harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|e^{i(t-a_{j+1})\Delta}w(a_{j+1})\|_{S(I_{j+1})}\leq 2C\epsilon+\frac13\sum_{k=1}^{j}\|w\|_{S(I_k)}+ C\sum_{k=1}^{j}(\eta^{p+1}\|w\|_{S(I_k)}^{p-2}+\eta\|w\|_{S(I_k)}^{2(p-1)}+\|w\|_{S(I_k)}^{2p-1}). \leq 2C\epsilon+4\sum_{k=1}^{j}\gamma_j.
H([\mu_m;f^x;e_m]) = \left[\mu_m;\gamma_{\mathcal N}\left(\Theta(H)(f);\lambda_{\vec a,x^{-1}(1)}^{-1},\dots,\lambda_{\vec a,x^{-1}(m)}^{-1}\right)^x;e_m\right] &= \left[\mu_m;\gamma_{\mathcal N}\left(\Theta(H)(f)^x;\lambda_{\vec a,1}^{-1},\dots,\lambda_{\vec a,m}^{-1}\right);e_m\right] &= [\mu_m;\Theta(H)(f)^x;e_m]\circ\lambda_{\vec a}^{-1}
I_\gamma(c) = \lim_{n\rightarrow \infty}E_\gamma(f_n) = \lim_{n\rightarrow \infty}E_\gamma(g_n) = E_\gamma(\phi) + \lim_{n\rightarrow \infty}E_\gamma(g_n-\phi) >\frac{\|\phi\|^2_{L^2}}{c}I_\gamma(c) + \frac{c-\|\phi\|^2_{L^2}}{c}I_\gamma(c) = I_\gamma(c)
D_2 \cos_k w = 2 \cos_k \left( \frac{w_1+w_2}2 \right) \cos_k \left( \frac{w_1-w_2}2 \right) - 2 \cos_k\left( \frac{w_{-1}+w_{-2}}2 \right)\cos_k\left( \frac{w_{-1}-w_{-2}}2 \right) &= 2 \cos_k \left( \frac{w_1+w_2}2 \right) \left[ \cos_k\left( \frac{w_1-w_2}2 \right) - (-1)^j \cos_k\left( \frac{w_{-1}-w_{-2}}2 \right) \right] .
\sum_{c>0}A^0(c)= \frac{(-1)^n \pi^2 \Gamma(4s-2n-1)^2}{4^{4s-2n-1}\Gamma(2s-2n) \Gamma(2s-n)^2 \Gamma(2s)}\frac{\zeta(4s-2n-1)}{\zeta(4s-2n)}(\Im (z_1)\Im (z_2))^{1+2n-4s}.
\begin{array}{llllllllllll}L_{3, 0}^{1}= \langle h_1, x_2, h_2\rangle, L_{3, 0}^{2}= \langle x_1, x_2, h_2 \rangle, L_{3, 0}^{3}= \langle x_1, h_1, h_2\rangle, L_{3, 0}^{4}= \langle x_1, h_1, x_2\rangle.\end{array}
\delta_t \rho^n + \sum\limits_{q=1}^{2}\mathcal{Q}_{n-1/2}^{(q)}(\mathbf{B}_q\rho) = R_1^{n-\frac{1}{2}}+ R_2^{n-\frac{1}{2}}, n\geq 1, \rho^0=0,
\frac{n^{2-q}}{2-q}S_{2-q}\left( \frac{k}{n},1-\frac{k}{n}\right) & =\frac{1}{2-q}\left( -k^{2-q}\ln_{2-q}\frac{k}{n}-\left( n-k\right)^{2-q}\ln_{2-q}\left( 1-\frac{k}{n}\right) \right) .
( ( \frac{\eta_{\mathrm{1c}}\cdot \omega}{\xi_{\mathrm{1c}}}\frac{\eta_{\mathrm{1c}}}{\xi_{\mathrm{1c}}}+\omega) \otimes ( \frac{\eta_{\mathrm{1c}}\cdot \omega}{\xi_{\mathrm{1c}}}\frac{\eta_{\mathrm{1c}}}{\xi_{\mathrm{1c}}}+\omega) ) \cdot v_{11}= 0.
\int_{\{u\leq\phi-k\}}-\chi(-k)\theta_{u_{k}}^n =\int_X-\chi(-k)\theta_{u_{k}}^n+\int_{\{u>\phi-k\}}\chi(-k)\theta_{u_{k}}^n &=\int_X-\chi(-k)\theta_{\phi}^n+\int_{\{u>\phi-k\}}\chi(-k)\theta_{u}^n.
0 ~<~ y \sqrt{b}- x \sqrt{x}~ = \sqrt{x}(\sqrt{x}\sqrt{b}- x) + \sqrt{b}(y-x) begin{align*}.5em] \leq~ \sqrt{b}(\sqrt{x}\sqrt{b}- x) + \sqrt{b}(y-x) ~\leq~ 2\sqrt{b}(y -x),
\left\langle \nabla_{T}\nabla_{X}\nu,T\right\rangle =T\left\langle\nabla_{X}\nu,T\right\rangle -\left\langle \nabla_{X}\nu,\nabla_{T}T\right\rangle & =T\Pi\left( T,X\right) -\alpha\left\langle \nabla_{X}\nu,\nu\right\rangle & =0.
\begin{array}{c}\phi_{k+1}:= \phi_{k}, \psi_{k+1}:= \psi_{k}, \gamma_{k+1}:= \frac{1}{2}\gamma_{k}, begin{align*}5pt] y_{k+1}:= \Pi_{C}\big( y_{k}- \frac{1}{\sigma_{k}}g(x_{k+1}) \big), Z_{k+1}:= \Pi_{D}\big([Z_{k}- \frac{1}{\sigma_{k}}X(x_{k+1})]_{+}\big),\end{array}
\liminf_{n \rightarrow+\infty}\left|v_n\right|_p^p \leq C\left(\liminf_{n \rightarrow+\infty}\left|v_n\right|_2\right)^{\left(1-\beta_p\right) p}K^{\beta_p p}\beta_p=N\left(\frac{1}{2}-\frac{1}{p}\right)
f_{i,j+k}(L,L+M) + f_{j,i+k}(M,L+M) + f_{k,i+j}(N,L+M) - 2f_n(L+M) \leq \dim (U_1+U_2) + \dim (U_1+U_2) + k + \dim (U_1+U_2)) - 2(\dim (U_1+U_2)) = k + \dim (U_1+U_2) = f_{i,j,k}(L,M,N).
\partial_i (g(a_1-1)-g(0)) [\hat{\imath}]=g'(0)\partial_i a_1[\hat{\imath}]+g''(0)(a_1-1)\,\partial_i a_1[\hat{\imath}]+\frac{g'''(0)}{2}(a_1-1)^2\,\partial_i a_1[\hat{\imath}]+\mathtt{T}(i_{\delta}, \hat{\imath})
\frac{A^{5n}}{(1-K)^{2n}}\left(A^n\sum_{r = 1}^n \frac{p_{nr}(K)}{(1-K)^{\rho(n,r)}}\theta_K^r\right) + \sum_{i = 1}^{n-1}\sum_{j = 0}^i a_{ij}\frac{A^{5n + i}}{(1-K)^{2n+j}}\left(A^{n-i}\sum_{r = 1}^{n-i}\frac{p_{(n-i)r}(K)}{(1-K)^{\rho(n-i,r)}}\theta_K^r\right) &+\sum_{j=0}^n a_{nj}\frac{A^{6n}}{(1-K)^{2n+j}}.
\varphi^{\circ \circ}(x^0;u,u) :=\limsup\limits_{\mathop{x \to x^0}\limits_{t \downarrow 0}}\frac{\langle\nabla \varphi(x+tu), u\rangle-\langle\nabla \varphi(x), u\rangle}{t}, \langle \xi^{*}, u\rangle :=\max \left\{\langle \xi, u\rangle\,:\, \xi \in \partial_S^2\varphi(x^0)(u)\right\}.
\left( 1 - \frac{\mu+\tau}{\mu_{k_0+1}(\lambda) + \tau}\right) \sum_{k=1}^\infty \tilde{c}_k^2 \; \; \leq \; \; \frac{(\mu+\tau)[ \mu_{k_0+1}(\lambda) - \mu_1(\lambda) ]}{(\mu_{k_0+1}(\lambda) + \tau)(\mu_1(\lambda) + \tau)}\, \sum_{k=1}^{k_0}\tilde{c}_k^2 \; + \; \ell(\hat{u}).
n^{-(3-2h(s))}\sum_{j=-\infty}^{nt}\left| \sum_{k=nu+1}^{nt}v_{k-j}(s) \right|^2\leq\left[ \frac{2}{(1-h(s))^2}+\frac{1}{2h(s)-1}\right] |t-u|^{3-2h(s)}.
\lambda^{\mu}\left( t \right) = \bar{X}^{\mu}+ \xi^{\mu}t - \frac{1}{2}\Gamma_{\nu \rho}^{\mu}\xi^{\nu}\xi^{\rho}t^{2}- \frac{1}{3 !}\Gamma_{\nu \rho \kappa}^{\mu}\xi^{\nu}\xi^{\rho}\xi^{\kappa}t^{3}+ \ldots ,
0= \langle \mu_{B^!}(e_i^*), x_0 y_m^*\rangle = \langle k_{i, 1}e_1^*+\cdots+k_{i, n}e_n^*+k_{i, n+1}y_1^*+k_{i, n+2}y_2^*, x_0 y_m^*\rangle = \langle k_{i, n+1}y_1^*+k_{i, n+2}y_2^*, x_0 y_m^*\rangle = k_{i, n+1}\langle y_1^*, x_0 y_m^*\rangle+k_{i, n+2}\langle y_2^*, x_0 y_m^*\rangle
\Omega'=\begin{pmatrix}0 \frac{1}{\prod_i(x-t_i)}\frac{\tilde{c}(x)}{(x-q)\prod_i(x-t_i)}\hskip0.5cm\frac{-x^3+\tilde{d}(x)}{\prod_i(x-t_i)}-\frac{1}{x-q}\end{pmatrix}dx
\frac{E_{+}-E_{-}}{2\hbar}t\simeq\frac{({m_{+}}^{2}-{m_{-}}^{2})c^{3}}{2p\hbar}t\simeq\frac{\Delta m^{2}c^{3}}{4E\hbar}x=\frac{2\pi}{\lambda_{osc}}x
t T = \frac{1}{p ! q !}T_{\nu_{1}. . . \nu_{q}\mu_{1}. . . \mu_{p}}d x^{\nu_{1}}\wedge . . . \wedge d x^{\nu_{q}}\otimes d x^{\mu_{1}}\wedge . . . \wedge d x^{\mu_{p}}\in X^{q , p}
d_U(\alpha(s_{k+1},x_{n_{k+1}}), \alpha(s_{k},x_{n_{k}}))=d_U(\alpha(s_{k+1}-s_k,x_{n_{k+1}}), x_{n_{k}})=d_U(\alpha(t_k,x_{n_{k+1}}), x_{n_k}< 2^{-k}\,.
(Ly_S,y_S)_V =\frac{1}{2}\sum\limits_{e\in E}\frac{\delta_E(e)}{|e|^2}\sum\limits_{u,v\in e}(y_S(u)-y_S(v))^2 \le\frac{1}{2}\sum\limits_{e\in \partial S}\frac{\delta_E(e)}{|e|^2}2|S\cap e||e-S|4 \le |\partial S|\delta_E(\max)=b(G)\delta_E(\max).
\mathrm{d}^{\parallel}_{\pm}=\sum_{n\neq0}p_n^{\pm}\ :c_{-n}\alpha_n^{\mp}:\;, \mathrm{d}^{\parallel}=\sum_{n,m\neq0\atop m+n\neq 0}:c_{-n}\left(\alpha^+_{-m}\alpha^-_{m+n}+\frac{1}{2}(m-n)c_{-m}b_{m+n}\right):\;, \mathrm{d}^{\perp}=\frac{1}{2}\sum_{i=2}^{q-1}\sum_{n\neq0}\ :c_{-n}a_m^ia_{n-m}^i:\;.
\sum_{j=1}^h \Big(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{P}^{Cont}_j}+ \frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{P}^{Cont}_j}\Big) & - \frac{1+\sum_{i=0}^h k_1(i)}{r+k_1(0)}a^{\widehat{P}^{Cont, +r}}- \frac{1+\sum_{i=0}^h k_2(i)}{s+k_2(0)}b^{\widehat{P}^{Cont, +s}}= - 1.
\{L(q)-tL(q^t) \mid t=2,4,5,8,10,20,40 \}, &\{E_{\chi_0,\chi_1}(q^t), E_{\chi_1,\chi_0}(q^t) \mid t=1,2,4,8\}, &\{E_{\chi_0,\chi_2}(q^t), E_{\chi_2,\chi_0}(q^t) \mid t=1,5\}, &\{E_{\chi_0,\chi_3}(q), E_{\chi_1,\chi_2}(q), E_{\chi_2,\chi_1}(q), E_{\chi_3,\chi_0}(q)\}
C_0\frac{f(\omega_0 h,\xi)}{g(\omega_0 h)}= C_0 e^{\frac{\kappa_n h}{2}(\xi+1)}+ C_0\frac{(-1)^{p+1}(\omega_0 h)^{p+1}}{2g(\omega_0 h)}\left[ R^-_{p+1}(\xi) + \sum_{k=1}^\infty \left(\frac{\omega_0 h}{2}\right)^k R^{-,(-k)}_{p+1}(\xi) \right]+ \mathcal{O}((\kappa_n h)^{2p+2}).
u^n_t(y) =\, P^n_{t}u^n_0(y) + \sum_e \Big(Y^e_t(\phi)+ Z^e_t(\phi)+E^{(1,e)}_t(\phi) + E^{(2,e)}_t(\phi)\Big) & + \sum_e \Big( E^{(3,e)}_t(\phi)+ E^{(4,e)}_t(\phi)\Big) +o(1).
\Delta_f (ue^h) =e^h\left[\Delta u+2\langle\nabla u,\nabla h\rangle+u(\Delta h+|\nabla h|^2)\right] &-e^h(\langle\nabla f,\nabla u\rangle+u\langle\nabla f,\nabla h\rangle) &=e^h[\Delta u-\langle\nabla(f-2 h),\nabla u\rangle+u(\Delta h+\langle\nabla h,\nabla (h-f)\rangle]
\nabla_{\rho}^{m_d}\left [\varphi(\rho) h_{m_d}(\rho) \right] = \sum_{n=1}^{m_d}\sum_{j=0}^{n}c_{n,j}^{(d)}\rho^{n-m_d}\varphi^{(n-j)}(\rho) h_{m_d}^{(j)}(\rho).
[\begin{matrix}y_{t}\\y_{t-1}\end{matrix}]=[\begin{matrix}c\\ 0\end{matrix}]+[\begin{matrix}A_{1}&A_{2}\\I&0\end{matrix}][\begin{matrix}y_{t-1}\\y_{t-2}\end{matrix}]+[\begin{matrix}e_{t}\\ 0\end{matrix}],
\left( \begin{array}{l}{\theta^{1}}{\theta^{2}}\end{array}\right) \rightarrow \left( \begin{array}{l l}{\operatorname{c o s}\frac{\alpha}{2}}{- \operatorname{s i n}\frac{\alpha}{2}}{\operatorname{s i n}\frac{\alpha}{2}}{\operatorname{c o s}\frac{\alpha}{2}}\end{array}\right) \left( \begin{array}{l}{\theta^{1}}{\theta^{2}}\end{array}\right) .
\|\omega\|_X^2-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx -(2_\alpha^*-1)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx=0,
V_{12}=-\varepsilon\bar{m}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})\sum_{i=1}^{d}\Theta_{p_{i}q}(t,x,\mu^{2},Dw^{2})\left(\frac{\partial(\mu^{1}-\mu^{2})}{\partial x_{i}}\right)\dx,
A_1A = [1:-1:0], A_1B_2 = [1:-1:y], A_1C_2 = [1:-1:-x], B_1B = [0:1:-1], B_1C_2 = [1:x:-x], B_1A_2 = [m:-1:1], C_1C = [1:0:1], C_1A_2 = [m:-1:m], C_1B_2 = [y:-1:y].
\begin{array}{rl}|[u^c]\left((1-u)^{b-1}L^b\right)| \leq \displaystyle\sum_{k=0}^b \binom{b}{k}a_L^b(c-k+1)^{b-1}q^{-b(c-k)}< \displaystyle\sum_{k=0}^b \binom{b}{k}a_L^b(c+1)^{b-1}q^{-b(c-b)}= a_L^b(c+1)^{b-1}q^{-b(c-b)}\left(\displaystyle\sum_{k=0}^b \binom{b}{k}\right) = 2^ba_L^bq^{b^2}(c+1)^{b-1}q^{-bc}.\end{array}
\left|\sum_{\substack{(\delta_1,\ldots,\delta_K,j_1,\ldots,j_K, n_1,\ldots,n_K)\in\mathcal{A}_{\{2,4,\ldots,K\}}, l_{H-1,1}\neq l_{H,1}, \Delta\in\mathcal{D}}}\frac{1}{2^K}\frac{1}{N^{K/2+1}}\prod_{k=1}^Ka'_{j_k}\right|=o(1)
D_{n}= \prod\limits_{k=0}^{n}b_{k}^{n +1 -k}\ , \ \D_{n+1}^{\, \prime}=\left(\sum\limits_{k=0}^{n}a_k\right) \prod\limits_{k=0}^{n}b_{k}^{n +1 -k}\ , \ \ \ \n \geq 0 \ .
V_{j-1}V_{j}=z_{j},V_{j}V_{j+1}=z_{j+1}, V_{j+1}W_{j+1}=z_{j+6}, W_{j+1}U_{j}=z_{j+7}, U_{j}W_{j-1}=z_{j+12},W_{j-1}V_{j-1}=z_{j+13},
(k+g)\frac{\partial}{\partial z_{i}}\Phi(v_{i},z_{i})=lim_{z\rightarrow z_{i}}[\sum_{s}X_{s}(z)\Phi(X_{s}v_{i},z_{i})-(z-z_{i})^{-1}\Phi(\sum_{s}X_{s}^{2}v_{i},z_{i})].
\mathcal{N}_{{\bf q},D_{V_\omega}(b)}\big( \vert \omega \vert r,\vert \omega \vert r_0 \big) \sim\frac{1}{d!}\Bigg( \frac{\vert \ln r \vert}{\ln \vert \ln r \vert}\Bigg)^d, r \searrow 0,
\big\Vert \big(1-C_{v^{(n+1)}_\Sigma}\big)^{-1}C_\Sigma^- \big(\tilde{\mu}^n \big(v_\Sigma^{(n)}- \tilde{v}_\Sigma^{(n)}\big) - \tilde{\mu}^{(n+1)}\big(v_\Sigma^{(n+1)}- \tilde{v}_\Sigma^{(n+1)}\big) \big)\big\Vert_{L^2}\qquad{}= O\left(\frac{1}{n^{3/2}\log^2 n}\right)
\Pi_{n}(\boldsymbol{\theta}|\mathbf{y})=\frac{\pi(\boldsymbol{\theta})f(\mathbf{y}| \boldsymbol{\theta})}{\int_{\boldsymbol{\Omega}}\pi(\boldsymbol{\theta})f(\mathbf{y}| \boldsymbol{\theta})d\boldsymbol{\theta}}=\frac{\pi(\boldsymbol{\theta})f(\mathbf{y}| \boldsymbol{\theta})}{m(\mathbf{y})}
\begin{array}{llr}x_1 x_3 = \mu_{14}x_3 x_1, x_3 x_4 = x_4 x_3, x_2 x_3 + x_3 x_2= - x_4^2, begin{align*}3mm]x_1 x_4 = -\mu_{14}x_4 x_1, x_4^2 = x_2^2, x_2 x_4 - x_4 x_2= - x_1^2, begin{align*}3mm]\end{array}
F(s+h)-F(s)= \int_X (\phi(s+h,x)-\phi(s,x)) \, \mu(dx) = \int_X \int_{s}^{s+h}\partial_r \phi(r,x) \, dr \, \mu(dx) &= \int_s^{s+h}\int_X \partial_r \phi(r,x) \, \mu(dx) \, dr
P_i(f)-P_0(U^if) = \mathbb E [f\mid T^i\mathcal M]-\mathbb E [U^if\mid \mathcal M] -\mathbb E [f\mid T^{i+1}\mathcal M]+\mathbb E [U^if\mid T\mathcal M] =(I-U^i)\mathbb E [f\mid T^i\mathcal M]-(I-U^i)\mathbb E [f\mid T^{i+1}\mathcal M]
W_i = - \frac{dz_i}{w_i}\Bigg( \frac{1-\frac{w_i\hbar}{2z_i}}{1+\frac{w_i\hbar}{2z_i}}\, e^{\frac{w_i\hbar}{z_i}}\Bigg)^{\frac 1\hbar}\exp\Big( \frac{(z_i-\frac{w_i\hbar}{2})^{N}-(z_i+\frac{w_i\hbar}{2})^{N}+Nw_i\hbar z_i^{N-1}}{\hbar N}\Big) ; D_x = d \; \frac{z^2}{((N-1)z^{N}-1)dz}.
\chi e^{-it_0h^{-1}P_h(z)}R_h(z)\chi= \chi e^{-it_0h^{-1}\tilde{P}_h(z)}\tilde{R}_h(z)\chi+\chi(R_h(z)-\tilde{R}_h(z))\chi -\frac{i}{h}\int_0^{t_0}\chi(e^{-ish^{-1}P_h(z)}-e^{-ish^{-1}\tilde{P}_h(z)})\chi ds.
\partial_t c_1 = \partial_x (\partial_x c_1+ q_1 c_1 \partial_x \psi) +f_1, \partial_t c_2 = \partial_x (\partial_x c_2+ q_2 c_2 \partial_x \psi) +f_2, - \partial_x^2 \psi = q_1c_1+ q_2c_2, \partial_x \psi(t,0) =0, \partial_x \psi(t, 1)=-e^{-t}/60,{\partial_x c_i}+q_ic_i{\partial_x \psi}=0, x=0, 1,
u(x) =\int_{\Omega}g(\omega_{\tau_D(\omega)})\,dP_x(\omega) &\quad+\int_\Omega\int_0^{\tau_D(\omega)}f(\omega_t,u(\omega_t))\,dt\,dP_x(\omega)+\int_\Omega A^\mu_{\tau_D(\omega)}(\omega)\,dP_x(\omega),
\left(2+\frac{4s}{N}\right)F(t) =\left(2+\frac{4s}{N}\right)\int_0^t \frac{f(\kappa)}{|\kappa|^{1+\frac{4s}{N}}}|\kappa|^{1+\frac{4s}{N}}\, d\kappa < \left(2+\frac{4s}{N}\right)\frac{f(t)}{|t|^{1+\frac{4s}{N}}}\int_0^t |\kappa|^{1+\frac{4s}{N}}\, d\kappa=f(t)t
V = \left( \begin{array}{l}{X ( u ) = 2 a ( u ) = \frac{2 \sqrt{2}}{\pi}\int_{- 1}^{1}\sqrt{\frac{u - x}{1 - x^{2}}}d x}\\{F_{A}( u ) = 2 a_{D}( u ) = 2 i \frac{\sqrt{2}}{\pi}\int_{1}^{u}\sqrt{\frac{u - x}{1 - x^{2}}}d x}\\ \end{array}\right) \ .
\omega(\Phi_t^1(x_1^0,x_2),0,t)= \omega\left(\Phi_t^1\left(\Phi_t^{-1}(\Phi_t^1(x_1^0,x_2),0),0\right),0,t\right) = \frac{\rho_0\left(\Phi_t^{1}(x^0_1,x_2)D(t),0\right)}{{\Phi_t^1(x_1^0,x_2)D(t)}}\int_0^tD(s)ds.
\begin{cases}L_N(\beta):=\max_{1\leq i\leq N}\sup_{[s]_i,[s']_i}(1+\beta)^i\frac{|\varphi_i([s]_i)-\varphi_i([s']_i)|}{|s_i-s_i'|}, begin{align*}5pt]F_N(\beta):=\max_{1\leq i\leq N}\sup_{[s]_i,[s']_i}(1+\beta)^i\Big|\varphi_i([s]_i)-\varphi_i([s']_i)\Big|.\end{cases}
{\cal G}_{\alpha_{1}, \ldots , \alpha_{n}}^{( A )}( x_{1}, \ldots , x_{n}) = \left({\frac{A}{\pi}}\right)^{( \sum \alpha_{i}- Q ) / b}{\frac{F_{\alpha_{1}, \ldots , \alpha_{n}}( x_{1}, \ldots , x_{n})}{\Gamma ( ( Q - \sum \alpha_{i}) / b )}}.
A' = \{x \in \textstyle\prod X_{n, k}\colon (\exists m)(\forall n >m)(\forall j)(\exists k >j)(x(n, k) \in A_{n,k})\}, B' =\{x \in \textstyle\prod X_{n, k}\colon (\exists m)(\forall n >m)(\forall j)(\exists k >j)(x(n , k) \in B_{n,k})\}.
J_{r}( Q_{j + 1}, Q_{j + 1}^{\prime}\mid Q_{j}, Q_{j}^{\prime}, \xi ) = ( \frac{i}{\cal N})^{1 / 2}\exp [ \frac{i}{\hbar}( - r_{j}R_{j + 1}- r_{j + 1}R_{j}+ \xi r_{j + 1}) ] .
\nabla_{\mu}\nabla^{\mu}\Phi = \frac{\kappa^{2}}{2}T_{3}e^{3 \Phi}V ( T ) \frac{3 + 2 e^{- 2 \Phi}\nabla_{\alpha}T \nabla^{\alpha}T}{\sqrt{1 + e^{- 2 \Phi}\nabla_{\alpha}T \nabla^{\alpha}T}},
R_{n,i}= \bigcup_{1\leq j_1,j_2\leq i}\left\{\begin{array}{c|c}(Q_{j_1}w_1Q_{j_2},w_2) (w_1,w_2)\in \Lambda_n\times \Lambda_n (w_1Q_{j_1},w_2Q_{j_2})(w_1Q_{j_1},Q_{j_2}w_2) &(w_1,Q_{j_1}w_2Q_{j_2}) \in R_n' \end{array}\right\}.
[B,K] = [B,C^2] - \sum_{i=1}^{L+1}\alpha_i[B,\{A^i,B\}] -\sum_{i=1}^L\beta\omega_i[B,\{A^i,B\}] - \beta[B,\{A,B^2\}]+\sum_{i=1}^{M+1}k_i[B,A^i].
S_{SYM}=\int d^{4}xTr[-\frac{1}{2g^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\vartheta}{16\pi^{2}}F_{\mu\nu}*F^{\mu\nu}-\frac{2i}{g^{2}}\lambda\sigma^{\mu}D_{\mu}\overline{\lambda}+\frac{1}{g^{2}}D^{2}]
\bar{\cal L}_{c}= - \frac{1 - \rho^{2}}4 \left( \bar{f}_{\mu \nu}^{i}\right)^{2}- \frac 1 4 \left( \bar{B}_{\mu \nu}^{i}\right)^{2}+ \frac i 2 \rho \bar{B}_{\mu \nu}^{i}{}^{\ast}\! \bar{f}^{\mu \nu i}- \frac 1{2 \beta}\left( \partial^{\mu}\bar{A}_{\mu}^{i}\right)^{2}.
\mathfrak p=\begin{array}{c}(\mathfrak p_1\leftrightarrow\cdots\leftrightarrow\mathfrak p_i) \updownarrow (\mathfrak q_1\leftrightarrow\cdots\leftrightarrow\mathfrak q_j)\end{array}\leftrightarrow\begin{array}{c}(\mathfrak p_{i+1}\leftrightarrow\cdots\leftrightarrow\mathfrak p_k) \updownarrow (\mathfrak q_{j+1}\leftrightarrow\cdots\leftrightarrow\mathfrak q_\ell)\end{array}
K_T(\zeta_1,\zeta_2):=\left\langle{\rm T}\exp \zeta_1\mid \exp \zeta_2\right\rangle =\left\langle \exp \zeta_1\mid{\rm T}^{\prime}\exp \zeta_2\right\rangle =\left( \exp \zeta_1\mid{\rm T}^{\prime}\exp \zeta_2\right) .
r(\alpha^{i+2}\times\alpha^{j})(x,y)=(\lambda_{\alpha^{i+2}(x)}\alpha^{j}(y),\rho_{\alpha^{j}(y)}\alpha^{i+2}(x)), (1\times\alpha^{2})r(\alpha^{i}\times\alpha^{j})(x,y)=(\lambda_{\alpha^{i}(x)}\alpha^{j}(y),\alpha^{2}\rho_{\alpha^{j}(y)}\alpha^{i}(x)).
\|x\|_{T}= \sup \bigg\{\theta \sum_{i=1}^d \big\|E_ix\big\| : d \leq E_1 < E_2<\cdots < E_d, \, d \in \mathbb{N},\, \{E_i\}_{i=1}^d \subset [\mathbb{N}]^{<\omega}\bigg\}.
\begin{aligned}[L] \circ [L'] = \int_{X}4 \beta (e^{L}_+ - e^{L}_-) \cdot (e^{L'}_+ - e^{L'}_-) = 4 \beta \int_{X}( e^{L}_+ \cdot e^{L'}_+ - e^{L}_- \cdot e^{L'}_+ - e^{L}_+ \cdot e^{L'}_+ + e^{L}_- \cdot e^{L'}_- ) . \end{aligned}
\int_{B_1}\left|u(x)-\ell(x)\right|^{\frac{n^2}{n-1}}\,dx\le C\left\|\nabla u-q\right\|_{L^n(B_1)}^{\frac{n^2}{n-1}}\le C\varepsilon^{\frac{n^2}{n-1}}a^{\frac{n^2}{n-1}},
\alpha_i := \frac{p-1}{2}\max(t^{p/2}, |s_i(t)|^{p/2}- \xi \gamma)^{2 - 4/p}~, \beta_i := p^2 \max(t^{p/2}, |s_i(t)|^{p/2}+ \xi \gamma)^{2 - 4/p}~.
\mathsf{L}_{d+1}-\mathsf{L}_{d}=\sum_{k=0}^{d+1}\binom{u}{k}\left(\binom{-u+d+1}{d+1-k}-\binom{-u+d}{d-k}\right)(1-y)^{k}=\sum_{k=0}^{d+1}\binom{u}{k}\binom{-u+d}{d+1-k}(1-y)^k &=\sum_{k=0}^{d+1}\binom{u}{d+1}\binom{d+1}{k}(-1)^{d-k+1}(1-y)^{k}=\binom{u}{d+1}(-y)^{d+1}.
\widetilde{D}_{G}^{a b}( x - y ) = \int \frac{d^{4}k}{\left( 2 \pi \right)^{4}}\delta^{a b}\left[ \frac{\left( - i \right)}{k^{2}}- C_{G}\delta \left( k \right) \right] \exp \left\{- i k \left( x - y \right) \right\}.
((b^{\mathbf{n}})^{g|_0}, (a^{\mathbf n \cdot \mathbf{e}_{{1}}})^{g|_0}, \dots, (a^{\mathbf n \cdot \mathbf{e}_{{p-1}}})^{g|_0})^{a^k}= \psi_1((b^{\mathbf{n}})^g) = \psi_1((b^{\mathbf{n}})^h) &= ((b^{\mathbf n})^{h|_0}, (a^{\mathbf n \cdot \mathbf{e}_{{1}}})^{h|_1}, \dots, (a^{\mathbf n \cdot \mathbf{e}_{{p-1}}})^{h|_{p-1}})^{h|^\varnothing}.
\begin{aligned}\|F(\cdot, u)-F(\cdot, v)\|_{H^{-1-s,q}}\stackrel{(i)}{\lesssim}\|F(\cdot, u)-F(\cdot, v)\|_{H^{-1,\xi}}&\lesssim (1+\|u\|_{L^{h \xi}}^{h-1}+\|v\|_{L^{h \xi}}^{h-1})\|u-v\|_{L^{h \xi}}&\stackrel{(ii)}{\lesssim}(1+\|u\|_{H^{\theta,q}}^{h-1}+\|v\|_{H^{\theta,q}}^{h-1})\|u-v\|_{H^{\theta,q}};\end{aligned}
\mathbf{y}=\mathcal{T}_{\mathbf{A}^{\top},\mathbf{A}}\left(\mathbf{x}\right)\Leftrightarrow\forall\;0\le k<n,\;y_{k}=\sqrt{\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top},\mathbf{x}\right)},
[z^{n}] \left. \frac{\partial^{r}F(z,u)}{\partial u^{r}}\right|_{u=1}= \left( 1 + O\left( n^{-1}\right) \right) \lambda_{1}\left( \frac{2}{3}\right)^{r}n^{r-\frac{3}{2}}\rho_{1}^{-n},
- \frac{1}{- \mu a}( 1 - \frac{r_{+}}{r})^{- 1}( 1 - \frac{r_{-}}{r})^{- \frac{a - \chi^{2}( 1 / a - 3 / 2 )}{a + \chi^{2}/ 2}}r^{\frac{2}{a}- 4}d r^{2}- r^{\frac{1 - a}{a}}( 1 - \frac{r_{-}}{r})^{\frac{1 - a}{a}\frac{\chi^{2}}{a + \chi^{2}/ 2}}d \Omega
\mathcal I=\Bigl\{\alpha : L_j\bigl(\frac 1{(1-r_j^2)^{1-2\delta}}-1\bigr)\leq \alpha_j\leq \frac{L_j}{\frac{(1+\epsilon_j)^2}{r_j^2}-1}, \j=1,\dots, n\Bigr\}
= \frac12 \int_{S^1}\begin{pmatrix}f_0 f_1 f_2\end{pmatrix}\begin{pmatrix}0 0 0 0 -D\frac1{k_0}D\frac1{k_0}D+D\frac{k_1}{k_0}+\frac{k_1}{k_0}D \frac{k_2}{k_0}D + D\frac{k_2}{k_0}0 \frac{k_2}{k_0}D + D\frac{k_2}{k_0}D\frac1{k_0}D\frac1{k_0}D - D\frac{k_1}{k_0}-\frac{k_1}{k_0}D\end{pmatrix}\begin{pmatrix}h_0 h_1 h_2\end{pmatrix}dx.
\xi_{i,j}^{r}(i) =(P_{i}-\pi_{i}(v_{r})^{*}\pi_{i}(v_{r}))(\pi_{i}(v_{i})\pi_{i}(v_{i})^{*}-\pi_{i}(v_{j})\pi_{i}(v_{j})^{*}) = (P_{i}-P_{i}T_{r}^{*}P_{i}T_{r}P_{i})(P_{i}T_{i}P_{i}T_{i}^{*}P_{i}-P_{i}T_{j}P_{i}T_{j}^{*}P_{i}) = (P_{i}-P_{i}T_{r}^{*}T_{r}P_{i-r}P_{i})(P_{i}T_{i}T_{i}^{*}P_{i}-0) = (P_{i}-P_{i-r})(P_{i}T_{i}T_{i}^{*}P_{i})
\hat{\textbf{X}}_{0}= \arg \mathop{\min}\limits_{\textbf{X}_{0}\in{\cal X}}\left\|{\bar{\textbf{Y}}_{0}- \sqrt \rho{\textbf{X}_{0}\sum\limits_{i=0}^{N_v-1}\textbf{H}_{0,i}\textbf{F}_{0,i}}}\right\|_F,
f(m_2 + e) = \f(m_2) + a^{(m_2)}= f(m_1 + re) + a^{(m_2)}= f(m_1) +a^{(m_1)}r + a^{(m_2)}= \f(m_1 + (r+1)e) = f(m_1) + a^{(m_1)}(r+1).
S_1(p^l,a)=\sum_{u=1}^{p^l}N_{d-2}(p,l,u)\left(\sum_{y=1}^{p^l}e\left(\frac{auy^2}{p^l}\right)\sum_{z=1}^{p^l}e\left(\frac{auz^2}{p^l}\right)\right).
\lim_{N\to\infty}Q_{N}\Big[\varepsilon_0 \Big(\frac{\gamma}{\beta+\gamma}\Big)^{1/D}N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}\leq R_{N,d,D}& \leq K_{0}\Big(\frac{\beta+\gamma}{\beta}\Big)^{1/2}N^{\frac{d}{2}+\frac{d-D}{D+2}}\Big] =1.
W^{\prime}= - \frac{1}{2}T r \beta^{\prime}\Delta^{- 1}+ \frac{\delta W_{S D}[ \beta_{0}]}{\delta \beta_{0}}\beta{'}+ W_{2}^{'}[ \beta_{0}] + \frac{1}{2}T r \beta^{\prime}\beta_{0}^{- 1}+ O ( \epsilon^{2}) .
\left( \hat{f}^{*}\hat{\bf \omega}^{n}\right) \left. \left( \hat{\bf A}_{1}, \ldots , \hat{\bf A}_{n}\right) \right|_{\Phi_{p}}= \hat{f}_{*}\hat{\bf \omega}^{n}\left. \left( \hat{f}_{*}\hat{\bf A}_{1}, \ldots , \hat{f}_{*}\hat{\bf A}_{n}\right) \right|_{f \left( \Phi_{p}\right)}.
\omega_{WP}=i\overline\partial{2\sum_{i,j=1}^h\omega_i''A_{ij}\overline \omega_j-3\left(\sum_{i,j=1}^h\omega_i'A_{ij}\overline\omega_j\right)^2\over 2\left(\sum_{i,j=1}^h\omega_iA_{ij}\overline\omega_j\right)^2}.
\begin{array}{c c c}{}&{h^{0 0}}&{}\\{h^{1 0}}&{}&{h^{0 1}}\\{}&{h^{1 1}}&{}\\ \end{array}\qquad = \qquad \begin{array}{c c c}{}&{1}&{}\\{n}&{}&{n}\\{}&{1}&{}\\ \end{array}
\int_{0}^{\infty}\|u(t, \cdot)\|_{X_{-1}}^{2}\mathrm{d}t = 2 \int_{0}^{\infty}E(t) \mathrm{d}t \leq 2 E(0) \int_{0}^{\infty}e^{-2 \omega t}\mathrm{d}t = -\tfrac{E(0)}{\omega}e^{-2 \omega t}\big|_{t = 0}^{t = \infty}= \tfrac{E(0)}{\omega},
nxn'=\left(\begin{matrix}1 0 0 0 0 1 A 0 0 0 1 0 0 0 0 1\end{matrix}\right) \left(\begin{matrix}1 a b c 0 1 d e 0 0 1 f 0 0 0 1\end{matrix}\right)\left(\begin{matrix}1 B C 0 0 1 D 0 0 0 1 0 0 0 0 1\end{matrix}\right) =\left(\begin{matrix}1 a' b' c' 0 1 d' e' 0 0 1 f' 0 0 0 1\end{matrix}\right)=x'
\frac{Q_\eta}{Q_n}=\frac{Q_{\eta-1}}{Q_n}+\frac{q_\eta}{Q_n}\leq\kappa+\frac{q_\eta}{Q_\eta}\frac{Q_\eta}{Q_n}=\kappa(1+o(1))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{Q_\eta}{Q_n}\geq\left(1+\frac{\gamma}{2}\right)=\frac{\kappa+1}{2}
\|v - \Pi_{h}v\|_{h,*}^2 = \|v - \Pi_{h}v\|_h^{2}+ (\theta h)^{-1}\|v - \Pi_{h}v\|_{L_2(Q)}^2 \leq \left( C_2 h^{2(s-1)}+ (\theta h)^{-1}C_s h^{2s}\right)\| v\|^2_{H^{s}(Q)}= \left(C_2 + \theta^{-1}h C_s\right) h^{2(s-1)}\| v\|^2_{H^{s}(Q)}= C_3 h^{2(s-1)}\| v\|^2_{H^{s}(Q)},
G^{2,0}_{2,2}\!\left(\!t~\vline\begin{array}{l}b_1,b_2 a_1,a_2\end{array}\!\!\right)=\frac{t^{a_2}(1-t)^{\psi_p-1}_{+}}{\Gamma(\psi_p)}{_2F_{1}}\!\left(\begin{array}{l}b_1-a_1,b_2-a_1 \psi_p\end{array}\!\!\vline\,1-t\right),
z^{p^2}-z =(y^p)^{p^2}-(T^{\lambda})^{p^2}-y^p+T^{\lambda}=(y^{p^2}-y)^p-T^{\lambda p^2}+T^{\lambda}&=(T^{\lambda p})^p-T^{\lambda p^2}+T^{\lambda}=T^{\lambda}=\nu,
\big\{c\big((x_1,\ldots,x_{a-1}, x_{a}= u^{\ast}, x_{a+1},\ldots, x_t);(u^{\ast}, v^{\ast})\big)\big\}_{(x_1,\ldots, x_{a-1}, x_{a+1},\ldots, x_t) \in [r]^{t-1}},
\kappa_\alpha(\nu^A)^*(x^*) =|x-y|^{n-\alpha}\kappa_\alpha\nu^A(x)=|x-y|^{n-\alpha}\kappa_\alpha\nu(x){}=\kappa_\alpha\nu^*(x^*)=\kappa_\alpha(\nu^*)^{A^*}(x^*)
\varphi(\Phi(x^{\ell(k)})\!-\!\Phi^*)=c(\Phi(x^{\ell(k)})\!-\!\Phi^*)^{1-\theta}\le c\big[bc(1\!-\!\theta)\|x^{\ell(k)}-x^{\ell(k)-1}\|\big]^{\frac{1-\theta}{\theta}}.
H(p+Dm,x,\omega)\leq\overline{H}(p)\;\;\textrm{on}\;\;\mathbb{R}^d\;\;\textrm{and}\;\; H(p+Dm,x,\omega)=\overline{H}(p)\;\;\textrm{on}\;\;\mathbb{R}^d\setminus\left\{y\right\}.
p = \frac12(4,4,4,4,1,1,0,0,0) +\frac12 (3,3,3,3,3,3,0,0,0) + &\frac12 (3,3,3,3,2,2,2,0,0) + \frac12 (2,2,2,2,2,2,2,2,2)
\boldsymbol\Sigma \triangleq{\bf I}_{N_{\rm T}}+ \gamma_{\rm d}\bar{\bf H}_{\rm d}\bar{\bf H}_{\rm d}^H + \gamma_{\rm r}\bar{\bf H}_{\rm r}\bar{\bf H}_{\rm r}^H + \sum_{k=1}^K\bar{\bf H}_{{\rm e},k}{\boldsymbol\Psi}_{{\rm e},k}\bar{\bf H}^H_{{\rm e},k}.
0\rightarrow\bigoplus_{i+j=k}H_{i}(X;R)\otimes_{R}H_{j}(Y;R)\rightarrow H_{k}(X\times Y;R)\rightarrow\bigoplus_{i+j=k-1}Tor_{1}^{R}(H_{i}(X;R),H_{j}(Y;R))\rightarrow0.
B^{a ( 0 , p_{a})}, \H_{\mu_{1}\ldots \mu_{p + 1}}^{a ( p_{a}+ 1 , 0 )}\mathrm{\ \a n d \ \}\partial_{\alpha_{1}\ldots \alpha_{k}}H_{\mu_{1}\ldots \mu_{p + 1}}^{a ( p_{a}+ 1 , 0 )}.
M(u(t),v(t)) := \|u(t)\|^2_{L^2}+ 2 \|v(t)\|^2_{L^2}= M(u_0,v_0), E(u(t),v(t)) := \frac{1}{2}(\|\nabla u(t)\|^2_{L^2}+ \kappa \|\nabla v(t)\|^2_{L^2}) - \emph{Re}(\langle v(t),u^2(t)\rangle) = E(u_0,v_0).
\left ( \begin{array}{c}\tilde{\varphi}_{0}\vdots \tilde{\varphi}_{p}\end{array}\right ) = \sum_{k \in \mathbb{Z}}\tilde{A}_{k}\left ( \begin{array}{c}\tilde{\varphi}_{0}(2 \cdot - k) \vdots \tilde{\varphi}_{p}(2 \cdot - k) \end{array}\right ).
\begin{array}{l}A= \left(\begin{array}{rrrr}\frac{2d}{b}1 0 0 0 \frac{b}{2d}0 0 0 0 1 0 0 0 0 1\end{array}\right) \in SL(2,\mathbb{C}) \times SL(2,\mathbb{C}).\end{array}
S_{\mathrm{q Y M}}^{\mathrm{r e d .}}= -{\frac{1}{4}}\left({\frac{2 \pi}{a}}\right)^{4}{\frac{N}{g_{\mathrm{Y M}}^{2}}}\mathrm{T r}{\mathrm{\boldmath{F}}}_{\mu \nu}{\mathrm{\boldmath{F}}}^{\mu \nu},