harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{T}(s\,,y\,,z) := \p_{t-s}(y-z) p_{t-s}(z-x)\left| u^{(\beta,m)}_s(y) u^{(\beta,m)}_s(z) \right|f_\beta(y-z) \le \p_{t-s}(y-z) p_{t-s}(z-x)\left| u^{(\beta,m)}_s(y) u^{(\beta,m)}_s(z) \right|f(0),
P_G\left(x\right) =x^{n-(t+1)}\left[\prod_{i=1}^{t+1}\left(x-\mu\right) -\sum_{i=1}^{t+1}\left(-\mu\right)\prod_{\substack{j=1 j\neq i}}^{t+1}\left(x-\mu\right) \right] &=x^{n-t-1}\left[\left(x-\mu\right)^{t+1}+\left(t+1\right) \mu\left(x-\mu\right)^{t}\right] &=x^{n-t-1}\left(x-\mu\right)^{t}\left(x+t\mu\right).
\begin{aligned}Z_(f) ~=~ \frac{V_1(f)}{I_1(f)}~ =~ \underbrace{\frac{1}{\textrm{j}2\pi f\frac{a}{cR}}}_{Z_C(f)}+ \underbrace{\frac{1}{\frac{1}{\textrm{j}2\pi f\frac{aR}{c}}+\frac{1}{R}}}_{Z_{\rm L}(f) \,\parallel\, Z_R(f)}=~ \frac{c^2R + \textrm{j}2\pi fcaR - (2\pi fa)^2R}{\textrm{j}2\pi f ca - (2 \pi fa)^2}\,\, [\Omega].\end{aligned}
r_L(u,v;t)= \left(G_L(u,v;t-i\eta)-G_L(u,v;t+i\eta)\right)G_L(v,u;t-i\eta) +G_L(u,v;t+i\eta)\left(G_L(v,u;t-i\eta)-G_L(v,u;t+i\eta)\right).
R_\psi(\xi+y) - R_\psi(\xi) = \frac{\xi+y -(1+O(x^{-\theta})) \xi}{\sqrt{\xi+y}}- \frac{\psi(\xi+y) - (1+O(x^{-\theta}))\psi(\xi)}{\sqrt{\xi+y}}&= O(x^{1/2-\theta}) + O(x^{-\theta - 1/2}) \psi(\xi) + O(x^{-1/2}) (\psi(\xi+y) - \psi(\xi)) &= O(x^{1/2-\theta})
P^\chi_r \propto M_\chi^2\cdot (\mu^\chi_h|g|\beta^\chi)^2 + M_\chi\cdot\big\{\sigma^\chi_h|g|^2(\beta^\chi)^2 + 2\mu^\chi_h|g|\beta^\chi \mathbb{E}[|h^R_d|] \big\}+\mathbb{E}[|h^R_d|^2].
\begin{aligned}\nabla_\xi\Psi=\sum_{w\in W}\biggl(\biggl(\partial(\xi)-\lambda(w^{-1}\xi)+\frac12\sum_{\alpha\in R^+}\mathbf k(\alpha) \alpha(\xi)\frac{1+e^{-\alpha}}{1-e^{-\alpha}}\biggr)\Psi_w - \sum_{\alpha \in wR^+}\mathbf k(\alpha)\alpha(\xi)\frac{e^{-\alpha}}{1-e^{-\alpha}}\Psi_{s_\alpha w}\biggr)w\end{aligned}
{\hat{\mathcal{X}}_k}=\Big\{\mathbf{x}(k): \mathbf{x}(k)-\mathbf{a}_d(\mathbf{x}(k-1),k-1) \in \mathcal{W}_{k-1}~with~ \mathbf{x}(k-1) \in \mathcal{X}_{k-1}\Big\}.
\int \nabla F_3 (x) \cdot H(x) \frac{b_3(x)}{|x^{\prime}|}dx = \; \int \frac 1{r^2}(- \partial_r F_3 \partial_z a_3 + \partial_r a_3 \partial_z F_3) \cdot (\partial_{zz}a_3 + \partial_{rr}a_3 - \frac 1 r \partial_r a_3) dr dz.
log(B)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(B-I)^{k}}{k}=(B-I)-\frac{(B-I)^{2}}{2}+\frac{(B-I)^{3}}{3}-\frac{(B-I)^{4}}{4}+\cdot\cdot\cdot
0<n_2-n_1\theta\leqslant\frac{3}{2}\frac{\det(\Lambda_d)n_1}{(\lambda_2-\alpha\mu_2)^2}\varepsilon B^{-\frac{1}{r}}\frac{\lambda_2-\alpha\mu_2}{\det(\Lambda_d)}KB-\frac{\alpha}{2}\leqslant n_1\leqslant \frac{\lambda_2-\alpha\mu_2}{\det(\Lambda_d)}KB+\frac{\alpha}{2}
\begin{aligned}D^{\beta}(|y|^{r_2}W(t) u_0)(x,y) &= W(t) (D^{\beta}|y|^{r_2}u_0)(x,y) + W(t)(D^{\beta}(\{\Phi_{2,t,r_2}(\widehat u_0)(\xi,\eta)\}^{\lor}))(x,y)\end{aligned}
| B >_{( \frac{1}{2}, \frac{1}{2})}= \sum_{k_{p}}\left( ( \tilde{A}_{\mu \nu}( k_{p}) \psi_{- 1 / 2}^{\mu}\psi_{- 1 / 2}^{\nu}+ \tilde{B}( k_{p}) ( \bar{\beta}_{- \frac{1}{2}}\gamma_{- \frac{1}{2}}- \beta_{- \frac{1}{2}}\bar{\gamma}_{- \frac{1}{2}}) \right) | \Omega , k > \ ,
\mathbb E\left[Y_{1,i}^{(2)}\right]^{\alpha/2}\leq Cn^\gamma\left[\max\{1,p^{\alpha/2-1}\frac{\rho(\alpha)}{\rho(2)^{\alpha/2}},p^{\frac{2^l-1}{2^{l+1}}\alpha}\frac{\rho(2^{l+1})^{\alpha/2^{l+1}}}{\rho(2)^{\alpha/2}},l=1,\cdots,m-1\}\right]^n,
\frac{d^{2}r}{dt^{2}}=\frac{1}{m}\frac{dp_{r}}{dt}=-\frac{1}{m}(\frac{\partial H}{\partial r})=\frac{p_{\varphi}^{2}}{m^{2}r^{3}}-\frac{1}{m}\frac{dU}{dr}=\frac{L^{2}}{m^{2}r^{3}}+\frac{1}{m}F(r)
b_+= \tilde\gamma^{-1}F+\tilde S-\frac{1}{2}\left(\tilde\gamma^{-1}-(\tilde\gamma^T)^{-1}\right)F-\frac{1}{2}(S-S^\prime), b_-= \frac{1}{2}\left(\tilde\gamma^{-1}-(\tilde\gamma^T)^{-1}\right)F+\frac{1}{2}(S-S^\prime),
\mathrm{P}(\mathcal{O})\approx \sum^{N}_{l=0}{N \choose l}\frac{\gamma^l}{\rho^l}e^{-3l\xi_1}\left[1-e^{-3\xi_1}\right]^{N-l}\approx \frac{1}{\rho^N}\sum^{N}_{l=0}{N \choose l}\gamma^l \left[\frac{3\epsilon_1}{\alpha_1^2 - \epsilon_1\alpha^2_2}\right]^{N-l}.
M:= \min\Big\{\frac{1}{\lambda_2}\big( (\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_+ + \chi_1\mu_1(\lambda_1-\lambda_2)_+ \big), & \frac{1}{\lambda_1}\big( (\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_+ + \chi_2\mu_2(\lambda_1-\lambda_2)_+ \big) \Big\}.
(\underline{s}_i^k \circ \widehat{\pi}_i)(x) = \lambda_k \bigl[ (\underline{s}_i \circ \widehat{\pi}_i)(x) + R \bigr] - R, (\overline{s}_i^k \circ \widehat{\pi}_i)(x) = \lambda_k \bigl[ (\overline{s}_i \circ \widehat{\pi}_i)(x) - R \bigr] + R.
f^{\ast}(u) =\inf\bigg\{\sum_{n\geq1}f_{n}^{\ast}(u_{n})\mid(u_{n})_{n\geq1}\subset\left( \operatorname*{dom}f_{n}^{\ast}\right)_{n\geq1},\u=\sum_{n\geq1}u_{n}\bigg\}& =\inf\bigg\{\sum_{n\geq1}u_{n}(\ln u_{n}-1)\mid(u_{n})_{n\geq1}\subset\mathbb{R}_{+},\u=\sum_{n\geq1}\sigma_{n}u_{n}\bigg\}.
I ( s , t ) = \frac{3^{9}}{2^{1 2}}\int_{0}^{1}\frac{d x}{x^{2}}\; \frac{1 - \frac{1 2 1}{7 2 9}x^{2}}{( 1 - \frac{2 5}{7 2 9}x^{2})^{1 3}}\; \exp \left( - \frac{1}{2}p_{i}Q_{i j}^{( 1 )}p_{j}\right) \, .
J_{lower}(D,D_1,u,v) = \frac{e^{-4d_{D_1}(u,v)}-1}{e^{-4d_D(u,v)}-1}\cdot \frac{\lambda_{D}(u)}{\lambda_{D_1}(u)}\ \ \ \ \J_{upper}(D,D_1,u,v) = \frac{e^{4d_D(u,v)}-1}{e^{4d_{D_1}(u,v)}-1}\cdot \frac{\lambda_{D_1}(u)}{\lambda_{D}(u)}.
\ell(w')-\ell(w)\geq\, \sum_{i=1}^k\left(\ell(u'^{(i)})-\ell(u^{(i)})\right)+2(p_1+\cdots+p_{j-1})(|L'_j|-|L_j|) \geq\, \ell(u'^{(j)})-\ell(u^{(j)})+2(p_1+\cdots+p_{j-1}) \geq\, 3.
\sum_{k=1}^K \left(\mathbf{W}_{a_k}^H\widehat{\widetilde{\mathbf{V}}}_k\left(\widehat{\widetilde{\boldsymbol{\Sigma}}}_k + \frac{KQ}{P_{T}}\mathbf{I}\right)\widehat{\widetilde{\mathbf{V}}}_k^H\mathbf{W}_{a_k}\right).
\iint_{\mathbb{D}}\frac{|m^{i}_t(1/\bar{z})|^2}{(1-|z|^2)^2}\, dA = \frac{1}{4}\iint_{\mathbb{D}}(1-|z|^2)^2 |\mathcal{S}(w_i(t,\cdot))(z)|^2\, dA.
\mathcal F_{1}(\mathcal C\mathbf f)(\boldsymbol\xi,s)=(2\pi)^{\frac{n-1}2}|\boldsymbol\xi|^\frac{3-n}2s^\frac{3-n}2\mathbf H_{\frac{n-3}2}\mathcal F_{1}\mathbf f(\boldsymbol\xi,s|\boldsymbol\xi|).
h''(x) = 0.5 \left(-x^{-2}\log(e^2/x)^{-3/2}+ x^{-1}\cdot 1.5x^{-1}\log(e^2/x)^{-5/2}\right) &= 0.5x^{-2}\log(e^2/x)^{-3/2}\left(-1 + 1.5 \log(e^2/x)^{-1}\right) < 0,
\left[ \pi_{l}^{\left( N \right)}\left( \bar{x}\right) , \, \pi_{k}^{\left( N \right)}( \bar{y}) \right]_{x^{0}= y^{0}}\, = \, - i \, \theta \, \left( - 1 \right)^{N - 1}\left( b + a \nabla^{2}\right) \epsilon_{i k}\partial_{l}\partial^{i}\delta^{2}\left( \bar{x}- \bar{y}\right) ,
\lim_{n\to\infty}\mathbb P\left[\sup_{t\in[0,T]}|(1)_n|>\epsilon\right] \leq \lim_{n\to\infty}\mathbb P\left[\sup_{t\in[0,T]}|(1.1)_{n,N}|>\epsilon\right]+\sup_{n\in \mathbb N}\mathbb P\left[\sup_{t\in[0,T]}|(1.2)_{n,N}|>\epsilon\right] = 0+ \frac 1{\epsilon}\sup_{n\in \mathbb N}\mathbb E\left[\sup_{t\in[0,T]}|(1.2)_{n,N}|\right]\leq \delta.
\sum_{n_1m_1=n_2m_2}= 2\sum_{\substack{m_1,m_2 \leq q^\theta (m_1m_2,p)=1}}\frac{a_{m_1}\overline{a}_{m_2}}{[m_1,m_2]}\sum_{\substack{r\ge 1 (r,p)=1}}\frac{1}{r}V\Big( \frac{r^2 m_1m_2}{q(m_1,m_2)^2}\Big).
X^* = \left( \begin{array}{rrrrr}0 0.1720 0 0.7280 0.7170 0 -0.0620 0 0 0 0 1.1990 -1.3444 0.1114 0 0 0.8020 0.4910 -1.1956 1.2779 0 1.0530 2.7535 0 -0.2525\end{array}\right)
\left(X_{\pi(i)}|X_{\pi(i-1)},X_{\pi(i-2)},\ldots,X_{\pi(1)}\right) &\geq\left(X_{\frac{r n}{2^k}}|X_{\frac{r n}{2^k}-\frac{n}{2^k}},X_{\frac{r n}{2^k}+\frac{n}{2^k}}\right) &=\left(X_m|X_{m-\frac{n}{2^k}},X_{m+\frac{n}{2^k}}\right) &\triangleq\left(\frac{n}{2^k}\right),
\int_{\Omega}(1-\sigma)D^2 v:D^2\varphi +\sigma\Delta v\Delta\varphi dx=\eta\int_{\partial\Omega}\frac{\partial v}{\partial\nu}\frac{\partial\varphi}{\partial\nu}d\sigma\,,\ \ \ \forall\varphi\in \mathcal{H}^2_{0,D}(\Omega),
\int_{\{2c_{o,p}^{f}(\varphi)\varphi<-t\}}|f_1|^2e^{-\varphi_1}\geq G(t) &\geq e^{-t}G(0) &=e^{-t}C_{f_1,\mathcal{I}(\varphi_1+2c_{o,p}^{f}(\varphi)\varphi)_o,\varphi_1}
\{\bar{q}_{i}^{\dot{\alpha}}, \bar{s}_{\dot{\beta}}^{j}\}\Phi = \left[ - \delta_{i}^{j}( \sigma^{\mu \nu})^{\dot{\alpha}}{}_{\dot{\beta}}m_{\mu \nu}- 2 \delta_{\dot{\beta}}^{\dot{\alpha}}\left( 2 t_{j}^{i}+ \delta_{i}^{j}( \ell + r ) \right) \right] \Phi = 0 .
[\begin{matrix}\epsilon_{xx}\\ \epsilon_{yy}\\ 2\epsilon_{xy}\end{matrix}]=[\begin{matrix}\frac{1}{E_{x}}&-\frac{\nu_{yx}}{E_{y}}&0\\ -\frac{\nu_{xy}}{E_{x}}&\frac{1}{E_{y}}&0\\ 0&0&\frac{1}{G_{xy}}\end{matrix}][\begin{matrix}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{matrix}].
\frac{\alpha_N(t^*)}{t^*}=\frac{(a_-)^N}{a_+t^*}\frac{1}{(a_+)^{N-1}}\int_0^{t^*}\frac{1}{(N-1)!}(a_+s)^{N-1}e^{-sa_+}d(a_+s) =a_-\left(\frac{a_-}{a_+}\right)^{N-1}\frac{1}{\tau}\int_0^{\tau}\frac{1}{(N-1)!}\tilde{s}^{N-1}e^{-\tilde{s}}d\tilde{s},
{\rm var}\{T(\psi)\}=E[\{T(\psi)\}^2]-[E\{T(\psi)\}]^2=E[\{T(\psi)\}^2]+O(n^{-1}) &=\eta E\{T_1^2+2T_1T_2+O_p(n^{-2})\}+O(n^{-1}) &=\eta E\{\lambda^{1r}\lambda^{1s}l_rl_s-2\lambda^{1r}\xi^{stu}l_rl_{st}l_u+2\lambda^{1r}\xi^{st}l_rl_sl_t+O_p(n^{-2})\}+O(n^{-1}) &=-\eta\{\lambda^{1r}\lambda^{1s}\lambda_{rs}+O(n^{-2})\}+O(n^{-1}) &=1+O(n^{-1}).
\overline{M}_3=(\mathop{q-2}\limits_{m-1},\underbrace{q-1, \ldots, q-1,}_{m-2-i}\mathop{q-2}_{i}, \underbrace{q-1, \ldots, q-1,}_{i-j-1}\mathop{q-2}\limits_{j},\underbrace{q-1, \ldots, q-1}_{j}),
I({K'}_B^C;W_1Y_2^{n_1}S^n\Theta_2U_C|1_B^C) \stackrel{}{=}I({K'}_B^C;Y_2^{n_1}S^{n_1}|1_B^C)\leq I({K'}_B^C1_B^C;Y_2^{n_1}S^{n_1})\leq I({K}_B;Y_2^{n_1}S^{n_1}).
\frac{\log q_{k_{n+1}(x)}(x)- bk_{n+1}(x)}{\sigma_1\sqrt{2k_n(x)\log\log k_n(x)}}= \frac{\log q_{k_{n+1}(x)}(x)- bk_{n+1}(x)}{\sigma_1\sqrt{2k_{n+1}(x)\log\log k_{n+1}(x)}}\cdot \frac{\sqrt{k_{n+1}(x)\log\log k_{n+1}(x)}}{\sqrt{k_n(x)\log\log k_n(x)}},
\begin{array}{l l}{\partial X_{c l}( z ) = a \omega ( z ) ,}{\partial \bar{X}_{c l}( z ) = \bar{a}\omega^{\prime}( z ) ,}{\bar{\partial}X_{c l}( \bar{z}) = b \bar{\omega}^{\prime}( \bar{z}) ,}{\bar{\partial}\bar{X}_{c l}( \bar{z}) = \bar{b}\bar{\omega}( \bar{z}) ,}\end{array}
a_\theta(0)-b(0)=\frac{1}{\beta_{N,m}}\left(\sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)J_{N,m}(\tau)-\sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)j_{N,m}(\tau)\right)=-\sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f).
\sum_{k=k_0+1+N}^\infty \sum_{j=1}^K |\varphi_j^k(x) - \varphi_j^k(y)| \leq \sum_{k=k_0+1+N}^\infty KR(M)^2 \frac{r_{k_0}(x)(M-2n+1)^{k-k_0}}{M^k}& \leq \left(\frac{2nMKR(M)^3}{1-(M-2n+1)/M}\right)d(x,y) \left(\frac{M-2n+1}{N}\right)^{N+1}
\phi''_\omega(t)=\|\omega\|_X^2-(2_\alpha^*-1)t^{2_\alpha^*-2}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx &-\lambda \left(\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx+\ln t\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx+\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx\right).
\begin{aligned}I_{r}(x,t)=I_{r_{1}}(x,t)+\overline{I}_{r_{1}}(x,t), &I_{r}(x_{0},t)=I_{r_{1}}(x_{0},t)+\overline{I}_{r_{1}}(x_{0},t), &I_{r}^{'}(x,t)=I_{r_{1}}^{'}(x,t)+\overline{I}_{r_{1}}^{'}(x,t),\end{aligned}
du\left( t \right) + \nu Au\left( t \right)dt + \lambda(r(t)) u\left( t \right)dt = \left({f\left({t,r(t),u(t)}\right) + g(r(t))}\right)dt +\varepsilon \sum\limits_{k = 1}^\infty{\left({h_k(r(t)) + \sigma_k \left({t,r(t),u\left( t \right)}\right)}\right)}dW_k \left( t \right), t>s,
\underline{{\cal H}}_c := \left[\begin{array}{cc}\underline{{\cal E}}_c 0 0{\overline{\sigma}}^{-1}\overline{\eta}^{-1}I_{n_3}\end{array}\right]\, , \overline{{\cal H}}_c := \left[\begin{array}{cc}\overline{{\cal E}}_c 0 0{\underline{\sigma}}^{-1}\underline{\eta}^{-1}I_{n_3}\end{array}\right]\, .
\dfrac{\rm d}{{\rm d}z}\bigl\{{\mathfrak T}^{\tau,\mu}\; f(z)\bigr\}= \dfrac{\rm d}{{\rm d}z}\left\{\frac{\Gamma(\mu)}{\Gamma(\tau)}\;z^{1-\mu}\; \mathfrak{D}^{\tau-\mu}_{z}\;z^{\tau-1}\;f(z)\right\}&=\frac{\tau}{\mu}- \sum_{\nu=k+1}^{\infty}\frac{\nu \Gamma(\nu+\tau)\Gamma(\mu)}{\Gamma(\nu+\mu)\Gamma(\tau\alpha)}\; a_{\nu}z^{\nu-1}(z\in \mathbb{U}),
+\sin\psi\cot\eta\cdot\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{1}[q_{0}](\frac{i}{\mu_{0}})w_{3}[q_{0}](\frac{i}{\mu_{0}})}{w_{2}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\#\{(n_1,n_2)\in\mathbb{Z}^2:\,n_1\mathbf{v}_1+n_2\mathbf{v}_2\in \phi(\mathcal{R})\}= \#\{\mathbf{x}\in\Lambda_1:\,\phi(x)\in\mathcal{R}\}-O(\delta N^2) &\ge \delta K N^2-O(\delta N^2).
\frac{1}{n^2}\sum_{i,j,k}(x_{i+j}x_{i+2j}x_{i-k}x_{i+k}-x_{i+j}x_{i+2j}y_{i-k}y_{i+k}) &= \frac{1}{n^2}\sum_{i,j,k}(x_{i+j}x_{i+2j}(x_{i-k}-y_{i-k}) x_{i+k}-x_{i+j}x_{i+2j}y_{i-k}(x_{i+k}-y_{i+k}))\, .
\partial_z\tilde{\varphi}(z,\bar{z})=\tilde{\Phi}=\partial_{\varphi}\bar{S}(z,\bar{z},\varphi=\tilde{\varphi}) \, ,\quad \partial_{\bar{z}}\tilde{\varphi}(z,\bar{z})=\tilde{\bar{\Phi}}=\partial_{\varphi}S(z,\bar{z},\varphi=\tilde{\varphi}).
={1 \over Z}\sum_{n=0}^{\infty}\big(-{1 \over 4 \pi \alpha '}\big){1 \over n!}\int_{{\cal H}_n^{\varepsilon}}d^d z d^d \bar{z}\langle \Phi_1 \Phi_2 \dots \Phi_N\prod_{a=1}^n B_{i_a j_a}J^{i_a}(z_a) \bar{J}^{i_a}(\bar{z}_a) \rangle
\begin{aligned}\operatorname{cardinality}\{y\in G\,: d(x,y)=1, d(y, x_0)=d(x, x_0)+1\}=b, \operatorname{cardinality}\{y\in G\,: d(x,y)=1, d(y, x_0)=d(x, x_0)-1\}=1\,.\end{aligned}
B_{0}=(P_{1}+\sum_{n\geq1}P_{n}^{\kappa\beta-\sigma})^{-1}B_{1}:=\sup_{n}\{P_{n}^{-\beta}(P_{1}^{\beta}+\sum_{1\leq i\leq n-1}P_{i}^{\kappa\beta})\;|\;n\geq1\}B^{2}:=\sup_{n}\left\{B_{1}P_{n}^{-\left( \kappa-1\right) \beta}+1\ |\n\geq1\right\}
Q_{0,0}^{(n)}= A_0 Q_{0,0}^{(n-1)}+ \sum_{m=2}^n A_1 D^{(m-1)}Q_{0,0}^{(n-m)}, &Q_{0,0}^{(n)}= Q_{0,0}^{(n-1)}A_0 + \sum_{m=0}^{n-2}Q_{0,0}^{(m)}A_1 D^{(m-n-1)},
\begin{array}{c}{{\Theta_{1}^{+}= \left( \pi_{a}+ i G_{, a}( z , \bar{z}) \right) \eta_{1}^{a}+ i{\bar{U}}_{, \bar{a}}({\bar{z}}){\bar{\eta}}_{2}^{\bar{a}},}}\\{{\Theta_{2}^{+}= \left( \pi_{a}- i G_{, a}( z , \bar{z}) \right) \eta_{2}^{a}- i{\bar{U}}_{,{\bar{a}}}({\bar{z}}){\bar{\eta}}_{1}^{\bar{a}},}}\\ \end{array}
\Upsilon_{\alpha \beta \gamma}^{g_{1}g_{2}}\: = \: \left( \Upsilon_{\alpha \beta \gamma}^{g_{2}}\right) \left( g_{2}^{*}\Upsilon_{\alpha \beta \gamma}^{g_{1}}\right) \left( \Omega_{\alpha \beta}^{g_{1}, g_{2}}\right) \left( \Omega_{\beta \gamma}^{g_{1}, g_{2}}\right) \left( \Omega_{\gamma \alpha}^{g_{1}, g_{2}}\right)
\Delta_{\underline x}h =-\partial_{\underline x}^2h=-\partial_{\underline x}\big(\underline\omega\,\partial_rh\big)=-\underline\omega^2\partial_r^2h-\big(\partial_{\underline x}\,\underline\omega\big)\big(\partial_rh\big) &=\partial_r^2h+\frac{p-1}{r}\,\partial_rh.
\lim_n \Vert g \cdot (p\xi_n) - (p\xi_n)\Vert_2 = \lim_n \Vert (g \cdot p)(g \cdot \xi_n) - (p\xi_n)\Vert_2 & \leqslant \lim_n \Vert (g \cdot p)\xi_n - (p\xi_n)\Vert_2 + \lim_n \Vert (g \cdot p)(g \cdot \xi_n) - (g \cdot p)\xi_n\Vert_2 & = \lim_n \Vert (g \cdot p - p)\xi_n\Vert_2 + \lim_n \Vert g \cdot \xi_n - \xi_n\Vert_{2,gB_0}
\Gamma_{1 1}^{1}\, = \, \sqrt{\frac{2 \tan \frac{2 \pi}{3}\, \tan \frac{8 \pi}{1 5}\, \tan \frac{1 1 \pi}{3 0}}{\tan \frac{2 \pi}{1 5}\, \tan \frac{3 \pi}{1 0}}}\, = \, 1 0 . 9 9 0 8 8 3 . .
E_{D}: = \{\frac{u_{( i + 1 , j )}- 2 u_{( i , j )}+ u_{( i - 1 , j )}}{{\Delta x_{1}}^{2}}+ \frac{u_{( i , j + 1 )}- 2 u_{( i , j )}+ u_{( i , j - 1 )}}{{\Delta x_{2}}^{2}}- f ( u_{( i , j )}) \}d u_{( i , j )},
\mathbb{E}[|J(s,t, x, y)|^p] = \mathbb{E}\biggl| \int_s^t\int_0^1 f(u\phi(s,r,x) + (1-u)\phi(s,r, y))dudB_r\biggr|^p \leq C(p) \mathbb{E}\biggl( \int_s^t\biggl\{\int_0^1 f(u\phi(s,r,x) + (1-u)\phi(s,r, y))du\biggr\}^2dr\biggr)^\frac{p}{2}\leq C(p, ||f||_{\infty}) |t-s|^{\frac{p}{2}}& \leq C(p, ||f||_{\infty}) |s-\tilde{s}|^{\frac{p}{2}}.
\sum_{a, b \in F}|T_{a b}|^2 = p^n \left( \sum_{b \in F}\tilde{N}_f (0, b)^2 + \sum_{a \in F^{\times}, b \in F}\tilde{N}_f (a, b)^2 \right) &= p^{3 n}+ p^n \sum_{a \in F^{\times}, b \in F}\tilde{N}_f (a, b)^2 \geq p^{3n}+ p^{n+1}\sum_{a \in F^{\times}, b \in F}\tilde{N}_f (a, b).
v_{1}^{1}=y_{1,[2]}^{1,d}-a_{1}^{1,1}(y_{1,[1]}^{1}-y_{1,[1]}^{1,d})-a_{1}^{1,0}(y_{1}^{1}-y_{1}^{1,d}) v_{2}^{1}=y_{2,[2]}^{1,d}-a_{2}^{1,1}(y_{2,[1]}^{1}-y_{2,[1]}^{1,d})-a_{2}^{1,0}(y_{2}^{1}-y_{2}^{1,d})
(e'_i)^p(G^{\mathrm{low}}(b))=\sum_{\tilde{b}\in B(\infty)}\left(\sum_{\substack{b^{1},\dots, b^{p-1}\in B(\infty)}}\mspace{-30mu}\widehat{d}_{b, b^1}^{i, 1}\widehat{d}_{b^1, b^2}^{i, 1}\cdots \widehat{d}_{b^{p-1}, \tilde{b}}^{i, 1}\right)G^{\mathrm{low}}(\tilde{b}).
\tilde t=t,\tilde r = r,\tilde\theta = \theta + \frac{f}{2}t,\tilde p=p, \tilde u^r = u^r,\tilde u^\theta = u^\theta + \frac{f}{2}r,\tilde\omega = \omega,\tilde\phi = \phi+\frac{f^2}{8}r^2,\tilde T=T,
\sum_{n=m2^{r-1}}^{m2^r-1}f_\mathcal{A}(n-k)=\sum_{n=m2^{r-1}}^{m2^r-1}f_{\mathcal{A}}(n)+\sum_{j=1}^k\left(f_{\mathcal{A}}(m2^{r-1}-j)-f_\mathcal{A}(m2^r-j)\right),
u_{m}=(2c+2)u_{m-1}-u_{m-2}\equiv(2c+2)\epsilon(1+(m-1)mc)-\epsilon(1+(m-2)(m-1)c) & \equiv\epsilon(1+2m(m-1)c^{2}+m(m+1)c)\equiv\epsilon(1+m(m+1)c)\pmod{4c^2},
\hat{T}= \exp \{- a_{0}[ \, \frac{g^{2}}{2 a}\sum_{\ell}\hat{E}_{\ell}^{2}+ \frac{1}{2 a g^{2}}\sum_{p}( \hat{W}_{p}+ \hat{W}_{p}^{\dagger}) \, ] \, + O ( a_{0}^{2}) \},
v_{\varepsilon}(\mathbf{x}_{\varepsilon}) = \sigma_3 \sqrt{\tilde{\lambda}_0 \underline{m}}\,|\mathbf{x}_{\varepsilon}-\mathbf{y}_{\varepsilon}| = \sigma_3 \sqrt{\tilde{\lambda}_0 \underline{m}}\,(|\mathbf{x}_{\varepsilon}- \bar{\mathbf{x}}_{\varepsilon}| + |\bar{\mathbf{x}}_{\varepsilon}- \mathbf{y}_{\varepsilon}|) \ge
S(m,Y,w_X) = \sum_{z^2-4xy=m}\chi_{Y}(xy)w_{X}(x)w_{X}(y) \leq \sum_{z^2-4xy=m}\Big( \sum_{d|\gcd(xy,\prod_{p<Y}p)}\lambda_{d}\Big)^2w_{X}(x)w_{X}(y) &=\sum_{d}\mu^{+}(d)\#_{w_X}A_{d},
C_{f_*\mathcal F}=\frac{\lambda_\Delta^+(f,x,\delta)C_{\mathcal F}+t(E_x,\Delta,\delta)t(f_*E_x,\Delta',\delta')\|Df\|_{C^{\theta}}}{\lambda_\Delta^-(f,x,\delta)\lambda_{\mathcal F}(f,x,\delta)^{\theta}},
\log \frac{|x_j|}{r_j}= \frac\lambda{n}\log\left(\frac1{\rho_j}\log \frac1{\rho_j}\right) &\ge \frac\lambda{n}\log \frac1{\rho_j}&= \frac\lambda{n}\rho_j\left(\frac{|x_j|}{r_j}\right)^{n/\lambda}&= \frac1A \frac{|x_j|^a}{\psi(|x_j|)^{\frac{\lambda-1}\lambda}}\left(\frac{|x_j|}{r_j}\right)^{n/\lambda}.
\deg_{E}\widetilde{\mathcal{T}}=\deg_{E}\left( \widetilde{T}_{j-i}\right) =\deg_{E}\left[ \mathcal{O}_{\widetilde{\mathcal{C}^{2}}}\left( \widetilde{C_{i}\times Z_{i}}+\widetilde{C_{j}\times Z_{j}}\right) \right] .
A=\begin{pmatrix}(12\lambda_i-45)\lambda_i(\lambda_i-4) -12\lambda_i(\lambda_i-4) 0 -12\lambda_i(\lambda_i-4) 2\lambda_i^2+\frac{25}{2}\lambda_i 20\lambda_i 0 20\lambda_i 20\lambda_i+245\end{pmatrix}.
\varphi= \begin{pmatrix}\phi_0+\phi_n -\pi_n -\pi_b -\pi_0 0 \omega^n \phi_0-\phi_n \gamma_b 0 -\pi_0 \omega^a \theta^a -\phi^a{}_b \gamma^a \pi^a \omega^0 0 \theta_b -\phi_0+\phi_n -\pi_n 0 \omega^0 -\omega_b \omega^n -\phi_0-\phi_n \end{pmatrix},
g_{h,k}(x,w) = \begin{cases}\left[ A_{h,k}(x, w) - A_{h,k}(x,0) \right]+\left[ B_{h,k}( x,w ) - \Psi_{h,k}(x,w) - \frac{1}{12k}\ln (1-x^k) \right] , k \geq 2 B_{0,1}(x,w) - \Psi_{0,1}(x,w) - \frac{1}{12}\ln(1 -x), k=1 .\end{cases}
m\rq_{2,n}(x,y)+\phi_n(m\rq_{2,0}(x,y))=m_{2,n}(x,y)+m_{2,0}(\phi_n(x),y)+m_{2,0}(x,\phi_n(y)), m\rq_{2,n}(x,y)-m_{2,n}(x,y)=m_{2}(\phi_n(x),y)+m_2(x,\phi_n(y))-\phi_n(m\rq_2(x,y))=\delta_2 \phi_n(x,y).
D_{\alpha}D_{\beta}\Omega - g_{\alpha \beta}D \cdot D \Omega+ \frac{\gamma}{8}\left[g_{\alpha \beta}\frac{(D \Omega )^2}{\Omega}- 2 \frac{D_{\alpha}\Omega D_{\beta}\Omega}{\Omega}\right] + \frac{\mu}{2}g_{\alpha \beta}\Omega^{1-\lambda}
J(X) = \sum_{i,j=1}^n \langle\Psi_{ij}(X)^2\rangle \ge \sum_{i,j=1}^n \langle\Psi_{ij}(X)\rangle^2 = \sum_{i,j=1}^n \left[\int_{\{f>0\}}\frac{\partial_i f\partial_j f}f \, dx\right]^2
U_1 =(a_{s-t+2})^*, U_2 =(a_{s-t+3}a_{s-t+2})^*(a_{s-t+3})^*, U_3 =(a_{s-t+4}a_{s-t+2})^*(a_{s-t+4}a_{s-t+3})^*(a_{s-t+4})^*, & \vdotswithin{=}U_{2t-1}=(a_{s+t}a_{s-t+2})^*(a_{s+t}a_{s-t+3})^*\dotsm(a_{s+t}a_{s+t-1})^*(a_{s+t})^*.
\int_{\widetilde{\Omega}_p}\left(e^U- V_p\right)\widetilde{\varphi}_{1,p}^2= \underbrace{\int_{\widetilde{\Omega}_p\cap\{|x|\leq R\}}\left(e^U- V_p\right)\widetilde{\varphi}_{1,p}^2}_{A_{p,R}}+\underbrace{\int_{\widetilde{\Omega}_p\cap\{|x|> R\}}\left(e^U- V_p\right)\widetilde{\varphi}_{1,p}^2}_{B_{p,R}}.
E_D:=\{\frac{u_{(i+1,j)}-2u_{(i,j)}+u_{(i-1,j)}}{{\Delta x_1}^2}+\frac{u_{(i,j+1)}-2u_{(i,j)}+u_{(i,j-1)}}{{\Delta x_2}^2}-f(u_{(i,j)})\}du_{(i,j)},
\sum_{r \in \mathbb{Z}}c^F_{-r}\big(\overline{\pi_E(w_n)}W,\sigma_F^{(l)}(w_{n-1})W'\big)q_F^{-\frac{r}{2}}X^{-lr}= r_l\big(\gamma\big(X, \pi_E, \widehat{\sigma_E}, \psi_E\big)\big) \sum_{r\in \mathbb{Z}}c^F_r\big(W, W'\big) q_F^{\frac{r}{2}}X^{lr},
\theta_i^2T_i^*(u) =(T_i^{\infty})^2\int_0^u\frac{e^{2\rho_i(v)}}{(T_i^{\infty}-T_i(v))^2}dv &=(T_i^{\infty})^2\left[\frac{1}{T_i^{\infty}-T_i(v)}\right]_0^u &=(T_i^{\infty})^2\left(\frac{1}{T_i^{\infty}-T_i(u)}-\frac{1}{T_i^{\infty}}\right).
\left[ \hat{L}_{z}, \, \hat{L}_{+}\right] = \hat{L}_{+}; \qquad \left[ \hat{L}_{z}, \, \hat{L}_{-}\right] = - \hat{L}_{-}; \qquad \left[ \hat{L}_{+}, \, \hat{L}_{-}\right] = 2 \hat{L}_{z}.
\begin{array}{r c l}{{U_{g}/ m^{4}}}&{{=}}&{{\displaystyle + 6 D_{m}( k_{1}) D ( k_{1}+ p ) D_{m}( k_{2}) D ( k_{2}+ p ) D_{m}( k_{1}- k_{2}) ( 1 + K_{1 2})}}\\{{}}&{{}}&{{\displaystyle + 4 D_{m}( k_{1}) D ( k_{1}+ p ) D ( k_{2}+ p ) D_{m}( k_{2}) D ( k_{1}+ k_{2}+ p ) K_{1 2},}}\\ \end{array}
[\dot x_i, \dot x_j]_\mathrm{Lie}= \sum_{k=1}^{d-a}c_{i,j,k}\, \dot y_k \text{for $1 \le i,j \le a$, where $\gamma(x_iZ,x_jZ) = [x_i,x_j] = \prod_{k=1}^{d-a}y_k^{\, c_{i,j,k}}$,}[\dot x_i, \dot y_j]_\mathrm{Lie}= [\dot y_j, \dot y_k]_\mathrm{Lie}= 0
\frac{1}{\epsilon}\int_0^t F(s+\epsilon,X_{s+\epsilon},X(s+\epsilon))-F(s,X_s,X(s))\, ds = &\frac{1}{\epsilon}\int_\epsilon^{t+\epsilon}F(s,X_{s},X(s)) ds - \frac{1}{\varepsilon}\int_0^t F(s,X_s,X(s))\, ds = &=\frac{1}{\epsilon}\int_t^{t+\epsilon}F(s,X_s,X(s)) ds - \frac{1}{\epsilon}\int_0^\epsilon F(s,X_s,X(s)) ds \, ,
\begin{aligned}f_i(x_i) -f_i(\breve{x}_i^k)+(x_i-\breve{x}_i^k)^T[ -A_i^T\widetilde{\lambda}^k &+\beta A_i^T\sum_{j=1}^{i}A_j(\widetilde{x}_j^{k}-x_j^k)+A_i^T(\widetilde{\lambda}^k-\lambda^k)]\geq 0,~\forall x_i\in\mathcal{X}_i. \end{aligned}
E_m \leq I(\rho(\omega_{\lambda})*\omega_{\lambda})=I(\rho(\omega_{\lambda})*u_{\delta})+I(\rho(\omega_{\lambda})*(\lambda*\tilde{v})) \leq I(\rho(u_{\delta})*u_{\delta})+I((\rho(\omega_{\lambda})+\lambda)*\tilde{v}) \leq I(u)+ \frac{\varepsilon}{4}+ \frac{\varepsilon}{4}\leq E_{m'}+\varepsilon
\Lambda_{\rm{4}}\left(a\right)=\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[a^2{\cal C}_{11}{\cal C}_{44}+a\left({\cal C}_{11}{\cal C}_{33}-2{\cal C}_{13}{\cal C}_{44}-{\cal C}_{13}^2\right)+{\cal C}_{33}{\cal C}_{44}\right]
f_k(Q_n\cdot(an,1)) =i^{nk}\bigg(\prod_{j=1}^n\frac1{a_j+a_j^{-1}}\bigg)^{(2n+1)(s+\frac12)}\!f_k(\left[\begin{smallmatrix}I_n &I_n &^tZ I_n Z I_n\end{smallmatrix}\right]\!\left[\begin{smallmatrix}C S &C S &-S C -S C\end{smallmatrix}\right]\!\left[\begin{smallmatrix}I_n -X &I_n & I_n & I_n\end{smallmatrix}\right]).
K_\lambda :=\min\big\{\eta_{1/2}^{\lambda/2}K_{0,\lambda},\, \eta_{1/2}^{\lambda/2}K_{1, \lambda}, \lambda^{-\lambda}(1 -\lambda)^{\lambda -1}2^{-5\lambda/2}(37-3\sqrt{65})^{\lambda/2}\big\}
V(x) = V\big(\rho(x)\big) = \left(\int_{\rho(x)}^\infty \frac{ds}{f_a^{n-1}(s)}\right) \left(\int_0^{\rho(x)}a_0(t) f_a^{n-1}(t) dt \right) - \int_0^{\rho(x)}\left( \int_t^\infty \frac{ds}{f_a^{n-1}(s)}\right) a_0(t) f_a^{n-1}(t) dt - H + ||\varphi||_\infty,
Y^*=\Big( \frac{(q^{n-1}-1)(q^{(n-1)/2}+1)(q^{(n+1)/2}-1)}{2(q-1)}, &\frac{(q^{n-1}-1)(q^{(n-1)/2}-1)(q^{(n+1)/2}-1)}{2(q-1)}, &(q^{2n-1}-1)-\frac{(q^{n-1}-1)}{q-1}\Big)
\mbox{There exists some $N$ such that $j>N\Rightarrow W_{i,j}=W_{i,j+1}$ for all $i$}. \mbox{There exists some $d$ such that $i>d\Rightarrow W_{i,j}=0$ for all $j$}. \textstyle \sum_{p\geq 0}W_{i+p,j+p}\geq \sum_{p\geq 0}W_{i+1+p,j+p}.
\frac{\left( -1\right)^k}{\Gamma \left( \frac 12\right)\Gamma\left( k\right)}\left( \frac t2\right)^{2k-1}a_{d+2k-1}\left[ 2\ln\left(\frac t2\right) -\left( \Psi \left( 1\right)-\sum\limits_{l=1}^{k-2}\frac1{l-k+1}\right) \right]
\psi_{\beta_+,t_+}(t) := \eta_{\beta_+,t_+}(t)\eta_{-\beta_+,t_+}(\alpha(t)), t\neq t_+\,, \psi_{\beta_-,t_-}(t) := \eta_{\beta_-,t_-}(t)\eta_{-\beta_-,t_-}(\alpha(t)), t\neq t_-\,.
\begin{cases}\nabla\cdot \mathcal B(f)=f\ \mbox{in}\ \Omega, \ \ \mathcal B(f)=0 \{\rm{on}}\ \partial\Omega, \big\|\mathcal B(f)\big\|_{W^{1,p}_{0}(\Omega)}\leq C(p)\big\|f\big\|_{L^p(\Omega)}.\end{cases}
{\cal H}= \int d^d x \Bigl[{1\over 2}\sum_i (\partial_\mu \Phi_{i})^2 +{1\over 2}\sum_i r_i \Phi_{i}^2 +{1\over 4!}\sum_{ijkl}u_{ijkl}\; \Phi_i\Phi_j\Phi_k\Phi_l \Bigr],