harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Vert \left(\left[\phi\,;\,\mathbb H\right]\Psi_{xx}\right)_x\Vert_{H^m(\mathbb T)}\le C\Vert \left[\phi\,;\,\mathbb H\right]\Psi_{xx}\Vert_{H^{m+1}(\mathbb T)}&\le C\Vert\partial^{m+1}_x\phi\Vert_{L^2(\mathbb T)}\Vert\Psi_{xx}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{x}\Vert_{H^m(\mathbb T)}\Vert\psi_x\Vert_{H^2(\mathbb T)}\,.
\left(A_1(m,k), A_2(m,k)\right) \quad:= \begin{cases}\left(27.640k\varphi(m)+40.160k+0.054, 0.017 k^2+ 0.033 k+0.011 \right) 3 \leq m \leq 60 \left(\left(0.001\varphi(m)+1.502\right)k, 0.373k^2+0.741 k+0.191\right) m \geq 61 \end{cases},
a_{11}-\frac{\sigma}{3}= \sum_k\overline{R}_{1k1k}-\frac{\sigma}{3}=\sum_k\overline{R}_{1k1k}-\frac{1}{3}\sum_{i,j}\overline{R}_{ijij}&=\frac{1}{3}\left(\overline{R}_{1212}+ \overline{R}_{1313}+ \overline{R}_{1414}\right)-\frac{2}{3}\left(\overline{R}_{2323}+ \overline{R}_{2424}+ \overline{R}_{3434}\right)\le 1
d =\sum_{i=1}^{n}\big(E^{(i)}_1 -A^{(i)}_1 +E^{(i)}_2 -A^{(i)}_2+1\big) &=\sum_{i=1}^{n}\big((B-v_{n-i+1}+1) +A-(u_{n-i+1}-1)+1\big) &=(A+B+3)n-\sum_{i=1}^{n}(u_i+v_i).
P(\lceil \frac{n}{\log n}\rceil<T_0\le n) = P(T_0>\frac{n}{\log n})-P(T_0> n) \le (\frac{\pi}{\log n-\log \log n}+\frac{C\log \log n}{(\log n-\log \log n)^2})-(\frac{\pi}{\log n}-\frac{C\log \log n}{(\log n)^2})\le \frac{C'\log \log n}{(\log n)^2},
\Gamma_{N \rightarrow \infty}[\sigma,\rho, \pi] = \int dx \,\sqrt{g}\, \frac{N}{2\lambda_B}(\sigma^2 + \rho^2+ \pi^2 ) -N \ln \det \left( \not{\!\! D}+ \sigma + i \tau^1\rho + i\gamma^5\pi \right).
2\left(n(i+q)-m\left(j+\frac q2\right)\right)d^1_{r,s}(m+n,i+j) =\left((n+r)(i+q)-m\left(j+s+\frac q2\right)\right)d^1_{r,s}(n,j), q(d^1_{r,s}(m,i)+d^1_{r,s}(n,j)) =0.
\begin{aligned}|\psi(x, t)| \lesssim \|f\|_{*, \beta, 2s+\alpha}\left(\sum_{j=1}^k\frac{t^{-\beta}}{1+|y_j|^{\alpha}}\right) &+ e^{-\delta(t-t_0)}\|h\|_{L^\infty(\mathbb{R}^n)}+ t^{-\beta}\|\tau^\beta g(x, \tau)\|_{L^\infty((\mathbb{R}^n\setminus\Omega)\times (t_0,\infty))},\end{aligned}
Hom_{[R\slash G^{\sigma}](\mathbb{F}_q)}\left( (r^{\prime}, \varepsilon^{\prime}) , (r^{\prime \prime}, \varepsilon^{\prime\prime}) \right) = \delta_{\varepsilon^{\prime},\varepsilon^{\prime\prime}}\mbox{Trans}_{G_e^{\sigma, \varepsilon^{\prime}}(\mathbb{F}_q)}(r^{\prime}, r^{\prime \prime}).
[\begin{matrix}M_{11}\\M_{22}\\M_{12}\end{matrix}]=-\{\int_{-h}^{h}x_{3}^{2}[\begin{matrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\ 0&0&C_{66}\end{matrix}]dx_{3}\}[\begin{matrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1})\end{matrix}]
f= \sum_{I<J, |I|<d}a_I\overline{Y}^I + a_J\overline{Y}^J + \sum_{J<I, |I|<d}a_I\overline{Y}^I & \sum_{|I|\geq d}\left\{\sum_{K<J, |I|<d}b_{IK}a_K + b_{IJ}a_J + \sum_{J<K, |K|<d}b_{IK}a_K\right\}\overline{Y}^I.
\pi_{1}(\phi_{\theta})Se_{n}^{2}= e^{-i \left( n + 1 + m + \frac{\lambda}{2}\right) \theta}Se_{n}^{2},\,\, n \geq 0;\,\, \pi_{1}(\phi_{\theta})Se_{-n}^{2}= e^{i \left( n - 1 + k - \frac{\lambda}{2}\right) \theta}Se_{-n}^{2},\,\, n\geq 1
E_\sigma = A_\sigma + B_{\sigma \nu}\ddot q^\nu - \frac{d}{dt}(2C_{\sigma \nu}\ddot q^\nu )= A_\sigma + \bar B_{\sigma \nu}\ddot q^\nu - 2 \frac{\partial C_{\sigma \nu}}{\partial \dot q^\rho}\ddot q^\nu \ddot q^\rho - 2 C_{\sigma \nu}\dddot q^\nu.
K_{p}( z ) \stackrel{( z \rightarrow 0 )}{\sim}\frac{1}{2}\, \left[ \Gamma ( p ) \left( \frac{z}{2}\right)^{- p}+ \Gamma ( - p ) \left( \frac{z}{2}\right)^{p}\right] \, \left[ 1 + O ( z^{2}) \right] \;
[L(z),L(x)]=(\frac{\partial}{\partial x}L(x))w^{-1}\delta(\frac{z}{x})-2L(x)x^{-1}\frac{\partial}{\partial z}\delta(\frac{z}{x})-\frac{1}{12}cx^{-1}(\frac{\partial}{\partial z})^{3}\delta(\frac{z}{x})
\Delta_{\frac{1}{2}}{w}^k=-Q_k^{-1}\Phi_{\mu_k}({w}^k), &\|\Delta_{\frac{1}{2}}{w}^k\|\le \|Q_k^{-1}\|_F\|\Phi_{\mu_k}({w}^k)\|=O(\|\Phi_{\mu_k}({w}^k)\|),
\frac{d}{dt}\left(\frac{\partial{\mathfrak l}}{\partial \dot{x}^i}\right)-\frac{\partial{\mathfrak l}}{\partial x^i}=0, \frac{d}{dt}\left(\frac{\partial{\mathfrak l}}{\partial \xi}\right)=\hbox{ad}^*_{\xi}\frac{\partial{\mathfrak l}}{\partial \xi}, \dot{x}^{i}= \frac{dx^{i}}{dt}.
\frac{p}{q}- \sum_{i=1}^{k+1}\frac{1}{a_i}= \frac{p}{q}- \sum_{i=1}^{k}\frac{1}{a_i}- \frac{1}{a_{k+1}}= \frac{1}{q\prod_{i=1}^k a_i}- \frac{1}{q\prod_{i=1}^k a_i + 1}= \frac{1}{q\prod_{i=1}^k a_i \left( q\prod_{i=1}^k a_i + 1\right)}& = \frac{1}{q \prod_{i=1}^{k+1}a_i}.
\big\langle \nu_{23}, \nu_{234}, \nu | Q^{(12)}|\nu_{12}, \nu_{123},\nu \big\rangle = \hat{\Lambda}_2^n \langle \nu_{23}, \nu_{234}, \nu |\nu_{12}, \nu_{123},\nu \rangle, \big\langle \nu_{23}, \nu_{234}, \nu | Q^{(123)}|\nu_{12}, \nu_{123},\nu \big\rangle = \hat{\Lambda}_1^n\langle \nu_{23}, \nu_{234}, \nu |\nu_{12}, \nu_{123}.\nu \rangle.
V_0 (1,(x_0,x_1),(m_0,m_1)) = \min_{\gamma \in \{0,1\}}\Big\{c_1 \gamma -x (m_1) & + \mathbb{E}_0 \left[ V_0(2, (x_0,x_1,\Psi(x_1,\gamma,\xi_2),(m_0,m_1,\mu_2))) | X^{(1)}=(x_0,x_1), \mu^{(1)}= (m_0,m_1)\right]\Big\}.
\psi_m(n) = n + \frac{1}{(m-2)!}f_m(n)\,, \intertext{with $f_m(n)$ given by}f_m(n) = \prod_{k=-1}^{m-2}(n + k) = \prod_{i=1}^m (n - \alpha_i) = \sum_{k=0}^m b_{m,m-k}n^k\,, \intertext{where the $\alpha_i$ are of course the integer roots of thispolynomial:}\alpha_i = -(i - 2), 1 \le i \le m\,.
\begin{aligned}(B_5)_j^j(0) =\frac{4}{3}c_2^2 \mathrm{i}\sum_{j_2\in S, j_2+j\in S}j_2^3\,\lvert j_2 \rvert \xi_{j_2}+8 c_2 c_3 \mathrm{i}\sum_{j_2\in S, j_2+j\in S}j_2 \,\lvert j_2 \rvert \xi_{j_2}&+12 c_3^2 \mathrm{i}\sum_{j_2\in S, j_2+j\in S}j_2^{-1}\,\lvert j_2 \rvert \xi_{j_2}.\end{aligned}
\begin{array}{r c l}{{T_{+}}}&{{=}}&{{-{\frac{\sqrt 3}{2}}( X_{( 1 1 )}+ X_{( - 1 - 1 )}) +{\frac{1}{2}}( X_{( 3 1 )}+ X_{( - 3 - 1 )}) \, ,}}\\{{T_{-}}}&{{=}}&{{{\frac{\sqrt 3}{2}}( X_{( 2 1 )}+ X_{( - 2 - 1 )}) -{\frac{1}{2}}( X_{( 0 1 )}+ X_{( 0 - 1 )}) \, ,}}\\{{T_{3}}}&{{=}}&{{-{\frac{\sqrt{3}}{2}}( X_{( 1 0 )}- X_{( - 1 0 )}) +{\frac{1}{2}}( X_{( 3 2 )}- X_{( - 3 - 2 )}) \, ,}}\\ \end{array}
D^{(m)}_n(x,y;a,b,c)=\frac{1}{(\ln a+\ln b)^n}\sum_{k=0}^m\binom{m}{k}{E}^{(-k)}_n(x;a,b,c)\left(\frac{y\ln c+\ln a}{\ln a +\ln b}\right)^{m-k}.
Y^b_t=Y^b_{(T-\varepsilon) \vee t}-\int_t^{(T-\varepsilon) \vee t}g^b(s,Y^b_s+\xi_s^a,Z^b_s+\gamma_s)-D_s+ \tilde\lambda_s Y^b_sds +\int_t^{(T-\varepsilon) \vee t}Z^b_s\cdot dW_s.
g((R(X,Y)A)Z,W)= c(3+||\xi^\perp||)\{g(Y,AZ)g(X,W)-g(X,AZ)g(Y,W) -g( Y,Z)g(AX,W)+g(X,Z)g(AY,W)\}+2c\{g(\phi Y,AZ)g(\phi X,W)-g(\phi X,AZ)g(\phi Y,W) -g(\phi Y,Z)g(A\phi X,W)+g(\phi X,Z)g(A\phi Y,W) & -4g(\phi X,Y)g(\phi AZ,W)\}
m(\xi,\eta) = O\big( f(|\xi|,|\xi-\eta|,|\eta|) \big) \, \Longleftrightarrow \,{\| m^{k,k_1,k_2}(\xi,\eta) \|}_{S^\infty}\lesssim f(2^k,2^{k_1},2^{k_2}) \mathbf{1}_{\mathcal{X}}(k,k_1,k_2)
\psi_{k}( z , \bar{z}) ={\frac{1}{\sqrt 2}}\left[ \left({\frac{f_{n , m}( z )}{\overline{f}_{n , m}( z )}}\right)^{k}+ \left({\frac{\overline{{f}}_{n , m}( z )}{\f_{n , m}( z )}}\right)^{k}\right] , \qquad k = 0 , 1 , 2 , \ldots ,
\mathcal{U}^{\llcorner}_{\rho_1, \rho_2}(\pi \vert \nu, \mu, \kappa) =\mathcal{U}^{\llcorner}_{\rho_1, \rho_2}(\pi \vert \nu, \kappa) = \frac{s_{\pi/\nu}(\rho_2) s_{\pi/\kappa}(\rho_1)}{\sum_{\lambda}s_{\lambda/\nu}(\rho_2) s_{\lambda/\kappa}(\rho_1)}.
= \int D \Phi_{a}D \Phi_{a}^{*}\exp \left\{- \int d^{4}x \sum_{a = 1}^{3}\left[ \frac 1 2 \left| \left( \partial_{\mu}- 2 i g_{m}{\bf e}_{a}{\bf B}_{\mu}\right) \Phi_{a}\right|^{2}+ \lambda \left( | \Phi_{a}|^{2}- \eta^{2}\right)^{2}\right] \right\},
\Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n})=\overrightarrow{\left(\prod_{\ell=n}^{3}\right)}Q_{j_{\ell-1}}(x_{\ell}, \underline{x}_{[\ell-1]}) m_{j_1}(x_2,x_1) v_{s},
\mathcal{B}[\beta b]^*\nabla_X\psi = b \nabla_X \nabla_X\cdot(b\nabla_X\psi) + b \nabla_X(b \nabla_X\cdot(h_b\nabla_X\psi)) + 2b\nabla_X(h_b\nabla_X b\cdot\nabla_X\psi).
\left\|d_G(\cdot,y_0)^{-2}E^{y_0,ij}_2\right\|_{L^\infty(\mathcal{B}_{1}^+(y_0))}+\left[d_G(\cdot,y_0)^{-(2-\epsilon)}E^{y_0,ij}_2\right]_{\dot{C}^{0,\epsilon}_\ast(\mathcal{B}_{1}^+(y_0))}\leq C.
3 (k-1){\rm e}^{18A}[ k r A'' + k^2 A' + 6 k r A'^2 - 18 r^2A'^3 ] + 2 b^2 r [ 3 r^2 A'' - 3 (2k-1) r A' + 18 r^2 A'^2 -k(k-1) ] = 0.
\;\|A^{(r_{k+1}-s)/2}w_k(t)\|_{L^2}^2 +\int_{t_k}^{t}\|A^{r_{k+1}/2}w_k\|_{L^2}^2 \leq \; C\|u_0\|_{D(A)}^2\left(1+\int_{t_k}^t \|A^{r_k/2}u_*\|_{L^2}^2 \,d\tau\right),
S_{E}=\int d^{d}{\bf x}\left\{-\bar{\psi}\left( \partial\!\!\!\slash+ g_{B}\left(\sigma+i\gamma_{S}\pi \right)\right)\psi+\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\pi)^{2}-\frac{1}{2}m_{B}^{2}\left({\sigma}^{2}+{\pi}^{2}\right) +\frac{\lambda_{B}}{4}\left({\sigma}^{2}+{\pi}^{2}\right)^{2}\right\}.
(1+\partial_t\mathrm L)^2 \leqslant 1+s^2 \frac{1}{2\pi}\oint_{S^1}\left(s^2-\partial_t\mathrm L^2\right)\,\mathrm dt <\varepsilon \frac{1}{2\pi}\oint_{S^1}\partial_t\mathrm L\,\mathrm dt = 0
\aligned(\ref{combine1775})= \left(\hat d_{2}^{i+1}\circ\left(-\frak h_{ab}^{i+1}\circ \psi_1^{i+1 i}+ \frak h_{a}^{i+1 i}- \frak h_{b}^{i+1 i}\right)\right)_{\alpha'_2\alpha_1}&\quad+\left(\left(-\frak h_{ab}^{i+1}\circ \psi_1^{i+1 i}+ \frak h_a^{i+1 i}- \frak h_b^{i+1 i}\right) \circ \hat d_{1}^{i}\right)_{\alpha'_2\alpha_1}.\endaligned
\|(\hat{Q}_r-Q_r)ER_r\|_2 =\sqrt{\sum_{s\neq r}\frac{\|\hat{Q}_r-Q_r)EQ_s\|_2^2}{(\mu_r-\mu_s)^2}}&\leq \sqrt{\sum_{\substack{s<r_0: s\neq r}}\frac{\|\hat{Q}_r-Q_r)EQ_s\|_2^2}{(\mu_r-\mu_s)^2}+\frac{\|\hat{Q}_r-Q_r)EQ_{\geq r_0}\|_2^2}{(\mu_r-\mu_{r_0})^2}}.
v(\psi\cdot g_{uuv}) =g_v\cdot g_{uuv}=(g_v\cdot g_{uv})_u-g_{uv}\cdot g_{uv}=\frac{(g_v\cdot g_{v})_{uu}}2-g_{uv}\cdot g_{uv}=\frac{(f_v\cdot f_{v})_{uu}}2-g_{uv}\cdot g_{uv}=v^2\frac{(\phi\cdot \phi)_{uu}}2-v^2 \psi_{u}\cdot \psi_{u},
u(z) = a\kern 1.5pt \hbox{Re\kern 1pt}(z^{-k}) +b\kern 1.5pt \hbox{Im\kern 1pt}(z^{-k}) ~~\Longrightarrow ~~ \nabla u(z) = -k\kern .7pt (a+bi\kern .7pt )\kern 1pt /\kern 1.5pt \overline{z}^{\kern 2pt k+1}.
p_\mu{\blacktriangleright}\big(\hat{f}\hat{g}\big)=\big[p_\mu,\hat{f}\hat{g}\big]{\blacktriangleright}1=\big(\big[p_\mu,\hat{f}\big]\hat{g}+\hat{f}[p_\mu,\hat{g}]\big){\blacktriangleright}1=\big[p_\mu,\hat{f}\big]{\blacktriangleright}\hat{g}+\hat{f}p_\mu{\blacktriangleright}\hat{g}.
\alpha =(\mathfrak{b}(u,u)+\delta\mathfrak{b}(v,v))+\eta(\mathfrak{b}(u,v)+\mathfrak{b}(v,u)+\mathfrak{b}(v,v)) &=(\mathfrak{b}(u,u)+\delta\mathfrak{b}(v,v))+\eta\mathfrak{b}(v,v).
k_{\beta}( \mathrm{\bf ~ 1}, e^{\varphi^{i}T_{i}}) = \frac{M}{( 2 \pi \beta )^{\frac{D}{2}}}\sum_{\nu}\prod_{\alpha > 0}\frac{< \varphi + 2 \pi \nu , \alpha > / 2}{\sin < \varphi + 2 \pi \nu , \alpha > / 2}e^{- \frac{1}{2 \beta}< \varphi + 2 \pi \nu , \, \varphi + 2 \pi \nu >}
{\frac{\ddot{a}}{a}}= -{\frac{1}{L^{2}}}\left[ p ( \bar{E}^{2}- \bar{P}_{\phi}^{2}) \left({\frac{L}{a}}\right)^{2 ( p + 1 )}- ( p - 1 ) \bar{E}^{2}\left({\frac{L}{a}}\right)^{2 p}+ ( p - 1 ) \bar{P}_{\phi}\left({\frac{L}{a}}\right)^{p + 1}\right] ,
e^{- i \sum_{j = 1}^{2}( P_{j}\cdot x_{j}- Q_{j}\cdot y_{j})}= e^{- i X \cdot \sum_{j = 1}^{2}( P_{j}- Q_{j})}e^{- i \sum_{j = 1}^{2}( P_{j}\cdot x_{j}^{\prime}- Q_{j}\cdot y_{j}^{\prime})}
\bar{\alpha}: = \left( \begin{matrix}{{\bar{d}_{1}}}\\{{\bar{d}_{2}}}\\ \end{matrix}\right) , \quad \bar{\beta}: = \left( \begin{matrix}{{\bar{d}_{1}}}\\{{- \bar{d}_{2}}}\\ \end{matrix}\right) , \quad \bar{D}_{0}: = 2 \left( \begin{matrix}{{\bar{d}_{1}}}&{{0}}\\{{0}}&{{\bar{d}_{2}}}\\ \end{matrix}\right) .
A ( p^{2}) = 1 + \frac{1}{4 p^{2}}\int \frac{d^{3}k}{( 2 \pi )^{3}}T r \lbrack ( i \gamma \cdot k ) \gamma_{\mu}G ( k ) \Gamma_{\nu}( p , k ) D_{\mu \nu}( p - k ) \rbrack .
\phi_0(t) =x_0(t-a)^{\gamma-1},t\in(a,a+l], \phi_n(t)=\phi_0(t)+ \int_{a}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,\phi_{n-1}(s))ds,t\in(a,a+l],n=1,2,\cdots.
\mathbb{E}\left[f\left(Z_{t}^{2}\right)\right]=\int_{\mathbb{R}_{\geqslant0}}f(y)\mu_{Z_{t}^{2}}(\mathrm{d}y)=e^{-\lambda_{t}}\sum_{n=0}^{\infty}\frac{\lambda_{t}^{n}}{n!}\int_{\mathbb{R}_{\geqslant0}}f(y)\rho_{t}^{\ast n}(\mathrm{d}y)<\infty.
\|g_0^{j,t}\|^2_{L^2(C^{j,t}_k, \mu_2)}\leq \frac{C}{\omega(B^g(\sqrt{t}))}\int_{B^g(x^t_j, 4\sqrt{t})}\int_{B^g(x^t_j, 4\sqrt{t})}|f(x)-f(y) |^2 d\mu_2(x)d\mu_2(y).
= \epsilon^{i j}({\frac{\mu}{2}}A^{j}- f^{j}) \dot{A}^{i}-{\frac{1}{2}}f^{i}f^{i}-{\frac{1}{2}}( \epsilon^{i j}\partial^{i}A^{j})^{2}+ A^{0}( \mu \epsilon^{i j}\partial^{i}A^{j}- \epsilon^{i j}\partial^{i}f^{j}) ,
\limsup_{\epsilon\to 0}\limsup_{L\to\infty}\sup_{\mu}\frac 1{L^d}\sum_{i\in\mathbb T_L^d}\sum_{k=1}^d \sum_{\eta\in\Omega_L}\mu(\eta)\;\!\Bigl|\hat\chi_{i,i+e_k}^{\lfloor \epsilon L\rfloor}(\delta_\eta) - \hat\chi_{i,i+e_k}(\nu_{\eta^{\lfloor \epsilon L\rfloor}(i)})\Bigr|= 0,
S = \int d^{2}x \mathrm{t r}\left( - \frac{1}{2}F_{\mu \nu}F^{\mu \nu}+ D_{\mu}\phi D^{\mu}\phi + i \bar{\Psi}\gamma^{\mu}D_{\mu}\Psi - i g \phi \bar{\Psi}\gamma_{5}\Psi \right) ,
\frac{1}{4\pi|z_\ell-z_j|}=\frac{1}{|\mathbf{B}_\ell|}\frac{1}{|\mathbf{B}_j|}\int_{\mathbf{B}_\ell}\int_{\mathbf{B}_j}\Phi_{0}(x, y)\,dx\,dy=\frac{36}{\pi^{2}\, d^{6}}\int_{\mathbf{B}_\ell}\int_{\mathbf{B}_j}\Phi_{0}(x, y)\,dx\,dy,
\prod_{i=1}^{\ell+L+1}\b{x}_i^{p_i}\times\mathcal{C}^{\lambda}_{\mu\nu}(x)=\oint_{w_1}\cdots\oint_{w_\ell}\frac{\prod_{1 \leq i<j \leq \ell}(w_j - w_i)}{\prod_{i=1}^{\ell}\prod_{j=1}^{\kappa_i}(w_i - \b{x}_j)}\prod_{i=1}^{\ell}w_i^{L+1}s_{\lambda/\mu}(\b{w}_1,\dots,\b{w}_\ell),
\begin{array}{l}(\epsilon_1,\epsilon_2,\epsilon_3) =\begin{cases}(\min\{n-r+k_2,n-2r+2k_1\}+\hat e_1, n-r+k_3, \mbox{if~}n-r+k_1>r, ~~~~~~~n-2r+k_1+k_2) (\min\{k_1, r-k_1+k_2\}+\hat e_1, r-k_1+k_3, k_2) \mbox{if~}n-r+k_1\le r,\end{cases}\end{array}
\beta_1=D' V^2\Bigl(\frac{a'}{q_1q_2}+\frac{b_1}{d_1d_2d_3}\Bigr),\beta_2=\frac{a'}{q_1q_2}+\frac{b_1}{d_1d_2d_3}+\frac{b_2}{d_2d_3}.
u_{i+1/2}(\omega,t) := \frac{1}{\Delta \theta}\int_{\theta_i}^{\theta_{i+1}}u(\theta,\omega,t)\,d\theta, \tilde\rho_{i+1/2}:= (1 - \delta_{i+1/2})\rho_{i+1}+ \delta_{i+1/2}\rho_i,
\begin{array}{r c l}{S_{3}}&{=}&{r_{1}( - \gamma ) r_{2}( - 2 \gamma ) r_{1}( - \gamma )}\\{\tilde{S}_{3}}&{=}&{r_{5}( - \gamma ) r_{4}( - 2 \gamma ) r_{5}( - \gamma ) .}\\ \end{array}
\{v\}= \frac{1}{2}(v^+ + v^-), [v] = v^+\boldsymbol{n}^+ + v^-\boldsymbol{n}^-, \{\boldsymbol{q}\}= \frac{1}{2}(\boldsymbol{q}^+ + \boldsymbol{q}^-), [\boldsymbol{q}] = \boldsymbol{q}^+\cdot \boldsymbol{n}^+ + \boldsymbol{q}^-\cdot \boldsymbol{n}^-.
v(y_0+\rho\eta) =v(y_0)+A_1\rho+A_2\rho^{2}+o(\rho^{2}), v(z_0+\rho\eta) =v(z_0)+B_1\rho+B_2\rho^{2}+o(\rho^{2}), u(x_0+\rho\eta) =u(x_0)+C_1\rho+C_2\rho^{2}+o(\rho^{2})
H_1(\tilde{\xi}) = \tilde{\xi}^3 + \frac{1}{2}(1- 3 \sqrt{2}+ 2 \sqrt{3}+ \sqrt{6}) \tilde{\xi}^2+ \frac{\sqrt{3}}{4}(8 +4 \sqrt{2}-5 \sqrt{3}-4 \sqrt{6})\tilde{\xi}& +\frac{3 \sqrt{3}}{8}(\sqrt{2}+1)(\sqrt{3}-\sqrt{2})(3 + 2 \sqrt{2}- 2 \sqrt{3}),
\| f_{hl}\|_{L^2}^2 \lesssim \ \sum_k \| 2^{(\sigma+s) k}R_k \|_{L^\infty}^2\| 2^{-\sigma k}|D|^\sigma w_{<k}\|_{L^2}^2\lesssim \sup_k \| 2^{(\sigma+s) k}R_k \|_{L^\infty}^2\sum_k \| 2^{-\sigma k}|D|^\sigma w_{<k}\|_{L^2}^2 \lesssim \ \Vert |D|^{\sigma+s}R\Vert_{BMO}^2 \Vert w\Vert_{L^2}^2.
\dfrac{p}{p+1}\displaystyle\sum_{n=1}^\infty \dfrac{2^{\omega(\gcd(n,p))}\cdot 2r_{Q^\ast}(n)^2}{n}\displaystyle\sum_{d=1}^\infty \psi \left( d\sqrt{\dfrac{n}{p}}\right) \leq \frac{1}{\min Q^{*}}+ 3216.6524 \frac{M(25.09p^{35/6})}{p^{1/4}}.
\int_I |S(\theta; X)|^2 \, d\theta = \sum_{p_1,p_2 \in \mathcal N_\gamma(X)}(\log p_1)(\log p_2) \int_I e(\theta(p_1^c - p_2^c)) \, d\theta &\ll L^2\sum_{n_1, n_2 \in \mathcal N_\gamma(X)}\min\big( |I|, |n_1^c - n_2^c|^{-1}\big) &\ll |I|X^{\gamma}L^2 + X^{1-c}L^2 \sum_{\substack{n_1, n_2 \in \mathcal N_\gamma(X) n_1 < n_2}}(n_2 - n_1)^{-1},
C^m m! \|Q(\cdot, y)\|_{\infty}^m \prod_{k=1}^d \sqrt{(m \alpha^{(k)})!}\int_{\Delta^{(m)}(0,t)}\prod_{k=1}^d |s_j - s_{j-1}|^{- H ( 1 + 2 \alpha^{(k)})}ds \leq \frac{C^m m! \prod_{k=1}^d \sqrt{(m \alpha^{(k)})!}\|Q(\cdot, y)\|_{\infty}^m}{\Gamma( m ( 1 - H(d + 2 |\alpha|)) + 1}
\sum_{( n_{j})_{0}^{k}}\sum_{( X_{i})_{0}^{n}}\prod_{i = 0}^{n - 1}b ( X_{i + 1}, X_{i}) \prod_{j = 0}^{k}\prod_{i = m_{j - 1}+ 1}^{m_{j}}\chi ( X_{i}\in L X_{j}^{\prime}) =
\vec{V}\cdot\vec{Z}_{i}=V_{i}=\vec{V}^{T}\vec{Z}_{i}=\vec{Z}_{i}^{T}\vec{V}=proj_{\vec{Z}^{i}}(\vec{V})\cdot\vec{Z}_{i}=proj_{\vec{V}}(\vec{Z}^{i})\cdot\vec{V}
{J}^{1}= i \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\{{0}}&{{1}}&{{0}}\\ \end{array}\right) , \quad \quad{J}^{2}= i \left( \begin{array}{c c c}{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}\\ \end{array}\right) , \quad \quad{J}^{3}= i \left( \begin{array}{c c c}{{0}}&{{- 1}}&{{0}}\\{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right) ~ .
\mathcal{H}^\mathsf{Rob}_{\hbar,\kappa,K}=-a_{\hbar,\kappa,K}^{-1}(\tau)\partial_{\tau}\left(a_{\hbar,\kappa,K}(\tau)\partial_{\tau}\right)=-\partial^2_{\tau}+\frac{\hbar^2 \kappa-2\hbar^4K\tau}{a_{\hbar,\kappa,K}(\tau)}\partial_{\tau}\,,
Z(\beta)=\sum_{\sigma_{1},...,\sigma_{L}}e^{\beta J\sigma_{1}\sigma_{2}}e^{\beta J\sigma_{2}\sigma_{3}}\cdot\cdot\cdot e^{\beta J\sigma_{L-1}\sigma_{L}}=2\prod_{j=2}^{L}\sum_{\sigma_{j}^{\prime}}e^{\beta J\sigma_{j}^{\prime}}=2[e^{\beta J}+e^{-\beta J}]^{L-1}.
\|g^n_M-g_M\|_{C([0,T]; H^s_r(\mathbb R^d))}= \|g^n\chi_M-g\chi_M\|_{C([0,T]; H^s_r(\mathbb R^d))}& = \sup_{0\leq t\leq T}\|(g^n(t, \cdot)-g(t, \cdot))\chi_M\|_{H^s_r(\mathbb R^d)}& \leq c(M) \sup_{0\leq t\leq T}\|g^n(t, \cdot)-g(t, \cdot)\|_{H^s_r(\mathbb R^d)}&= c(M) \|g^n-g\|_{C([0,T]; H^s_r(\mathbb R^d))},
\mathcal{J}=\left(\begin{array}[c]{cc}\mathcal{J}_{+}0 0 \mathcal{J}_{-}\end{array}\right) ,\ \mathcal{L=}\left(\begin{array}[c]{cc}\mathcal{L}_{+}-\mathcal{B}\mathcal{B}\mathcal{B}\mathcal{L}_{-}-\mathcal{B}\end{array}\right)
{\rm Corr}_{(\frak X_{23},\widehat{\frak S}^{\epsilon}_{23}),P_1}\circ{\rm Corr}_{(\frak X_{12},\widehat{\frak S}^{\epsilon}_{12}),P_2}={\rm Corr}_{(\frak X_{13},\widehat{\frak S}^{\epsilon}_{13})}
\|I_\alpha f\|_{L^{p_\alpha}(\mathbb{H}^n)}=\|I_\alpha f_\epsilon\|_{L^{p_\alpha}(\mathbb{H}^n)},\quad\|f\|_{L^{q_\alpha}(\mathbb{H}^n)}=\|f_\epsilon\|_{L^{q_\alpha}(\mathbb{H}^n)},
{\frac{d}{d u}}W_{\gamma_{u}}^{R}= \oint d v \gamma_{u}^{' \mu}( v ) \dot{\gamma}_{u}^{\nu}( v ) U_{\gamma_{u}}^{R}( s , v ) F_{\mu \nu}( \gamma ( v ) ) U_{\gamma_{u}}^{R}( v , s ) ,
+ \sum_{i = 1}^{2 n}m_{i}a_{- m_{i}- 1}\prod_{j \neq i}a_{- m_{j}}a_{- m_{2 n + 1}}o d d ( k ) + k \prod_{i : = 1}^{2 n + 1}a_{- m_{i}}a_{- 1}e v e n ( k ) \Bigr ) \quad .
\varphi ( - \vec{r}_{1}, - \vec{r}_{2}, - t ) \longrightarrow \chi ( \vec{r}_{1}, \vec{r}_{2}, t ) , \chi ( - \vec{r}_{1}, - \vec{r}_{2}, - t ) \longrightarrow \varphi ( \vec{r}, \vec{r}, t )
M_{1}=\int_{-t/2}^{t/2}[-x_{3}\sigma_{12}e_{1}+x_{3}\sigma_{11}e_{2}]dx_{3}andM_{2}=\int_{-t/2}^{t/2}[-x_{3}\sigma_{22}e_{1}+x_{3}\sigma_{12}e_{2}]dx_{3}
\lim_{\theta\to 1}\Bigl(\Omega,S_{\theta ({\cal L}_0+\epsilon{\cal L}_1)}(\lambda f)\Omega\Bigr)=\lim_{\theta\to 1}\frac{\Bigl(\Omega,S_{\theta{\cal L}_0}(\theta\epsilon{\cal L}_1+\lambda f)\Omega\Bigr)}{\Bigl(\Omega,S_{\theta{\cal L}_0}(\theta\epsilon{\cal L}_1)\Omega\Bigr)}
\lim_{s\to 0}\mathbb K^{PIII}(u,v;s)=\mathbb{J}_\alpha(-u,-v), &\lim_{s\to +\infty}\frac{s^{\frac{2\eta}{3}}}{c_2}K^{PIII}\left(s^{\eta}\left(z_0 + s^{-\frac{\eta}{3}}\frac{u}{c_2}\right),s^{\eta}\left(z_0 + s^{-\frac{\eta}{3}}\frac{v}{c_2}\right);s\right)= \mathbb{A}(u,v),
\sum_{k=0}^{n}\binom{n}{k}\dbinom{m+k}{p}y^{n-k}f_{m-p+1+k}^{\left(\alpha \right)}\left( x\right) =\sum_{k=0}^{m}\binom{m}{k}\dbinom{n+k}{p}\left( -y\right)^{m-k}\left(f_{n-p+1+k}^{\left( \alpha \right)}\left( x+y\right) -yf_{n-p+k}^{\left( \alpha \right)}\left( x+y\right)\right).
(g^{\mu \nu}) = (g_{\mu \nu})^{-1}= \left(\begin{array}{cccc}1 0 0 0 0 \frac{1}{a(t)^2}0 0 0 0 \frac{\csc^2(\eta )}{a(t)^2}0 0 0 0 \frac{\sec^2(\eta )}{a(t)^2}\end{array}\right).
S_{e n / I , e^{\prime}n^{\prime}/ I}= i V \mathrm{e x p}\frac{2 i \pi}{k}\left[ e^{\prime}( \widetilde{n}- 1 / 2 ) + e ( \widetilde{n^{\prime}}- 1 / 2 ) \right] = \S_{e n / I , e^{\prime}n^{\prime}/ I I}
\mathbb{P}\left( f(M_\ell)=0, \M_\ell \in \mathcal{M}_\ell \right) \leq d\left(3 \sqrt{\ell}\epsilon_\ell\right)^{m n-1}\left( \frac{c_3}{\ell^{mn/2}}\right)^m \leq c_4 \frac{1}{\sqrt{\ell}},
\Gamma_{t}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) - g_{t}\sim g_{t}^{2}\gamma^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) + O ( g_{t}^{3}) .
V_t^i := e^{\int_0^t\Lambda\big (s,\tilde \xi^{i,N}_{s},u^{S^N(\tilde{\xi})}_{s}(\tilde \xi^{i,N}_{s})\big )ds}\textrm{and}\tilde V_t^i := e^{\int_0^t\Lambda\big (r(s),\tilde \xi^{i,N}_{r(s)},\tilde{u}_{r(s)}(\tilde \xi^{i,N}_{r(s)})\big )ds}\ ,
q_{A_1B_1A_2B_2}(a_1,b_1,a_2,b_2)=\sum_{x_1,x_2,y_1,y_2}\Big(q_{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) &=p_{AB}(a_1,b_1)p_{AB}(a_2,b_2)
\min_j nF_j \rightarrow \infty \ \ \ \ \ \max_{j=1,\ldots,m}\|\psi_j\|_{\infty}\lesssim \max_{j=1,\ldots,m}\frac{\alpha \Delta_j^\beta}{f_0(x_{j-1})}\lesssim \max_{j=1,\ldots,m}\frac{\alpha}{\sqrt{nF_j}}\ll \alpha.
\begin{pmatrix}x_2^{k+1}\lambda^{k+1}\end{pmatrix}=\begin{pmatrix}x_2^{k}\lambda^{k}\end{pmatrix}-\begin{pmatrix}I_{n_2}0 -s\beta A_2 (r+s)I_l\end{pmatrix}\begin{pmatrix}x_2^{k}-\widetilde{x}_2^k \lambda^{k}-\widetilde{\lambda}^k\end{pmatrix}.
\begin{aligned}\int_{Q\left(R_i,R_i^2\right)}\left|\eta_iv^{\ast}_i\right|^2\,dxdt \geq \left(l_{i+1}-l_i\right)^2\int_{-R_i^2}^0\left|\left\{(x,t)\in B_{R_{i+1}}:v^m<l_{i+1}\right\}\right|\,dt &=\left(\frac{\omega}{2^{i+s_0+2}}\right)^2Z_{i+1}.\end{aligned}
S_k(a) = -\frac{u^{i-1}\exp{(-u^2)}}{2}\Big|_{-a}^{\infty}+\frac{1}{2}(i-1)\int_{-a}^{\infty}u^{k-2}\exp{(-u^2)}du =\frac{a^{k-1}\exp{(-a^2)}}{2}+\frac{1}{2}(k-1)S_{k-2}(a).
0=\mathrm{div}_w\tilde{E}^i+\epsilon^i_{\phantom{i}jk}\imath_{\tilde{E}^k}A^j=\big(\mathrm{div}_v\hat{E}^i+\epsilon^i_{\phantom{i}jk}\imath_{\hat{E}^k}A^j\big)\left(\frac{v}{w}\right)\,.
\Phi_1(x) = \frac{(1-q^2)^{\alpha}x^{\alpha-\nu-1}}{\Gamma_{q^2}(1-\alpha )}D_{q,x}\left[x^{-2\alpha}\int_0^xg_1 (\rho) \rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{-\alpha}\,d_q\rho\right] &=(1-q^2)^{\alpha}x^{\alpha-\nu-1}D_{q^2,x}^{\alpha}\left(t^{\nu/2}g_1(\sqrt{t})\right)(x),\;x\in A_{q,b},
F^{\hat{\nabla}}= \left( \begin{array}{c c c}{{0}}&{{f_{1 3}\vartheta}}&{{- f_{1 2}\vartheta}}\\{{- f_{1 3}\vartheta}}&{{0}}&{{f_{2 3}\vartheta^{2}}}\\{{f_{1 2}\vartheta}}&{{- f_{2 3}\vartheta}}&{{0}}\\ \end{array}\right) + \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\vartheta}}\\{{0}}&{{- \vartheta}}&{{0}}\\ \end{array}\right) \, .
\hat{P}_{i,k+1}(\sqrt{x},T) =P_{i,k+1}(x,T), \hat{P}_{i+2^k,k+1}(\sqrt{x},T) =P_{i+2^k,k+1}(x,T), \hat{Q}_{i,k+1}(\sqrt{x},T) =Q_{i,k+1}(x,T), \hat{Q}_{i+2^k,k+1}(\sqrt{x},T) =Q_{i+2^k,k+1}(x,T)
\|\omega_z\|_{q/2,u^o}\le\bigg(2q'z^{1-q}\sup_{y\ge z}\frac 1y\int_0^yu(t)\,dt\bigg)^{2/q}= x^2\bigg(2q'\sup_{1/x\le y}\frac 1{xy}\int_0^yu(t)\,dt\bigg)^{2/q}.
\displaystyle \lambda^{2}_1 \leq \left(\frac{1}{6}\sum_{i=1}^{m}r^2_i(r^2_i -1)-\frac{15}{8}m_2-\frac{3}{4}m_3-\frac{1}{2}m_4\right)\left(\frac{n-1}{n}\right) \displaystyle \lambda_1 \leq \sqrt{\left(\frac{1}{6}\sum_{i=1}^{m}r^2_i(r^2_i -1)-\frac{15}{8}m_2-\frac{3}{4}m_3-\frac{1}{2}m_4\right)\left(\frac{n-1}{n}\right)}
\varphi(b_0a_1b_1a_2b_2\cdots a_nb_n) = \varphi(b_0a_1(c_1+\tau(b_1))a_2\cdots (c_{n-1}+\tau(b_{n-1}))a_nb_n) = \varphi(b_0a_1c_1a_2\cdots c_{n-1}a_nb_n) &+\sum_{r=1}^n\sum_{1\leq k_1<\cdots<k_r\leq n-1}\varphi(b_0a_1c_1a_2\cdots \hat{c}_{k_1}\cdots \hat{c}_{k_r}\cdots c_{n-1}a_nb_n)\tau(b_{k_1})\cdots\tau(b_{k_r})
(x+b)*y+(x+b)*c =((-u)(xb))*y+((-u)(xb))*c =(-u)*((xb)*y)+(-u)*y+b*y &\qquad\mbox{}+(-u)*((xb)*c)+(-u)*c+b*c =(xb)*y+(xb)*c =x*(b*y)+x*y+b*y &\qquad\mbox{}+x*(b*c)+x*c+b*c =x*y+x*c,
\begin{pmatrix}A_{11}A_{12}A_{21}A_{22}\end{pmatrix}= Q_{4m}W_{4m}B(a) W_{4m}^* Q_{4m}^* \begin{pmatrix}B_{11}B_{12}B_{21}B_{22}\end{pmatrix}= Q_{4n}W_{4n}B(b) W_{4n}^* Q_{4n}^* \; .
dL_{x}^{-1}(T_{\pi_{1}(x)}(K_{2}\pi_{1}(x))) = \sum_{\lambda \in \Sigma^{+}}\mathfrak{m}_{\lambda}\oplus V(\mathfrak{m}_{1}\cap \mathfrak{k}_{2})\oplus \sum_{\alpha \in W^{+}}V_{\alpha}^{\perp}(\mathfrak{m}_{1}\cap \mathfrak{k}_{2}), dL_{x}^{-1}(T_{\pi_{1}(x)}^{\perp}(K_{2}\pi_{1}(x))) = \mathfrak{a}.
UDU^*\;=\;\left( \begin{array}{cc}\nabla\!\!\!\!/ k\,\gamma^5\,e^{-Re\,\sigma}{0 \choose 1}k^* \gamma^5\,e^{-Re\,\sigma}\,(0,1)\quad \nabla\!\!\!\!/\end{array}\right)\; +\;\left( \begin{array}{cc}0 0 0 i\,\partial\!\!\!/\, Im\,\sigma_2\end{array}\right)
\begin{matrix}{{\tilde{h}^{0}= h^{0}}}\\{{\tilde{h}^{1}= - h^{2}\sin ( 2 t ) + h^{1}\cos ( 2 t )}}\\{{\tilde{h}^{2}= h^{2}\cos ( 2 t ) + h^{1}\sin ( 2 t )}}\\ \end{matrix}