harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
nR_3=H(M_3) &=H(M_3 \mid Y_3^{(n)},M_2)+I(M_3;Y_3^{(n)}\mid M_2) &\leq n\epsilon_n+h(Y_3^{(n)}\mid M_2)-h(Y_3^{(n)}\mid M_2,M_3) &\leq n\epsilon_n+\frac{n}{2}\log2\pi e(P+N_3)-h(Y_3^{(n)}\mid M_2,M_3) &\overset{(f)}{\leq}n\epsilon_n+\frac{n}{2}\log2\pi e(P+N_3)-\frac{n}{2}\log2\pi e(\alpha P+N_3),
( \bar \nabla_\mu +2 \partial_\mu \bar \phi ) ({\cal G}^{\mu \nu}- \bar G_\alpha^{~~\nu}h^{\alpha \mu}- \bar G^{\mu}_{~~\beta}h^{\beta \nu}) + \bar G^{\mu \nu}(\delta \Gamma^\sigma_{\sigma \mu}(h) +2 (\partial_\mu \varphi)) = 0,
\int_0^T \sum_{i=1}^{q}\theta_i (t) f_i (\xi(t)) dt \geq \int_0^T \sum_{i=1}^{q}\theta_i (t) f_i (u(t)) dt + \int_0^T \left\langle -\dot{u}(t) - \eta (t), \xi(t) - u(t) \right\rangle dt.
\chi_{n}(x) =0x\in(-\infty,n)\cup(2n,\infty), \chi_{n}(x) =1x\in(n+1,2n-1), \widetilde{\chi}_{n}(y) =0y\in(-\infty,-an)\cup(an,\infty), \widetilde{\chi}_{n}(y) =1y\in(-an+1,an-1).
\left[ - \frac{1}{2 L}\left( \frac{\partial^{2}}{\partial ( a^{3})^{2}}+ \frac{\partial^{2}}{\partial ( a^{8})^{2}}\right) + U_{e f f}( a^{3}, a^{8}) \right] \tilde{\Phi}( a^{3}, a^{8}) = E \tilde{\Phi}( a^{3}, a^{8})
\mathcal{L}\phi(s,x)=\Gamma x\frac{\partial \phi}{\partial x}(s,x) +\frac{1}{2}\sigma^2x^2\frac{\partial^2 \phi}{\partial x^2}(s,x) &\begin{aligned}+ \int_\mathbb{R}\Big\{\phi(s,x(1+\gamma(z))-\phi(s,x)-x\gamma(z)\frac{\partial\phi}{\partial x}\Big\}\nu(dz), \forall (s,x)\in\mathbb{R}_{>0}\times\mathbb{R}. \end{aligned}
\begin{aligned}f(Y_t)=f(y_0) +\sum_{k=1}^d \int_0^t \frac{\partial f}{\partial y_k}(Y_{s-}) d(Y^c_s)^k + \frac{1}{2}\sum_{k,j=1}^d \int_0^t \frac{\partial^2 f}{\partial y_k\partial y_j}(Y_{s-}) d[(Y^c_s)^k,(Y^c_s)^j] &+ \sum_{0<s\le t}\big( f(Y_s)-f(Y_{s-}) \big). \end{aligned}
e_1=\left(\begin{array}{cccccc}1 0 0 -r_3 -r_2 -r_1\end{array}\right) &e_2=\left(\begin{array}{cccccc}0 1 0 -p_3 -p_2 -p_1\end{array}\right) &e_3=\left(\begin{array}{cccccc}0 0 1 -q_3 -q_2 -q_1\end{array}\right).
\mathcal{S}_{1,1}= -(N-k)s(z_{2})+O_{L^1}((d_{N}N^{-1})^{1/2}) \mathcal{S}_{1,3}= -s(z_{1})\mathcal{S}_{1}+O_{L^1}(d_{N}/N) \mathbb{E}|\mathcal{S}_{1,4}| \leq c\sqrt{d_{N}^{3}N^{-1}}\mathbb{E}|\mathcal{S}_{1,6}| \leq c\sqrt{d_{N}^{3}N^{-1}}
\sum_{\substack{\textrm{$\O$ cut connected}\textrm{fourientation of $G$}}}k^{|\O^o|}l^{|\O^u|}m^{|\O^b|}= (k+m)^{n-1}(k+l)^g\cdot T_G\left(\frac{k + m}{k+m},\frac{2 k + l + m}{k+l}\right) &= (k+m)^{n-1}(k+l)^g\cdot T_G\left(1,\frac{2k + l + m}{k+l}\right) &= (1-p)^{n-1}p^g\cdot T_G\left(1,\frac{1}{p}\right) &= R_G(p).
(n+1) \Delta_{R, n}^{(s; \, \ell+1)}(x) + \ell \Delta_{R, n+1}^{(s; \, \ell)}(x) + \ell (x-1) \Delta_{R, n + 1}^{(s+1; \, \ell-1)}(x) + ( n + \frac{s}{2}+ 1) (x-1)\Delta_{R, n}^{(s +1; \, \ell)}(x) = s \Delta_{R, n}^{(s; \, \ell)}(x) .
\tau\cdot \alpha\|x^\natural\|_2 >\tau\cdot \|\Phi x^\natural\|_2 = \sup_{\|x\|_\sharp\leq 1}\|\Phi x\|_2 \cdot \|\Phi x^\natural\|_2 \geq \sup_{\|x\|_\sharp\leq 1}\langle \Phi x, \Phi x^\natural\rangle = \sup_{\|x\|_\sharp\leq 1}\langle x, \Phi^T \Phi x^\natural\rangle = \|\Phi^T \Phi x^\natural\|_\diamond.
\bigl[P^*(x^*\chi_{B_r})\bigr](f)= \int_{\Omega}x^*\bigl(P(f)(t)\bigr) \chi_{B_r}(t)\,d\mu(t) &=\int_{\Omega}\sum_{i=1}^{\infty}\frac{1}{\mu(B_i)}x^*\left(\int_{B_i}f d\mu\right)\, \chi_{B_i}(t)\chi_{B_r}(t)\,d\mu(t) = \frac{1}{\mu(B_r)}x^*\left(\int_{B_r}f d\mu\right)\,\int_{\Omega}\chi_{B_r}(t)\,d\mu(t)=\bigl[x^*\chi_{B_r}\bigr](f)
\mathbf{P}^{n}\equiv \int d^{3}\mathbf{x}\left( \nabla \varphi_{n}^{\alpha}\right) \left\{\frac{\partial \mathcal{F}^{n}}{\partial \varphi_{n}^{\alpha}}- \nabla \mathbf{\cdot}\left( \frac{\partial \mathcal{F}^{n}}{\partial \nabla \varphi_{n}^{\alpha}}\right) \right\}.
V(R)=\lim_{T\rightarrow\infty}\,\frac{1}{T}\,\left[\int\limits_{0}^{T}\!\!dt\int\limits_{0}^{R}\!\!dr\,M_{0}^{2}\,+\,\frac{D-2}{2}\,\mbox{Tr}\ln G^{-1}\right]\,{.}
\mathcal{A}^{\mu}[\beta b]\bullet = \frac{\beta}{2}\big(\mathrm{F}_3(b\bullet) + b\mathrm{F}_3\bullet\big) + \frac{\mu\beta}{2}\big( \mathrm{F}_4 \Delta_X(b\bullet) + b\mathrm{F}_4\Delta_X\bullet\big)- \frac{\mu\beta^3}{12}\big( b^3\Delta_X\mathrm{F}_3\bullet + \Delta_X\mathrm{F}_3(b^3\bullet)\Big) ,
\zeta_{\bar{\cal V}}(s)=\frac{1}{\Gamma(s)}\frac{m_d L}{\sqrt{4\pi}}\sum_{a=1}^N\int_0^1 d\beta \beta^{s-\frac{3}{2}}e^{-\beta\bar{v}_a^2}+B_{\bar{\cal V}}(s)\quad ,
\Omega_i = \frac{l_i}{(l_i^2 + r_H^2)}\frac{r_0^{3/2}}{(\pi N)^{1/2}}\sp J_i = \frac{4}{3}\frac{V_5 (\pi N)^{1/2}}{(2 \pi)^6}r_0^{3/2}l_i
\zeta_{2}^{\prime}=\frac{3}{8}-\frac{1}{8}e^{-2y/\delta}-e^{-y/\delta}[sin\frac{y}{\delta}+\frac{1}{4}cos\frac{y}{\delta}+\frac{y}{4\delta}(sin\frac{y}{\delta}-cos\frac{y}{\delta})].
\mathbf{S}_{a}^{-1}=\left(\begin{array}{cccccc}1/a_{1}0 0 \cdots 0 0 -1/a_{1}1/a_{2}0 \cdots 0 0 0 -1/a_{2}1/a_{3}\cdots 0 0 \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots -1/a_{N-1}1/a_{N}\end{array}\right).
\tilde{B}^{( 0 , 0 )}( R ) = \frac{1}{2 ( d - 3 ) y_{0}}[ 2 ( d - 3 ) y_{0}^{2}+ ( d - 2 - x ) y_{0}y_{1}+ ( 1 - x ) y_{0}y_{2}- ( 1 - x ) y_{1}^{2}]
D_{\vec{x}}\vec{f}\left({\vec{F}_0,\vec{z},0}\right)\vec{F}_1 = D_{\vec{z}}\vec{F}_0 \vec{g}\left({\vec{F}_0, \vec{z},0}\right)-\dfrac{\partial \vec{f}}{\partial\varepsilon}\left({\vec{F}_0, \vec{z},0}\right).
\mathrm{C}_{M}(\cdot\mid \alpha):=\sum_{d=1}^{\infty}M(d)\bigg{[}\mathop{\otimes}_{i=1}^{d}\mathcal{N}(0,i^{-2\alpha-1})\bigg{]}\otimes\bigg{[}\mathop{\otimes}_{i=d+1}^{\infty}\mathcal{N}(0,0)\bigg{]},
\left ( M_r \Phi (V) \right )_S = \sum_{T \subseteq [r]}M_r(S,T) (\Phi (V))_T =\sum_{|T|=0}^r \sum_{T}M_r(S,T) (\Phi (V))_T =\sum_{|T|=0}^r N_r(|S|,|T|)V_T = ( N_r V)(|S|) = \lambda V(|S|) = \lambda (\Phi (V))_S
u(x,t)= \frac{1}{2 \pi}\int_{-\infty}^{+\infty}e^{-i \beta x}E_{\nu, 1+\frac{2H-1}{\nu}, \frac{2H-1}{\nu}}\left(- H \beta^2 t^{\nu+2H-1}\right) \mathrm{d}\beta.
\begin{aligned}\min_{y_1,y_2}y_1 \;\; \eta_1y_2 - L_1 y_1 \le (2\epsilon_f + r)/\delta_k - L_1\delta_k/2 \;\; ~ |y_1| \le \|x_k\| y_2 \ge \min\{10^{-6}, 10^{-2}\|L_1x_k\|\}, \end{aligned}
Z_{Henn}(S\#\overline{S},H)=\sum\limits_{(\Lambda)}\lambda\left(S(\Lambda'_{(3)})\Lambda_{(1)}\Lambda'_{(5)}S^{-1}(\Lambda_{(3)})\Lambda'_{(1)}S^{-1}(\Lambda_{(5)})\right) &\cdot\lambda\left(\Lambda_{(4)}S(\Lambda'_{(4)})S(\Lambda'_{(2)})\Lambda_{(2)}\Lambda'_{(6)}\Lambda'_{(7)}\Lambda'_{(8)}\right)
\sigma\left((S,\underline{S})^*\right):=\begin{cases}(\sigma(S),\sigma(\underline{S}))^{\mathsf{sgn}(\sigma(\hat{S}),\sigma(\underline{\hat{S}}))}\mbox{if}* = +, (\sigma(S),\sigma(\underline{S}))^{-\mathsf{sgn}(\sigma(\hat{S}),\sigma(\underline{\hat{S}}))}\mbox{if}* = -.\end{cases}
Z^{(1)}= \binom{a_1,a_2,\ldots,a_k,d_{k'+1},\ldots,d_{m'}}{b_1,b_2,\ldots,b_l,c_{l'+1},\ldots,c_{m'}}, Z'^{(1)}=\binom{c_1-1,c_2-1,\ldots,c_{l'-1}-1,b_{l+1},\ldots,b_m}{d_1-1,d_2-1,\ldots,d_{k'-1}-1,a_{k+1},\ldots,a_{m+1}}.
\begin{array}{rcl}\sigma_{t_i}^{t_i,t_j}(x) = k(x) c_{t_i}^{t_i,t_j}- d_{t_i}(t_i) (m(x)-m(t_i)), \sigma_{t_j}^{t_i,t_j}(x) = k(x)c_{t_j}^{t_i,t_j}+ d_{t_j}(t_j) (m(x)- m(t_j)), \sigma_s^{t_i,t_j}(x) = k(x) c_s^{t_i,t_j}, s \in T\setminus \{t_i, t_j\}, \end{array}
\sum_{i=1}^l\lambda_i\nabla f_i(x^0)+\sum_{j\in J(x^0; u)}\mu_j\nabla g_j(x^0)=0, &\sum_{i=1}^l\lambda_i\xi^{*i}+\sum_{j\in J}\mu_j\zeta^{*j}-\nu= 0, &\lambda_i> 0, i=1, 2, ..., l, \mu_j\geq 0, j\in J(x^0; u), \nu\geq 0,
\mathfrak{Q}_{1}= \mathfrak{B}_{1}; \mathfrak{Q}_{2}= \mathfrak{B}_{2}; \mathfrak{Q}_{3}= \mathfrak{B}_{3}; \mathfrak{Q}_{4}= [\mathfrak{B}_{4}, \mathfrak{B}_{5}] = (x^{3}, y^{2}, x^{2}y, z).
D_2(\iota_{\bar{y}*}1) =\sum\limits_{w}\frac{(D_2(\iota_{\bar{y}*}1), \iota_{w*}1)}{e(T_wT^*\mathcal{B})}\iota_{w*}1 &=\sum\limits_{\bar{w}=\bar{y}}\frac{1}{\prod\limits_{\alpha\in R^+_P}(-w\alpha)}\iota_{w*}1.
g^k = (g_m\cdots g_0) (g_m\cdots g_0)\cdots (g_m\cdots g_0) &= ((g_m\cdots g_1)\cdot (g_0)) ((g_m\cdots g_2)\cdot (g_1 g_0))\cdots ((g_m\cdots g_k)\cdot (g_{k-1}\cdots g_0)) &=(g_m\cdots g_1)(g_m\cdots g_2)\cdots (g_m\cdots g_k)\cdot (g_0) (g_1g_0)\cdots (g_{k-1}\cdots g_0) &= (u\wedge_L g^k) (g_0)(g_1g_0)\cdots (g_{k-1}\cdots g_0)
V^{\ast}\Psi A^{\dagger}=V^{\ast}A^{\dagger}-(S_{A}-I)(I+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}K^{\ast}A^{\dagger}, V^{\ast}\Psi KS_{A}^{\dagger}H =(S_{A}-I)S_{A}^{\dagger}H-(S_{A}-I)(I+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}K^{\ast}KS_{A}^{\dagger}H.
\mathrm{P}({E_2},\bar{E}_1,|\mathcal{S}_r|>0) = \mathrm{P}\left(\log(1+\rho|g_{n^*,2}|^2\alpha_2^2)<2R_2, \right. \left. \log(1+\rho|h_{n^*}|^2\alpha_2^2)>2R_2,|\mathcal{S}_r|>0\right) .
\| \sum_{j=1}^2 ( A_j (u_j \theta) - u_j A_j \theta) \|_p \lesssim_{p,p_1,p_2}\|D u\|_{p_1}\|\theta \|_{p_2}& \lesssim_{p,p_1,p_2}\| \nabla u \|_{p_1}\| \theta \|_{p_2}.
\sum_{k=1}^{n-1}\frac{[3k]}{[2k]^2}{2k\brack k}q^{-{k\choose 2}}=[n]{2n-1\brack n-1}\sum_{k=1}^{n-1}\frac{q^{-{n-2k\choose 2}}}{[2k]^2{n-1\brack k}_{q^2}^2}.
\varrho_{*}\Big ( \sum_{k = 1}^{n}\, f_{k}\, d^{\prime}f_{k}^{\prime}\Big ) \, = \, \sum_{k = 1}^{n}\, \big ( r_{U_{\iota}\cap U}^{U}( f_{k}) \, \# \, 1 \big ) \, d \big ( r_{U_{\iota}\cap U}^{U}( f_{k}^{\prime}) \, \# \, 1 \big ) ,
\sum_{n=0}^{\infty}\frac{q^{2n^{2}}\left(q;q^{2}\right)_{n}S_{2n}\left(xq^{-2n-1/2};q\right)}{\left(c;q^{2}\right)_{n}}\left(\frac{c}{x^{2}q}\right)^{n}=\int_{-\infty}^{\infty}\frac{\left(ce^{iy}/\left(qx\right);q\right)_{\infty}}{\left(c,ce^{2iy}/\left(x^{2}q\right);q^{2}\right)_{\infty}}\frac{\exp\left(\frac{y^{2}}{\log q^{2}}\right)dy}{\sqrt{\pi\log q^{-2}}}.
\chi_{0, 0, 2}(q, \lambda, \omega) = f_{2, 2}(q, q^{-2\omega}) q^\lambda (q^{-2\lambda + 2}; q^{-2\omega})(q^{2\lambda + 2}q^{-2\omega}; q^{-2\omega}) (q^{-2\omega + 2}; q^{-2\omega}).
\frac{d}{d\lambda}S_{\omega,\mathbf{c}}(\Phi_0^{\lambda})=\frac{5}{2}N(\Phi_0)\lambda^{\frac{3}{2}}+2\lambda L(\Phi_0)+\mathbf{c}\cdot \mathbf{P}(\Phi_0)=\frac{I_{\omega,\mathbf{c}}(\Phi_0^{\lambda})}{\lambda}.
\Big(x\frac{d}{dx}\Big)_{p,\lambda}\frac{1}{1-x}=\sum_{n=0}^{\infty}\Big(x\frac{d}{dx}\Big)_{p,\lambda}x^{n}=\sum_{n=0}^{\infty}(n)_{p,\lambda}x^{n}=\frac{1}{1-x}F_{p,\lambda}\Big(\frac{x}{1-x}\Big).
= \Omega_{d}\left({\frac{m_{B}^{2}}{2}}\varphi_{B}^{2}+{\frac{\lambda_{B}}{2 4}}\varphi_{B}^{4}\right) +{\frac{\hbar}{2}}\log \det ( \triangle + M^{2}) + O ( \hbar^{2}) + \Sigma_{d - 2}\left( \mu_{B}+ \sigma_{B}\varphi_{B}^{2}\right)
\lim_{\lambda\to 0}\left\|\frac{\displaystyle \int_0^1K_1(x,\tau,y_\lambda)d\tau}{\sqrt{x^2+\lambda^2}\log^+(x^2+\lambda^2)}-\frac{\displaystyle \int_0^1 K_1(x,\tau,y_0)d\tau}{x\log^+(x^2)}\right\|_{L^\infty[0,1]}=0
|b_{n,r,s}| \le C\sum_{j, k=1}^n(-j-r+|k+s|)^{-\beta}L(-j-r+|k+s|) &=C\sum_{j, k=1}^n(-j+n+R+|k+s|)^{-\beta}L(-j+n+R+|k+s|) & \propto n^2(|r|+|s|)^{-\beta}L(|r|+|s|).
\sum_{i=i^\ast+1}^j\ln \left( 1-\frac{\theta}{32}[\omega (r_i)]^{1+\frac{\kappa}{\kappa-1}}\right) \leq -\frac{\theta}{32}\sum_{i=i^\ast+1}^j [\omega (r_i)]^{1+\frac{\kappa}{\kappa-1}}& \leq -\frac{\theta}{32}\frac1{\ln 32}\int_{r_j}^{r_{i^\ast}}[\omega (\rho)]^{\frac1\alpha}\frac{d \rho}{\rho}\,,
\begin{aligned}q_k = -2\nu_0\xi_k + k \ln 4\nu_0^2 - i (\delta_k+\delta_0 + k\pi-2\mu_0 \xi_k), \nu_0 ={1 \over N}\sum_{s=1}^{N}\nu_s, \mu_0 ={1 \over N}\sum_{s=1}^{N}\mu_s, \delta_0 ={1 \over N}\sum_{s=1}^{N}\delta_s.\end{aligned}
\lim_{s\to\infty}Q({\tilde{\Sigma}}_s)=(\int_{S^{n-1}}e^{(n-1)f}\mu_{\sigma})^{-\frac{n-5}{n-1}}\int_{S^{n-1}}e^{(n-3)f}|\mathring{D^2}e^{-f}|^2_{\sigma}\mu_{\sigma}
\beta_1(e^{\lambda_1}\cdots e^{\lambda_k})=\beta_1(e^{\lambda_1}) \cdots \beta_1(e^{\lambda_k}) = [L_{\lambda_1}]\cdots [L_{\lambda_k}]= [L_{\lambda_1}\otimes \cdots \otimes L_{\lambda_k}]=[L_{{\lambda_1}+\cdots +{\lambda_k}}]\,,
\sigma^{l}= \pi \Omega_{n + 1}\left[{\frac{\Gamma ({\frac{n}{2}}+ 1 )}{\Gamma ({\frac{n}{2}}+ l + 1 ) \Gamma ({\frac{n + 2 l}{2 b}})}}\right]^{2}{\frac{2 l + n}{n}}{\binom{l + n - 1}{l}}S \left({\frac{T}{b}}\right)^{{\frac{n + 2 l}{b}}- 1}\left({\frac{\omega}{2}}\right)^{{\frac{n + 2 l}{b}}+ 2 l - 1}.
D^{(BC_n,C_n)}_x=\sum_{\sigma_1,\cdots,\sigma_n=\pm 1}\prod_{i=1}^n{(1-a b^{1/2}x_i^{\sigma_i})(1+b^{1/2}x_i^{\sigma_i})\over 1-x_i^{2\sigma_i}}\prod_{1\leq i<j\leq n}{1-t x_i^{\sigma_i}x_i^{\sigma_j}\over 1-x_i^{\sigma_i}x_i^{\sigma_j}}T_{q^{\sigma_1/2},x_1}\cdots T_{q^{\sigma_n/2},x_n},
\operatorname{Lin}:\Omega^1(M)\oplus\Omega^0(M)\to\Omega^1(M)\oplus\Omega^0(M),\begin{pmatrix}v f\end{pmatrix}\mapsto\begin{pmatrix}\delta\mathrm{d}-2\,\mathrm{Ric}\mathrm{d}\delta 0\end{pmatrix}\begin{pmatrix}v f\end{pmatrix}.
h(x)=q^{x}\bigg(\frac{x}{r}-\frac{\beta+n}{q-1}\bigg) + \frac{n}{q-1}\geq q\bigg(\frac{1}{r}-\frac{\beta+n}{q-1}\bigg)+ \frac{n}{q-1}&\geq q\bigg(\frac{1}{r}+ \frac{1}{q-1}-\frac{n+1}{q(q-1)}-\frac{1}{r}\bigg)+ \frac{n}{q-1}&=1.
{\rho}_{(1), \infty}^{(\rm cJ)}(x;\beta,p,q) |_{\beta = 2 \atop p=2, q=0}= 1 -{2 \over \pi^2 x^2}-{3 \over 2 \pi^4 x^4}+{3 - 2 \pi^2 x^2 \over 2 \pi^4 x^4}\cos 2 \pi x +{3 \over \pi^3 x^3}\sin 2 \pi x.
\|\boldsymbol{\lambda}+ \boldsymbol{\rho}_{G}\|^{2}- \|\boldsymbol{\nu}+ \boldsymbol{\rho}_{G}\|^{2}= \|\boldsymbol{\lambda}+ \boldsymbol{\rho}_{G}\|^{2}+ 2 \cdot n \cdot (k+ h^\vee) - \|\nu + \rho_G\|^2.
[\begin{matrix}Y^{\prime}\\P_{B}\\P_{R}\end{matrix}]=[\begin{matrix}K_{R}&K_{G}&K_{B}\\ -\frac{1}{2}\cdot\frac{K_{R}}{1-K_{B}}&-\frac{1}{2}\cdot\frac{K_{G}}{1-K_{B}}&\frac{1}{2}\\ \frac{1}{2}&-\frac{1}{2}\cdot\frac{K_{G}}{1-K_{R}}&-\frac{1}{2}\cdot\frac{K_{B}}{1-K_{R}}\end{matrix}][\begin{matrix}R^{\prime}\\G^{\prime}\\B^{\prime}\end{matrix}]
0 \longleftarrow A^0 \buildrel d_\omega^* \over\longleftarrow A^1 \buildrel{\cal D}^{1*}\over \longleftarrow A^2\buildrel{\cal D}^{2*}\over \longleftarrow \cdots \buildrel{\cal D}^{s-2*}\over \longleftarrow A^{s-1}\buildrel{\cal D}^{s-1*}\over \longleftarrow A^s \longleftarrow 0
\sigma^{n}( \lambda ) = \sum_{j = q_{+}, q_{-}}Z_{n j}( \lambda -{\frac{( - )^{j}}{\alpha_{0}}}) - \sum_{j = - , +}( A_{n j}* \sigma^{j}) ( \lambda ) -{\frac{1}{N}}A_{n l}( \lambda - \lambda_{0}) \, ,
q_{-1,-2,3,-1}=\frac{\zeta(\zeta+2)(\zeta^3+3\zeta^2+3\zeta+5)(5\zeta^3+15\zeta^2+7\zeta+1)}{(2\zeta+1)(5\zeta^3+3\zeta^2+3\zeta+1)(\zeta^3+7\zeta^2+15\zeta+5)},
\Phi(r)=\int\frac{\rho(r^{\prime})}{|r-r^{\prime}|}d^{3}r^{\prime}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}(\frac{4\pi}{2l+1})q_{lm}\frac{Y_{lm}(\theta,\varphi)}{r^{l+1}}
\gamma_{0}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \gamma_{1}= \left( \begin{array}{c c}{{0}}&{{i}}\\{{- i}}&{{0}}\\ \end{array}\right) , \gamma_{5}= i \gamma_{0}\gamma_{1}.
\frac{\Pr^y\{T_z < \infty\}}{\Pr^w\{T_z < \infty\}}\bigg /\frac{\Pr^x\{T_z < \infty\}}{\Pr^w\{T_z < \infty\}}=\lim_{u \downarrow \ell}\frac{s(y) - s(u)}{s(x) - s(u)}.
\nu*\nu(H^{(\rho)})=\mu^{*2n}(H^{(\rho)})\leqslant\sup_{x\in (\mu^{*(2n-m)})}\mu^{*m}(xH^{(\rho)})=\sup_{x\in (\mu^{*(2n-m)})}\mu^{*m}(A_x).
\begin{aligned}\mu_c^{-}(Y_1,Y_2,\dots,Y_m) =\sum\limits_{k=1}^{m-1}\mu(Y_k,Y_{k+1}) + \mu(Y_m,Y_1)=\sum\limits_{k=1}^{m-1}\mu^*(Y_{k+1},Y_{k}) + \mu^*(Y_1,Y_m)+s_m=\mu^{+}(Y_m,Y_{m-1},\dots,Y_1)+s_m,\end{aligned}
2(\#S_1) + \#S_2-4 + (\#S_1+\#S_2-1) s_1^{-1}s_2 + (\#S_2-1) s_1 s_2^{-1}+ \sum_{i \not\in S_1 \cup S_2}s_1 t_i^{-1}+ s_2 t_i^{-1}+ s_1^{-1}t_i.
A_{\theta}\left(h+c\left(e^{-x}+e^{i\theta}e^{x-1}\right)\right)=-i\, h'+i\, c\left(e^{-x}-e^{i\theta}e^{x-1}\right),\;\mbox{where}dom\left(A_{\theta}\right):=\left\{h+c\left(e^{-x}+e^{i\theta}e^{x-1}\right)\:\big|\: h\in dom\left(D_{F}\right),c\in\mathbb{C}\right\}.
h^1(N_{C/S}(-p_1-\cdots-p_8 - q_1 - \cdots - q_4)) = h^1(\O_C(2H+L)(-p_1-\cdots-p_8 - q_1 - \cdots - q_4)) &=h^1(\O_C(H+L)(p_9- q_1 - \cdots - q_4)) &= h^1(K_C(p_9)(-q_1 - \cdots - q_4)).
\left( 1- q^n \right) \gamma_n =\sum_{kj+r=n}\frac{(-1)^j q^{\frac{r(r+1)}2 + krj+ \frac{k(k+1)}2 j^2}}{\left( q^{k}; q^{k}\right)_j (q;q)_{r}}\Big( \left(1-q^r\right) + q^r \left( 1-q^{kj}\right)\Big) & = q^n \gamma_{n-1}- q^{(k+1)(n-k) + \frac{k(k+1)}2}\gamma_{n-k},
I_1=\frac{i{e}^{\pi i\Theta}-\frac{2\sqrt{\pi}a}{\Gamma(1+2iab)}{e}^{\frac{i}{4}\pi}{e}^{-\pi ab}2^{6iab}}{i{e}^{-\pi i\Theta}-\frac{2\sqrt{\pi}a}{\Gamma(1+2iab)}{e}^{\frac{i}{4}\pi}{e}^{-\pi ab}2^{6iab}}.
\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-1}(q^{1-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}&\equiv \frac{(1-q)(1-q^{d-1})(q^d;q^d)_{n-1-(n+1)/d}}{-(q^d;q^d)_{(n+1)/d}^{d-1}}q^{(d(d+n)(n+1)-(n+1)^2)/(2d)-1}\pmod{\Phi_n(q)^2}.
G_{m,n}= \langle\rho,\sigma,\tau\mid\rho^4=\sigma^{2^{n+1}}=\tau^{2^{m+1}}=1,\rho^2=\sigma^{2^n},\lbrack\rho,\sigma\rbrack=\sigma^2,\lbrack\rho,\tau\rbrack=\tau^2,\lbrack\sigma,\tau\rbrack=1\rangle,
l_R(xy)l_R(pqa) =l_R(xy)l_R(p)+l_R(xy)l_R(q)+l_R(xy)l_R(a) =l_R(y)l_R(p)+l_R(x)l_R(q)+l_R(x)l_R(a)+l_R(y)l_R(a) =l_R(y)l_R(pa)+l_R(x)l_R(qa) =l_R(y)l_R(x)+l_R(x)l_R(y)=0.
\tilde{p}_{k,\varphi}(\omega,(-1)^k \xi_n) = p_k (x,Y+(-1)^k \xi_n \nu_k+i\tau d\varphi_k(x)) = p_k (x, \xi +i\tau d\varphi_k(x)) = p_{k,\varphi}(x,\xi,\tau),
\rho \left( \sum_{i, j=1, i < j}^{k}\langle x_i, x_j \rangle \right) \geq \prod_{i=1}^k \rho^{\frac{1}{k}}\Bigl( \frac{k(k-1)}{2}\, \left| e^{y_i - y}\, x_i \right|^2 \Bigr)
\hat{\phi}_X(u) = e^{\frac{1}{K}\hat{\psi}(u)}= e^{\frac{1}{K}\Re \hat{\psi}(u)}e^{\frac{1}{K}i \Im \hat{\psi}(u)}=|\hat{\phi}(u)|^{\frac{1}{K}}e^{\frac{1}{K}i \Im \hat{\psi}(u)},
\gamma_2 = \frac{(1 - \rho)P_S|\mathbf{h}_{D,R}^T\mathbf{F}\mathbf{h}_{R,S}|^2}{\mathbf{h}_{D,R}^T\mathbf{F}\mathbf{F}^H\mathbf{h}_{D,R}^\ast\sigma_n^2 + \sigma_n^2}\,.
\bar{\theta}^k =\Pi_{\mathcal{K},G}\big\{\bar{\theta}^k-G^{-1}[T(\tilde{\theta}^k)+G(\bar{\theta}^k-\tilde{\theta}^k)]\big\}&=\Pi_{\mathcal{K}}\big\{\bar{\theta}^k-[T(\tilde{\theta}^k)+G(\bar{\theta}^k-\tilde{\theta}^k)]\big\}.
(\begin{matrix}(mc^{2}-E+e\phi)&c\sigma\cdot(p-\frac{e}{c}A)\\ -c\sigma\cdot(p-\frac{e}{c}A)&(mc^{2}+E-e\phi)\end{matrix})(\begin{matrix}\psi_{+}\\ \psi_{-}\end{matrix})=(\begin{matrix}0\\ 0\end{matrix}).
\begin{array}{c}ds^2 = -\lambda dt^2 + \lambda^{-1}dr^2 + R^2 d\Omega, e^{2\phi}= e^{2\phi_0}\frac{r+\Sigma}{r-\Sigma}, F = \frac{Qe^{\phi_0}}{(r-\Sigma)^2}dt \wedge dr,\; G=Pe^{\phi_0}\sin \theta d\theta \wedge d\varphi, \end{array}
\mathfrak{f}_1(x) = (x+5)^{41}(x-6) (x-9)^2 (x-11)^7 (x-13)^8, \mathfrak{f}_2(x) = (x+5)^{41}(x-9)^3 (x-11)^5 (x-13)^8 (x^2-19x+82).
(\operatorname{id}\otimes\varphi)\bigl((c\otimes1)\Delta((\theta\otimes\operatorname{id})[(y\otimes1)(\Delta a)])\bigr) =(\operatorname{id}\otimes\varphi)\bigl(Q_{\lambda}(c\otimes(\theta\otimes\operatorname{id})[(y\otimes1)(\Delta a)])\bigr).
\frac{1}{2iT}\int_{\frac{1}{2}-iT}^{\frac{1}{2}+iT}\chi(1-s)\zeta_N(s+ia)\zeta(s)ds = \frac{2\pi}{T}D_{i(x-y)}(T/2\pi;N) & +\mathcal{O}\Big(T^{-1/2+\varepsilon}\big( 1+ |a|^{1/6}\big)\Big))\Big) =\zeta_N(1+ia) \;+\; \mathcal{O}\big(T^{-1/3}(1+|a|^{1/6})\big)
\frac 1{p_0}= \frac 1 2 +\frac{2\mu-m}{\mu(2\mu_\ast -m)}, \frac 1{p_1}=\frac 1 2+ \frac{2\mu -m}{\mu (m+2)(n-1)}, \frac 1{p_2}=\frac 2{p_0}-\frac 1{p_1}.
\mathcal{I}_{2,\beta_1,\beta_2,\beta_3}\!\!=\!\!3\!\!\int_{[0,1]^2}\! \check{f}_{\mathbf{t}}(x,y)^s x^{3\beta_1+4\beta_2+12\beta_3+24s}y^{3\beta_1+4\beta_2+13\beta_3+1+26s}q(y) \frac{dx}{x}\frac{dy}{y},
\big | \frac{d}{da}\, M(a,b,z) \big | \ \lesssim \ \frac{1}{1+|a|}\, \sum_{n=1}^\infty\, \frac{(|a|)_n}{(b)_n}\, \frac{n\, z^n}{n!}\ \leq \ \frac{1}{1+|a|}\, \sum_{n=1}^\infty\, \frac{(|a|)_n}{(b)_n}\, \frac{(2z)^n}{n!}& \ \leq \ \frac{1}{1+|a|}\M(|a|, b, 2z).
{\hat{q^{\prime}}}_{l}( \zeta ) = f ( \theta_{+ l}^{\prime}( \zeta ) , \theta_{+ k}( \zeta ) ){\hat{q}}_{k}( \zeta ) , \quad{\bar{\hat{q}}}_{l}^{\prime}( \zeta ) ={\bar{\hat{q}}}_{k}( \zeta ) f^{*}( \theta_{- l}^{\prime}( \zeta ) , \theta_{- k}( \zeta ) ) ,
m^+(L'')=\pi^{-4}\cdot 2\cdot \frac{1}{6}\cdot 2\cdot \frac{3^\frac{13}{2}}{2^4}\cdot (1-2^{-2})^2\cdot (1-3^{-2})^2\cdot \zeta(2)^2=\frac{3^{3/2}}{2^3}\approx 0.6495
\int_{M\setminus E}|B(z,r)\cap D|\,dM(z) \ge \int_{B\setminus E}|B(z,r)\cap D|\,dM(z) &=\int_{B\setminus E}|B(z,r)|\,dM(z)\ge|B\setminus E|\,V_{\delta,n}(r) &\ge \big(|B|-|E|\big)\,\,V_{\delta,n}(r),
r_{0}m_{0^{+ +}}= 3 . 8 3 ( 1 5 ) - ( 3 . 4 5 \pm 0 . 7 5 ) \biggl ({\frac{a_{s}}{r_{0}}}\biggr )^{2}+ ( 4 . 5 0 \pm 0 . 8 0 ) \biggl ({\frac{a_{s}}{r_{0}}}\biggr )^{4}
2\overline\square_B\phi = -\partial_B H^{1,0}\lrcorner\,\phi - H^{1,0}\lrcorner\,\partial_B\phi + H^{0,1}\lrcorner\,\bar\partial_B\phi, &\bar\partial_B H^{1,0}\lrcorner\,\phi + H^{1,0}\lrcorner\,\bar\partial_B\phi =0.
g_{\mu\nu}= v_0h_{\mu\nu}+ \kappa v_0(\varphi h_{\mu\nu}+ \varphi_{\mu\nu})\; , \qquad \mbox{and}\qquad I_\mu^{\ \nu}= \left[\delta_\mu^{\ \nu}+ \kappa\frac{\varphi_{\mu\alpha}h^{\alpha\nu}}{(1 +\kappa\varphi)}\right] \; .
\left[\L_{\infty}\phi\right](x) = A\triangle\phi(x)+\left\langle Sx,\nabla\phi(x)\right\rangle-B\phi(x) = A\sum_{i=1}^{d}D_i^2\phi(x) + \sum_{i=1}^{d}\sum_{j=1}^{d}S_{ij}x_j D_i\phi(x)-B\phi(x)
\bar{\Phi}(z,v;h,k,n) =\frac{\Phi_{h,k}(z)}{s_0(2\pi)^{s_0}(\alpha_n-iv)^{s_0}}+2\pi n(\alpha_n-iv) -\frac{s_0+1}{s_0}n^\frac{s_0}{s_0+1}\Re L_{p,q}(z) &+ g_{h,k}(z,2\pi(\alpha_n -iv)) &:= A(z,v;h,k,n) + g_{h,k}(z,2\pi(\alpha_n -iv))
\widetilde{\psi}_{n}^{( n^{2})}(x;N,M,\lambda_{1}) =1/\Psi_{n}^{(n)}(x;N,M,\lambda_{1}) \hphantom{\widetilde{\psi}_{n}^{( n^{2})}(x;N,M,\lambda_{1})}{}=(1-z)^{( N+1/2) /2}(1+z)^{(M+1/2) /2}\frac{P_{n}^{(N,M)}(z)}{\lambda_{1}+Q_{n}^{(N,M)}(z)},
\langle a, b \rangle = (\vert b \vert + 1) ba - (\vert a \vert + 1) ab, \langle a_{i}; b,c \rangle = i(i+1) a_i[c,b] \langle a_i, a_j ; b,c \rangle = i(i+1)(i+2j+1) a_ia_j[c,b] - i(i+1)(j+1) a_ja_i[c,b].
J_{r}(x,t):=\int_{0}^{t}\int_{\partial\Omega\setminus\partial\Omega_{r}}\big\langle\nabla_{x}\varepsilon_{n, a}(x-\xi,b(t,\tau)),\nu(x_{0})\big\rangle\varphi(\xi,\tau)a(\tau)dS_{\xi}d\tau.
\vartheta(z;\tau) = \prod_{k \geq 1}\left(1 - e^{2 k \pi i \tau}\right) \left(1 + e^{(2k -1) \pi i \tau}e^{2 \pi i z}\right) \left(1 + e^{(2k -1) \pi i \tau}e^{-2 \pi i z}\right) & = \prod_{k \geq 1}\left(1 - e^{2 k \pi i \tau}\right) \left(1 + 2 \cos(2 \pi z) e^{(2k-1)\pi i \tau}+ e^{2 (2k-1) \pi i \tau}\right).
f \rightarrow f + \sum_{i}N_{i}( \det M_{i})^{- 2}[ 2 ( a_{i}T + b_{i})^{2}+ ( c_{i}T + d_{i})^{2}]^{2}\equiv f + \sum_{n = 0}^{4}c_{n}T^{n}\quad N_{i}\in{\bf Z}\ .
|e^{+}e^{-}\rangle\rightarrow\frac{1}{4}(3|c^{+}\rangle|c^{-}\rangle+|c^{+}\rangle|d^{-}\rangle+|d^{+}\rangle|c^{-}\rangle-|d^{+}\rangle|d^{-}\rangle-2|\gamma\rangle|\gamma\rangle).
S^t \le \sum_{\substack{{1 \le a < b \le m}p^t_a < j(t)}}c_ad(x_{p^{t+1}_a}, x_{p^{t+1}_b}) + \sum_{\substack{{1 \le a < b \le m}p^t_a = j(t)}}c_ad(x_{p^{t+1}_a}, x_{p^{t+1}_b}) - R_1 - R_2,
-\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{1}[q_{0}](\frac{i}{\mu_{0}})w_{2}[q_{0}](\frac{i}{\mu_{0}})}{w_{3}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{3}\gamma^{0}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\Phi^*ds^2 = \bigg( \dfrac{||f||^2e^{p\sum_{j=1}^qu_j}|W(f_0,f_1)|^p}{\Pi_{j=1}^q|F_j|^{(1-\delta)p}}\bigg)^2|dw|^2 &= \bigg( \dfrac{||f||^{\gamma-q\delta}e^{\sum_{j=1}^qu_j}|W(f_0,f_1)|}{\Pi_{j=1}^q|F_j|^{1-\delta}}\bigg)^{2p}|dw|^2.
B_j^{s_i}x =\sum_{a=0}^{l/s_i-1}\Big(\prod_{q=a_j}^{a_j\oplus(s_i-1)}\lambda_{j,q}\Big)x_{a(j,a_j\oplus s_i)}e_a^{(l/s_i)}, B_nx =\sum_{a=0}^{l/s_i-1}\Big(\prod_{q=s_ia_n}^{s_ia_n+s_i-1}\lambda_{n,q}\Big)x_{a(n,(s_ia_n\oplus s_i)/s_i)}e_a^{(l/s_i)}.
P_{n}^{(\alpha,\beta-1)}= \frac{(n+\alpha+\beta)_2}{(2n+\alpha+\beta)_2}P_n^{(\alpha,\beta+1)}+\frac{2(n+\alpha)(n+\alpha+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta+1)}+\frac{(n+\alpha-1)_2}{(2n+\alpha+\beta-1)_2}P_{n-2}^{(\alpha,\beta+1)}.
\langle 0 | \mathrm{T r}( P^{n}P^{n}) | 0 \rangle = \frac{M^{6}}{8 \pi^{2}}\sum_{m , n}| \langle 0 | \mathrm{T r}( [ X^{m}, X^{n}]^{2}) | 0 \rangle | \ .
(A.1) \, \|v(t,\cdot)\|_{W_h^{\rho, \infty}}\le A' \varepsilon \,, (B.1) \,\|v(t,\cdot)\|_{H^s_h}\le B'\varepsilon \, h^{-\sigma}\, , (B.2) \,\|\mathcal{L}v(t,\cdot)\|_{L^2}\le B'\varepsilon \, h^{-\sigma}\, ,
\bar{Q}_{\alpha \alpha^{\prime}I}\equiv ( Q_{I}^{\beta \beta^{\prime}})^{\dagger}( \gamma^{0})_{\alpha}^{\beta}\delta_{\alpha^{\prime}}^{\beta^{\prime}}= - Q_{I}^{\beta \beta^{\prime}}C_{\beta \alpha}C_{\beta^{\prime}\alpha^{\prime}}
\sum_{k=1}^{M}(-1)^{k}\frac{1}{k!(M-k)!}=\sum_{k=0}^{M}(-1)^{k}\frac{1}{k!(M-k)!}-\frac{1}{M!}= &=\frac{1}{M!}\left(\sum_{k=0}^{M}(-1)^{k}\binom{M}{k}-1\right)=-\frac{1}{M!},
\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big(2\alpha^2(h)[[x,y],\alpha(z)]\Big) =\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big([\alpha(h),[x,y],\alpha^2(z)] &+\varepsilon(h,x+y)[\alpha([x,y]),\alpha(h\cdot z)]\Big).
\omega = \frac{1}{\gamma g_{\phi \phi}}\left( g_{\phi \phi}( \dot{f}+ A_{a}\dot{x}^{a}) - g_{a \phi}\dot{x}^{a}( n + A_{\phi}) \right)^{2}+ \frac{( n + A_{\phi})^{2}}{g_{\phi \phi}}.