harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U=[\begin{matrix}\rho\\u\\v\\p\\ \end{matrix}],F=[\begin{matrix}\rho_{0}u+\rho u_{0}\\u_{0}u+p/\rho_{0}\\u_{0}v\\u_{0}p+\gamma p_{0}u\\ \end{matrix}],G=[\begin{matrix}\rho_{0}v\\ 0\\p/\rho_{0}\\ \gamma p_{0}v\\ \end{matrix}],
g^{2}( \mu ) ={\frac{g^{2}( \mu_{0})}{1 +{\frac{2 2}{3}}{\frac{C_{2}( G )}{( 4 \pi )^{2}}}g^{2}( \mu_{0}) \ln{\frac{\mu}{\mu_{0}}}}}={\frac{1}{{\frac{2 2}{3}}{\frac{C_{2}( G )}{( 4 \pi )^{2}}}\ln{\frac{\mu}{\Lambda_{\mathrm{Q C D}}}}}},
\frac12\frac{d}{dt}\big(\|\nabla u^{(m)}\|^2_{L^2(B)}\big) =\int_B\nabla u^{(m)}\nabla u^{(m)}_t\;dz=(u^{(m)}_r,u^{(m)}_t)_{L^2(S^1)}=-(u^{(m)}_r,(\varepsilon+d\pi_N(v)) u^{(m)}_r)_{L^2(S^1)}=-\varepsilon\|u^{(m)}_r\|^2_{L^2(S^1)}-\|d\pi_N(v)u^{(m)}_r\|^2_{L^2(S^1)}\le-\frac12\|u^{(m)}_t\|^2_{L^2(S^1)}\le 0,
h_R(x_i^{2k+2}) =h_R(x_i^{2k+1})x_i+(-1)^{|x_i^{2k+1}||x_i|}h_R(x_i)x_i^{2k+1}&=\bigtriangleup (2k+1)y_ix_i^{2k}x_i+(-1)^{|x_i^{2k+1}||x_i|}y_ix_i^{2k+1}&=(k+1+(k+1)(-1)^{(2k+1)|x_i|^2})y_ix_i^{2k+1}&=\bigtriangleup (n+1)y_ix_i^{n},
\Psi_jh =\sum_{k\in\mathbb{L}_j}\langle \Psi_jh,e_{j,k}\rangle e_{j,k}=\sum_{k\in\mathbb{L}_j}\langle h,v_{j,k}\rangle e_{j,k}=\sum_{k\in\mathbb{L}_j}\langle h,c_ju_{j,k}\rangle e_{j,k}=c_j\sum_{k\in\mathbb{L}_j}\langle h,u_{j,k}\rangle e_{j,k}=c_j\sum_{k\in\mathbb{L}_j}\langle h,A_j^*e_{j,k}\rangle e_{j,k}=c_jA_jh, ~ \forall j \in \mathbb{J}.
\{\tilde{L}_{\alpha}, \tilde{\theta}^{\beta}\}= k_{2}{p^{+}}_{\alpha}{}^{\beta}+ O^{2}( \tilde{\theta}, \tilde{L}) + O^{1}( T ){p^{-}}_{\alpha}{}^{\beta}+ O^{1}( T ){p^{+}}_{\alpha}{}^{\beta}.
L_G(u)^i_j = E_0 + \sum_{\alpha \neq (0, 0)}E_\alpha (I_\alpha)^i_j =\sum_k \phi(u, z)^i_k \phi^{-1}(u, z)^k_j \partial_u - l \sum_k \partial_u\phi(u, z)^i_k \phi^{-1}(u, z)^k_j,
V_{F}^{( d i v )}( H ) = - \frac{N}{6 4 \pi^{2}\epsilon}\left( m_{t}( H )^{4}+ \sum_{k = 1}^{\infty}\left( \frac{2 k}{R}+ m_{t}( H ) \right)^{4}+ \sum_{k = 1}^{\infty}\left( - \frac{2 k}{R}+ m_{t}( H ) \right)^{4}\right)
0 \leq \widetilde L \int_{B_R}\nabla Q_\infty \cdot\nabla P \, dx + \frac{\widetilde L}{2}\int_{B_R}|\nabla P|^2 &+ \frac{1}{2}\int_{B_R}(|Q_\infty+P|^2-1)^2 -\frac 12 \int_{B_R}(|Q_\infty|^2-1)^2 & = E(Q_\infty + P;B_R)-E(Q_\infty;B_R).
\sum_{n\in \mathcal F_2^c}\min\left\{\left[\frac{1}{\theta_{1,K_2^c}}\sqrt{\frac{a_n\theta_{2,K_2^c}}{\nu^*}}-\frac{\theta_{2,K_2^c}}{\theta_{1,K_2^c}}\right]^+,1\right\}=K_2^c.
\begin{cases}(\partial_t^2+ia\partial_{\alpha})\bar{z}_t=-ia_t\bar{z}_{\alpha}\dot{z}_j(t)=(v-\frac{\lambda_j i}{2\pi}\frac{1}{\overline{z(\alpha,t)-z_j(t)}})\Big|_{z=z_j(t)}(I-\mathfrak{H})(\bar{z}_t+\sum_{j=1}^N\frac{\lambda_j i}{2\pi}\frac{1}{z(\alpha,t)-z_j(t)})=0.\end{cases}
b=\begin{cases}\varepsilon \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \delta \leq 0, \ \mbox{and}\ 0<\alpha <1 &\frac{1}{\log x}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \delta <0 \ \mbox{and}\ 1\leq \alpha.\end{cases}
B_{p}^{\top}\eta'_{p}=\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\left(-\lambda_{p}\hat{A}_{p}v''_{p}+\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]\right).
N\big( \left\| ax - \|u\|x_i \right\|, \left\| by - \|v\|y_i \right\| \big) \geq N\big( a + \|u\| - \varepsilon/\nu, b + \|v\| - \varepsilon/\nu \big) \geq N\big( a + \|u\|, b + \|v\| \big) - N \big( \varepsilon/\nu, \varepsilon/\nu \big) = 2 - \varepsilon.
p=\frac{RT}{V}(1+\frac{B}{V_{r}}+\frac{C}{V_{r}^{2}}+\frac{D}{V_{r}^{5}}+\frac{c_{4}}{T_{r}^{3}V_{r}^{2}}(\beta+\frac{\gamma}{V_{r}^{2}})exp(\frac{-\gamma}{V_{r}^{2}}))
\lim_{t\to\infty}\frac{\bar{\mathcal{M}}(t)}{t}\stackrel{d}{=}\sum_{j=1}^{\infty}j\lim_{t\to\infty}\frac{N_{j}(t)}{t}&=\frac{\lambda r\rho}{(1-\rho)(1-(1-\rho)^{r})},\ ,
\partial_{s_j}(a_1\sharp a_2) = a_1\sharp(\partial_{s_j}a_2) , \partial_{\xi_j}(a_1\sharp a_2) = (\partial_{\xi_j}a_1)\sharp a_2+a_1\sharp(\partial_{\xi_j}a_2) , \delta_j(a_1\sharp a_2) = (\delta_j a_1)\sharp a_2+a_1\sharp(\delta_j a_2) .
\vec{\chi}( x^{\mu}) = \left( \begin{array}{c}{{\vec{\chi}_{1}( x^{\mu})}}\\{{\vec{\chi}_{2}( x^{\mu})}}\\ \end{array}\right) \hspace{0 . 5 c m}, \hspace{0 . 5 c m}\vec{\chi}_{\alpha}^{*}= \vec{\chi}_{\alpha}\quad , \quad \alpha = 1 , 2
\;\;\;\;\; + 2 K^b_a K^a_b \: s \partial_s \phi^{(2)}+ 4 (K^b_a K^a_b)^2 \: s^3 \partial_s \phi^{(0)}_{+}+ 4 K^a_c K^b_a K^c_b \: s^2 \partial_s \phi^{(1)}.
\epsilon_j := \begin{cases}\tau_j^{1/2+\gamma_0}\gamma_0 < 1/4, \sqrt{1+\ln(t_j/\tau_j)}\, \tau_j^{1-\gamma_0}\gamma_0 = 1/4, \tau_j^{1-\gamma_0}\gamma_0 > 1/4, \end{cases}
\sum_{k=1}^n\{H(Y_k|Y_1,\ldots,Y_{k-1})-H(Y_k|X_0,Y_1,\ldots,Y_{k-1})\}=H(Y_1,\ldots,Y_n)-H(Y_1,\ldots,Y_n|X_0)\phantom{\sum_{k=1}^n}&= H(X_0)-H(X_0|Y_1,\ldots,Y_n) \le H(X_0),
\frac{1}{2}\frac{d}{d \tau}\|\varepsilon_-\|_{L^2_{\rho}}^2 = \left\langle \partial_\tau \varepsilon_-, \varepsilon_- \right\rangle_{L^2_{\rho_\beta}}+ \beta' \left\langle \varepsilon_- , \varepsilon_- \left( \frac{d+2}{2\beta}-\frac{y^2}{2}\right) \right\rangle_{L^2_{\rho_\beta}}.
\begin{array}{ll}\mathbb{E}[ Z_1 ] = 0 \textsf{Var}[ Z_1 ] = \mathbb{E}[G]{\sigma^2_N}/{2}\mathbb{E}[ Z_0 ] = \sigma^2_N \textsf{Var}\left[ Z_0 \right] = \sigma^4_N \Delta \mathbb{E}\left[ Z_1 (Z_0-\mathbb{E}[Z_0]) \right] = 0.\end{array}
\tilde{S}= \pi t \left[ \left( m_{i}+ C_{i j k}n^{j}p^{k}\right)^{2}+ \left( n^{i}\right)^{2}\left( p^{i}\right)^{2}- \left( n^{i}p^{i}\right)^{2}\right] + 2 \pi i m_{i}\left( \tau_{1}n^{i}+ \omega_{1}p^{i}\right) ,
\int_{0}^{*}\frac{1}{t}\frac{1}{\log_{(1)}(\frac{1}{t})}\cdots \frac{1}{\log_{(n+2)}(\frac{1}{t})}\, \mathrm{d}t =\ \int_{*}^{+\infty}\frac{1}{s}\frac{1}{\log_{(1)}(s)}\cdots \frac{1}{\log_{(n+2)}(s)}\, \mathrm{d}s =\ \cdots \ = \ \int_{*}^{+\infty}\frac{1}{\log(x)}\, \mathrm{d}x\ =\ \infty.
\frac{(u_{+}-v_{-})^{2}}{(u_{+}-v_{-})^{2}-h^{2}}k_{2}^{+}(v)^{-1}k_{1}^{-}(u)=\frac{(u_{-}-v_{+})^{2}}{(u_{-}-v_{+})^{2}-h^{2}}k_{1}^{-}(u)k_{2}^{+}(v)^{-1}
S(s) = C U{(s I - \Lambda)}^{-1}U^T B = C \begin{pmatrix}U_1 U_2 \end{pmatrix}\begin{pmatrix}s^{-1}I 0 0{(s I- \Lambda_2)}^{-1}\end{pmatrix}\begin{pmatrix}U_1^T U_2^T \end{pmatrix}B = C U_2{(s I - \Lambda_2)}^{-1}U_2^T B.
{\rm Re}\left\{1+z\frac{T_g''(z)}{T_g'(z)}\right\}={\rm Re}\left\{z\frac{f'(z)}{f(z)}+z\frac{g''(z)}{g'(z)}+1\right\}>\frac{1+\alpha-2\gamma r+(1-\alpha)r^2}{1-r^2}>0,
V_N(0,it,k)\ll \begin{cases}\frac{l^{1/2}}{N}d_{0}^{\epsilon}\max{\left( \frac{\sqrt{T}}{k},\frac{1}{\sqrt{k}}\right)}, d_0\geq 1 \frac{1}{\sqrt{lT}}\left( \frac{d_0}{2}\right)^k, d_0<1. \end{cases}
0 =\int_M \langle\delta_{g_0}\delta_{g_0}^*Y,Y\rangle_{g_0}dvol_{g_0}&=\int_M|\delta_{g_0}^*Y|^2-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,Y)-\int_{\infty}\delta^*_{g_0}Y(\mathbf n,Y) &=\int_M|\delta_{g_0}^*Y|^2-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,Y^T)-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,\mathbf n) Y^{\perp}.
\frac{K_{b}^{\pm}( \theta )}{K_{f}^{\pm}( \theta )}= \frac{\operatorname{c o s h}( \frac{\theta}{2}\pm \frac{i \pi}{4})}{\operatorname{c o s h}( \frac{\theta}{2}\mp \frac{i \pi}{4})}= \frac{1 \pm i \operatorname{s i n h}\theta}{\operatorname{c o s h}\theta},
\frac{|r|^{n-1}}{(n-1)!}|c_{n-1,n}| - \sum_{i=2}^n \frac{|r|^{n-i}}{(n-i)!}|c_{n-i,n}| > \sum_{i=1}^j |c_{j-i,j}|\frac{|r|^{j-i}}{(j-i)!}
[y_1, y_1]=\frac{\alpha_2\gamma^2_3}{\alpha^2_5}y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_5}y_4+\theta_1y_5, [y_2, y_1]=y_4+\theta_2y_5, [y_2, y_2]=\frac{\beta_2\beta^2_3}{\alpha^2_5}y_5, [y_1, y_3]=y_4+\theta_3y_5, [y_3, y_1]=\frac{\beta_6\gamma_3}{\alpha_5}y_5, [y_2, y_3]=\frac{\beta_3\gamma_2}{\alpha_5}y_5, [y_3, y_2]=y_4, [y_3, y_3]=\gamma_6y_5.
g\det(P^i_j) =\det(gP^i_j) &=\det\left(\rho_{\chi}(g)(P^1_1,\ldots,P^1_{\chi(1)}),\ldots,\rho_{\chi}(g)(P^{\chi(1)}_1,\ldots,P^{\chi(1)}_{\chi(1)})\right) &=\det(\rho_{\chi}(g))\det(P^i_j)=\det\chi(g)\det(P^i_j),
\frac{P(R)-P_{\infty}}{\rho_{L}}=[-\frac{1}{r}(2R(\frac{dR}{dt})^{2}+R^{2}\frac{d^{2}R}{dt^{2}})+\frac{R^{4}}{2r^{4}}(\frac{dR}{dt})^{2}]_{R}^{\infty}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}(\frac{dR}{dt})^{2}
{\rm GL}(\gamma_1,\gamma_2) ={\textstyle{1\over 4 \pi}}\oint_{\gamma_1}ds \oint_{\gamma_2}dt dot{\gamma_1}^a(s) \dot{\gamma_2}^b(t) \epsilon_{abc}{(\gamma_1^c(s) -\gamma_2^c(t)) \over |\gamma_1(s) -\gamma_2(t)|^3}
w_{tt}- A(x,\partial_x) w =0, \,\, \,w(x,0;b)= f (x,b), \,\, w_t(x,0;b)=0 ,\,\, t \in [ 0,T_1]\subseteq{\mathbb R}, \,\, x \in \widetilde{\Omega}\subseteq{\mathbb R}^n,
\varphi_{g_k}([g_i,h_j])={g_k}[g_i,h_j]{g_k}^{-1}=[g_k g_i,h_j][h_j,g_k] \varphi_{h_k}([g_i,h_j])={h_k}[g_i,h_j]{h_k}^{-1}=[h_k,g_i][g_i,h_k h_j].
W=W(x_0,t_0,u_0) - \frac{1}{2}\epsilon^\beta{u_0}^2{\tau}+\epsilon^\gamma u_0{{y}}+\epsilon^{\alpha+\gamma}{{y}}{\upsilon}+\epsilon^{\alpha+\beta}u_0{\tau}{\upsilon}+ \frac{1}{2}\epsilon^{\beta+2\alpha}{\tau}{\upsilon}^2 + \epsilon^{\alpha(k+2)}A_{k+2}{\upsilon}^{k+2}+\dots\, ,
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}( \partial_{\mu}\phi )^{2}+ e ( g^{\mu \nu}- \epsilon^{\mu \nu}) ( \partial_{\mu}\phi ) A_{\nu}+ \frac{1}{2}e^{2}\alpha A_{\mu}^{2}
\eta_{1,n}(\delta) ~ :=~ 1 - \mathbb{P}\left(\theta\mapsto\mathbb{M}_n(\theta)\mbox{is convex on}[\theta_0 - \delta, \theta_0 + \delta]\right), \eta_{2n}(\delta) ~ :=~ \mathbb{P}(|\widehat{\theta}_n - \theta_0| > \delta).
r_{ab}= r_{ba}, [r_{ab},r_{cd}] = 0 \; \mbox{for}\{a,b\}\cap\{c,d\}=\emptyset \, , & [r_{ab}+r_{ac},r_{bc}] = 0 = [r_{ab},r_{ac}+r_{bc}] \, \textrm{(called the ``four term relations'')}.
\frac{\varepsilon}{2}= \sum_{\ell = 0}^L c_3 \varphi(\triangle u_\ell) \left(\frac{\mathcal C_\ell}{\lambda c_3\varphi(\triangle u_\ell)}\right)^{\frac{\mu_2}{\mu_2+1}}\geq \sum_{\ell = 0}^L \varphi(\triangle u_\ell)
- \, \, < < G_{1}^{0}\, G_{2}^{0}\, G_{3}^{0}\, K_{2 3}^{L}\, G_{2}^{0}\, G_{3}^{0}> > g^{0}< < G_{1}^{0}\, G_{2}^{0}\, K_{1 2}^{L}\, G_{1}^{0}\, G_{2}^{0}\, G_{3}^{0}\, > > .
\prod_{j=1}^{n}|\xi_{j}-\xi_{j+1}|^{\lambda}\leq c\,\prod_{j=1}^{n}(|\xi_{j}|^{k}+|\xi_{j+1}|^{\lambda}) &=\sum_{S_1}\prod_{j=1}^{n}(|\xi_{j}|^{\lambda\gamma_{j}}|\xi_{j+1}|^{\lambda\overline{\gamma_{j}}}) &\leq\sum_{S_2}\prod_{j=1}^{n}(|\xi_{j}|^{\lambda\alpha_{j}}),
{\overline{\phi}}( p , z ) = - \int{\frac{d^{2}k}{( 2 \pi )^{2}}}e^{- i k \cdot z}\left[ b L^{2}{\frac{{\cal O}_{4}( p )}{k^{2}+ p^{2}}}+ ( 2 \pi )^{4}c L^{- 2}{\frac{\delta^{( 4 )}( p )}{k^{2}}}\right] .
e^{\psi}=|s|_{b}^2:=|t_{1}|_{b_{1}}^{2a_{1}}|t_{2}|_{b_{2}}^{2a_{2}}\cdots|t_{N}|_{b_{N}}^{2a_{N}}~~~~~~-\gamma=\Theta_{b}(D):=\sum_{i=1}^{N}a_{i}\Theta_{b_{i}}(D_{i}).
\alpha_{\mu}=|E(uw)|^{2(1+k)}E(|w|^2) -2\mu^n|E(uw)|^{2(k-1)}\chi_{S_0}(E(|w|^2))^2E(|u|^2) &+\mu^{2}|E(uw)|^{2(k-1)}\chi_{S_0}E(|w|^2).
\Bigl(~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d}}\Bigr)^{(-1)}=\Bigl(\sum_{n\geq 0}\prod^{d-1}_{l=1}(\theta^{q^{n+d}}-t)^{c(l)q^{d-l}}\alpha^{q^{n+d}}\Bigr)^{(-1)}&=\prod_{l=1}^{d-1}(\theta^{q^{d-1}}-t)^{c(l)q^{d-l}}\alpha^{q^{d-1}}+~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d-1}}.
D(x,x')=\sum_{n=-\infty}^{\infty}\int{d^4 k\over (2\pi)^4}\int_0^\infty{q dq\over 2\pi\alpha}{e^{ik_\mu (x^\mu-x'^\mu)}e^{in(\theta-\theta')/\alpha}\over k_4^2 + q^2}J_{\left|n/ \alpha\right|}(q r) J_{\left|n/ \alpha \right|}(q r'),
\langle \hat u_j,(E-\lambda_jI)\hat u_j\rangle=\sum_{k\neq j}\langle \hat u_j,u_k\rangle\langle (\Sigma-\lambda_jI)\hat u_j,u_k\rangle=\sum_{k\neq j}(\lambda_k-\lambda_j)\langle \hat u_j,u_k\rangle^2.
N^{B,\xi}(x,t) := \sum_{z \notin B}\; \sum_{1 \le i \le \eta(z)}\mathbf{1}_{\{S^{z,i}_t = x\}}+ \sum_{z \in B}\; \sum_{\xi(z) < i \le \eta(z)}\mathbf{1}_{\{S^{z,i}_t = x\}}.
\kappa_{(K, \widehat{K})}(f):= \begin{cases}0 \xi_{K}= \emptyset \sum_{x \in \xi_{K}\cap[0,1]}f(\phi({\cal M}^{(x,0)}), \phi^\prime(\widehat{\cal M}^{(x,0)})) 0 < \# \xi_{K}< \infty \infty .\end{cases}
||\tilde{h}_n (x,x')-h_0(x,x')||_{C^r}=||Q_n(x')||_{C^r}, &\leq{q_n}^{-2}(||u_n(q_nx')||_{C^r}+||v_n(q_nx')||_{C^r}), &\leq{q_n}^{-2}({q_n}^{-a}(2\pi)^r{q_n}^r+C_1{q_n}^{-a}{q_n}^r), &\leq C_2{q_n}^{r-a-2},
u(x, \lambda) = \sum_{k=1}^\infty \phi_k \frac{1}{\lambda_k - \lambda}\Big(\int_{\partial B_R}\partial_\nu\phi_k f{\rm d}s(y) + \int_{\partial B_R}\partial_\nu(\Delta\phi_k)g{\rm d}s(y)\Big), x\in B_R.
|F(\psi(s))(y)|\lesssim y^{-2}\left|U_{\alpha/\delta}(ye^{\omega_l s})-\frac{\pi}{2}\right|^3\lesssim y^{-2}\lesssim y^{-\gamma-2}(\Gamma e^{-\omega_{l}s})^{\gamma}=y^{-\gamma-2}e^{-\lambda_{l}s}\Gamma^{\gamma}.
\partial_1 \left(\frac{h_2h_3}{h_1}\right) = \partial_2 \left(\frac{A_1h_2h_3}{h_1}\right)~,~~~~ \partial_3 \left(\frac{h_2h_1}{h_3}\right) = \partial_2 \left(\frac{A_3h_2h_1}{h_3}\right)~.
tNM(A,B)=(\sum_{C\in H_{\tau_{\Phi,\epsilon}}(Z)}|C|)^{-1}\cdot\sum_{C\in H_{\tau_{\Phi,\epsilon}}(Z)}|C|\frac{min(|C\cap A|,|[C\cap B|)}{max(|C\cap A|,|C\cap B|)}.
\left( -\hslash^{2}\Delta\right)^{\alpha/2}\Psi(x,t)=\frac{1}{2\pi\hslash}\int_{-\infty}^{+\infty}dpe^{ipx/\hslash}\left\vert p\right\vert^{\alpha}\Phi(p,t),1<\alpha\leq2,
0\le \eta_i\le 1, \eta_i\equiv 1 \mbox{on}B_{r_{i+1}}, \eta_i\equiv 0 \mbox{on}\mathbb{R}^N\setminus B_\frac{r_i+r_{i+1}}{2}, |\nabla \eta_i|\le \frac{c_N}{r_{i}-r_{i+1}}=4\,c_N\,i_0 \mbox{and}|D^2\eta|\leq \frac{c_N}{(r_i-r_{i+1})^2}=16\,c_N\,i_0^2.
\Phi_{\mathrm{GA}}(t;i,R_{\mathrm{GA},u}^*) = \frac{e^{-jt(2^{R_{\mathrm{GA},u}^*}-1)}}{t}\cdot \left[ \sqrt{- \frac{4jst}{\sigma_{Z}^2}}\cdot K_{1}\left( \sqrt{- \frac{4jst}{\sigma_{Z}^2}}\right) \right]^i.
S_{\mu\nu}=\left(L_{\mu\nu}\gamma_\mu \gamma_\nu+\frac{1}{2}\right) S_{\mu\nu\rho\sigma}=\left(L_{\mu\nu}\gamma_\mu \gamma_\nu+L_{\mu\rho}\gamma_\mu \gamma_\rho+L_{\mu\sigma}\gamma_\mu \gamma_\sigma+L_{\nu\rho}\gamma_\nu \gamma_\rho+L_{\nu\sigma}\gamma_\nu \gamma_\sigma+L_{\rho\sigma}\gamma_\rho \gamma_\sigma+\frac{3}{2}\right)
\begin{aligned}\widehat{g}_{\mathcal{R}}^m(z_{R_i},z_{R_j}) =\left(4\delta\left( \frac{R_i+R_j}{2}\right)-\delta(R_i)-\delta(R_j)\right)-\left(\delta(R_i)+\delta(R_j)\right)= &=4\delta\left( \frac{R_i+R_j}{2}\right)-2\delta(R_i)-2\delta(R_j),\end{aligned}
T_{i}^{(-)}(x).f(y)=T_{w_{x}}^{-1}(T_{\sigma_{x}(i)-1}^{-1}\cdots T_{\sigma_{x}(i)-d_{i}^{-}(x)}^{-1})(T_{\sigma_{x}(i)-d_{i}^{-}(x)}^{-1}\cdots T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}^{-1}).f(w_{y}y).
(f_0+f_1(s-1))(g_0+(s+1){}^sg_1)=f_0g_0 + f_1({}^sg_0-g_0) + f_0(g_1+{}^sg_1) \hphantom{(f_0+f_1(s-1))(g_0+(s+1){}^sg_1)=}{}+(f_1{}^s g_0 + f_0 g_1)(s-1),
0\ \leq\ \frac{1}{2}\iint_{\Omega^2}(u_*-u)(u_*^{p-1}-u^{p-1})\mu_*\mu \ = \ \iint_{\Omega^2}u(u^{p-1}-u_*^{p-1})\mu_*\mu & = \ \iint_{\Omega^2}u^{p-1}(u-u_*)\mu_*\mu & = \ \int_\Omega u^{p-1}(u-\langle u\rangle_{\mu,\Omega})\mu.
\begin{cases}u_k \to u \mbox{a.e. on}\Omega, u_k \to u \mbox{in}L^r(\Omega) \mbox{for every}r \in \left(0,\frac{n(p-1)}{n-p}\right), \nabla u_k \to \nabla u \mbox{in}L^{s}(\Omega) \mbox{for every}s \in \left(0,\frac{n(p-1)}{n-1}\right). \end{cases}
\left| f \right|_{\operatorname{Lip}^k( E_1, E_2 )}= \sup_{\substack{x, y \in E_1 , x \neq y}}\tfrac{\left\| f^{(k)}( x ) - f^{(k)}( y ) \right\|_{L^{(k)}( E_1, E_2 )}}{\left\| x - y \right\|_{E_1}},
H(\cdot,\overline{Y})\le \overline{Y},\underline{Y}\le\overline{Y},\sum_{j=1}^NE\big(\int_0^T|f^j(r,\underline{Y}_r;\overline{Y}^j_r)|\,dr+\int_0^T|f^j(r,\overline{Y}_r)|\,dr\Big)<\infty,
T(z) = \begin{pmatrix}1 0 i \frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}r^{\frac{\sigma_{3}}{4}}G_{0}(rz;r\vec{x},\vec{s})(rz)^{\frac{\alpha}{2}\sigma_{3}}e^{-\sqrt{r}g(z)\sigma_{3}}.
m_A^{\phantom{A}B}= \frac{1}{2!}({\cal P}_+ \Gamma^{MN})_A^{\phantom{A}B}C_{MN}+ \frac{1}{6!}({\cal P}_+ \Gamma^{M_1 \ldots M_6})_A^{\phantom{A}B}Z^+_{M_1 \ldots M_6}\;.
X^{y}_{h L_{k+1}}-\pi_{1}(y) = \sum_{j=0}^{\ell_k-1}\left[X^{Y^{y}_{jhL_{k}}}_{h L_{k}}-X^{y}_{jh L_{k}}\right] \leq \left[\ell_k - N\right]hL_k v_{\min}+ N h L_{k}v_{\max}& = h L_{k+1}\left\{v_{\min}+ (v_{\max}- v_{\min}) \frac{N}{\ell_k}\right\}< h L_{k+1}\bar{v},
\sum_{k=0}^{\infty}\alpha_{2}(k)\frac{1}{(j_{6A}(\tau)+4)^{k+\frac{1}{2}}}=\sum_{k=0}^{\infty}\alpha_{2}^{\prime}(k)\frac{1}{(j_{6A}(\tau)-4)^{k+\frac{1}{2}}}
\begin{cases}\Psi^*_t-\Delta \Psi^*=\delta^* \sigma e^{\delta^* t},\ x\in D_{\hat c_kT}, 0<t\leq \tilde T, \Psi^*\geq 0, x\in\partial D_{\hat c_k T}, 0<t\leq \tilde T, \Psi^*\geq 0, x\in D_{\hat c_k T}, \t=0.\end{cases}
\liminf_{i \to \infty}\delta(\widetilde{C}_i) \ge \frac{1}{2}-\frac{1}{2}\Bigl(\frac{1}{\ell^{\lceil r/2\rceil}-1}+\frac{1}{\ell^{\lfloor r/2\rfloor}-1}\Bigr) q = \ell^r, r>1
\mathbf{w}^0_1 =\sum_{\mathbf{k}\in\mathbb{Z}^k}\mathbf{V}(i \langle \mathbf{k},\Omega\rangle \mathbf{I}-\mathbf{\Lambda})^{-1}\mathbf{V}^{-1}\mathbf{g}_{ext}^{\mathbf{k}}e^{i\langle\mathbf{k},\phi\rangle}, \mathbf{r}^0_1 =\mathbf{0}.
\eta_{X}(x) =\left(\prod_{i}u^{*}_i \right)x \left( \prod_{i}u_{i}\right) &=\left(\prod_{i\in J_{F}}u^{*}_{i}\right) x \left(\prod_{i\in J_{F}}u^{*}_{i}\right) &=\left(\prod_{i\in J_{G}}u^{*}_{i}\right) x \left(\prod_{i\in J_{G}}u^{*}_{i}\right)
= e x p \mathrm{}[ \mathrm{}( \frac{1}{4 N_{0}}) \sum_{{\bf{q}}}f_{{\bf{q}}}( -{\bf{x}}, - t ) \mathrm{}] \langle \psi ({\bf{x}}, t ) \psi^{\dagger}({\bf{0}}, 0 ) \rangle_{f r e e}
\vartheta(H_n) =\begin{cases}\phantom{-}1, \mbox{if{\;}$n = 4s$;}begin{align*}.5mm]\phantom{-}(2s+1)\lambda, \mbox{if{\;}$n = 4s+1$;}begin{align*}.5mm]-1, \mbox{if{\;}$n = 4s+2$;}begin{align*}.5mm]-(2s+2)\lambda, \mbox{if{\;}$n = 4s+3$.}begin{align*}.3mm]\end{cases}
\widetilde{\Pi}^{(2)}(k^{2},m) = - \frac{g^{2}}{4\pi}\int_{0}^{1}d\alpha \frac{m}{\sqrt{m^{2}-k^{2}\alpha(1-\alpha)}}= - \frac{g^{2}}{4\pi}\frac{m}{\sqrt{k^{2}}}\ln{\left( \frac{2\sqrt{m^{2}}+\sqrt{k^{2}}}{2\sqrt{m^{2}}-\sqrt{k^{2}}}\right)}\, .
R_1(v)=(k-1)(M_{12}(e_1)+M_{12}(f_1))=(k-1)(M_{21}(e_1^{-1})+M_{21}(f_1^{-1}))= =m_1(k-1)(M_{22}(e_1^{-1})+M_{22}(f_1^{-1})+M_{22}(h_1^{-1}))= =m_1(k-1)(M_{22}(e_1)+M_{22}(f_1)+M_{22}(h_1)),
\frac{|-1 + z_{j,n}|}{|z_{j,n}|}< \kappa(1,a) =\frac{\left|1 -\exp\bigl(\frac{\pi}{a}\bigr)\right|\exp\bigl(\frac{-\pi}{a}\bigr)}{\exp\bigl(\frac{\pi}{a}\bigr) +\left|1 -\exp\bigl(\frac{\pi}{a}\bigr)\right|}.
\begin{array}{l l}{{\sigma^{0}= -{\bf 1}_{2 \times 2}={\bar{\sigma}}^{0},}}&{{\sigma^{1}= \tau_{z}= -{\bar{\sigma}}^{1},}}\\{{\sigma^{2}= \tau_{x}= -{\bar{\sigma}}^{2},}}&{{\sigma^{3}= \tau_{y}= -{\bar{\sigma}}^{3}, \quad ( \mathrm{o r}\quad \sigma^{y}= \tau_{y}= -{\bar{\sigma}}^{y}) ,}}\\ \end{array}
\delta_{n,k}^{1}V(0,\alpha_{n,k}^{2},\ldots,\alpha_{n,k}^{k})=\det\begin{pmatrix}1 1 \ldots 1 \Delta_{n,k}\alpha_{n,k}^{2}\ldots \alpha_{n,k}^{k}0 (\alpha_{n,k}^{2})^{2}\ldots (\alpha_{n,k}^{k})^2 \vdots \vdots \ldots \vdots 0 (\alpha_{n,k}^{2})^{k-1}\ldots (\alpha_{n,k}^{k})^{k-1}\end{pmatrix}.
\frac{\partial g}{\partial z_i}(z) = 2a f^{a-1}(z) z_i \,, \frac{\partial^2 g}{\partial z_i\, \partial z_j}(z) = 4a(a-1)f^{a-2}(z) z_i z_j + 2a f^{a-1}(z) \delta_{i,j}\,,
\langle [ A_{1}]_{N}({\bf x}) \rangle = \frac{i}{\sqrt{2 k \omega}}\left\{\frac{1}{2}( 1 - \frac{m}{\omega}) J_{N + 1}( k r ) e^{i ( N + 1 ) \varphi}+ \frac{1}{2}( 1 + \frac{m}{\omega}) J_{N - 1}( k r ) e^{i ( N - 1 ) \varphi}\right\}
{\cal L}_{F}^{( 2 )}= \frac{\mathrm{i}}{2}( \theta_{i}\tilde{{\cal D}}_{0}\theta^{i}+ \eta_{i}\tilde{{\cal D}}_{0}\eta^{i}- 2 \mathrm{i}\eta_{i}e_{0}^{i}{}_{j}\eta^{j}) + e^{2 \phi}\eta^{i}C_{i j}^{\prime}\tilde{{\cal D}}_{1}\theta^{j}+ h . c . \ ,
e^{G(z)}=\sum_{k=0}^{\infty}\frac{G(z)^k}{k!}=\sum_{k=0}^{\infty}\frac{\left(\sum_{n=1}^{\infty}q_n z^n \right)^k}{k!}=1+\sum_{n=1}^{\infty}q_n z^n+\sum_{k=2}^{\infty}\frac{\left(\sum_{n=1}^{\infty}q_n z^n \right)^k}{k!}
PW_{\delta}(x) =W_{\delta}(x)-\delta^{-\frac{n-4}{2}}\Big(\varphi_0\Big(\frac{e_n-x}{\delta}\Big)-\varphi_0\Big(\frac{e_n+x}{\delta}\Big)\Big)+\zeta_\delta(x),\zeta_\delta(x)=O(\delta^{\frac{6-n}{2}}(\log \delta)^\tau),
p_{\varphi}^{A}= \frac{6 \, \sqrt 2 \, p_{y}^{M}}{8 + p_{y}^{M}}\quad , \quad p_{i}^{A}= \frac{8 \, p_{i}^{M}+ p_{y}^{M}}{8 + p_{y}^{M}}\, .
\begin{array}{c c}{{w = \textstyle{\frac{1}{4}}w_{\mu \nu}\tilde{\sigma}^{\mu}\sigma^{\nu}\, , ~}}&{{~ \tilde{w}= \textstyle{\frac{1}{4}}w_{\mu \nu}\sigma^{\mu}\tilde{\sigma}^{\nu}\, .}}\\ \end{array}
\lim_{\epsilon \to 0}\, \left\{\frac{d}{d r}G \left( r , r^{\prime}\right) \vert_{r = r^{\prime}+ \epsilon}- \frac{d}{d r}G \left( r , r^{\prime}\right) \vert_{r = r^{\prime}- \epsilon}\right\}= \frac{1}{r^{3}g ( r )^{2}}
M_2(\rho_h^N) - M_2(\rho^0) \ \leq\C hN + C h \sum \limits_{k=1}^N M_2(\rho_h^k) + \sum \limits_{k=1}^N W_2^2(\rho_h^k, \rho_h^{k-1}) &\ \stackrel{\mathclap{\eqref{estimate_for_the_distance_sum_refined}}}{\leq}\C T + C h \sum \limits_{k=1}^N M_2(\rho_h^k) .
J_1^0 = \int_0^1G_t(s,y)\xi(y) dy, & J_2^0 = 0, &J_3^0 = -\int_0^t\int_0^1 \partial_yG_{t-s}(x,y) g(s,V^{{0}, v}_{\eta}(s))(y) dy ds, & J_4^0 = \int_0^t\int_0^1 G_{t-s}(x,y) f(s,V^{{0}, v}_{\eta}(s))(y) dy ds, & J_5^0 = \int_0^t\int_0^1 G_{t-s}(x,y) \sigma(s,V^{{0}, v}_{\eta}(s))(y) v(s,y) dy ds.
\begin{matrix}L&=&(D_{\mu}\phi)^{*}D^{\mu}\phi-V(\phi^{*}\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\\ &=&(\partial_{\mu}\phi)^{*}(\partial^{\mu}\phi)-ie((\partial_{\mu}\phi)^{*}\phi-\phi^{*}(\partial_{\mu}\phi))A^{\mu}+e^{2}A_{\mu}A^{\mu}\phi^{*}\phi\end{matrix},
\frac d{dt}\sum\limits_{\beta=0}^{1}\sum\limits_{|\alpha|=0}^{3-\beta}\left(\|\sqrt{B}\partial_\tau^\alpha\partial_y^\beta u\|_{L^2}^2+\|\sqrt{A}\partial_\tau^\alpha\partial_y^\beta\tilde{h}\|_{L^2}^2\right)+\frac{C_0}{2}D(t)\le CE(t)^{\frac53}.
[c^*,c]=\sum_{i=1}^m[p_i,r_i^{(1)}]+\sum_{i=1}^m[q_i,r_i^{(2)}]+\sum_{i=1}^m\sum_{k=1}^{M_{i}}[[p_i,r_k^{(3)}],s_k^{(1)}]+\sum_{i=1}^m\sum_{k=1}^{M_{i}}[[q_i,r_k^{(4)}],s_k^{(2)}].
V_{\mathrm{n p}}\rightarrow \frac{A^{2}N_{\mathrm{c r}}^{2}\Sigma_{0}^{3}}{3 2}\left[ -{\frac{1 0}{3}}- 2 \log \frac{\Lambda_{\mathrm{n p}}e^{\delta_{0}}}{\Sigma_{0}}+{\frac{3}{2}}\log^{2}\frac{\Lambda_{\mathrm{n p}}e^{\delta_{0}}}{\Sigma_{0}}\right] \rightarrow \infty
\overline{{{n}}}_{k}= \sum_{n = 0}^{\infty}n \theta \left( \mu_{k}- n \omega_{k}\right) = \frac{1}{2}\left( 1 + \left[ \frac{\mu_{k}}{\omega_{k}}\right] \right) \, \left[ \frac{\mu_{k}}{\omega_{k}}\right] \,
(K+a)^2 d_{K+1}+ \sum_{k=2}^{K}a\left( 2k +a -2\right)d_k &\leq s \left( (a+1)^2 \theta_1 - (K+a)^2 \theta_{K+1}+\sum_{k=2}^{K}\left( 2k +2a -1 \right)\theta_k +\sum_{k=1}^{K}(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| \right) .
\alpha_R \ = \ \int_{R_{i}^{(k)}\cap \Omega_{k+1}^{c}}\alpha_{R}\cdot \frac{\textbf{1}_{R_{i}^{(k)}\cap \Omega_{k+1}^{c}}(y)}{|R_{i}^{(k)}\cap \Omega_{k+1}^{c}|}\dy \leq 2\int_{R_{i}^{(k)}\cap \Omega_{k+1}^{c}}\alpha_R \frac{\textbf{1}_{R}(y)}{|R|}\dy \leq 2 \int_{R_{i}^{(k)}\cap \Omega_{k+1}^{c}}Sf(x) Sg(x) dx.
{\textstyle\sum_{j=k}^{\nu}}\,\sqrt{\Phi(x^{\ell(j)})\!-\!\Phi(x^j)}\le \sqrt{b^2/2}\,\Xi_k+m\sqrt{(\rho\!+\!1)/2}\,{\textstyle\sum_{j=k-\widehat{k}_1}^{\nu}}\,\|x^{j+1}\!-\!x^{j}\|.
\left(\int_Gu^6dsdt\right)^{\frac 13}\le \left(\frac1{2}\int_{\mathbb R^3}w^6dsd\tau d\lambda\right)^{\frac 13}\le C\int_{\mathbb R^3}|\widetilde{\nabla}w|^2dsd\tau d\lambda &\le C'\int_{\widetilde{T(G)}}|\widetilde \nabla w|^2dsd\tau d\lambda=2C'\int_G(tu_t^2+u_s^2)dsdt,
(\omega_{\psi^{-1}}(\omega_2t(a)n(u))\phi)(x)=|a|^{1/2}\frac{\gamma(\psi^{-1})}{\gamma(\psi^{-1}_a)}\int_F\psi(ua^2y^2)\phi(ay)\psi(2xy)dy=|a|^{1/2}\frac{\gamma(\psi^{-1})}{\gamma(\psi^{-1}_a)}\hat{\Phi}_{a,u}(x)
g_i(y_i)\coloneqq\prod_{k=0}^\infty \left(1-a_k^{(i)}y_i\right),h_i(y_i)\coloneqq \sum_{k=0}^\infty\frac{a_k^{(i)}}{1-a_k^{(i)}y_i}\quad\;(1\leq i\leq s).
\ \dot{s}\varepsilon(z)\dot{s}(e-\varepsilon(1))\lambda_i = \sum_{j,\mu}g^i_{j,\mu}\dot{s}(\varepsilon(z)-\varepsilon(x^i_{j,\mu}+z))\lambda_j \ = \sum_{j,\mu}g^i_{j,\mu}\sum_{k,\nu}g^j_{k,\nu}(\varepsilon(z^{-1}x^j_{k,\nu})- \varepsilon((x^i_{j,\mu}+z)^{-1}x^j_{k,\nu})) \lambda_k
\frac{1}{2 \pi i}\int_{C_1}\frac{(1-z^{\alpha})zqw}{(1-z)(1-qwz^2)}\, dz = \frac{1}{\pi}\sum_{j=0}^{\alpha-1}\sum_{m=0}^{\infty}(qw)^{m+1}\frac{\sin (2 \pi \theta_0(2m+j+2))}{2m+j+2}= \frac{1}{\pi}\sum_{j=0}^{\alpha-1}\sum_{m=1}^{\infty}(qw)^{m}\frac{\sin (2 \pi \theta_0(2m+j))}{2m+j}.
{\psi}^{D - K}= \left[ \begin{array}{c}{{\sum_{j , m}\xi_{m}^{j +}< j m |{\psi}_{-}^{( j )}>}}\\{{\sum_{j , m}\xi_{m}^{j -}< j m |{\psi}_{+}^{( j )}>}}\\ \end{array}\right] .
P_\lambda\Big(\eta_t^{(C,D)}(x)=2 x\in A\eta_t^{(C,D)}(y)\neq 0 y\in B\Big) &=P_\lambda\Big(\xi_t^{(B,A)}(x)=2 x\in D\xi_t^{(B,A)}(y)\neq 0 y\in C\Big).
\zeta:=\tfrac{1}{2}\left(\overline{f}^1_{12}+\overline{f}^2_{22}-\overline{f}^1_{21}-\epsilon t^2_{1\overline{1}}\right)\eta^{\overline{2}}-\tfrac{1}{2}\left(f^1_{12}+f^2_{22}-f^1_{21}-\epsilon\overline{t}^2_{1\overline{1}}\right)\eta^{2},