harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Lambda := \nu_{0}^{-1}\left( \int_{0}^{\|\nabla u_{0}\|_{L^{2}}^{2}}\varphi(\rho) d\rho + \| \partial_{t}u_{1}\|_{L^{2}}^{2}\right) , M := \sup_{\rho \in [0, \Lambda]}\varphi(\rho) , L := \sup_{\rho_{1}, \rho_{2}\in [0, \Lambda]}\frac{|\varphi(\rho_{2}) - \varphi(\rho_{1})|}{|\rho_{2}- \rho_{1}|},
\sum_{w,z \in X\setminus Y_1}\langle K(w,z)\xi(z) | \xi(w)\rangle = \sum_{w,z \in X\setminus Y_1}\langle V_w^*V_z \xi(z) | \xi(w)\rangle & = \sum_{w,z \in X\setminus Y_1}\langle V_z \xi(z) | V_w\xi(w)\rangle &= \Big|\Big|\sum_{w \in X\setminus Y_1}V_w \xi(w)\Big|\Big|^2
F_j(\omega, \vec{k}) \;=\; \int \frac{d\omega^\prime}{2 \pi}\int\frac{d\vec{k}^\prime}{(2 \pi)^3}\; \tilde{B}(\omega - \omega^\prime, \vec{k}-\vec{k}^\prime) \:C_{j-1}(\omega^\prime, \vec{k}^\prime) \:F_{j-1}(\omega^\prime, \vec{k}^\prime)
e^{\textstyle - J \alpha_{-}\Phi ( \sigma , \tau )}={\frac{1}{\sqrt{\varpi}}}\sum_{M = - J}^{J}( - 1 )^{J - M}e^{i h ( J - M )}\xi_{M}^{( J )}( x_{+}) \,{\overline{{\xi}}_{- M}^{( J )}}( x_{-}) \sqrt{\varpi}
\sum_{t=0}^T a_t{(b_t - b_{t+1})}=\sum_{t=0}^T a_t{[(b_{t}-b^*)- (b_{t+1}-b^*)]}&= a_0 (b_0-b^*) - a_T (b_{T+1}-b^*) + \sum_{t=0}^{T-1}\left(a_{t+1}- a_t \right)(b_{t+1}- b^*) &\leq a_0 (b_0- b^*),
h_{r}^{\varepsilon}=h-\frac{\varepsilon}{\tau}(\pi_{2\#}\gamma_{\overline{\Omega}}^{B_{r}(y_{1})})+\frac{\varepsilon}{\tau}(\mathcal{T}_{y_{1}}^{y_{2}}\circ\pi_{2\#}\gamma_{\overline{\Omega}}^{B_{r}(y_{1})}).
H_r(n,p) \ge (p/q)^n e^{-2/p}H(T_r(n,k))(1-o(1)) & \ge (p/q)^n \frac{1}{2}\left(\frac{k-r+1}{k}\right)^n (n-1)! (k-r+1)^{-\Theta(\sqrt{n})}& = E(n,p) \left(\frac{k-r+1}{qk}\right)^n (k-r+1)^{-\Theta(\sqrt{n})}& = E(n,p) \left(\frac{(k-r+1)!k^{r-1}}{k!}\right)^n (k-r+1)^{-\Theta(\sqrt{n})}& = E(n,p) \left(\frac{(k-r+1)!k^{r-1}}{k!}\right)^{n-o(n)}\;.
\P\left[ \Phi_k (G_N)>(1+\varepsilon)\lambda \right]\leq \frac{\log N}{\left( \binom{SN}{k}-\binom{S(N-1)}{k}\right)^{\gamma}}\leq \frac{\log N}{\left( \binom{SN}{k}-\binom{SN-1}{k}\right)^{\gamma}}= \frac{\log N}{\binom{SN-1}{k-1}^{\gamma}}.
\left( \begin{array}{c c}{1}&{k}\\{0}&{1}\\ \end{array}\right) \otimes \left( \begin{array}{c c c c}{1}&{p}&{q}&{r}\\{0}&{1}&{0}&{q}\\{0}&{0}&{1}&{p}\\{0}&{0}&{0}&{1}\\ \end{array}\right)
\begin{cases}(D_t^2-iA\partial_{\alpha})Z=-i \frac{d}{dt}z_j(t)=(v-\frac{\lambda_j i}{2\pi(\overline{z-z_j})})\Big |_{z=z_j}(I-\mathbb{H})(D_t\bar{Z}+\sum_{j=1}^N \frac{\lambda_j i}{2\pi(Z(\alpha,t)-z_j(t))})=0.\end{cases}
\det(\phi_h'(x,y))\mu(\phi_h(x,y)) =\det(\phi_h'(x,y))\rho(\psi_h(x))\ \nu(\chi_h(d(x),y+he(x)) + h c(x)) &= \det(\psi_h'(x))\rho(\psi_h(x))\cdot\det\Bigl(\partial_y\chi_h(d(x),y+he(x)) \Bigr)\ \nu\Bigl(\chi_h(d(x),y+he(x)) + h c(x)\Bigr).
\|u'\|_{d'}\leq \int_{0}^{t_0}\|u_t\|_{d'}\,dt &= \int_{0}^{t_0}(\mathrm{volume}(\{|u|>t\})^{1/d'}\,dt &\leq \frac{1}{c}\int_{0}^{t_0}(\mathrm{volume}(\{|u|=t\})\,dt &= \frac{1}{c}\|du'\|_1.
\left(f^{\otimes2}\Delta\right)\,(K\otimes H_k) = f^{\otimes2}\sum_{\substack{I+J=K i+j=k}}\left(I\otimes H_i \right) \otimes \left( J \otimes H_j\right) &= \sum_{\substack{I\sqcup J = K i+j=k}}\delta_{|I|,i}\, H_i \otimes \delta_{|J|,j}\, H_j &= \delta_{|K|,k}\sum_{i+j=k}H_i \otimes H_j \ = \left(\Delta f\right) (K \otimes H_k) .
P_g[X_1+\cdots+X_k=n] = \sum_{\pi_1+\cdots+\pi_k=n}P[X_1=\pi_1]\cdots P[X_k=\pi_k] = \sum_{\pi_1+\cdots+\pi_k=n}g(\pi_1)\cdots g(\pi_k) &= \frac{1}{\bar{F}^n}\sum_{\pi_1+\cdots+\pi_k=n}f(\pi_1)\cdots f(\pi_k) = \frac{1}{\bar{F}^n}\binom{k}{n}_f.
(a_0+t_1)(1-t\xi^2)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}=(a_0+t_1)\left(\frac{(1-t\xi_1^2)(u_1^2-t)}{(t\xi_1+u_1)^2}\right)\frac{(t\xi_1+u_1)^2}{u_1^2-t}\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}=(a_0+t_1)(1-t\xi_1^2)\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}
\int DV\exp (\frac 12\int d^4x\varepsilon^{\mu \nu \tau \lambda}{}^{+}W_{\mu \nu}^{\alpha \beta}{}^{+}G_{\tau \lambda}^{\sigma \rho}{}\epsilon_{\alpha \beta \sigma \rho}-\frac 12\int d^4x\varepsilon^{\mu\nu \tau \lambda}{}^{-}W_{\mu \nu}^{\alpha \beta}{}^{-}G_{\tau \lambda}^{\sigma \rho}{}\epsilon_{\alpha \beta \sigma \rho}).
\begin{array}{rcl}{\cal R}(u,c) = \displaystyle\frac{u-icP}{u+ic}begin{align*}4mm] = \displaystyle\frac{1}{u+ic}\left[ \begin{array}{cccc}u-ic 0 0 0 0 u -ic 0 0 -ic u 0 0 0 0 u-ic \end{array}\right]~,\end{array}
a_{+}(k)=a^*_{-}(k), b_{+}(k)=-b^*_{-}(k), k\in I_{R}, a_{+}(k)=b_{-}^{*}(k), b_{+}(k)=a^*_{-}(k), k\in I_{L}\backslash I_{R}, a_{+}(k)=a_{-}(k), b_{+}(k)=b_{-}(k), k\in \mathbb{R}\backslash(I_L\cup I_R).
\sum_{k=0}^{M}(-1)^k[4k-1]\frac{(q^{-1};q^2)_k^5}{(q^2;q^2)_k^5}q^{k^2+5k}\equiv\frac{\Omega_q(n)}{[2]^3[4]}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3}{(q^2;q^2)_k(q^6;q^2)_k^2}q^{2k}.
4i(\Delta n - 2i)d^2(dn-6) - 2id^3\Delta^2 n - 16id^4\Delta n \geq 4i\Delta d^3 n^2 - 8i^2 d^3 n - 24 i \Delta d^2 n - 2id^3\Delta^2 n - 16id^4\Delta n = 4i\Delta d^3 n^2 - 2i(dn)^3\,\left( 4i/n^2 + 12 \Delta/(dn^2) + \Delta^2/n^2 + 8d\Delta/n^2 \right) \geq 4i\Delta d^3 n^2 - 2i(dn)^3\,\left( 11 d\Delta/n^2 + \Delta^2/n^2 + 12/n\right)
\Phi(A^q) \le\Phi(I_n)^{1/2}\bigl(\Phi(I_n)^{-1/2}\Phi(A)\Phi(I_n)^{-1/2}\bigr)^q\Phi(I_n)^{1/2}&=\Phi(I_n)\,\#_q\,\Phi(A)\le(\|\Phi(I_n)\|_\infty I_n)\,\#_q\,\Phi(A) &=\|\Phi(I_n)\|_\infty^{1-q}\Phi(A)^q
[ H_{\beta_i}, H_{\beta_j}] = 0,[ H_{\beta_i}, X_{\pm \beta_j}] = \pm \beta_j(H_{\beta_i}) X_{\pm \beta_j}, [ X_{\beta_i}, X_{-\beta_j}] = \delta_{i,j}H_{\beta_i},[ X_{\pm \beta_i}, X_{\pm \beta_i}] = 0
F^{}(x_{i+1/2}) = \sum_{k=1}^{i}(xn)_k \Delta x_k \Bigg(\sum_{j=\alpha_{i,k}}^{I}(xn)_j \int_{\Lambda_j}\frac{\beta(x,x_k)}{x}dx+ (xn)_{\alpha_{i,k}-1}\int_{x_{i+1/2}-x_k}^{x_{\alpha_{i,k}-1/2}}\frac{\beta(x,x_k)}{x}dx \Bigg) &+{\cal O}(\Delta x^2).
\sigma_{cr}=\sigma_{y}-\frac{1}{E}{(\frac{\sigma_{y}}{2\pi})}^{2}{(\frac{l}{k})}^{2}=324\times10^{6}-\frac{1}{73.1\times10^{9}}{(\frac{324\times10^{6}}{2\pi})}^{2}{(\frac{l}{k})}^{2}
\alpha_1=q^{h_1+1/2}t_1, \alpha_2=q^{h_2+3/2}t_2, q^{\lambda}=q^{(h_1+h_2-l_1-l_2-k_1+k_2+1)/2}, &\beta_1=q^{(h_1+h_2+l_1-l_2-k_1+k_2)/2+1}t_1, \beta_2=q^{(h_1+h_2-l_1+l_2-k_1+k_2)/2+1}t_2.
g_1\cdot^\vee (g_2 \cdot^\vee g_3) = Z^{d(g_1) \, d(g_2 g_3)}(g_1 \cdot (g_2 \cdot^\vee g_3 )) & =Z^{d(g_1) d(g_2) + d(g_1) d(g_3) + d(g_2)d(g_3)}g_1 g_2 g_3 &= Z^{d(g_1 g_2) \, d(g_3)}(g_1 \cdot^\vee g_2) \cdot g_3= (g_1 \cdot^\vee g_2) \cdot^\vee g_3 .
Mf(u_n,2kv_n) =\frac{1}{4kv_n}\int_{u_n-2kv_n}^{u_n+2kv_n}f(t)dt=\frac{1}{4kv_n}\int_{x+(u_n-x)-2kv_n}^{x+(u_n-x)+2kv_n}f(t)dt &\geq \frac{1}{4kv_n}\int_{x}^{x+kv_n}f(t)dt+\frac{1}{4kv_n}\int_{x-kv_n}^{x}f(t)dt &\geq \frac{1}{4}(M_Rf(x,kv_n)+M_Lf(x,kv_n))
e(R_{33}) = \, \left(\frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- 2(q-1)^r+3^r \right) \big( (q^{2r}-q^{r+1})(q^4+q^3+q^2) + (q^2-q^{r+1})(q^3+q^2+q)\big) + \left(- \frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- (q-1)^r+3^r \right) \big((q^{2r}-q^{r+1}) (q^3+q^2+q)+ (q^2-q^{r+1})(q^4+q^3+q^2)\big) .
\mathcal{S}(h,k) = \frac{1}{2}\sum_{\substack{1\leq q<\infty (q,hk)=1}}W\left(\frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty (mn,q)=1}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right) \sum_{\substack{c,d\geq 1 cd=q d|mh\pm nk}}\phi(d)\mu(c).
b_{-+}(\xi,\eta) = - \frac{i}{4}\frac{|\xi| \widetilde{m}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|}- \frac{i}{4}\frac{|\xi|^{1/2}\widetilde{q}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|^{1/2}},
{f_{{X_{is}}}}({x_{is}}) = \frac{1}{{\Gamma ({m_i})}}{(\frac{{{m_i}}}{{\sigma_{is}^2}})^{{m_i}}}x_{is}^{{m_i}- 1}\exp ( - \frac{{{m_i}{x_{is}}}}{{\sigma_{is}^2}}) &{f_{{X_{ie}}}}({x_{ie}}) = \frac{1}{{\Gamma ({k_i})}}{(\frac{{{k_i}}}{{\sigma_{ie}^2}})^{{k_i}}}x_{ie}^{{k_i}- 1}\exp ( - \frac{{{k_i}{x_{ie}}}}{{\sigma_{ie}^2}}),
\begin{cases}(2r(2r-1)+(1-r+2rm)2r)[\lambda]+ (2r+(1-r+2rm))[\mu] =p[\lambda]+q[\mu] (2r(2r-1)+(1-3r-2rm)2r)[\lambda] +(2r+(1-3r-2rm))[\mu]=p[\lambda]+q'[\mu]. \end{cases}
\delta\varphi^{rs}= \left[\eta^\mu_i\eta^\nu_j\delta^{ir}\delta^{js}-(\eta^\mu_o\eta^\nu_j-\eta^\nu_o\eta^\mu_j)\frac{\delta^{ir}\delta^{js}p^j}{p^o+\sqrt{p^2}}\right]\delta\omega_{\mu\nu}.
\lim_{l\to\infty}\nu_{F_l}\left(\bigg\{\sum_{i=1}^{k_l}\left( G_{l,i}- \mathbb E_{F_l}(G_{l,i})\right)\leqslant t \hat{s}_{l}\bigg\}\right)=\mathcal{N}(t), \forall t\in\mathbb R.
\bar{u}_{\sigma_{1}}^{L F}( k_{1}) = \sum_{\sigma_{1}^{\prime}= \pm \frac{1}{2}}\beta_{\sigma_{1}\sigma_{1}^{\prime}}^{*}( k_{1}) \bar{u}_{\sigma_{1}^{\prime}}( k_{1}) , \quad \bar{u}_{\sigma_{2}}^{L F}( k_{2}) = \sum_{\sigma_{2}^{\prime}= \pm \frac{1}{2}}\beta_{\sigma_{2}\sigma_{2}^{\prime}}^{*}( k_{2}) \bar{u}_{\sigma_{2}^{\prime}}( k_{2}) ,
\rho_{\mathrm{m i n}}= \vert \beta \vert \biggl ( \frac{\beta}{1 + \beta}\biggr )^{1 / 4}\sqrt{\biggl ( 1 + \frac{K_{+}}{4 k_{\mathrm{c}}}\biggr )^{2}+ \frac{1 + \beta}{\beta}}\, , \quad \cos \Phi_{\mathrm{m i n}}\, = \, \frac{1 + K_{+}/ ( 4 k_{\mathrm{c}})}{\sqrt{\left[ 1 + K_{+}/ ( 4 k_{\mathrm{c}}) \right]^{2}+ ( 1 + \beta ) / \beta}}.
\sum_{i=0}^{j-1}\frac{1}{s_{i}}=\sum_{i=0}^{j-1}(\frac{1}{s_{i}-1}-\frac{1}{s_{i+1}-1})=\frac{1}{s_{0}-1}-\frac{1}{s_{j}-1}=1-\frac{1}{s_{j}-1}.
T ( z ) = - \frac{1}{2}: \! \partial X \partial X \! : + \, i \frac{\kappa}{2}\partial^{2}X , \qquad \tilde{T}( \bar{z}) = - \frac{1}{2}: \! \bar{\partial}X \bar{\partial}X \! : - \, i \frac{\kappa}{2}\bar{\partial}^{2}X ,
\prod_{\substack{\gcd(j_1,j_2,j_3,k)=1 j_1,j_2,j_3<k j_1,j_2,j_3 \geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3}z^k}\right)^{\frac{1}{k}}= \exp\left[ \sum_{k=1}^{\infty}\left( \frac{y}{1}+ \frac{y^2}{2}+ \cdots + \frac{y^k}{k}\right)^{3}\frac{z^{k+1}}{(k+1)^2}\right].
{\cal L}_{m}( g ) = \frac{1}{2^{m}}\sqrt{- g}\delta_{c_{1}d_{1}\cdots c_{m}d_{m}}^{a_{1}b_{1}\cdots a_{m}b_{m}}R_{a_{1}b_{1}}{}^{c_{1}d_{1}}\cdots R_{a_{m}b_{m}}{}^{c_{m}d_{m}}
\psi_{1}= \int d^{D}x \left( - \mu_{3 a}^{( \varphi )}\partial^{\alpha}\mu_{2 \alpha}^{a}+ \left( \partial^{\alpha}\mu_{3 \alpha}^{a}\right) \mu_{2 a}^{( \varphi )}+ \left( \partial^{\alpha}A_{\alpha}^{a}\right) \nu_{1 a}^{( \varphi )}- \left( \partial^{\alpha}\varphi_{a}\right) \nu_{1 \alpha}^{a}\right) ,
\mathfrak{n}_{\Phi}:= \left\{x \in M\mid \Phi(x^*x)< +\infty \right\}, \mathfrak{m}_{\Phi}:=(\mathfrak{n}_{\Phi})^*\mathfrak{n}_{\Phi}= \left\{\sum_{i=1}^nx_i^*y_i \mid n \geq 1, x_i, y_i\in \mathfrak{n}_{\Phi}1 \leq i \leq n \right\}.
\frac{1}{8 \pi}\int^{r}d^{3}x (K - K_{0}) =-\frac{\pi}{\kappa}\Biggl[ r - M + \frac{r(r-r_+)(r-r_-)}{R^2}- 2[(r-r_+)(r-r_-)]^{\frac{1}{2}}\Biggr] \ .
f \,\otimes\, g &\,=\, \sum\limits_{i,\, j \,=\, 1}^{\,\infty}\,\left<\,f \,\otimes\, g,\, e_{\,i}\,\otimes\, h_{\,j}\,|\, a_{\,2}\,\otimes\, b_{\,2}, \,\cdots,\, a_{\,n}\,\otimes\, b_{\,n}\,\right>\,\left(\,C_{1}\,\otimes\, C_{2}\,\right)\,\left(\,f_{\,i}\,\otimes\, g_{\,j}\,\right),
{1\over w_{n-1}+ (1-u_{n-1}) v_{n-1}}={1\over w_{n-2}(1-u_{n-2})+a_{n-1}v_{n-2}(1-u_{n-2})-a_{n-1}^2 v_{n-2}}&={1\over -v_{n-2}(a_{n-1}-a_{n-1}^+)(a_{n-1}-a_{n-1}^-)}={1\over- v_{n-2}(a_{n-1}^{+}-a_{n-1}^{-})}({1\over a_{n-1}-a_{n-1}^{+}}-{1\over a_{n-1}-a_{n-1}^{-}})
p_{1}=\frac{n}{\alpha}<\frac{ln}{\alpha},p_{2}<\frac{l^{\prime}n}{\alpha},v>q_{2},\quad\frac{v}{u}=\frac{q_{1}}{p_{1}},\quad\frac{1}{u}=\frac{1}{p_{1}}-\frac{\alpha}{ln}=\frac{\alpha}{l^{\prime}n}.
\sum_{n=1}^{\infty}\frac{1}{n^{2}+t^{2}}=\frac{\pi(te^{2\pi t}+t)-e^{2\pi t}+1}{2(t^{2}e^{2\pi t}-t^{2})}=-\frac{1}{2t^{2}}+\frac{\pi}{2t}coth(\pi t).
\times \prod_{j = r_{1}}^{r_{1}+ l_{2}- 1}d_{j}( \lambda_{1}) \prod_{j = l_{1}+ l_{2}}^{r_{1}+ l_{2}- 1}d_{j}( \lambda_{1}) \prod_{j = r_{2}- l_{2}}^{r_{2}- 1}d_{j}( \lambda_{2}) \prod_{j = r_{2}- l_{2}}^{r_{1}+ r_{2}- l - 1}d_{j}( \lambda_{2})
w_{\kappa}(x,t) := \begin{cases}\max( \hat{w}(x,t), w(x,t) ) (x,t) \in N_{7r/8,s}(0,t_0), begin{align*}1.2mm]w(x,t) (x,t) \notin N_{7r/8,s}(0,t_0).\end{cases}
M ( p ) = \frac{8}{\pi^{2}N}\left[ \int_{0}^{p}d q \; \left( \frac{1}{p}+ \frac{\xi \alpha}{2 p^{2}}\right) \frac{q^{2}M ( q )}{q^{2}+ M^{2}( q )}+ \int_{p}^{\alpha}d q \; \left( q + \frac{\xi \alpha}{2}\right) \frac{M ( q )}{q^{2}+ M^{2}( q )}\right] .
\sup_{t\in(0,T)}t^{\frac{\beta}{\alpha}}\|B(t)\|_{H^{\frac{5}{2}-\alpha+\beta}}\leq \C(T) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}+CC(T)\left(\sup_{t\in(0,T)}t^{\frac{\beta}{\alpha}}\|B(t)\|_{H^{\frac{5}{2}-\alpha+\beta}}\right)^2
{{f}_{I}}\left( I \right)\approx A \prod\limits_{\begin{smallmatrix}j=1 j\ne k\end{smallmatrix}}^{p+q}{\Gamma \left({{b}_{j}}-{{b}_{k}}\right)}{{I}^{p{{\gamma}_{2}}\min \left\{\frac{{{m}_{1}}}{q},\frac{{{m}_{2}}}{p}\right\}-1}}
g_{\gamma,n,a}\left(x\right)=f_{\gamma,n,a}\left(x\right)+V_{\gamma-1}\left(np^{-\gamma}+ap^{-\gamma}\right)\sum_{i=1}^{\infty}\dfrac{1}{V_{\gamma+i-1}\left(n_{i-1}p^{-\gamma-i+1}\right)}f_{\gamma+i,n_{i},\left[n_{i-1}p\right]}\left(x\right).
z(t_1) =1+(d_1+it_1)N_1^{-1/3}\,\,,\,\,t_1\in\mathbb{R}, \zeta(s_1) =(1-d_2N_1^{-1/3})e^{is_1N_1^{-1/3}}\,\,,\,\,s_1\in I_1=[-\pi N_1^{1/3},\pi N_1^{1/3}], w(t_2) =1+(d_3+it_2)N_2^{-1/3}\,\,,\,\,t_2\in\mathbb{R}, \omega(s_2) =(1-d_4N_2^{-1/3})e^{is_2N_2^{-1/3}}\,\,,\,\,s_2\in I_1=[-\pi N_2^{1/3},\pi N_2^{1/3}].
r_{n+1,n+1}(T) = \frac{\eta_0}{(T^2 -T_n^2)}\Big(K_n+ \frac{J_n}{M_n}\Big) & \le \frac{\eta_0 T_{n+1}^{\ast \,2}}{M_n(T_{n+1}^{\ast \,2}-T_n^2)}\Big(K_n+ \frac{J_n}{M_n}\Big) \frac{1}{T^2}=\frac{H_{3,n+1}}{T^2}\,.
\sup_{t \in [0,T]}\left\|m(t)\right\|_\infty \leq C(\|m_0\|_\infty,\{\|m^{\beta_k}b_k\|_{r/(r-1)}\}_{k=1}^N), \sup_{t \in [0,T]}\left\|\frac{1}{m(t)}\right\|_\infty \leq C(\|m_0^{-1}\|_\infty,\{\|m^{\beta_k}b_k\|_{r/(r-1)}\}_{k=1}^N).
\dot{\epsilon}_{\delta \sigma}= \epsilon_{\delta \sigma}\left[ 3{\frac{\dot{a}}{a}}( 1 + w ) + \left\{2{\frac{\dot{G}_{5}}{G_{5}}}-{\frac{6 + 8 \epsilon}{1 + \epsilon}}{\frac{\dot{c}}{c}}\right\}( 1 + \epsilon_{\delta \sigma}) +{\frac{9 C c^{6}}{3 2 \pi^{2}G_{5}^{2}\sigma a_{0}^{4}\varrho_{m}}}{\frac{\dot{c}}{c}}\right] .
\bar{g}_{a b}= \left( \begin{array}{c c}{\displaystyle{\cosh \chi + \cos \phi \sinh \chi}}&{\displaystyle{\sin \phi \sinh \chi}}\\{\displaystyle{\sin \phi \sinh \chi}}&{\displaystyle{\cosh \chi - \cos \phi \sinh \chi}}\\ \end{array}\right) \qquad .
\begin{array}{rl}= -\frac{2}{y}+1-\sqrt{1-z}+\{\strut \displaystyle z y \over\displaystyle 3(2-y)-{\strut \displaystyle (3+z)y^2 \over\displaystyle 5(2-y)-{\strut \displaystyle (8+z)y^2 \over\displaystyle 7(2-y)}}}& \ddots\end{array}
f_{r a d}( x - y ) = \left( \begin{array}{c c c c}{{0}}&{{{- 1}}}&{{0}}&{{\partial_{x}}}\\{{1}}&{{0}}&{{{- \partial_{x}}}}&{{0}}\\{{0}}&{{{- \partial_{x}}}}&{{0}}&{{0}}\\{{\partial_{x}}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \delta ( x - y )
\widetilde{N}_S=\{(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \widetilde{N}\; | \; \epsilon<\tau_2, \;\; \epsilon< \frac{\tau_2}{|\tau_1|^2+|\tau_2^2|},\;\; t^a>K, \;\; |\tau|t^a>K\}\,.
A= \{e^{i\pi/4}, e^{i5\pi/4}\}, P_1= \{1, -1\}, L(P_1) = \{i, -i\}, P_2= \{e^{3\pi i/4}\}, L(P_2) = \{\}, P_3=\{e^{7\pi i/4}\}, L(P_3) = \{\}.
\left(f_{n}\left(g_{s^{\prime},j,k}\right)\right)\left(T_{b_{j}}\left(v\right)\right)=T_{b_{k}}\left(v\right)\quad\quad\left(f_{n}\left(g_{s^{\prime\prime},k,i}\right)\right)\left(T_{b_{k}}\left(v\right)\right)=T_{b_{i}}\left(v\right).
\Big<\sum_{l \leq j \epsilon}T_a^{j,l}f, g\Big> \lesssim_{\epsilon}\sum_{j\ge 0}2^{(-j\rho+j\epsilon)n(1/r-1/s)}2^{jm}2^{-jn(\frac{1}{s}-\frac{1}{r})}2^{j(n(1-\rho)/2 + \varepsilon)(2/s-1)}\times\sum_{\substack{Q: \ell(Q)= 2^{\lfloor{-j\rho+j\epsilon+10}\rfloor}}}|Q|\left<f\right>_{r, Q}\left<g\right>_{s', Q}.
F(\vec{X}):= \dfrac{1}{2}\|\vec{Y - X}\|_F^2 + \lambda_0\sum_{i=1}^{m}\phi(\sigma_i(\vec{X});a_0) &+ \lambda_1\sum_{i=1}^{m}\sum_{j=1}^{n}\phi(\vec{X}_{i,j};a_1),
\lambda ({\mathrm{\boldmath \rho}},{\mathrm{\boldmath \dot{\rho}}}) = ({\mathrm{\boldmath \rho}}{\mathrm{\boldmath \dot{\rho}}}) \left( \frac{\mu{\mathrm{\boldmath \dot{\rho}}}^{2}}{8}\left( 1 - \frac{3 \mu}{m}\right) + \frac{1}{4}\left( 1 - \frac{\mu}{m}- 4 B \right) U ( \rho ) \right) ,
K_N(H_N,e_1)^{-1}= 2\begin{bmatrix}\frac12 T_{0,0}T_{1,0}\cdots T_{N-1,0}0 T_{1,1}T_{N-1,1}\vdots \ddots \vdots 0 \cdots T_{N-1,N-1}\end{bmatrix},
\begin{aligned}d(x,x') \leq d(x, xH_2 \cap x' H_1)+ d( xH_2 \cap x' H_1, x')\leq 2 d( x', xH_2 \cap x' H_1) & \leq 2d(x', x' H_1) \leq 2 d(1, \pi_{H_1}(x^{-1}x') ). \end{aligned}
G_{0,1}(x,T)=\sum_{m=0}^{\infty}h_{0,1,m}(x)T^{m}=\sum_{m=0}^{\infty}\frac{1}{2}((1+\sqrt{x})^{m}+(1-\sqrt{x})^{m})T^{m}=\frac{T-1}{(x-1)T^2+2T-1}, &G_{1,1}(x,T)=\sum_{m=0}^{\infty}h_{1,1,m}(x)T^{m}=\sum_{m=0}^{\infty}\frac{1}{2\sqrt{x}}((1+\sqrt{x})^{m}-(1-\sqrt{x})^{m})T^{m}=-\frac{T}{(x-1)T^2+2T-1}.
q^2\gamma\Omega_\psi(m) x^-_{n+1}- \Omega_\psi(m+1)x^-_n &=(q^2\gamma-1)\delta_{m,-n-1}+ \gamma x^-_{n+1}\Omega_\psi(m)-q^2x^-_n\Omega_\psi(m+1) q^2 \Omega_\psi(k+1) \Omega_\psi(l) -\Omega_\psi(l)\Omega_\psi(k+1) = \Omega_\psi(k)\Omega_\psi(l+1) - q^2\Omega_\psi(l+1)\Omega_\psi(k)
|c_{i,j}^{(k-2s,l-2r-1)}| \leq \dfrac{1}{2j}\left(|c_{i,j-1}^{(k-2s,l-2r)}| + |c_{i,j+1}^{(k-2s,l-2r)}|\right) \leq \dfrac{1}{2j}\frac{4V_{k,l}}{\pi^2}\Gamma_{0,0}[s](i)\left(\Gamma_{0,-1}[r](j)+\Gamma_{0,1}[r](j)\right)= \dfrac{4V_{k,l}}{\pi^2}\Gamma_{0,0}[s](i)\Gamma_{1,-1}[r](j).
\Phi_{\nu}(\phi) = N_{\nu}(\cos\phi)^{\alpha+\frac{1}{2}}(\sin\phi)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{\nu}(-\cos 2\phi), N_{\nu}= (-1)^{\nu}\left(\frac{2 (\alpha+\beta+1+2\nu) \nu! \Gamma(\alpha+\beta+1+\nu)}{\Gamma(\alpha+1+\nu) \Gamma(\beta+1+\nu)}\right)^{1/2},
0\leq \int_{D_r}\left(Q_{g}e^{4\Phi}+ \frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}\right)dV_{\tilde{g}}=\int_{S_r}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}+2\int_{S_r}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\omega_{\tilde{g}}
\iint\left| \frac{g(z)-g(w)}{z-w}\right|^{2}\,\alpha(dz)\,\alpha(dw) +2r\iint \Re\left(\frac{g(z)-g(w)}{z-w}\right)\,\alpha(dz)\,\alpha(dw)+r^{2}\le \int |g'|^{2}d\mu-2r\int \Re(ig'(z)\bar{z})\mu(dz)+r^{2}.
\hat{w}_{-}\left(z,\,\tau_{0}\right)\,=\,\hat{\psi}_{\lambda_{-}\left(\tau_{0}\right)\,k}\left(z\right)\,=\,\hat{\psi}_{\left(1-\beta^{\alpha-3}\right)\left(1+o\left(1\right)\right)}\left(z\right)<\,\hat{w}\left(z,\,\tau_{0}\right)
\Lambda_1({\rm diag}(a,b,a,b)) =\nu^{1/2}(a)\nu^{-1/2}(b)\sigma_1(ab)\omega_{\pi_1}(a), \Lambda_2({\rm diag}(a,b,a,b)) =\nu^{-1/2}(a)\nu^{1/2}(b)\sigma_1(ab)\omega_{\pi_1}(b),
F^{{\mu}{\nu}{\sigma}}=-G^{{\mu}{\nu}{\sigma}}+\frac{1}{({\partial}_{\lambda}a)({\partial}^{\lambda}a)}\left[{\partial}^{[{\mu}}a{\cal G}^{{\nu}{\sigma}]{\rho}}{\partial}_{\rho}a+\frac{1}{3!}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}a{\cal G}_{{\eta}{\delta}]{\theta}}{\partial}^{\theta}a\right].
\langle y\rangle_{T^{*3}}= ~\{y, T^{*3}y ,\ldots, T^{*3(j-2)}y\}~,~~~\langle y\rangle_{T^{*3}}=j-1, \langle z\rangle_{T^{*3}}= ~\{z, T^{*3}z ,\ldots, T^{*3(n-2j-1)}z\}~,~~~\langle z\rangle_{T^{*3}}=n-2j,
\sum_{i=1}^3 \mu_i \begin{pmatrix}v_1(t_i)\begin{pmatrix}\overline{v_1(t_i)}\overline{v_2(t_i)}\overline{v_3(t_i)}\end{pmatrix}v_2(t_i) \begin{pmatrix}0 \overline{v_2(t_i)}\overline{v_3(t_i)}\end{pmatrix}v_3(t_i) \begin{pmatrix}0 0 \overline{v_3(t_i)}\end{pmatrix}\end{pmatrix}
\left| \int_{\mathbb S^1}[D^m, B]J_x D^mB\, dx \right|\lesssim \|J_x\|_{L^\infty}\|D^mB\|^2_{L^2}+ \|B_x\|_{L^\infty}\|D^{m-1}J_x\|_{L^2}\|D^mB\|_{L^2}, \left| \int_{\mathbb S^1}[D^m, J]B_x D^mB\, dx \right|\lesssim \|B_x\|_{L^\infty}\|D^mJ\|_{L^2}\|D^mB\|_{L^2}+\|J_x\|_{L^\infty}\|D^{m-1}B_x\|_{L^2}\|D^mB\|_{L^2}.
\Phi_{\mathbf h_p}(\nu \beta_1\nu') = \frac{\chi_{\psi}(p)\underline{\chi}_p(\gamma) p^{3/2}}{p-1}\cdot \mathcal G\left(\frac{-\gamma}{p}\right)\left(\mathcal G\left(\frac{\gamma^{-1}}{p}\right) - \frac{1}{p}\right).
\mathcal{T}_K'(X^*;-X^*) =\lim_{\eta\searrow0}\frac{\mathcal{T}_K(X^*-\eta X^*)-\mathcal{T}_K(X^*)}{\eta}&=\lim_{\eta\searrow0}\frac{(1-\eta)\mathcal{T}_K(X^*)-\mathcal{T}_K(X^*)}{\eta}&=\lim_{\eta\searrow0}\frac{-\eta\mathcal{T}_K(X^*)}{\eta}&=-\mathcal{T}_K(X^*).
\epsilon^{(1)}=e^{{g\eta t\over\sqrt{2}}\epsilon_{ij}}{\cal H}^{-1/8}\left[\sqrt{f^{1/2}+1}-\sqrt{f^{1/2}-1}\gamma_{\overline{r}}\right]e^{{\eta\over2}\gamma_{\overline{0\theta r}}\epsilon^{ij}}e^{-{1\over2}\gamma_{\overline{\phi\theta}}}(1-\eta\gamma_{\overline{0}}\epsilon^{ij})\epsilon_0^{(1)}.
D_s^{\ell}\alpha_s(a) =(-i\partial_s)^{\ell}\sum_me^{is\cdot m}a_m\prod_{j=1}^nU_j^{m_j}&=\sum_m m^{\ell}e^{is\cdot m}a_m\prod_{j=1}^nU_j^{m_j}&=\sum_m\delta^{\ell}e^{is\cdot m}a_m\prod_{j=1}^nU_j^{m_j}&=\delta^{\ell}\alpha_s(a)
a(x,y)\frac{\partial^{2}u}{\partial x^{2}}+2b(x,y)\frac{\partial^{2}u}{\partial x\partial y}+c(x,y)\frac{\partial^{2}u}{\partial y^{2}}+d(x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y})=0
\sum_{L=0}^{-J-K}\xi_{L+1}p^L \frac{(-1)^{-J-K-L}}{(-J-K-L)!}= [z^{-J-K}] \prod_{j\ge 0}\frac{e^{qp^jz}-1}{qp^j z}e^{-z}= [z^{-J-K}] e^{z/2 + O(qz^2) - z}
\widetilde{M}_{n}(t)=\sup_{s\in [0, t]}M_{n}(s)+ (1+t)^{n^3}\lesssim \big( \widetilde{M}_{n}( t) \big)^{1- \epsilon}+ (1+t)^{n^3}, \Longrightarrow \widetilde{M}_{n}(t)\lesssim (1+t)^{n^3}.
\sum_{t\mid x^{m-w}}\mu_p\left(\frac{x^{m-w}}{t}\right)\deg\left(\frac{t}{\gcd(t,\ell)}\right)p^{\deg(\gcd(t,\ell))}= \deg\left(\frac{x^{m-w}}{\gcd(x^{m-w},\ell)}\right)p^{\deg(\gcd(x^{m-w},\ell))}-\deg\left(\frac{x^{m-w-1}}{\gcd(x^{m-w-1},\ell)}\right)p^{\deg(\gcd(x^{m-w-1},\ell))}= (m-w-k)p^k-(m-w-k-1)p^k=p^k.
\alpha_{k,j}=\begin{cases}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{w}^+(\phi_i)\geqslant1-c t_{k,j}=f^\sharp_{k,j}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{w}^+(\phi_i)<-c \end{cases}
\frac{(c, axt, byt; q)_\infty}{(ab, tx, ty; q)_\infty}{_3\phi_2 \left({{d, xt, yt}\atop{axt, byt}}; q, c \right)}=\sum_{n=0}^\infty \lambda_n \Phi_n^{(a, b)}(x, y|q).
\frac{\partial}{\partial\bar z}\left(\varphi^{-1}\frac{\partial\varphi}{\partial z}\right)+\frac{\partial}{\partial z}\left(\varphi^{-1}\frac{\partial\varphi}{\partial\bar z}\right)-2U\left(\varphi^{-1}\frac{\partial\varphi}{\partial z},\varphi^{-1}\frac{\partial\varphi}{\partial\bar z}\right)=0.
\, \sum_{j=1}^{n}\theta_\alpha \left(x - \frac{j}{n}+ \varepsilon_j \right) = A(x)+g_1(x) + g_2(x) + h(x) = \, \sum_{j=1}^{n}\theta_\alpha \left(x - \frac{j}{n}\right) + \mathcal{O}\left(n e^{-\pi \alpha n^2}\sum_{j=1}^{n}\varepsilon_j^2\right) + g_1(x) + g_2(x) + h(x).
\int \Gamma(f) d\mu = \int fLf d\mu = \int f \hat{L}g d\mu = \int \hat{\Gamma}(f, g) d\mu \leq{\left(\int\hat{\Gamma}(f) d\mu\right)}^{1/2}{\left(\int\hat{\Gamma}(g) d\mu\right)}^{1/2}\leq{\left(\int\Gamma(f) d\mu\right)}^{1/2}{\left(\int\hat{\Gamma}(g) d\mu\right)}^{1/2}.
G_{b b}^{- 2}( x , y ) = \left[ - \frac{{N^{2}}}{g^{2}}\frac{{4}}{f^{4}({\hat{\phi}})}\left({1 - \frac{{3}}{2}\frac{{\hbar G_{a a}( z , z )}}{f ({\hat{\phi}})}}\right) \nabla_{x}^{2}+ \frac{{1}}{2}\frac{{\hbar}}{f ({\hat{\phi}})}G_{a a}^{- 1}( z , z ) \right] \delta ( x - y ) ,
\mu_1(G,w) =\int_0^1 6t(1-t)\ln\left(\frac{t}{1-t}\right)\mathrm{d}t=0, \mu_2(G,w) =\int_0^1 6t(1-t)\ln^2\left(\frac{t}{1-t}\right)\mathrm{d}t=\frac{\pi^2}{3}-2\ .
\int_\Omega \nabla u_0\cdot \nabla vdx=\int_\Omega f_0vdx\;\; \mathrm{for\, all}\; v\in H_0^1(\Omega ), &\int_\Omega \nabla u_1\cdot \nabla vdx=\int_\Omega f_1vdx\;\; \mathrm{for\, all}\; v\in H^1(\Omega ).
\mathfrak{M}\left[{}^{M\!L\!\!}\textrm{F}_{p}(a,b;c;z):s\right]= \frac{{}^{M\!L\!}\Gamma(s)B(b+s,c+s-b)}{B(b,\c-b)}{}_{2}\textrm{F}_{1}(a,b+s;c+2s;z).
\eta(\tau_i,\tau_n^q\tau_j) =\eta(\tau_i,\tau_n)\eta(\eta(\tau_n,\tau_i),\tau_n^{q-1}\tau_j) = \tau_n^q\tau_1 \eta(\tau_n^{q-1+D(i)}\tau_{i+1}, \tau_n^{q-1}\tau_j)=\tau_n^q\tau_1 \eta(\tau_n^{D(i)}\tau_{i+1},\tau_j).
4\eta^2(1-\eta)\dfrac{d^2\psi(\eta)}{d\eta^2}+\left[4\eta-6\eta^2\right]\frac{d\psi(\eta)}{d\eta}+\left(\varepsilon d^2+ d^2 U_0{\eta^{(q-p)/2}(1-\eta)^{p/2}}\right)\psi(\eta)=0,0<\eta<1, p\geq 0,q> p.
c =[u_1,[u_1,[u_1,z]]]/6+[u_2,[u_1,z]] &=[u_1,[u_1,[u_1,z]]]/6+[u_2,a] &=[u_1,[u_1,[u_1,z]]]/6+[u_1,[u_2,z]] &=-[u_1,[u_1,[u_1,z]]]/3+[u_1,b] &=-[u_1,[u_1,a]]/3+[u_1,b].
\sum_{n\in L^x\cap{\rm L}_{A^y}}\frac{1}{n}= \sum_{a\in A^y}\frac{1}{a}\prod_{P(a)\le p\le x}(1-\tfrac{1}{p})^{-1}= \sum_{a\in A^y}\frac{1}{a}\prod_{p<P(a)}(1-\tfrac{1}{p})\prod_{p\le x}(1-\tfrac{1}{p})^{-1}&={\rm d}({\rm L}_{A^y})\prod_{p\le x}(1-\tfrac{1}{p})^{-1}.
I_g^{\gamma,\alpha}(u)=\frac{1}{2}\int_{X^{n+1}}\rho^{1-2\gamma}|\nabla u|^2_g\,dv_g+\frac{1}{2}\int_{M^n}Q^\gamma_\alpha u^2\,d\sigma_{\hat{h}}-\frac{n-2\gamma}{2n}\int_{M^n}|u|^{\frac{2n}{n-2\gamma}}d\sigma_{\hat{h}}.
{\it L}= \sqrt{- g}\biggl [ \phi \biggl ( R - \omega \frac{\phi_{; \rho}\phi^{; \rho}}{\phi^{2}}- \frac{\Psi_{; \rho}\Psi^{; \rho}}{\phi^{2}}\biggr ) - \frac{1}{2}\xi_{; \rho}\xi^{; \rho}\biggr ] + L_{\mathrm{r}},