harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\theta(W)| = |w_0|+ |w_1|+ |w_2|+ |w_3|+ 2|a_1|+2|a_2| &= (|w_0|+ |w_1|+ |w_2|+ |w_3|+ 2|a_1|+2|t_1|)+2|a_2|-2|t_1| \ge |\psi(W')|+2|a_2|-2|t_1|.
f(x)= \frac{a_0}{\sqrt{\sinh 1}}\sinh\left(x-\frac{1}{2}\right)+b_0 &+\sum_{k=1}^\infty\sqrt{\frac{2}{1+4\pi^2 k^2}}\big(a_k\sin(2\pi k x)+b_k\cos(2\pi k x)\big)\,,
\Psi(K_i\otimes 1) = K_i\otimes 1, \Psi(1\otimes K_i) = 1\otimes K_i, \Psi(L_i\otimes 1) = L_i\otimes 1, \Psi(1\otimes L_i) = 1\otimes L_i, \Psi(E_i\otimes 1) = E_i\otimes L_i^{-1}, \Psi(1\otimes E_i) = K_i^{-1}\otimes E_i, \Psi(F_i\otimes 1) = F_i\otimes L_i, \Psi(1\otimes F_i) = K_i\otimes F_i.
\# \mathrm{F Z M ( I I B )}= 8 \left( \# \mathrm{m o n o p o l e}- ( \widetilde{N}- 1 ) \right) + 4 \widetilde{N}= \# \mathrm{m o n o p o l e}\, \times 8 - 4 \widetilde{N}+ 8 \, ,
\Big\langle \prod_{a=1}^N\prod_{i=1}^{n_a}\overline{\psi}^{(a)}(x^{(a)}_i )P_+ \psi^{(a)}(x^{(a)}_i )\prod_{i=1}^{m_a}\overline{\psi}^{(a)}(y^{(a)}_i )P_- \psi^{(a)}(y^{(a)}_i ) \Big\rangle
\chi (S;a,b) \cdot V = (-1)^{r-|S|+a-b}|T_1||T_2| = (-1)^{r-|S|+a-b}\binom{|S \backslash U|}{a - |S \cap U|}\binom{|\bar{S}\backslash U|}{b}= (-1)^{r-|S|+a-b}\binom{|S|-|S \cap U|}{a-|S \cap U|}\binom{r-|S|-|U|+|S \cap U|}{b}
\sigma_c^{\rm np}=-\frac{2N_f}{\pi^2}\int\limits_0^{\Lambda_{\rm np}}dp\frac{p^2\Sigma(p)}{p^2+\Sigma^2(p)}=\frac{N_f^2}{4}\Lambda_{\rm np}^2\frac{d\Sigma}{dp}\bigg|_{p=\Lambda_{\rm np}}.
{\rm Coeff}\left[\prod_{1 \leq i<j \leq n}\left(\frac{1-z_j/z_i}{1- t z_j/z_i}\right)S_{\lambda/\mu}(z_1,\dots,z_n;t),z_1^{\nu_1}\dots z_n^{\nu_n}\right]=\b{K}^{\lambda}_{\mu\nu}(t)b_{\nu}(t).
\lim\limits_{t\rightarrow \infty}\mathbb{E}^{m_0}[<\mu_t^e,\hat{\psi}_0>]=\lim\limits_{t\rightarrow \infty}\mathbb{E}^{\hat{\psi}_0}[<m_0,\psi_t^{t,e}>]=\lim\limits_{t\rightarrow \infty}\mathbb{E}^{\hat{\psi}_0}[\psi_t^{t,e}],
\sigma(0,2k+1) =1+\frac{1}{p}-\frac{1}{p^{k+1}}-\frac{1}{p^{k+2}}, \sigma(1,2k+1) =2-1/p^{k}-1/p^{k+1}, \sigma(2,2k+1) =1+p-\frac{1}{p^{k-1}}-\frac{1}{p^{k}}.
\left.\frac{\partial}{\partial\varepsilon}\left( (f^{\varepsilon\nu})^{-1}\overline{f^{\varepsilon\nu}}\right)\right|_{\varepsilon=0},\left.\frac{\partial}{\partial\bar{\varepsilon}}\left( (f^{\varepsilon\nu})^{-1}\overline{f^{\varepsilon\nu}}\right)\right|_{\varepsilon=0},
n=[\begin{matrix}\frac{yz}{\sqrt{yz^{2}+xz^{2}+xy^{2}}}\\ \frac{xz}{\sqrt{yz^{2}+xz^{2}+xy^{2}}}\\ \frac{xy}{\sqrt{yz^{2}+xz^{2}+xy^{2}}}\end{matrix}]
S^{\pm}_{j}=\theta_{s^\delta}(|\eta|)\theta_{s^\delta}(|\xi-\eta|) \mathbb{I}_{\partial_{\eta}\phi\sim \pm 2^{-j}}(1-\varphi^{s})\frac{\partial_{\xi}\phi}{\partial_{\eta}\phi}\frac{\partial_{\eta}^2\phi}{\partial_{\eta}\phi}.
S(x,y) = \begin{cases}(x+\eta,y+\alpha) x\in [0,1-\eta), (x+\eta,1-y) x\in [1-\eta,1); \end{cases}R(x,a) = \begin{cases}(x+\eta,a) x\in [0,1-\eta), (x+\eta,a+1) x\in [1-\eta,1). \end{cases}
\hbar \sum_{r \geq 0}X_{i,r}^{\pm}u^{-r-1}= x_i^{\pm}(u - \dfrac{1}{2}i(\varepsilon_1 - \varepsilon_2)), 1 + \hbar \sum_{r \geq 0}H_{i,r}u^{-r-1}= h_i(u - \dfrac{1}{2}i(\varepsilon_1 - \varepsilon_2))
X_k=T_k(k+1)\cup (\bigcup_{i=1}^{k+1}S_k(i)) k\geq 1, &Y_1=T_1(1), Y_k=T_k(k)\cup (\bigcup_{i=1}^{k-1}S_k(i)) k\geq 2, &Z_2=T_2(1), Z_k=T_k(k-1)\cup( \bigcup_{i=1}^{k-2}S_k(i)) k\geq 3.
\delta \psi_{\mu}= \partial_{\mu}\epsilon -{1\over 4}\omega_{\mu}^{ab}\gamma_{ab}\epsilon +{(-1)^p \over 8(p+2)!}e^{\phi}F_{\mu_1 ... \mu_{p+2}}\gamma^{\mu_1 ...\mu_{p+2}}\gamma_{\mu}\epsilon^{'}_{(p)}
{\cal A}={\cal A}_0+{\cal A}^{\rm int},~~~~~~{\cal A}_0=\int_{t_a}^{t_b}dt\left[\frac{M}{2}\dot x^2-\frac{\Omega^2}{2}x^2\right] ,~~~~{\cal A}^{\rm int}=\int_{t_a}^{t_b}dt\,V^{\rm int}_{\Omega}\equiv\int_{t_a}^{t_b}dt\left[-\frac{( \omega^2- \Omega^2)}{2}x^2-V^{\rm int}(x)\right] ,
\frac{\partial \mathbf H^{(1)}}{\partial Z_{jk}^{(2)}}=\mathbf e_{j+n}\mathbf e_{k+n}^T\mathbf L_t -\mathbf M_t\mathbf L_t\mathbf e_{k+n}\mathbf e_{j+n}^T\mathbf M_t\mathbf L_t -\mathbf M_t\mathbf L_t\mathbf M_t^*\mathbf e_{j+n}\mathbf e_{k+n}^T\mathbf L_t
J_1(b,c)= \frac{2b-1}{2}\left( \log\left(\binom{b+1}{2}+c \right) -\log\left(\binom{b}{2}+c \right) -2\log\left(b+c \right)+2\log\left(b+c-1 \right) \right), J_2(b,c)= c \left( \log(c+1)-\log(c)-\log(b+c)+\log(b+c-1) \right), J_3(b,c)= \log\left(\binom{b+1}{2}+c \right) +\log(c+1) -2\log(b+c).
D_{s\chi_0\eta_0}=(1+\varrho^0)D+s((\chi_0)_z\eta_0+(\chi_0)_{\bar{z}}\bar{\eta}_0)e_1\partial_t+\Theta_s^0+\mathcal{H}^0_s+\mathcal{F}^0_s+\mathcal{R}_s^0.
\int_{s}^{t}S_{t-\tau}Y_{\tau}\circ\mathrm{d}\mathbf{X}_{\tau}:=\lim_{\substack{|\pi|\rightarrow 0, \pi=\lbrace \tau_0=s,\tau_{1},...\tau_{m}=t \rbrace}}\sum_{0\leq j<m}\big{[}S_{t-\tau_j}Y_{\tau_j}\circ (\delta X)_{\tau_j,\tau_{j+1}}+S_{t-\tau_j}Y^{\prime}_{\tau_j}\circ\mathbb{X}_{\tau_j,\tau_{j+1}}\big{]}.
\mu_{\theta}(x_{t},t)=\tilde{\mu}_{t}(x_{t},\frac{x_{t}-\sqrt{1-\overline{\alpha}_{t}}\epsilon_{\theta}(x_{t},t)}{\sqrt{\overline{\alpha}_{t}}})=\frac{x_{t}-\epsilon_{\theta}(x_{t},t)\beta_{t}/\sqrt{1-\overline{\alpha}_{t}}}{\sqrt{\alpha_{t}}}
s_{k,l}= \tilde s_k + lq_k,0\le l\le L_k, L_k = [(1-\varepsilon)/ q_k],q_k = \theta \frac{\Delta(x_k)}{x_k}, \tau_{k,n}= \tau(x_k)+ nq_k, 0\le |n| \le N_k,N_k = [\tau^*(x_k)/q_k].
\frac{(x_1 \cdots x_{\rho})^k}{(1 - x_1 - \ldots - x_{\rho})^{k + 1}}= \sum_{n_1 \geq 0}\cdots \sum_{n_{\rho}\geq 0}\binom{n_1 + \ldots + n_{\rho}- (\rho - 1) k}{n_1 - k, \ldots, n_{\rho}- k, k}x_1^{n_1}\cdots x_{\rho}^{n_{\rho}}.
\mathbf{w}_\varepsilon (\cdot ,t):=e^{-\mathcal{A}_N^0 \varepsilon^2}\mathbf{u}_0(\cdot ,t-\varepsilon^2)=e^{-\mathcal{A}_N^0 t}\boldsymbol{\varphi}(\cdot)+\int_0^{t-\varepsilon^2}e^{-\mathcal{A}_N^0 (t-\widetilde{t})}\mathbf{F}(\cdot ,\widetilde{t})\,d\widetilde{t}.
A(\sqrt{p}(\sum_{n=b+1}^{N}x_{n}^{2})^{1/2}+\sum_{n=1}^{b}x_{n})\le(E|\sum_{n=1}^{N}\epsilon_{n}x_{n}|^{p})^{1/p}\le B(\sqrt{p}(\sum_{n=b+1}^{N}x_{n}^{2})^{1/2}+\sum_{n=1}^{b}x_{n})
\begin{aligned}\sum_{r=1}^{k-1}(-1)^{r-i}\binom{k-i}{r-i}\binom{m-r}{s-r}=\sum_{r=i}^{s}(-1)^{r-i}\binom{k-i}{r-i}\binom{m-r}{s-r}=\sum_{r=0}^{s-i}(-1)^r\binom{k-i}{r}\binom{m-i-r}{s-i-r}&=(-1)^{s-i}\binom{k-m+s-i-1}{s-i}.\end{aligned}
A_{a}^{-}= i \omega_{0 a}-{\frac{1}{2}}\epsilon_{a}^{b c}\omega_{b c}- ( 2 \pi G ) \epsilon_{a b c}\left[{\frac{1}{2}}\epsilon^{b c}_{A B}\psi^{A}e^{B}- i \psi^{b}e^{c}\right]
L_{j,k,k-1}={t_{j,k+\widetilde{p}}-t_{j,k+\widetilde{p}-1}\over t_{j,k+\widetilde{p}}-t_{j,k}}\,\,\,\mbox{and}\,\,\,L_{j,k,k+1}={t_{j,k+1}-t_{j,k}\over t_{j,k+\widetilde{p}}-t_{j,k}}.
\begin{array}{rclcrcl}\mbox{Tr}[T_a , \cdots] = 0,{\qquad}\mbox{Tr}[\Sigma_\alpha, \cdots] = 0, \mbox{Str}[T_a , \cdots] = 0,{\qquad}\mbox{Str}\{\Sigma_\alpha, \cdots\}= 0,\end{array}
1^T_{2n+2}\exp_2\left(\omega_{2n+2|0}\right) = 1^T_{2n}\left[2\exp_2\left(\omega_{2n|0}\right) + 2\exp_2\left(\omega_{2n|0}-2\cdot 1_{2n}\right)\right] = \left(2 + 2\cdot2^{-2}\right) 1^T_{2n}\exp_2\left(\omega_{2n|0}\right)
w_1(M) = \begin{cases}O(\geq 0)-\frac{1}{2}\beta_n-\frac{1}{2}\gamma_n (a,b) = (n,n), O(\geq 0)-\beta_n (a,b)=(n+1,n), O(\geq 0)-\frac{1}{2}(\beta_{n-1}+\beta_n+\gamma_{n-1}+\gamma_n) (a,b)=(n+1,n+1). \end{cases}
\mathcal{O}^{}_{\overline{\mathcal{M}}_{g,n}(X,0)}= \sum_{i \geq 0}(-1)^i \bigwedge \nolimits^i \left( pr_1^* R^1\pi_* \mathcal{O}_\mathcal{C}\otimes \textnormal{ev}^* T_X \right)^\vee
{\cal K}^L(\sigma=2)= \left( \begin{array}{cc}-\frac{d^2}{dx^2}+4-\frac{3}{\cosh^2x}\frac{3}{\cosh^2x}\frac{3}{\cosh^2x}-\frac{d^2}{dx^2}+4-\frac{3}{\cosh^2x}\end{array}\right)\qquad ,
=\int_0^\infty\ldots\int_0^\infty\mathrm{d}s_{12}\ldots\mathrm{d}s_{(n-1)n}\tilde{F}(s_{12},\ldots,s_{(n-1)n},g^2,d)e^{-(s_{12}r_{12}^2+\ldots+s_{(n-1)n}r_{(n-1)n}^2)}
F^{(2m)}_n :=\bar{g}_n^{(2m)}\sum_{j=j_0+1}^{n-1}\dfrac{f_j^{(2m)}}{\bar{g}_{j+1}^{(2m)}},H_n^{(2m)}:=\bar{g}_n^{(2m)}\sum_{j=j_0+1}^{n-1}\dfrac{h_j^{(2m)}}{\bar{g}_{j+1}^{(2m)}}.
\dot C_t(V_t)(\varphi)\dot C_t - \frac{1}{2}\ddot{C}_t \geq\Big[\frac{C_t}{t}\big(C_t^{-1}- \chi_tC_t^{-2}\big)\frac{C_t}{t}+\frac{1}{t}AC_t \Big]\dot C_t &= \Big[\frac{1}{t}\Big(\frac{1}{t}+A\Big)C_t - \frac{\chi_t}{t^2}\Big]\dot C_t &= \Big[\frac{1}{t}- \frac{\chi_t}{t^2}\Big]\dot C_t= \dot \kappa_t \dot C_t.
I(z,-N,c,d,r_{1},r_{2}) =2\pi ik^{N}\sum\limits_{m+n=N+2}B_{m}\left(\frac{c\mu+j-\left\{R_{1}\right\}}{ck}\right) B_{n}\left( \frac{v+\left\{(dj+\rho)/c\right\}}{k}\right) \frac{\left( -(cz+d)\right)^{m-1}}{m!n!}.
-\Delta g=|S|^{\frac{4}{N-2}}S-|Q|^{\frac{4}{N-2}}Q=|Q+g|^{\frac{4}{N-2}}(Q+g)-|Q|^{\frac{4}{N-2}}Q=\frac{N+2}{N-2}|Q|^{\frac{4}{N-2}}g+R_{Q}(g),
I(i,j) =\left( \sum_{s}g(s) \mu(C_s)\right)\left( \sum_{s'}g(s') \mu(C_{s'})\right) \left(1+O\left(\tau^{\sqrt{j-i-k}}\right)\right) &=\left( \int_0^1 \log f_k(x) \d\mu\right)\left( \int_0^1 \log f_k(x) \d\mu\right)\cdot \left(1+O\left(\tau^{\sqrt{j-i-k}}\right)\right) &= \lambda_k^2 \left(1+O\left(\tau^{\sqrt{j-i-k}}\right)\right).
m_i = x_{i-1}, 1 \le i \le n-d-1, m_{n-d + k}= \left( \prod_{j=0}^{k-1}x_{n-d-1 + j}^{a_{d - j}}\right) x_{n-d-1 + k}^{a_{d-k}+ 1}\,, 0 \le k \le d - \ell -2, m_{n-\ell-1}= \prod_{j=0}^{d-\ell-1}x_{n-d-1 + j}^{a_{d - j}}.
F_n =\left( \begin{array}{ccccccc}F^{(n)}_{B_1}* \cdots * * * F^{(n)}_{B_2}\cdots * * \vdots \vdots \ddots \vdots \vdots * * \cdots F^{(n)}_{B_l}* * * \cdots * F^{(n)}_{B_0}\end{array}\right),
\sum_{\mathbf{i}\in \mathcal{J}^2}\varphi_2(\mathbf i_{1:4})^3\varphi_2(\mathbf i_{5:8})^2=\sum_{\mathbf{i}\in \mathcal{I}_2^2}\varphi_2(\mathbf i_{1:4})^3\varphi_2(\mathbf i_{5:8})^2 + o(n^4)=\frac{n^4}{4}\cdot \frac{1}{90^5}(1+o(1)),
\leq C_1\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}\left(1+|u|^{\alpha}\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx &\leq 2C_1\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}|u|^{\alpha}\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx &\leq C\frac{(\beta+1)^2}{4\beta}|u|_{2_s^*}^{\alpha}|(u_k+\delta)^{\beta}|_q
\ell \left( \frac{\tau_{0}}{2^{k-1}}\right) \, < \, \sqrt{\frac{\tau_{0}}{\varepsilon 2^{k-1}}}\, = \, \left( \frac{1}{\sqrt{2}}\right)^{k_0 + k-1}\, \leq \, \left( \frac{1}{\sqrt{2}}\right)^{k},
L^{2d}\langle | (\Delta_j \Gamma)^k(Z,Z',Z'')|^{2q}\rangle^{1/q}= L^{2d}\langle | \Delta_j \Gamma(Z,Z')|^{2q}\rangle^{1/q}\leq C \langle \hat \Phi_0^{4q}\rangle^{\frac{1}{q}}.
h(f,\Gamma_n)= h\big(f|_{\Gamma_n},\hat{X}(f^{\ell_n}|_{\Gamma_n},\varphi|_{\Gamma_n})\big)\le h\big(f,\hat{X}(f,\varphi)\big)\le h(f,\hat{X}(f))\le h(f,X).
\xi_{\pm m}^{(s)}=\,\frac{\texttt{t}^{\pm m}-1}{\texttt{t}^{\pm m}}\texttt{p}_{\pm m}^{*(s)}= (-1)^{m}\texttt{n}_s\frac{\texttt{q}_s^{\frac m2}\texttt{t}_s^{\pm\texttt{a}_sm}}{1-\texttt{q}_s^m}\,\, s=1,\dots,k,m>0
= \frac{px^2}{(1-px)(1-x)}\sum_{r = 1}^{\infty}\bigg((p^3x^3)^r - (p^3x^4)^r\bigg) - \frac{p^2x^3}{(1-px)(1-px^2)}\sum_{r = 1}^{\infty}\bigg((p^4x^4)^r - (p^5x^6)^r\bigg)
\frac{\partial Y}{\partial x}= \left(A_\infty+\frac{A_0}{x}+\frac{A_1}{x-1}\right)Y ,\frac{\partial Y}{\partial t_1}=(E_3x+B_1)Y,\frac{\partial Y}{\partial t_2}=\left( \frac{1}{t_2}Nx+B_2 \right)Y.
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{d b_3}{b_3}= \frac{d x_1}{x_1}\wedge \frac{d x_2}{x_2}+ \frac{d y_1}{y_1}\wedge \frac{d y_2}{y_2}+ \frac{d z_1}{z_1}\wedge \frac{d z_2}{z_2}
(N_r V)_s = \sum_t \left[ \binom{s}{2}\binom{r-2}{r-t}+ \binom{r-s}{2}\binom{r-2}{t}\right] \left[ \binom{t}{2}- \binom{r-t}{2}\right] =\binom{s}{2}P + \binom{r-s}{2}Q,
\begin{array}{lll}[J_0,J_1]=J_2, [J_0,J_2]=-J_1, [J_1, J_2]= -J_0, begin{align*}2pt][J_0,P_0]=0 , [J_0,P_1]=P_2 , [J_0, P_2]=-P_1, begin{align*}2pt][J_1,P_0]=-P_2 , [J_1,P_1]=0 , [J_1, P_2]=- P_0, begin{align*}2pt][J_2,P_0]=P_1 , \quad[J_2,P_1]=P_0 , \quad[J_2, P_2]=0, begin{align*}2pt][P_0,P_1]= 0, \quad[P_0, P_2]=0 , [P_1,P_2]= 0 , \end{array}
S_{e f f}^{2 + 1}= - 2 \pi \int d \tau \sqrt{g_{\tau \tau}}, g_{\tau \tau}= ( \frac{d \xi^{0}}{d \tau})^{2}- h_{\alpha \beta}( \xi ) \frac{d \xi^{\alpha}}{d \tau}\frac{d \xi^{\beta}}{d \tau},
\begin{aligned}- 4 \pi \left( C_{2}\mathfrak{c}_{p}\frac{m}{a}+C_{1}\right)- 2C_{1}\int_{\Sigma}H |\nabla u|+ \left( C_{2}\mathfrak{c}_{p}\frac{m}{a}+C_{1}\right) 2^{4a}(I_{a}(1))^{2}\int_{\Sigma}|\nabla u|^2 \ge -C_{2}\mathfrak{c}_{p}8\pi(\mathfrak{m}_{ADM}-m).\end{aligned}
J_{\rho_n}(v_n)=\left(\frac{\lambda}{p}-\frac{\lambda}{r}\right)\int_\Omega |v_n|^r\, dx +\left(\frac{\mu}{p}-\frac{\mu}{q_n}\right)\int_{\Omega}\frac{|v_n|^{q_n}}{|x|^\alpha}\, dx.
\prod_{a=1}^{n}\frac{f(z-\frac{1}{2}hk-u_{a})}{f(z+\frac{1}{2}hk-u_{a})}=\frac{1+\frac{h}{z+\frac{1}{2}hk-u}}{1+\frac{h}{z-\frac{1}{2}hk-u}}.
J_{\mathrm{a s y m}}= - \int_{R_{*}^{2}}d^{2}x \epsilon_{i j}x^{i}\biggl \{\dot{f}\partial_{j}f - \kappa \bar{A}_{j}\epsilon_{k l}\partial_{k}\bar{A}_{l}\biggr \}- n \pi \rho_{e}| \vec{q}|^{2}.
\sum_{\lambda\in \Lambda_{m,\mu}}\mathrm{CT}_{s=0}\left[\int_1^\infty e^{4\pi v Q(\lambda_z)}\, \frac{dv}{v^{s+1}}\right] =\sum_{\lambda\in \Lambda_{m,\mu}}\int_1^\infty e^{4\pi v Q(\lambda_z)}\, \frac{dv}{v}& =\sum_{\lambda\in \Lambda_{m,\mu}}\Gamma(0 , -4\pi Q(\lambda_z)),
\int_{0}^{\infty}R dR \left(\frac{\cos(2Q_{1}(R))-1}{R^{2}\lambda(t)^{2}}\right) \phi_{0}(R) v_{2}(t,R\lambda(t)) =-2c_{b}\int_{0}^{\infty}\xi^{2}d\xi \sin(t \xi)\frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}K_{1}(\xi\lambda(t))
\frac{A_{bypass}}{A_{tube}}=[\frac{\gamma-1}{\gamma+1}]^{\frac{1}{2}}[\frac{2\gamma}{\gamma+1}]^{\frac{1}{\gamma-1}}[1+\frac{2}{\gamma-1}\frac{1}{M^{2}}]^{\frac{1}{2}}[1-\frac{\gamma-1}{2\gamma}\frac{1}{M^{2}}]^{\frac{1}{\gamma-1}}
\left(\frac{\pi}{\sqrt{c}}\right)^{-s}\Gamma(s)\,\tilde{\zeta}_{p,p^{\prime}}\left(s,c\right)=\left(\frac{\pi}{\sqrt{c}}\right)^{-(1-s)}\Gamma\left(1-s\right)\,\tilde{\zeta}_{p^{\prime},p}\left(1-s,c\right).
{}\sum_{v_{\mathfrak{p}}(f^n(\alpha))>0}N_{\mathfrak{p}}> d(1-\epsilon_1)(h(f^{n-1}(\alpha)-h(\gamma)-\log2)-h(f^{n-1}(\alpha))-h(\gamma)-\log2 &-(1-\epsilon_1)(h(c)-\log2)-2h(c)-h(\alpha) = (d-d\epsilon_1-1)h(f^{n-1}(\alpha))-(d-d\epsilon_1+1)(h(\gamma)+\log2) -(1-\epsilon_1)(h(c)-\log2)-2h(c)-h(\alpha)
\delta^i_n(x_1\otimes \cdots\otimes x_n) = \begin{cases}1\otimes x_1\otimes\cdots\otimes x_n, i=0 x_1\otimes\cdots\otimes x_{i-1}\otimes\Delta(x_i)\otimes x_{i+1}\otimes\cdots\otimes x_n, 1\leq i \leq n x_1\otimes \cdots\otimes x_{n}\otimes 1, i=n+1.\end{cases}
\zeta(j,s)' = \left(\prod_{m=m(j,s)+1}^{N_{k(j,s)}}b_{k(j,s),m-1}^{-1}p_{k(j,s),m}b_{k(j,s),m}\right)\left(\prod_{k=k(j,s)+1}^{j}\prod_{l=1}^{N_k}b_{k,l-1}^{-1}p_{k,l}b_{k,l}\right) &= b_{k(j,s),m(j,s)}^{-1}\zeta(j,s) b_{j, N_j}.
\bigl ( S_{\mu \nu}\bigr )^{- 1}= \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \, \Bigl ( \eta_{\mu \nu}+ ( \alpha - 1 ) \, \eta_{0 \mu}\eta_{0 \nu}\Bigr ) \, \delta ( \vec{x}- \vec{y}) \, .
{{M}_{\gamma}}\left( s \right)=\frac{m^m\mu^\mu \left( 1+K \right)^\mu}{\left( \mu \left( 1+K \right)-\overline{\gamma}s \right)^{\mu-m}}\frac{1}{\pi}\underset{I_1}{\underbrace{\int_{0}^{\pi}\frac{d\theta}{\left ( a(s)+b(s) \cos{\theta}\right )^{m}}}},
{\cal A}= N \int_{z_{-}}^{z_{1}}\! d z \,{\frac{1 - z}{z ( 1 + z / N )^{2}}}{\frac{\bigl [ 1 - 1 / N - z / N^{2}\bigr ] \bigl [ 1 + z / ( N ( 1 - z ) ) \bigr ]}{\displaystyle \sqrt{1 - N^{2}( 1 + z / N )^{2 N}/ ( z^{2}r^{2 N})}}}\, \cdotp
\begin{aligned}R_E(\theta)=-\frac{1}{\theta NT}\log_{}\mathbb{E}_{\widehat{\mathbf{H}}_{t}}\left\{e^{-\theta T\mathcal{I}(\boldsymbol{\mathrm{x}}_{t};\boldsymbol{\mathrm{y}}_{t})}\right\}.\end{aligned}
T_{\mu \nu}^{b r a n e , i}= V_{i}\, \omega ( x , y ) \, \delta ( \vec{n}_{i}\cdot \vec{x}) \left( \begin{array}{c c c c c c}{{1}}\\{{}}&{{- 1}}\\{{}}&{{}}&{{- 1}}\\{{}}&{{}}&{{}}&{{- 1}}\\{{}}&{{}}&{{}}&{{}}&{{- \cos^{2}\varphi_{i}}}&{{- \sin \varphi_{i}\cos \varphi_{i}}}\\{{}}&{{}}&{{}}&{{}}&{{- \sin \varphi_{i}\cos \varphi_{i}}}&{{- \sin^{2}\varphi_{i}}}\\ \end{array}\right) .
F_{}(A) =\sum_{\tilde{g}\subset\tilde{\mathcal{G}}:A\cap \tilde{g}\neq\emptyset}\eta_{\tilde{g}}=\sum_{\substack{\tilde{g}\subset\tilde{\mathcal{G}}_{s^{-1}(A)}: A\cap \tilde{g}\neq\emptyset}}\eta_{\tilde{g}}=\frac{K-s^{-1}(A)}{s^{-1}(A)+1}\left(1-\left(1-P(A)\right)^{s^{-1}(A)+1}\right)
P_1 \left( n \right) = \delta^{4n - 8}\left({\frac{1}{{n - 3}}\bar B\left({\frac{1}{2},n - 1,n - 2}\right) + \frac{1}{{n - 1}}\bar B\left({\frac{1}{2},n - 3,n}\right) - \frac{2}{{n - 2}}\bar B\left({\frac{1}{2},n - 1,n - 2}\right)}\right)
\begin{cases}D^{1-\epsilon}_0\tilde{y}(t) = \tilde{z}(t), \tilde{y}(0) = \frac{\gamma_1-b_1s}{a_1}, D^{\alpha_2-1}_0\tilde{z}(t) = f(t,\tilde{y}(t),\tilde{z}(t)). \tilde{z}(0) = s.\end{cases}
\begin{array}{ll}\vec{L}_{0}(x) = \vec{l}(x) \;\;\;\;\;{;}\;\;\;\;\;\;\vec{L}_{1}(x) = \gamma \partial_{x}\vec{n}\wedge \vec{n}\vec{J}_{0}(x) = \vec{l}(x) -\gamma \partial_{x}\vec{n}\;\;\;{;}\;\;\; \vec{J}_{1}(x) = \gamma \partial_{x}\vec{n}\wedge \vec{n}- i\vec{\pi}= -i ( \vec{\pi}+ i\gamma \partial_{x}\vec{n}\wedge \vec{n}) \end{array}
H ={\frac{4 \pi r}{\lambda}}\left[ - \left({\frac{2 \pi r}{\lambda}}+{\frac{\overline{{l}}}{r}}\right) \pm \sqrt{\vec{p}^{\ 2}+ \left({\frac{2 \pi r}{\lambda}}+{\frac{\overline{{l}}}{r}}\right)^{2}}\ \right] .
C_{i j k l}= \lambda \delta_{i j}\delta_{k l}+ \mu(\delta_{i k}\delta_{j l}+ \delta_{i l}\delta_{j k}) + \{\delta_{i j}D_{k l}^1 + \delta_{k l}D_{i j}^1 \}+ \{\delta_{i k}D_{j l}^2 + \delta_{i l}D_{j k}^2 + \delta_{j k}D_{i l}^2 + \delta_{j l}D_{i k}^2\}+ D_{i j k l}
\begin{cases}\Psi_{-1}(X,Y) = -1; \Psi_{0}(X,Y) = 0; \Psi_{1}(X,Y) = 1; \Psi_{2}(X,Y) = 2Y; \Psi_{3}(X,Y) = 3X^4+6 AX^2+12 B X - A^2; \Psi_{4}(X,Y) = 4Y(X^6 + 5AX^4 + 20BX^3 - 5A^2 X^2 - 4AB X - 8 B^2 -A^3).\end{cases}
\frac{\mathbb{P}\left( \frac{\log \| G_n \| - n \lambda}{\sigma \sqrt{n}}\geq y \right)}{1-\Phi(y)}= e^{\frac{y^3}{\sqrt{n}}\zeta ( \frac{y}{\sqrt{n}})}\left[ 1 + O \left( \frac{y+1}{\sqrt{n}}\right) \right],
(\bar{w}, z_\Gamma)_{H_\Gamma}= \langle \partial_{\boldsymbol{\nu}}\bar{u}+\bar{\eta}, z_\Gamma \rangle_{Z_\Gamma^*,Z_\Gamma}+ \bigl( \pi_\Gamma(v_1)-\pi(v_2)-\bar{g},z_\Gamma \bigr)_{\!H_\Gamma},
\partial_{q_{N_{r}}}\ln Z_{L}^{( n )}( \{q_{N_{s}}\}) = \sum_{p = a , b}\int_{C_{p}^{( r )}}d t W_{p , N_{r}}^{( o )}( t ) + \sum_{p = a , b}\int_{C_{p}^{( r )}}d t \tilde{W}_{p , N_{r}}( t ) + \int_{C_{r}}d t \hat{W}_{N_{r}}( t ) + I_{( a n )}
\ln J = \alpha \left( \mbox{Tr}_{V}e^{t ((D_\mu)^2\delta_{\mu\nu}-i F_{\rho\sigma}(M^{\rho\sigma})_{\mu\nu})}- \mbox{Tr}_{\lambda}e^{t{\not D}^{2}}- \mbox{Tr}_{\bar{\lambda}}e^{t{\not D}^{2}}\right) .
\frac{Q_{i1}N_{i^\prime}+Q_{i2}(N_1+N_2)}{N_{i\prime}(Q_{i2}+N_{i\prime})}<{}\frac{(N_{i^\prime}+Q_{i2})(N_1+N_2)}{N_{i\prime}(Q_{i2}+N_{i\prime})}={}\frac{N_1+N_2}{N_{i\prime}}.
(g-1) x_i =\sum_{j=1}^i (-1)^{j-1}f_{j,i-j}+\sum_{j=1}^i (-1)^{j-1}f_{j-1,i+1-j}&=\sum_{j=1}^i (-1)^{j-1}f_{j,i-j}+\sum_{j=0}^{i-1}(-1)^j f_{j,i-j}&=f_{i,0}+f_{0,i}&=0.
\left( \frac{x^{-}}{x^{+}}\right)^{2}F ( x^{-}, x^{+}) = \left. \left|{\frac{L}{\pi}}\langle 0 | T^{+ +}( k ) | n \rangle \right|^{2}{\frac{M_{n}^{4}}{8 \pi^{2}k^{3}}}K_{4}( M_{n}r ) \right|_{M_{n}= 0}= \left|{\frac{L}{\pi}}\langle 0 | T^{+ +}( k ) | n \rangle \right|_{M_{n}= 0}^{2}{\frac{6}{k^{3}\pi^{2}r^{4}}}.
T_{P}D_{0}T_{P}\, = \, \begin{pmatrix}1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 \end{pmatrix}T_{P}D_{1}T_{P}\, = \, \begin{pmatrix}0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 -1 \end{pmatrix},
{\cal Q}_{N}(\vartheta )=\mathop{\lim}_{\eta \rightarrow 0}\frac{1}{i}\log \frac{1+(-1)^{\delta}e^{iZ_{N}(\vartheta +i\eta )}}{(-1)^{\delta}+e^{-iZ_{N}(\vartheta -i\eta )}}=Z_{N}(\vartheta )\bmod 2\pi \: ,
S_{1}^{\bullet}= \beta \left.{\frac{\partial{\tilde{W}_{1}^{\bullet}( \beta , \alpha , y , \ldots )}}{\partial \beta}}\right|_{\alpha , y , \ldots}+ \alpha \left.{\frac{\partial{\tilde{W}_{1}^{\bullet}( \beta , \alpha , y , \ldots )}}{\partial \alpha}}\right|_{\beta , y , \ldots}- W_{1}^{\bullet}.
(|\eta_\delta^\perp|^2*\chi_{B_{n}})(x)=\int\limits_{B_{n}}|\eta_\delta^\perp(y-x)|^2\,dy\leq C\int\limits_{B_{n}}\|x-y\|^{-\ell}\,dy\leq C\int\limits_{B_{n}}(2n-\|y\|)^{-\ell}\,dy \leq C\int\limits_{0}^n(2n-r)^{-\ell}r^{d-k-1}\,dr\leq Cn^{-\ell+d-k}
[y_1, y_1]=\frac{\alpha_2}{\beta_5}y_4, [y_1, y_2]=y_3+\frac{\alpha_5}{\beta_5}y_4, [y_2, y_1]=\frac{\beta_1}{\alpha_4}y_3, [y_2, y_2]=\frac{\beta_4}{\alpha_4}y_3+y_4, [y_2, y_3]=y_5, [y_1, y_4]=y_5.
i\big(\gamma (g + q h)\big) = (g + q h) \otimes (g + q h) & =g \otimes g + q (g\otimes h) + q (h \otimes g) + q^2 (h \otimes h) &=g \otimes g + \{[g, h]\}+ \{[h, g]\}+ q \{[h, h]\}& =g \otimes g + \{[g, h] + [h, g]+ q[h, h]\}=g \otimes g = i\big(\gamma(g)\big) .
S_{S U S Y}\left( \left. \begin{array}{c c}{a}&{d}\\{b}&{c}\\ \end{array}\right| \theta_{1}- \theta_{2}\right) \times S_{S G}\left( \theta_{1}- \theta_{2}\right)_{\epsilon_{1}\epsilon_{2}}^{\epsilon_{1}^{\prime}\epsilon_{2}^{\prime}}
f\left(\sum_{l=1}^n \|w_l-w\|_\infty \right) \leq f\left(\sum_{l=1}^n k\|x_l-x\|_\infty+k\|y_l-y\|_\infty+k\|z_l-z\|_\infty\right) \leq kf\left(\sum_{l=1}^n \|x_l-x\|_\infty \right)+kf\left(\sum_{l=1}^n \|y_l-y\|_\infty \right) &+kf\left(\sum_{l=1}^n \|z_l-z\|_\infty \right),
\begin{array}{l}u_1 (t) = v_1 + a_{12}\cos\left(\frac{2\pi k_{12}}{\varepsilon}t \right), u_2 (t) = v_2 + a_{12}\sin\left(\frac{2\pi k_{12}}{\varepsilon}t \right),k_{12}\in\mathbb Z\setminus \{0\}.\end{array}
\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x) = \int_{S_1(a)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x) &+\sum^{n_a}_{n=2}\int_{S_n(a)\backslash S_{n-1}(a)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x) =: J_1(a)+J_2(a).
T_nE_{o_+}(\lambda n^{-1}) =E_{o_-\cdot w_{P'}}(n)E_{o_-\cdot w_{P'}n}(\lambda n^{-1}) &=E_{o_-\cdot w_{P'}}(n\lambda n^{-1}) &=j_{P'}^+(E^{P'}_{o_{-,P'}\cdot w_{P'}}(n\lambda n^{-1})) &=j_{P'}^+(E^{P'}_{o_{+,P'}}(n\lambda n^{-1})).
\tilde{\chi}_A(n_1, n_2) = \chi_A M_A(n_1, n_2), \tilde{\chi}_{B, C}(n_1, n_2) = \chi_{B, C}M_{B, C}(n_1, n_2), \tilde{\chi}_K(n_1, n_2) = \chi_K M_K(n_1, n_2), \tilde{\chi}_N(n_1, n_2) = \chi_N M_N(n_1, n_2).
\mathcal{H}_{l}(A):=\{h\in\mathcal{A}\mid h\,\mathrm{k}\langle A\rangle\subseteq\mathrm{k}\langle A\rangle\}\quad\quad\mathcal{H}_{r}(A):=\{h\in\mathcal{A}\mid\mathrm{k}\langle A\rangle\,h\subseteq\mathrm{k}\langle A\rangle\}
\gamma^{\textrm{I}}_{app}= \frac{(1-\lambda) \beta_{1,k}^2}{\frac{(1-\lambda)\alpha}{\lambda}\frac{1}{\tau}b_1 + \frac{1}{M}\Big(b_2 + \sigma_n^2\beta_{1,k}\Big)}, \gamma^{\textrm{II}}_{app}= \frac{(1-\lambda)\beta_{1,k}^2}{\frac{(1-\lambda)^2\alpha}{\lambda}\frac{1}{\tau}b_1 + \frac{1}{M}\Big((1-\lambda)b_2 + \sigma_n^2\beta_{1,k}\Big)},
|E(it)| \leq\frac{C_3}{\sqrt{|t|}}\prod_{\kappa_{1,n}\ge\alpha_n}\frac{\kappa_{1,n}}{\alpha_n}\prod_{\kappa_{2,n}\ge\beta_n}\frac{\kappa_{2,n}}{\beta_n}&\le\frac{C_3}{\sqrt{|t|}}\prod_{n=0}^\infty\left[1+\frac{(\kappa_{1,n}-\alpha_n)_+}{\alpha_n}\right]\prod_{n=0}^\infty\left[1+\frac{(\kappa_{2,n}-\beta_n)_+}{\beta_n}\right].
F(x,y) = a_1 + a_2 y + a_3 x^2 y x+ a_4 x y x^2 + a_5 x+ a_6 yx +a_7 x^2 y x^2 + a_8 xy + a_9 x^2 +a_{10}y x^2 + a_{11}x^2 y + a_{12}x y x,
s_\lambda(\varphi)_i(x,y) = \varphi_i(s^\lambda(x),y),1\leq i\leq \lambda, s_\lambda(\varphi)_{\lambda+1}(x,y) = \sigma_\lambda(x)\varphi_\lambda(s^\lambda(x),y) + (1 - \sigma_\lambda(x))\varphi_{\lambda+1}(s^\lambda x,y), s_\lambda(\varphi)_i(x,y) = \varphi_{i-1}(s^\lambda(x)), \lambda +1 < i \leq k+1.
(m \cdot \Psi_n(\xi))(\eta) = m \cdot U_{n,m}(\xi \otimes \eta) = U_{n,m}(m \cdot \xi \otimes \eta) = \Psi_n(m \cdot \xi) (\Psi_n(\xi) \cdot m)(\eta) = \Psi_n(\xi)(m \cdot \eta) = U_{n,m}(\xi \otimes m\cdot \eta) = U_{n,m}(\xi \cdot m \otimes \eta) = (\Psi_n(\xi \cdot m))(\eta)
B_{f}= \lim_{\alpha \rightarrow \infty}q^{-\alpha(m+2)}\sum_{\beta = 0}^{\alpha-1}(q-1)q^{\alpha-\beta-1}(\beta+1)^{m+1}(q-1)^{m+1}q^{(\alpha-1)(m+1)}&= (1-q^{-1})^{m+2}\sum_{\beta=0}^{\infty}(\beta+1)^{m+1}q^{-\beta}= q^{-m}E_{m+1}(q),
\lVert \mathtt{R}_k \rVert_s^{Lip(\gamma)}\le_s\varepsilon^3+ \varepsilon \lVert \mathfrak{I}_{\delta}\rVert_{s+\sigma}^{Lip(\gamma)}, \lVert \partial_i \mathtt{R}_k [\hat{\imath}] \rVert_s\le_s \varepsilon (\lVert \hat{\imath}\rVert_{s+\sigma}+\lVert \mathfrak{I}_{\delta}\rVert_{s+\sigma}\lVert \hat{\imath}\rVert_{s_0+\sigma}), k=0,1.
{\cal L}={\frac{m}{2}}\epsilon^{\mu \nu \lambda}A_{\mu}^{a}( \partial_{\nu}A_{\lambda}^{a}+{\frac{g}{3}}f^{a b c}A_{\nu}^{b}A_{\lambda}^{c}) - F^{a}( \partial_{\mu}A^{\mu a}) + \partial^{\mu}\overline{{c}}^{a}( D_{\mu}c^{a})
\mathcal T_k(\Pi_y) - \mathcal T_k(\Pi) =\ \sqrt{2k}\cdot l(\partial \Pi_y) + \int_{\Pi_y}\sigma -\sqrt{2k}\cdot l(\partial \Pi) - \int_\Pi \sigma &=\ \sqrt{2k}\cdot \Big [l(y)-l(\gamma)\Big ] + \int_{\Pi_y\setminus \Pi}\sigma\,.