harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tau_{m e}= \frac{1}{1 + b \chi + \chi^{2}}\left( \begin{array}{c c c}{{( b + \chi )^{2}}}&{{- 2}}&{{b + \chi}}\\{{- \frac{1}{2}}}&{{\chi^{2}}}&{{\frac{\chi}{2}}}\\{{- 2 ( b + \chi )}}&{{- 4 \chi}}&{{- 1 + b \chi + \chi^{2}}}\\ \end{array}\right)
w(0) = \sum_{i=1}^{s-1}\frac{3^{s-1-i}4^i}{i!}(-s+1)(-s+2)\cdots(-s+i) &=\sum_{i=1}^{s-1}\frac{(s-1)(s-2)\cdots(s-i)}{i!}3^{s-1-i}(-4)^i &=\sum_{i=1}^{s-1}\binom{s-1}{i}3^{s-1-i}(-4)^i= (3+(-4))^{s-1}-3^{s-1}=(-1)^{s-1}-3^{s-1}.
\left( \begin{array}{c c}{{S^{i}\sigma_{i}}}&{{0}}\\{{0}}&{{S^{i}\sigma_{i}}}\\ \end{array}\right) \equiv \frac{1}{2}h^{- 1}\left( \begin{array}{c c}{{\sigma_{3}}}&{{0}}\\{{0}}&{{\sigma_{3}}}\\ \end{array}\right) h \; ,
x=\frac{k_1}{t+c}, y=\frac{k_2}{t+c}, z=\frac{k_3}{t+c},w=\frac{k_4}{t+c}\intertext{which, in turn, imply}x'=\frac{-k_1}{(t+c)^2}, y'=\frac{-k_2}{(t+c)^2}, z'=\frac{-k_3}{(t+c)^2},w'=\frac{-k_4}{(t+c)^2}
[X\oplus f, [Y\oplus g, Z\oplus h]]_{_{Der(\mathcal{F})\oplus\mathcal{F}}}+[Y\oplus g, [Z\oplus h, X\oplus f]]_{_{Der(\mathcal{F})\oplus\mathcal{F}}}&+[Z\oplus h, [X\oplus f, Y\oplus g]]_{_{Der(\mathcal{F})\oplus\mathcal{F}}}=0_{Der(\mathcal{F})\oplus\mathcal{F}}.
{\rm Coeff}[f_{\delta},p+m_1]={\rm Coeff}[E_{\delta},p+m_1]=\lim_{q\rightarrow t^{-p-m_1}}(1-qt^{p+m_1})\left(\sum_{\nu \in \sigma(\epsilon)}c_{\nu}(q,t) E_{\nu}(z;q,t)\right),
\sup_{\sigma\in\mathcal{V}_n}\sum_{j_1=N_1}^{J_{n,1}-1}\cdots\sum_{j_d=N_d}^{J_{n,d}-1}\sum_{k_1=0}^{2^{j_1}-1}\cdots\sum_{k_d=0}^{2^{j_d}-1}n^{-\frac{C_1\overline{\gamma}^2}{16\sigma^2}}\sigma\lesssim(\sigma_0+o(1))n^{-\left[\frac{C_1}{16(\sigma_0^2+o(1))}\overline{\gamma}^2-1\right]},
1)\,[m,[X,Y]]_{R}=[[m,X]_{R},Y]_{R}+[X,[m,Y]_{R}]_{L}, & 2)\,[X,[m,Y]_{R}]_{L}=[[X,m]_{L},Y]_{R}+[m,[X,Y]]_{R}, & 3)\,[X,[Y,m]_{L}]_{L}=[[X,Y],m]_{L}+[Y,[X,m]_{L}]_{L},\quad\forall X,Y\in \mathfrak{g},\quad\forall m\in M.
\sum_{n=0}^\infty\sum_{\ell=0}^n \frac{t^n(s+\ell)!}{n!}B_{n, \ell}(x_1, x_2, \ldots) = \sum_{i=0}^{\infty}\frac{(s+i)!}{i!}\sum_{j=1}^{\infty}\left(\frac{x_jt^j}{j!}\right)^i & = s!\sum_{i=0}^{\infty}\binom{s+i}{i}C(t)^i & = s!\sum_{i=0}^{\infty}\binom{-(s+1)}{i}(-C(t))^i & = s!(1-C(t))^{-(s+1)}
U_5(FZ\rho t^i)=F\Big((\sum_{j=\left\lceil \frac{i+2}{5}\right\rceil}^\infty z_0(i,j)t^j)+\rho(\sum_{j=\left\lceil \frac{i+2}{5}\right\rceil}^\infty z_1(i,j)t^j) \Big),
u(z) \L^* v(z) - v(z) \L u(z) = \div_x \big( u(z) A(z) \nabla_x v(z) - v(z) A(z) \nabla_x u(z) \big) - \div_x \big( u(z) v(z) b(z) \big) + \partial_t (u(z)v(z))
\delta = 2(1-p)[ \alpha_1 -\alpha_2 + \alpha_1\alpha_3 - \alpha_1\alpha_2 + \alpha_2\alpha_3] + 4\alpha_1p - 2\alpha_3 + 2 \kappa_1 = \frac{(1-\alpha_3)(1-\alpha_2 (1-p) )}{\delta}\kappa_2 = \frac{\alpha_1(p + \alpha_3 (1-p))}{\delta}\kappa_3 = \frac{\alpha_1(1- \alpha_2(1-p))}{\delta}.
\displaystyle{\frac{8}{\kappa}}V_{\mu \nu , \, \alpha \beta}^{^2 g h o s t}( k_{1}, k_{2}; \, p_{1}, p_{2}) = - 4 \, p_{_1 \mu}\, p_{_2 \nu}\, \eta_{\alpha \beta}+ 8 \, p_{_1 \beta}\, p_{_2 \nu}\, \eta_{\alpha \mu}- 2 \, p_{1}\cdot p_{2}\, \eta_{\alpha \mu}\, \eta_{\beta \nu}+ p_{1}\cdot p_{2}\, \eta_{\alpha \beta}\, \eta_{\mu \nu}.
0 \longrightarrow R \xrightarrow{\begin{pmatrix}-y -x z\end{pmatrix}}R^3 \xrightarrow{\begin{pmatrix}z 0 y -x y 0 0 -xz -x^2 \end{pmatrix}}R^3 \xrightarrow{\begin{pmatrix}x^2 xz y\end{pmatrix}}R \longrightarrow R/I \longrightarrow 0
P(\tau_1\geq \rho)\geq P(\tau_1^*\geq \rho)\rightarrow1 \mbox{as}\rho\rightarrow0, P(n_T\leq K)=P(\tau_{k+1}>T)\geq P(\tau_{K+1}^*>T)\rightarrow 1 \mbox{as}K\rightarrow\infty.
\frac{1}{4}+Y_0^2=s.(1-s)=\frac{-s.\Gamma (s)}{\frac{\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.\Gamma (s+2.k+1)}{x_0}-\Gamma (s).\underset{n=1}{\overset{n=N}{\sum}}\frac{n^{-s}}{x_0^{1-s}}}s=\frac{1}{2}\pm iY_0
\nu_x x^2 = q^{-1}x^2 - q^{-1}+q^{-2}\nu_y + q^2\nu_z - q\nu_y\nu_z, \nu_y y^2 = q^{-1}y^2 - q^{-1}+q^{-2}\nu_z + q^2\nu_x - q\nu_z\nu_x, \nu_z z^2 = q^{-1}z^2 - q^{-1}+q^{-2}\nu_x + q^2\nu_y - q\nu_x\nu_y.
\Phi[f]=\frac{2}{(2\pi\hbar)^{3/2}}\iint\iint dqdpd\tilde{x}d\tilde{p}e^{\frac{i}{\hbar}(\tilde{x}\tilde{p}-2(\tilde{p}-p)(\tilde{x}-q))}f(q,p)|\tilde{x}\rangle\langle\tilde{p}|.
\stackrel{( 3 )}{\varphi}_{\mu}= p_{1 \mu}- m \frac{q_{2 \mu}}{\sqrt{q_{3}^{2}}}- \beta \frac{\varepsilon_{\mu \nu \lambda}q_{2}^{\nu}q_{3}^{\lambda}}{q_{2}^{2}\sqrt{q_{2}^{2}}}= 0 , \quad \mu = 0 , 1 , 2{.}
df=\frac{\partial f}{\partial t}dt+\frac{1}{2}\frac{\partial^{2}f}{\partial t^{2}}dt^{2}+\cdot\cdot\cdot+\frac{\partial f}{\partial x}dx+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}dx^{2}+\cdot\cdot\cdot.
\hat{b}_k=\frac{\beta X_0}{\sigma}\theta_k+\frac{\beta^2 X_0}{\sigma}\sum_{j=k+1}^T\theta_j\alpha^{j-k-1}=\frac{\beta X_0}{\sigma}(1-(T-k)\beta)+\frac{\beta^2 X_0}{\sigma}(T-k)=\frac{\beta X_0}{\sigma}.
\frac{\zeta^{\prime}}{\zeta}(s)+\frac{\zeta^{\prime}}{\zeta}(1-s)=log\pi-\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}(\frac{s}{2})-\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}(\frac{1-s}{2})
-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq-\frac{1}{\pi^{2}}\frac{1}{2}\lambda_{1,n}^{r,s}.\leq-\frac{9}{4\pi^{2}}\lambda_{1,n}^{r,s}\sum_{i=1}^{3}\Big(\frac{\theta_{i}}{2\pi}-1/3\Big)^{2}.
N_1^{1/3}Q'_1(0)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=0}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\frac 1{N_1^{(r+s)/3}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^{r}\mathbf{d'}\in [n_1+2,n_2]_<^t}}\frac 1{N_1^{(r+t)/3}}\det \left(N_1^{1/3}V\right).
\Theta ( t ) = \left[ \sum_{\delta \neq{\bf 1}}\sum_{p = 1}^{\infty}{\frac{l ( \delta ) / 2}{\mathrm{S i n h}( p l ( \delta ) / 2 )}}e^{- p^{2}l^{2}/ 4 t}\right]{\frac{e^{- M^{2}t}}{( \pi t )^{1 / 2}}}\quad .
{e(B^m_5)\over e(B^m_2)}={d_1^{d_1-1}H^{-1}\over (d_1-1)!(\lambda_1-\lambda_2)^{d_1}}\prod_{k=1}^{d_1-1}\left(\lambda_2-2\lambda_1+{k\over d_1}(\lambda_1-\lambda_2)\right).
\P_x(Z_d(n_1) \geq \xi_1, \ldots, Z_d(n_m) \geq \xi_m) & = \frac{1}{h(x^0)}\int_{I_{\xi_1}}\ldots \int_{I_{\xi_m}}\det(f_{n_1-1+i}(x_j^1 - x_i^0))\det(f_{n_2 - n_1}(x_j^2 - x_i^1)) & \ldots \det(f_{n_m - n_{m-1}}(x_j^m - x_i^{m-1})) \Delta(x^m) \prod_{k=1}^m \prod_{j=1}^d dx_j^k.
\sum_{-\lfloor m/2 \rfloor \leq j \leq - m^{1/4}}b_j + \sum_{m^{1/4}\leq j \leq \lfloor \frac{m-1}{2}\rfloor}b_j = O \left( \sum_{j \geq m^{1/4}}1/j^2 \right) = O(m^{-1/4}).
\vartheta\leq J\left( u_{0},v_{0},w_{0}\right) \leq\liminf\limits_{k\rightarrow\infty}J\left( u_{k},v_{k},w_{k}\right) =\inf\limits_{\left( u,v,w\right) \in\mathcal{D}}J\left( u,v,w\right)=\vartheta.
E(t) = \frac{1}{2\pi\rm i}\int_\Gamma e^{zt}z^{\alpha-1}(z^\alpha+A)^{-1}\d z \quad\mbox{and}\bar E(t)= \frac{1}{2\pi\rm i}\int_\Gamma e^{zt}z^{-\gamma}(z^\alpha+A)^{-1}\d z.
\tilde{C}( p ) = - 2 \gamma g^{2}\delta_{a}^{b}\delta_{J}^{I}\int \frac{d^{4}k}{( 2 \pi )^{4}}d^{2}\theta d^{2}{\overline{\theta}}\frac{1}{k^{4}}\left\{\Phi_{I}^{a \dagger}( p , \theta ,{\overline{\theta}}){\Phi}_{b}^{J}( - p , \theta ,{\overline{\theta}}) \right\},
\left( \sum_{j_{1}=1}^{n}\left( \sum_{j_{2}=1}^{n}\left( \cdots \left(\sum_{j_{m}=1}^{n}\left\vert T(e_{j_{1}},...,e_{j_{m}})\right\vert^{q_{m}}\right)^{\frac{q_{m-1}}{q_{m}}}\cdots \right)^{\frac{q_{2}}{q_{3}}}\right)^{\frac{q_{1}}{q_{2}}}\right)^{\frac{1}{q_{1}}}\leq C_{m,p,\mathbf{q}}^{\mathbb{K}}\left\Vert T\right\Vert
[\alpha_{i}(z),x_{i}^{\pm}(w)]=\pm(\alpha_{i},\alpha_{i})x_{i}^{\pm}(w)\delta(z-w), &[x_{m+1}^{+}(z),x_{m+1}^{-}(w)] &=\big(:\delta_{1}(z)\delta_{1}^{*}(z):-:X(\varepsilon_{m+1},z)X(-\varepsilon_{m+1},z):\big)\delta(z-w)-\partial_{w}\delta(z-w) &=-\big(\alpha_{m+1}(z)\delta(z-w)+\partial_{w}\delta(z-w)\cdot1\big),
\left< \hat{u}^{n}_{t},\varphi\right> - \left< \hat{u}^{n}_{0},\varphi\right> = \int_0^t \left< \hat{u}^{n}_{z}, \hat{L}_z^n \varphi\right> dz + \hat{W}^n_t(R\varphi) + r^n_t(\varphi),
dX_i(t)=\sqrt{2(X_i^2(t)+1)}dW_i(t)+\left[\left(2-2N-2\Re(s)\right)X_i(t)+2\Im(s)+\sum_{j\ne i}^{}\frac{2(X^2_i(t)+1)}{X_i(t)-X_j(t)}\right]dt,
\left ( \begin{array}{c}(\mathcal{F}\tilde{f_{1}})(\xi) \vdots (\mathcal{F}\tilde{f_{N}})(\xi) \end{array}\right ) = \tilde{\mathcal{A}}(z) \left ( \begin{array}{c}(\mathcal{F}\tilde{f_{1}})(\frac{\xi}{2}) \vdots (\mathcal{F}\tilde{f_{N}})(\frac{\xi}{2}) \end{array}\right ) .
u =B^{\ast}T_{q}^{\ast}(t_k-t)\mathcal{R}(\lambda, \Psi^{t_k}_{t_{k-1},1})P_{1}^{k}(x(\cdot)), v =D^{\ast}S_{q}^{\ast}(t_k-t_{k-1})\mathcal{R}(\lambda, \Psi^{t_k}_{t_{k-1},2})P_{2}^{k}(x(\cdot)),
I(t) = i \frac{\Gamma(\pm t + 1/8)\Gamma(\pm t + 5/8)\Gamma(\pm t + \sigma/2 + 1/8)\Gamma(\pm t + \sigma/2 + 5/8)\Gamma(\pm t + \tau/4) \Gamma(\pm t + \tau/4 + 1/2)}{\Gamma(\pm 2t)},
\begin{aligned}V_{\mathcal S}= \|e_{\mathcal S}\|_{q}^{q},V_{\mathcal S^c}=\|e_{\mathcal S^c}\|_{q}^{q}, \eta_{\mathcal S}= \|e_{\mathcal S}\|_2^2/\|e_{\mathcal S}\|_{q}^2 ,\eta_{\mathcal S^c}=\|e_{\mathcal S^c}\|_2^2/\|e_{\mathcal S^c}\|_{q}^2,\end{aligned}
\begin{cases}\dot{\bf m}= (A + G){\bf m}+BR^{-1}B^T{\bf y}+\gamma{\bf p}, \dot{\bf p}= -(A+G)^T{\bf p}-(I-\Gamma)^T Q [{\bf m}- (\Gamma{\bf m}+\eta)], \dot{\bf y}=-A^T{\bf y}+Q[{\bf m}-(\Gamma{\bf m}+\eta)],\end{cases}
s_{\textrm{dir}_{n_r}}(t) = \sum_{n_t=1}^{N_t}\alpha_{1}x_{n_t}(t-\tau_{\textrm{dir},n_r,n_t})e^{\mathrm{j}2\pi{f_c}(t-\tau_{\textrm{dir},n_r,n_t})}e^{\mathrm{j}2\pi{f_{D,1}}(t-\tau_{\textrm{dir},n_r,n_t})}.
I_0 = \left\{n \middle|\, x_n >\delta\right\}, I_k = \left\{n \middle|\, \delta^{m^k}< x_n \leq \delta^{m^{k-1}}\right\}(1\leq k \leq m), I_{m+1}= \left\{n \middle|\, x_n \leq \delta^{m^m}\right\}.
L^{-}= q \left( \begin{array}{c c c}{{q^{- h_{2}}}}&{{0}}&{{0}}\\{{( q - q^{- 1}) \alpha^{-}}}&{{q^{- h_{1}- h_{2}}}}&{{0}}\\{{( q^{- 1}- q ) b^{-}}}&{{( q^{- 1}- q ) \beta^{-}}}&{{( - 1 )^{F}q^{- h_{1}- 2 h_{2}}}}\\ \end{array}\right)
a b l_\sigma(a) = (w , \beta_{1,1}, \beta_{2,1}) (z , r_b , s_b)(xy^{-1}w(x^{-1},y^{-1}) y x^{-1}, |w|_x + \beta_{1,1}, |w|_y + \beta_{2,1}) & = ( w z xy^{-1}w(x^{-1},y^{-1}) y x^{-1}, 2 \beta_{1,1}+ |w|_x + r_b, 2 \beta_{2,1}+ |w|_y + s_b) .
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{III}^-}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d,s,k}T \eta^d\end{aligned}
(\hat \mu \circ \hat T^n)(\Delta) \ge 1 - \sum_{k=0}^n \binom{n}{k}(\tfrac12)^{n-k}\cdot \begin{cases}1 2 \cdot (\tfrac13 \cdot \tfrac16)^{k/2}(\tfrac16 + \tfrac13) \cdot (\tfrac13 \cdot \tfrac16)^{(k-1)/2}\end{cases}&\ge 1 - 2 \cdot (\tfrac12 + \tfrac16\sqrt2)^n
f^{(j)}(x) = \sum_{k=0}^j \binom{j}{k}(p)_k (x - \lambda)^{p-k}g^{(j-k)}(x) = \sum_{k=0}^p \binom{j}{k}(p)_k (x - \lambda)^{p-k}g^{(j-k)}(x) = (x - \lambda) \sum_{k=0}^{p-1}\binom{j}{k}(p)_k (x - \lambda)^{p-1-k}g^{(j-k)}(x) +\binom{j}{p}p! g^{(j-p)}(x)
\left\langle \frac{1}{2 N}\mathrm{T r}\, H_{S}^{2}\right\rangle - \left\langle \frac{1}{2 N}\mathrm{T r}\, H_{S}\right\rangle^{2}= \left\langle \frac{1}{N}\mathrm{T r}\, H_{0}^{2}\right\rangle - \left\langle \frac{1}{N}\mathrm{T r}\, H_{0}\right\rangle^{2}+ \frac{3}{8}a^{4}= k^{2}a^{2}+ \frac{9}{1 6}a^{4}
X_{4}(g_{1}^{k},g_{2}^{l})=\sum_{n=0}^{N-2}\sum_{m=0}^{M-2}x(g_{1}^{-n},g_{2}^{-m})cas(\frac{2\pi g_{1}^{(-n+k)}}{N}+\frac{2\pi g_{2}^{(-m+l)}}{M}),
|Y_n(t,\omega) - Y_n(t,0)| = |u_n(t,\omega/\sqrt{n}) - u_n(t,0)| &\le \left\|F\left(\frac{1}{\sqrt{n}}(\omega \otimes_t \cdot)\right) - F\left(\frac{1}{\sqrt{n}}(0 \otimes_t \cdot)\right)\right\|_\infty,
d^2\!f(\overline{x}|\overline{v})(w) \le\inf_{z\in\mathbb{R}^n}\big\{d^2\vartheta(F(\overline{x})) (dF(\overline{x})(w)|d^2F(\overline{x})(w|z))\!-\! \langle \overline{v},z\rangle \}\le d^2\vartheta(F(\overline{x}))(dF(\overline{x})(w)\,|\,d^2F(\overline{x})(w|0))=0.
R_{G,opt}(N,t) \geq 1-\frac{\log \left[\prod\limits_{k=0}^{\min\{8t,N-1\}}(N-k)\right]}{\log N!}&>1-\frac{(8t+1)\log N}{(N+\frac{1}{2})\log N -(\log e)N}&>1-\frac{(8t+1)\log N}{N\log N -(\log e)N}&>1-\frac{8t+1}{N}\left(1+\frac{2\log e}{\log N}\right),
K[H_E,\rho =1]( \vec{u}\sigma ;\vec{u}_0 0) = \exp(i4e^2\sigma / \hbar ){\bf k}^{osc}\langle u_1\sigma \vert u_{10}0 \rangle{\bf k}^{osc}\langle u_2 \sigma \vert u_{20}0 \rangle
\omega_R - t^+_R = \int_{t^+_R}^{\omega_R}\frac{\dot r_R(t)}{\dot r_R(t)}\,dt \geq \frac{1}{\sqrt{2C_+}}\int_{t^+_R}^{\omega_R}\frac{\dot r_R(t)}{r_R(t)^{-\alpha/2}}\,dt & = \frac{1}{\left(1+\alpha/2 \right)\sqrt{2C_+}}\left(R^{1+\alpha/2}-K^{1+\alpha/2}\right),
\frac{1}{k}=\frac{1}{b}+\frac{(b-k)^{1}}{b^{2}}+\frac{(b-k)^{2}}{b^{3}}+\frac{(b-k)^{3}}{b^{4}}+\cdot\cdot\cdot+\frac{(b-k)^{N-1}}{b^{N}}+\cdot\cdot\cdot=\frac{1}{b}\frac{1}{1-\frac{b-k}{b}}.
\left. + \frac{1}{2}\left( \{\psi_{+}( x ) , \psi_{+}( y ) \}+ \{\psi_{-}( x ) , \psi_{-}( y ) \}\right) \left( \frac{1}{2}\phi^{\prime}( x ) U ( y ) + \frac{1}{2}U ( x ) \phi^{\prime}( y ) \right) \right] \ \ .
\frac{\varphi_x(w_1,\dots,w_k)}{\phi(w_1)\cdots\phi(w_k)}= &\left(\frac{d}{2}\right)^{-\frac{p k}{2}}\prod_{i=1}^k \frac{\Gamma \left( (d-i+1)/2\right)}{\Gamma \left((d-p-i+1)/2\right)}\det(S_k)^{-\frac{p}{2}}\left( 1-\frac{\|x\|^2}{d}\iota'S_k^{-1}\iota\right)^{\frac{d-p-k-1}{2}}e^{\frac{k}2 \|x\|^2}
(1+s^-+s^+)^\alpha = (s^-)^{\alpha}\left[\left(1+\tfrac{1+s^+}{s^-}\right)^{\alpha}- \left(\tfrac{1+s^+}{s^-}\right)^{\alpha}\right] + (1+s^+)^{\alpha}&\leq \epsilon\left(\alpha,\tfrac{1+s^+}{s^-}\right)(1+s^-)^\alpha + (1+s^+)^\alpha \leq \epsilon\left(\alpha,\tfrac{s^+}{s^-}\right)(1+s^-)^\alpha + (1+s^+)^\alpha
g_{\lambda,\rho}^1\! (w_1) = \frac{n_0}{4 k_{\ell}}\log \left( 1 + \lambda \rho (1 - \lambda \rho) w_1 P' \right) - \frac{|A^{\ast}|}{k_{\ell}}H_2 \Big( \!\frac{w_1}{|A^{\ast}|}\!\Big)
b_2^p = b_1^p [ a_1,b_2]^{-1}& b_{2i+1}^p = b_{2i}^p [a_2, b_{2i+1}]^{-1}, \,\,\,\, b_{2i+2}^p =b_{2i+1}^p[a_1, b_{2i+2}]^{-1}, \hskip5mm i=1,\ldots, n-1.
A = -5 x_0^2 + 4 x_0x_2 -4 x_1^2 + 2x_1x_2 -4 x_2^2, B = 5x_0^2 + 2x_0x_1 -2 x_0x_2 + 2x_1^2 + 2x_1x_2 + 4x_2^2, C = 4x_0^2 + 2x_0x_1 -4x_0x_2 +2x_1^2 -2x_1x_2 + 5x_2^2, D = -4x_0^2 -2x_0x_1 - x_1^2 -2x_1x_2 -4 x_2^2, E = 4x_0^2 +3 x_1^2 + 4x_2^2, F = -4x_0^2 + 4x_0x_1 +2x_0x_2 -2 x_1^2 -4x_1x_2 -5 x_2^2.
u ( \mathbf{p ,}i ) \simeq \left( 1 + \frac{\mathbf{p}^{2}}{8 m^{2}}\right) \left[ \begin{array}{c}{1}\\{\frac{( \overrightarrow{\mathbf{\sigma}}\mathbf{\cdot p})}{2 m}}\\ \end{array}\right] \varphi_{i}= \left[ \begin{array}{c}{\left( 1 + \frac{\mathbf{p}^{2}}{8 m^{2}}\right)}\\{\frac{( \overrightarrow{\mathbf{\sigma}}\mathbf{\cdot p})}{2 m}}\\ \end{array}\right] \varphi_{i}
\begin{aligned}\|u_I-Q_F u_I\|_{0,e}\lesssim \epsilon_F^{-1}\big(h_F^{-1/2}\|u_I-Q_F u_I\|_{0,F}+h_F^{1/2}|u_I-Q_F u_I|_{1,F}\big) & \lesssim \epsilon_F^{-1}\big( h_F^{-1/2}\|u_I-u\|_{0,F}+h_F^{-1/2}\|u-Q_F u_I\|_{0,F}& +h_F^{1/2}|u_I-u|_{1,F}+h_F^{1/2}|u-Q_F u_I|_{1,F}\big)\end{aligned}
\mathcal{G}_\sharp(E):=\mathcal{H}^{n-1}(\Omega\cap\partial E)+\sigma\,\mathcal{H}^{n-1}(\partial\Omega\cap\partial E)+c\,\mathcal{H}^{n-2}\left((\overline{\partial E\cap\Omega})\cap\partial\Omega\right),
\left(x^2 \sqrt{l}\Phi_1^{'}\right)^{'}=\frac{l^{3/2}x^2 \Phi_1}{f}\left[\beta_2^2 \left(\Phi_1^2 + \Phi_3^2 + \Phi_4^2-1 \right) + \beta_3^2 \left(\frac{\Phi_1^2}{2}+ \frac{27}{30}\Phi_4^2 - \frac{7}{30}\right) \right] + 2 K^2 \sqrt{l}\Phi_1
\bar\pi_{n,g}( \bar\lambda_1,\ldots,\bar\lambda_n;a_0,a_1,\ldots,a_g)=\sum_{\substack{k_2,\ldots,k_n\geq 0 \sum ik_i=g}}\bar e_2^{k_2}\dots\bar e_n^{k_n}U_{k_2,\ldots,k_n}.
G\left( f\left( a\right) ,f\left( b\right) \right) \left( \underset{a}{\overset{b}{\int}}f\left( u\right)^{du}\right)^{\frac{1}{a-b}}=\left( \underset{0}{\overset{1}{\int}}\left( f^{\ast}\left( \left( 1-t\right)a+tb\right)^{\left( 2t-1\right)}\right)^{dt}\right)^{\frac{b-a}{2}},
||f-x_n(f)](t)| =|a^{-n}(t)f(t)-f(t)| &=a^{-n}(t)f(t)-f(t) & \leq [(1-r)^{-in}-1]f(t) & \leq [(1-r)^{-in}-1]f_0(t) & \leq [(1-r)^{-in}-1]\frac{\epsilon}{K}e^{-i^2}&\leq\epsilon.
\Gamma_{i j}^{k}\Gamma_{i j}^{k}- \Gamma_{i j}^{k}\Gamma_{i k}^{j}- \Gamma_{i j}^{j}\Gamma_{i k}^{k}+ 2 \Gamma_{i i}^{j}\Gamma_{j k}^{k}- \Gamma_{i i}^{k}\Gamma_{j j}^{k}= 0
A_{\mu \nu \eta \sigma \rho}^{( 2 )}( x ) = \frac{e^{2}}{2}( A_{\mu}A_{\eta}\partial_{\sigma}F_{\nu \rho}^{0}- \partial_{\nu}A_{\rho}\partial_{\eta}A_{\mu}A_{\sigma}+ A_{\mu}F_{\nu \eta}^{0}F_{\sigma \rho}^{0}) \, \, \, ,
\leqslant d_5C_{\star}\delta + C_{9}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\big(|u_k(y)|^{p(x,y)}+ |v_k(y)|^{p(x,y)}\big)\frac{|\Phi_{R}(x)-\Phi_{R}(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy\,dx.
\Delta_{k}' = \displaystyle \sum_{sa_{1},b_{1}t}\displaystyle \sum_{v,v_{1}\in L(k)}(v_{1}^{-1})^{\ast}(s^{-1}t^{-1})v^{\ast}(ts)a_{1}^{\ast}v_{1}^{-1}(^{\ast}b_{1})[b_{1}va_{1}] &=\displaystyle \sum_{sa_{1},b_{1}}\displaystyle \sum_{v,v_{1},t \in L(k)}(v_{1}^{-1})^{\ast}(s^{-1}t^{-1})v^{\ast}(ts)a_{1}^{\ast}v_{1}^{-1}(^{\ast}b_{1})[b_{1}va_{1}]
\sum_{u \in A}d_{\mathcal{P}'}(u) = 2e_{\mathcal{P}'}(A) + e_{\mathcal{P}'}(A,B) + e_{\mathcal{P}'}(A,\overline{U}) = 2e_{\mathcal{P}}(A) + e_{\mathcal{P}'}(A,B) + e_{\mathcal{P}}(A,\overline{U}),
{\cal P}_{x^{-}}e^{i \int_{- L}^{L}d y^{-}\big [ A_{-}( x^{+}, y^{-},{\vec{x}}_{\perp}) - i{\vec{\alpha}}_{\perp}\big ({\vec{\partial}}_{\perp}+ i{\vec{A}}_{\perp}( x^{+}, y^{-},{\vec{x}}_{\perp}) \big ) \big ]}= \pm 1 \, \, ,
f(x') - a_{n,k}\geq f_n(x') - a_{n,k}= f_n (x') - f_{n-1}(x') + f_{n-1}(x') - a_{n,k}\geq 2^{- \gamma m n}g_n(x') - |f_{n-1}(x') - a_{n,k}| \geq \frac{1}{4}2^{-\gamma m n}- \frac{1}{8}2^{-\gamma m n}= \frac{1}{8}2^{- \gamma m n},
\begin{array}{rcl}Y = \frac{1}{2}( x + y ) = \frac{1}{2}(z_1^{2n-4}+ z_2^{2n-4}) X = \frac{1}{2}z ( x - y ) = \frac{1}{2}z_1 z_2(z_1^{2n-4}- z_2^{2n-4}) Z = z^2=z_1^2 z_2^2\end{array}
{{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}=\sum\nolimits_{m=1}^{M}\sum\nolimits_{\mu=0}^{K'}\frac{(-1/s_{m}^{\star})^{K'-\mu}}{(K'-\mu)!\ln{2}}\left(-{\rm{e}}^{-\frac{1}{s_{m}^{\star}}}{\rm{Ei}}\left(\frac{1}{s_{m}^{\star}}\right)\right. &+\left.\sum\nolimits_{i=1}^{K'-\mu}(i-1)!\left(-1/s_{m}^{\star}\right)^{-i}\right),
\begin{aligned}534\,824 , 2\,661\,365 , 2\,733\,965 , 3\,194\,013 , 3\,268\,781, 4\,006\,033 , 5\,180\,081 , 5\,250\,941 , 5\,489\,661 , 6\,115\,852, 6\,290\,549 , 7\,712\,184 , 7\,857\,048 , 7\,943\,761 , 8\,243\,113, 8\,747\,997 , 8\,899\,661 , 9\,907\,837 , \end{aligned}
Z = \int [ d \Phi^{A}] [ \det \omega ]^{-{\frac{1}{2}}}[ \det f ]^{-{\frac{1}{2}}}\exp \{\frac{i}{\hbar}\, S_{n m}[ \Phi^{A}, \Phi_{A}^{*}-{\frac{\partial \Psi}{\partial \Phi^{A}}}] \}
=- (2 \pi \sqrt{\alpha'})^2\,\left(\frac{1}{2}+ \frac{1}{(2 \pi\sqrt{\alpha'})^2}\tilde{b}\right) + (2 \pi \sqrt{\alpha'})^2 = (2\pi \sqrt{\alpha'})^2 \,\left(\frac{1}{2}- \frac{1}{(2 \pi\sqrt{\alpha'})^2}\tilde{b}\right)
\mathbf{L}_{b}^{a}=\delta_{b}^{a}-\frac{1}{2}(\alpha_{b}^{a}+i\lambda_{2b}^{a})\mathbf{m-}\frac{1}{2}(\alpha_{b}^{a}-i\lambda_{2b}^{a})\overline{\mathbf{m}}+\frac{1}{2}(\beta_{b}^{a}-\overline{\beta}_{b}^{a}-id\lambda_{2b}^{a}-i\lambda_{2c}^{a}\alpha_{b}^{c})\mathbf{m}\overline{\mathbf{m}},
\sigma_{i}(t,\omega)=\begin{cases}\sigma_{i}(t-1,\omega), I(t,\omega) \neq i; +1, I(t,\omega)=i; \U(t,\omega) < \frac{1}{2}; -1, I(t,\omega)=i; \U(t,\omega) \geq \frac{1}{2}.\end{cases}
\Gamma_{p,\ell}\cap B_h =\big\{x\in \Lambda[1/p]:\,\, N(x)=1,\, x= I\,(\hbox{mod}\, \ell),\, \|x\|_p\le h \big\}&=\big\{h^{-1}y:\, y\in\Lambda,\, N(y)=h^2,\, y= hI\,(\hbox{mod}\, \ell)\big\},
\begin{array}{l}|h=\frac{kD}{2(k+\check{g})}, q=\frac{kD}{(k+\check{g})}\rangle =|a_{1},....,a_{D}\rangle \ |h=\frac{kD}{2(k+\check{g})}, q=-\frac{kD}{(k+\check{g})}\rangle =|\bar{a}_{1},..., \bar{a}_{D}\rangle\end{array}
a_{e_1+\dots+e_n}\, = \, \frac{a_{e_1+\dots+ e_{n-1}}\cdot a_{{e_1}+\dots+ e_{n-2}+ e_n}\cdot a_{e_{n-1}+e_n}\cdot a_{{\bf 0}}}{a_{e_1+\dots+e_{n-2}}\cdot a_{e_{n-1}}\cdot a_{e_n}},
\frac{\int\limits_{S}\sum\limits^\infty_{n_{1}=1}\left|\sum\limits_{\gamma\in\Gamma^{2}_{\infty}\backslash\Gamma^{2}_{1}}\nabla_{\tau}\phi^0_{n_{1}}(\tau)\big|_{\gamma\circ\tau}\right|^2 dV}{\int\limits_S\sum\limits^\infty_{n=1}\left|\sum\limits_{\gamma\in\Gamma^{2}_{\infty}\backslash\Gamma^{2}_{1}}\phi^0_{n_{1}}(\tau)\big|_{\gamma\circ\tau}\right|^2 dV}< 10 \lambda.
\delta \eta_{\alpha_{2k+1}}^{*}=\Phi_{\alpha_{2k}}^{*}Z_{\;\;\alpha_{2k+1}}^{\alpha_{2k}}+\Phi_{\alpha_{2k+2}}^{*}A_{\alpha_{2k+1}}^{\;\;\alpha_{2k+2}},\;k=0,\cdots ,b,
\int_{0}^{\pi}d \theta{\frac{( \sin \theta )^{( d - 4 )}}{( a^{2}+ b^{2}\sin^{2}\theta )^{( d - 2 ) / 2}}}={\frac{( d - 3 ) ! !}{2^{( d - 2 ) / 2}\left( ( d - 2 ) / 2 ) \right) !}}{\frac{1}{a ( b^{2}+ a^{2})^{( d - 3 ) / 2}}}
(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})=(\begin{matrix}j_{1}&\frac{j_{2}+j_{3}-m_{1}}{2}&\frac{j_{2}+j_{3}+m_{1}}{2}\\j_{3}-j_{2}&\frac{j_{2}-j_{3}-m_{1}}{2}-m_{3}&\frac{j_{2}-j_{3}+m_{1}}{2}+m_{3}\end{matrix}),
\left(P_{\cal E}\right)_{\mu\nu}=\frac{1}{2}\left(\bar{n}_{\mu}n_{\nu}+n_{\mu}\bar{n}_{\nu}\right),\qquad\left(P_{\cal H}\right)_{\mu\nu}=\frac{1}{2}\left(\bar{m}_{\mu}m_{\nu}+m_{\mu}\bar{m}_{\nu}\right)
\lim_{k\to \infty}\frac{c_{\langle \alpha_{k}\rangle}(Y,\lambda)^{2}}{I(\alpha_{k}^{\Gamma})}=\lim_{k\to \infty}\frac{A(\alpha_{k}^{\Gamma})^{2}}{2|\Lambda(A(\alpha_{k}^{\Gamma}),\Gamma)|}=\mathrm{Vol}(Y,\lambda).
\hat{Q}\left( z \right)\Gamma_{0}^{+}=\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}\Gamma_{0}^{+}\left\{A(I-P)\left( \tilde{A}-z \right)^{-1}(I-P)A-\left( A-zI \right) \right\}\Gamma_{0}\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}\Gamma_{0}^{+}
S(z) = m(z), |z| < \rho_N S(z) = m(z)\begin{pmatrix}1 -(1 - \varphi_{n,t}(z)^{-1})z^n 0 1\end{pmatrix}^{-1}, \rho_N < |z| < 1 S(z) = m(z)\begin{pmatrix}1 0 (1 - \varphi_{n,t}(z)^{-1})z^{-n}1\end{pmatrix}, 1 < |z| < \rho_N^{-1}S(z) = m(z), \rho_{N}^{-1}< |z| .
\tau_{k , \ell +1}^{(\alpha,\beta-1)}\tau_{k-1, \ell}^{(\alpha,\beta)}= \tau_{k , \ell + 1}^{(\alpha,\beta )}\tau_{k-1, \ell}^{(\alpha,\beta-1)}-\tau_{k,\ell}^{(\alpha,\beta)}\tau_{k-1, \ell + 1}^{(\alpha,\beta-1)},
\zeta_{e = - 2 p k , n}= - \sum_{j = 0}^{\infty}\zeta_{e = - 2 p k , n - j}, \mathrm{a n d}\zeta_{e = 2 p k , n}= \sum_{j = 0}^{\infty}\zeta_{e = 2 p k , n + 1 + j}
d\Phi_a(X) = (dL_{a^2{x_1}})|_m(X_0) + a_0T + a_1(a^2|T|^2 + 1)I_1T, d\Phi_a(Y) = (dL_{a^2{x_1}})|_m(Y_0) + b_0T + b_1(a^2|T|^2 + 1)I_1T.
\int_{\mathbb{R}^{2N}}\frac{|w(x)-w(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy=\int_{E\times E}\frac{|w(x)-w(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy+2\int_{\Omega\times E^c}\frac{|w(x)|^p}{|x-y|^{N+s\,p}}\, dx\, dy,
\textbf{N}_{\mathbb{Q}(\alpha, \beta) / \mathbb{Q}}J(\alpha, \beta) = \textbf{N}_{\mathbb{Q}(\alpha, \beta) / \mathbb{Q}}\left(\frac{1}{\overline{\alpha}}+ p K(\alpha, \beta)\right) \equiv \frac{1}{\overline{\alpha}}\pmod{p}.
i A^{+}= \left[ \begin{array}{l l}{< \psi_{1}^{( 0 )}| d \psi_{1}^{( 0 )}>}&{< \psi_{2}^{( 0 )}| d \psi_{1}^{( 0 )}>}\\{< \psi_{1}^{( 0 )}| d \psi_{2}^{( 0 )}>}&{< \psi_{2}^{( 0 )}| d \psi_{2}^{( 0 )}>}\\ \end{array}\right] = i A_{a}d x^{a},
\epsilon_{i}^{a}( \theta ) \approx r \, i \, m_{a}\cosh \theta - \sum_{j = 1}^{k - 1}\left( \delta_{i j}- 2 \left( A_{i j}^{s u ( k )}\right)^{- 1}\right) L_{j}^{a}( \theta ) - \sum_{j = 1}^{k - 1}\left( A_{i j}^{s u ( k )}\right)^{- 1}L_{j}^{b}( \theta - \sigma ) \; .
Z_{\scriptscriptstyle M C S}= \frac{1}{N_{\scriptscriptstyle M C S}}\int_{\Omega^{\scriptscriptstyle 1}( M )}\mathcal{D}A \hspace{4 p t}e^{- i \bigl \langle A , \hspace{4 p t}\left( \ast d_{\scriptscriptstyle 1}+ ( \ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\right) A \bigr \rangle}
I_{\epsilon}( z ) = \sum_{n = 0}^{\infty}f^{( n )}( 0 ) \, \frac{z^{n + 1}}{( n + 1 ) !}\Gamma \left[ 1 + ( n + 1 ) \epsilon \right] \, \cos \left[ ( n + 1 ) \epsilon \frac{\pi}{2}\right] .
\sum_{m=0}^{\infty}\frac{\lambda_{u}^m\cdot m^{d_u}}{\lambda^m}\frac{1}{\lambda_{u}^m\cdot m^{d_u}}M^m\vec{u}=\sum_{m=0}^{\infty}\left(\frac{\lambda_u}{\lambda}\right)^m\cdot m^{d_u}\frac{1}{\lambda_{u}^m\cdot m^{d_u}}M^m\vec{u}
\begin{array}{r c l}{{Y^{k}}}&{{=}}&{{\alpha^{- \frac{k}{2}}S ( A ) w^{- k}A w^{k}}}\\{{}}&{{=}}&{{\alpha^{- \frac{k}{2}}< ( w^{- k})_{( 1 )}, A > ( w^{- k})_{( 2 )}w^{k},}}\\ \end{array}
\hat{{F}}_{s+1}\left( z\right) =-{\dfrac{1}{2f^{1/2}\left( z\right)}}\frac{d\hat{{F}}_{s}\left( z\right)}{dz}-{\dfrac{1}{2}}\sum\limits_{j=1}^{s-1}{\hat{{F}}_{j}\left( z\right) \hat{{F}}_{s-j}\left( z\right)}\ \left({s=2,3,4\cdots}\right) .
k_{1}\cdot(PA)^{t}+k_{2}\cdot(PB)^{t}+k_{3}\cdot(PC)^{t}\ge2^{t}\sqrt{k_{1}k_{2}k_{3}}(\frac{(PD)^{t}}{\sqrt{k_{1}}}+\frac{(PE)^{t}}{\sqrt{k_{2}}}+\frac{(PF)^{t}}{\sqrt{k_{3}}}),
\phi(e_\lambda)x\phi(e_\lambda)x_{e_\lambda,g_i}=x_{e_\lambda,g_i}\beta_{g_i}(x)\alpha_{g_i}(\phi(\alpha_{k_0}^{-1}(e_\lambda^2 ))),\ \ \ \x\in Q,\ \lambda\geq\lambda_0,\i=1,2.