harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A'_2:= - s_+^2 \sup_{\substack{\psi \in \mathcal V_{01}\|\psi\|_{\ell^2(\square)}= 1}}\sup_{\substack{\varphi \in \mathcal V_0^\perp \|\varphi\|_{\ell^2(\square)}= 1}}\frac{\left| \left\langle \psi, V^\square \varphi \right\rangle \right|^2}{\left\langle H^\square_0(\theta) \varphi, \varphi \right\rangle}\leq 0.
g_{\mu \nu}^{( 6 )}= \left( \begin{array}{c c}{g_{\alpha \beta}^{( 5 )}+ e^{\psi_{1}}A_{\alpha}A_{\beta}}&{e^{\psi_{1}}A_{\alpha}}\\{e^{\psi_{1}}A_{\beta}}&{e^{\psi_{1}}}\\ \end{array}\right) ,
\log Z_{\sigma}(t) = \int_0^{\infty}\bigl(\mathrm{e}^{\mathrm{i}tx}-1 \bigr)N_{\sigma}(\mathrm{d}x), N_{\sigma}(\mathrm{d}x):=\sum_{p}\sum_{r=1}^{\infty}\frac{p^{-r\sigma}}{r}\delta_{r\log p}(\mathrm{d}x).
[1-\frac{n-d}{n2^{d-1}}]\mathbb{E}[N_c(d,n)] \le\log n+\frac{d}{n}\mathbb{E}[N_c(d-1,n)] &\quad+\frac{n-d}{n2^d}\sum_{j=1}^{d-1}\binom{d}{j}\big{(}\mathbb{E}[N_{c}(j,n)]+\mathbb{E}[N_{c}(d-j,n)]\big{)}&\le \log n+\frac{d}{n}2(d-2)\log n+\frac{n-d}{n2^d}\sum_{j=1}^{d-1}\binom{d}{j}2(d-2)\log n,
y \in\partial V(u-\sum_{t=0}^{T-1}x_t\Delta s_{t+1}-\bar c\cdot\bar x+S_0(\bar x)), p_t+y\Delta s_{t+1}\in N_{D_t}(x_t)t=0,\dots,T, p_{-1}+y\bar c \in\partial(yS_0)(\bar x)
K_{i \Theta_{l}}( z ) \stackrel{( z \rightarrow 0 )}{\sim}- \sqrt{\frac{\pi}{\Theta_{l}\sinh \left( \pi \Theta_{l}\right)}}\sin \left[ \Theta_{l}\ln \left( \frac{z}{2}\right) - \delta_{\Theta_{l}}\right] \left[ 1 + O \left( z^{2}\right) \right] ,
f(\hat{x}_k)-f(x^*)\leq \frac{C_k}{S_k}\leq \frac{\frac{1}{2}{\rm Diam}(Q)^2\sum_{i=0}^kL(x_i)\frac{\lambda_i^2}{S_i}}{S_k}+\frac{\sum_{i=0}^kS_i\delta(x_i,\hat{x}_i)}{S_k}
\begin{aligned}\sum_{j=2}^{k}\left\{m_{l}\hat{\theta}_{j}\right\}=\sum_{j=2}^{k_1}\left\{m_{l}\hat{\theta}_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{m_{l}\hat{\theta}_{j}\right\}&=\sum_{j=2}^{k_1}\left\{p_{j}\{m_{l}\theta\}+\xi_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{p_{j}\{m_{l}\theta\}\right\}, \end{aligned}
v(t) = \int^t_0 e^{(t-s)A}f(s)ds + \int^t_0\Big(\int^{t-s}_0 \alpha e^{(t-s-\tau)A}A_1G(\tau-r)d\tau\Big)f(s)ds &\,\,\,\,\,\, + \int^t_0\Big(\int^{t-s}_0 e^{(t-s-\tau)A}\int^0_{-r}\beta(\theta)A_2G(\tau+\theta)d\theta d\tau\Big)f(s)ds &=: I_1 +I_2+I_3\in Z,\hskip 20pt t\in [0, T].
\Sigma^{+}=\{e_{1}\pm e_{2}, e_{1}, e_{2}, 2e_{1}, 2e_{2}\},\W^{+}=\{e_{1}, e_{2}\}, \Pi =\{\alpha_{1}=e_{1}-e_{2}, \alpha_{2}= e_{2}\}, tilde{\alpha}=\alpha_{1}+\alpha_{2}=e_{1}
[0,1]^{|V(F)|-||S^k(F)||}\ni(z_{v_1},\cdots,z_{v_p})\in\Delta_F\setminus S^k(F) :\Longleftrightarrow \left(\{v_1,\cdots,v_p\}=V(F)\setminus \{v'\in s_v|s_v \in S^k(F)\}~\wedge~(v_i\preceq v_j \Leftrightarrow z_{v_j}\leq z_{v_i})\right)
[X, [Y, Z]] + (-1)^{\langle \deg(X),\, \deg(Y) \rangle + \langle \deg(X),\, \deg(Z) \rangle}[Y, [Z, X]] + (-1)^{\langle \deg(X),\, \deg(Z) \rangle + \langle \deg(Y),\, \deg(Z) \rangle}[Z, [X, Y]] = 0\;.
\int_{|x|\le R}\Big( \frac{\Phi(|x|)}{|x|}\Big)^2 \, dx = 2\pi \int_0^R \frac{r^2}{4}\Big(\frac{2\Phi(r)}{r^2}\Big)^2 rdr \le \pi \int_0^R \int_0^r B(s)^2 s ds\, rdr = \frac{\pi}{2}\int_0^R B(s)^2 (R^2-s^2) sds <\infty
f_1(g_1+g_2) = f_1 \mu_B (g_1 \otimes g_2) \Delta_A = \mu_C (f_1 \otimes f_1)(g_1 \otimes g_2) \Delta_A = \mu_C (f_1 g_1 \otimes f_1 g_2) \Delta_A = f_1 g_1 + f_1 g_2,
f_{t_0}(z) = \sum_{n=0}^\infty \xi_n z^n = z^0+ \xi_1 z^1 + \xi_2 z^2 + \ldots + z^{2q}+ \xi_1z^{2q+1}+ \xi_2z^{2q+2}+\ldots =\frac{\sum_{n=0}^{2q-1}\xi_n z^n}{1-z^{2q}}
\left|\int_{0}^{T}\widetilde{V}_{N,\epsilon}^{\mathbf{g}}(t,\,x(t))dt\right|<\delta\,\,\,\,\,\,\left|\int_{0}^{T}V_{N,\epsilon}^{(c)}(t,\,x(t))dt\right|<\delta\,\,\,c=1,\,2,\,\cdots,\,m
e_4(k,N) ={8N^4\over C_{(1^k;0)}}\left[{1\over 2 C_{(1;0)}}+ \sum_{i=1}^4{1 \over C_{(1^k;0)}- 3 C_{(1;0)}- C_{r_i}}{d_{r_i}\over N d_{(1^k;0)}}\right].
a_{m+1}^{-\ell}\geq a_m^{-\ell}(1-\frac{r_{m+1}}{r}a_m^\ell)^{-\ell}\geq a_m^{-\ell}(1+\frac{r_{m+1}}{r}a_m^\ell)=a_m^{-\ell}+\frac{r_{m+1}}r.
0 = f_v(u,0) = \tilde f_v(u,0) = \xi_v(u,0)f_\xi\bigl(\xi(u,0),\eta(u,0)\bigr) + \eta_v(u,0)f_\eta\bigl(\xi(u,0),\eta(u,0)\bigr) =\xi_v(u,0)f_\xi\bigl(\xi(u,0),0\bigr).
-\int_{B}\psi^{2}\tilde{u}^{-1}\partial_{t}(g_{1-\alpha,m}*\tilde{u})dx + \frac{c_{2}}{r^{2\beta}}\int_{B}(w-W)^{2}\psi^{2}dx \leq\frac{C_{1}\mu_{n}(B)}{r^{2\beta}}+ R_{m}(t),
\mathbb{P}(\phi(f)\leq 0)=\frac{1}{\sqrt{2\pi}\varsigma(f)}\int_{-\infty}^0ds\ \exp\left(-\frac{(s-\phi_0)^2}{2\varsigma^2(f)}\right)=N\left(-\frac{\phi_0}{\varsigma(f)},0,1\right),
d(\mathcal{U}((\overline{w}_{\i,\widehat{\eta}})_{\i=1}^{\infty}),\mathcal{U}((\overline{v}_{\i,\widehat{\eta}})_{\i=1}^{\infty}))\leq~cF_{\eta}((\overline{w}_{\i,\widehat{\eta}})_{\i=1}^{\infty},(\overline{v}_{\i,\widehat{\eta}})_{\i=1}^{\infty})),
P_{F\neq \hat{F}}(\rho)=1-\sqrt{\frac{\pi}{2}}\sum\limits_{k=0}^{N_{\mathrm{F}}-1}C^k_{N_{\mathrm{F}}-1}(-1)^{N_{\mathrm{F}}-k-1}\frac{\left[1-\Phi\left(\sqrt{\frac{N_{\mathrm{F}}-k}{2\rho (N_{\mathrm{F}}-k-1)}}\right) \right]}{\sqrt{\rho(N_{\mathrm{F}}-k-1)(N_{\mathrm{F}}-k)}}\exp\left( \frac{N_{\mathrm{F}}-k}{2\rho (N_{\mathrm{F}}-k-1)}\right),
[ \chi_{N}Z + \bar{\chi}_{N}Z^{- 1}]_{\alpha_{1}\alpha_{2}}^{\beta_{1}\beta_{2}}= z_{\alpha_{1}\alpha_{2}}\, \delta_{\alpha_{1}}^{\beta_{1}}\delta_{\alpha_{2}}^{\beta_{2}}\quad , \quad [ \chi_{N}Z^{- 1}+ \bar{\chi}_{N}Z ]_{\alpha_{1}\alpha_{2}}^{\beta_{1}\beta_{2}}= z_{\alpha_{1}\alpha_{2}}^{- 1}\, \delta_{\alpha_{1}}^{\beta_{1}}\delta_{\alpha_{2}}^{\beta_{2}}\, , \nonumber
\begin{aligned}D_N(x_1,\ldots,x_N)\:\:\: = \:\:\: \sup_{\mathcal{A}\in\mathcal{B}}\left|\frac{1}{N}\sum_{n=1}^N\mathbf{1}_{\mathcal{A}}(x_n)-\lambda(\mathcal{A})\right|, D^*_N(x_1,\ldots,x_N)\:\:\: =\:\:\: \sup_{\mathcal{A}\in\mathcal{B}^*}\left|\frac{1}{N}\sum_{n=1}^N\mathbf{1}_{\mathcal{A}}(x_n)-\lambda(\mathcal{A})\right|, \end{aligned}
f(x_1,x_1,x_2)=0, f(x_1,x_2,x_3)+f(x_2,x_3,x_1)+f(x_3,x_1,x_2)=0, f(x_1,x_2,[y_1,y_2,y_3])+D(x_1,x_2)f(y_1,y_2,y_3) =f([x_1,x_2,y_1],y_2,y_3)+f(y_1,[x_1,x_2,y_2],y_3)+f(y_1,y_2,[x_1,x_2,y_3]) +\theta(y_2,y_3)f(x_1,x_2,y_1)-\theta(y_1,y_3)f(x_1,x_2,y_2)+D(y_1,y_2)f(x_1,x_2,y_3).
\pi_2(x)= 2\prod_{p\ge 3}\frac{\left(1-\frac{2}{p}\right)}{\left(1-\frac{1}{p}\right)^2}\left\{\frac{x}{(\log x)^2}+\frac{2x}{(\log x)^3}\right\}+O\left(\frac{x}{(\log x)^4}\right),
\begin{array}{l l l}{{{}[ J_{1}, J_{2}] = i J_{3}\, ,}}&{{{}[ J_{2}, J_{3}] = i J_{1}\, ,}}&{{{}[ J_{3}, J_{1}] = i J_{2}\, ,}}\\{{{}}}&{{{}}}&{{{}}}\\{{{}[ K_{1}, K_{2}] = - i K_{3}\, ,}}&{{{}[ K_{2}, K_{3}] = i K_{1}\, ,}}&{{{}[ K_{3}, K_{1}] = i K_{2}\, ,}}\\ \end{array}
\log L(1,{\rm sym}^m f) =\sum_{\substack{p\leq z p\nmid N}}\sum_{0\leq j\leq m}\log\bigg(1-\frac{\alpha_f(p)^{m-2j}}{p}\bigg)^{-1}& +\sum_{\substack{p\leq z p\mid N}}\log\bigg(1-\frac{\lambda_f(p)^{m}}{p}\bigg)^{-1}+O\bigg(\frac{1}{\log_2(kN)}\bigg),
X(g_{1}, g_{2})=\prod_{(a,b)\in I(g_{1}^{-1})\backslash I((g_{1}g_{2})^{-1})}X_{\{a,\,b\}}\left(=\prod_{(a,b)\in I(g_{1})\cap I(g_{2}^{-1})}X_{\{g_{1}(a),\,g_{1}(b)\}}\right).
d^\ast\alpha_1 =0, d^\ast\beta_2 -(\lambda+4)\alpha_1=0, &d\alpha_1 + d^\ast\alpha_3 - (\lambda+2)\beta_2 =0, d\beta_2 + d^\ast\beta_4 - (\lambda+2)\alpha_3 =0, &d\alpha_3 - (\lambda+4)\beta_4 =0, d\beta_4=0.
\Phi^{r}(z^{n}t^{k})|_{t=z}= \frac{z^{n + (2^{r+1}-1)k}F_{2^{r+1}-1}(-z)^{2n-1}}{F_{2^{r+1}}(-z)^{2n+2k-1}}= \frac{1 - u^{2^{r+1}}}{(1-u^{2^{r+1}-1})(1+u)}\Big(\frac{u (1 - u^{2^{r+1}-1})^{2}}{(1 - u^{2^{r+1}})^{2}}\Big)^{n}\Big(\frac{u^{2^{r+1}-1}(1-u)^{2}}{(1-u^{2^{r+1}})^{2}}\Big)^{k},
g(x_{L,\epsilon}(t)) =x_1(\epsilon)\exp\left(-C_1(\epsilon)\int_0^t \frac{1}{\sigma(s)}\,ds\right), g(x_{U,\epsilon}(t)) =x_2(\epsilon)\exp\left(-C_2(\epsilon)\int_0^t \frac{1}{\sigma(s)}\,ds\right).
x_i =y_i, -N\leq i\leq N-1, x_{i}\in C_1, N\leq i\leq N+a-1, x_{i}\not\in C_1, N+a\leq i\leq N+a+b-1, x_{i}=z_{i-2N-a-b}, N+a+b\leq i\leq 3N+a+b-1.
H_{j}= \begin{pmatrix}1 0 i\frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}r^{\frac{\sigma_{3}}{4}}G_{j}(-rx_{j};r\vec{x},\vec{s}) = R(-x_{j})E_{-x_{j}}(-x_{j})\begin{pmatrix}\Psi_{j,11}\Psi_{j,12}\Psi_{j,21}\Psi_{j,22}\end{pmatrix}.
\frac{d}{dt}\frac{u(t)}{v(t)}=\frac{u^{\prime}(t)v(t)-v^{\prime}(t)u(t)}{v^{2}(t)}=\frac{u^{\prime}(t)v(t)-\beta(t)v(t)u(t)}{v^{2}(t)}\le0,t\in I^{\circ},
\prod_{\nu\in F(uv)}uv\varphi_\nu = \prod_{\substack{\nu>0 uv\nu<0}}uv\varphi_\nu = \prod_{\substack{\nu>0 v\nu>0 uv\nu<0}}uv\varphi_\nu \cdot \prod_{\substack{\nu>0 v\nu<0 uv\nu<0}}uv\varphi_\nu,
{\bf E}\Bigl[M_\mu(0,1)^n\Bigr] = \sum\limits_{k=0}^n \binom{n}{k}{\bf E}\Bigl[M_\mu\bigl(0,\frac{1}{2}\bigr)^{k}\,M_\mu\bigl(\frac{1}{2}, 1\bigr)^{n-k}\Bigr]
\mathbb{P}(\xi^{(n)}([y,y+G_n(x)/S(I)])=0)-D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2) = \mathbb{P}(\xi^{(n)}([y,y+\delta_n/\rho_{sc}(y)])=0)-D_n(\pi\delta_n) = \mathbb{P}^{GUE(n)}(\lambda_i\not\in [y,y+\delta_n/\rho_{sc}(y)],1\leq i\leq n) &-\mathbb{P}^{CUE(n)}(\theta_i\not\in [0,2\pi\delta_n],1\leq i\leq n)\leq O((n\ln n)^{-1}),
K_{N}(x, y) = \int_{c-i\infty}^{c + i\infty}\frac{ds}{2\pi i}\oint_{\Sigma}\frac{dt}{2 \pi i}\frac{x^{t}y^{-s-1}}{s-t}\frac{\Gamma(t)}{\Gamma(s)}\left( \frac{\Gamma(s+N)}{\Gamma(t+N)}\right)^{M+1},
{\left.{\frac{{\partial U}}{{\partial \xi}}}\right|_{nml}}= \sum\limits_{i,j,k = 0}^N{{{{U}}_{ijk}}{\ell'_i}({\xi_n}){\ell_j}({\eta_m}){\ell_k}({\zeta_l})}= \sum\limits_{i = 0}^N{{U_{iml}}{{\ell '}_i}({\xi_n})}\equiv \sum\limits_{i = 0}^N{{U_{iml}}{\mathcal{D}_{ni}}},
\vartheta \colon G \cong \prod_{i=1}^\infty \mathbb{Z}/p^i\mathbb{Z}\longrightarrow G \cong \prod_{i=1}^\infty \mathbb{Z}/p^i\mathbb{Z}, (x_i + p^i\mathbb{Z})_i \mapsto (x_i-x_{i+1}+ p^i\mathbb{Z})_i
=\left(1+O\left(\frac{2k\ell^2\log n}{n}\right)\right)(t!)^{r-1}\left((t-1)!\right)^{\ell-r}e^{-2i+O\left(\frac{1}{k^3\left(\log n\right)^4}\right)}\leq \left(1+O\left(\frac{1}{k^3\left(\log n\right)^4}\right)\right)(t!)^{r-1}\left((t-1)!\right)^{\ell-r}e^{-2i}.
\hat{g}_m(\zeta_*) = -\frac{1}{4}\log |q| \left(1-\frac{2}{m^2-1}\cdot \frac{m^2+m}{2}\right) +\frac{2}{m^2-1}\cdot \frac{m^2-2m-3}{16}\log|q| &=-\frac{1}{4}\log|q| \left(\frac{-1-m}{m^2-1}\right) + \frac{1}{8}\log|q| \cdot \frac{m^2-2m-3}{m^2-1}& = \frac{1}{8}\log|q| \left(\frac{2m+2+m^2-2m-3}{m^2-1}\right) = \frac{1}{8}\log|q|\ .
k_{\alpha\beta}(x):=\frac{2\mu}{3\mu+\lambda}\frac{\delta_{\alpha\beta}}{\omega_{n-1}}\frac{1}{|x|^{n}}+\frac{\mu+\lambda}{3\mu+\lambda}\frac{n}{\omega_{n-1}}\frac{x_\alpha x_\beta}{|x|^{n+2}},x\in{\mathbb{R}}^{n}\setminus\{0\}.
\Theta \left[ \begin{array}{l l l}{{1 / 2}}&{{1 / 2}}&{{1 / 2}}\\{{1 / 2}}&{{1 / 2}}&{{0}}\\ \end{array}\right] \stackrel{D_{\alpha_{2}^{- 1}\alpha_{3}}}{\longrightarrow}\Theta \left[ \begin{array}{l l l}{{1 / 2}}&{{1 / 2}}&{{1 / 2}}\\{{1 / 2}}&{{0}}&{{1 / 2}}\\ \end{array}\right] ,
\lambda_i P |h_i|^2 + \mu_i = \sum_{k<i}\lambda_k (2^t-1)P|h_k|^2 + 1, &\beta_i P |h_i|^2 \ge (2^t-1)\left(P|h_i|^2\sum_{l=i+1}^N\beta_l+\sigma_n^2\right), &\beta_i\ge0,~\lambda_i\ge0, ~\mu_i\ge0, &\lambda_i\left((2^t-1)\left(P|h_i|^2\sum_{l=i+1}^N\beta_l+\sigma_n^2\right) - \beta_i P |h_i|^2\right)=0, &\mu_i\beta_i=0,
K=\frac{w_{yy}w_{zz}-w_{yz}^{2}}{w_{z}^{4}},H=\frac{w_{z}^{2}w_{yy}-2w_{y}w_{z}w_{yz}+\left( 1+w_{y}^{2}\right) w_{zz}}{2w_{z}^{3}},
u_{2j}(x_1,x_2)|_{|x|=R_0}&=(-1)^j \cos2j\theta+2\sum_{k=1}^\infty\sum_{l=0}^\infty \frac{(-\alpha)^k{{2l+2j-1}\choose{2j-1}}(-1)^l\cos (2l\theta)}{(kR_0)^{2l+2j}},
\mathcal{D}_{z}= \left( \begin{array}{c c}{\mathbf{a}+ \mathbf{b}\mathbf{x}}&{\mathbf{0}}\\{\mathbf{0}}&{\mathbf{a}^{\prime}+ \mathbf{b}^{\prime}\mathbf{x}^{\prime}}\\ \end{array}\right) \equiv \left( \begin{array}{c c}{D_{z}}&{\mathbf{0}}\\{\mathbf{0}}&{D_{z}^{\prime}}\\ \end{array}\right) ,
\widetilde{\nabla}_X(\gamma(\alpha)\phi) =\widetilde{\nabla}_X(\alpha_kR_k\phi)=\widehat{\nabla}_X(\alpha_kR_k\phi)+L(X)\gamma(\alpha)\phi &=(X\alpha_k)(R_k\phi)+\gamma(\alpha)\widehat{\nabla}_X\phi+L(X)\gamma(\alpha)\phi
T_{\alpha \beta}= - \left( \nabla_{\alpha}\phi \nabla_{\beta}\phi - \frac 1 2 ( ( \nabla \phi )^{2}+ m^{2}+ \xi \tilde{R}~ \phi^{2}) \gamma_{\alpha \beta}+ \xi ( \gamma_{\alpha \beta}\nabla^{2}- \nabla_{\alpha}\nabla_{\beta}) \phi^{2}\right) ~ ,
\theta_{s,a^2}^2=\bar{r}^2(s,a^2)+\beta \sum_{s'\in S}p^2(s'|s,a^2) u^{2*}(s')-u^{2*}(s), \ \forall \s\in S, a^2\in A^2(s), a^2\neq a_s^2.
{}~{\displaystyle \sum_{\{k_{i}\}, \atop i = 1 , \cdots , 6}}R ( u_{3}, u_{5}, u_{6})_{i_{3}, i_{5}, i_{6}}^{k_{3}, k_{5}, k_{6}}R ( u_{2}, u_{4}, u_{6})_{i_{2}i_{4}k_{6}}^{k_{2}k_{4}j_{6}}R ( u_{1}, u_{4}, u_{5})_{i_{1}k_{4}k_{5}}^{k_{1}j_{4}j_{5}}R ( u_{1}, u_{2}, u_{3})_{k_{1}k_{2}k_{3}}^{j_{1}j_{2}j_{3}}
(1-q)\left(\sum_{0\le i\le \frac{b-3}{3}}q^{6i+2}\binom{d-4i-1}{1}_q\binom{2d-8i-1}{1}_q + q^{6i}\binom{3d-12i+1}{1}_q\right)\leq (1-q)\sum F_{\lambda}(q),
\Pi^{*}({\vec{x}}) = \sum_{n_{1}, n_{2}, n_{3}= - N}^{+ N}\frac{1}{X^{3 / 2}}p_{n_{1}, n_{2}, n_{3}}^{*}e x p \left( - 2 \pi i ( \sum_{j = 1}^{3}n_{j}x_{j}) / X \right) ,
\frac{x}{2\pi i}\int_{b-i \tau}^{b+i \tau}R_{k, \beta}(z) \frac{(\alpha x^\delta)^{z}}{z(1-z+\delta z)}dz=R_{k, \beta}\left(\frac{1}{(1-\delta)}\right)y^{\frac{1}{(1-\delta)}}+E_2,
|s(n,k)| \le\frac{1}{k!}\sum_{l_{1}+\dots+l_{k}=n}\frac{n!}{l_{1}!\dots l_{k}!}|f^{(l_{1})}(0)|\dots|f^{(l_{k})}(0)| \leq\frac{1}{k!}\sum_{l_{1}+\dots+l_{k}=n}\frac{n!l_{1}!\dots l_{k}!}{l_{1}!\dots l_{k}!}F_{\varepsilon}^{k}\varepsilon^{-n}\leq\frac{n!}{k!}F_{\varepsilon}^{k}2^{n}\varepsilon^{-n}.
\,-\,\frac{1}{3!}\int d^3x_1 d^3x_2 d^3x_3{\mathcal W}^{(3)}_{\mu_1\mu_2\mu_3}(x_1,x_2,x_3) b_{\mu_1}(x_1) b_{\mu_2}(x_2) b_{\mu_3}(x_3) \,+\, \ldots
tr(AB)=\sum_{i=1}^{m}(AB)_{ii}=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}b_{ji}=\sum_{j=1}^{n}\sum_{i=1}^{m}b_{ji}a_{ij}=\sum_{j=1}^{n}(BA)_{jj}=tr(BA).
\gamma^{0}_{\alpha \beta}= \left(\begin{array}{cc}1 0 0 1\end{array}\right)\,, \quad \gamma^{1}_{\alpha \beta}=\left(\begin{array}{cc}0 1 1 0\end{array}\right)\, , \quad \gamma^2_{\alpha \beta}=\left(\begin{array}{cc}-1 0 0 1\end{array}\right) \,.
\begin{aligned}\|z_1 - z_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}+\|w_1 - w_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}&\leq C_3\big(\|h_1-h_2\|_{C^1([0,T_0])}+ \|g_1-g_2\|_{C^1([0,T_0])}+ \|z_{01}-z_{02}\|_{C^{2}([0,1])}+ \|w_{01}-w_{02}\|_{C^{2}([0,1])}\big) \end{aligned}
T^{\mu\nu}_0=-\left(\frac{g_{str}\alpha'}{2}\right)\prod_u\frac{\delta(\sqrt{\alpha'}k_u)}{\sqrt{\tau_2}}\frac{\eta^{\mu\nu}}{2\pi\alpha'}\left( \prod_o \frac{\delta(\sqrt{\alpha'}k_o)}{\sqrt{\tau_2}}X_{1,1}+2^{-(d+1)}X_{1,0}\right)
N^{k}(F)(x) = N^{0}(F)(x) + \frac{ik}{4\pi}\int_{D}F(y) dy + \frac{1}{4 \pi}\; \sum_{n \geq 1}\frac{(ik)^{n+1}}{(n+1)!}\; \int_{D}\; \left\Vert x - y \right\Vert^{n}F(y) \, dy .
f^\infty(x,A) := \lim_{\substack{t \to \infty}}\,\frac{f(x,tA)}{t}, f^\#(x,A) := \limsup_{\substack{t \to \infty}}\, \frac{f(x,tA)}{t}, f_\#(x,A) := \liminf_{\substack{t \to \infty}}\, \frac{f(x,tA)}{t}.
T_t^\alpha f(x) = \int_{\mathbb R_+^d}e^{-|z|^2t}\phi_z^\alpha(x)h_\alpha f(z) z^{2\alpha}dz =\int_{\mathbb R_+^d}\left(\int_{\mathbb R_+^d}e^{-|z|^2t}\phi_z^\alpha(x)\phi_z^\alpha(y)z^{2\alpha}dz\right)f(y)y^{2\alpha}dy =\int_{\mathbb R^d_+}\mathcal{W}_t^\alpha (x,y)f(y)y^{2\alpha}dy,\,\,\, x\in \mathbb{R}_+^d,
\textrm{Var}\Big[\big(1+\eta^\ell(0)\big)\Big(\psi^{n,\ell}(0)-\lambda_n- \frac{1}{(1+\rho_n)^2}\big(\eta^{n,\ell}(0)-\rho_n\big)\Big)\Big] = \mathcal O(n^{-2\delta})
s_{n}^{2}=\frac{n-2}{n-1}s_{n-1}^{2}+\frac{(x_{n}-\overline{x}_{n-1})^{2}}{n}=s_{n-1}^{2}+\frac{(x_{n}-\overline{x}_{n-1})^{2}}{n}-\frac{s_{n-1}^{2}}{n-1},n>1
\left( \prod_{i = 1}^{5}{\frac{\lambda_{i}/ 2}{\mathrm{s i n h}\lambda_{i}/ 2}}\right) \left( 2 \sum \mathrm{c o s h}\lambda_{i}- 2 \right) -{\frac{1}{8}}\prod_{i = 1}^{5}{\frac{\lambda_{i}}{\mathrm{t a n h}\lambda_{i}}}
\underset{l\rightarrow \infty}{\lim}J_{\mathrm{d},l}^{(\omega ,\mathcal{A})}(t)=\underset{l\rightarrow \infty}{\lim}\left( \left. \partial_{\eta}\mathbb{J}_{\mathrm{d}}^{(\omega ,\eta \mathbf{\bar{A}}_{l})}\left( t\right)\right\vert_{\eta =0}\right) =\left( \mathbf{\Xi}_{\mathrm{d}}\vec{w}\right) \int_{t_{0}}^{t}\mathcal{E}_{s}\mathrm{d}s\ .
\underbrace{\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)}(z-w)^{-\frac{\alpha\beta}{k}}= \boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)(z-w)^{-\frac{\alpha\beta}{k}}-\boldsymbol{:}\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)\boldsymbol{:}(z-w)^{-\frac{\alpha\beta}{k}}
\alpha(\mathcal{K}^\pm (n,k)) \leq \frac{2\binom{n}{k}}{1-\frac{2\binom{n-k}{k}}{-2\binom{n-k-1}{k-1}}}= \frac{2\binom{n}{k}}{1+\frac{n-k}{k}}= 2\binom{n-1}{k-1}= |\mathcal{S}|.
S_{\psi}^{( 2 )}= \sum_{n \leq p + 1}\int_{n \mathrm{\tiny - b r a n e}\subseteq \mathrm{\tiny D - b r a n e}}d y^{m_{1}}\cdots d y^{m_{n}}A_{m_{1}, \cdots , m_{n}}( y_{1}, \cdots , y_{p}) ,
(\begin{matrix}-1&\alpha^{-1}+\alpha^{6}x\\ \alpha^{3}+\alpha^{1}x&-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2})\end{matrix})(\begin{matrix}\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2}&\alpha^{-1}+\alpha^{6}x\\ \alpha^{3}+\alpha^{1}x&1\end{matrix})=(\begin{matrix}1&0\\ 0&1\end{matrix}),
\rho^2 \dot{\Xi}= -\frac{\partial}{\partial x}(\rho^2 p) + \frac{\partial}{\partial x}(\rho^2) p &=R_{a,b}(x,\tau,\Phi,\Psi) \Delta^{-2}+ \frac{\partial}{\partial x}(\rho^2) p - 4(2x+a^2+b^2-r_s^2) \Xi^2.
\sphericalangle\big(P(y,r_{x,i}),P_y\big) \leq \sphericalangle\big(P(y,r_{x,i}),P(y,r_{x,k})\big) + \sphericalangle\big(P(y,r_{x,k}), P_y\big) \leq \sum \limits_{l=0}^{k-i-1}\sphericalangle\big(P(y,r_{x,i+l}),P(y,r_{x,i+l+1})\big) + \varepsilon \leq \sum \limits_{l=0}^{\infty}\sphericalangle\big(P(y,r_{x,i+l}),P(y,r_{x,i+l+1})\big) + \varepsilon.
\varphi \big( g_2 g^{i}_1 \big) = \varphi (g_2) \varphi \big( g^{i}_1 \big) \mbox{and}\varphi \big( g_3 g^{i}_1 \big) = \varphi \big( g^{i}_1 \big) \varphi (g_3) \,,
C_{\theta}( \tau ) = \langle{\cal A}( \tau ){\cal B}\rangle \; - \; \langle{\cal A}\rangle \lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal B}}){\cal B}\rangle \stackrel{\tau \rightarrow \infty}{\longrightarrow}0 \; ,
Z^{\ast}( \theta_{2}) Z^{\ast}( \theta_{1}) \Omega = \frac{1}{\sqrt{2}}( \chi_{2 1}a^{\ast}( \theta_{2}) a^{\ast}( \theta_{1}) \Omega + \chi_{1 2}S ( \theta_{2}- \theta_{1}) a^{\ast}( \theta_{1}) a^{\ast}( \theta_{2}) \Omega )
\sum_{n=0}^{\infty}\frac{q^{2n^{2}}}{\left(q^{2},q^{2};q^{2}\right)_{n}}=\frac{1}{\left(q;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{\left(q;q\right)_{n}}A_{q}^{2}\left(q^{n}\right),
\mathbb{E}[\lambda_{-s}^{\dagger,2}(t)] =\overline{\gamma}_{0}\int_{-\infty}^{t}\gamma_{2}(t-u)H_{1}(u+s)du &=\overline{\gamma}_{0}\int_{-\infty}^{t+s}\gamma_{2}(t+s-u)H_{1}(u)du=\overline{\gamma}_{0}(\gamma_{2}\ast H_{1})(t+s).
{\cal{J}}_{1}= \frac{x^{2}}{\varrho}; \quad{\cal{J}}_{2}= n \varrho ; \quad{\cal{J}}_{3}= 2 \alpha x^{2}- v^{2}\varrho + \frac{x v}{2}\varrho_{z}; \quad \varrho = \left( 1 - 2 \alpha z^{2}\right) .
\begin{aligned}(A+BK^{(i)})^{\top}P^{(i+1)}+P^{(i+1)}(A+BK^{(i)})+(C+DK^{(i)})^{\top}P^{(i+1)}(C+DK^{(i)}) &~~~=-[K^{(i)\top}RK^{(i)}+S^{\top}K^{(i)}+K^{(i)\top}S+Q].\end{aligned}
\mathbb{A}^Ix= \mathbb{A}_{-1}x+\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x = \mathbb{A}_{-1}\big(x-R(\lambda,\mathbb{A}_{-1})\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x\big)+\lambda R(\lambda,\mathbb{A}_{-1})\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x\in X.
h_{X}(t) = \cos^2\left(\frac{\pi t}{2}\right) h_{X}\left(\frac{t}{4}\right)+\sin^2\left(\frac{\pi t}{2}\right)\frac{|1+ \overline{\rho}|^2}{2}h_{X}\left(\frac{t-1}{4}\right)+\sin^2\left(\frac{\pi t}{2}\right)\frac{|1- \overline{\rho}|^2}{2}h_{X}\left(\frac{t-3}{4}\right) .
\begin{aligned}\Vert \partial_x (v^1_p,g^1_p)\Vert_{L^2}^2 \leq C(L,\zeta)\epsilon^{-1/2-2\zeta}, \Vert \partial_y (v^1_p,g^1_p)\Vert_{L^2}^2 \leq C(L,\zeta)\epsilon^{-1/2-2\zeta}, \Vert \partial_{xx}(v^1_p,g^1_p)\Vert_{L^2}^2 \leq C(L)\epsilon^{-5/2}.\end{aligned}
\mathbb{P}\left\{\mathcal{M}_i^{1, j}\right\}\leq \sum_{t=0}^{T-1}\mathbb{P}\left\{f^{(i)}_t V^{(i)}_t \leq \bar{t}\right\}+\mathbb{P}\left\{g^{(i)}_t V^{(i)}_t > \bar{t}\right\},
\sup_{\{v\in \Phi_{-p}:\Vert v\Vert_{-p}\leq m_{T}\}}\sum_{\mathbf{j}\in \mathbb{N}_{0}^{d}}|D(b_{0}(v)[\phi_{\mathbf{j}}^{q_{1}}])[\eta ]|^{2}leq C_{3}\Vert \eta \Vert_{-q}^{2}\sum_{i=1}^{\ell}\sum_{\mathbf{j}\in \mathbb{N}_{0}^{d}}|\zeta_{i}[\phi_{\mathbf{j}}^{q_{1}}]|^{2}.
\psi([x,y]_{\mathfrak{g}}) = \pi([x,y]_{\mathfrak{g}}) = \pi(x\otimes y- \varepsilon(x,y) y\otimes x) = \pi(x\odot y- \varepsilon(x,y)y\odot x)= \pi(x)*\pi(y)-\varepsilon(x,y)\pi(y)*\pi(x) = \psi(x)*\psi(y)-\varepsilon(x,y)\psi(y)*\psi(x) =[\psi(x),\psi(y)]_{\mathfrak{g}},
{\cal Z}= \sum_{\alpha}\mbox{Tr}( q^{H^L_{\overline{\alpha V}}}~{\overline q}^{H^R_{\overline{\alpha V}}}e^{2\pi i \alpha s}\xi ({\overline{\alpha V}},{\bf n}) \eta({\overline{\alpha V}},{\cal N}_{\overline{\alpha V}},{\bf n}) )~.
\left|\times_{j=1}^{d}\overline{\overline K}_j\right|\ll_d \prod_{j=1}^{d}(c_6\,r_j)^{15r_j^2/2}\, m_j\le\prod_{j=1}^{d}(c_6\,r_j)^{15r_j^2/2}\, w_j ,
\Omega'_{\omega, m}= \{\omega\}\times [0,T] \times \cup_{k=1}^m Z_k, \mathcal{F}'_{\omega,m}= \{(\omega, A\times B), A\in \mathcal{B}([0,T]), B\in \mathcal{B}(\cup_{j=1}^m Z_k)\}, \mathbb{Q}_{\omega, m}(A \times B) = \frac{\mu(A \times B, \omega)}{\mu([0,T]\times \cup_{k=1}^m Z_k,\omega)},
a(n)-a(n+1) = \sqrt{\frac{\pi_{n+1}}{\pi_n}}- \sqrt{\frac{\pi_{n+2}}{\pi_{n+1}}}\ge \lambda_h \sqrt{\frac{\pi_{n+1}}{\pi_{n}}}, b(n+1)-b(n) = \sqrt{\frac{\pi_{n}}{\pi_{n+1}}}- \sqrt{\frac{\pi_{n-1}}{\pi_{n}}}\ge \lambda_h\sqrt{\frac{\pi_{n}}{\pi_{n+1}}}.
-{\beta f}= \ln{\{2 \sqrt 3 \exp ( -{\frac{\Delta}{4}}) [ 1 + \exp ( - 2{\Delta}) ] \}}+{\frac{1}{8{\pi}^{2}}}\int_{0}^{2 \pi}\d{\vartheta}\int_{0}^{2 \pi}\d{\varphi}\ln [ c^{2}+ s ( \cos{\vartheta}+ \cos{\varphi}) ] ,
\frac{\partial y}{\partial x}=\frac{\partial y}{\partial w_{2}}\frac{\partial w_{2}}{\partial w_{1}}\frac{\partial w_{1}}{\partial x}=\frac{\partial f(w_{2})}{\partial w_{2}}\frac{\partial g(w_{1})}{\partial w_{1}}\frac{\partial h(w_{0})}{\partial x}
\big\{f, g\big\}= \Big(\frac{\partial f}{\partial x_{i}}\frac{\partial g}{\partial p_{i}}- \frac{\partial g}{\partial x_{i}}\frac{\partial f}{\partial p_{i}}\Big) + \theta\Big( \frac{\partial f}{\partial x_{1}}\frac{\partial g}{\partial x_{2}}- \frac{\partial g}{\partial x_{1}}\frac{\partial f}{\partial x_{2}}\Big).
|Q_{-\nu,m}|^{\frac{1}{u}-\frac{1}{p}}\left(\sum_{k:Q_{0,k}\subset Q_{-\nu,m}}\!\!\!|f(e^{(k)})|^p\right)^{\frac{1}{p}}= \| \{2^{\nu d(\frac{1}{p}-\frac{1}{u})}f(e^{(k)})\}_k|\ell^{2^{d\nu}}_p(Q_{-\nu,m})\| \le \|f\|\, .
v_1 =\frac{p_1+p_2+1}{p_1+p_2-1},\v_2=\frac{1+2p_1+p_1^2-p_2^2}{1-2p_1+p_1^2-p_2^2}, v_3 =-\frac{2(3p_1+p_1^3+p_2+p_1^2p_2-p_1p_2^2-p_2^3)p_3}{(p_1-p_2-1)^2(-3p_1+3p_1^2-p_1^3+p_1^4+p_2+2p_1p_2+p_1^2p_2-p_2^2+p_1p_2^2-2p_1^2p_2^2-p_2^3+p_2^4)}
\Omega= k+g_1(z)-zg_1^\prime(z)+\dfrac{1}{2}\displaystyle \int^{r^2-s^2}_0g_6(\xi)d\xi>0,\;\forall z\in\mathbb{R},\;r\geq|s|; \Lambda= \Omega \phi_{zz}+(r^2-s^2)g_6(r^2-s^2)\phi_{zz}>0,\;\forall z\in\mathbb{R},\;r\geq|s|.
\frac{[\beta_{1}, \beta_{2},\beta_{3}]}{[ \beta_{23}, \beta_{13}, \beta_{12}]}= \pm\frac{\eta_1 \eta_2 \eta_3}{\eta_{23}\eta_{13}\eta_{12}}\cdot \frac{\vartheta[q_S](\tau)}{\vartheta[q_T](\tau)}.
[g(e^{i\vartheta})\frac{d}{d\vartheta}, f(e^{i\vartheta})\frac{d}{d\vartheta}]= \left((\frac{d}{d\vartheta}g(e^{i\vartheta}))f(e^{i\vartheta})-(\frac{d}{d\vartheta}f(e^{i\vartheta}))g(e^{i\vartheta})\right)\frac{d}{d\vartheta}.
\zeta_d = \int_0^1\Big((2\pi)^{d/2}\tilde{J}_{\frac{d}{2}-1}(2\pi s) - \sigma_{d-1}\Big)\,\frac{ds}{s}+ \int_1^{\infty}(2\pi)^{d/2}\tilde{J}_{\frac{d}{2}-1}(2\pi s)\,\frac{ds}{s}
\tau_{\mu,\mu}^{1}= (1-q^{-1})\dfrac{\mathbf{z}^C}{1 - q^{-1}\mathbf{z}^{n_{\alpha}\alpha^{\vee}}}\, , \tau_{s_{\alpha}(\mu) + \alpha, \mu}^{2}= q^{-1}D \mathbf{z}^{- \alpha^{\vee}}\dfrac{1 - \mathbf{z}^{n_{\alpha}\alpha^{\vee}}}{1 - q^{-1}\mathbf{z}^{n_{\alpha}\alpha^{\vee}}}\, .
d_q:C^q(U,A) \to \mathcal C^{q+1}(U,A) \varphi \mapsto((x_0,\ldots,x_{q+1})\mapsto\sum_{i=0}^{q+1}(-1)^i\varphi(x_0,\ldots,\hat x_i,\ldots,x_{q+1}).
F_{\mu \nu \lambda \rho}= \partial_{\mu}A_{\nu \lambda \rho}- \partial_{\nu}A_{\mu \lambda \rho}+ \partial_{\lambda}A_{\rho \mu \nu}- \partial_{\rho}A_{\lambda \mu \nu}\equiv \partial_{\left[ \mu \right.}A_{\left. \nu \lambda \rho \right]}.
\begin{aligned}{}A_{n,1}(z) =z^n(z-1)^{n+1}(z-a)^{-(n+1)}a^{(n+1)}, A_{n,2}(z) =z^{n+1}(z-1)^{n}(z-a)^{-(n+1)}(1-a)^{(n+1)}\,,\end{aligned}
\int d^{p}x \, \sqrt{g}\, C^{i}\Psi_{\omega}^{*}\partial_{i}\Psi_{\omega}= \Omega_{H}\int d^{p}x \, \sqrt{g}\, \Psi_{\omega}^{*}\partial_{\phi}\Psi_{\omega}\nonumber ={\cal O}( \omega^{2})