harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{I}_{\mathrm{D E}}= \frac{\Omega_{\tilde{d}+ 1}}{1 6 \pi}[ ( \tilde{d}+ 1 ) r_{+}^{\tilde{d}}- r_{-}^{\tilde{d}}] \beta_{H}- \frac{1}{4}\Sigma - \frac{\pi Q}{\tilde{d}\Omega_{\tilde{d}+ 1}r_{+}^{\tilde{d}}}Q \beta_{H}
E=-\frac{1}{2}\frac{a_{1}a_{2}}{4\pi r}\vec{v}_{1}\cdot[1+\hat{r}\hat{r}]\cdot\vec{v}_{2}e^{-\omega_{p}r}\{\frac{2}{(\omega_{p}r)^{2}}(e^{\omega_{p}r}-1)-\frac{2}{\omega_{p}r}\}.
F(\tau) :=1+\pi\sum_{c\geq1}\frac{(-1)^{\left\lfloor \frac{c+1}{2}\right\rfloor}}{c}&\times\sum_{d\pmod{2c}^\ast}\omega_{-d,2c}\exp\left(2\pi i\left(-\frac{d}{8}\left(1+(-1)^c\right)+\frac{d}{2c}+\tau\right)\right)\Phi_{c,d}(\tau),
\mathcal{A}_{shG}=\int^{\infty}_{-\infty}dt\left( \int^{0}_{\infty}dx\left[ \frac{1}{2}\partial_{\mu}\Phi \partial^{\mu}\Phi -\frac{m^{2}_{0}}{b^{2}}\cosh b\Phi \right] -M_{0}\cosh \frac{b}{2}\left( \Phi -\phi_{0}\right) \right)\,,
\xi_{0}= \xi_{0}^{\mu}\partial_{\mu}+ \xi_{0}^{\alpha}D_{\alpha}- \Lambda^{i}{}_{j}\phi^{j}\partial_{i}- \Big ( ( \alpha + \phi^{j}\Gamma_{j}D \alpha + D \xi ) \psi \Big )^{\alpha^{\prime}}\partial_{\alpha^{\prime}},
f_{k - \ell^+ - 1}(Q/X^+) \geq{\ell^- + 1 \choose k - \ell^+}={\ell^- + 1 \choose d - k - 1}, f_{k - \ell^- - 1}(Q/X^-) \geq{\ell^+ + 1 \choose k - \ell^-}={\ell^+ + 1 \choose d - k - 1}.
e_{\mu}^{( 0 )}= \frac{p_{\mu}}{\sqrt{- p^{2}}}, \quad e_{\mu}^{( \alpha )}\eta_{\alpha \beta}e_{\nu}^{( \beta )}= \eta_{\mu \nu}, \quad \varepsilon_{\mu \nu \lambda}e^{( 0 ) \mu}e^{( i ) \nu}e^{( j ) \lambda}= \varepsilon^{0 i j}.
\begin{cases}A_{\alpha,\mu}^-: =\chi_{(-\alpha, \alpha)}P_\mu \chi_{(-\infty,-\alpha)}P_\mu \chi_{(-\alpha, \alpha)}, begin{align*}0.2cm]A_{\alpha,\mu}^+: = \chi_{(-\alpha, \alpha)}P_\mu \chi_{(\alpha,\infty)}P_\mu \chi_{(-\alpha, \alpha)}.\end{cases}
\Gamma_{ij}^{2}=\begin{cases}-\tfrac12{x_2}L, (i,j)= (1,1), \tfrac14{x_1}L, (i,j)\in \{(1,2),(2,1)\}, -\tfrac12{L}, (i,j)\in \{(1,3),(3,1)\}, 0, (i,j)\in \{(2,2), (2,3), (3,2), (3,3)\}, \end{cases}
I^{( 1 )}= - 8 \pi \alpha \epsilon_{\lambda \kappa \sigma}\int \! \int_{S}d S_{\lambda}^{x}\int \! \int_{S_{f}}d S_{\kappa}^{y}\, \frac{( x - y )_{\sigma}}{| \overline{{{x}}}- \overline{{{y}}}|^{4}}\, ,
\widetilde{\eta}(t,x) = \frac{1}{\sqrt{\cosh(2t)}}e^{-\frac{\tanh(2t)}{2}x^2}, &\widetilde{K}(t, x, \xi) = \sqrt{\frac{\coth(2t)}{2\pi}}e^{-\frac{\coth(2t)}{2}\left(\xi- \frac{1}{\cosh(2t)}x\right)^2}.
\hat{\eta}=\bigl(\mathbf{h}_{\mathrm{c}}^{\mathrm{T}}\,\mathbf{h}_{\mathrm{c}}\bigr)^{-1}\,\mathbf{h}_{\mathrm{c}}^{\mathrm{T}}\,\bigl(\mathbf{y}-\mathbf{1}^{\mathrm{T}}L_{0}\bigr).
u_{1}+\Delta((c_{1}^{2}+a)\Delta u_{1}-a\Delta f_{1})=f_{1}+g_{1}\quad\;L^{2}(\Omega_{1}), u_{2}+c_{2}^{2}\Delta^{2}u_{2}=f_{2}+g_{2}\quad\;L^{2}(\Omega_{2}).
\int_{\mathbb R}g(x) \mathrm m_{t}({\mathrm d}x) \, =\, \int_{\mathbb R}g(x) \mathrm m_{0}({\mathrm d}x) + \int^{t}_{0}[ \mathcal A_{s}(\mathrm M) g] \,{\mathrm d}s \, ; 0 \le t \le T
0\leq\kappa_{\eta_1}(l)-\kappa_{\eta_2}(l)=\dot\alpha_2(l)\sin(\alpha_2(l))-\dot\alpha_1(l)\sin(\alpha_1(l))-\frac{2l}{1+l^2}\bigl(\cos(\alpha_2(l))-\cos(\alpha_1(l))\bigr).
S ={\frac{1}{1 6 \pi G_{1 0}}}\int d^{1 0}x \sqrt{- G^{d u a l}}\left[ e^{-{\frac{6}{7}}\phi}({\cal R}^{d u a l}+{\frac{1 6}{4 9}}\partial_{M}\phi \partial^{M}\phi ) -{\frac{1}{4}}e^{{\frac{6}{7}}\phi}F_{M N}F^{M N}\right] .
\begin{aligned}z_jz_i=qz_iz_j, \;\;\; \; i<j, \ \z_iz_j^*=qz_j^*z_i, \;\;\; \; i\neq j, &z_i^*z_i=z_iz_i^*+(1-q^2)\sum_{j=i+1}^n z_jz_j^*, \; \i=0,...,n, \sum_{j=0}^n z_jz_j^*=1,\end{aligned}
\langle{\mathcal L}Z,\phi\rangle \!=\!\int_U\! \left( (a_{ij}-\delta_{ij})\partial_i\partial_j Z\phi-\partial_iZ\,\partial_i\phi+b_k\partial_kZ\, \phi\right)dx \quad\hbox{for}\ \phi\in C^1_0(U).
\aligned\|W_{t_i,\varphi_i}\|_{W^{2,p}}\le c_2t_i^{-N}\big\|\varphi_i^{N}d\tau^{a_0/t_i}\big\|_{L^p}\le a_0t_i^{-N-1}\big\|\tau\|_{L^\infty}^{\frac{a_0-t_i}{t_i}}\|\varphi_i^Nd\tau\big\|_{L^p}\to 0.\endaligned
A(\varphi): h(x)=\sum_{j\in\mathbb{Z}}h_j\,e^{\mathrm{i}\,j\,x}\mapsto A(\varphi) h(x)=\sum_{j_1, j_2\in\mathbb{Z}}A_{j_1}^{j_2}(\varphi)h_{j_2}\,e^{\mathrm{i}\,j_1\,x}.
J^{w}(X) =-\frac{1}{2}\int_{0}^{\infty}w(x) f^2(x)dx =-\frac{1}{2}\int_{0}^{\infty}w(F^{-1}(u))f(F^{-1}(u))du=-\frac{1}{2}E(\Lambda_X^{w}(U)),
\begin{array}{l}{\partial_{\tau}X^{2}( 0 ) = \partial_{\sigma}X^{1}( 0 ) = 0 ,}\\{\partial_{\tau}X^{1}( \pi ) + \partial_{\tau}X^{2}( \pi ) \cot ( \pi \vartheta ) = 0 ,}\\{\partial_{\sigma}X^{2}( \pi ) - \partial_{\sigma}X^{1}( \pi ) \cot ( \pi \vartheta ) = 0 ,}\\ \end{array}
\{f_1,f_2\}= -L_{x_1}f_2 +(-1)^{(|x_1|-1)(|x_2|-1)}L_{x_2}f_1 &= -(-1)^{(|x_1|-1)(|x_2|-1)}(-L_{x_2}f_1 +(-1)^{(|x_1|-1)(|x_2|-1)}L_{x_1}f_2) &=-(-1)^{(|x_1|-1)(|x_2|-1)}\{f_2,f_1\}&=-(-1)^{(|f_1|+n-1)(|f_2|+n-1)}\{f_2,f_1\}.
{\cal L}_{H E}^{( 1 )}= \sum_{k = 2}^{\infty}{\frac{( - 1 )^{k}}{8 \pi^{2}}}{\frac{2^{2 k}\vert B_{2 k}\vert \; m^{4}}{2 k ( 2 k - 1 ) ( 2 k - 2 )}}{\frac{B^{2 k}}{B_{\mathrm{c r}}^{2 k}}}\, ,
C(j) -C(1)= \biggl(\bigl( g(jx)-g(x) \bigr)h(x)^{2k+1}\biggr)_{2k}= \biggl( \bigl( g(jx)-g(x) \bigr) h(x) h(x)^{2k}\biggr)_{2k}= \biggl( \sum_{i_1\ge 1}b_{i_1}(j^{2i_1}-1) x^{2i_1}\sum_{i_2\ge 0}a_{i_2}x^{2i_2}\sum_{i_3\ge 0}a^{(2k)}_{i_3}x^{2i_3}\biggr)_{2k}=\sum_{i_1+i_2+i_3=k, i_1\ge 1}(j^{2i_1}-1)b_{i_1}a_{i_2}a^{(2k)}_{i_3}.
\sup_{0\leq s\leq t\leq T}E\left[\left|J^m_{s,t}-\int_s^t B^{i_1,...,i_k}(r,q^m_r) \left(\int h(r,q^m_r,z) z_{i_1}...z_{i_k}dz \right) dr\right|^p\right]^{1/p}\leq \tilde C( m^{1/2}T+ m(2+T)+m^{1/2}T^{1/2}) \sup_{r\in[0,T]}E\left[ (1+\|q_r^m\|^{\tilde p})^{2p}\right]^{1/(2p)}E\left[(1+\|z_r^m\|^{k+1})^{2p}\right]^{1/(2p)}= O(m^{1/2}).
\int_{\{z\in M:-\Psi(z)\in U\}}|F_{t_0,\tilde{c}}|^2_h = \int_{\{z\in M:-\Psi(z)\in U\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in U\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h &+\int_{\{z\in M: -\Psi(z)\in U\cap N\}}|F_{t_0,\tilde{c}}|^2_h
R = \frac{1}{2}\left( R_1 + R_2\right) = \frac{1}{2}\left( H( \frac{B_1}{M})+ \frac{B_1}{M}H( \frac{B_2^1}{B_1}) + (1 - \frac{B_1}{M}) H( \frac{B_2^0}{M - B_1}) \right),
f_{k;0}^{+\pm\pm}= f_{k;0}^{-\pm\pm}(\stackrel{}{=:}f_{k;0}^{\circ\pm\pm}),f_{k;-kp}^{\pm+\pm}= f_{k;-kp}^{\pm-\pm}(\stackrel{}{=:}f_{k;0}^{\pm\circ\pm}),f_{k;kd}^{\pm\pm+}= f_{k;kd}^{\pm\pm-}(\stackrel{}{=:}f_{k;0}^{\pm\pm\circ}),
\Delta S_{b}^{\prime}= \int d^{D}x h \Phi_{b}^{\dagger}\left( T + \frac{1}{2}\stackrel{\sim}{V}+ \frac{1}{2}V + \frac{1}{2}b \Phi_{b}^{\dagger}\Phi_{b}- \mu \right) \Psi_{s}+ c o n j .
G\left(Z,W\right)= \left(g_{kl}\left(Z,W\right)\right)_{1\leq k,l\leq q'},\quad F\left(Z,W\right)= \left(f_{kl}\left(Z,W\right)\right)_{1\leq l\leq p'-q' \atop 1\leq k \leq q'}.
\{\epsilon^{0},\epsilon^{1},\epsilon^{2},\epsilon^{3}\}=\Bigg{\{}\begin{pmatrix}1 0 0 0 \end{pmatrix},\begin{pmatrix}0 1 0 0 \end{pmatrix},\begin{pmatrix}0 0 1 0 \end{pmatrix},\begin{pmatrix}0 0 0 1 \end{pmatrix}\Bigg{\}}\subset\mathbb{R}^{3,1}
\min_{x=(\widetilde{x},x_0)\in\mathbb{R}^{p+1}}\Big\{F_{\lambda,\mu}(x):=\sum_{i=1}^n\log\big(1+\exp(-b_i(a_i^{\mathbb{T}}\widetilde{x}+x_0))\big)+\frac{\mu}{2}\|x\|^2 +\lambda\|\widetilde{x}\|_0\Big\},
S(\omega)=\sqrt{(\frac{\pi\cdot T}{\Delta\Omega})}\cdot exp[-j((\omega-\Delta\Omega)^{2}\cdot\frac{T}{2\cdot\Delta\Omega})]\cdot[C(X_{1})+j\cdot S(X_{1})+C(X_{2})+j\cdot S(X_{2})]
\begin{aligned}\big(\,H_{\widehat{V}}u,u\,\big) \ = \ \|\widehat{D}u\|^2\ + \ (\widehat{V}u,u) \ \leq\ 2 \|Ds\|^2 \ + \ 2a^2\|s\|^2 \ + \ (Vs,s) \leq \ 2\,\big(\, \|Ds\|^2+ (Vs,s)\,\big) \ + \ 2a^2\|s\|^2 \ = \ 2\,(H_Vs,s) \ + \ 2a^2\,\|s\|^2.\end{aligned}
\begin{aligned}\mathrm{tr}(df_{1})\wedge\ast\boldsymbol{n}(\mu\wedge df_{2}) =\langle df_{1},i_{\mathcal{N}}(\mu\wedge df_{2})\rangle_{\Lambda^{1}}v_{\partial\Omega}=\langle df_{1},i_{\mathcal{N}}\mu\wedge df_{2}-\mu\wedge i_{\mathcal{N}}df_{2}\rangle_{\Lambda^{1}}v_{\partial\Omega}. \end{aligned}
\Gamma [ V ] = \int_{0}^{\infty}{\frac{\mathrm{d}T}{T}}[ 4 \pi T ]^{- d / 2}\mathrm{e}^{- m^{2}T}{\sum_{n = 1}^{\infty}}{\frac{( - T )^{n}}{n !}}\! \! \int \mathrm{d}x_{0}\mathrm{t r}O_{n}.
\begin{aligned}|u(z)-u(w)| \leq\left|u(z)-m_u(B(z, R))\right|+\left|u(w)-m_u(B(w, R))\right| &+\left|m_u(B(z, R))-m_u(B(w, R))\right| & \leq 2 \widetilde{A}[u]_{W^{\alpha, G}_{s}(X,d,\mu)}d(z, w)^{\alpha-\frac{s}{p_{0}}}&+\left|m_u(B(z, R))-m_u(B(w, R))\right| .\end{aligned}
\begin{array}{l}{{H \equiv P_{+}= \int d x^{-}\big [ ( 1 / 8 ) \mathrm{t r}~ E^{2}- ( m / 2 ) \big ( \psi_{-}^{\dagger}( x ) \psi_{+}( x ) + \psi_{+}^{\dagger}( x ) \psi_{-}( x ) \big ) \big ]}}\\{{\left[ A_{-}( x^{-}, x^{+}) , E ( y^{-}, x^{+}) \right] = i \delta ( x^{-}- y^{-})}}\\ \end{array}
\hat{A}[ C ] \hat{B}[ C^{\prime}] = \hat{B}[ C^{\prime}] \hat{A}[ C ] \times e^{2 \pi i L ( C , C^{\prime}) / N}, \mathrm{a n d}[ \hat{A}[ C ] , \hat{A}[ C^{\prime}] ] = [ \hat{B}[ C ] , \hat{B}[ C^{\prime}] ] = 0 ,
w(g(A,u),\xi) = \langle *F_{gA}+ \mu(gu), \xi \rangle_{L^2}+ \int_0^{\infty}||\mathcal{L}_{(e^{\textbf{i}t\xi}gA, e^{\textbf{i}t\xi}gu)}\xi||_{L^2}^2\, dt
H_{\mathrm{{n e w}}}\propto \sum_{m = \frac{1}{2}}^{\Lambda}b_{m}^{\dag}b_{m}\sum_{l = 1}^{\Lambda + \frac{1}{2}}\sum_{n = \frac{1}{2}}^{\Lambda}R^{l m n}( \zeta ) C_{1}^{l m n}( \zeta ) \delta_{m + l}^{n}. \nonumber
R_{\mu \nu}- \frac{1}{2}\, G_{\mu \nu}\,{\cal R}- \frac{1}{2}\left( \partial_{\mu}\varphi \, \partial_{\nu}\varphi - \frac{1}{2}\, G_{\mu \nu}\, \partial_{\rho}\varphi \, \partial^{\rho}\varphi \right) = 0
\displaystyle - \frac{1}{5 7 6 0}( 1 3 3 7 7 + 1 6 7 7 3 \xi + 1 2 0 9 1 \xi^{2}+ 9 4 6 3 \xi^{3}) + \frac{1}{3 0}( 1 + \xi )^{2}( - 3 8 + 7 \xi ) \zeta ( 3 )
R_{\kappa , i}( x ; q ) = \sum_{n \geq 0}\frac{x^{\kappa n}q^{\kappa n + \kappa n^{2}+ n - i n}( 1 - x^{i}q^{2 n i + i}) ( - 1 )^{n}q^{\frac{n ( n - 1 )}{2}}}{( q )_{n}( x q^{n + 1})_{\infty}}.
\sum_{m_1=0}^\infty \ldots \sum_{n_N=0}^\infty \frac{(2k-1)!!}{\prod_{i=1}^M (m_i)!\prod_{j=1}^N (n_j)!}C^k=\sum_{k=0}^\infty \frac{(2k-1)!!C^k(M+N)^{2k}}{(2k)!}<\infty,
\sup_{\lambda\in B^d}|\max_{j\in\mathcal J^*}(u^*_{n,j,\theta_{n}}(\lambda)-c^{*}_{n})-\max_{j\in\mathcal J^*}(\mathfrak{w}_{j}^*(\lambda)-c_{\pi^*})|\le \sup_{\lambda\in B^d}\max_{j\in\mathcal J^*}|r_{n,j,\theta_{n}}(\lambda)|=o_{\mathcal P}(1).
a_1^2 d_T(1,2) + a_1 \left [ \sum_{j \geq 3}(d_T(1,2)-d_T(1,j)+d_T(2,j))a_j \right ] + \sum_{j,k \geq 3}(d_T(2,j)-2 d_T(j,k)) a_j a_k = 0,
\begin{array}{l c c c l c c}{{{\mathrm R e}\Gamma_{+ -}^{T}( p^{2}) \big |_{p^{2}= M_{W}^{2}}}}&{{=}}&{{0}}&{{\qquad}}&{{{\mathrm R e}\Gamma_{Z Z}^{T}( p^{2}) \big |_{p^{2}= M_{Z}^{2}}}}&{{=}}&{{0}}\\{{\phantom{\mathrm R e}\Gamma_{A A}^{T}( p^{2}) \big |_{p^{2}= 0}}}&{{=}}&{{0}}&{{\qquad}}&{{{\mathrm R e}\Gamma_{H H}( p^{2}) \big |_{p^{2}= m_{H}^{2}}}}&{{=}}&{{0}}\\ \end{array}
(b^1_{ij})^{-1}= \left(\begin{array}{cccc}2 1 \cdots 1 1 2 \cdots 1 \vdots \vdots \vdots \vdots 1 1 \cdots 2\end{array}\right)^{-1}=\frac{1}{n-1}\left(\begin{array}{cccc}n-2 -1 \cdots -1 -1 n-2 \cdots -1 \vdots \vdots \vdots \vdots -1 -1 \cdots n-2\end{array}\right).
\tilde{\nabla}_ET_{\theta}e_4=-\omega_{34}(E)T_{\theta}e_3+\frac{i\kappa_2}{\kappa_1}T_{\theta}e_5+\frac{\mu_2}{\kappa_1}T_{\theta}e_6+i\kappa_1 e^{i\theta}dg_{\theta}(\overline{E}),
\mathcal{H}(\mathcal{L};q,t)=\sum_{g\geq 0}h_{2g-L}^{\mathcal{L}}(t)z^{2g-L}, P(\mathcal{L};q,t)=\sum_{g\geq0}p_{2g+1-L}^{\mathcal{L}}(t)z^{2g+1-L},
\frac{1}{A_{1}\cdot\cdot\cdot A_{n}}=(n-1)!\int_{0}^{1}d\alpha_{1}\cdot\cdot\cdot d\alpha_{n-1}\frac{1}{[\alpha_{1}A_{1}+\cdot\cdot\cdot+\alpha_{n-1}A_{n-1}+(1-\alpha_{1}-\cdot\cdot\cdot-\alpha_{n-1})A_{n}]^{n}},
S_{n}(t)=\frac{1}{(n\sigma^2)^{1/2}}\left( \log \|M_{\lfloor tn \rfloor}\|-nt\Lambda + (nt-\lfloor nt \rfloor) (\log \|M_{\lfloor tn \rfloor +1}\| - \log\|M_{\lfloor tn \rfloor}\|) \right)
E_{n_1}=\frac{\pi}{2\lambda_L^2e^2}\frac{V}{L_1^2}\left(\frac{n_1}{N_f}\!+\!\frac{\Phi}{\Phi_0}\!+\!\frac{Q}{2}\right)^{\!2}\!\!+\!2\pi N_f^2\lambda_L^2 e^2\frac{Q^2}{V}.
F(\widetilde{V'})\subseteq\widetilde{V}, G(\widetilde{V})\subseteq\widetilde{V'},A(\widetilde{V}),\mbox{}B(\widetilde{V})\subseteq\widetilde{V}, A'(\widetilde{V'}),\mbox{}B'(\widetilde{V'})\subseteq\widetilde{V'},J(\widetilde{V})=0.
\left\{\left( \begin{array}{c}{{u}}\\{{0}}\\ \end{array}\right) , \left( \begin{array}{c}{{v}}\\{{0}}\\ \end{array}\right) \right\}= \left\{\left( \begin{array}{c}{{0}}\\{{u}}\\ \end{array}\right) , \left( \begin{array}{c}{{0}}\\{{v}}\\ \end{array}\right) \right\}= 0
\rho(x)=|e|\sum_{i=1}^{N}\int_{\mathbb{R}^{3(N-1)}}P(x_{1},...,x_{i-1},x,x_{i+1},...,x_{N})d^{3}x_{1}\cdot\cdot\cdot d^{3}x_{i-1}d^{3}x_{i+1}\cdot\cdot\cdot d^{3}x_{N}
\||x|^{\gamma_1}\hat{v}\|_{L^s(K)}\le C\left(\||x|^{\gamma_2}\nabla v\|_{L^p(K)}^{a'}\||x|^{\gamma_3}v\|_{L^q(K)}^{1-a'}+\||x|^{\gamma_2}\nabla v\|_{L^p(K)}^{a''}\||x|^{\gamma_3}v\|_{L^q(K)}^{1-a''}\right) =2C\||x|^{\gamma_2}\nabla \hat{v}\|_{L^p(K)}^a\||x|^{\gamma_3}\hat{v}\|_{L^{q}(K)}^{1-a}.
M_\nu(\alpha\Vert\beta):=\left\{[P_{\nu}(2\beta-1)]^2[P_\nu(1-2\alpha)]^2-P_\nu(1-2\beta)P_\nu(2\beta-1)P_\nu(1-2\alpha)P_\nu(2\alpha-1)\right\}\log\left( 1-\frac{\beta}{\alpha}\right)
(\begin{matrix}u\\v\end{matrix})=(\begin{matrix}\frac{\partial\varphi}{\partial x}\\ \frac{\partial\varphi}{\partial y}\end{matrix})=(\begin{matrix}+\frac{\partial\psi}{\partial y}\\ -\frac{\partial\psi}{\partial x}\end{matrix})=(\begin{matrix}+2Ax\\ -2Ay\end{matrix}).
w_{tt}- \gamma \triangle w_{tt}+ a(-\triangle w) \triangle^{2}w + \alpha \triangle \theta = f(-\triangle w, -\nabla \triangle w)\phantom{0}(0, \infty) \times \Omega, \beta \theta_{t}- \eta \triangle \theta + \sigma \theta - \alpha \triangle w_{t}= 0\phantom{f(-\triangle w, -\nabla \triangle w)}(0, \infty) \times \Omega
0 \le \sum_{x \in (y,ut]}G^\omega_{(-\epsilon t,x)}(x-1,y)(1+\eta(\epsilon))^{x-y}-c^\omega_{-\epsilon t}(y) & \le \sum_{x \in (y,\infty)}c_1e^{-c_2(x-y)}((1+\eta(\epsilon))^{x-y}-1)\to 0.
\widehat{\mathcal A}_{s}( \widehat{\mathrm M}) g \, :=\, u \int_{\mathbb R^{2}}\widetilde{b}(s, y_{1}, y_{2}) \cdot \frac{\,m_{s}(y_{1}) \,}{\,m_{s}(y_{2})\,}g^{\prime}(y_{1}) \widehat{\mathrm M}_{s}({\mathrm d}y_{1}{\mathrm d}y_{2}) + (1-u) \int_{\mathbb R^{2}}\widetilde{b}(s, y_{1}, y_{2}) g^{\prime}(y_{1}) \mathrm m_{s}({\mathrm d}y_{1}) \mathrm m_{s}({\mathrm d}y_{2})
\Lambda_\mathcal{R}(n)=\sum_{\substack{p_1,\dots,p_{\ell}p_1\cdots p_\ell=n (\frac{\log{p}_1}{\log{X}},\dots,\frac{\log{p_\ell}}{\log{X}})\in\mathcal{R}}}\prod_{i=1}^\ell \log{p_i}.
\sum_{n,k,\ell,j\ge 0}A(n,k,\ell,j)x^k y^{\ell}z^j q^n = \sum_{k,\ell\ge 0}\frac{(-zq;q)_{\ell}}{(q;q)_k(q^2;q^2)_{\ell}}x^k y^{\ell}q^{\binom{k+\ell+1}{2}+\binom{\ell+1}{2}}.
\sum_{i=0}^{s}\frac{{s\choose i}f(i)}{(m+n+s-i+1){m+n+s-i\choose m}}=\sum_{i=0}^{s}\frac{{s\choose i}(-1)^{s-i}F(i)}{(m+n+s-i+1){m+n+s-i\choose n}},
| B ; \pm \rangle_{I}= \int [ d P d \chi ] e^{\frac{1}{2}\int d \sigma ( i P^{i}\partial_{\sigma}P^{j}+ \chi^{i}\chi^{j}) \omega_{i j}- \int d \sigma ( i p_{i}P^{i}- \pi_{i}\chi^{i})}| B ; \pm \rangle_{- 1 , I},
m=\sum_{j=1}^mp_j(e_j)=\sum_{j=1}^mp_j\left(\sum_{k=1}^nf_k(e_j)\tau_k\right)=\sum_{k=1}^n\sum_{j=1}^mf_k(e_j)p_j(\tau_k)=\sum_{k=1}^nf_k\left(\sum_{j=1}^mp_j(\tau_k)e_j\right)=\sum_{k=1}^nf_k(\tau_k).
\Delta\Phi^T(v) = \Phi_{T(v_{(1)})}v_{(2)}\otimes v_{(3)}+ (-1)^{(|v_{(1)}|+1)|v_{(2)}|}v_{(2)}\otimes \Phi_{T(v_{(1)})}v_{(3)}&=\Phi_{T(v_{(1)})}v_{(2)}\otimes v_{(3)}+ (-1)^{|v_{(1)}|}v_{(1)}\otimes \Phi_{T(v_{(2)})}v_{(3)}&= \left(\Phi^T\otimes \mathrm{id}+ \mathrm{id}\otimes \Phi^T\right)\Delta(v).
1 + \frac{1}{4 \beta^{2}}F_{\mu \nu}F^{\mu \nu}- \frac{1}{3 2 \beta^{4}}[ ( F_{\mu \nu}F^{\mu \nu})^{2}+ ( F_{\mu \nu}\tilde{F}^{\mu \nu})^{2}] \pm O ( \beta^{- 6})
\Omega\left[f^{\not\ominus}(z,t)\right] = \left(\frac{z}{1-z}\right)\left(f(z,t)-1\right)\left(\frac{f(z,1)f^{\not\ominus}(z,f(z,1)) - tf^{\not\ominus}(z,t)}{f(z,1)-t}\right).
\bar{P}_{r_{1}...r_{t}}(ik_{1},...,ik_{t})=\bar{P}_{r_{1}...r_{t}}(ik_{1},...,ik_{t-1},-i(k_1+...+k_{t-1}))+ik_{\mu}\tilde Q_{r_{1}...r_{t}}^{\mu}(k_{1},...,k_{t})
\begin{aligned}\mu_1:=3\mu+9, \qquad\alpha:=3\mu_1+1, \alpha_1:=(\alpha-3 \mu)/2, &k:=3(\mu_1+\rho^{-1})+1, \beta_1:=6 \mu_1+3 \rho^{-1}+3, 0<\rho<\frac{1-3 a}{C_1 (1+a)}.\end{aligned}
\widehat{\mu}_{(m)}=C_{(2)}(\vec{\varepsilon}) \int_{-\infty}^{\infty}p_{G}(x) x^{m}\left(1+\sum_{k\neq 2}^{M}\varepsilon_{k}x^{k}+ \frac{1}{2}\left(\sum_{k\neq 2}^{M}\varepsilon_{k}x^{k}\right)^{2}\right)dx,
\frac{\ell+1}{\sin^4 \phi}\big[ (1+2 \cos^2 \phi ) \, p_{1,\ell}( \phi)-5 \cos \phi \, p_{1,\ell+1}( \phi)+2 p_{1,\ell+2}( \phi) \big] &= \phi^{-4-1/2}O(\ell^{-1/2})+O(\phi^{-2}).
Z_m \leq C T^{1/2}(T Z_0^2 V_m + T Z_m Z_0 V_0 + T^{1/2}Z_0 V_m + T^{1/2}Z_m V_0 + V_m) &+ C T^{3/2}\sum_{0 < |(j,k)| < m}{m\choose j~k}Z_j Z_k V_{m-j-k}+ C T \sum_{j=1}^{m-1}{m \choose j}Z_j V_{m-j}
- A_s(t) = \int_0^s \frac{\partial}{\partial r}(\sigma^* \omega)(\dot \gamma_r) \, dr = \int_0^s d(\sigma^* \omega)\left( \frac{\partial}{\partial r}\gamma_r, \dot \gamma_r \right) \, dr & = \int_0^s \frac{1}{r}(\sigma^* \Omega)\left( S(\gamma_r), \dot \gamma_r \right) \, dr .
H_{C o l}= - e^{2}\int d^{3}x \left( \rho_{e l}( x_{i}) - \rho_{i o n}( x_{i}) \right) \left( - \frac{1}{4 \pi | \vec{x}- \vec{y}|}\right) \left( \rho_{e l}( y_{i}) - \rho_{i o n}( y_{i}) \right) .
S_{{\cal M}_{6}}= T_{5}\int_{{\cal M}_{6}}d^{6}y \big [ - \sqrt{- d e t ( g_{m n}+{}^{*}{H}_{m n p}v^{p})}+ \sqrt{- g_{6}}{\frac{1}{4}}v_{l}{~}^{*}H^{l m n}H_{m n p}v^{p}\big ] ,
\int_y \Delta_\varphi(x_1,y) \frac{\delta^2 S_B}{\delta\varphi(y)\delta\varphi(x_2)}=\int_y \frac{\delta^2 S_B}{\delta\varphi(x_1)\delta\varphi(y)}\Delta_\varphi(y,x_2)=\delta(x_1-x_2) \;.
a \leq a^2 t a^2 &\leq a^2 ta^2 ta^3 &\leq a^2 ta^2 ta^2 xb &\leq (a^2 ta^2 ta^2x b^2 sb^2 sb^2 )b &\leq hb, \;\textrm{where}\;h = a^2 ta^2 ta^2x b^2 sb^2 sb^2 \in S.
\bf{V}= \left( \begin{array}{cc}\frac{-i -2 u(t) \dot{u}}{2(x^2 -u(t)^2)}ix \frac{4l^2x^2+u(t)^2 (1-4l^2-4x^4 \omega^2 + 4 x^2 (\omega^2 u(t)^2 + \dot{u}(t)^2))}{4xi u(t)^2 (x^2 -u(t)^2 )^2}\frac{i +2 u(t) \dot{u}}{2(x^2 -u(t)^2)}\end{array}\right).
0< h_1 < \widetilde{h}:= \Bigl(\frac{c_L}{1+C_{\cal L}+\eta C_{A_1, B_2}}+\frac{\eta^2 C_{A_1, B_2}^2}{4(1+C_{\cal L}+\eta C_{A_1, B_2})}\Bigr)^{1/2}- \frac{\eta C_{A_1, B_2}}{2(1+C_{\cal L}+\eta C_{A_1, B_2})}
\begin{aligned}{\alpha}({\alpha}^{-1}-2\sqrt{2}r)\leq Q(\Lambda, r) \leq{\alpha}({\alpha}^{-1}-\sqrt{2}r) \sqrt{2}r\leq{\alpha}^{-1}; Q(\Lambda,r)=0 \sqrt{2}r\geq{\alpha}^{-1}.\end{aligned}
w_n=\mathbf{1}_{\mathcal{A}{}}(n)-\frac{\kappa_\mathcal{A}\#\mathcal{A}{}}{\#\mathcal{B}{}}=\mathbf{1}_{\mathcal{A}{}}(n)-\frac{\kappa_\mathcal{A}\#\mathcal{A}{}}{X}\ge -\frac{\kappa_\mathcal{A}\#\mathcal{A}{}}{X}.
\sum_{\ell=4}^{n-1}\sum_{s=1}^{n-\ell}\sum_{t=0}^{n-\ell-s}(\ell-3) =\sum_{s=1}^{n-4}\sum_{t=0}^{n-4-s}\sum_{\ell=4}^{n-s-t}(\ell-3)=\sum_{s=1}^{n-4}\sum_{t=0}^{n-4-s}\binom{n-2-s-t}{2}&=\sum_{s=1}^{n-4}\binom{n-1-s}{3}=\binom{n-1}{4}
\tau = \left( \begin{array}{l l}{i T_{1 1}}{i T_{1 2}}{i T_{2 1}}{i T_{2 2}}\end{array}\right) ~ ~ ~ , ~ ~ ~ I = \left( \begin{array}{l l}{- 1_{2}}{0_{2}}{0_{2}}{1_{2}}\end{array}\right) ~ ,
E_{o_-}(\lambda)T_{n_v}\in T_{n_v}E_{o_-}(n_v^{-1}\cdot\lambda) + \left(\sum_{v_3<v,v_3\in W^P_0,x\in W_P(1)}C[q_s^{\pm 1}]E_-(n_{v_3}x)\cap \mathcal{H}\right) & = T_{n_v}E_{o_-}(n_v^{-1}\cdot\lambda) + \sum_{v_3<v,v_3\in W^P_0,x\in W_P(1)}C[q_s]E_-(n_{v_3}x).
\int_X u(x) d \mu(x) \leq \sum_{p \in N(X, \varepsilon)}\int_{B(p, 4 \varepsilon)}u(x) d \mu(x) = \sum_{p \in N(X, \varepsilon)}u_{4B(p,4\varepsilon)}\mu(B(p,4 \varepsilon)) \leq C(2 \varepsilon)C(\varepsilon)\sum_{p \in N(X, \varepsilon)}u_{4B(p, \varepsilon)}\mu(B(p, \varepsilon))
\int_{\Omega}D_t c_{ih}^n v_i dx =- A_{c_{ih}^n}(p_{ih}^n, v_{i}), \int_{\Omega}p_{ih}^n r_i dx =\int_{\Omega}(q_i \psi_h^n + \log c_{ih}^n )r_{i}dx, A_1(\psi_{h}^n, \eta) = \int_{\Omega}\rho^n_h \eta dx + \int_{\partial \Omega}\sigma \eta ds.
\phi_{(123),1}(e_1,e_2,e_3) =2a_1(\kappa^L_{(123)}[e_1,e_2-e_3]+\kappa^L_{(123)}[e_2,e_3-e_1]+\kappa^L_{(123)}[e_3,e_1-e_2]) &=4a_1(\kappa^L_{(123)}[e_1,e_2]+\kappa^L_{(123)}[e_2,e_3]+\kappa^L_{(123)}[e_3,e_1]) &=12a_1[a(e_1+e_2+e_3)+b(e_4+\cdots+e_n)]
\begin{array}{rcl}\Xi_A(S) = \bigwedge \{\theta \in Co_{\bf K}(A) \mid S \subseteq i_A(\theta) \}= \bigwedge \{\theta \in Co_{\bf K}(A) \mid S \subseteq \{a \in A \mid \delta(\bar{a})=\epsilon(\bar{a}) A/\theta \}\}&= \bigwedge \{\theta \in Co_{\bf K}(A) \mid \{\langle \delta(s), \epsilon(s)\rangle \mid s \in S \}\subseteq \theta\}\end{array}
|R_N^1(s,n)| \leqslant C \log(n) \int_0^n z^{2\alpha-2s-1}\leq C, \ \textup{for}\N = M+1, |R_{N'}^2(s,n)| \leqslant C \log(n) \int_0^n z^{2\alpha-2s-1}\leq C, \ \textup{for}\N'=2(M+1).
c_{1}= \alpha - 1 \hspace{0 . 3 c m}, \hspace{0 . 3 c m}c_{2}= - \alpha \hspace{0 . 3 c m}, \hspace{0 . 3 c m}c_{j}= 0 \, \, \, ; \, \, \, j > 2 \, \, ,
\phi_{t}^\ell(S) =|\{m\in S\ :\F_t(m)\ge \ell\}| \alpha_{t}^\ell(S) =|\{m\in S\ :\A_t(m)\ge \ell\}| \beta_{t}^\ell(S) =|\{m\in S\ :\B_t(m)\ge \ell\}|,
U(z)+iV(z) = \displaystyle\frac{1}{2\pi}\int_0^\pi \frac{e^{it}+z}{e^{it}-z}\,u_+(t)\,dt +\displaystyle\frac{1}{2\pi}\int_{-\pi}^0 \frac{e^{it}+z}{e^{it}-z}\,u_-(t)\,dt,z \in{\mathbb D}\,.
\alpha_1(e_1)=e_1, \alpha_1(e_2)=\frac{2 a}{b-1}e_1-e_2, \beta_1(e_1)=e_1, \beta_1(e_2)=-ae_1+b e_2, \mu_1(e_1,e_1)=e_1, \mu_1(e_1,e_2)= -ae_1+b e_2, \mu_1(e_2,e_1)=\frac{2 a}{b-1}e_1-e_2, \mu_1(e_2,e_2)=-\frac{a^2(b-2)}{(b-1)^2}e_1+ae_2,
\begin{array}{c c c c c c c c c}{\cite{b h k}:}&{x}&{A}&{N}&{K}&{H}&{J}&{\alpha}&{\beta^{2}}\\{\cite{b f m 2}:}&{\alpha r}&{e^{\lambda + \nu}}&{e^{- 2 \lambda}}&{W}&{\alpha H}&{J}&{\alpha}&{\frac{\beta^{2}}{2}}\\ \end{array}
\Delta(1_H)=1_H\otimes 1_H,\varepsilon_H(1_H)=1, \psi(1_H)=1_H, \omega (1_H)=1_H, \varepsilon_H\circ \alpha =\varepsilon_H, \varepsilon_H\circ\beta =\varepsilon_H, \varepsilon_H(hh^{\prime})=\varepsilon_H(h)\varepsilon_H(h^{\prime}), \forall \,h, h^{\prime}\in H.
\exp \bigg( 2 \sum_{m=1}^M \sigma_{m,W}^2 \bigg) \prod_{m=1}^M \frac{1}{\sqrt{2 \pi \sigma_{m,W}^2}}\idotsint \limits_{\mathcal{R}'}\exp \bigg( - \sum_{m=1}^M \frac{z_m'^2}{2 \sigma_{m,W}^2}\bigg) \, d\mathbf{z}',
\lambda_{11}- i \lambda_{12}=1+2\rho \nu +2\rho^2 \nu^2 \frac{e_2}{\pi}+ 2\rho^3 \nu^3 \sum_{n=2}^\infty (-1)^n (n-1)e_{nn}\frac{\nu^{n-2}}{\pi^n}+ O(\rho^4).
\displaystyle B(|v-v_{\ast}|, \cos\theta) = \phi(|v-v_{\ast}|)\, b(\cos\theta), \phi(\xi) = C_{\phi}\, \xi^{\gamma}, \, \gamma\in[0,1], begin{align*}2pt] \displaystyle \forall\eta\in[-1,1], |b(\eta)|\leq C_b, |b^{\prime}(\eta)|\leq C_b\,,
P_{\lambda}(1,t,t^2, \ldots, t^{n-1};q,t) = t^{n(\lambda)}\prod_{b\in \lambda}\frac{1-q^{\mathrm{coarm}_\lambda(b)}t^{n-\mathrm{coleg}_\lambda(b)}}{1-q^{\mathrm{arm}_\lambda(b)}t^{\mathrm{leg}_\lambda(b)+1}}.
{\cal L}_2 = e^2\left(-f^{abc}f^{ade}A_\mu^bA^{d\mu}A_\nu^cA^{e\nu}-\frac{\mu^2}{e^2v^2}\left|\Phi^\prime\right|^4 +\frac{1}{2}\,A_\mu^aA^{b\mu}\Phi^{\prime\dagger}\{\lambda^a,\lambda^b\}\Phi^\prime\right).
\exp\left\{-\sum_{i=0}^{n}\frac 1{\left(i+e_{[k]}^1\right)\ln \left(i+e_{[k]}^1\right)\dots \ln_k \left(i+e_{[k]}^1\right)}\right\}\le \frac{1}{\ln_{k}(n+1+e_{[k]}^1)}.
\displaystyle{\frac{m_{\pi}^{2}f_{\pi}j_{\pi}( q^{2})}{m_{\pi}^{2}- q^{2}}= \partial^{\mu}J_{\mu}( q ) = \frac{q^{2}f_{\pi}j_{\pi}( q^{2})}{m_{\pi}^{2}- q^{2}}+ \partial^{\mu}\widehat J_{\mu}( q ) .}
\Big ( \partial_{\sigma}\partial^{\sigma}H_{\mu \nu}+ \partial_{\mu}\partial_{\nu}H - \partial_{\mu}\partial^{\sigma}H_{\sigma \nu}- \partial_{\nu}\partial^{\sigma}H_{\sigma \mu}- \eta_{\mu \nu}\left[ \partial_{\sigma}\partial^{\sigma}H - \partial^{\sigma}\partial^{\rho}H_{\sigma \rho}\right] +{\widehat M}_{P}^{3 - D}T_{\mu \nu}\Big ) \delta ( z ) = 0 ~ .
(i=1) \,\,0.25030, 0.24960, 0.24953, 0.25056, (i=2) \,\,0.25118, 0.24711, 0.24956, 0.25216, (i=3) \,\,0.25013, 0.25103, 0.24831, 0.25053, (i=4) \,\,0.24914, 0.25041, 0.25020, 0.25025, (i=5) \,\,0.24941, 0.24956, 0.25081, 0.25023, (i=6) \,\,0.24950, 0.25072, 0.24924, 0.25054, (i=7) \,\,0.25039, 0.24860, 0.25156, 0.24946.
\int_{|x_T| \le \tau^\theta,\, |x_\perp| \le R}(1 + |x_T|^2)^{-1}(1 + |x|^2)^{\delta+1}\, dx\le \mathcal O(1) \int_{|x_T| \le \tau^\theta}(1 + |x_T|^2)^{\delta}\, dx_T = \mathcal O(\tau^{2(\delta+1)\theta})\quad\quad\tau \to \infty.
r:=\left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{h}\right\rceil,\,\,\,s:=\left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{w}\right\rceil,
\begin{aligned}\tau_h(\boldsymbol{D}^{\mu}u)=\boldsymbol{D}^{\mu}(\tau_hu) ,\tau_h(\boldsymbol{D}^{\mu*}u)=\boldsymbol{D}^{\mu*}(\tau_hu) \Pi_\kappa(\boldsymbol{D}^{\mu}u)=\kappa^{-\mu}\boldsymbol{D}^{\mu}(\Pi_\kappa u) ,\Pi_\kappa(\boldsymbol{D}^{\mu*}u)=\kappa^{-\mu}\boldsymbol{D}^{\mu*}(\Pi_\kappa u).\end{aligned}