harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pi_{_h \rho ,}^{\rho}^{\mu \nu}\left( k \right) = - \kappa \Gamma_{_h}^{\mu \nu}= - \kappa^{2}\omega \frac{\pi^{2}T^{4}}{1 8 0}\left( 4 \eta^{\mu 0}\eta^{\nu 0}- \eta^{\mu \nu}\right) ,
A^{(\epsilon)}(t) =\int_0^t\bigg\{2\sum_{i=1}^dX^{(\epsilon)}_i(s)f^{(\epsilon)}_i(X^{(\epsilon)}(s))+|\sigma|^2\bigg\}\,ds, M^{(\epsilon)}(t) =2\sum_{i=1}^d\sum_{k=1}^d\int_0^tX^{(\epsilon)}_i(s)\sigma_{ik}\,dW_k(s).
F(x)=\frac{\Gamma(p+1,\frac{1}{2}\chi^{2}(1-\frac{x^{2}}{c^{2}}))-\Gamma(p+1,\frac{1}{2}\chi^{2})}{\Gamma(p+1)-\Gamma(p+1,\frac{1}{2}\chi^{2})},0\le x\le c,c>0,\chi>0,p>-1
p :=\mu \left(\bigwedge_{((i, l_2,\ldots, l_k),v) \in [m]^k \times \{0,1\}}E^{v,t^v_{(i, l_2,\ldots, l_k)}}_{(a^1_{i}, a^2_{l_2}\ldots, a^k_{l_k})}\right) < q := \mu \left(\bigwedge_{((i, l_2,\ldots, l_k),v) \in [m]^k \times \{0,1\}}E^{v,t^v_{(i, l_2,\ldots, l_k)}}_{(a^1_{\sigma(i)}, a^2_{l_2}, \ldots, a^k_{l_k})}\right)
E_{X/AN}=M\times_{M\cap K}\big(S_{\mathfrak{p}\cap \mathfrak{m}}\otimes S_{\mathfrak{k}/(\mathfrak{k}\cap \mathfrak{m})}\otimes W|_S\big)\cong M\times_{M\cap K}\big( S_{\mathfrak{p}\cap \mathfrak{m}}\otimes \widetilde{W}\big),
\log F(A,\!B)\!=\!-\!T_2\!\left(\alpha l\!+\!A_0\right)/\alpha\!-\!T_2\!\left(\beta l\!+\!B_0\right)/\beta\!+\!T_2\!\left((\alpha\!+\!\beta)l\!+\!D_0\right)/(\alpha\!+\!\beta),
{\mathbb E}\|\hat P^{(n)}-P^{(n)}\|_2^2={\mathbb E}\Bigl(\|\hat P^{(n)}\|_2^2+\|P^{(n)}\|_2^2-2\langle \hat P^{(n)}, P^{(n)}\rangle\Bigr)={\mathbb E}\Bigl(2-2\langle \hat P^{(n)}, P^{(n)}\rangle\Bigr) &=2-2\langle{\mathbb E}\hat P^{(n)}, P^{(n)}\rangle=2-2(1+b^{(n)})\langle P^{(n)}, P^{(n)}\rangle =-2b^{(n)}
\frac{d}{dt}\frac{\partial q_{\epsilon}^{u,s}}{\partial\epsilon}|_{\epsilon=0}=JD^{2}H(q_{0}(t-t_{0}))\frac{\partial q_{\epsilon}^{u,s}}{\partial\epsilon}|_{\epsilon=0}+g(q_{0}(t-t_{0}),t,0)(9)
1 - F_\rho(k) = \sum_{t = k + 1}^\infty f_\rho(t) = \mu_\rho \sum_{t = k + 1}^\infty \frac{f_\delta^\ast(t)}{t}= \mu_\rho \sum_{t = k + 1}^\infty \frac{f^\ast(t)}{t}= \frac{\mu_\rho}{\mu}\sum_{t = k + 1}^\infty f(t),
\tau_j(p^1) =(\cos \varphi^1_j,\sin \varphi^1_j) \varphi^1_j\in (\pi,\pi + 2\epsilon), \tau_j(p^2) =(\cos \varphi^2_j,\sin \varphi^2_j) \varphi^2_j\in (-2\epsilon,0),
\textstyle\bigl\{\binom{a_{m+1}}{b_m},\binom{c_{m'}}{d_{m'}}\bigr\},\bigl\{\binom{a_m}{b_m},\binom{c_{m'}}{d_{m'-1}}\bigr\},\bigl\{\binom{a_m}{b_{m-1}},\binom{c_{m'-1}}{d_{m'-1}}\bigr\},\bigl\{\binom{a_{m-1}}{b_{m-1}},\binom{c_{m'-1}}{d_{m'-2}}\bigr\},\ldots.
G_2(x,y,z;q):= -2 \frac{J_1^3J_2^3}{j(x;q)j(y;q)j(z;q)}\frac{j(xy;q^2)j(xz;q^2)j(yz;q^2)}{j(-x;q^2)j(-y;q^2)j(-z;q^2)}.
\ddot{\lambda}_{H_r,d_r}(\rho) = \frac{\cosh(\rho)\sinh^{\frac{2(n-r)}{r}-1}(\rho)\left(nH_r\frac{\sinh^n(\rho)}{\cosh^r(\rho)}-(n-r)(nH_rI_{n,r}(\rho)+d_r)\right)}{r(nH_rI_{n,r}(\rho)+d_r)^{\frac{r-1}{r}}\left(\sinh^{\frac{2(n-r)}{r}}(\rho)-(nH_rI_{n,r}(\rho)+d_r)^{\frac{2}{r}}\right)^{\frac{3}{2}}}.
\lambda^{- 2 n}H , E_{- \alpha}, \lambda^{- 2 n}E_{\pm \alpha}, F_{- \alpha_{1}}, \lambda^{- 2 n}F_{\pm \alpha_{1}}, \lambda^{- 2 n}F_{\alpha_{2}}, F_{- \alpha_{2}}, \lambda^{- 2 n}F_{- \alpha_{2}}, \lambda^{- 1}H^{\prime}, \lambda^{- ( 2 n + 1 )}H^{\prime}, ~ ~ n > 0
V_{\mathrm{e f f}}( \phi_{R}) = \pi^{2}\zeta^{2}\phi_{R}^{4}\left( \ln{\frac{\phi_{R}^{2}}{v_{R}^{2}}}-{\frac{1}{2}}\right) +{\frac{1}{4}}( \zeta - 1 ) m_{h}^{2}\phi_{R}^{2}( 1 -{\frac{\phi_{R}^{2}}{2 v_{R}^{2}}})
|I(t)| \le \sum_{j=0}^{N+1}c_j \left\vert \int_0^{\frac{\pi}{|t|}}e^{-it\tau}F\left( \tau +\frac{j\pi}{|t|}\right) \, d\tau \right\vert + \frac{1}{2^{N+1}}\left\vert \int_{0}^{\infty}e^{-it\tau}F_{N+1}(\tau)\,d\tau \right\vert.
\int_\Omega\langle\nabla v_0,\nabla\varphi\rangle\,dx+\left(\int_\Omega |v_0|^{2p/(p-1)}\,dx\right)^{-1/p}\int_\Omega|v_0|^{2/(p-1)}\,v_0\,\varphi\,dx-\langle f,\varphi\rangle=0,
\begin{array}[c]{l}Y_{0}:=L^{2}( ( 0,T) \times( 0,L) )\cap C^{0}( \left[ 0,T\right] ;H^{-1}( 0,L) ) Y_{1}:=L^{2}( 0,T;H^{4}( 0,L) ) \cap C^{0}(\left[ 0,T\right] ;H^{3}( 0,L) ) \end{array}
\langle{\cal O}\rangle_{R}= \sum_{n = 0}^{\infty}\frac{1}{n !}\frac{1}{( 2 \pi )^{n}}\int \left[ \prod_{i = 1}^{n}d \theta_{i}g ( \theta_{i}, R ) e^{- m R \cosh \theta_{i}}\right]{\cal F}_{2 n}^{\cal O}( \theta_{1}, \ldots , \theta_{n}) \, \, \, ,
\hat{k}_d(\Delta)=\sum_{\Upsilon\in\mathcal{T}_d(\Delta)}\prod_{\tau \in \Upsilon_d}\left(\prod_{v \in \tau}x_{\tau}\right){\mathbf{t}_{d-1}(\Upsilon)^2}=\sum_{\Upsilon\in\mathcal{T}_d(\Delta)}\left(\prod_{v \in V}x_{v}^{\deg_\Upsilon v}\right){\mathbf{t}_{d-1}(\Upsilon)^2}.
E_{F_{n}}[Y|X] =\gamma_{0}(X)+\left( \frac{\mu}{\sqrt{n}}\right)[\bar{\alpha}(x)-\alpha_{0}(x)], \theta_{n}=E[ZE_{F_{n}}[Y|X]]=E[\alpha_{0}(X)E_{F_{n}}[Y|X]]=\theta_{0}-\left( \frac{\mu}{\sqrt{n}}\right) \bar{\sigma}^{2}.
L_{0}= \mathrm{D i a g o n a l}( 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 6 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , \ldots ) - \frac{1}{5}
\begin{aligned}{q}^{T}_{i}(x_{i+1/2})=\bar{q}_{min}+\dfrac{\bar{q}_{max}}{2}\left(1+\theta \dfrac{\tanh(\beta)+A}{1+A~\tanh(\beta)}\right) &{q}^{T}_{i}(x_{i-1/2})=\bar{q}_{min}+\dfrac{\bar{q}_{max}}{2}\left(1+\theta~ A\right)\end{aligned}
\mathbf{P}_{0,i}^{\theta}(H^+(t) \in{\rm d}x, J^+(t)= j) = \frac{v_j(\theta)}{v_i(\theta)}{\rm e}^{\theta x}\mathbf{P}_{0,i}(H^+(t) \in{\rm d}x, J^+(t)= j), i,j=\pm1, x\geq 0.
\lim_{\epsilon\to0}\delta_{(X,\Delta)}(\epsilon A-(K_X+\Delta))= 2\frac{(\max_{p\in\mathbb{P}^1}\mathrm{ord}_p(B_{\mathbb{P}^1})-1)}{u}=\delta_{(\mathbb{P}^1,B_{\mathbb{P}^1})}(-K_{\mathbb{P}^1}-B_{\mathbb{P}^1}-M_{\mathbb{P}^1})
\gamma_{\mu}= \left( \begin{array}{c c}{\sigma_{\mu}}&{0}\\{0}&{- \sigma_{\mu}}\\ \end{array}\right) , \quad \sigma_{\mu \nu}={\frac{1}{2 i}}[ \gamma_{\mu}, \gamma_{\nu}] = \epsilon_{\mu \nu \rho}\sigma_{\rho}; \quad \mu , \nu , \rho = 1 , 2 , 3 .
\phi_{\triangle \omega}( t , \rho , z ) ={\frac{1}{2 \pi}}\int_{0}^{\infty}d \omega \int d^{2}k u_{\omega , k}( t , \rho ) e^{- i k_{j}z^{j}}\tilde{\phi}_{\triangle \omega}( \omega ) \tilde{\varphi}( k ) .
D^{(2)}_n = \begin{cases}8(n+2)(2n+1)(2n+5)(4n+3)(4n+5)(4n+7)(4n+9)f_0^3f_8^3 , 24(n+2)(2n+1)(3n+7)(3n+8)(4n+3)(4n+5)f_0^3f_7^2 , 3(n+2)(3n+2)(3n+4)(3n+5)(3n+7)f_1^2f_7^2 ,\end{cases}
\begin{cases}\left\langle \nu^H,\xi^H\right\rangle= |\xi^H|^2 k(\xi), \left\langle \eta,\xi^H\right\rangle= 2|\xi^H|^2 \frac{\left\langle \nu_\xi, T\right\rangle}{|\mathcal{P}_H(\nu_\xi)|}, \left\langle V_j,\xi^H\right\rangle = \left\langle W_j,\xi^H\right\rangle=0\quad\mbox{for}j\in\{1,\ldots,n-1\}.\end{cases}
\Delta_{G}=\{\widehat{x}_i-\widehat{x}_{i+1}\,,\;1\leq i\leq p+q-1\,, \widehat{x}_{p+q}\}\intertext{and}\Delta_{K}=\left\{\begin{cases}\widehat{x}_i-\widehat{x}_{i+1}\,,\;1\leq i\leq p-1\,,\; \widehat{x}_{p-1}+\widehat{x}_{p}\,, \widehat{x}_i-\widehat{x}_{i+1}\,,\;p+1\leq i\leq p+q-1\,,\; \widehat{x}_{p+q}\,,\end{cases}\right\}\,,
f(t)=\left\Vert v_{2}\right\Vert^{2}-\cos\varphi\,\sqrt{t^{2}+\left\Vert v_{2}\right\Vert^{2}}\,\left\Vert v_{2}\right\Vert \mbox{and}f(0)=\left\Vert v_{2}\right\Vert^{2}\left(1-\cos\varphi\right)\mbox{.}
\begin{aligned}R_{{O1}}\triangleq\max_{{P}_i}\sum_{i=1}^K r_i \mathrm{s.t.}r_i=\frac{1}{K}\ln \Big(1+\frac{KP_i|h_i|^2}{N_0}\Big), &\sum_{i=1}^K P_i\leq P, \min_i r_i\geq r^*. \end{aligned}
\det(\mathbf{Q}_{i,j}) = \begin{vmatrix}2(d_{i,1}-d_{j,1}) & 2(d_{i,2}-d_{j,2}) & \ddots & 2(d_{i,m}-d_{j,m})\end{vmatrix}& = 2^m \prod_{k=1}^{m}(d_{i,k}-d_{j,k}).
\begin{aligned}u_0, u_3, u_4 \in \mathcal{S}, u_1 = \partial_x sn( 2 K(k)x/\pi,k) \in \mathcal{A}^+, u_2 = \partial_x cn( 2 K(k)x/\pi,k) \in \mathcal{A}^-. \end{aligned}
\begin{cases}\tilde{L}_y-\tilde{M}_x=\varepsilon\tilde{L}_{xx}-\tilde{\Gamma}^2_{22}\tilde{L}+2\tilde{\Gamma}^2_{12}\tilde{M}-\tilde{\Gamma}^2_{11}\tilde{N}, & \tilde{M}_y-\tilde{N}_x=\varepsilon\tilde{M}_{xx}+\tilde{\Gamma}^1_{22}\tilde{L}-2\tilde{\Gamma}^1_{12}\tilde{M}+\tilde{\Gamma}^1_{11}\tilde{N}, \tilde{L}\tilde{N}-\tilde{M}^2=-1,\end{cases}
R_{\beta \mu \nu}^{\alpha}= \partial_{\mu}\Gamma_{\nu \beta}^{\alpha}- \partial_{\nu}\Gamma_{\mu \beta}^{\alpha}+ \Gamma_{\mu \gamma}^{\alpha}\Gamma_{\nu \beta}^{\gamma}- \Gamma_{\nu \gamma}^{\alpha}\Gamma_{\mu \beta}^{\gamma}
F(t): =\int_{0}^{l(\gamma_{1})}\varphi^0(\bar{\gamma}(s),\dot{\bar{\gamma}}(s))ds &=\int_{0}^{a}\varphi^0(\gamma_1(s),\dot{\gamma_1}(s))ds+\int_{a}^{l(\gamma_1)}\varphi^0(\gamma_1(s)+ty(s),\dot{\gamma_1}(s)+t\dot{y}(s))ds
W_{\mu}({\bf p}) \cdot D \left\{V^{- 1}( \Lambda_{p}, k ) \right\}= D \left\{V^{- 1}( \Lambda_{p}, k ) \right\}\cdot \left[ W_{\mu}({\bf k}) + \frac{p_{\mu}+ k_{\mu}}{M ( \Delta_{0}+ M )}p_{\nu}W_{\nu}({\bf k}) \right] ,
A=\frac{\left(z^{-1}\xi;q\right)_{\infty}}{(q;q)_{\infty}}\left(\frac{z^{2}}{z^{2}-1}{}_{1}\phi_{1}\left(q;qz^{-2};q,qz^{-1}\xi^{-1}\right) +{}_{1}\phi_{1}\left(q;z^{-1}\xi;q,z\xi\right)-1\right)
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{{r}(\cdot)}(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{{r}(\cdot)}(\mathbb{R}^n)};
\sum_{m=0}^{\infty}\sum_{\ell=0}^{\min\{2p,2m\}}A_{m,\ell}=\sum_{m=0}^{p}\sum_{\ell=0}^{2m}A_{m,\ell}+\sum_{m=p+1}^{\infty}\sum_{\ell=0}^{2p}A_{m,\ell}.
2\xi_2 = \frac{2\big(9(\ln3)^a-8 (\ln4)^a\big)}{3}< \xi_1= \frac{2\big(3 (\ln2)^a-2 (\ln4)^a\big)}{3}< \frac{\xi_2}{2}=\frac{9(\ln3)^a-8 (\ln4)^a}{6}.
\# \big\lbrace z \in \sigma_{\mathrm{disc}}( H(b_0,e V)) : -r_0 e^2 \leq z < -r e^2 \big\rbrace &= \# \big\lbrace z \in \sigma( p_{0}Up_{0}) : z \geq \big( 2 r b_0 \big)^\frac{1}{2}\big\rbrace \big( 1 + o(1) \big).
- \frac{1}{2}\left( \frac{\delta S_{0}}{\delta \phi}\frac{\delta S_{1}}{\delta \rho}+ \frac{\delta S_{0}}{\delta \rho}\frac{\delta S_{1}}{\delta \phi}\right) - \frac{1}{2 i}\frac{\delta^{2}S_{0}}{\delta \rho \delta \phi}+ \frac{1}{2}\left( e^{- i S_{1}}\frac{- \delta^{2}( e^{i S_{1}})}{\delta f^{2}}+{f^{\prime}}^{2}\right) = 0 .
(h_{\lambda,\mu},h_{0,0})_{\mathbb{Z}(\mathcal{T})_\xi\rtimes \mathcal{S}}=\sum_{(\lambda',\mu')\in \overline{C}}\frac{\dim^-(L_\xi(\lambda',\mu'))}{\dim^+(L_\xi(\lambda',\mu'))}S_{(\lambda,\mu),(\lambda',\mu')}S_{(\lambda',\mu'),(0,0)}\neq 0.
|\widehat{N_{2}(f_{v_{5}})}(t,\xi)| \leq \frac{C}{\xi t^{4}\log^{3b}(t)}+ \frac{C}{\xi^{3}t^{6}\log^{3b}(t)}+ \frac{C \log^{3}(t)}{\sqrt{\xi}t^{5/2}}, 1 \leq \frac{1}{\xi}\leq \frac{t}{2}
-m\frac{2ln(H^{\frac{1}{3}}-k^{\frac{1}{3}}v)-ln(H^{\frac{2}{3}}+(Hk)^{\frac{1}{3}}v+k^{\frac{2}{3}}v)+2\sqrt{3}arctan\frac{2(\frac{k}{H})^{\frac{1}{3}}v}{\sqrt{3}}}{6(Hk^{2})^{\frac{1}{3}}}=t
X_{n,r}:= \begin{dcases}\delta^{-1}_n(3)\cdot \sum_{q =1}^{r-1}\left[\frac{\delta_n(3)}{\delta_n(2)}\cdot \tilde M_{n,\{q,r \}}+\sum_{p=1}^{q-1}\tilde M_{n,\{p,q,r \}}\right] , r \ge 2 0 , r=1 ,\end{dcases}
R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}= \frac{1}{l^{4}}\big [ 4 0 + \frac{4 8 M^{2}z^{4}}{\Sigma^{6}}( r^{2}- a^{2}c o s^{2}{\theta}) ( r^{4}- 1 4 a^{2}r^{2}c o s{\theta}+ a^{4}c o s^{4}{\theta}) \big ]
{\bf p r}_{0}\{\vec{v}( \tau , x ) \}= \frac{d{\bf T}_{f}}{d a}[ \vec{f}_{0}, 0 ] ( \tau , x ) \cdot a ( \tau ) + \frac{d{\bf R}_{f}}{d \alpha}[ \vec{f}_{0}, 0 ] ( \tau , x ) \cdot \alpha ( \tau ) .
\Theta_1 (f):= \Big\{(f_1,g_1,\dots,f_d,g_d,a): \sum_{j=1}^d \big(f_j(x_j)-g_j(x_j))+\sum_{i\in I}a^i1_{A^i}(x)\geq f(x), \forall x\in\mathbb{R}^d \Big\},
r_u \left( \frac{\hat x + C_2 \delta}{t + \delta}\right) - r_u \left( \frac{\hat x}{t}\right) \ge - 2 \kappa C_1 \frac{\tilde \epsilon \delta}{t + \delta}+C_1 \frac{\tilde \epsilon}{2}= \frac{C_1 \tilde \epsilon}{2}\left( 1- \frac{4 \kappa \delta}{t+\delta}\right) \ge \frac{C_1 \tilde \epsilon}{2}\left( 1- 4 \kappa \right).
\begin{aligned}\Delta f_{\varepsilon}=\alpha(\alpha-1)(B+\varepsilon)^{\alpha-2}|\nabla B|^2+\alpha (B+\varepsilon)^{\alpha-1}\Delta B &\geq \alpha(\alpha-1)(B+\varepsilon)^{\alpha-2}|\nabla B|^2 &+2\alpha (B+\varepsilon)^{\alpha-1}\biggl(\sum_{i,j,k=1}^nh_{ijk}^2+B(n+nH^2-B)-nH\dfrac{n-2}{\sqrt{n(n-1)}}B^{\frac32}\biggl).\end{aligned}
P_{q}^{( \alpha , \beta )}( \cos \theta ) \approx \frac{\Gamma ( q + \alpha + 1 )}{N^{\alpha}n !}\left( \sin \frac{\theta}2 \right)^{- \alpha}\left( \cos \frac{\theta}2 \right)^{- \beta}\left( \frac{\theta}{\sin \theta}\right)^{\frac 1 2}J_{\alpha}( N \theta ) ,
\hat{J}_a^{(q,r)}(x,\theta ) := Z^{(q,r)}_a(x,\theta ) - \frac{Z^{(q)}(-a, \theta )}{W^{(q)}(-a)}W^{(q,r)}_a(x) = \mathcal{M}_a^{(q,r)}\Big( Z^{(q)}(x,\theta ) - \frac{Z^{(q)}(-a, \theta )}{W^{(q)}(-a)}W^{(q)}(x) \Big),
X^i_{;jk}= \frac{1}{2}g^{i\ell}\left[ (\mathcal D_g X)_{\ell j;k}+ (\mathcal D_g X)_{k\ell;j}- (\mathcal D_g X)_{jk;\ell})\right] -\frac{1}{2}g^{i\ell}\widetilde{C}^{p}_{\ell jk}X_{p}&= \widetilde{A}^i_{jk}f + \widetilde{B}^{i\ell}_{jk}f_{; \ell}-\frac{1}{2}g^{i\ell}\widetilde{C}^{p}_{\ell jk}X_{p},
D_{2}^{\scriptscriptstyle SE}(x_{1},x_{2})=\bigl[ T_{1}^{h+u}(x_{1}), T_{1}^{h+u}(x_{2})\bigr]\bigr|_{\scriptscriptstyle SE}=:h^{\alpha\beta}(x_{1})h^{\mu\nu}(x_{2}):d_{2}(x_{1}-x_{2})_{\alpha\beta\mu\nu},
|\Delta_j(1\otimes f-f\otimes 1) |_{C^\alpha(X\times X)}=|\Delta_j(1\otimes f-f\otimes 1)|_{C^\alpha(U_j)}\leq \leq |\Delta_j|_{C^\alpha(U_j)}\|1\otimes f-f\otimes 1\|_{C^0(U_j)}+\|\Delta_j\|_{C^0(U_j)}|1\otimes f-f\otimes 1)|_{C^\alpha(U_j)}\lesssim \lesssim \|f\|_{C^\beta(X)}j^{-\gamma},
\mu_{0}(t):=\sum_{i=1}^{d}\|A^{i}\|_{\bar{s}}^{2}+\sum_{i,j=1}^{d}\|B^{ij}\|_{\bar{s}}^{2}+\sum_{i=1}^{d}\|A^{i}\|_{\bar{s}}+\|D\|_{\bar{s}}^{2}+\|D\|_{\bar{s}}+1
C_0:= A'_2-2A'_1-\bigg(\frac{\varepsilon_2}{2}+M_1+\frac{\varepsilon_3M_2}{2}\bigg)>0, & \bigg(\frac{b_1}{b_2}+\frac{\varepsilon_1}{2}\bigg)\mu \leq A'_1,\ \frac{M_2}{2\varepsilon_3}\leq A'_3.
\left(\begin{array}{cccccccc}0 0 0 -x 0 0 -w_1 w_0 0 0 0 q 0 0 2w_2 -w_1 0 0 0 0 0 0 -6w_3 2w_2 0 0 0 0 0 0 h -u 0 0 0 0 0 0 q -x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y 0 0 0 0 0 0 0 0 \end{array}\right).
\frac{d^{2}z_{a}^{\pm}}{d \lambda_{a}^{2}}+ 2 \frac{\partial \rho ( z_{a})}{\partial z_{a}^{\pm}}\frac{d z_{a}^{\pm}}{d \lambda_{a}}\frac{d z_{a}^{\pm}}{d \lambda_{a}}= 0 ,
u(s,x_0) = \mathbb E \bigg[ \Phi( x_0+W_{T -s}) &+ \int_s^T f( r,W_r +x_0, u(r, W_r+x_0), \nabla u (r, W_r +x_0) ) \mathrm dr &+ A_T^{W ,W}((\nabla u^* b)(x_0 + \cdot ) ) -A_s^{W ,W}((\nabla u^* b)(x_0 + \cdot ) ) \bigg]
\int g(x) \, d ( R_\eta \mu) = \int g(x \cdot \eta^{-1}) \, d \mu = \int d \nu (C) \, \int g(x \cdot \eta^{-1}) \, d \mu_C (x) = \int d \nu (C) \, \int g(x) \, d (R_\eta \mu_C ) (x),
r^{2s-1}{{\Gamma (1-s) \alpha}\over{\Gamma (s) 8\pi}}2^{2-2s}\int_{-\infty}^{\infty}B (1-s +iy,1-s-iy) [{{\sinh \pi y \cosh \pi\alpha y - \sinh \pi\alpha y \cosh \pi y}\over{\sinh \alpha\pi y}}] dy
\times \theta ({\frac{k^{+}\Delta}{p^{+}}}- k_{\perp}^{2}) \bigl [{\frac{1}{{( k_{\perp}^{2}+{\frac{m^{2}k^{+ 2}}{p^{+ 2}}}})}}\bigr ] \biggl [ 1 -{\frac{\frac{m^{2}k{+ 2}}{p^{+ 2}}}{k_{\perp}^{2}+{\frac{m^{2}k^{+ 2}}{p^{+ 2}}}}}\biggr ] \; .
g_m(0)= \left( \frac{\partial_x^m \tilde{u}_0}{u^s_{0,y}+ \tilde u_{0, y}}\right)_y= \frac{\partial_y\partial_x^m \tilde{u}_0}{u^s_{0,y}+ \tilde u_{0, y}}- \frac{\partial_x^m \tilde{u}_0}{u^s_{0,y}+ \tilde u_{0, y}}\eta_2(0),
g(i,i^\prime,J_1,J_2,p,n)= I(i,J_{1},p,n)I(i,J_{2},p,n) \delta_{0}(\sum_{q=1}^{2k_1}(-1)^{q}j_{q}) \delta_{0}(\sum_{q=1}^{2k_2}(-1)^{q}j_{q}^\prime).
Q_{1313}=\frac{{{e}^{q_3-q_1}}}{\Delta_3},Q_{1213}=-\frac{{{e}^{2 q_3 - q_2-q_1}}}{\Delta_3},Q_{1223}=\frac{{{e}^{2 q_3-2q_1}}}{\Delta_3},Q_{1323}=-\frac{{{e}^{q_3+q_2-2q_1}}}{\Delta_3},
f = \frac{\theta_{3}( u | \tau / 2 )}{\theta_{4}( u | \tau / 2 )}={\frac{\theta_{3}( 2 u | 2 \tau ) + \theta_{2}( 2 u | 2 \tau )}{\theta_{3}( 2 u | 2 \tau ) - \theta_{2}( 2 u | 2 \tau )}}.
\Delta^2_{\tau}h(x)(w') :=\frac{2[h(x\!+\!\tau w')-h(x)-\tau dh(x)(w')]}{\tau^2}\quad{\rm for}\ \tau>0, \Delta^2_{\tau}h(x|v)(w') :=\frac{2[h(x\!+\!\tau w')-h(x)-\tau\langle v,w'\rangle]}{\tau^2}\quad{\rm for}\ \tau>0,
{\frac{\sinh \left({\frac{\theta}{2}}-{\frac{i \pi}{2 n}}\alpha \right) \sinh \left({\frac{\theta}{2}}-{\frac{i \pi}{2 n}}( 2 - \alpha ) \right)}{\sinh \left({\frac{\theta}{2}}+{\frac{i \pi}{2 n}}\alpha \right) \sinh \left({\frac{\theta}{2}}+{\frac{i \pi}{2 n}}( 2 - \alpha ) \right)}}\ ,
J_{n,v,\varphi}:(-1,\infty) \rightarrow \mathbb{R}; J_{n,v,\varphi}(\lambda) :=\int\limits_{\mathbb{R}^n}\left(\frac{|K_n\varphi(\mathbf{x})|^2}{1+\lambda -v(\mathbf{x})}+\big(1-\lambda+v(\mathbf{x})\big)|\varphi(\mathbf{x})|^2\right)\mathrm{d}\mathbf{x}.
\langle 0|O_i^{V}O_j^{V}|0\rangle \sim \langle 0|\Theta_i^{\partial V}\Theta_j^{\partial V}|0\rangle_{d-1}= \alpha^d_{ij}\int\limits_{\partial V}\!\! d^{d-1}z_1 \int\limits_{\partial V}\!\! d^{d-1}z_2~\hat n_1 \cdot\hat n_2 \langle 0|\hbox{\large$\vartheta$}_i(\vec{z}_1) \hbox{\large$\vartheta$}_j(\vec{z}_2)|0\rangle_{d-1}.
G^{ij}_\psi(\nu) \equiv (-1)^{i+j}\Sigma((-1)^i \bar R_i)\Sigma((-1)^j \bar R_j) \bar\gamma_i^{-2}\bar\gamma_j^{-2}\sum_{n=1}^{n_{{\rm max}}}\frac{1}{\chi_n - \nu^2}\frac{s_n'((-1)^i \bar R_i) s_n'((-1)^i \bar R_j)}{\int_{-\bar R_1}^{\bar R_2}dr'' \Sigma(r'') s_n^2(r'')}\,.
0 = \left( \frac{\partial U}{\partial u_{\nu}}\right)_{u_{\mu}= e_{\mu}}\rightarrow \frac{1}{g}+ \sum_{\mu = 1 ( \neq \nu )}^{N}\frac{2}{e_{\mu}- e_{\nu}}= \sum_{j = 1}^{\Omega}\frac{1}{z_{j}- e_{\nu}}\; ,
\begin{array}{r c l}{\Phi}&{=}&{\phi + \phi_{0},}\\{\phi}&{=}&{\left( \pi_{1}, \dots , \pi_{N - 1}, ( \sigma + i \pi_{0}) / \sqrt{2}\right)^{T},}\\{\phi_{0}}&{=}&{( 0 , \dots , 0 , v / \sqrt{2})^{T},}\\ \end{array}
\lbrack A(s)u(s)]_{i}=\sum_{j=1}^{d}\langle G_{i}(X^{0}(s),\cdot),e_{j}(\cdot ,s)\rangle_{L^{2}(\nu )}u_{j}(s) & =\left\langle G_{i}(X^{0}(s),\cdot ),\sum_{j=1}^{d}e_{j}(\cdot,s)u_{j}(s)\right\rangle_{L^{2}(\nu )}& =\langle G_{i}(X^{0}(s),\cdot ),\psi (\cdot ,s)\rangle_{L^{2}(\nu )},
z(e_{k+1}\psi_k e_k) = \sum_{j=0}^{k-1}g_{k+1,j}.y_{k+1}^j e_{k+1}\psi_k e_k &= \sum_{j=0}^{k-1}g_{k+1,j}.\psi_k y_k^j e_k &= \sum_{j=0}^{k-2}(g_{k+1,j}+ (-1)^{k-j}E_{k-1-j}(x_1, \ldots, x_{k-1}) g_{k+1,k-1}) .\psi_k y_k^j e_k
4 f ( k , z ) ={\frac{\alpha}{\pi}}\int_{- 1}^{1}d z^{\prime}\int_{0}^{1}d \gamma A ( \theta , \theta^{\prime}, \gamma ) \left\{\gamma^{3 / 2}f ( \gamma k , z^{\prime}) + \gamma^{- 5 / 2}f ({\frac{k}{\gamma}}, z^{\prime}) \right\}
\sup_{\mathbf{\hat{x}}\in \Psi(k_0),\,\mathbf{k}\in \Pi(k_0)}\ \ \liminf_{T \to \infty}\ \sum_{t=0}^T \beta_t\,U_t\left(\hat{x}_t\right) begin{align*}5pt] \hat{x}_t + k_{t+1}\leq f(k_t), t=0,1,\ldots, begin{align*}5pt] k_0 > 0\ .
B'(t)\geq\frac{1-C_1e^{-K/2\log B}}{(1+C_2e^{-(k-1)\frac{K}{3}\log B})^2}B^{K-1}C_3=\frac{1-C_1B^{-K/2}}{(1+C_2 B^{-(k-1)K/3})^2}B^{K-1}C_3\geq C_4B^{K-1}.
\mathcal F_\alpha^V(\pi):= \sup_{h\in C(\Lambda;\mathbb R)}\biggl[ \langle \pi, h\rangle - \int_\Lambda \log\biggl(\frac{Z_1( f'(a) + h(u) - V(u) )}{Z_1( f'(a) - V(u))}\biggr) \;\!\mathrm du\biggr].
C_{m_d}(\xi) \; = \; \frac{1}{2\pi}\int d^2p \; \hat{C}_{m_d}(p) \;e^{ip\xi}\; = \;\frac{1}{(2\pi)^2}\int d^2p \; \frac{1}{p^2+m_d^2}\; e^{ip\xi}\; ,
(\tau/\tau_\lambda)(\exp(x)) = \tau(\exp(x) \cdot v_\lambda)/\tau_\lambda(\exp(x) \cdot v_\lambda) &({\rm by\the\definition\of\the\isomorphism}\ \rho_\lambda^\ast: V(\lambda)^\ast \xrightarrow{\sim}H^0(G/B, \mathcal{L}_\lambda)\{\rm in}\ \S\S 3.1) &= \tau(\exp(x) \cdot v_\lambda)\quad({\rm since}\ \tau_\lambda(\exp(x) \cdot v_\lambda) = 1).
\sum_{n=0}^{\infty}|q_n(t,x,y)|\,\le\, \sum_{n=0}^{\infty}\gamma_n \big(\rho_{(n+1){\beta_2}}^0+\rho^{\beta_2}_{n{\beta_2}}\big)(t,x-y) &\le \,\sum_{n=0}^{\infty}\gamma_n \Phi^{-1}(T^{-1})^{-(n+1){\beta_2}}\big(\rho_{{\beta_2}}^0+\rho_0^{{\beta_2}}\big)(t,x-y)=C_2 \big(\rho_{{\beta_2}}^0+\rho_0^{{\beta_2}}\big)(t,x-y)\, .
h_{i}(x_{i})=\frac{\alpha (\theta_{i}+\theta_{3})(a+bx_{i})\eta^{\alpha-1}(x_{i})e^{-\eta^{\alpha}(x_{i})}\left( \Psi (x_{i})\right)^{\theta_{i}+\theta_{3}-1}}{1-\left( \Psi (x_{i})\right)^{\theta_{i}+\theta_{3}}},\i=1,2.
- i \Sigma ( p ; \Omega , s ) = \int{\frac{d^{4}k}{( 2 \pi )^{4}}}[ i D_{\mu \nu}( k ; \Omega , s ) ] ( - i e ) \gamma^{\mu}{\frac{i}{\rlap{/}p - \rlap{/}k - m}}( - i e ) \gamma^{\nu}.
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=be_3,[e_2,e_{n+1}]=be_2,[e_{i},e_{n+1}]=be_{i}+\sum_{k=i+2}^n{a_{k-i+3,3}e_k}, \displaystyle [e_{n+1},e_1]=-be_3,[e_{n+1},e_i]=-be_i-\sum_{k=i+2}^n{a_{k-i+3,3}e_k},(3\leq i\leq n,n\geq4).\end{array}
ie(\varphi \partial^{\mu}\varphi^{\ast}-\varphi^{\ast}\partial^{\mu}\varphi )+2e^{2}A^{\mu}|\varphi |^{2}+\mu \varepsilon^{\mu \nu \kappa\lambda}v_{\nu}\partial_{\kappa}A_{\lambda}=\partial_{\nu}F^{\mu \nu},
g_{\mathrm{mj}}\; \ddot x^{\mathrm{j}}+ \biggl({\partial g_{\mathrm{ml}}\over\partial x^{\mathrm{k}}}-{1\over 2}{\partial g_{\mathrm{kl}}\over\partial x^{\mathrm{m}}}\biggr) \dot x^{\mathrm{k}}\dot x^{\mathrm{l}}= -{\partial V\over \partial x^{\mathrm{m}}}
Y_1(t) =d \exp_{\gamma_2(s)}(t(V_1)_{t\dot\sigma_s(0)}),\,\, Y_2(t)=d \exp_{\gamma_3(s)}(t(V_2)_{t\dot\sigma_s^{-}(0)}), Y_1^{N}(t) =d \exp_{\gamma_2(s)}( t(V_1^N)_{t\dot\sigma_s(0)}),\,\, Y_2^{N}(t)=d \exp_{\gamma_3(s)}(t(V_2^N)_{t\dot\sigma_s^{-}(0)}),
L\left(\left(\prod_{s=0}^3Y_{1,-4s}^{u_{1, -4s}}\right)\cdot Y_{1,-6}^{u_{1, -6}}\cdot \left(\prod_{s=0}^2Y_{2,-2-4s}^{u_{2, -2-4s}}\right)\cdot \left(\prod_{s=0}^1Y_{2,-4-4s}^{u_{2, -4-4s}}\right)\cdot \left(\prod_{s=0}^4Y_{3,-1-2s}^{u_{3, -1-2s}}\right)\right)
-\ln\frac{|\xi-z|}{|\xi-z^*||z|}= -\frac{1}{2}\ln\left(1-\frac{(1-|\xi|^2)(1-|z|^2)}{|\xi-z^*|^2|z|^2}\right) \geq \frac{1}{2}\frac{(1-|\xi|^2)(1-|z|^2)}{|\xi-z^*|^2|z|^2},
\frac{F''(z)F(z)-(F'(z))^2}{(F(z))^2}=\sum_{n=0}^\infty \frac{q^{2n+1}(-2z^{-3}+z^{-4}q^{2n+1})}{(1-z^{-1}q^{2n+1})^2}-\sum_{n=0}^\infty \frac{q^{4n+2}}{(1-zq^{2n+1})^2}.
c(D_{i-1}^\perp/D_{i-1}- D_i/D_{i-1}- F_{q_i}/D_{i-1}) = c(V-D_i-F_{q_i})\cdot c(D_{i-1}-D_{i-1}^*) = c(i) \prod_{j=1}^{i-1}\frac{(1-t_j)(1-\beta t_j)}{1+t_j-\beta t_j},
\left\lbrace \begin{array}{cc}\partial_{\tau}A_1 +c_1\cdot\nabla A_1 = i \gamma_1\overline{A_2A_3}, \partial_{\tau}A_2 +c_2\cdot\nabla A_2 = i \gamma_2\overline{A_1A_3}, \partial_{\tau}A_3 +c_3\cdot\nabla A_3 = i \gamma_3\overline{A_1A_2}, \end{array}\right. \Omega,
\gamma \alpha_{0}= \partial_{\mu}\left( \varepsilon^{\mu \nu \lambda \rho}f_{\; \; b c}^{a}\left( C_{\rho a}A_{\nu}^{b}A_{\lambda}^{c}+ C_{a}\partial_{\rho}\left( A_{\nu}^{b}A_{\lambda}^{c}\right) \right) \right) .
\begin{aligned}\frac{\partial}{\partial t}h_{ij}= ~{\Phi}'\dot{F}^{kl}\nabla_k\nabla_lh_{ij}+{\Phi}'\ddot{F}^{kl,mn}\nabla_ih_{kl}\nabla_jh_{mn}+{\Phi}''\nabla_iF\nabla_jF &+{\Phi}'\left(\dot{F}^{kl}h^p_kh_{pl}-K_N\dot{F}^{kl}g_{kl}\right)h_{ij}&\quad-(\Phi+\Phi'F)h^k_ih_{kj}+K_N(\Phi+\Phi'F)g_{ij},\end{aligned}
I(K_B;Y_2^{n_1}S^{n_1})=I(X_1^BG_{K_B};X_1^{BC}S^n) &=H(X_1^BG_{K_B})-H(X_1^BG_{K_B}|X_1^{BC}S^n) =k_B-H(X_1^BG_{K_B}|X_1^{BC}S^n) &=k_B-H(\left[X_1^{B\emptyset}G_{K_B}^{I_{B\emptyset}}X_1^{BC}G_{K_B}^{I_{BC}}\right]|X_1^{BC}S^n) &=k_B - H(X_1^{B\emptyset}G_{K_B}^{I_{B\emptyset}}|X_1^{BC}S^n) &=k_B-H(X_1^{B\emptyset}G_{K_B}^{I_{B\emptyset}}|S^n),
\|u\|_{L^p_tL^q_z(I_t \times \mathbb{R}^m\times \mathbb{T}^n)}=\left(\int_{I_t}\left(\int_{\mathbb{R}^m\times \mathbb{T}^n}|u(t,z)|^q \, dz \right)^{\frac{p}{q}}\, dt\right)^{\frac{1}{p}}.
\| E_1 \|_p \lesssim\begin{cases}\| D^{s_1}f\|_{p_1}\cdot \| D^{s_2}g\|_{p_2}, ; \| D^{s_1}f\|_p \| D^{s_2}g \|_{\dot B^0_{\infty,\infty}}, ; \|D^{s_1}f \|_{\dot B^0_{\infty,\infty}}\cdot \|D^{s_2}g \|_p, .\end{cases}
J_n (k_1, \cdots, k_n) = \int_{0}^{1}d \tau_1 \cdots \int_{0}^{1}d \tau_n \; \exp \left[ \frac{i}{2}\sum_{a<b}^n(k_a \times k_b) (2 \tau_{ab}- \epsilon (\tau_{ab})) \right].
{\cal Q}_{N}( \vartheta ) = \mathop{\operatorname*{l i m}}_{\eta \rightarrow 0}\frac{1}{i}\operatorname{l o g}\frac{1 + ( - 1 )^{\delta}e^{i Z_{N}( \vartheta + i \eta )}}{1 + ( - 1 )^{\delta}e^{- i Z_{N}( \vartheta - i \eta )}}\: .
(x_1 - q y_1) z_1 = (q-1)(q^d-1) \big( (q+1) y_1^{-1}- q \tau \big), (x_1- q^{2d-3}y_1) z_d = q^{d-2}(q-1)(q^d-1) \big( (q+1) x_1 - \tau \big).
n_{t+1}\in\bigcap_{\ell=0}^d\left[B_1^\ell\cap \bigcap_{s\in\{1,...,\min\{2^\ell,t\}\}\setminus\{2^\ell\}}\bigcap_{1\leq j_1<\cdots<j_s\leq t}B_{s+1}^\ell(n_{j_1},...,n_{j_s})\right]\in p.
\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}\lambda^{2n}g^0(\lambda)+\lambda^{2n}C^{\mu,0}_{N}\big(e^{i\lambda}\big)\big|^2 &=\alpha_1|\lambda|^{2n}\big|1-e^{i\lambda\mu}\big|^{2n}\bigl(f^0(\lambda)-f_1(\lambda)\bigr)\bigl(f^0(\lambda)+\lambda^{2n}g^0(\lambda)\bigr)^2,