harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sup_{n \in \mathbb{N}}\sup_{0 \le t \le T}\left| \sum_{k=K}^\infty (k-2){\bar{B}}^n_k(t) \right| \le \sup_{n \in \mathbb{N}}\sum_{k=K}^\infty \frac{kn_k}{n}\le K^{-\varepsilon_{\boldsymbol{p}}}\sup_{n \in \mathbb{N}}\sum_{k=K}^\infty \frac{n_k}{n}k^{1+\varepsilon_{\boldsymbol{p}}}\to 0.
\delta^n_i|_U\colon \Gamma(g_{n-1}^{-1}(U), g_{n-1}^* F) = \Gamma(U, F^{n-1}) \to \Gamma(g_{n}^{-1}(U), g_n^* F) = \Gamma(U, F^n), \sigma \mapsto \sigma \circ d^n_i,
\nabla_{\mathsf{R}_z}\left( \begin{matrix}\alpha_1 \alpha_2 \alpha_3\end{matrix}\right) = \left( \begin{matrix}0 0 0 -1 0 0 0 0 0\end{matrix}\right)\left( \begin{matrix}\alpha_1 \alpha_2 \alpha_3\end{matrix}\right) ,
b_{\kappa,\alpha}=\frac{1}{N^{1/2}}\sum_{\kappa_{p},\alpha}e^{-i(\kappa_{p}\cdot x)}s_{\alpha}(\kappa_{p})\cdot[(\frac{m\omega_{p,\alpha}}{2\hbar})^{1/2}d(x)+i(\frac{1}{2\hbar m\omega_{p,\alpha}})^{1/2}p(x)],
Y^\kappa_t = L_{0, t}\, Y^\kappa_0 + \int_0^t L_{s,t}\, e^{\kappa A}F( Y^\kappa_{\varPi(s)}) \, ds + \int_0^t L_{s,t}\, e^{\kappa A}B( Y^\kappa_{\varPi(s)}) \, dW_s .
I_{7}\leq -\varepsilon\sum^{3}_{j=1}(\frac{1}{\rho}\mu(\theta)(\partial^{\alpha}\partial_{x_{j}}\widetilde{u}_{i}+\partial^{\alpha}\partial_{x_{i}}\widetilde{u}_{j}-\frac{2}{3}\delta_{ij}\nabla_{x}\cdot \partial^{\alpha}\widetilde{u}),\partial^{\alpha}\partial_{x_{j}}\widetilde{u}_{i}) &+C[\eta_0(1+t)^{-\vartheta}+\varepsilon^{\frac{1}{2}-a}]\varepsilon^{2-2a}.
\phi(p^{*}_1 \bigl( \{x_m \}\bigr)) = p^{*}_1 \phi(\bigl( \{x_m \}\bigr)) = p^{*}_1( \{j^{*}_my\})=p^{*}_2( \{j^{*}_my \})= \phi(p^{*}_2 \bigl( \{x_m \}\bigr)).
6 \frac{t_1^4}{4!}+ t_4 \frac{t_0^3}{3!}+ 4 \frac{t_0^2}{2!}\frac{t_2^2}{2!}+ 6 t_0 t_2 \frac{t_1^2}{2!}+ 3 t_1 t_3\frac{t_0^2}{2!}
\mathcal{U}_+=\Bigg\{u\in\mathcal{U}:\sum\limits_{y\in\mathcal{Y},s\in\mathcal{S}}p_S(s)p_{Y|X,S}(y|\psi(u,\ast),s) &\times\log\frac{p_{Y|X,S}(y|\psi(u,\ast),s)}{\sum_{x\in\mathcal{X}}p_{\hat{X}}(x)p_{Y|X,S}(y|x,s)}=\underline{C}(p_{Y|X,S},p_S)\Bigg\}.
\tilde{a}(n)=\min_{n}\Biggl(2^n \cdot \left(2 \cdot \cos\left( \frac{\pi \cdot (n-1)}{2}\right) -4 \cdot \sin \left( \frac{\pi \cdot (n-1)}{2}\right) +5 \right)+1, 2^n \cdot \left(4 \cdot \sin \left( \frac{\pi \cdot (n-1)}{2}\right) -2 \cdot \cos \left( \frac{\pi \cdot (n-1)}{2}\right) +5 \right)-1 \Biggr).
V_G^{(b)}(\Phi_0) = V_{GEP}^{2+1}(\Phi_0) + \frac{3{\hat{\lambda}}^2}{4 \pi^3}(\sqrt{x}-1) ( 1- \ln{4\over 3}) -{3\over{4 \pi^3}}{\hat{\lambda}}^2 \sqrt{x}\ln \sqrt{x}\; .
b_k(x,s)=\begin{cases}(s-\overline{u}_k(x))^{q_k(x)-1}s>\overline{u}_k(x), 0 \underline{u}_k(x)\leq s\leq \overline{u}_k(x), -(\underline{u}_k(x)-s)^{q_k(x)-1}s<\underline{u}_k(x).\end{cases}
\psi\leqslant\phi\Leftrightarrow\phi=\psi+\bigg(\sum_{\gamma\in\Phi^+_e\backslash\{\frac{\theta}{2},\theta\}}r_\gamma\gamma\bigg)_{|\mathfrak{h}^e},r_\gamma\in\mathbb{Z}_+\qquad(\forall\phi,\psi\in(\mathfrak{h}^e)^*).
\begin{bmatrix}B_k(\lambda) B_{k-1}(\lambda) \cdots B_{1}(\lambda)\end{bmatrix}(\Lambda_{k-1}(\lambda^\ell)\otimes I_n)= &(\Lambda_{k-1}(\lambda^\ell)^T\otimes I_m)\begin{bmatrix}B_s(\lambda) B_{k-1}(\lambda) \vdots B_1(\lambda)\end{bmatrix}=P(\lambda),
\textbf{H}=- \frac{1}{2}((d_{X}d_{X}x + d_{Y}d_{Y}x,g)\tilde{g}+ (d_{X}d_{X}x + d_{Y}d_{Y}x,\tilde{g})g)= \frac{1}{2}(\alpha + \beta)\tilde{g}\, mod\, G.
\left( \begin{matrix}{0}&{2 \partial + i ( T_{1}+ i T_{2})}\\{2 \bar{\partial}+ i ( T_{1}- i T_{2})}&{0}\\ \end{matrix}\right) \left( \begin{matrix}{c}\\{\bar{b}}\\ \end{matrix}\right) = ( \partial \! \! \! / + i T \! \! \! \! / ) \psi = 0 \ .
C_{N}(z_{1},\overline{z_{2}}) := \sum_{k=1}^{N}\mathbb{E}_{k-1}[Y_{k}(z_{1})Y_{k}(\overline{z_{2}})] \to \frac{1}{(i(\tau_1-\tau_2+i(\eta_1+\eta_2)))^{2}}, N \to \infty .
\pi_{1}^{\prime \prime}+ \pi_{1}^{\prime}\Bigl ({\frac{6}{\lambda \phi^{5}}}+{\frac{9}{\phi}}\Bigr ) + \pi_{1}\Bigl ({\frac{6}{\lambda \phi^{6}}}+{\frac{1 5}{\phi^{2}}}+{\frac{3 6 \pi}{\lambda \phi^{4}}}-{\frac{\lambda_{1}}{\pi \phi^{6}}}\Bigl ({\frac{6 \pi}{\lambda}}\Bigr )^{3 / 2}\Bigr ) = 0 \ .
I(M_1; Y_0^T|M_2) =\frac{1}{2}E\left[\int_0^T E[(X_1(t)+X_2(t)-\hat{X}_1(t)-\hat{X}_2(t))^2|M_2] dt \right] &=\frac{1}{2}\int_0^T E[(X_1(t)+X_2(t)-\hat{X}_1(t)-\hat{X}_2(t))^2] dt,
\begin{array}{c}P^s(b,c)\, \vec{\hat U}^{(s)\,a}_{\upsilon}(b,c)= \hat P^{(s)\,a}(b,c)\, \vec{\hat U}^{(s)\,a}_{\upsilon}(b,c)=\vec{\hat U}^{(s)\,a}_{\upsilon}(b,c)\, ; \hat P^{(s)\,f}(b,c)\, \vec{\hat U}^{(s)\,a}_{\upsilon}(b,c)=\vec 0,\qquad\quad a\neq f.\end{array}
{\frac{1}{g^{2}( q^{2})}}={\frac{R}{g_{5}^{2}}}+ \lambda_{k}( 2 k ) + \lambda_{T}( 2 T ) +{\frac{1}{1 6 \pi^{2}}}\int_{0}^{1}d x x \sqrt{1 - x^{2}}\ln N_{+ +}\left({\frac{x \sqrt{q^{2}}}{2}}\right) ,
W(u)_{aebcfd}=\sum_{\alpha,\beta,\gamma,\delta}S_{ae\alpha}S_{be\beta}\left(\delta_{\alpha\gamma}\delta_{ef}\delta_{\beta\delta}+Q^{-1/2}f_{0}\delta_{\alpha\beta}\delta_{\gamma\delta}+f_{1}\delta_{\alpha\beta\gamma\delta}+Q^{-1/2}f_{2}\delta_{\alpha\gamma}\delta_{\beta\delta}\right)S_{\gamma fc}S_{\delta fd}
\frac{\delta <H_A^{\theta}>}{\delta G(k)}=\frac{1}{4}[k^2-G^{-2}(k)+\frac{4\pi^2}{g^2}(2zk^{-2}G^{-3}(k)-\frac{\delta z}{\delta G(k)}4\pi^2 \int \frac{d^2p}{(2\pi)^2}p^{-2}G^{-2}(p))]
{\langle 0 \rangle}_{\Sigma}\left[{\bf x}_0^{\partial\Sigma}\right]=\int_{{\bf x}\nolinebreak\begin{picture}(10,3)(0,0)\setlength{\unitlength}{.5pt}\put(1,-7){\line(0,1){14}}\put(2,-7){$\scriptscriptstyle\partial\Sigma$}\end{picture}={\bf x}_0}\exp{\left(-S\left[{\bf x}^{\Sigma}\right]\right)}d\!{\bf x}^{\Sigma}\ .
\lim_{n \rightarrow \infty}\frac{\lambda_{j}^{n}}{\lambda_{k}^{n}}+ \frac{\lambda_{k}^{n}}{\lambda_{j}^{n}}+ (\lambda_{n}^{j})^{1/2}(\lambda_{n}^{k})^{1/2}|t_{j}^{n}- t_{k}^{n}| = \infty.
\mathbf A = \left( \frac{25}{12}\mathbf I - k\mathbf L \right), \tilde{\mathbf y^{n+4,(\ell)}}= k\tilde{\mathbf b}^{n+4}+ 4\tilde{\mathbf V}^{n+3,(\ell)}- 3\tilde{\mathbf V}^{n+2,(\ell)}+ \frac{4}{3}\tilde{\mathbf V}^{n+1,(\ell)}- \frac{1}{4}\tilde{\mathbf V}^{n,(\ell)}.
A_{j k}( x + y ) ={\frac{\biggl | \begin{matrix}{{c_{2}A_{m k}( x )}}&{{c_{2}A_{m k}( y )}}\\{{C_{m k}( x )}}&{{C_{m k}( y )}}\\ \end{matrix}\biggr |}{\biggl | \begin{matrix}{{G_{m k}( x )}}&{{G_{m k}( y )}}\\{{1}}&{{1}}\\ \end{matrix}\biggr |}},
{\boldsymbol S}_{D_k}(i)\bar{\boldsymbol \omega_k}(i) ={\boldsymbol S}_{D_k}(i) \Lambda_D^{-1}{\boldsymbol \Phi}_D^H{\boldsymbol p}_k(i) & ={\boldsymbol \Phi}_D \Lambda_D^{-1}{\boldsymbol \Phi}_D^H{\boldsymbol p}_k(i) & ={\boldsymbol \omega_k}^{(D)}(i),
\|R^j_+\|^{\mathbb{P}_{\mathcal{GS}^+_n}}_1 = \sum_{G \in \mathcal{GS}^+_n}R_+^j (G) \frac1{|\mathcal{GS}^+_n|}\le \left(U_n + |\mathcal{CGS}^+_n|\right) \frac1{|\mathcal{GS}^+_n|}= |\mathcal{CGS}^+_n|(1 + 2^{-n/2 + o(n)}) \frac1{|\mathcal{GS}^+_n|}= \|R_+\|^{\mathbb{P}_{\mathcal{GS}^+_n}}_1(1 + 2^{-n/2 + o(n)}).
I:= \frac{\alpha}{4\pi^2}\int_{\mathbb{R}^3}\mathrm{d}k \; \frac{\chi^2_{\Lambda}(k)}{|k|}(-1-\hat{k}_1^2+\hat{k}_2^2+\hat{k}_3^2) \cos(2k_1y)\left\langle x u_{\alpha}\,\bigg|\, \frac{(h_{\alpha}-e_{\alpha})|k|}{3(h_{\alpha}-e_{\alpha}+|k|)}\,\bigg|\, x u_{\alpha}\right\rangle.
\frac{\pi}{2}=\prod_{n=1}^{\infty}(1+\frac{1}{n})^{(-1)^{n+1}}=\prod_{n=1}^{\infty}(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1})=(\frac{2}{1}\cdot\frac{2}{3})(\frac{4}{3}\cdot\frac{4}{5})(\frac{6}{5}\cdot\frac{6}{7})\cdot\cdot\cdot
c_{l,m}\;=\;q^{m-l}\,i\tau_l^2\tau_l^1 \;+\;q^{l-m}\,i\tau_m^2\tau_m^1 \;-\;\eta^{m-l}\,i\tau_l^2\tau_m^1 \;-\;\eta^{l-m}\,i\tau_m^2\tau_l^1 \,
\begin{aligned}\int_{B_{2\Lambda_{n,j,d}^{+}}(0)}\rho_n \overline{h}_j(z)e^{\overline{\tilde{u}_n^{(1)}}(z)}e^{-\lambda_{n,j}^{(1)}}\Big(\frac{1-e^{\overline{\tilde{u}_n^{(2)}}-\overline{\tilde{u}_n^{(1)}}}}{\|\tilde{u}_n^{(1)}-\tilde{u}_n^{(2)}\|_{L^\infty(M)}}\Big)\mathrm{d}z =o\Big(\frac{1}{\lambda_{n,j}^{(1)}}\Big).\end{aligned}
\begin{array}{r c l}{{{\hat{\cal K}}^{( 2 )}}}&{{=}}&{{d{\hat{\omega}}^{( 1 )}+ l_{p}^{- 2}\, \partial{\hat{X}}^{M}\partial{\hat{X}}^{N}( i_{\hat{k}}{\hat{C}}^{( 3 )})_{M N}\, ,}}\\ \end{array}
-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{(z+c)^{2}}{c^{2}}=1\leftrightarrow-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}+\frac{2z}{c}=0.
\mathcal{F}( P_{W_{p,0}}(f)) = \sum_{q = 0}^{p}\frac{[ \mathcal{F}f , \mathcal{F}\psi_{q}^{\star}]}{[ \mathcal{F}\psi_{q}^{\star}, \mathcal{F}\psi_{q}^{\star}]}\mathcal{F}\psi_{q}^{\star}.
-m^2(\rho_1^2\partial^2_{\rho_1}\hat{\phi}+\rho_1\partial_{\rho_1}\hat{\phi}+\rho_2^2\partial^2_{\rho_2}\hat{\phi}+ \rho_2\partial_{\rho_2}\hat{\phi})+ 2(k_{1,0}k_{2,0}-{\bf k_1}.{\bf k_2})\rho_1\rho_2\partial_{\rho_1}\partial_{\rho_2}\hat{\phi}+m^2\hat{\phi}+\lambda\hat{\phi}^3=0
\begin{cases}\vec{r}_{k\pm 1}=-\frac{1}{2}\vec{r}_k\mp \vec{\rho}_k, \vec{\rho}_{k\pm 1}=\pm\frac{3}{4}\vec{r}_k-\frac{1}{2}\vec{\rho}_k, \end{cases}k\in\mathbb{Z}\mspace{1.5mu}\diagup_{\mspace{-4.5mu}\{3\}}.
\|u_{B}- u_D\|_{L^H(D)}= |u_{B}- u_D|\, \|1 \|_{L^H(D)}\le\frac{\|1 \|_{L^H(D)}}{|D|}\|u-u_{B}\|_{L^1(D)}&\le C \frac{\|1 \|_{L^H(D)}\|1 \|_{L^{H^*}(D)}}{|D|}\|u-u_{B}\|_{L^H(D)}
\frac{\sigma(Q_0)}{|Q_0|}\|\chi_{Q_0}\|_{\mathcal{M}^p_q(dx,w)}\lesssim_{[\sigma]_{A_\infty}}\|\chi_{Q_0\setminus\Omega_1}\cdot\sigma\|_{\mathcal{M}^p_q(dx,w)}=\|\chi_{Q_0\setminus\Omega_1}\|_{\mathcal{M}^p_q(dx,\sigma)}.
\big(p_{G,r}(x)\big)^{|B|{-}k}\big(p_{G,\emptyset}(x){-}xp_{G,r}(x)\big)^{k}{=}(x-q)^{\ell|B|}\big(\underbrace{\ell(q^2-qx-1)/(x-q)}_{=Y}\big)^k=(x-q)^{\ell|B|-1}Y^k.
a(n) \bigl\lbrack \hat{a}(n-1) + \hat{b}(n-1) \bigr\rbrack + b(n)\bigl\lbrack \hat{a}(n) +\hat{b}(n) \bigr\rbrack = d(x){\hat{d}}(x)- \hat{\Lambda}( x), \hat{a}(n-1) \bigl\lbrack a(n-1) + b(n-1) \bigr\rbrack +\hat{b}(n-1) \bigl\lbrack a(n) + b(n)\bigr\rbrack ={\hat{d}}(x) d(x)- \Lambda ( x).
\Bigl|\Bigl(\frac{V( \xi_1+\lambda^{-1}y)}{\lambda^{\frac{N-2}2}}\Bigr)^{2^*-1}-\Bigl(\frac{P_1( \xi_1+\lambda^{-1}y)}{\lambda^{\frac{N-2}2}}\Bigr)^{2^*-1}\Bigr| \le C \Bigl( \frac1{(1+|y|)^4}\frac{|\ln k|^{\sigma_N}}{\lambda^{\frac{N-2}2}}+ \frac{|\ln k|^{(2^*-1)\sigma_N}}{\lambda^{\frac{N+2}2}}\Bigr).
\overline{F}(k,t+1;p,n) = \begin{dcases}\frac{(n-k)p}{n}\overline{F}(k-1,t;p,n) + \frac{n - (n-k)p}{n}\overline{F}(k,t;p,n) \textnormal{if}\; k \le t; 0 \textnormal{if}\; k > t. \end{dcases}
\left\|u\right\|_{\mathcal{C}^{k,\alpha}_{\boldsymbol{\beta}}}:=\sum_{j=0}^{k}\sup_{\dot{M}}\rho^{-\boldsymbol{\beta}+j}|\nabla^{j}u| + \sup_{x\neq y \in \dot{M}}\frac{|\rho^{-\boldsymbol{\beta}+k}(x)\nabla^{k}u(x)-\rho^{-\boldsymbol{\beta}+k}(y)\nabla^{k}u(y)|}{d(x,y)^{\alpha}}.
(e_g \, e_h) (\phi) = (e_g \otimes e_h) (\phi^{(1)}\otimes \phi^{(2)}) = e_g( \phi^{(1)}) e_h( \phi^{(2)}) = \langle g, \phi^{(1)}\rangle_{\mathcal{C}}\, \langle h, \phi^{(2)}\rangle_{\mathcal{C}}.
(-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}\left(y_s(u)-y_s(v)\right)^2 =\frac{1}{2}\sum\limits_{e\in \partial S}\frac{\delta_E(e)}{|e|^2}2|S\cap e||e-S|\frac{1}{|S|}\le\frac{1}{4}\sum\limits_{e\in \partial S}\delta_E(e)\frac{1}{|S|}\le \frac{1}{4}\delta_E(\max)\frac{|\partial S|}{|S|}.
\Hat{\mathrm{D}}_k = \mu_k \, e^{-\frac{\mu_k^{-1}\theta \mathrm{W}}{\mathrm{G}_{\mathrm{S,TX}}\,\mathrm{G}_{\mathrm{RX}}}}\prod_{j = 1, j \neq k}^a \left(\frac{1}{\mu_j}- \frac{1}{\mu_k}\right)^{-1}.
\frac{\hat{N}_1(L_1,S_1,U_1)}{D_1(L_1,S_1,U_1)}-\frac{N_0(L,S,U)}{D_0(L,S,U)}&=\frac{\hat{N}_1(L_1,S_1,U_1)D_0(L,S,U)-N_0(L,S,U)D_1(L_1,S_1,U_1)}{D_1(L_1,S_1,U_1)D_0(L,S,U)}
{\cal L}_{\mathrm{Y - M}}^{\prime}= G_{\alpha \beta}^{I}G_{\gamma \delta}^{I}\epsilon^{\alpha \beta \mu \nu}\epsilon^{\gamma \delta \rho \tau}\nabla_{\mu}\phi^{a}\nabla_{\nu}\phi^{b}\nabla_{\rho}\phi^{c}\nabla_{\tau}\phi^{d}\eta_{a c}\eta_{b d}.
\int d{\bf k}\frac{|{\bf k}|}{(k_{0}^{\mbox{\scriptsize eff}})}\Psi({\bf k},{\bf q};\omega)=\frac{\alpha}{\left\{1-\left(g^{2}/4\pi\right)(\xi+1)\int \frac{d{\bf k}'}{(k_{0}^{\mbox{\scriptsize eff}})'}\frac{({\bf k}')^{2}}{[\{(k_{0}^{\mbox{\scriptsize eff}})'\}^{2}-\omega^{2}/4+i\epsilon]}\right\}}
\lim_{M\to\infty}\mathrm{Pr}\left(\zeta>\!\tau\right)\le\lim_{M\to\infty}\mathrm{Pr}\!\left(\!\!\frac{\sigma_{\theta}^2N}{4M}\frac{\max_i\frac{d_i^{\alpha}}{\sigma_{v,i}^2}}{\min_i d_i^{\alpha}}\chi_{(2N)}^2\!>\!\tau\!\!\right)=0\;,
{\rm i)}\ \Delta U = a(a-1) r^{a-2}|\nabla r|^2 + a r^{a-1}\Delta r \leq C (1 + r^a){\rm ii)}\ \epsilon U^{1+\epsilon}= \epsilon r^{a(1+\epsilon)}\leq 1 + a^2 r^{2a-2}|\nabla r|^2{\rm iii)}\ \nabla^2 U = a r^{a-1}\nabla^2 r + a (a-1) r^{a-2}\mathbf{d}r^2 \geq -B.
J_{dyn}(W)=\frac{1}{2}\mathcal{B}_{10}\left((\tilde{F}_W, \tilde{G}_W, \tilde{H}_W), (\tilde{F}_W, \tilde{G}_W, \tilde{H}_W)\right)=\mathcal{J}_1(W, \tilde{F}_W, \tilde{G}_W, \tilde{H}_W)
[y_{(1\,p)},y_{(1\,l)}]_c=-3\xi^2 \lambda_{112}\chi_{(1\,l)}(g_{(2\,p)})y_{(3\,l)}y_{(2\,p)}+3\lambda_{112}\chi_{1}(g_{(3\,p)})y_{(3\,p)}y_{(2\,l)}.
V_{i}( k_{i}, \epsilon_{i}; z_{i}, \bar{z}_{i}) = ( \epsilon_{i}\cdot \partial X ( z_{i}) + i k_{i}\cdot \psi ( z_{i}) \epsilon_{i}\cdot \psi ( z_{i}) ) \mathrm{e}^{i k_{i}\cdot X ( z_{i}, \bar{z}_{i})}.
J_{T_{*}}^{(1)}= (-1)^{1+\tfrac{3}{m}}\frac{2 \pi \sigma^{3}}{m T_{*}^{3/m}}\sum_{k=0}^{\infty}\frac{1}{k!}\Gamma \left( \frac{nk-3}{m}\right) (-1)^{(m-n)k/m}T_{*}^{(n-m)k/m},
\sum_{k=p_{n}}^{q_{n}}<P_{k}x^{*}_{n},x^{(n)}_{k}>=(\sum_{k=p_{n}}^{q_{n}}\|P_{k}x^{*}_{n}\|^{p^{*}})^{\frac{1}{p^{*}}}>\epsilon_{0}^{\frac{1}{p^{*}}}.
L_{10}(q) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{6n^2 + 1- 4j^2 -j}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 6n + 2- 4j^2 -3 j}
\int q_u(x_1,x_2) \; dP_0 (x_1)= \underset{y \in \mathcal{Y}}{\overset{}{\sum}}\frac{K(y|x_2,u)}{\P_0^{Y|U=u}(y)}\int K(y|x_1,u) \; dP_0 (x_1)=\underset{y \in \mathcal{Y}}{\overset{}{\sum}}K(y|x_2,u)=1,
H=\frac{z_A(z_A-1)(z_A - z_3)}{z_3(z_3-1)}\left\{b_A^2 - (\frac{\mu_1}{z_A}+\frac{\mu_2}{z_A-1}+\frac{\mu_3-1}{z_A-z_3})b_A +\frac{\kappa}{z_A(z_A-1)}\right\}.
S(\alpha)= \sum_{\substack{1\le\mathbf{x}\le X 1\le\mathbf{y}\le X 1\le\mathbf{z}\le X}}\Lambda(\mathbf{x})\Lambda(\mathbf{y})\Lambda(\mathbf{z})e\Big(\alpha\big(\mathbf{x}^TA_1\mathbf{x}+2\mathbf{x}^TB\mathbf{y}+\mathbf{y}^TA_2\mathbf{y}+2\mathbf{y}^TC\mathbf{z}+\mathbf{z}^TA_3\mathbf{z}\big)\Big),
\frac{\partial^{2}\psi}{\partial r^{2}}-\frac{1}{r}\frac{\partial\psi}{\partial r}+\frac{\partial^{2}\psi}{\partial z^{2}}=\frac{2\Omega^{2}}{U}r^{2}-\frac{4\Omega^{2}}{U^{2}}\psi
T_{\mu \nu}\sp{\underline{{a}}}= \partial_{\mu}E_{\nu}^{\underline{{a}}}- \partial_{\nu}E_{\mu}^{\underline{{a}}}= -{\frac{\lambda}{2}}\epsilon_{\mu \nu \rho}\sum_{\beta}\xi_{\beta}\sp{\underline{{a}}}\dot{x}\sp{\rho , \beta}\delta ( \vec{x}- \vec{x}\sp{\beta}) .
q_i \circ g =(p_iq_{i-1}q_0) \circ g =( p_i \circ g ) (q_{i-1}\circ g ) (q_0 \circ g)= p_iq_{i-1}q_0 &=q_i, i=1,\ldots,r, q_{r+1}\circ g =q_0^s \circ g = (q_0 \circ g)^s=q_0^s &=q_{r+1},
f_{i}^{\sigma}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})-f_{i}^{\sigma}(\vec{x},t)+F_{i}=\frac{1}{\tau_{f}^{\sigma}}(f_{i}^{\sigma,eq}(\rho^{\sigma},v^{\sigma})-f_{i}^{\sigma})
A_i =(Z_{1i},{(Z_{1i}^2-1)}/{2})^\tau, B_i =((Z_{2i}^2-1)/2, (Z_{1i}^2-1)(Z_{2i}^2-1)/2,Z_{1i}(Z_{2i}^2-1)/2, - (Z_{2i}^4-6Z_{2i}^2+3)/12)^\tau.
\begin{array}{l}\partial_{t}\phi - \nabla \cdot (A(t,x) \nabla \phi) - eA(t,x)e \partial_{zz}\phi -2e A(t,x)\partial_{z}\nabla \phi +q(t,x)\cdot\nabla\phi+q(t,x)\cdot e\partial_{z}\phi+c\partial_{z}\phi=f(t,x,\phi), \end{array}
n^{-1/2}\sum_{r = \lceil pn \rceil}^n \exp(-n\epsilon (r/n - p)^2) \le n^{-1/2}\sum_{r=\lceil pn \rceil}^{\lceil pn \rceil + \ell-1}\sum_{m=0}^\infty \exp(-n\epsilon ((r + m\ell)/n - p)^2).
(T^{(1)}_{n \times p})' e_i= \sum_{j=0}^{p-1}a^{(1)}_{-j}\chi_{[1, p]}(i+j) e_{i+j}+ \sum_{j=1}^{n-1}a^{(1)}_{j}\chi_{[1, p]}(i-j) e_{i-j}= \sum_{j=-(p-1)}^{n-1}a^{(1)}_{j}\chi_{[1, p]}(i-j) e_{i-j}.
\delta U=\int_{L}\int_{A}(\sigma_{xx}\delta\epsilon_{xx}+2\sigma_{xz}\delta\epsilon_{xz})dAdL=\int_{L}\int_{A}[-z\sigma_{xx}\frac{\partial(\delta\varphi)}{\partial x}+\sigma_{xz}\kappa(-\delta\varphi+\frac{\partial(\delta w)}{\partial x})]dAdL
\sum_Y \int_{\Delta'_Y}u^*P_{DH}'d\sigma = \int_{\Delta'}\Big{(}u^*u^{*,i,j}_{\mathrm{ref},i,j}P_{DH}'+u^*2u^{*,i,j}_{\mathrm{ref},j}P_{DH,i}' +u^*u^{*,i,j}_{\mathrm{ref}}P_{DH,i,j}'-u^{*,i,j}_{\mathrm{ref}}u^*_{i,j}P_{DH}'\Big{)}.dp
d(o, s_1 h s_2 o) = d(o, s_1 o) + d(s_1 o, s_1 h o) + d(s_1 h o, s_1 h s_2 o) - (o, s_1 h o)_{s_1 o}- (s_1 o, s_1 h s_2 o)_{s_1 h o}+ O(\delta) & \geq 3 D - 4 C + O(\delta) \geq D
RD_{A}=\frac{\frac{\rho_{s}}{\rho_{w}}-\frac{\rho_{a}}{\rho_{w}}}{1-\frac{\rho_{a}}{\rho_{w}}}=\frac{RD_{V}-\frac{\rho_{a}}{\rho_{w}}}{1-\frac{\rho_{a}}{\rho_{w}}}.
p(Q) = \sum_{P:Q\subset P\subset R}\frac{\ell(Q)}{\ell(P)}\,\Theta(P) + \sum_{P\supsetneq R}\frac{\ell(Q)}{\ell(P)}\,\Theta(P) & \leq 2B\Theta(R) + p(R) \leq 2B\Theta(R) + c_{db}\,\Theta(R).
\Delta_{2,1}^\mu(t) = i P_\mu \int_0^t \mathbf S(t-s) \left[\mathbf N\left( \Theta_R^{\mathbf u^\mu}(s) P_\mu \mathbf u^\mu(s) \right) - \mathbf N\left( \Theta_R^{\mathbf u}(s) \mathbf u(s) \right) \right] \, ds, \Delta_{2,2}^\mu(t) = i (1-P_\mu) \int_0^t \mathbf S(t-s) \mathbf N\left( \Theta_R^{\mathbf u}(s) \mathbf u(s) \right) \, ds.
\det\left(\begin{array}{cccc}\cos\theta e^{i\alpha}-R\sin\theta e^{i\alpha}iR\cos\theta e^{i\alpha}\sin\theta e^{i\beta}R\cos\theta e^{i\beta}iR\sin\theta e^{i\beta}\cos\theta e^{-i\alpha}-R\sin\theta e^{-i\alpha}-iR\cos\theta e^{-i\alpha}\sin\theta e^{-i\beta}R\cos\theta e^{-i\beta}-iR\sin\theta e^{-i\beta}\end{array}\right)=-R^3\sin2\theta.
\int\limits_{\lambda<p^2, q^2 \leq n}\!\!\!\!\!\!\!\!\! dp \, dq \,\: a_q^* \: \frac{1}{(H_f + p^2 + q^2 - \mu)^2}\: a_p \qquad= \int\limits_0^\infty \! ds \int\limits_0^\infty \! dt \! \int\limits_{\lambda<p^2, q^2 \leq n}\!\!\!\!\!\!\!\!\! dp \, dq \, \: a_q^* \: e^{-(s+t)q^2}e^{-(s+t)(H_f - \mu)}e^{-(s+t)p^2}\: a_p.
m_{R}(T^{-1}A) =\int_{K_{u}}m_{0}\bigl(R(T^{-1}A-x) \bigr)\,d\nu_{u,T}(x)= \int_{K_{u}}m_{0}\bigl(R(T^{-1}(A-Tx)) \bigr)\,d\nu_{u,T}(x) &=\int_{T(K_{u})}m_{0}\bigl(R(A-x) \bigr)\,d\nu_{u,T}(x)=m_{R}(A).
i_1 <... < i_k, \; \; \; j_1 <... < j_{b-k}\{i_1 ,..., i_k\}\cup \{j_1 ,... , j_{b-k}\}=\{2,...,b+1\}\{i_1 ,..., i_k\}\cap \{j_1 ,... , j_{b-k}\}= \emptyset.
S_{i n t}= \frac{g}{4 ! ( 2 \pi )^{d}}\int d p_{1}d p_{2}d p_{3}d p_{4}\, e^{- i p_{1}\wedge p_{2}- i p_{3}\wedge p_{4}}\varphi ( p_{1}) \varphi ( p_{2}) \varphi ( p_{3}) \varphi ( p_{4}) \delta ( p_{1}+ p_{2}+ p_{3}+ p_{4}) .
\kappa^{s}\Omega(sp)-\kappa\Omega(p)= (\kappa^{s-1}-1)\cdot\kappa\Omega(sp)+\kappa\cdot\big(\Omega(sp)-\Omega(p)\big) \equiv \frac{s-1}{2}\cdot\big((\kappa^{2}-1)\cdot\kappa\Omega(3p)+\kappa\cdot(\Omega(3p)-\Omega(p))\big) = \frac{s-1}{2}\cdot\big(\kappa^{3}\Omega(3p)-\kappa\Omega(p)\big)\pmod{p^2}.
\begin{gathered}\forall v\in E_s^*(X,x),\ \|(d\varphi_{t}^X(x)^T)^{-1}v\|\leq Ce^{-\lambda t}\|v\|, \forall v\in E_u^*(X,x),\ \|(d\varphi_{-t}^X(x)^T)^{-1}v\|\leq Ce^{-\lambda t}\|v\|,\end{gathered}
J(w) =J(w^+)+J(w^-) & +\int_{(w^-)}\int_{(w^+)}G\left( \frac{w^+(x)-w^-(y)}{\vert x-y\vert^{\alpha}}\right)\frac{dxdy}{\vert x-y\vert^{d}}&+ \int_{(w^+)}\int_{(w^-)}G\left( \frac{w^-(x)-w^+(y)}{\vert x-y\vert^{\alpha}}\right)\frac{dxdy}{\vert x-y\vert^{d}}.
\frac{{\rm d}}{{\rm d}t}\|\beta_n(t)\|^q_{L^q}=q \langle |\beta_n(t)|^{q-2}\beta_n(t), \Delta \beta_n(t)\rangle-q\langle |\beta_n(t)|^{q-2}\beta_n(t)v_n(t), \nabla \xi_n(t)\rangle &=-q(q-1)\||\beta_n(t)|^{\frac{q-2}{2}}\nabla \beta_n(t)\|^2_{L^2}-q\langle |\beta_n(t)|^{q-2}\beta_n(t)v_n(t), \nabla \xi_n(t)\rangle.
\mathcal{C}' =\{\langle g\rangle\in\mathcal{C}(M)\mid \langle g\rangle\notin \mathcal{C}_i^p \textrm{for every}p\textrm{and}i,\,\mathrm{fix}_\Delta(g)\geq c\}, \mathcal{C}'' =\{\langle g\rangle\in\mathcal{C}(M)\mid\mathrm{fix}_\Delta(g)<c\}.
\frac{\partial}{\partial t}\eta_{\alpha \bar{\beta}}=\Delta_{b}\eta_{\alpha \bar{\beta}}+2R_{\alpha \bar{\gamma}\mu \bar{\beta}}\eta_{\gamma \bar{\mu}}-(R_{\gamma \bar{\beta}}\eta_{\alpha \bar{\gamma}}+R_{\alpha \bar{\gamma}}\eta_{\gamma \bar{\beta}})
\left[ \frac{d^2}{dt^2}+3H\frac{d}{dt}+\left[\frac{k^2}{a_o^2}+m^2(T_o)+\frac{\lambda_R T^2_c}{24}\right]e^{-2Ht}-\frac{\lambda_R T^2_c}{24}\right]\varphi^H_k(t) = 0
[e_1,e_2]+J[Je_1,e_2] + J[e_1,Je_2]-[Je_1,Je_2] = [e_1,e_2]+J[Xu,e_2] + J[e_1,Au]-[Xu,Be_1+Ce_2]-[Ye_1+Ze_2,Au] = -X\theta J e_2+AJe_1+XBe_1+XC\theta e_2 -AYe_1-AZ\theta e_2 = (-X\theta B+AY+XB-AY)e_1+(-X\theta C+AZ+X\theta C-AZ\theta)e_2+ &+(-XA\theta+AX )u
\Delta_{1}= \Delta_{1}^{*}= \widetilde{\Delta_{6}}= \Delta_{6}^\dagger &\Delta_{2}^{*}= \widetilde{\Delta_{2}}=\Delta_{5}= \Delta_{5}^\dagger, \Delta_{3}^\dagger =\Delta_{3}^{*}= \Delta_{4}= \widetilde{\Delta_{4}}, .
\Pi_{\beta_i}=\{\delta_1-\delta_2,\ldots, \delta_{k'-2}-\delta_{k'-1}, \cr \delta_{k'-1}-\epsilon_1(=\beta_{i+1}), \cr \epsilon_1-\epsilon_2,\ldots,\epsilon_{n-1}-\epsilon_n; \cr \epsilon_n-\delta_{k'}(=-\beta_i),\cr \delta_{k'}-\delta_{k'+1},\ldots,\delta_{m-1}-\delta_m\}k<m,
N_P(r)\geqslant \sum_{i=s,t,l}(N(r,\min\{\nu_{(f^u,H_i),\leqslant k_i};1\leqslant u\leqslant 3\})-N^{(1)}_{(f,H_i),\leqslant k_i}(r)) &+ 2\sum_{i=1}^qN^{(1)}_{(f,H_i),\leqslant k_i}(r)+o(T(r)).
R_j(u) =(u_0,u_1,\dots,u_{j-1}, u_{j+1},u_{j+2},\dots,u_d),j=0,1,\dots,d; G_z(u) =\left(u_0z_1+u_1,u_0z_2+u_2,\dots,u_0z_d+u_d\right).
\mathcal{T}(*^V,L) =-\tfrac14\sum_{i<j}\left(\lambda_i\lambda_j\, w_iw_j\otimes 1+1\otimes v_iv_j\right)\circ\left(\mu_{ij}\, w_iw_j\otimes 1+1\otimes v_iv_j\right) &=\tfrac14\sum_{i<j}\left(-\lambda_i\lambda_j+w_iw_j\otimes v_iv_j\right)\circ\left(-\mu_{ij}+w_iw_j\otimes v_iv_j\right),
G_{\mu \nu}= \left( \begin{array}{c c c}{- \frac{2 \pi \alpha^{\prime}}{\theta_{0}}}&{0}&{- \frac{2 \pi \alpha^{\prime}}{\theta_{1}}}\\{0}&{\frac{2 \pi \alpha^{\prime}}{\theta_{0}}+ \biggl ( \frac{2 \pi \alpha^{\prime}}{\theta_{1}}\biggr )^{2}}&{0}\\{- \frac{2 \pi \alpha^{\prime}}{\theta_{1}}}&{0}&{1}\\ \end{array}\right) ,
K_{I}=\frac{1}{2\sqrt{\pi a}}\int_{-a}^{a}F_{y}(x)\sqrt{\frac{a+x}{a-x}}dx,K_{II}=-\frac{1}{2\sqrt{\pi a}}(\frac{\kappa-1}{\kappa+1})\int_{-a}^{a}F_{y}(x)dx,.
\sum_{h \geq 1}\frac{\rho_j(h)}{h^{s + w - \frac{1}{2}}}= L(s + w - \tfrac{1}{2}, \mu_j) \sum \frac{\sigma_{1-2w}(h)}{h^{s + \frac{1}{2}- w}}= L(s, E(\cdot, w)) = \zeta(s + w - \tfrac{1}{2})\zeta(s - w + \tfrac{1}{2}),
\xi''(x) +\xi'(x)(2 -m)\tanh x -\xi(x)(m-1)\big(\tanh^2x-\frac{1}{\cosh^2x}\big) &+\frac{p-2}{\cosh x}\frac{d}{dx}\big(\frac{\cosh x}{m}(\xi'(x)-(m-1)\tanh x \xi(x))\big) &+\frac{\hat{\lambda}}{\cosh^{2}x}\xi(x)=0,
[\beta_n,\alpha_{-m}] = \frac{\gamma_{n,-m+1}[\beta_{-1},\alpha_{n-m+1}]-(n+1) \gamma_{n-1,-m+1}\alpha_{n-m}}{1-m}&= \frac{(n-m+1)\gamma_{n,-m+1}-(n+1) \gamma_{n-1,-m+1}}{1-m}\alpha_{n-m}.
X_{n+1}^{\delta,\tau,\Delta t}=S_{\Delta t}X_{n}^{\delta,\tau,\Delta t}+\Delta t S_{\Delta t}G_\delta\bigl(X_{n}^{\delta,\tau,\Delta t}\bigr)+e^{\tau A}S_{\Delta t}\sigma_\delta\bigl(X_{n}^{\delta,\tau,\Delta t}\bigr)\Delta W_n,
\begin{cases}t_0 := \max \bigl\{t \in [0,1] \mid H_{j}^{\eta}(t) = m_{j}^{\eta}\bigr\}, begin{align*}1.5mm]t_1 := \min \bigl\{t \in [t_0,1] \mid H_{j}^{\eta}(t) = m_{j}^{\eta}+ 1 \bigr\};\end{cases}
\langle 0|\varphi^{2}(x)|0\rangle =\langle0_{{\mathrm{m}}}|\varphi^{2}(x)|0_{{\mathrm{m}}}\rangle +\langle\varphi^{2}(x)\rangle_b^{(a)}+ \langle \varphi^{2}(x)\rangle_b^{(b)}+\langle \varphi^{2}(x)\rangle^{(ab)},
\deg_{X_1}F_i^{(n+1)}- \deg_{X_1}F_i^{(n)}& = \sum_{j=1, j\ne i}^m s_{ij}\deg_{X_1}F_j^{(n)}+ (s_{ii}-1) \deg_{X_1}F_i^{(n)}& < d^n \big(\sum_{j=1}^{m}s_{ij}-1 \big) < d^n (d-1).
\begin{aligned}H_2(u):=\frac{1}{2}\int_{\mathbb{T}}u_x^2\,dx, H_3(u):=\int_{\mathbb{T}}c_1\,u_x^3+c_2\,u_x^2\,u+c_3\,u^3 \,dx, &H_4(u):=\int_{\mathbb{T}}c_4\, u_x^4+c_5\,u_x^3\,u+c_6\,u_x^2\,u^2+c_7 \,u^4 \,dx, H_{\geq 5}(u):=\int_{\mathbb{T}}f_{\geq 5}(x, u, u_x) \,dx.\end{aligned}
C^{i}=\frac{1}{4}\left(S_{2i-1,2i}^{2}-S_{2i-1,2i}-\frac{3}{4}\right), C^{ij}=\frac{1}{4}\left(S_{2i-1,2i,2j-1,2j}^{2}-S_{2i-1,2i,2j-1,2j}-\frac{3}{4}\right).
\mathcal{Z=}\int DA_{i}D\varphi^{+}D\varphi \exp \left[ -\int d^{3}x\left( \frac{1}{4}F_{ij}F_{ij}+\left( \mathcal{D}_{i}{\varphi}\right)^{+}\left( \mathcal{D}_{i}{\varphi}\right) +V(\varphi )\;\right) \right] \;,
\begin{array}{r c l}\psi(p) = <p|\psi>=\int_{-\infty}^\infty\,dq<p|q><q|\psi>=\int_{-\infty}^\infty\,\frac{dq}{\sqrt{2\pi\hbar}}\,e^{-\frac{i}{\hbar}qp}\,\psi(q)\ \ ,\ \ \psi(q) = <q|\psi>=\int_{-\infty}^\infty\,dp<q|p><p|\psi>=\int_{-\infty}^\infty\,\frac{dp}{\sqrt{2\pi\hbar}}\,e^{\frac{i}{\hbar}qp}\,\psi(p)\ .\end{array}
\widehat{D}\theta = d \theta + \frac{1}{4}\hat{\omega}^{\hat{r}\hat{s}}\Gamma_{\hat{r}\hat{s}}+ \hat{e}^{\hat{r}}T_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}\widehat{F}_{\hat{s}\hat{t}\hat{u}\hat{v}}\theta