+\left\lfloor\dfrac{5k-i-3}{2}\right\rfloor\right) \geq j+1+\left\lfloor\dfrac{5k-5}{2}\right\rfloor.
\int_{\partial\Omega}\{(\nu_1+i\nu_2)\Phi'(\mathcal P_1 \mbox{\bf a},\mathcal P_2 \mbox{\bf b})+(\nu_1-i\nu_2)\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})\}d\sigma=0.
a_n=\sum_{k=0}^na_k-\sum_{k=0}^{n-1}a_k =\sum_{k=0}^nf_{n-k}f_k-\sum_{k=0}^{n-1}f_{n-1-k}f_k &=f_{n-1}+\sum_{k=0}^{n-1}(f_{n-k}-f_{n-1-k})f_k &=f_{n-1}+\sum_{k=0}^{n-2}f_{n-2-k}f_k.
\langle\langle B, \bar{V}\rangle\rangle_{k+1}=\langle b_q, v_{M+k+1}\rangle= b^+_q p_{(M+k+1)}+b^-_q \bar{p}_{-(M+k+1)}= 0 \langle\langle B^*, \bar{V}\rangle\rangle_{k+1}=\langle \hat{b}_1, v_{M+k+1}\rangle= \hat{b}^-_1 p_{(M+k+1)}+\bar{b}^+_1 \bar{p}_{-(M+k+1)}= 0.
\leq C\left({\displaystyle\sum\limits_{i=1}^{N-1}}\left\langle D_{y_{i}}^{m}u\right\rangle_{y_{i},\overline{Q}_{3/4}}^{(\gamma)}+\left\langle D_{y_{N}}^{m}v\right\rangle_{y_{N},\overline{Q}_{1/4}}^{(\gamma)}+|v|_{\overline{Q}_{1/4}}^{(0)}\right) .
\begin{aligned}K_{0,2}(x,z) = \sum_{\alpha \in \mathbf N^{d_2}}\sum_{\beta \in \mathbf N^{d_1}}b_{\alpha ,\beta}h_{d_2,\alpha}(x)h_{d_1,\beta}(z), begin{align*}1ex]K_{0,1}(z,y) = \sum_{\alpha ,\beta \in \mathbf N^{d_1}}c_{\alpha ,\beta}h_{d_1,\alpha}(z)h_{d_1,\beta}(y),\end{aligned}
g_1(x_1, y_1, z_1) =x^p_1+(tz_1+y_1)z^l_1\sum_{i=0}^{k}{\dbinom{k+1}{i+1}(tz_1+y_1)^i(-y_1)^{k-i}}&=x^p_1+z^l_1\sum_{i=0}^{k}{\dbinom{k+1}{i+1}(tz_1+y_1)^{i+1}(-y_1)^{k-i}}&=x^p_1+z^l_1\sum_{i=1}^{k+1}{\dbinom{k+1}{i}(tz_1+y_1)^{i}(-y_1)^{k+1-i}}&=x^p_1+t^{k+1}z^{k+1+l}_1+(-1)^{k}y^{k+1}_1z^l_1.
\binom{k}{n}_f = \sum_{s\ge 0}f(s)\binom{k-1}{n-s}_f, \binom{k}{n}_f = \frac{k}{n}\sum_{s\ge 1}sf(s)\binom{k-1}{n-s}_f, \binom{k}{n}_f = \sum_{i\ge 0}f(r)^i\binom{k}{i}\binom{k-i}{n-ri}_{\tilde{f}}.
\psi_f(M)=\displaystyle{\bigoplus_{-1<\alpha \leq 0}}Gr_V^{\alpha}(\tilde{M}), \phi_f(M)=\displaystyle{\bigoplus_{-1<\alpha < 0}}Gr_V^{\alpha}(\tilde{M}) \oplus Gr_V^{-1}(\tilde{M})
\begin{gathered}\left\{(0,n,n)\mid n\in\mathbb{N}\right\}\cup\left\{(n,0,n)\mid n\in\mathbb{N}\right\}, \left\{(1,n,1)\mid n\in\mathbb{N}\right\}\cup\left\{(n,1,1)\mid n\in\mathbb{N}\right\}, \left\{(2,2,0) \right\}. \end{gathered}
\int_{\mathbb{R}^n}\frac{1}{|x|^4}|P^\perp f(x)|^2 \, \mathrm{d}x & = \frac{1}{\sigma(\mathbb{S}^{n-1})^2}\int_0^{\infty}\int_{\mathbb{S}^{n-1}}\bigg| \int_{\mathbb{S}^{n-1}}(f(r\omega) - f(r\omega')) \, \mathrm{d}\sigma(\omega') \bigg|^2 \, \mathrm{d}\sigma(\omega) \, r^{n-5}\mathrm{d}r.
\left(\frac{\partial h_{\tau}}{\partial \tau}(x_{0})\right)_{|\tau=0}=\frac{\partial}{\partial \tau}\varphi^{\tau}_{\ell(\gamma_{0})}(x_{0})_{|\tau=0}+\partial_{\tau}\ell(\gamma_{\tau})_{|\tau=0}X_{0}(x_{0})+d\varphi^{0}_{\ell(\gamma_{0})}(x_{0})\cdot \left(\frac{\partial h_{\tau}}{\partial \tau}(x_{0})\right)_{|\tau=0}.
\nabla f^* = (\nabla \mathbf{y})\cdot (\nabla f)^{-1}+\mathbf{y}\cdot\nabla(\nabla f)^{-1}-(\nabla f\circ (\nabla f)^{-1})\cdot\nabla(\nabla f)^{-1}= (\nabla f)^{-1}
d[\begin{matrix}P\\T\\t\\u\end{matrix}]=[\begin{matrix}0&ds&0&0\\ 0&0&\kappa_{g}ds&\kappa_{n}ds\\ 0&-\kappa_{g}ds&0&\tau_{r}ds\\ 0&-\kappa_{n}ds&-\tau_{r}ds&0\end{matrix}][\begin{matrix}P\\T\\t\\u\end{matrix}].
\lambda(s,t) =\begin{cases}\dfrac{1}{4\pi t}\left( e^{-4\pi \sin^{-1}(s) t}-e^{-4\pi (\pi-\sin^{-1}(s)) t}\right), \; t\neq 0 \pi - 2\sin^{-1}s, t=0. \end{cases}
f\big( k^2 + (k-1)^2 + 2^2(k-3) + 3^2 \big) &= f(k^2) + f\big((k-1)^2\big) + f(4)(k-3) + f(9) = 2(k^2 + k - 1) &= f\big(2(k^2+k-1)\big) = f(2)\,f(k^2+k-1) = f(2)\,(k^2 + k-1),
\int_{0}^{\infty}x^{k}\, f_{Z_{n}}(x) dx = n \sum_{j=0}^{n-1}\binom{n-1}{j}(-1)^{j}\int_{0}^{\infty}x^{k}e^{-(1+j)x}dx & = n \sum_{j=0}^{n-1}\binom{n-1}{j}(-1)^{j}\frac{1}{(1+j)^{k+1}}\int_{0}^{\infty}t^{k}e^{-t}dt = k! \sum_{j=1}^{n}\binom{n}{j}(-1)^{j-1}\frac{1}{j^{k}},
-R_0^3\left[\left(1-\frac{N_f}{N_c}\right)\left(\ln R_0^3-1\right)+\frac{N_f}{N_c}\ln\rho_0^2\right]-\frac{N_f}{N_c}\rho_0^2=\cos\left(\pi\frac{N_f}{N_c}\right).
\left( \gamma_{n-1}U_{n-1}U_{n}-xU_{n}+1\right) y=\gamma_{n-1}A_{n-1}A_{n}y^{\prime\prime}+\gamma_{n-1}\left[ A_{n-1}\left( \partial_{x}A_{n}-B_{n}\right)-A_{n}B_{n-1}\right] y^{\prime}-xA_{n}y^{\prime}+\gamma_{n-1}\left( B_{n}B_{n-1}-A_{n-1}\partial_{x}B_{n}\right)y+xB_{n}y+y.
U(\beta,t) = E_{\nu, 0, -1}\left(- H \beta^2 \right)=1+\sum_{k=1}^\infty \left(-H\beta^2\right)^k \prod_{j=0}^{k-1}\frac{\Gamma \left(1-\nu \right)}{\Gamma \left(1\right)}= \sum_{k=0}^\infty \left(-H\beta^2\Gamma(1-\nu)\right)^k=\frac{1}{1+H\beta^2\Gamma(1-\nu)},
S = \int d^{3}x \left( \psi^{\dagger}\gamma \cdot ( \partial - i A ) \psi + \frac{G}{2}( \psi^{\dagger}\psi )^{2}+ \frac{i}{4 \pi \alpha}\epsilon^{\mu \nu \lambda}A_{\mu}\partial_{\nu}A_{\lambda}\right) .
\sqrt{\mathcal{L}}_{G}^z\in \Psi^{\textnormal{Re}(z)}(G), \textnormal{and}\sqrt{\mathcal{L}}_{G}^z=\left(\sqrt{\mathcal{L}}_{G}+\textnormal{Proj}_{\textnormal{Ker}(\sqrt{\mathcal{L}}_{G})}\right)^z- \textnormal{Proj}_{\textnormal{Ker}(\sqrt{\mathcal{L}}_{G})}.
\Pr_{p_{X^n}p_{E^{m+n}}}\left\{\bigcup_{i=1}^n \left\{\sum_{\ell=1}^i X_\ell \ge \sum_{\ell=1}^{m+i}E_\ell\right\}\right\}\le \left(\frac{e^{0.4}}{\ln n}\right) e^{2\ln n -\frac{mP}{2}\sqrt{\frac{\ln n}{a n}}}.
\sum_{i\ge 4}C^i_n\| (\partial^{-1}_y g_{i}) \partial^{n + 1 -i}_x w\|_{L^2_{\ell'}(\mathbb{R}^2_+)}\le \sum_{i\ge 4}C^i_n\| \langle y \rangle^{\ell' - \ell - 1}(\partial^{-1}_y g_{i}) \|_{L^2(\mathbb{R}^2_+)}\| \langle y \rangle^{\ell + 1}\partial^{n + 1 -i}_x w\|_{L^\infty}& \le C \sum_{i = 1}^{n}\|g_{i}\|_{L^2_{\ell'}(\mathbb{R}^2_+)}.
\left[ \left( 1-r_{i}\left( k\right) \right) \right] =\frac{1}{N}\sum_{k=1}^{N}\left( 1-r_{i}\left( k\right) \right)^{2}-\left( \frac{1}{N}\sum_{k=1}^{N}\left( 1-r_{i}\left( k\right)\right) \right)^{2}& =\frac{1}{N}\sum_{k=1}^{N}\left( x_{k}\right)_{i}^{4}\left(\frac{\lambda_{k}^{2}(A)}{d_{i}}+1\right)^{2}-\frac{4}{N^{2}}
F^{a,b}_1(x):= \frac{2\left( \tanh^{-1}\left(\sqrt{1 - \frac{2}{b}+ \frac{2}{b}\sqrt{1-\frac{4x}{b}}}\right) - \tanh^{-1}\left(\sqrt{1 - \frac{2}{a}+\frac{2}{a}\sqrt{1-\frac{4 x}{b}}}\right)\right)}{\sqrt{1-\frac{4x}{b}}}
[\psi^{(\ell)}(e),x] = \left[ \sum_{j=1}^{s(\ell)}g_j^{(\ell)^{\frac{1}{2}}}\varphi_j^{(\ell)}(e) g_j^{(\ell)^{\frac{1}{2}}}, x \right] &= \sum_{j=1}^{s(\ell)}g_j^{(\ell)^{\frac{1}{2}}}[ \varphi_j^{(\ell)}(e), x] g_j^{(\ell)^{\frac{1}{2}}}.
\int_0^1 (1 - x^2) \, e^{-\frac{1}{2\sigma^2}\left(x^2-1\right)^2}\, dx \leq \int_0^1 (1 - x^2) \, dx = \left[x - \frac{1}{3}x^3 \right]_0^1 = \frac{2}{3}
\Big|f(t)-\sum_{n=1}^N \phi_n t^{-\mu_n}\Big|_{\alpha,\sigma}\le \Big|f(t)-\sum_{n=1}^{N_*}\phi_n t^{-\mu_n}\Big|_{\alpha,\sigma}+ \Big|\sum_{n=N+1}^{N_*}\phi_n t^{-\mu_n}\Big| &=\mathcal O(t^{-\mu_{N_*}-\delta})+\mathcal O(t^{-\mu_{N+1}}) t\to\infty,
\left\{\begin{array}{c}{\mathbf H}^{\rm aff-aff}{\mathbf H}^{\rm met-aff}{\mathbf H}^{\rm aff-met}\end{array}\right\}=\left\{\begin{array}{c}{\mathbf T}^{\rm aff-aff}{\mathbf T}^{\rm met-aff}{\mathbf T}^{\rm aff-met}\end{array}\right\}+{\mathbf V}\left(q^{1}, \ldots , q^{n}\right).
E_\alpha(t)= \sum_{|\alpha|\leq N-1}(\sum^{3}_{i,j=1}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\big\{\partial^{\alpha}[\frac{1}{\rho}\partial_{x_j}(R\theta B_{ij}(\frac{v-u}{\sqrt{R\theta}})\frac{\varepsilon \sqrt{\mu}}{M}f)]\partial^{\alpha}\widetilde{u}_i\big\}\,dv\,dx-\widetilde{E}(t)).
\left( \begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{- \sqrt{2}\eta}&{\sqrt{2}\eta}\\{0}&{\sqrt{2}\eta}&{1 - \eta^{2}}&{\eta^{2}}\\{0}&{\sqrt{2}\eta}&{- \eta^{2}}&{1 + \eta^{2}}\\ \end{matrix}\right) .
g_i(x^*) \leq 0,~\forall^{m}_{i=1}\lambda^*_i \geq 0,~\forall^{m}_{i=1}\lambda^*_i g_i(x^*) =0,~\forall^{m}_{i=1}\nabla f(x^*)+\lambda^{*T}\nabla g(x^*) =0.
H^{(n,m)}_{\texttt{b};j,k}:=\partial_{\xi_j}\partial_{\xi_k}V^{(n,m)}_{\texttt{b};\lambda}(\boldsymbol{\xi}) &=\begin{cases}2(m+1) + u_{q_0}(\xi_j) + u_{q_1}(\xi_j) + \sum_{\substack{1\leq l\leq n l\neq j}}\bigl (u_q(\xi_j+\xi_l) +u_q(\xi_j-\xi_l) \bigr) u_q(\xi_j+\xi_k) -u_q(\xi_j-\xi_k) \end{cases}
a^{4}+b^{4}=\frac{(a^{2})^{2}}{1}+\frac{(b^{2})^{2}}{1}\ge\frac{(a^{2}+b^{2})^{2}}{2}\ge\frac{(\frac{(a+b)^{2}}{2})^{2}}{2}=\frac{(a+b)^{4}}{8}.
1_{A^c}\hat{v}(r, X^{s,x; \alpha,v}_r)= 1_{A^c}(\psi+\gamma)(r, X^{s,x; \alpha,v}_r) \leq \leq \mathbb{E}[1_{A^c}\hat{v}(t, X^{s,x; \alpha[r,\hat{\xi}],v}_t)|\mathcal{F}_r]+2\|\hat{v}\|_{\infty}\mathbb{P}[A^c\cap B^c|\mathcal{F}_r], a.s.
\beta_{0 0}= \sum_{\rho = 0}^{N}\frac{\left( \eta_{0 \rho}\right)^{2}}{4}\left[ \left( \frac{\tilde{\omega}}{\Omega_{\rho}}- \frac{\Omega_{\rho}}{\omega}\right) \mathrm{e}^{i \Omega_{\rho}t}+ \left( \frac{\Omega_{\rho}}{\tilde{\omega}}- \frac{\tilde{\omega}}{\Omega_{\rho}}\right) \mathrm{e}^{- i \Omega_{\rho}t}\right] .
\chi_{isos.}=\frac{<p_{g.s.}|M|p_{g.s.}>}{<p_{g.s.}|p_{g.s.}>}=\frac{<g.s.|M|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|M|p_{g.s.}^{1}>+\epsilon^{4}<p_{g.s.}^{2}|M|p_{g.s.}^{2}>+\ldots}{<g.s.|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|p_{g.s.}^{1}>+\epsilon^{4}<p_{g.s.}^{2}|p_{g.s.}^{2}>+\ldots}\quad .
f_{m}( k ) \ = \ \frac{2 ( \partial_{r}J_{m}( k r ) |_{r = R}H_{m - \beta}^{( 1 )}( k R ) - \partial_{r}H_{m - \beta}^{( 1 )}( k r ) |_{r = R}J_{m}( k R ) )}{H_{m - \beta}^{( 2 )}( k R ) \partial_{r}H_{m - \beta}^{( 1 )}( k r ) |_{r = R}- H_{m - \beta}^{( 1 )}( k R ) \partial_{r}H_{m - \beta}^{( 2 )}( k r ) |_{r = R}}\ ,
\prod_{q\leq H}\left(1-\frac{\chi_{-3}(q)}{q}\right)^{-1}= L(1, \chi_{-3}) + O\left(\exp\left(-\frac{c}{2}\sqrt{\log H}\right)\right) &= \frac{\pi}{3\sqrt{3}}+O\left(\exp\left(-\frac{c}{2}\sqrt{\log H}\right)\right).
L_{k j_1 \cdots j_s l_1\cdots l_{s-1}}D_{l_1 \cdots l_{s-1}}= \frac{2s-1}{s-1}\delta_{\hat{kj_1}}D_{\hat{j}_2 \cdots \hat{j}_s}- \delta_{\hat{j}_1\hat{j}_2}D_{\hat{j}_3 \cdots \hat{j}_sk}
\begin{array}{c}{u_{k}= u_{0}+ k ( u_{0}- u_{- 1})}\\{U_{k}( z ) = U_{0}( z ) + k ( u_{0}- u_{- 1})}\\{\bar{W}_{k}= U_{0}( \sigma^{+}) + U_{0}( \sigma^{-}) + ( k + 1 ) ( u_{0}- u_{- 1}) - u_{0}}\\ \end{array}
k_2(X) =\sigma^2 k_4(X) =\sigma^4\left(\frac{\Gamma(1/\alpha)\Gamma(5/\alpha)}{\Gamma(3/\alpha)^2}-3\right) k_6(X) =\sigma^6\left(\frac{\Gamma(1/\alpha)^2\Gamma(7/\alpha)}{\Gamma(3/\alpha)^3}-15\frac{\Gamma(1/\alpha)\Gamma(5/\alpha)}{\Gamma(3/\alpha)^2}+30\right).
\int_{\Delta_{\theta ,t}^{m}}\varkappa (s)D^{\alpha}f(B_{s}^{H})ds=\int_{(\mathbb{R}^{d})^m}D^{\alpha}f(z)L_{\varkappa}(t,z)dz=(-1)^{\left\vert \alpha\right\vert}\int_{(\mathbb{R}^{d})^m}f(z)D^{\alpha}L_{\varkappa}(t,z)dz,
{\cal Z}_{\nu}^{( N_{f})}( \mu ) = \int_{0}^{2 \pi}\prod_{j = 1}^{N_{f}}\left( \frac{d \phi_{j}}{2 \pi}\exp \left[ \mu \cos ( \phi_{j}) - i \nu \phi_{j}- i j \phi_{j}\right] \right) \varepsilon_{i_{1}\ldots i_{N_{f}}}e^{i \phi_{i_{1}}}\ldots e^{i N_{f}\phi_{i_{N_{f}}}}\; ,
\frac{d z}{d\lambda}=\mathcal{B}\frac{\omega(\zeta,-1)^2}{\prod_{j=1}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})}\prod_{j=0}^{M}{\prod_{k=1}^{n_j}\left[\omega(\zeta,a_{k}^{(j)})\right]^{\beta_k^{(j)}}},
2 \frac{\ddot{b}}{b}+ \frac{{\dot{b}}^{2}- 1}{b^{2}}+ \frac{f'''(R)}{f'(R)}\dot{R}^{2}+ \frac{f''(R)}{f'(R)}\ddot{R}+ 2 \frac{f''(R)}{f'(R)}\frac{\dot{b}}{b}\dot{R}- \frac{1}{2}\frac{f(R)}{f'(R)}+ \frac{1}{2}R = 0 .
U_v(N,\alpha,T) = \inf\{\nu_v(T(x)-f(x)):f\in S_v(\alpha,N)\}= \inf\{\nu_v(T(x) - (T(x) - T(\alpha) +f(x))):f\in S_v(\alpha,N)\}= \inf\{\nu_v(T(\alpha) - f(x)):f\in S_v(\alpha,N)\}= \inf\{\nu_v(T(\alpha)(1 - f(x))):f\in S_v(\alpha,N)\}.
H \coloneqq \sum_{i, j\ge 0}(-1)^{i+j}\sum_{\substack{P\subseteq \{2,3,\dots,n-1\}, \#P=i}}\sum_{\substack{Q\subseteq \{1,n\}, \#Q=j}}\binom{d-\sum_{p\in P}(\alpha_{p}-1)-\sum_{q\in Q}\alpha_{q}}{n-1}
p_{3Du}\left( xy|\mu_{x}\sigma_{x}\sigma_{y}\right) \overset{}{=}\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[ -\frac{1}{2\sigma_{x}^{2}}\left( x-\mu_{x}\right)^{2}-\frac{1}{2\sigma_{y}^{2}}y^{2}\right] ,
A_2 = A_1\left(1 + \log^{1 - B - \alpha}x_0 + \left(\frac{2}{\log 2}+ 5.43 + 7.87\cdot 10^{12}\right)\frac{(u(x_0))^C\log^{1 - B}x_0}{A_1x_0}\right) &\le 0.028,
|H|_{D^2}\leq C(\sigma)(|H_t|_2+1),\ |\rho_t|_2\leq C(\sigma)(1+|u|_\infty)\leq C(\sigma)(1+\|\nabla u\|_1), |u|_{D^2}\leq C(\sigma)(|\sqrt{\rho}u_t|_2+|\nabla \theta|_2+1), |\theta|_{D^2}\leq C(\sigma)(|\sqrt{\rho}\theta_t|_2+|\nabla \theta|_2+|\nabla u|_6|\nabla u|_3+|\nabla H|_6|\nabla H|_3+1).
G(t-t^{\prime}, \tau) = \frac{1}{a^D}\int^{\infty}_{1/\Lambda^2}ds \frac{e^{-\frac{t-t^{\prime}}{4s}}}{(4\pi s)^{\frac{1}{2}}}\int_0^{\infty}\phi_r (\tau) |C(r)|^2e^{-\frac{s}{a^2}(r^2 + \rho_D^2)}dr\,,
\begin{cases}u_t(t,x)+au_{xxxx}(t,x)=h(t,x), (t,x)\in (0,T) \times (0,1), u_{xx}(t,0)=u_{xx}(t,1)=0, t\in (0,T), u_{xxx}(t,0)=u_{xxx}(t,1)=0, t\in (0,T), u(0,x)=u_0(x), x\in(0,1),\end{cases}
A_\cdot^{W,W}(l) = \lim_{n\to\infty}A_\cdot^{W,W}(l_n) &= \lim_{n\to\infty}\left( \phi_n(\cdot, W_\cdot) - \phi_n(0, W_0) - \int_0^\cdot \nabla \phi_n^* (r, W_r) \mathrm d W_r\right) = \phi(\cdot, W_t) - \phi(0, W_0) - \int_0^\cdot \nabla \phi^* (r, W_r) \mathrm d W_r
g_{k}(x,y_{k})=\exp\left[\sum_{i=1}^{n}\left\{\sum_{j=1}^{p}y_{k}^{(i)}z_{k}^{(i,j)}x^{(j)}-\psi\left(\sum_{j=1}^{p}z_{k}^{(i,j)}x^{(j)}\right)+\log\phi(y_{k}^{(i)})\right\}\right],
A_1(s, t) = \int\limits_0^{\infty}\! db \, b \, J_0 (b\sqrt{-t}) \int\limits_0^{\infty}\! dq_{\bot}q_{\bot}J_0(q_{\bot}b) A_B(s,q_{\bot}) &= \int\limits_0^{\infty}\! dq_{\bot}q_{\bot}A_B(s, q_{\bot})\int\limits_0^{\infty}\! db \, b \, J_0(q_{\bot}b) J_0(b\sqrt{-t}) = A_B(s,\sqrt{-t}) \;.
\begin{aligned}\hat{\mathbf{R}}_1 = \{(t,s) \in [0,T]^2: \beta^{1/3}N^{1/3}(t-s) > C_0\}\hat{\mathbf{R}}_2 = \{(t,s) \in [0,T]^2: \beta^{1/3}N^{1/3}(t-s) \leq C_0\}. \end{aligned}
\frac{(-a-p)_{p+k}}{(1)_{p+k}}= \frac{\Gamma_p(-a+k)\Gamma_p(1)}{\Gamma_p(-a-p)\Gamma_p(p+k+1)}\cdot\frac{-p}{p}\equiv -\frac{\Gamma_p(-a+k)\Gamma_p(1)}{\Gamma_p(-a)\Gamma_p(k+1)}\bigg(1+p\cdot\frac{\Gamma_p'(-a)}{\Gamma_p(-a)}-p\cdot\frac{\Gamma_p'(k+1)}{\Gamma_p(k+1)}\bigg) \equiv -\frac{(-a)_k}{(1)_k}-p\cdot\frac{(-a)_k}{(1)_k}\cdot(H_{p-a-1}-H_k)\pmod{p^2}.
A_1(a) = \{r \in [R-aD, R+aD] \mid \theta_a^X( r),\theta^H( a^{-1}(r-R)) \in [\omega,\beta]\}, A_2(a) = [R-aD,R+aD]\backslash ( A_1(a)\cup \{r \in [R-aD,R+aD] \mid \theta_a^X(r),\theta^H( a^{-1}(r-R)) \notin [\omega,\beta]\}).
\prod_{i=m+k+1}^n a_i \leq \frac{\prod_{i=m+1}^n x_i}{\prod_{i=m+1}^{m+k}a_i}= \left( \frac{\prod_{i=m+1}^{m+k}x_i}{\prod_{i=m+1}^{m+k}a_i}\right) \prod_{i=m+k+1}^n x_i < \prod_{i=m+k+1}^n x_i
n_{s^*}- k_{s^*}= n_{s^*}- \lfloor m_{s^*}\rfloor r_{s^*}\geq n_{s^*}- m_{s^*}r_{s^*}= m_{s^*}( \delta_{s^*}- 1 ) \geq \lfloor m_{s^*}\rfloor( \delta_{s^*}- 1 ) = \frac{k_{s^*}}{r_{s^*}}( \delta_{s^*}- 1 )
\sigma_{1}^{j - 1}\lambda_{a_{1, 1}, k}^{j - 1}c^{1}_{1, b(a_{1, 1}, k)}+ \sum_{q_{i}\in Q}\sigma_{q_i}^{j - 1}\lambda_{a_{1, 1}, k}^{j - 1}c^{q_i}_{1, b(a_{1, 1}, k)}+ \sum_{i = 1}^{|V|}\sigma_{v_i}^{j - 1}\lambda_{a_{v_{i, 1}}, b_{\gamma_{i}}}^{j - 1}c^{v_i}_{1, b(a_{1, 1}, k)}= 0 \textbf{\space}\forall j \in [r], k \in [0, s - 1],
R_{\alpha,e_d}=1,~R_{\alpha, e_d+e_1}=1, ~R_{\alpha,e_d+2e_1}=0, R_{\alpha, e_d+me_1}-G(e_d+me_1) = -\frac{1}{m}\theta_{11}(R_{\alpha, e_d+(m-2)e_1}-G(e_d+(m-2)e_1)).
\left([\alpha,\beta]_{\Lambda}\right)_+ = \alpha_+ \cdot \beta_0 - (-1)^{l+1}\beta_0 \cdot \alpha_+ -{\alpha_+}\cdot \beta_{\mp}- (-1)^{l+1}\beta_{\mp}\cdot \alpha_+ +(-1)^{k+1}\alpha_0 \cdot \beta_+ - \beta_+ \cdot \alpha_0 &+ (-1)^{k+1}\alpha_{\mp}\cdot \beta_+ + \beta_+ \cdot \alpha_{\mp}.
dx=dx(z_*) =\frac{\partial x(z_*)}{\partial z_*}dz_*,D_x(\omega)=d\frac{\omega}{dx}=\frac{\partial\frac{\omega}{dx(z_*)}}{\partial z_*}dz_*, dy=dy(z_*) =\frac{\partial y(z_*)}{\partial z_*}dz_*,D_y(\omega)=d\frac{\omega}{dy}=\frac{\partial\frac{\omega}{dy(z_*)}}{\partial z_*}dz_*.
S = \int d^{4}x \sqrt{| g_{E}|}\left[ \hat{R}( \hat{g}_{E}) -{\textstyle \frac{1}{2}}\frac{\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\bar{\hat{\lambda}}}{\left( \Im \mathrm{m}\hat{\lambda}\right)^{2}}+{\textstyle \frac{1}{4}}\hat{F}^{I}\{}^{\star}\tilde{\hat{F}^{I}}\right] \, .
\langle \phi ( z , \bar{z}) \phi ( z^{\prime}, \bar{z}^{\prime}) \rangle ={\frac{1}{4 \pi}}\int_{0}^{\infty}{\frac{d s}{s}}e^{-{\frac{d^{2}}{4 s}}- m^{2}s}={\frac{1}{2 \pi}}K_{0}( d^{2}m^{2}) ~ ,
u^{q^2+q}v^q\beta - u^{q^2+2q+1}+ u^{q+1}v^{q^2}\beta^q-v^{q^2+q}\beta^{q+1}=\, v^{q^2+q+1}u^q-\beta^{q^2}vu^{2q}-v^{q^2+q}\beta^{q+1}+\beta^{q^2+q+1}u^q.
\Big{(}\mathcal{D}_t(u-U_h)^{n-\theta},v_h\Big{)}-\gamma(\nabla (\sigma-\Sigma_h)^{n-\theta},\nabla v_h)+(\nabla (u-U_h)^{n-\theta},\nabla v_h) &+(f^{n-\theta}(u)-(1-\theta)\mathfrak{F}(U_{h}^{n},u_{H}^{n})-\theta f(U_{h}^{n-1}),v_h)=0,
e(A,B)=\frac{1}{2}\sum_{x\in A\cup B}\deg_{G[A,B]}(x) &\geq \frac{1}{2}\left(\left(\frac{1}{2}-\frac{1}{2h}\right)n-h\right)(|A|+|B|) &\geq \frac{1}{2}\left(\left(\frac{1}{2}-\frac{1}{2h}\right)10h-h\right)(|A|+|B|) &> \frac{h}{2}(|A|+|B|) &> \frac{l-1}{2}(|A|+|B|).
\int_{[0,1)^3}g(x,y)g(y,z) ~ dx dy dz =\frac{1}{4}\cdot \frac{1}{2}((2p_d-1)^2+1)+\frac{3}{4}\cdot \frac{1}{6}((2p_d-1)^2+4(2p_d-1)+1) =p_d^2.
\mathfrak{M}[{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z):s ]= \int_{0}^{\infty}p^{s-1}{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z) dp = \frac{1}{B(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\left[\int_{0}^{\infty}p^{s-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}dp\right)\right]dt.
\begin{aligned}G(\xi-\eta,\eta)= t_0e^{-t_0|\xi|^2}\sum_{k=1}^\infty\frac{(-1)^{k}t^{k}_0}{(k+1)!}\sum_{m=0}^{k-\ell}\sum_{\ell=0}^{k}C^\ell_kC^{m}_{k-\ell}|\xi-\eta|^{2\ell}|\eta|^{2m}(-|\xi|^2)^{k-\ell-m}.\end{aligned}
(q',10)=(g_1',10)=(b_1',d_0q' g_2)=(b_2',d_0d_1q' g_1')=1, X(b_1'/d_0 q' g_2+\nu_1)\in\mathbb{Z},X(b_2'/d_0 d_1 q' g_1'+\nu_2)\in\mathbb{Z},
\varphi^{B}(t, g, \omega)=\psi_{t-\sum_{j=1}^{i-1}t_{j}}^{\omega_{i}}(\psi_{t_{i-1}}^{\omega_{i-1}}(\cdots(\psi_{t_{1}}^{\omega_{1}}(g))\cdots)),\; \; \;t\in \left( \sum_{j=0}^{i-1}t_{j},\sum_{j=0}^{i}t_{j}\right] .