harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|N(s) \backslash N(S-s \cup \{a\})| = |N(S \cup \{a\}) \backslash N(S-s \cup \{a\})| = |N(S \cup \{a\})| - |N(S-s \cup \{a\})| &\geq |N(S \cup \{a\})| - |N(S)| &\geq k.
g_1 =3e_0-e_1-2e_2-e_3-e_4-e_5, g_2 =3e_0-e_1-e_2-2e_3-e_4-e_5, g_3 =3e_0-e_1-2e_2-e_4-e_5, g_4 =3e_0-e_1-2e_3-e_4-e_5.
R_{\mathcal{BS}(\alpha)}(\mathcal{S}^{*}) =\begin{dcases}\dfrac{1}{3-2\alpha}, 0\leq \alpha\leq \frac{1}{9}, \dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{1+16\alpha}}, \frac{1}{9}\leq\alpha \leq 1.\end{dcases}
\sum_{\|y\|_1 \leq R}(1 - c(\xi^*_{x+y})) = O_{B,R_1}(1), \sum_{\|y\|_1 > R}(1 - c(\xi^*_{x+y})) = O_{B,R_1}\left( R^{-2}\right), \left| \sum_{\|y\|_1 \leq R}s(\xi^*_{x+y}) \right| \leq \sum_{\|y\|_1 \leq R}\left| s(\xi^*_{x+y}) \right| = O_{B,R_1}(\log R).
W^{gen}\kappa \partial_{\kappa}\Gamma = [W^{gen}, \kappa \partial_{\kappa}] \Gamma + \kappa \partial_{\kappa}W^{gen}\Gamma = - (\kappa \partial_{\kappa}W^{gen}) \Gamma + \kappa \partial_{\kappa}(\chi \Delta_{br}+ \chi_A \Delta_{br_A})
\begin{array}{l l}{{\hat{\gamma}^{\alpha}= \gamma^{\alpha}\otimes 1 \! \! 1_{3 2}}}&{{\alpha = 0 , 1 , 2 , 3}}\\{{\hat{\gamma}^{a}= \gamma_{\parallel}\otimes \gamma^{a}}}&{{a = 4 , 5 , . . . , 1 3}}\\ \end{array}\hspace{5 . 5 c m}
\begin{array}{l}\overline{\bf{y}}=\left[{\begin{array}{*{20}{c}}{{a_1\lambda_1}}{}{}{}\ddots{}{}{}{{a_4 \lambda_4}}\end{array}}\right] \! \! \left[{\begin{array}{*{20}{c}}{{V_{11}}}\hdots{{V_{14}}}\vdots \ddots \vdots{{V_{41}}}\hdots{{V_{44}}}\end{array}}\right] \mathbf{d}+{{\bf{n}}}\end{array}
\begin{array}{c c c}{\pi}&{:}&{\Omega^{*}({\cal A}) \longrightarrow{\cal L}({\cal H})}\\{\pi_{p}( ( D F_{1}) . . . ( D F_{p}) F )}&{=}&{\prod_{i = 1}^{p}[ D , \pi_{0}( F_{i}) ] \pi_{0}( F ) ,}\\ \end{array}
\frac{\#\mathcal{A}(x)}{V\tilde{\lambda}_1^2}\sum_{\substack{d_0,e_0 (d_0,e_0,W_0)=1}}\frac{\tilde{\lambda}_{d_0}\tilde{\lambda}_{e_0}}{[d_0,e_0]}\sideset{}{'}\sum_{\substack{\mathbf{d},\mathbf{e}\in\mathcal{D}_k (de,d_0e_0)=1}}\frac{\lambda_{\mathbf{d}}\lambda_{\mathbf{e}}}{[\mathbf{d},\mathbf{e}]}.
M_{np}=\begin{cases}\delta_{p,n+1}\{\rm if}\ \zeta_{n+1}=0\ , \sqrt{\mu_{n+1}}\gamma_{n+1}\frac{\langle f_p,1 \rangle}{(\lambda_p-\lambda_n-1) \langle f_{n+1}| 1\rangle}\{\rm if}\ \zeta_{n+1}\ne 0\ .\end{cases}
L_{\mathrm{f l u c}}^{\mathrm{s t r i n g}}= \frac{\pi}{1 2 R_{p}}- \sum_{i = 1}^{2}\frac{\omega v_{i}\bar{\gamma}_{i}}{\pi}\left[ \ln \left( \frac{M m_{i}}{\sigma}\right) + 1 \right] + \frac{1}{2}\omega + \omega f ( v_{1}, v_{2}) \, .
F \epsilon^{i j}={\frac{1}{2}}E^{{\alpha}i}\wedge E^{{\beta}j}\epsilon_{{\alpha}{\beta}}+{\frac{1}{2}}\bar{E}^{\dot{\alpha}i}\wedge \bar{E}^{\dot{\beta}j}\epsilon_{\dot{\alpha}\dot{\beta}}+{\frac{1}{2}}E^{a}\wedge E^{b}f_{a b}\epsilon^{i j}, \quad d F \equiv 0 .
2(g_{7}\!+\!g_{17})\!=\!0, 2(g_{18}\!-\!g_{3})\!=\!3, &2(g_{19}\!+\!g_{2})\!=\!0, -2(g_{26}\!+\!g_{16})\!=\!1, &2(g_{5}\!+\!g_{25})\!=\!0, 2(g_{6}\!-\!g_{22})\!=\!1, &2(g_{15}\!-\!g_{30})\!=\!1, 2(g_{8}\!+\!g_{28})\!=\!2, &2(g_{21}\!+\!2g_{10})\!=\!0, -2(g_{31}\!+\!g_{14})\!=\!1, & 2(g_{11}\!-\!g_{27})\!=\!1, 2(g_{23}\!+\!g_{12})\!=\!2,
\left[ a\epsilon -b \delta\right]^2[\epsilon^2+\delta^2 ]+ \left[ c\epsilon -d \delta \right]^2[\epsilon^2+\delta^2 ] = \left[a\epsilon^2 - (b+c)\epsilon\delta +d\delta^2\right]^2 + \left[c\epsilon^2+(a-d)\epsilon\delta-b\delta^2\right]^2.
\sum_{\substack{J\subseteq I |J|=l}}w_S(\varepsilon(J)) =\sum_{j=0}^{l}\sum_{\substack{J\subseteq I |J|=l,\,|J\cap S|=j}}(-1)^{j}=\sum_{j=0}^{l}(-1)^{j}\binom{|S|}{j}\binom{m-|S|}{l-j}.
Y_{1}= \left( \begin{array}{c c c c}{{x_{1}}}&{{0}}&{{0}}&{{\cdots}}\\{{0}}&{{x_{1}}}&{{0}}&{{\cdots}}\\{{\vdots}}&{{0}}&{{x_{1}}}&{{\cdots}}\\{{\vdots}}&{{\vdots}}&{{0}}&{{\ddots}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) .
w ( \epsilon ) \, = \, \frac{1}{2}\, \sum_{n = 1}^{\infty}\, \sqrt{n^{2}\, + \, \epsilon^{- 1}}\, = \, \frac{1}{4}\, \left( \sum_{n = - \infty}^{+ \infty}\, \sqrt{n^{2}\, + \, \epsilon^{- 1}}\, - \, \frac{1}{\sqrt{\epsilon}}\right) \,{,}
A_{> R}^{(l)}= \bigoplus_{i\in I^{(l)}}\overline{f_i^{\frac{1}{2}}C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}}}A_{> R}= \bigoplus_{l = 0}^{5(d+1)-1}A_{> R}^{(l)}= \bigoplus_{i\in I}\overline{f_i^{\frac{1}{2}}C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}}}
\int \frac{d^dq_1}{N_d \pi^{d/2}}\int \frac{d^dq_2}{N_d \pi^{d/2}}\frac{\partial}{\partial q_1^{\mu}}\frac{q_1^{\mu}}{(q_1^2 + 1)^a (q_2^2 + 1)^b ((q_1+q_2)^2 + 1)^c}= 0
\widehat{Y}_{i}(x) = \int^{\xi \infty}_{0}\frac{P_{\lambda}(x, s)}{s-b_{i}}Y(s) \,{\rm d}_{q}s, i=0,\dots,N, \widehat{Y}(x) = \begin{pmatrix}\widehat{Y}_{0}(x) \vdots \widehat{Y}_{N}(x) \end{pmatrix}\! .
\Delta K ( p_{1 j}^{2}; m_{0}, M , \epsilon ) = \frac{- i}{3 2 \pi^{4}}\frac{1}{M^{2 \epsilon}}\left[ \Delta I ( \frac{p_{1 2}^{2}}{M^{2}}, \frac{m_{0}}{M}, \epsilon ) + \Delta I ( \frac{p_{1 3}^{2}}{M^{2}}, \frac{m_{0}}{M}, \epsilon ) + \Delta I ( \frac{p_{1 4}^{2}}{M^{2}}, \frac{m_{0}}{M}, \epsilon ) \right] .
-\frac{O_A^{(1)}O_A^{(ga^{-1})}O_A^{(1)}}{\big(Q^2-Q^{-2}\big)^2}=D_A^{-1}O_A^{(g)}(p/s,s)D_A, \frac{QO_A^{(1)}O_B^{(ga^{-1})}-Q^{-1}O_B^{(ga^{-1})}O_A^{(1)}}{Q^2-Q^{-2}}=D_A^{-1}O_B^{(g)}(p/s,s)D_A.
\sum_{|\lambda/\mu|=n}\frac{2^{|\lambda|-\ell(\lambda)-|\mu|+\ell(\mu)}f_{\lambda/\mu}H_\mu}{H_{\lambda}}\left(\sum_{\square\in\lambda}\binom{c_{\square}}{2}-\sum_{\square\in\mu}\binom{c_{\square}}{2}\right)=\binom{n}{2}+n|\mu|.
\pi_{a_{0},a_{1}}\left( \mathbf{Z}_{0},\mathbf{Z}_{1};P\right) \equiv P\left[A_{1}=a_{1}|A_{0}=a_{0},\mathbf{Z}_{0},\mathbf{Z}_{1}\right], &\pi_{a_{0}}\left( \mathbf{Z}_{0};P\right) \equiv P\left[ A_{0}=a_{0}|\mathbf{Z}_{0}\right].
| r \int_{6r}^{\infty}dw \lambda''(t+w) w \partial_{t}\left(\frac{1}{(\lambda(t)^{2-2\alpha}+w^{2})}- \frac{1}{\lambda(t+w)^{2-2\alpha}+w^{2}}\right)| &\leq \frac{Cr}{t^{3}\log^{b+2}(t)}
\sqrt{- g}L \equiv{\frac{1}{4 \pi{\cal A}_{P}}}\left( \sqrt{- g}{\cal G}\right) ={\frac{1}{4 \pi{\cal A}_{P}}}\left( \sqrt{- g}g^{i k}\left( \Gamma_{i \ell}^{m}\Gamma_{k m}^{\ell}- \Gamma_{i k}^{\ell}\Gamma_{\ell m}^{m}\right) \right) .
{{\cal I}}_{\mathrm{1 - l o o p}}^{\mathrm{h e t}}= \frac{V^{( 8 )}T_{2}}{2^{8}\pi^{4}}\Delta_{F_{+}^{4 - l}F_{-}^{l}}t_{8}F_{+}^{4 - l}F_{-}^{l},
\langle \Phi | C_{\alpha}| \Psi \rangle = \int_{\alpha}{\cal D}^{4}\! A \, \Phi^{*}[{\bf A}^{\prime \prime}, t^{\prime \prime}) \delta [ G ] \Delta_{G}[ A ] e^{i S [ A ]}\Psi [{\bf A}^{\prime}, t^{\prime}) ,
\lim_{t\to 0^+}\left[\sup_{x}\int_{|z-x|< \sqrt{t}}\frac{|q(z)|}{|z-x|^{d-2}}\,dz\right]=0\,, \mbox{for}d\geq 3\,, \lim_{t\to 0^+}\left[\sup_{x}\int_{|z-x|< \sqrt{t}}|q(z)|\ln \frac{t}{|z-x|^2}\,dz\right]=0\,, \mbox{for}d= 2\,, \left[\sup_{x}\int_{|z-x|< 1}|q(z)|\,dz\right]<\infty\,, \mbox{for}d= 1\,.
H_2^{'}(p) = \int \frac{d^3q}{(2\pi)^3}\, \frac{1}{(p - q)^2}\frac{\left[ H_1^{'}(q) \sin^2\theta+H_2^{'}(q)\cos^2\theta \right]}{\left|-\frac{1}{2}\delta+iq_3 + \frac{\left|{\bf q}\right|^2}{2m}+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|) \right|^2}.
-\int_{\Omega}(u-\bar{u})\Delta \varphi \leq \int_{\Omega}\left(\bar{g}(x,u)-fh(\bar{u})\right)\varphi &=\int_{\Omega}\chi_{\{u\leq \bar{u}\}}\left(\bar{g}(x,u)-fh(\bar{u})\right)\varphi.
i\frac{\partial h}{\partial z}=-\frac{1}{z-\lambda}+\frac{\alpha}{z}-\frac{\lambda}{1-\lambda z}, i\frac{\partial^{2}h}{\partial z^{2}}=\frac{1}{(z-\lambda)^{2}}-\frac{\alpha}{z^{2}}-\frac{\lambda^{2}}{(1-\lambda z)^{2}}, i\frac{\partial^{3}h}{\partial z^{3}}=-\frac{2}{(z-\lambda)^{3}}+\frac{2\alpha}{z^{3}}-\frac{2\lambda^{3}}{(1-\lambda z)^{3}}.
\begin{aligned}p-\alpha \tilde{p}= p(0)((1-\alpha A)\det(I-DP(z)) - \alpha BP(z) (I-DP(z))C) &= p(0)(1-\alpha A)\det( I-DP(z) -\frac{\alpha}{1-\alpha A}CB P(z)) &= p(0)(1-\alpha A) \det( I - V_{\alpha}P(z)).\end{aligned}
{\cal L}_{G}= - \frac{1}{4}( B_{\mu \nu}- Q_{\mu \nu})^{2}+ \frac{1}{2}( A_{\mu}+ \partial_{\mu}\theta )^{2}+ \frac{1}{2 m}\epsilon_{\mu \nu \rho \sigma}( B^{\mu \nu}- Q^{\mu \nu}) \partial^{\rho}( A^{\sigma}+ \partial^{\sigma}\theta ) ,
b(m)=b(3k+1) =\sqrt{2}\cdot b(k)+b(k+1) &\leqslant (\sqrt{2}+1)\cdot\max\{b(k),b(k+1)\}&\leqslant (\sqrt{2}+1)\left(h(k+1)+\lfloor \log_3(k+1)\rfloor\right),
0\leq C_\infty(0) - C_t(0) =:\eta_t \leq c\log\Big(1+\frac{1}{m^2t}\Big){\bf 1}_{d=2}+ c m\Big(\sqrt{1 + \frac{1}{tm^2}}- 1\Big){\bf 1}_{d=3}, &0\leq \|C^3_\infty\|_{L^1}- \|C^3_t\|_{L^1}:= \gamma_t \leq \frac{c}{m^2(m^2t+1)}{\bf 1}_{d=2}+ c\log\Big(1+\frac{1}{m^2t}\Big){\bf 1}_{d=3}.
g_{-}= \left( \begin{array}{c c}{1}&{0}\\{\frac{x_{-}}{\sqrt{2}}}&{1}\\ \end{array}\right) g_{0}= \left( \begin{array}{c c}{e^{\frac{x}{2}}}&{0}\\{0}&{e^{- \frac{x}{2}}}\\ \end{array}\right) g_{+}= \left( \begin{array}{c c}{1}&{\frac{x_{+}}{\sqrt{2}}}\\{0}&{1}\\ \end{array}\right) .
\mathcal{B}_{-}(\lambda )=\left( -1\right)^{\mathsf{N}}(-a_{-}(\lambda)A(\lambda )B(1/\lambda )+b_{-}(\lambda )A(\lambda )A(1/\lambda)-c_{-}(\lambda )B(\lambda )B(1/\lambda )+d_{-}(\lambda )B(\lambda)A(1/\lambda )),
Z_{in}=\frac{s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}=\frac{L_{1}}{L_{2}}Z(\frac{1}{1+\frac{Z}{sL_{2}}})(1+\frac{1-k^{2}}{\frac{Z}{sL_{2}}})
\xi(x,t) \ge \sup_{\varepsilon>0}(g(x,0)-2\varepsilon-(L_g+N_2+\|f\|_\infty)t) &=g(x,0)-(L_g+\|f\|_\infty)t-\inf_{\varepsilon>0}2\varepsilon-N_2t &=g(x,0)-(L_g+\|f\|_\infty+N_2)t.
\|f-\mathcal{L}_{n}f\|_{\ell^2(X)}=\min_{p\in \mathbb{P}_n^d(K)}{\|f-p\|_{\ell^2(X)}}=\min_{p\in \mathbb{P}^d_{n}(X)}{\|f-p\|_{\ell^2(X)}}\;.
\Omega_{75}\equiv 8\pi \overline{G}\, \rho^{\rm matter}_0 /[3 (75\ \hbox{km s}^{-1}\hbox{Mpc}^{-1})^2] = \rho^{\rm matter}_0 / 1.0568 \times 10^{-29}\hbox{g cm}^{-3}\ .
0\geq -\frac{\partial G}{\partial \nu}(x_0, t_0)= \frac{\partial G}{\partial x_n}(x_0, t_0) = \frac{2w_kw_{kn}}{|Dw|^2}+ah_n=\frac{2w_{1n}}{w_1}- a = \frac{2u_{1n}}{w_1}- \frac{2\varphi_{1}h_n+2\varphi_{n}h_1+2\varphi h_{1n}}{w_1}- a = \frac{2u_{1n}}{w_1}-\frac{-2\varphi_{1}+2\varphi h_{1n}}{w_1}- a = 2{\nu^1}_1 - a \geq 2 a_0 - a > 0,
\int_{-t_0}^{t_0}t\exp\left(-\frac{t^2}{2}I''(\xi)\log^2 y\right) dt = 0, \int_{-t_0}^{t_0}|t|^{k}\exp\left(-\frac{t^2}{2}I''(\xi)\log^2 y\right) dt \ll_k (I''(\xi)\log^2 y)^{-\frac{k+1}{2}}\ll_k (\log x \log y)^{-\frac{k+1}{2}},
\left|A_{\nu,K}(z,\delta z)\right| \leq 2\sum_{s=K}^\infty \left(\frac{e\,|\delta z|}{2s}\right)^s \leq 2\sum_{s=K}^\infty s^{-s}\sum_{s=K}^\infty \left(\frac{e\,|\delta z|}{2}\right)^s \leq 5.2\left(\frac{e\,|\delta z|}{2}\right)^K,
\mu_\circ(B(x ,\rho))= \lambda^{\frac{d-\alpha}{2}}|S\cap B(x ,\rho)|\lesssim \lambda^{\frac{d-\alpha}{2}}(j+1)\lambda^{-\frac12}\rho^{d-1}\lesssim \lambda^{-\frac{\alpha - (d-1)}{2}}T \rho^{d}\le \rho^d \le \rho^\alpha.
P_{\rm modout}=\frac{\sum\limits_{l=1}^{N_{\rm b}}(\sum\limits_{m=1}^{\beta}x_{lm}^{2}T_{x}+\sum\limits_{m=1}^{\beta}a^{2}x_{lm}^{2}T_{x})}{N_{\rm b}T_{b}}& \thickspace =\frac{(1+a^{2})\sum\limits_{l=1}^{N_{\rm b}}\sum\limits_{m=1}^{\beta}x_{lm}^{2}}{2\beta N_{\rm b}},
\tilde{\varphi_t}^\star\big((\gamma^i_{jk})_{_{F_0}}\big)=(\gamma^i_{jk})_{\tilde{\varphi_t}^\star(F_0)}, &\tilde{\varphi_t}^\star\big(G^i_{_{F_0}}\big)=G^i_{\tilde{\varphi_t}^\star(F_0)}.
\mho_\Lambda(a) :=4\pi\bigg(\frac{6(1-({1\wedge\Lambda})^{1-2|a|})}{1-2|a|}+\frac{8}{(1+2|a|)(1\vee\Lambda)^{1+2|a|}}+\frac{1}{|a|(1\vee\Lambda)^{2|a|}}\bigg).
c_\nu\; =\;p^+{\textstyle\Big(\frac{\Gamma(2B)}{\Gamma(B)}\,\frac{1+\nu+B}{1+\nu}\Big)^{\!-1}}d_\nu\; =\;q^+{\textstyle\Big(\frac{\Gamma(2B)}{\Gamma(B)}\,\frac{1+\nu+B}{1+\nu}\Big)^{\!-1}},
\begin{array}{r c l}{{\varphi_{1}^{+}\varphi_{2}^{+}\varphi_{4}^{-}}}&{{=}}&{{\varphi_{4}^{-}\varphi_{1}^{+}\varphi_{2}^{+}+ D_{2 4}\varphi_{1}^{+}+ D_{1 4}\varphi_{2}^{+},}}\\ \end{array}
\widetilde{A}_{\mu}( x ) = 4 \pi \int \delta \xi d s \epsilon_{\mu \nu \rho \sigma}{\Omega}_{\xi}( s , 0 ) T^{i}{\Omega}_{\xi}^{- 1}( s , 0 ) L_{i}^{\rho \sigma}[ \xi | s ] \dot{\xi}^{\nu}( s ) \delta^{4}( x - \xi ( s ) ) .
\eta e^{\displaystyle(\frac{h}{\eta})^\gamma}\Gamma\big( 1+\frac{1}{\gamma},(\frac{h}{\eta})^\gamma\big)=h\int_0^{+\infty}e^{-u}\big( 1+(\frac{\eta}{h})^\gamma u\big )^{\frac{1}{\gamma}}\,du.
\tau_n(\xi)^* \tau_n(\eta) = \tau^\infty(j_n(\xi))^*\tau^\infty(j_n(\eta)) = \pi^\infty(\langle j_n(\xi), j_n(\eta) \rangle) = \pi^\infty( j_n(\langle \xi, \eta \rangle)) = \pi_n(\langle \xi, \eta \rangle),
N_{2 2}^{+}( x , y , a , b ) = - \frac{1}{8 a b}\sum_{n = 1}^{\infty}\cos ( \frac{2 n \pi x}{a}) \sum_{k = 0}^{\infty}\frac{e^{- 2 \left( 1 - \frac{y}{b}+ k \right) \frac{n \pi}{a}b}}{\left( 1 - \frac{y}{b}+ k \right)}.
e^{- 2 \hat{\phi}}= e^{- 2 \hat{\phi}_{0}}+{\frac{2 \tilde{m}_{k}}{r}}\ , \qquad A_{u}= ( A_{u})_{0}+{\frac{2 \hat{m}_{k}}{r}}\ , \qquad A_{i}= ( A_{i})_{0}+{\frac{2 ( q_{k})_{i}}{r}}\ , \qquad i = 4 , \dots 9 .
U_i = \max_{t\in[m]}\frac{\sup_{x\in \mathbb{R}^m:~\hat{c}_i^T x=1}\nabla_t H_i (x)}{c_{i,t}}L_i = \min_{t\in[m]}\frac{\inf_{x\in \mathbb{R}^m:~\hat{c}_i^T x\leq 1}\nabla_t H_i (x)}{c_{i,t}}
\int_{0}^{T}\rho^2(t,1)dt =-\int_{0}^{1}x(\rho^2(T,x)-\rho_0^2(x))dx+\int_{0}^{T}\int_{0}^{1}\rho^2dxdt-2\int_{0}^{T}\int_{0}^{1}x\rho u_xdxdt &\leq 2\int_{0}^{T}\int_{0}^{1}\rho^2dxdt+\int_{0}^{T}\int_{0}^{1}u_x^2dxdt+\int_{0}^{1}\rho_0^2(x)dx.
\delta_{p}(M,\eta) = \P\left( \big\|\big((\Sigma +M)^{1/2}- \Sigma^{1/2}\big) Z\big\|_p \geq \eta \right) \leq \frac{\phi_p(d)}{\eta}\big\|(\Sigma +M)^{1/2}- \Sigma^{1/2}\big\|_2.
M(p) = \frac{e^2}{\pi}m\left[\frac 34\ln\left(1 + \frac{L^{\prime\,2}}{m^2}\right) + \frac{L^\prime}{2m}\arctan\frac m{L^\prime}-\frac 18\frac{L^{\prime\,2}/m^2}{1 + L^{\prime\,2}/m^2}\right].
-[\alpha_j,[f_0^x,f_i^y]]+[\alpha_j,[f_0^y,f_i^x]] =[\alpha_j,-[f_0^x,f_i^y]+[f_0^y,f_i^x]] =[\alpha_j,\sum_{m+n=i,~m,n>0}[f_m^x,f_n^y]-f_0^{\lambda_i(x,y)}-f_i^{[x,y]}-\sum_{m+n=i,~m,n>0}f_m^{\lambda_n(x,y)}].
c^{-4}\alpha_{T}=d^{2}\left( a^{2}+16ab+16b^{2}\right) ,c^{-6}\beta_{T}=d^{3}\left( a+8b\right) \left( -a^{2}-16ab+8b^{2}\right), c^{-12}\gamma_{T}=b^{4}c^{2}d^{7}\left( a+16b\right) .
\frac{\int_{S}\sum_{n_{1}= 1}^{\infty}\left| \sum_{\gamma \in \Gamma_{\infty}^{2}\backslash \Gamma_{1}^{2}}\nabla_{\tau}\phi_{n_{1}}^{0}( \tau ) \big |_{\gamma \circ \tau}\right|^{2}d V}{\int_{S}\sum_{n = 1}^{\infty}\left| \sum_{\gamma \in \Gamma_{\infty}^{2}\backslash \Gamma_{1}^{2}}\phi_{n_{1}}^{0}( \tau ) \big |_{\gamma \circ \tau}\right|^{2}d V}< 1 0 \lambda .
\iota_{n}(t_{\mu}) T_{(s(\mu), s(\mu))}= \Big(\sum_{\lambda \in s(\mu) \Lambda^{<n}}T_{(\mu, \lambda)}\Big) T_{(s(\mu), s(\mu))}= \Big(\sum_{\lambda \in s(\mu) \Lambda^{<n}}T_{(\mu, \lambda)}T_{(s(\mu), \lambda)}\Big) T_{(s(\mu), s(\mu))}= T_{(\mu, s(\mu))}.
\begin{aligned}\int_{\Omega_\ell}(A(X_2)\nabla u_\ell).\nabla u_\ell \geq \int_{\Omega_\ell}(A_{22}(X_2)\nabla_{X_2}u_\ell).\nabla_{X_2}u_\ell-\frac{|A_{12}(X_2)\nabla_{X_2}u_\ell|^2}{a_{11}(X_2)}&\geq \Lambda^1\int_{\Omega_\ell}u_\ell^2\,.\end{aligned}
\chi_\tau(t)=\sum_{j=k}^\infty(b_j-a_{j+1})=\sum_{j=k}^\infty (\tau_{j}-\tau_{j+1})\left(1-\frac{1}{j+1}\right)= \tau_k-\sum_{j=k}^\infty\frac{\tau_j-\tau_{j+1}}{j+1},
U^{\prime}= \left( \begin{array}{c c}{U_{o o}^{\prime}}&{U_{o e}^{\prime}}\\{U_{e o}^{\prime}}&{U_{e e}^{\prime}}\\ \end{array}\right) , \bar{U}^{\prime}= \left( \begin{array}{c c}{U_{o o}^{\prime}}&{- U_{o e}^{\prime}}\\{- U_{e o}^{\prime}}&{U_{e e}^{\prime}}\\ \end{array}\right) .
G_C=\begin{bmatrix}I_k\otimes M\tilde{D}_\alpha M^T+(A\widetilde{\tilde{\otimes}}M)(I_{N-k}\otimes\tilde{D}_\alpha)(A\widetilde{\tilde{\otimes}}M)^T (A\widetilde{\tilde{\otimes}}M)(I_{N-k}\otimes \tilde{D}_\alpha M_p^T) (I_{N-k}\otimes M_p\tilde{D}_\alpha)(A\widetilde{\tilde{\otimes}}M)^T I_{N-k}\otimes M_p\tilde{D}_\alpha M_p^T\end{bmatrix}.
F(k)=F(ab^2)= \sum_{a=r\ell_1'n_1'\ell_2'n_2'}h(r) f_1(\ell_1')f_2(\ell_2')g_1(n_1')g_2(n_2') \hskip 120pt \cdot \sum_{b=b_1b_1'=b_2b_2'}f_1(b_1)g_1(b_1')f_2(b_2)g_2(b_2') \prod_{p|(b_1,b_2)}t(p).
\aligned K(t,u) = \frac{1}{2}(|t|+|u|-|t-u|) = \begin{cases}~u, ~t>0,~u>0,~t-u>0, ~t, ~t>0,~u>0,~t-u<0, ~0, ~t<0,~u>0,~t-u<0, ~-u, ~t<0,~u<0,~t-u<0, ~-t, ~t<0,~u<0,~t-u>0, ~0, ~t>0,~u<0,~t-u>0, \end{cases}\endaligned
x\otimes E_{o_-}(\lambda n_a) =x\sigma(E_{o_{-,Q}}^Q(\lambda_Q^+))^{-1}\otimes E_{o_-}(\lambda_Q^+)E_{o_-}(\lambda n_a) & \in Cx\sigma(E_{o_{-,Q}}^Q(\lambda_Q^+))^{-1}\otimes E_{o_-}(\lambda_Q^+\lambda n_a).
E_{f}= -{\frac{\cos{\pi s}}{\pi}}\mu^{2 s}\sum_{l}( l + \frac 1 2 ) \int_{m}^{\infty}d k [ k^{2}- m^{2}]^{1 / 2 - s}\frac{\partial}{\partial k}[ \ln f_{l}( i k ) - \ln f_{l}^{a s}( i k ) ]
E(\varphi_{\varepsilon,k};k,R) = E(\varepsilon \psi;k,R) + E( \xi u_{k};k,R) + \varepsilon^{2}\int_{B_{R}(0)}\left( (\xi u_{k})^{2}\cdot \chi_{B_{R}(0)}* Y_{a_{k}}\right) \psi^{2}.
F^0(\gamma_1) = q_2 \frac{1}{2}Y^{(1)}+ q_2^2 \left[\left(\frac{q_1^4}{\eta^{96}}\right)P_{46}(E_4,E_6) + k E_2 \left(\frac{1}{2}Y^{(1)}\right)^2\right] + \mathcal{O}(q_2^3) ,
\mathbf{H}_{k,}= \frac{1}{N}\sqrt{\frac{\kappa}{\kappa+1}}\sum_{p=1}^{L_k}\alpha_{k,p}\sum_{n_1=0}^{N_1-1}\sum_{n_2=0}^{N_2-1}\chi_{k,p,n_1\cdot N_2+n_2}\mathbf{a}^H(\phi^).
\widehat{\lambda}_l (t, \boldsymbol{\theta}) = \widehat{\gamma}_l (\boldsymbol{\theta}) t^2 + \widehat{\mu}_l (\boldsymbol{\theta}) t^3 + \ldots, l = 1, \ldots, n, \widehat{\varphi}_l (t, \boldsymbol{\theta}) = \widehat{\omega}_l (\boldsymbol{\theta}) + t \widehat{\psi}^{(1)}_l (\boldsymbol{\theta}) + \ldots, l = 1, \ldots, n.
\Phi_{2, t, \alpha}(f)(x_1, x_2) = \lim_{\epsilon \to 0}\frac{1}{d_{\alpha}}\int_{|y|\geq \epsilon}\frac{e^{it(\varphi(x_1,x_2+y)- \varphi(x_1,x_2))}- 1}{|y|^{1 + \alpha}}\, f(x_1, x_2+y)\, dy
(\rho_i)_t - \sigma(\rho_i)_{xx}- ((\Psi_i)_x \rho_i)_x + \sum\limits_{j,\j\neq i}^{}\nu_{ji}\rho_i = \sum\limits_{j,\j\neq i}^{}\nu_{ij}\rho_j, \i=1,\dots, N,
Q_P(x,x) = \sum_{i=1}^{n}w'(v_i) c_i^2 + \sum_{i=1}^{n-1}(c_i - c_{i+1})^2 &\geq w'(v_j) c_j^2+c_1^2 + (c_1 - c_{2})^2 + \dots + (c_{n-1}- c_n)^2 + c_n^2 &\geq |c_1|+ |c_1 - c_{2}| + \dots + |c_{n-1}- c_n| + |c_n| &\geq w'(v_j) |c_j|+2|c_j| &\geq w(v_j) |c_j|
\zeta_{\bar{{\cal V}}}(s)=\frac{mL}{\Gamma(s) \sqrt{4 \pi}}\int_0^\infty \!\! d\beta \, \beta^{s-\frac{3}{2}}e^{-\beta}=\frac{mL}{\sqrt{4 \pi}}\frac{\Gamma(s-\frac{1}{2})}{\Gamma(s)}\,.
E_{n,k}=E_{n,0}+\frac{\hbar^{2}k^{2}}{2m}+\frac{\hbar^{2}}{m^{2}}\sum_{n^{\prime}\ne n}\frac{|\langle u_{n,0}|k\cdot p|u_{n^{\prime},0}\rangle|^{2}}{E_{n,0}-E_{n^{\prime},0}}
\left\{x^\alpha,p_\beta \right\}_{D(\Phi^{(1)}_{3n})}= \delta^\alpha_\beta \,,\,\,\,\,\,\,\, \left\{\psi^n,\psi^m \right\}_{D(\Phi^{(1)}_{3n})}=\frac{i}{2}\eta^{nm}\,,
\psi_{1}( p^{\mu}) ={\cal C}\psi_{2}^{\ast}( p^{\mu}) = \left( \begin{matrix}{{- i ( E^{0}+ i B^{0})}}\\{{E^{1}+ i B^{1}}}\\{{E^{2}+ i B^{2}}}\\{{E^{3}+ i B^{3}}}\\ \end{matrix}\right) \, \, , \, \, \psi_{2}( p^{\mu}) ={\cal C}\psi_{1}^{\ast}( p^{\mu}) = \left( \begin{matrix}{{- i ( E^{0}- i B^{0})}}\\{{E^{1}- i B^{1}}}\\{{E^{2}- i B^{2}}}\\{{E^{3}- i B^{3}}}\\ \end{matrix}\right) \, ,
\left|D_{x}^{m}\left(|x|^{-d}\textnormal{H}_{k,0}^{p}(R)\right)\right|\leq N|x|^{-d-m}\sum_{l=1}^{m}\left|d\textnormal{H}_{k,l-1}^{p}(R)+2\textnormal{H}_{k,l}^{p}(R)\right|.
2\beta_i = 2(\alpha_i + \cdots +\alpha_{n-1})+\alpha_n (i=1,\dots, n), \beta_i-\beta_j= \alpha_i + \dots +\alpha_{j-1}(i<j) , \beta_1+\beta_j = \alpha_i + \dots +\alpha_{j-1}+2(\alpha_j+\cdots +\alpha_{n-1})+\alpha_n (i=1,\dots,n) .
E\left[(M_{i+1}- M_i)^2 \mid \mathcal{F}^X_i \right] = E[(X_{i+1}-X_i - E[X_{i+1}-X_i \mid \mathcal{F}^X_i ] )^2 \mid \mathcal{F}^X_i ] &= 1 - E[X_{i+1}-X_i \mid \mathcal{F}^X_i ]^2.
W [{\Phi}] = \frac{1}{2}\int_{- \infty}^{\infty}d x^{+}\int_{0}^{L}d x_{1}^{-}\int_{0}^{L}d x_{2}^{-}{\Phi}^{\star , + +}( x_{1}^{-}, x^{+}| x_{2}^{-}, x^{+}) ({\pi}_{( 1 )}^{-}+{\pi}_{( 2 )}^{-}
\left\{\stackrel{\wedge}{\psi}^{\dagger}, \stackrel{\wedge}{\psi}\right\}= 1 , \quad \left\{\stackrel{\wedge}{\psi}, \stackrel{\wedge}{\psi}\right\}= \left\{\stackrel{\wedge}{\psi}^{\dagger}, \stackrel{\wedge}{\psi}^{\dagger}\right\}= 0
d|_{\phi=\tilde{\phi}}(\Lambda_{F,s})[\psi]=-\lambda_F^{s+2}[\tilde{\phi}]\binom{|F|-1}{s-1}\sum_{l=1}^{|F|}\mathcal P^{\mathcal S}_{\tilde{\phi}}\left[d|_{\phi=\tilde{\phi}}W^{\mathcal S}_{\phi}[\psi][p(u_l)]\right]\left[p(u_l)\right].
0 \ge\frac{c_{1}\rho_0^{n-1}}{2}\int_{-1}^1 \phi(\tau)\,d\tau \int_{\partial B_1}G(\rho_0 e,\rho_0\omega)\,dH^{n-1}_\omega-C &\ge\frac{c\,c_{1}}{2\rho_0^{1-2s}}\int_{-1}^1 \phi(\tau)\,d\tau \int_{\partial B_1\cap\{|e-\omega|<\eta\}}\,\frac{dH^{n-1}_\omega}{|e-\omega|^{n-2s}}-C &=+\infty.
\Big(\sum\limits_{i=1}^{n}\sum\limits_{s=1}^{m}\Phi_{1i}(x_{is})\xi_{is},\dots,\sum\limits_{i=1}^{n}\sum\limits_{s=1}^{m}\Phi_{ni}(x_{is})\xi_{is}\Big); x_{is}\in\mathcal{M};\, \xi_{is}\in \mathcal{H};\, 1\leq i\leq n;\, 1\leq s\leq m;\, m\in\Bbb{N},
G ( \omega , \mathrm{\boldmath ~ k ~}, \rho \Lambda ; v , e^{2}) = \exp{\left\{\int^{\rho}\frac{d \rho^{\prime}}{\rho^{\prime}}\; \gamma \right\}}\; \; G ( \omega , \mathrm{\boldmath ~ k ~}, \Lambda ; v_{e f f}( \rho ) , e_{e f f}^{2}( \rho ) )
\begin{aligned}\partial_s w - L w = Q( w, w) - Q(u_\mathrm{app},w) - Q(w,u_\mathrm{app}) + \varepsilon(R_1(w)+ R_2(w,w)) & - \varepsilon (R_2(u_\mathrm{app},w) + R_2(w,u_\mathrm{app})) - R_\mathrm{app}\end{aligned}
|e^{i\delta_n}|^2=\frac{(\cosh\frac{\pi r_{n}^{2}}{v}-\cosh\frac{\pi c}{v})^4}{\cosh^6 \frac{\pi r_{n}^{2}}{2v}(\cosh\frac{\pi r_{n}^{2}}{v}+\cosh\frac{\pi 2c}{v})}
\langle c | \mathrm{e}^{- \pi \ell H}| c \rangle_{\mathrm{R - R}}\sim \sum_{m , k}\left[ 1 - ( - 1 )^{m}\right] \left[ 1 + ( - 1 )^{k}\right] \mathrm{e}^{-{\frac{\pi \alpha^{\prime}\ell}{2 R^{2}}}( k - \gamma m )^{2}}
\left\{\begin{array}{ll}\frac{\partial^2u}{\partial t^2}-{\rm div}\,(a(t, z)|Du_t|^{p-2}Du_t) + \beta(z)u_t - \Delta u \in \left[f_l(t, z, u), f_u(t, z, u)\right] \mbox{in}\T\times\Omega, u|_{T\times\partial\Omega}= 0,\u(0, z)=u_0(z),\u_t(0, z)=u_1(z).\end{array}\right\}
S'_{r}\coloneqq \begin{cases}G_{r-1,0}\mbox{if}(\varepsilon,d) = (+1,1), begin{align*}-3pt]G_{r-2}\mbox{if}(\varepsilon,d) = (-1,0), begin{align*}-3pt]G_{r-2,1}\mbox{if}(\varepsilon,d) = (+1,0).\end{cases}
\| g(t_{n+1})g(t_{n})^{-1}. v_0 - v_0 \| = \| g(t_{n+1})^{-1}. v_0 - g(t_{n})^{-1}. v_0 \| & \le \| g(t_{n+1})^{-1}. v_0 - v(t_{n+1}) \| + \| v(t_{n+1}) - v(t_{n}) \| + \| v(t_{n}) - g(t_{n})^{-1}. v_0 \| & \le \| g(t_{n+1}) . v(t_{n+1}) - v_0 \| + d(t_{n}) + \| g(t_{n}) . v(t_{n}) - v_0 \| \to 0.
\begin{array}[c]{l}e_{n}^{m}\left( \gamma\right) \left\{{w_{2}\left({\gamma,\alpha,\zeta}\right) -\left({-1}\right)^{m+n}w_{4}\left({\gamma,\alpha,\zeta}\right)}\right\}=o\left( 1\right) A{\operatorname{env}}U\left({-{\tfrac{1}{2}}\gamma\alpha^{2},\zeta\sqrt{2\gamma}}\right) ,\end{array}
\alpha^b :=\alpha_1^{b_1}\alpha_2^{b_2}\dots \alpha_n^{b_n}\in L_n, \alpha^M :=(\alpha^{M[-,1]},\alpha^{M[-,2]},\dots,\alpha^{M[-,n]})\in L_n^n, \alpha \beta :=(\alpha_1\beta_1,\alpha_2\beta_2,\dots,\alpha_n\beta_n)\in L_n^n.
h^{(\epsilon),a}_{mn}(r,s) =\langle\Xi^{(\epsilon)}_{\bullet mn}(r,s),a^{(\epsilon),\prime}(X(s))\Xi^{(\epsilon)}_{\bullet mn}(r,s)\rangle, h^{(\epsilon),b}_{mn}(r,s) =\langle\Xi^{(\epsilon)}_{\bullet mn}(r,s),b'(X(s))\Xi^{(\epsilon)}_{\bullet mn}(r,s)\rangle.
{\cal L}= \frac{1}{2}h_{ij}(y) \dot{y}^i \dot{y}^j + \frac{1}{2}L_{AB}(y) (\dot{\xi}^A + w^A_i(y) \dot{y}^i) (\dot{\xi}^B + w^B_j(y) \dot{y}^j)\,,
\begin{array}{ll}\displaystyle{{\overline J}_\infty(u,v)= \int_\Omega \mathcal{CQ}f(x,u,v,\nabla u) dx + \int_{J_u}K_{\infty}(x,0, u^+, u^-,\nu_u)d\mathcal{H}^{N-1}}\displaystyle{+ \int_\Omega (\mathcal{CQ}f)^\infty\left(x,u,0,\tfrac{dD^cu}{d|D^c u|}\right) d|D^c u|}, \end{array}
p_{OB,j,i}^{(q)}= \frac{{{{\left(x_{n,j}^{(\alpha )}- z_{m,j,i}^{(\alpha_j^* )}\right)}^{q - 1}}}}{{(q - 1)!}}p_{OB,j,i}^{(1)},j = 0, \ldots ,n; i = 0, \ldots, m.
\frac{\partial^{2}u}{\partial t^{2}}=c^{2}(\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}})for0\le r<a,0\le\theta\le2\pi
f_{\theta}(z)-\mathcal{E}_2(z) = -\sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)-\sum_{m=1}^\infty \biggl( \sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)J_{N,m}(\tau)\biggr) q^m.
\zeta[\sigma_1,\sigma_2,\rho](u) \stackrel{def}{=}\exp \left\{\ - \frac{u^2}{2 (1 - \rho^2)}\ \frac{\sigma_1^2 + \sigma_2^2 - 2 \rho \ \sigma_1 \ \sigma_2}{\sigma_1^2 \sigma_2^2}\ \right\}.
\begin{array}{c}{{\alpha ( a ( f ) ) = a ( U f ) + a^{*}( V f )}}\\{{U U^{*}+ V V^{*}= 1 = U^{*}U + V^{*}V , \quad V^{*}U + U^{*}V = 0 = U V^{*}+ V U^{*}}}\\ \end{array}
\int_{\tilde\Gamma}2\mbox{Re}((\nu_1-i\nu_2)\overline{\Phi_0\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})})+2\mbox{Re}((\nu_1+i\nu_2) \overline{\bar\Phi_0\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})})d\sigma=0.