harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J\coloneqq\begin{bmatrix}0 0 \cdots 0 1 0 0 \cdots 1 0 \vdots \vdots \ddots \vdots \vdots 0 1 \cdots 0 0 1 0 \cdots 0 0\end{bmatrix}\in \mathbb{C}^{n \times n}.
L_m(\varepsilon_*\Omega_{(X, D)_{\bullet}/Y})^n=\bigoplus_{k \ge -m}(\varepsilon_k)_*\Omega^{n-k}_{(X, D)_k/Y}F^p(\varepsilon_*\Omega_{(X, D)_{\bullet}/Y})^n=\bigoplus_{0 \le k \le n-p}(\varepsilon_k)_*\Omega^{n-k}_{(X, D)_k/Y},
V^{( 1 )}( \Phi ) = \frac{\lambda}{\alpha^{2}}\Phi^{4}+ \frac{1}{2}\left[ \frac{\gamma^{2}( 4 \pi )^{2}}{2 \theta^{4}}- \frac{8 \lambda}{\theta^{2}}\right] \Phi^{4}\left( \ln \frac{\Phi^{2}}{\mu^{2}}- \frac{2 5}{6}\right) .
c=c_{1}^{1}c_{1}^{2}\cdot\cdot\cdot c_{1}^{n_{T}}c_{2}^{1}c_{2}^{2}\cdot\cdot\cdot c_{2}^{n_{T}}\cdot\cdot\cdot c_{T}^{1}c_{T}^{2}\cdot\cdot\cdot c_{T}^{n_{T}}
- \alpha^{-1}\big(\int_0^s U(u,x)f'(u)\mathrm{d}u+f(s)U(s,x)-f(0)U(0,x)\big) =- \int_0^s \int_0^1 f(u)A(x,y)U(u,y)\mathrm{d}y\mathrm{d}u+ \beta \int_0^s f(u) \mathrm{d}\xi_2(u,x) 1_{\{\gamma \sigma d^{-1}T^{-\frac{1}{2}}\to\beta\}}+\zeta \int_0^s f(u) \mathrm{d}\xi_3(u,x) .
A_{E(n)}^t \delta_{\mu,n}= \sum_{\nu \in E^{<n}}A_{E(n)}^t(\nu,\mu) \delta_{\nu,n}= \sum_{\nu \in E^{<n}}|\mu E(n)^1 \nu| \delta_{\nu,n}&= \sum_{\nu \in E^{<n}, e \in E^1 r(\nu), [e \nu]_n = \mu}\delta_{\nu,n}= \delta_{\mu_2 \dots \mu_{|\mu|},n}.
\begin{array}{l}\beta^2 T_1=-\partial_0(1)'^2T_1-2\partial_0(1)'T_1\partial_0(1)-T_1\partial_0(1)^2 \beta^2 T_2=-\partial_0(1)'^2T_2-2\partial_0(1)'T_2\partial_0(1)-T_2\partial_0(1)^2. \end{array}
\omega^{(1)}= D^{\ell_1}(u - u_0 - u_\beta x^\beta) \mbox{with}\, 0 \le \ell_1 \le \ell+4, \omega^{(2)}\in \left\{u_0, D^{\ell_2}(u-u_0)\right\}\mbox{with}\, 0 \le \ell_2 \le \ell+4, \omega^{(j)}\in \left\{1, u_0, D^{\ell_j}(u-u_0)\right\}\mbox{with}\, 0 \le \ell_j \le \frac{\ell+4}{2}\, \mbox{for}\, j \ge 3.
\sup_{(a,b) \in \Sigma}\mathbb{E}[|\hat{\pi}_{h,n}(x) - \pi (x)|^2] \le c \sum_{j = 1}^d h_j^{* \, 2 \beta_j}+ \frac{c}{T_n}\frac{\sum_{l = 1}^d |\log(h^*_l)|}{\prod_{l \ge 3}h^*_l}& = (\frac{\log T_n}{T_n})^{\frac{\bar{\beta}_3}{(2 \bar{\beta}_3 + d - 2)}}
\begin{cases}\begin{array}{ll}\dot{v}_m-\log\det (v_m)_{\alpha\bar{\beta}}+Av_m-f(z,t)\geq 0 \forall (z,t)\in\Omega\times (0,\epsilon_A(\nu)), v_m(z,0)\geq u_0(z), \forall z\in\bar{\Omega}, v_m(z,t)\geq \varphi (z,t), \forall (z,t)\in\partial\Omega\times [0,\epsilon_A(\nu)). \end{array}\end{cases}
\begin{array}{c c}{\mathrm{v}_{\alpha \beta}= \textstyle{\frac{1}{2}}\epsilon_{\alpha \beta \gamma \delta}\tilde{\mathrm{v}}^{\gamma \delta}~ ~ ~ ~}{~ ~ ~ ~ \tilde{\mathrm{v}}^{\alpha \beta}= \textstyle{\frac{1}{2}}\epsilon^{\alpha \beta \gamma \delta}\mathrm{v}_{\gamma \delta}}\end{array}
\xi = (g-1)l_1 + gl_2, & \alpha_{1,0}(\xi) = -\left[2g(l_1 + l_2) + 4gl_1\right], & \alpha_{1,1}(\xi) = (g - 1)(g - 2)l_1^2 + g(g - 1)l_2^2 - 2(g^2 + 2g - 1)l_1l_2.
M(\alpha,\varepsilon,Z,T,f) =\lim_{n\to\infty}M(n,\alpha,\varepsilon,Z,T,f), \underline{R}(\alpha,\varepsilon,Z,T,f) =\liminf_{n\to\infty}R(n,\alpha,\varepsilon,Z,T,f), \overline{R}(\alpha,\varepsilon,Z,T,f) =\limsup_{n\to\infty}R(n,\alpha,\varepsilon,Z,T,f), M^P(\alpha,\varepsilon,Z,T,f) =\lim_{n\to\infty}M^P(n,\alpha,\varepsilon,Z,T,f).
\begin{aligned}E_4^{(2)}E_6^{(2)}=~ (1+240\tilde{q}+240q+\cdots+13440\tilde{q}pq+ 30240\tilde{q}q+\cdots) \times &(1-504\tilde{q}-504q +\cdots+44352\tilde{q}p q+166320\tilde{q}q+\cdots) =~ 1-264\tilde{q}-264q +\cdots+57792 \tilde{q}pq-45360 \tilde{q}q+\cdots . \end{aligned}
D^{1\;\alpha}\, J^{[a_1,a_2,a_3]}_{\alpha\alpha_2\ldots\alpha_{2j_1};\ \dot\alpha_1\ldots\dot\alpha_{2j_2}}= \bar D_4^{\dot\alpha}\, J^{[a_1,a_2,a_3]}_{\alpha_1\ldots\alpha_{2j_1};\ \dot\alpha\dot\alpha_2\ldots\dot\alpha_{2j_2}}=0
\tan \mathrm{\boldmath \Large \left( \right.}\! \! \! \delta_{0}^{( D_{0})}( k ) - ( D / 2 - 1 ) \pi / 2 \mathrm{\boldmath \Large \left. \right)}\! \! = \frac{\pi}{\ln \left( k^{2}/ | E_{_\mathrm{( g s )}}| \right)}.
= \exp\left\{\frac{y Li_2(z)}{(1-y)^2}- \frac{y Li_2(yz)}{(1-y)^2}- \frac{y}{(1-y)^2}\log\left(\frac{1}{1-yz}\right) + \frac{y^2}{(1-y)^2}\log\left(\frac{1}{1-yz}\right) \right\}
\left| \left. a , 1 / 2 \right| n \right\rangle = \frac{1}{g_{\left| \left. a , 1 / 2 \right| n \right\rangle}^{\left| 1 / 2 \right\rangle}}K_{a \frac{1}{2}}\left( i \, \nu_{n}\right) \left| \frac{1}{2}\right\rangle \; , \quad \mathrm{w h e r e}\; \left| \frac{1}{2}\right\rangle \equiv \left| B_{\frac{1}{2}}\right\rangle \; ,
W_{\psi_{i,1,*},\varphi_{i,2,*}}f(z'_{[j]}) = \psi_{i,1,*}(z'_{[j]})f\big(G^i z'_{[j]}+ (b^{i,1})'_{[j]}\big) = W_{\psi_{i,1,*},\varphi_{i,1,*}}f(z),
A_4E_i =\frac{1}{|C|}\sum_{j=0}^4 K_i(\beta_j)A_4A_j=\frac{1}{|C|}\sum_{j=0}^4 K_i(\beta_j)A_{4-j}=\frac{1}{|C|}\sum_{j=0}^4 K_i(\beta_{4-j})A_{j}&=\frac{1}{|C|}\sum_{j=0}^4 K_i(n-\beta_{j})A_{j}=\frac{1}{|C|}\sum_{j=0}^4(-1)^i K_i(\beta_{j})A_{j}=(-1)^iE_i.
{\frac{d \tilde{m}_{0}}{d \tilde{\lambda}}}={\frac{\tilde{m}_{0}}{2}}\partial_{y}({\cal W}g_{\mu \nu}){\frac{d x^{\mu}}{d \tilde{\lambda}}}{\frac{d x^{\nu}}{d \tilde{\lambda}}}{\frac{d y}{d \tilde{\lambda}}}.
P_{\kappa}(x)\!=\!\frac{\int_{0}^{x}\!h(t)\exp_{\kappa}[-f(t)]\,dt}{\int_{0}^{x}\!h(t)\exp_{\kappa}[-f(t)]\,dt \!+\!\lambda \int_{x}^{\infty}\!h(t)\exp_{\kappa}[-f(t)]\,dt}\ ,
\begin{aligned}I_{\nu+3}(z)-3I_{\nu+2}(z)+3I_{\nu+1}(z)-I_\nu(z)= \frac{z^{\nu}}{\sqrt{\pi}\,2^{\nu}\Gamma(\nu+1/2)}* \quad\times \Big(\frac{3}{z^2}\int_{-1}^1e^{-zs}s(1-s^2)^{\nu-1/2}\,ds+ \frac{3}{z}\int_{-1}^1e^{-zs}s(1+s)(1-s^2)^{\nu-1/2}\,ds &+ \int_{-1}^1e^{-zs}(1+s)^3(1-s^2)^{\nu-1/2}\,ds \Big).\end{aligned}
\sum_{\mu\in \mathcal{P}_j}t^{h_{2,0}(\mu)}q^{|\mu|}=\sum_{\tilde{\mu}\in \mathcal{P}_j\cap \mathcal{D}}q^{|\tilde{\mu}|}\sum_{\lambda\in \mathcal{P}_e}t^{l(\lambda)}q^{|\lambda|}=\frac{q^{2j\choose 2}}{\prod_{i\geq 1}(1-q^{2i})(1-tq^{2i})}
(-\Delta)^{n+1}\tilde{f}=0\,\, B_{r_1}(0,r_1), &\tilde{f}=f,\,\, D_\nu^1\tilde{f}=D_\nu^1f,\ldots,\,D_\nu^{n}\tilde{f}=D_\nu^{n}f\partial B_{r_1}(0,r_1),
e_\lambda(x, y) ~=~2^{7/2}\int_r^\infty \frac{\sinh s}{\sqrt{\cosh 2s -\cosh 2r}}\cdot \frac{-1}{2 \pi \sinh s}\partial_s \left(\frac{-1}{2 \pi \sinh 2s}\partial_s\right)^3 \cdot \cdot \left( \frac{-1}{2 \pi \sinh s}\partial_s \right)^{4}\frac{\sin (s \cdot \sqrt{\lambda - 11^2})}{\pi s}\,ds\,.
\Upsilon_c:=\begin{bmatrix}\cdots \cdots \cdots \cdots \Gamma_{ij}^1 \Gamma_{ij}^2 \cdots \Gamma_{ij}^n \cdots \cdots \cdots \cdots \Gamma_{ii}^1 \Gamma_{ii}^2 \cdots \Gamma_{ii}^n \cdots \cdots \cdots \cdots \end{bmatrix}, i\neq j,
\prod_{k = 1}^{n_{c}- \tilde{n_{c}}}( x - \Lambda^{2}\omega^{2 k - 1}) = x^{n_{c}- \tilde{n}_{c}}+ \Lambda^{2 ( n_{c}- \tilde{n}_{c})}\simeq \Lambda^{2 ( n_{c}- \tilde{n}_{c})},
R \, \dot{z}_{\mu}\, T^{\mu \nu}\, n_{\nu}= - \epsilon \, \epsilon_{1}\, ( \ddot{z}^{\alpha}\ddot{z}_{\alpha}+ \epsilon a^{2}) \, \mathrm{T r}\, ( \beta - \sigma_{, a})^{2}+ O ( 1 / R ) \; .
\begin{array}{l l}{\begin{array}{l l}{\varphi ( r ) = \eta}&{r \rightarrow \infty}\\{\varphi ( r ) = 0}&{r = 0}\\ \end{array}}&{\begin{array}{l l}{P ( r ) = 0}&{r \rightarrow \infty}\\{P ( r ) = 1}&{r = 0 .}\\ \end{array}}\\ \end{array}
\bigl(\widehat{S}_\chi^{\rm stiff(soft)}(z)\bigr)^* = \widehat{P}_\chi \bigl(S_\chi^{\rm stiff(soft)}\bigr)^* , \bigl(\widehat{\Pi}_\chi^{\rm stiff(soft)}\bigr)^* = \widehat{P}_\chi\bigl(\Pi_\chi^{\rm stiff(soft)}\bigr)^*.
\sup_{\beta\in\mathbb{R}}\sum_{\mathbf{t}\in\{0,\dots,9\}^k}F_Y\Bigl(\sum_{i=1}^{k}\frac{t_i}{10^i}+\beta\Bigr)=\sup_{\eta\in[0,Y^{-1}]}\sum_{\mathbf{t}\in\{0,\dots,9\}^k}F_Y\Bigl(\sum_{i=1}^{k}\frac{t_i}{10^i}+\eta\Bigr),
\limsup_{n \rightarrow \infty}\| Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}Q_{n,\textup{\textbf{m}}}F_\alpha-Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}P_{n,\textup{\textbf{m}},\alpha}\|_K^{1/n}\leq \frac{\|\Phi\|_K}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})},\alpha=1,2,\ldots,d,
\pi^b_{0b(n-b)}= \pi^{b+1}_{0b(n-b)}-\frac{n-1-2b}{2}c_b &=z_b - \frac{n-1-2b}{n+1-2b}\left(z_b - \frac{1}{n-1-2b}\sum_{i=b+1}^{n-b-1}z_i\right) & = \frac{\sum_{i=b}^{n-b}z_i}{n+1-2b},
\varphi(0, w) = \sqrt \pi \frac{\Gamma(w - \frac{1}{2})}{\Gamma(w)}\sum_c c^{-2w}S_\mathfrak{a}(0,0;c) \varphi(m, w) = \frac{\pi^w}{\Gamma(w)}\lvert m \rvert^{w-1}\sum_c c^{-2w}S_\mathfrak{a}(0, m; c)
\limsup_{n \rightarrow \infty}\|\sum_{\nu=n+|\textup{\textbf{m}}|-m_\alpha+1}^{\infty}b_{\nu,n}^{(\alpha)}\Phi_\nu\|_K^{1/n}\leq \frac{\|\Phi\|_K}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})}.
\int_{\bigcup_{\gamma\in\Psi_{k}}\gamma^{-1}V}\left\vert \varphi\left(x\right) \right\vert^{2}dx\leq\int_{G}\left\vert \varphi\left( x\right)\right\vert^{2}dx=\left\Vert \varphi\right\Vert^{2}<\infty.
r_{a}^{t}+\delta r_{a\rightarrow i}^{t}=y_{a}-\sum_{j\in[n]}A_{aj}\left(x_{j}^{t}+\delta x_{j\rightarrow a}^{t}\right)+A_{ai}x_{i}^{t}, x_{i}^{t+1}+\delta x_{i\rightarrow a}^{t+1}=\frac{1}{\sigma_{i}^{2}}\eta_{t}\left(\sum_{b\in[m]}A_{bi}\left(r_{b}^{t}+\delta r_{b\rightarrow i}^{t}\right)-A_{ai}r_{a}^{t}\right).
f(x)= f\bigg(m\bigg(1-\frac{x-ma}{b-ma}\bigg)a+\bigg(\frac{x-ma}{b-ma}\bigg)b\bigg) \leq m \bigg(1-\frac{x-ma}{b-ma}\bigg)^s f(a)+ \bigg(\frac{x-ma}{b-ma}\bigg)f(b).
\pi\mu(B(t,r)) \le C\pi\mu_n(B(t,C 2^{-n}))+C 2^{n(\theta-k)}&\le C 2^{n(\theta-k)}+C\int_{t\in B(t,C 2^{-n})}\int\mu_n\,\mathrm{d}\eta_t\,\mathrm{d}t &\le C 2^{n(\theta-k)}\,,
V^{u}\bigl(t, x\bigr) = \xi_{t,T}\bigl(u\bigr) + \int_t^T G\bigl(s, Y_s^{t,x;u}, Z_s^{t,x;u}\bigr) ds - \int_t^T Z_s^{t,x;u}dB_s = \Psi(X_T^{t,x;u}) + \int_t^T \Bigl\{c\bigl(s, X_s^{t,x;u}, u_s\bigr) + G\bigl(s, Y_s^{t,x;u}, Z_s^{t,x;u}\bigr) \Bigr\}ds - \int_t^T Z_s^{t,x;u}dB_s,
\theta_{\beta}= \Psi_{\beta \alpha}( z_{\alpha}, \theta_{\alpha}) = \psi_{\beta \alpha}( z_{\alpha}) + \theta_{\alpha}\sqrt{f_{\beta \alpha}^{\prime}( z_{\alpha}) + \psi_{\beta \alpha}( z_{\alpha}) \psi_{\beta \alpha}^{\prime}( z_{\alpha})}.
K(x,y): = \gamma(x,y) -\gamma(\xi_{0}, y) -p(x)\gamma(\xi_{1}, y)-\gamma(\xi_{0}, x) -p(y)\gamma(\xi_{1}, x) &+ \gamma(\xi_{0}, \xi_{0}) + (p(x) + p(y))\gamma(\xi_{0}, \xi_{1}) + p(x)p(y)\gamma(\xi_{1}, \xi_{1})
\sum_{n\geq0}\sum_{M\geq1}h_r(n,M) z^Mq^n= \sum_{n\geq 1}z^{n-1}(zq^n+z^2q^{2n}+\cdots+ z^{r-1}q^{(r-1)n})\prod_{j=1}^{n-1}(1+zq^j+z^2q^{2j}+\cdots +z^{r-1}q^{(r-1)j}).
\left(\int_\Omega U\,u^2\,dx\right)^{p/(p+1)}-\int_\Omega|u|^{2p/(p+1)}\,dx &\ge\frac{1}{2^p(p+1)}\left(\int_\Omega U\,u^2\, dx\right)^{p/(p+1)}\,\left\|\frac{|u|^{2/(p+1)}\,U^{1/(p+1)}}{\displaystyle\left(\int_\Omega U\,u^2\,dx\right)^{1/(p+1)}}-U^{-p/(p+1)}\right\|^{p+1}_{L^{p+1}(\Omega)}.
\Lambda_{j,n}^{(k)}(x;\ell)=(-1)^{k}\det \begin{pmatrix}C_{k,n}^{\ell}(x) A_{\nu,n}^{j,\ell}(x) begin{align*}3mm] D_{k,n}^{\ell}(x) B_{\nu,n}^{j,\ell}(x)\end{pmatrix}, \ \ \nu=1,2,\ \k=1,2,\ell=1,2.
\exp \left( - \bigtriangleup S_{1}[ \overline{{X}}+ \xi ] \right) = i g \pi \alpha^{\prime}\frac{1}{2}N_{d i v}\int_{0}^{2 \pi}d \theta ~ \mathrm{T r}~{\cal P}\left( U [ A ] ~ \left[ ( D_{a}D^{c}F_{c b}- D_{b}D^{c}F_{c a}) \xi^{a}\partial_{\theta}\xi^{b}\right. \right.
A(z)=2q^{-1/12}\sqrt{z}\,\exp\!\left(-\frac{\ln^{2}(z)}{2\ln(q)}+\frac{\pi^{2}}{3\ln(q)}\right) \left|\left(\tilde{q}\, e^{-2i\beta(z)};\tilde{q}\right)_{\infty}\right|^{2}.
f_{n}\left(g_{s^{\prime\prime},k,i}\right)\circ f_{n}\left(g_{s^{\prime},j,k}\right)\circ f_{n}\left(g_{s,i,j}\right)\restriction_{T_{b_{i}}\left(V^{\prime\prime}\right)}=\mathrm{id}_{T_{b_{i}}\left(V^{\prime\prime}\right)},
F^{(2)}=\frac{1}{\prod_{i=1}^r d_i}(\partial_{t^{n-1}}F^{(1)},\dots,\partial_{t^{0}}F^{(1)}) \Phi^{-1}\begin{pmatrix}0 \partial_{t^1}\partial_{t^1}F^{(1)}\dots \partial_{t^1}\partial_{t^{n-1}}F^{(1)}\end{pmatrix},
a \rightarrow \left( \begin{matrix}{a_{+ , i j}}&{\varrho a_{- , i k}}\\{\varrho a_{- , \ell j}^{\dagger}}&{\tilde{a}_{+ , \ell k}}\\ \end{matrix}\right) , \quad \psi \rightarrow \left( \begin{matrix}{\psi_{+ , i j}}&{\varrho \psi_{- , i k}}\\{\varrho \psi_{- , \ell j}^{\dagger}}&{\tilde{\psi}_{+ , \ell k}}\\ \end{matrix}\right) .
Y_{m_1,m_2}(A) := \sum_{\sigma \in \mathrm{Sym}(\mathbb{T}_{m_1,m_2})}\prod_{x \in \mathbb{T}_{m_1,m_2}}\left( \alpha_x \mathbf{1}_{\sigma(x) = x}+ \beta_x \mathbf{1}_{\sigma(x) = x + \mathbf{e}^1}+ \gamma_x \mathbf{1}_{\sigma(x) = x + \mathbf{e}^2}\right).
u(x,t)= \frac{1}{2 \pi}\int_{-\infty}^{+\infty}e^{-i \beta x}E_{\nu, 1+\frac{2H-1}{\nu}, \frac{2H-1}{\nu}}\left(- H \beta^2 t^{\nu+2H-1}\right) \textrm d\beta.
\mathbf{v}_0 (\cdot, \tau) = \cos(\tau (\widehat{\mathcal{A}}^0)^{1/2}) \boldsymbol{\phi}+ (\widehat{\mathcal{A}}^0)^{-1/2}\sin(\tau (\widehat{\mathcal{A}}^0)^{1/2}) \boldsymbol{\psi}+ \int_{0}^{\tau}(\widehat{\mathcal{A}}^0)^{-1/2}\sin((\tau - \tilde{\tau}) (\widehat{\mathcal{A}}^0)^{1/2}) \mathbf{F}(\cdot, \tilde{\tau}) \, d \tilde{\tau}.
\rho_0^2(x^2)= \left|\frac{\psi_0(z_0) - \psi_0(\overline z_0)}{z_0 - \overline z_0}\right| \le \frac 2{|z_0^3|}\int_0^\varepsilon d\sigma(t) + \frac{|A|}{|z_0^2|}\le 2\frac{\sigma(\varepsilon-)-\sigma(0)}{|x|^3}+ \frac{|A|}{x^2},
-\int_{B_{1}}\partial_{s}(g_{1-\alpha,m}*[\phi\psi^{2}w^{2}])dx +\frac{1}{2}c_{1}\leq \int_{0}^{s}\dot{g}_{1-\alpha,m}(s-\tau)(\phi(s)-\phi(\tau))\left( \int_{B_{1}}\psi^{2}\tilde{u}^{1-q}dx \right)(\tau)d\tau & + c_{4}(\rho-\rho')^{-2\beta}\phi(s)\int_{\rho B_{1}}w^{2}(s,x)dx + R_{m}(s),
\overline{w}_0e_{i(m)}e_{-i(m)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}=\overline{u^c}\overline{u}e_{i(m)}e_{-i(m)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}\in B_+u^cB_+,
u_{(m-\ell)n}\left(\underline{b_\ell}\right)t(h,g)= \begin{pmatrix}I_{(m-\ell)n-1}0 0 &1 \underline{b_\ell}I_{\ell n}\end{pmatrix}\begin{pmatrix}h_{1,1}g \cdots h_{1,\ell}g \vdots \vdots h_{\ell,1}g \cdots h_{\ell,\ell}g\end{pmatrix}.
|S_{n,r}| \le \frac{c_1n^{r-1}}{(\log n)^{1/(r-2)}}\left((\log n)^{1/(r-2)}\right)^{\frac{c_1n^{r-1}}{(\log n)^{1/(r-2)}}}&= 2^{\log\left(\frac{c_1n^{r-1}}{(\log n)^{1/(r-2)}}\right)+\frac{c_1n^{r-1}\log(\log n)^{1/(r-2)}}{(\log n)^{1/(r-2)}}}&\le2^{c_1n^{r-1}}.
b_{ij}=(-1)^{j+1}\lhd\, T^+(u_i), u_{j+(-1)^{j+1}}\,\rhd=(-1)^{j+1}(-1) \lhd\, T(u_{j+(-1)^{j+1}}), u_{i}\,\rhd &=(-1)^j\lhd\, T(u_{j+(-1)^{j+1}}), u_{i}\,\rhd = (-1)^{j+i}a_{j+(-1)^{j+1}\,i+(-1)^{i+1}}
A^{\mu}( x ) = e \int_{- \lambda_{m}}^{\infty}d \lambda \, \delta ( \lambda ) \, \frac{v^{\mu}}{R \cdot v}= e \int_{- \lambda_{m}}^{\infty}d \lambda \, \delta ( \lambda ) \, \frac{\partial}{\partial x^{\mu}}\log ({R \cdot v}) .
B = - \frac{2}{3 \pi}\frac{1}{\beta^{4}}( 4 + \beta^{2}m^{2})^{3 / 2}+ \frac{{\vert e \vert}^{2}H^{2}}{1 2 \pi}\frac{1}{( 4 + \beta^{2}m^{2})^{1 / 2}},
T_{\nu}^{\mu}= \left[ \begin{array}{c c c}{- ( \rho + \delta \rho )}&{}&{a ( \rho + P ) ( v + B )_{| j}}\\{}&{}&{}\\{- a^{- 1}( \rho + P ) v^{| i}}&{}&{( P + \delta P ) \delta_{j}^{i}+ \delta \pi_{j}^{i}}\\ \end{array}\right] ,
\{\theta_0 = -t\}\cap \{\theta^n_0 = -t^n_0\}\cap E_t(i,n)\cap O_n = & (F_t)^c \cap\{\widehat{\pi}_0(s),\widehat{\pi}_1(s) \in [i/n,(i+1)/n)s\in [-t^n_0,-t]\}\cap E_t(i,n,-t^n_0)\cap O_n.
\Theta_{k,\Lambda}t_{n_k, \mu}\theta_{n_k, \alpha}t^*_{n_k,\nu}\Theta_{k,\Lambda}= \sum_{\eta,\zeta \in E^{<n_k}, s(\eta),s(\zeta) \in \Lambda}(q_\eta t_{(\mu,\alpha)}t^*_{(\nu,\alpha)}q_{\zeta}) = t_{(\mu,\alpha)}t^*_{(\nu,\alpha)}= t_{n_k, \mu}\theta_{n_k, \alpha}t^*_{n_k,\nu},
\delta_{L}V_{k}^{R}= i \epsilon_{i}[ K_{i}^{R}, V_{k}^{R}] = \epsilon_{i}( x_{i}\partial_{t}+ t \partial_{i}) V_{k}^{R}- \partial_{k}{\frac{1}{\vec{\partial}^{2}- M^{2}}}\epsilon_{i}\dot{V}_{i}^{R}~ ,
\begin{array}{c c c c c c c}{{\cal L}_{\mathbf{I}}K^{x}}{=}{\epsilon^{x y z}K^{y}W_{\mathbf{I}}^{z}}{;}{{\cal L}_{\mathbf{I}}\omega^{x}}{=}{\nabla W_{\mathbf{I}}^{x}}\end{array}
\widetilde{A}^{DVD}:=\left(\begin{array}{c}\delta_n^+u_{m,n}+\delta_m^{(2)}\mu_nv_{m,n}+\mu_n(u_{m,n}^2+v_{m,n}^2)\mu_nv_{m,n}-\delta_n^+v_{m,n}+\delta_m^{(2)}\mu_nu_{m,n}+\mu_n(u_{m,n}^2+v_{m,n}^2)\mu_nu_{m,n}\end{array}\right)=\mathbf{0}.
\xi=\tau\sigma.{r^2}\tau{r^{-2}}.r\sigma^{-1}\tau^{-1}r^{-1}=\beta^{-1}.r^{-1}\tau{r}.\sigma.{r^2}\tau{r^{-2}}.r\sigma^{-1}r^{-1},~[[\,,\,],[\,,\,]]=1\rangle,
\begin{array}{c}\displaystyle \sum \limits_{i=1}^{n}\left[ \langle \mathbf{g}_{i}(\mathbf{x}),\mathbf{y}_{i}-\mathbf{y}_{i}(\mathbf{x}) \rangle +h_{i}(\mathbf{y}_{i})-h_{i}(\mathbf{y}_{i}(\mathbf{x})) \right] \geq 0 \displaystyle \forall \mathbf{y}_{i}\in X_{i}, \mbox{for}\i=1, \dots, n.\end{array}
(\sum_{k=1}^{\nu}Y_{k}X_{k}^{j+\nu})+\Lambda_{1}(\sum_{k=1}^{\nu}Y_{k}X_{k}^{j+\nu-1})+\Lambda_{2}(\sum_{k=1}^{\nu}Y_{k}X_{k}^{j+\nu-2})+\cdot\cdot\cdot+\Lambda_{\nu}(\sum_{k=1}^{\nu}Y_{k}X_{k}^{j})=0
w_{\max}= &\sqrt{\frac{\bar{\gamma}\cdot \sigma_{n}^2}{N\cdot (\sigma_{\rm u}^2 \cdot \sigma_{\rm a}^2 + \sigma_{\rm a}^2\cdot m_{\rm u}^2 + \sigma_{\rm u}^2 \cdot m_{\rm a}^2 ) + N^2\cdot m_{\rm u}^2 \cdot m_{\rm a}^2}}.
\theta \left( \begin{array}{c}\psi_{+}\psi_{-}\end{array}\right) =D^{(s)}(i\sigma_{2})\left( \begin{array}{c}\psi_{-}^{*}\psi_{+}^{*}\end{array}\right) ,\,\,\,\,\,\pm \mbox{=(anti)particle\thinspace \thinspace doubling}
[|D|^m,x] = |D|^mx-\sum_{k=0}^m(-1)^{m-k}\binom{m}{k}|D|^k\delta^{m-k}(x) = \sum_{k=0}^{m-1}(-1)^{m-k-1}\binom{m}{k}|D|^k\delta^{m-k}(x).
\int\limits_Y \left(\int\limits_0^T f(g_t y) \, dt \right)^2\,d\mu(y) \leqslant QT \|f\|_B^2 C> 1, \int\limits_Y \left(\int\limits_0^T f(g_t y) \, dt \right)^2\,d\mu(y) \leqslant QT \log T \|f\|_B^2 C=1, \int\limits_Y \left(\int\limits_0^T f(g_t y) \, dt \right)^2\,d\mu(y) \leqslant QT^{2-C}\|f\|_B^2 C< 1.
y(x) =\sum_{n,j\geq0}\sum_{k\geq1}(-1)^{k}a_{k,j}B_{n,k}(f_{1,0},\ldots,f_{n-k+1,0})\frac{x^{n+j}}{n!j!}=\sum_{m\geq n\geq 0}\binom{m}{n}\left\{\sum_{k\geq1}(-1)^{k}a_{k,m-n}B_{n,k}(f_{1,0},\ldots,f_{n-k+1,0})\right\}\frac{x^m}{m!}.
\delta \leq |T(e_\xi)(x_\xi,y_\xi)|=|T(e_\xi)(x_\xi,y_\xi)-T(e_\eta)(x_\xi,y_\eta)| &=|R(e_\xi\restriction_{\alpha\times \{y_\xi\}})(x_\xi)-R(e_\eta\restriction_{\alpha\times \{y_\eta\}})(x_\xi)| &\leq\|R(e_\xi\restriction_{\alpha\times \{y_\xi\}})-R(e_\eta\restriction_{\alpha\times \{y_\eta\}})\|<\delta,
\|a=b\| \wedge \|t(a)=s(a)\| \leq \|a=b\| \wedge \|t(a)=t(b)\|\wedge \|t(a)=s(a)\| \wedge \|s(a)=s(b)\| & \leq \|a=b\| \wedge \|t(b)=t(a)\|\wedge \|t(a)=s(a)\| \wedge \|s(a)=s(b)\| \leq \|t(b)=s(b)\|
\frac{1}{4 \pi^{2}}A ( \vec{x}) \wedge F ( \vec{x}) = \frac{1}{2 \pi^{2}}J^{3}( \vec{x}) \wedge J^{1}( \vec{x}) \wedge J^{2}( \vec{x}) = \frac{1}{2 4 \pi^{2}}\mathrm{t r}( g^{\dagger}( \vec{x}) d g ( \vec{x}) )^{3},
A_5^2 =kA_0+\frac{1}{2}(k+s+t+st)A_2+\frac{1}{2}(k+st)A_3, A_5A_6 =A_6A_5=kA_1+\frac{1}{2}(k+s+t+st)A_2+\frac{1}{2}(k+st)A_3, A_6^2 =kA_0+\frac{1}{2}(k+s+t+st)A_2+\frac{1}{2}(k+st)A_3.
\frac12\frac d{dt}{\|w\|_{L^2}^2}=\int w\nabla\cdot(w \nabla(-\Delta)^{-s}u)+\int w\nabla \tilde{u}\cdot\nabla(-\Delta)^{-s}w-\int w \tilde{u}(-\Delta)^{1-s}w\triangleq I_1+I_2+I_3.
[\Delta\vartheta_a^{(1)},P_{\rm F}]P_{\rm F}=(\Delta\vartheta_a^{(1)}P_{\rm F}-P_{\rm F}\vartheta_a^{(1)})P_{\rm F}&=(\Delta\vartheta_a^{(1)}-P_{\rm F}\vartheta_a^{(1)})P_{\rm F}=(1-P_{\rm F})\vartheta_a^{(1)}P_{\rm F}.
{\bf R}_u(p,e) =-\sum_{\alpha,\beta}{\rm tr}(R_{\alpha\beta}T_{\alpha_{\flat}(u)})e_{\beta},{\bf H}_u(p,e) =\sum_{\alpha}\sigma_{\alpha_{\flat}(u)}e_{\alpha},{\bf S}_u(p,e) =\sum_{\alpha}{\rm tr}(A_{\alpha}T_u)e_{\alpha},{\bf W}_u(p,e) =\sum_{\alpha}{\rm div}_E({\rm div}_E T_{\alpha_{\flat}(u)})e_{\alpha},
\Re \langle \Lambda_\varphi(b), \Lambda_\varphi(a) \rangle_\varphi = \lim_{n \to \omega}\Re \langle \Lambda_\varphi(u_n^* a u_n), \Lambda_\varphi(a)\rangle_\varphi &= \lim_{n \to \omega}\frac12(\|u_n^* a u_n\|_\varphi^2 + \|a \|_\varphi^2 - \|u_n^* a u_n - a\|_\varphi^2) &= \|a\|_\varphi^2 - \frac12 \varepsilon^2 < \|a\|_\varphi^2.
N|\psi_{n}\rangle =(\nu_0 + n)|\psi_{n}\rangle, \quad a^{\dagger}|\psi_n \rangle =|\psi_{n+1}\rangle, \quad a |\psi_{n}\rangle =\lambda_{n}|\psi_{n-1}\rangle, \quad \lambda_{n}= [n+\nu_0]_q - [\nu_0]_q
z_{2,\delta}= O\left(\frac{\delta^\frac{4-n}{2}}{(1+|\frac{x'}{\delta}|)^{n-3}}\right) \partial B_1(0)\cap B_1(-e_n),z_{2,\delta}= O\left(\delta^\frac{n-2}{2}\right) \partial B_1(0) \cap B_1^c(-e_n).
_{m^{\prime}}\langle b d | \hat{H}| b d \rangle_{m}= \Big [ \frac{J_{m}( \zeta )}{m + \zeta}+ \frac{J_{K - m}( - \zeta )}{K - m - \zeta}\Big ] \delta_{m}^{m^{\prime}}+ S_{b d}( \zeta )
{v_{T}}\left( \lambda - \int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{\sqrt{{\bf k}^{2}+ v_{T}^{2}}}\frac{\sinh ( \beta \sqrt{{\bf k}^{2}+ v_{T}^{2}})}{\cosh ( \beta \sqrt{{\bf k}^{2}+ v_{T}^{2})}+ 1}\right) = 0 .
||\delta\dot{{\mathbf V}_i}||_{s,\ast,T}+||\delta\psi_i||_{H^s(\Gamma_T)}&\leq C\Big(||{f}_i||_{s+2,\ast,T}+||{g}_i||_{H^{s+2}(\Gamma_T)}&\quad+(||{f}_i||_{8,\ast,T}+||{g}_i||_{H^{8}(\Gamma_T)})||(\tilde{\mathbf U}^a+{\mathbf V}_{i+\frac{1}{2}}, \nabla(\Psi^a+\Psi_{i+\frac{1}{2}}))||_{s+4,\ast,T}\Big).
\int_{M\setminus\Sigma}\langle \tilde{\mathfrak{h}},\hat{\mathfrak{u}}_l \rangle=0=|\hat{h}^-_{0,l}|\int_0^{r_0}e^{-4|l|r}dr+\int_{M\setminus N_r}\langle \tilde{\mathfrak{h}},\hat{\mathfrak{u}}_l \rangle.
K_{\mu}\left( \begin{matrix}{{r_{k}}}\\{{s_{k}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{r_{k}}}\\{{s_{k}}}\\ \end{matrix}\right) + ( \, r_{k}\, q_{\mu}\, \, - s_{k}\, p_{\mu}\, ) \left( \begin{matrix}{{p_{\mu}}}\\{{q_{\mu}}}\\ \end{matrix}\right) \, .
\frac{1}{2}\left( \Upsilon_{N}(f,g)-2\Gamma_{N}(f,g)\right) =-\frac{1}{2}\sum_{k=1}^{N}\Phi_{k}^{(N)}(f)^{t}\cdot\mathbb{V}\left( \mathbb{G}[\mathcal{A}^{(k)}]\right) \cdot\Phi_{k}^{(N)}(g),
\frac{\partial}{\partial y}\Re{f_{\tau, \theta}(r + iy)}= \frac{2y \big(r - \frac{1}{2}(\tau + \frac{1}{\tau})\big) \big(\frac{1}{\tau}- \tau\big)}{\big((r - \frac{1}{\tau})^{2}+ y^{2}\big) \big((r - \tau)^{2}+ y^{2}\big)},
\widehat{L}( \lambda ) = \frac{1}{\sin \gamma}\left( \begin{array}{c c}{{\sinh [ \gamma ( \lambda + \mathrm{i}S^{3}) ]}}&{{\mathrm{i}\, S^{-}\sin \gamma}}\\{{\mathrm{i}\, S^{+}\sin \gamma}}&{{\sinh [ \gamma ( \lambda - \mathrm{i}S^{3}) ]}}\\ \end{array}\right) \, .
\chi^+_\lambda(gx)=\begin{cases}\chi_{\lambda_0}^+(g x_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}^+(x's')+\chi_{\lambda_0}^-(g x_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}^-(x's') \text{if}x's'\in\tilde{A}_{p(t-t_0)}, 0 \end{cases}
l_\sigma(u^{r+1};0,0) = l_\sigma(u;0,0) l_\sigma(u^r;0,0) = ( B u^{-1}B^{-1}; 1,0 ) ((Bu^{-1})^r B^{-r};r,0) & = ( B u^{-1}B^{-1}\left( \theta(1,0)(B) \theta(1,0)(u)^{-1}\right)^r \theta(1,0)(B)^{-r}; r+1,0) & = (B u^{-1}B^{-1}\left( B (BuB^{-1})^{-1}\right)^r B^{-r}; r+1,0) = ( (Bu^{-1})^{r+1}B^{-(r+1)}; r,0).
\sum_{j=1}^{\infty}(2^{j}\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\mathbf{K}^{\varepsilon,n}_{e,\rho}(r\theta)|d\mathcal{H}^{d-1}(\theta)\lesssim 1.
\int d U \Upsilon_{\sigma}(U) \Upsilon_{\tau}(U^{\dagger}) = \delta_{T_\sigma, T_{\tau}}C_\sigma= \prod_{l}\delta_{\sigma^{(l)},\tau^{(l)}}C_{\sigma^{(l)}}=\prod_{l}\int d U_l\Upsilon_{\sigma^{(l)}}(U_l)\Upsilon_{\tau^{(l)}}(U_l^{\dagger}).
\begin{array}{c}\lambda_1 \rightarrow e^{i ( \phi_1 - \phi_2 - \phi_3)/2}\lambda_1, \lambda_2 \rightarrow e^{i ( -\phi_1 + \phi_2 - \phi_3)/2}\lambda_2, \lambda_3 \rightarrow e^{i ( -\phi_1 - \phi_2 + \phi_3)/2}\lambda_3, \lambda_4 \rightarrow e^{i ( \phi_1 + \phi_2 + \phi_3)/2}\lambda_4.\end{array}
|S_{a,b,c,x',y',z'}(\psi_p,\psi_p';\tilde{c},w_{G_4})| \leq C p^{d-h+f+s+\frac{a+b+c}{2}+3m}=C p^{d+f+s+(t-h)/2+3m}\leq C p^{r+s+t/2+3m}.
L(x^{k+1},\lambda)-L(x,\lambda^{k+1}) = L(x^{k+1},\lambda)-L(x^{k+1},\lambda^{k+1})+ L(x^{k+1},\lambda^{k+1})-L(x,\lambda^{k+1}) \leq \langle F^{k+1}, z^{k+1}-z\rangle+\frac{L}{2}\|z^{k}-z^{k+1}\|_2^2 \ ,
\lim_{m\rightarrow 0}Im\,T_{1}(t)=-\lim_{m\rightarrow 0}\frac{m^{2}}{4t^{2}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}\frac{1}{2}}\theta(t-4m^{2})=-\frac{1}{24}\delta(t) \, .
\mathbb{E}\left[ K' \right] \stackrel{(a)}{=}\lambda_{\mathrm{u}}\mathbb{E}\left[\left| \mathcal{C}_{i}^{\mathrm{E}}\right| \right] - \lambda_{\mathrm{u}}\mathbb{E}\left[\left| \mathcal{C}_{i}\right| \right]= K
\tilde{x}_{k+1}^{(i)}= \overline{x}_{k+1}^{(i)}\hat x_{k+1}^{(i)}= \hat x_{k}^{(i)}- \frac{1-\theta_k/\theta_0}{c_k}(\tilde{x}_{k+1}^{(i)}- \tilde{x}_{k}^{(i)})
T \approx (m t^{3(1-\delta)/2})^{-1/2}\left( D_1(k) \exp\left[-i\frac{2m t^{3(1-\delta)/2}}{3(1-\delta)}\right]+D_2(k) \exp\left[i \frac{2m t^{3(1-\delta)/2}}{3(1-\delta)}\right] \right),
\tau_0 := \eta(\mathsf{A}_0,\mathsf{A}_1)\geq 2^{-5}\,\mathsf{q}^{-1}\,\pmb{\eta}\,,\quad\qquad\tau_{\mathsf{M}+1}:= \eta(\mathsf{A}_{\mathsf{M}+1},\mathsf{A}_{\mathsf{M}+2})\geq 2^{-5}\,\mathsf{q}^{-1}\,\pmb{\eta}\,,