harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d s^{2}= d t^{2}+{\frac{1}{4}}t^{2}( \Sigma_{1}^{2}+ \Sigma_{2}^{2}+ \Sigma_{3}^{2}) + 1 6 ( A_{1}^{2}\alpha_{1}^{2}+ A_{2}^{2}\alpha_{2}^{2}+ A_{3}^{2}\alpha_{3}^{2}) .
|u_m|+|u_{m-1}|=(u_{m-1}-u_m)+2u_m \leq (u_{m-1}-u_m)+2\ell & \leq (u_{m-1}-u_m)+2(u_{m-1}-u_m) &=3(u_{m-1}-u_m) &=3|u_m-u_{m-1}|.
\Gamma_H~=~\frac14 \pi^2 r_0^2 (a+b) [\frac14 + \frac{9}{16}(a+b)^2]~\left(e^{\frac{\pi}{2}(a+b)}-1 \right)^{-1}\left( e^{\frac32 \pi(a+b)}+ 1\right)^{-1}~\frac{d^3k}{(2\pi)^3}
e^{\textstyle - J \alpha_{-}\Phi ( \sigma , \tau )}= c_{J}\sum_{m = - J}^{J}{\frac{( - 1 )^{J - m}\lambda_{m}^{( J )}( \varpi )}{\sqrt{\lfloor \varpi \rfloor}\sqrt{\lfloor \varpi + 2 m \rfloor}}}\psi_{m}^{( J )}( x_{+}){\overline{\psi}_{m}^{( J )}}( x_{-})
a_s \partial_1^2 \partial_s u_s^{\tau}- E_s [ a_s, (\cdot)_{\tau}] \diamond \partial_1^2 \partial_s v_s - \partial_s a_s E_s [ a_s, (\cdot)_{\tau}] \diamond \partial_1^2 \partial_{a_0}v_s = a_s \diamond \partial_1^2 \partial_s u_s^{\tau},
\triangle (x)=\frac{1}{C_{\alpha,\beta}}\{|B_{R_0, R_{\alpha}}^{0,\alpha}\cup B_{R_0, R_{\beta}}^{0,\beta}(x)|-|B_{R_0, R_{\alpha}}^{0,\alpha}\cup B_{R_0, R_{\beta}}^{0,\beta}|\},x\in{\Bbb R^2}.
{\bar{\eta}^{a}}_{\mu \nu}= -{\bar{\eta}^{a}}_{\nu \mu}, \qquad{\bar{\eta}^{a}}_{\mu \nu}{\bar{\eta}^{b}}_{\mu \sigma}={\delta}_{a b}{\delta}_{\nu \sigma}+{\epsilon}_{a b c}{\bar{\eta}^{c}}_{\nu \sigma}.
\begin{array}{l}{2 m \le \sum_{i}\deg_{i}( A ) = \sum_{\mu}| \alpha_{\mu}| + \sum_{\nu}( 2 + | \beta_{\nu}| ) = n + 2 v}\\{\le n + \sum_{\nu}| \beta_{\nu}| = 2 n - \sum_{\mu}| \alpha_{\mu}| \le 2 n - u \le 2 n .}\\ \end{array}
(S\circ T)_{z, x}(v) =(\pi_z\circ S\circ T\circ\iota_x)(v) &=\Big(\pi_z\circ S\Big)\Big(\sum_{y\in Y}\iota_y\big(\pi_y(T(\iota_x(v)))\big)\Big) &=\sum_{y\in Y}\big(\pi_z\circ S\circ\iota_y\circ\pi_y\circ T\circ\iota_x\big)(v) &=\sum_{y\in Y}(S_{z, y}\circ T_{y, x})(v)
\begin{matrix}&\sigma^{*}&=&H(B^{*})\\ \Rightarrow&R\cdot\sigma\cdot R^{T}&=&H(F^{*}\cdot(F^{*})^{T})\\ \Rightarrow&R\cdot H(B)\cdot R^{T}&=&H(R\cdot F\cdot F^{T}\cdot R^{T})\\ \Rightarrow&R\cdot H(B)\cdot R^{T}&=&H(R\cdot B\cdot R^{T}).\end{matrix}
\psi(s)=\left (n_1\,s\,\left [K^{2}\,\log\left (1+\frac{K^{q}\,n_2}{s}\right )\right ]^{1-q/2}\right )\vee\left ( n_1\,s^{2/q}\right )\vee \left (n_1\,n_2\,K^{2}\right ).
\sum_{a\leq k<b}\frac 1k =\sum_{\lfloor a\rfloor+1\leq k<\lfloor b\rfloor}\frac 1k+O(1)=\log(\lfloor b\rfloor)-\log(\lfloor a\rfloor+1)+O(1) &\leq\log b-\log a+O(1).
\alpha_j\alpha_k + \alpha_k\alpha_j = 2\delta_{jk}I_4,j,k\in\{0,1,2,3\}, \alpha_0 := \beta, \gamma_5 \alpha_j = \alpha_j \gamma_5, ~j \in \{1, 2, 3\}, \gamma_5 \beta = -\beta \gamma_5.
T_\alpha^\Delta \Big(k(t)v(t)K^p(t)\Big) \leq\beta k^\sigma(t)v^\sigma(t)r(t)f(t)K^{p-\alpha}(t)+(p-\alpha+1)k^\sigma(t)v^\sigma(t)r(t)f(t)\big(K^\sigma(t)\big)^{p-\alpha}&\leq(p-\alpha+\beta+1)k^\sigma(t)v^\sigma(t)r(t)f(t)\big(K^\sigma(t)\big)^{p-\alpha}.
D(g_1 v_2,g_1 v_5,g_1v_7)D(g_1v_4,g_1v_6,g_1v_8)(g_1v_1\times g_1v_3)\times (g_1X\times g_1Y), = c(g_1)^3 D(v_2,v_5,v_7)D(v_4,v_6,v_8)g_1((v_1\times v_3)\times (X\times Y)).
Y_n(x)= \frac{\sigma_1\sqrt{2k_n(x)\log\log k_n(x)}}{b\sigma\sqrt{2n\log\log n}}\ \ \ \ \ \ \ \ \ \ \ \Z_n(x)= \frac{\log q_{k_n(x)}(x)-abn}{b\sigma\sqrt{2n\log\log n}}.
\Big\| \sup_{\epsilon > 0}\Big| \mathop{\mathop{\sum_{Q \in \mathcal{D}:\, Q \subset P}}_{Q^a = F}}_{\ell(Q) > \epsilon}\epsilon_Q D_Q f \Big|\, \Big\|_{L^p(\mu)}^p\lesssim \int_P \Phi_P^p\,d\mu \lesssim \mu(P).
M\models \forall \; x \; \theta_{m}(x, \overline{b})\leftrightarrow \theta_{n}(x, \overline{a}) \leftrightarrow \theta_{n}(x, \overline{a}^{\prime})\leftrightarrow \theta_{m}(x, \overline{b}^{\prime}) & c=g_{m}(\overline{b}) & c^{\prime}= g_{m}(\overline{b}^{\prime})
\mathbf{T^2_{k}}= \int_{B_{1/2}}\nabla( \eta_r(U_k-U_0)) \cdot \nabla (U_0+V) \,dx = \int_{B_{1/2}}\eta_r \nabla (U_0+V)\cdot \nabla (U_k-U_0)\,dx & + \int_{B_{1/2}}(U_k-U_0)\nabla \eta_r\cdot \nabla (U_0+V)\,dx.
u(t)=\sqrt{-\frac{t}{2}}\left(1 -\frac{1}{8}(-t)^{-3}-\frac{73}{128}(-t)^{-6}-\frac{10657}{1024}(-t)^{-9}\right. \left.\hphantom{u(t)=}{}-\frac{13912277}{32768}(-t)^{-12}-\frac{8045883943}{262144}(-t)^{-15}-\frac{14518451390349}{4194304}(-t)^{-18}+\cdots \right),
\begin{aligned}|h(x(t)+\delta u(t))-h(z(t)+\delta u(t))| \le \, L_r |x(t)-z(t)| |h(z(t)+\delta u(t))-h(x_a(t)+\delta u(t))|\le \,\gamma L_r |\epsilon(x_a(t),t)|.\end{aligned}
\sum_{i=1}^n \lambda_i ||\partial^{\alpha}\phi_i||_{L^2}^2 = (\mathcal{E}_{k_n}^{\alpha}) = \sum_{j=1}^{+\infty}\langle \mathcal{E}_{k_n}^{\alpha}\phi_j,\phi_j\rangle_{L^2}\leq \sum_{j=1}^{+\infty}\langle \mathcal{E}_{k}^{\alpha}\phi_j,\phi_j\rangle_{L^2}= (\mathcal{E}_{k}^{\alpha})
\mathbb{\hat{E}}[\mathbf{1}_O(B_t)\mathbf{1}_O(B_{t+s}-B_t)]=\lim\limits_{n\rightarrow\infty}\mathbb{\hat{E}}[\psi^n(B_t)\psi^n(B_{t+s}-B_t)]= \lim\limits_{n\rightarrow\infty}\mathbb{\hat{E}}[\psi^n(B_t)]\lim\limits_{n\rightarrow\infty}\mathbb{\hat{E}}[\psi^n(B_{t+s}-B_t)] = \mathbb{\hat{E}}[\mathbf{1}_O(B_t)]\mathbb{\hat{E}}[\mathbf{1}_O(B_{t+s}-B_t)],
F_{a b}~ = ~ - \,{\frac{\mathrm{\s l ~ g}_{1}}{4 \! \mathrm{\s l ~ g}}}\left( E_{a a}+ E_{\theta ( a ) \theta ( a )}\right) - \,{\frac{\mathrm{\s l ~ g}}{2 \! \mathrm{\s l ~ g}_{1}}}\, E_{b b}\, + \,{\frac{1}{n}}\left({\frac{\mathrm{\s l ~ g}_{1}}{2 \! \mathrm{\s l ~ g}}}\, + \,{\frac{\mathrm{\s l ~ g}}{2 \! \mathrm{\s l ~ g}_{1}}}\right) 1
u(\mathbf{p,}i)\simeq \left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) \left[ \begin{array}{c}1 \frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdot p})}{2m}\end{array}\right] \varphi_{i}=\left[ \begin{array}{c}\left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) \frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdot p})}{2m}\end{array}\right] \varphi_{i}
\| u^{*}(\varphi(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\sigma (a)}^m) u & - \psi(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\sigma (a)}^m\| < \varepsilon
{\hat{\gamma}}_{( \lambda , \mu , \alpha )}^{r}\Rightarrow{\hat{\gamma}}_{( \lambda , \mu , \alpha )}^{r}\eta_{r}^{\lambda \mu}, \quad{\hat{\gamma}}_{r}^{( \lambda , \mu , \alpha )}\Rightarrow \eta_{r}^{\lambda \mu}{\hat{\gamma}}_{r}^{( \lambda , \mu , \alpha )}={\hat{\gamma}}_{r}^{( \lambda , \mu , \alpha )}\eta_{r}^{\lambda \mu}, \quad{( \eta_{r}^{\lambda \mu})}^{+}= \eta_{r}^{\lambda \mu},
\tilde{A}_0^* : = \sup_{t > 0}\bigg( \int_t^{\infty}U_1^q(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}V_1^{- \frac{1}{p}}(t), \tilde{A}_1^* : = \sup_{t > 0}W^{\frac{1}{q}}(t) \bigg( \int_t^{\infty}U_1^{p'}(\tau) V_1^{-(2+p')}(\tau) V^{-2}(\tau)v(\tau)\,d\tau \bigg)^{\frac{1}{p'}};
F_{2}=\frac{x}{b}T^{-1}(H_{2})=\frac{b}{x}T^{-1}(F_{1}\oplus F_{2})=\frac{b}{x}T^{-1}(F_{1})\oplus\frac{b}{x}T^{-1}(F_{2})=H_{1}\oplus H_{2}
\begin{array}{cl}L_{\mu\nu}=-ig\frac{\tau_i}{2}W^i_{\mu\nu}+i\frac{g'}{2}B_{\mu\nu};~~~R_{\mu\nu}=ig'B_{\mu\nu}begin{align*}4mm] D_{\mu}\phi=({\partial}_{\mu}-ig\frac{{\tau}_{i}}{2}W^{i}_{\mu}-i\frac{g'}{2}B_{\mu})\phi.\end{array}
\frac{\sum_{k=1}^m T^{N,1}_k}{\sum_{k=1}^m(T^{N,1}_k+T^{N,0}_k)}\ge\frac{\sum_{k=1}^m (N+\tau^{N,1}_k)(c)_k}{\sum_{k=1}^m(N+\tau^{N,1}_k)(c)_k+\tau^{N,0}_k}
\frac{1}{2}(\phi(2\xi)-\phi(2\beta))+\beta\phi'(\beta)-\phi(\beta+\xi-x)-x\phi'(\beta)+\phi(\beta) &=\frac{1}{2}\phi(2\xi)+\frac{1}{2}\phi(2\beta)-x\phi'(\beta)-\phi(\beta+\xi-x) &\geq \phi(\xi+\beta)-x\phi'(\beta)-\phi(\beta+\xi-x)=:F_2(x,\xi),
\int_{-\infty}^{\infty}x^{2}e^{-\frac{1}{2}ax^{2}}dx=-2\frac{d}{da}\int_{-\infty}^{\infty}e^{-\frac{1}{2}ax^{2}}dx=-2\frac{d}{da}(\frac{2\pi}{a})^{\frac{1}{2}}=(\frac{2\pi}{a})^{\frac{1}{2}}\frac{1}{a}
(\zeta) = \frac{4l}{k}\underset{i=1}{\overset{k}{\sum}}(p_i - \frac{1}{2})^2 p_i(1-p_i) \leq \frac{l}{k}\underset{i=1}{\overset{k}{\sum}}\left(p_i - \frac{1}{2}\right)^2.
S_2 \leq \zeta(2-2\theta_3-0.01)^{35}BX\prod_{i=1}^{R}\exp\Big(\frac{k^2-1}{2}\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big)\sum_{p\in \cup P_j}\frac{A(p,1)^2\log p}{p}&\leq \zeta(2-2\theta_3-0.01)^{35}BX(\frac{\log X}{10^{2M}}+O(1))\exp\Big(\frac{k^2-1}{2}\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big).
S(k) :=\sum_{i=0}^k\frac{1}{q_i},T(k):=\sum_{i=0}^k\frac{i}{q_{i}}begin{align*}2mm]M(k) :=\prod_{i=1}^k \left(\frac{\rho_{i-1}}{\rho_i}\right)^{\frac{1}{q_i}}.
\widehat{(\mathfrak{a}^i)}_{\mathfrak{a}}=\varphi_{\mathfrak{a}^i}(\mathfrak{a}^i)\cdot \hat \Lambda_{\mathfrak{a}}=\hat \Lambda_{\mathfrak{a}}\cdot \varphi(\mathfrak{a}^i)= \hat{\mathfrak{a}}^i.
\operatorname{Var}\left(\frac{S^x_n}{n}\right) = E\left(\left(\frac{S^x_n}{n}\right)^2\right) - \left( E\left(\frac{S^x_n}{n}\right) \right)^2 & = \frac{1}{n^2}\left( \sum_{j, k = 1 \atop k \neq j}^n E(X^x_j) E(X^x_k) + \sum_{k=1}^n E\left((X^x_k)^2\right) \right) - x^2 & = \frac{1}{n}x (1-x),
h^{\prime}(\theta) =\frac{\pi}{2}\sum_{i=1}^{n}F_{i}(\tilde e\cos(\pi\theta/2)+e\sin(\pi \theta/2))(e_{j}\cos(\pi \theta/2)-\tilde e_{j}\sin(\pi\theta/2)) &=\frac{\pi}{2}(\nabla F(\widetilde{W}_{\theta}),W_{\theta}),
z\,(q/z;q)_{\infty}\,\frac{\partial}{\partial z}\frac{1}{(q/z;q)_{\infty}}=-z\,\frac{\partial}{\partial z}\ln((q/z;q)_{\infty})=-\frac{1}{z}\sum_{j=1}^{\infty}\frac{q^{j}}{1-q^{j}z^{-1}}.
L ( \phi_{\mathrm{m}}, - \phi_{\mathrm{m}}) = 2 \ln \left[ \sinh \chi_{\mathrm{m}}\sin \phi_{\mathrm{m}}+ ( \sinh^{2}\chi_{\mathrm{m}}\sin^{2}\phi_{\mathrm{m}}+ 1 )^{1 / 2}\right] .
J_{2,3}=\int_{\frac{3\pi}{2}}^{3\pi -\theta_{0}}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho \sin\theta+1|^{\tau}\rho^{q-1}}=\int_{\theta_{0}}^{\frac{3\pi}{2}}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\varphi}{|\rho \sin\varphi+1|^{\tau}\rho^{q-1}}=J_{2,2}.
\begin{array}{l}{{\Phi_{2}\bar{\Phi}_{2}- q^{2}\bar{\Phi}_{2}\Phi_{2}= 1 ,}}\\{{\Phi_{1}\bar{\Phi}_{1}- q^{2}\bar{\Phi}_{1}\Phi_{1}= 1 + ( q^{2}- 1 ) \bar{\Phi}_{2}\Phi_{2},}}\\{{\Phi_{2}\Phi_{1}= q \Phi_{1}\Phi_{2},}}\\{{\Phi_{2}\bar{\Phi}_{1}= q \bar{\Phi}_{1}\Phi_{2},}}\\ \end{array}
R_{m}=\left(r_{\pi_{c}^{-1}\left(\pi_{d}\left(m\right)\right)}\negmedspace+\negmedspace\frac{1}{2}\log\negmedspace\left(\frac{p_{\pi_{s}^{-1}\left(\pi_{e}\left(m\right)\right)}}{p_{\pi_{c}^{-1}\left(\pi_{d}\left(m\right)\right)}}\right)\right)^{+}.
\widetilde{Z}( R ( S ) , \langle \cdot , \cdot \rangle_{H}, \langle \cdot , \cdot \rangle ) = \widetilde{Z}( R ( S ) , \langle \cdot , \cdot \rangle ){\cdot}\Psi ( R ( S ) , \langle \cdot , \cdot \rangle_{H}, \langle \cdot , \cdot \rangle )
(z_{R-k}-2)+\cdots+(z_1-2) +\chi(p'_1=-1)+\chi(p'_{R-k+1}=-1) &=(z_1+\cdots+z_{R-k})-2(R-k)+\chi(\alpha_R\neq 1)+\chi(\alpha_1\neq 1) &=(N-k)-2(R-k)+(2-\delta_\alpha)=N-2R+k+2-\delta_\alpha
\int_G \nu((g^{-1}\cdot \phi_1) \phi_2) \, d\eta(g) = \int_G \lambda((g^{-1}\cdot T\phi_1) T\phi_2) \, d\eta(g) =\lambda(\phi_1) \, \lambda(\phi_2) = \nu(\phi_1) \nu(\phi_2).
S_n = \lim_{m \to \infty}\sum_{r, s\in [-m, m]}\bigg(\sum_{(j,k)\in\Gamma_n}a_{j+r,k+s}\bigg)\xi_{-r, -s}&=\sum_{r, s\in \mathbb{Z}}\bigg(\sum_{(j,k)\in\Gamma_n}a_{j+r,k+s}\bigg)\xi_{-r, -s}=\sum_{r,s\in\mathbb{Z}}b_{n,r,s}\xi_{-r, -s}.
f_{\delta}=\prod_{i=1}^{r-1}(1-q^i)\times\sum_{\mu[1] \in \sigma(\delta^{*})}\cdots\sum_{\mu[r-1] \in \sigma(\delta^{*})}z^{\delta^{*}}\left(\prod_{j=1}^{r-1}C_{j}\Big(\mu[j+1],\mu[j];q,t\Big)z^{\mu[j]}\right)
\int_{|z+1| > e/|x|}[ f(x + x z) - f(x) - f'(x) x z I(|z| \leq 1)] \nu_U (\d z)= \int_{|z+1| > e/|x|}[ \log |1 + z| - z I (|z| \leq 1) ]\nu_U (\d z),
0 = \lVert \eta(x^*x) \rVert = \lVert \eta(xx^*) \rVert = \limsup_{n \to \infty}\lVert \alpha_n(a) \otimes \beta_n(b) \rVert &= \limsup_{n \to \infty}\left(\lVert \alpha_n(a) \rVert\,\lVert \beta_n(b) \rVert\right) = \left(\limsup_{n \to \infty}\lVert \alpha_n(a) \rVert\right) \lVert b \rVert = \lVert a \rVert\,\lVert b \rVert\ .
[M_{0j},X_k] = \frac{1}{2}[\big( d\Gamma(X_jP_0)+d\Gamma(P_0X_j)\big),d\Gamma (X_k)] &=\frac{1}{2}\big( d\Gamma(X_j[P_0,X_k] )+d\Gamma( [P_0,X_k]X_j)\big) &=\frac{i}{2}\big( d\Gamma(X_jV_k )+d\Gamma( V_kX_j)\big) .
\Pi_{00}(0,k) \approx \frac{e^2}{8 \beta}\left[ \beta k + \frac{8\ln [2 + 2 \cos (e_{N}\beta \tilde{A}_0 )]}{\pi}\exp \left\{{-\frac{\pi \beta k}{8 \ln [2 + 2 \cos (e_{N}\beta \tilde{A}_0 )]}}\right\}\right]
\nabla_q(h_{pk}P_{ijk \nu \nu}) = (\nabla_k h_{pq}) P_{ijk \nu \nu}+h_{pk}h_{q \ell}(P_{ijk \ell \nu}+ P_{ijk \nu \ell}) & +h_{pk}(D_q P_{ijk \nu\nu}-h_{qi}P_{\nu j k \nu\nu}-h_{qj}P_{i \nu k \nu\nu}-h_{qk}P_{ij \nu \nu \nu}),
\mathbb{E}\left( ||x^{k+1}(\theta) - x^*(\theta)||_\pi^2 |\mathcal{F}_k \right) \leq \mathbb{E}||x^1(\theta) - x^*(\theta)||_\pi^2 -\sum_{i=1}^k 2\alpha_i \mathbb{E}\left(\mathbb{E}_\pi\left(f(x_i(\theta),\theta) - f(x^*(\theta),\theta) | \mathcal{F}_i\right) \right) + \sum_{i=1}^k \alpha_i^2 \left(G^2 + V^2\right)
|f(X_1,\dots, X_n)-f(X_1',\dots, X_n')| \leq D m_r \|C_r\|_{\infty}^3 \delta^2\frac{\|\Sigma\|_{\infty}^{1/2}+\sqrt{\delta}}{\sqrt{n}}\biggl(\sum_{j=1}^n \|X_j-X_j'\|^2\biggr)^{1/2}.
\sum_j \partial f_{<j-2}\cdot J^{s-2}\partial g_j = \partial f \cdot J^{s-2}\partial g - \sum_j \partial f_{\ge j-2}\cdot J^{s-2}\partial g_j = \partial f \cdot J^{s-2}\partial g - \sum_j \partial f_j \cdot J^{s-2}\partial g_{\le j+2}.
{\bf \tau}(E)=\frac{1}{\det{\bf \tau}^{-1}}\left( \begin{array}{cc}-\frac{C}{C_2^2}- I_5 - 2 \mu E I_3 - 4 \mu^2 E^2 I(E) I_3 + 2 \mu E \, I(E) -\frac{1}{C_2}I_3 + 2 \mu E \, I(E) - \frac{1}{C_2}-I(E)\end{array}\right),
\mu^{M_\alpha}(\sigma\otimes\cdot)=c_{s_{\alpha}}'\,\frac{(1-X_{\alpha})(1-X_{\alpha}^{-1})}{(1-q_{\alpha}^{-1}X_{\alpha})(1-q_{\alpha}^{-1}X_{\alpha}^{-1})}\cdot\frac{(1+X_{\alpha})(1+X_{\alpha}^{-1})}{(1+q_{\alpha^*}^{-1}X_{\alpha})(1+q_{\alpha^*}^{-1}X_{\alpha}^{-1})}.
(\mathbb E \|F\|^p \star \nu)^{\frac 1p}= \sup_{G\in Y^*:\|G\|\leq 1}\mathbb E \langle F, G\rangle\star \nu = \sup_{G\in Y^*:\|G\|\leq 1}\mathbb E \langle F \star \bar{\mu}, G \star \bar{\mu}\rangle \lesssim_{p}\sup_{\xi \in L^{p'}(\Omega; X^*):\|\xi\|\leq 1}\mathbb E\langle F \star \bar{\mu},\xi\rangle = (\mathbb E \|F\star \bar{\mu}\|^p)^{\frac 1p}.
\Delta^{(\ell)}= \frac{1}{\ell}\sum_{i=1}^{\ell}\mathbb{E}[ d( \underline S_i, \hat{\underline S}_i ) ]\geq \frac{1}{\ell}\sum_{i=1}^{\ell}\mathbb{E}[d( \underline S_i, \hat{\underline s}^{\star}( \underline X_i^L, Y_i^L))],
\iint_{\Omega\times\mathbb{R}^3}F(t_*,x,v)\varphi(x,v) dxdv \leq \;\; \|\psi\|_{L^\infty(Q_{t_*})}\iint_{\Omega\times\mathbb{R}^3}\big|f_0(x,v)\big| dxdv \leq \;\; \|f_0\|_{L^1(\Omega\times\mathbb{R}^3)}.
f_{j-1}\left(u,\sum_{i=1}^N\frac{1}{N}\int f_j(u,\eta_j, x)K(\frac{x-X_i}{h_N})\frac{1}{h_N}dx, X \right) \geq f_{j-1}\left(u, \frac{1}{N}\sum_{i=1}^N f_j(u,\eta_j, X_i), X \right).
I_{4,r}=\sum_{\xi=0}^{r}\frac{r!}{(r-\xi)!}\left[\frac{(-1)^{r-\xi-1}}{g_0^{r-\xi}}e^{1/g_0}\textrm{Ei}\left(\frac{-1}{g_0}\right)+\right. &\left.\sum_{\nu=1}^{r-\xi}\frac{(-1)^{r-\xi-\nu}(\nu-1)!}{g_0^{r-\xi-\nu}}\right]
(h^{2}- 2h-\mu)(h-4)^{n}=h^{n+2}-2(2n+1)h^{n+1}&+\sum_{k=0}^{n-2}\bigg(-\mu{n\choose k}(-4)^{k}-2{n\choose k+1}(-4)^{k+1}&+{n\choose k+2}(-4)^{k+2}\bigg)h^{n-k}&+(-\mu n(-4)^{n-1}-2(-4)^{n})h-\mu(-4)^{n}.
\pi_{A}^{a \, \prime}\, = \, \phi_{A}^{\ast \, a}\, \, , \, \, \, \, \phi_{A}^{\prime}\, = \,{\bar{\phi}}_{A}\, \, \, \, , \, \, \, \, \phi_{A}^{\ast \, a \, \prime}\, \, = \, \, \, \, \pi_{A}^{\, a}\, \, \, , \, \, \, \,{\bar{\phi}}^{A \, \prime}\, = \, \phi^{A}\, \, ,
\min_{(\mathbf{U},\mathbf{C})\in \Delta\times\mathbb{R}^{sN}}\sum_{i=1}^N\left\{\int_\Omega\left\|\nabla u_i(x)\right\|dx + \lambda\int_\Omega \left|I(x) - c_i\right|^2u_i(x) \dx\right\},
2m_{2}\partial(a_{2}e^{2m_{2}(\varphi-\eta)})\wedge\bar\partial(\varphi-\eta) &=2m_{2}(e^{2m_{2}(\varphi-\eta)}\partial a_{2}\wedge\bar\partial(\varphi-\eta)+2m_{2}a_{2}e^{2m_{2}(\varphi-\eta)}\partial(\varphi-\eta)\wedge\bar\partial(\varphi-\eta) ).
L U + \partial_s U = (P_{t-s}f ) \, \Gamma^{\mathrm{hori}}(\log(P_{t-s}f)) \leq (P_{t-s}f)\,\Gamma^{\mathrm{elli}}(\log(P_{t-s}f ))=V(s) L V + \partial_s V =2 (P_{t-s}f ) \, \Gamma_2^{\mathrm{mix}}( \log(P_{t-s}f )) \geq -\frac{2}{\nu}V(s).
E[(e^{\alpha \overline{X(t)}}- e^{\alpha \overline{X^n(t)}})^4] = E[e^{4 \alpha \overline{X(t)}}- 4 e^{- 3 \alpha \overline{X(t)}- \alpha \overline{X^n(t)}}+ 6 e^{- 2 \alpha \overline{X(t)}- 2 \alpha \overline{X^n(t)}}- 4 e^{- \alpha \overline{X(t)}- 3 \alpha \overline{X^n(t)}}+ e^{- 4 \alpha \overline{X^n(t)}}].
\{l\}\times\big[-\lfloor q/2\rfloor+1,\nu+\lfloor q/2\rfloor\big] \hbox{for}\,\, l=-\lfloor p/2\rfloor+1,...,1, &\hbox{and}\,\, l=\mu,...,\mu+\lfloor p/2\rfloor, \big[-\lfloor p/2\rfloor+1,\mu+\lfloor p/2\rfloor\big]\times\{l\}\hbox{for}\,\, l=-\lfloor q/2\rfloor+1,...,1, &\hbox{and}\,\, l=\nu,...,\nu+\lfloor q/2\rfloor,
\sum_{m=0}^{\infty}\frac{q^{m(m+\sigma)}}{(q)_m}=\frac{1}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left\{q^{10j^2+(1-4\sigma)j}-q^{10j^2+(11-4\sigma)j+3-2\sigma}\right\}\qquad \sigma=0,1,
S_{p}^{\left( 1\right)}\left( b,c:\chi\right) =\sum\limits_{n=1}^{ck}\left( -1\right)^{n}\overline{E}_{p,\overline{\chi}}\left( \frac{bn}{c}\right) , S_{p}^{\left( 2\right)}\left( b,c:\chi\right) =\sum\limits_{n=1}^{ck}\left( -1\right)^{n}\chi\left( n\right) \overline{E}_{p}\left(\frac{bn}{c}\right)
\frac{B(z)}{A(z)}= \frac{B'(s_j)}{A'(s_j)}- \frac{B(s_j)A''(s_j)}{2A'(s_j)^2}+ \frac{B(s_j)}{A'(s_j)(z-s_j)}+ \sum_{n\neq j}\frac{B(s_n)}{A'(s_n)}\bigg( \frac{1}{z-s_n}+ \frac{1}{s_n-s_j}\bigg).
-\log T+\sum_{k=0}^{T}\frac{1}{k+d}=-(\log T-\sum_{k=1}^{T}k^{-1})-(\sum_{k=0}^{T-1}(k+1)^{-1}-\sum_{k=0}^{T}(k+d)^{-1}) & \rightarrow\gamma-\sum_{k=0}^{\infty}((k+1)^{-1}-(k+d)^{-1})=-\psi(d).
\mathfrak{B}=\{a\in A:a^*a\preceq1\}. \tfrac{1}{2}\mathfrak{F}=\{a\in A:a^*a\preceq a\}. \mathfrak{F}=\{a\in A:a^*a\preceq2a\}. \mathfrak{c}=\{a\in A:a^*a\preceq na,n\in\mathbb{N}\}.
R(z) = S(z) \begin{pmatrix}1 0 0 0 z^{-\beta}0 0 0 z^{-\beta}\end{pmatrix}D_0^{-n}(z) \Psi_{\alpha}^T(n^3f(z)) \frac{-1}{4\pi^2}\begin{pmatrix}1 0 0 -2\alpha - \frac{1}{2}-1 0 \alpha(\alpha+\frac{1}{2}) 2 \alpha + \frac{1}{2}1 \end{pmatrix}E_{in}^{-1}(z)
f(v_{1},...,v_{n})=\sum_{j_{1}=1}^{d_{1}}\cdot\cdot\cdot\sum_{j_{n}=1}^{d_{n}}\sum_{k=1}^{d}A_{j_{1}\cdot\cdot\cdot j_{n}}^{k}v_{1j_{1}}\cdot\cdot\cdot v_{nj_{n}}b_{k}.
x = [u^*_0,\dots,u^*_{N-2},u^*_{N-1}]^* \in \mathbb{R}^{rN}\varepsilon = [v_{2T}^*,\dots,v^*_N]^* \in \mathbb{R}^{p\bar{N}}y = [x^*,\varepsilon^*]^* \in \mathbb{R}^{rN+p\bar{N}}U_l = [u_{2T-l},u_{2T+1-l},\dots,u_{N-l}] \in \mathcal{M}_{p \times \bar{N}}(\mathbb{R}).
d\alpha_n =\sum_{l=1}^{n-1}\left(\frac{-\bar w_l e^{i\psi}}{2\sqrt{1-|w|^2}}d w_l+ \frac{- w_l e^{i\psi}}{2\sqrt{1-|w|^2}}d \bar w_l\right)+\sqrt{1-|w|^2}ie^{i\psi}d\psi d\bar\alpha_n =\sum_{l=1}^{n-1}\left(\frac{-\bar w_l e^{-i\psi}}{2\sqrt{1-|w|^2}}d w_l+ \frac{- w_l e^{-i\psi}}{2\sqrt{1-|w|^2}}d \bar w_l\right)-\sqrt{1-|w|^2}ie^{-i\psi}d\psi\ .
1 \simeq e^{2}( 4 | e H | ) \int \frac{d^{4}\hat{q}}{( 2 \pi )^{4}}\frac{\mathrm{e}^{- \hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{1}{2 | e H | \hat{q}_{\parallel}^{2}+ m^{2}},
f(\mathbf{j}_0,\mathbf{j}_1,\dots, \mathbf{j}_{s-1}) := \mathbb{P}\left( \mathbf{J}^{t}=\mathbf{j}_0;\mathbf{J}^{t+1}=\mathbf{j}_1;\dots,\mathbf{J}^{t+s-1}=\mathbf{j}_{s-1}\right).
\|f_\ell\| =\|f*\int_0^1\omega^{t}_\ell*\psi_{t\ell^3}dt\|\leq \int_0^1 \|\omega^{t}_\ell*f_{t\ell^3}\| dt \leq \int_0^1 \|f_{t\ell^3}\| \|\omega^{t}_\ell\|_{L^1}\, dt\lesssim [f]_\beta \int_0^1 (t^{\frac13}\ell)^\beta \, dt\lesssim [f]_\beta \ell^\beta,
\psi(x+\mu+\lambda) =\psi(x+\mu)+A_\lambda(x+\mu) &=\psi(x)+A_\mu(x)+A_\lambda(x+\mu) &=\psi(x)+A_\lambda(x)+A_\mu(x+\lambda) &=\psi(x)+A_{\mu+\lambda}(x).
Tr (\gamma_i)=0,i=1,2,3,4, Tr (\gamma_i\,\gamma_j)=\,4 \,\eta_{ij},i,j=1,2,3,4, Tr \left(\gamma_i\,\gamma_j\,\gamma_k\right)=0,i,j,k=1,2,3,4, Tr \left(\gamma_i\,\gamma_j\,\gamma_k\,\gamma_l\right)=\,4\left(\eta_{ij}\,\eta_{kl}\,-\eta_{ik}\,\eta_{jl}+\,\eta_{il}\,\eta_{jk}\right),i,j,k,l=\,1,2,3,4, ...
\Gamma \left( p , P_{o s}\right) = - \int \frac{d^{4}p^{\prime}}{\left( 2 \pi \right)^{4}}K \left( p , p^{\prime}, P_{o s}\right) G_{1}\left( p_{1}^{\prime}\right) G_{2}\left( p_{2}^{\prime}\right) \Gamma \left( p^{\prime}, P_{o s}\right) .
\sum \limits_{k=1}^{\infty}\int \limits_0^{2 \pi}\left( \frac{k a_k- b_k}{1+k^2}\right) R(t) \sin k t\dt - \sum \limits_{k=1}^{\infty}\int \limits_0^{2 \pi}\left( \frac{a_k+ k b_k}{1+k^2}\right) R(t) \cos k t \dt.
X_1=\begin{pmatrix}n j\cr0 m\end{pmatrix},X_2=\begin{pmatrix}0 (m-n)z_2\cr0 0\end{pmatrix},\varphi(X_1)=\begin{pmatrix}z_1 z_2\cr0 z_4\end{pmatrix},\varphi(X_2)=\begin{pmatrix}y_4 0\cr0 y_4\end{pmatrix}.
\mathcal{F}_2''(\bar X, t)= \left| \int_{S^N(t)}d y \, dw\,\frac{f_0^N( y, w)}{|\bar X- Y^N(t)|^2}\ -\int_{S^{N+1}(t)}dy \, dw\,\frac{f_0^{N+1}(y,w)}{|\bar X-Y^{N+1}(t)|^{2}}\right|
\partial^{(a_{n+1}}K^{a_1a_2\ldots a_{n})}=0,n=2r+1, &\partial^{(a_{2s}}K^{a_1 a_2...a_{2s-1})}+ M\sum\limits_{k=s}^{r}(-1)^{s+k+1}{2(2k+1)!\over(2k-2s+1)!(2s)!}U_k^{a_1 a_2...a_{2s}}=0, &s=0, 1,2,...,r.
[a_l,a_m,a_k,a_{\infty}]=\epsilon(k,l,m)\left( \frac{\theta[\eta_{S_l \circ U_\eta}](\Omega)\theta[\eta_{T_m \circ U_\eta}](\Omega)}{\theta[\eta_{S_m \circ U_\eta}](\Omega)\theta[\eta_{T_l \circ U_\eta}](\Omega)}\right)^2,
N \Big( \frac{N_1}{N_+}\Big)^{(5-2\alpha)+\varepsilon}\Big | \int (f_{N_1,L_1}\ast g_{N_2,L_2})\cdot h_{N,L}\Big| \lesssim N_1^{-c_1(\varepsilon)}[\min(N, 1)]^{c_2(\varepsilon)}~(L_1L_2L^{1-})^{\frac{1}{2}}&\|f_{N_1,L_1}\|_{L^2}\|g_{N_2,L_2}\|_{L^2}\|h_{N,L}\|_{L^2},
d(\mathcal{U}((\overline{w}_{\i,\widehat{\eta}})_{\i=1}^{\infty}),\mathcal{U}((\overline{v}_{\i,\widehat{\eta}})_{\i=1}^{\infty}))\leq~cK_{\eta}((\overline{w}_{\i,\widehat{\eta}})_{\i=1}^{\infty},(\overline{v}_{\i,\widehat{\eta}})_{\i=1}^{\infty})),
\lim_{n\rightarrow +\infty}\mathbb E\left[e^{i\sum_{j=1}^m\theta_{j}a_n^{-1}U_{\lfloor nt_j\rfloor}}\right] =\mathbb E\left[\Phi_{(c_0+c_1)G^+,(c_0-c_1)G^-,\beta}(1)\right],
\ell(a,b,\theta)= n[\log(2\theta) - \log \{\operatorname{B}(a,b) \}]-3 \sum^n_{i=1}\log (x_{i})-a\theta\sum^n_{i=1}\frac{1}{x^2_i}&+(b-1)\sum^{n}_{i=1}\log\left\{1- \exp\left(-\frac{\theta}{x_i^2}\right)\right\}.
\Vert f\Vert_{U^{d}(N)}^{2^d}= \frac{1}{N^{d+1}}\sum\limits_{x,h_1,\dots,h_d}\prod\limits_{\boldsymbol{\omega}\in \{0,1\}^d}\mathcal{C}^{\vert \boldsymbol{\omega}\vert}f(x+\mathbf{h}\cdot\boldsymbol{\omega}),
\|u\|_{Z}:=\sup_{t>0}(1+t)^{-\frac{1}{2}}e^{\delta t}\left(\|\Delta^{\frac{1}{2}}u(\cdot,t)\|_{L^{2}(X)}+\|u(\cdot,t)\|_{L^{2}(X)}+\|u_{t}(\cdot,t)\|_{L^{2}(X)}\right),
\hat{\lambda}( z ) \sigma^{\mu}- \tilde{\hat{w}}( z ) \sigma^{\mu}+ \sigma^{\mu}\hat{w}( z ) = \hat{\lambda}( z ) \sigma^{\mu}- \sigma^{\nu}\hat{w}_{\nu}^{~ \mu}( z ) = \sigma^{\nu}\partial_{\nu}h^{\mu}( z ) \, ,
\Delta=\nabla\xi^{m}\cdot\nabla\xi^{n}\frac{\partial^{2}}{\partial\xi^{m}\partial\xi^{n}}+\nabla^{2}\xi^{m}\frac{\partial}{\partial\xi^{m}}=g^{mn}(\frac{\partial^{2}}{\partial\xi^{m}\partial\xi^{n}}-\Gamma_{mn}^{l}\frac{\partial}{\partial\xi^{l}}),
X_{k+1,n}=\begin{cases}X_{k,n}+Y_{k+1,n}-1 X_{k,n}>0 k = 0, \dots, g-1, 0 X_{k,n}=0 k = 0, \dots, g-1, X_{k,n}+Y_{k+1,n}k = g, \dots, c-1.\end{cases}
\langle E\rangle=\sum_{s}E_{s}P_{s}=\frac{1}{Z}\sum_{s}E_{s}e^{-\beta E_{s}}=-\frac{1}{Z}\frac{\partial}{\partial\beta}Z(\beta,E_{1},E_{2},\cdot\cdot\cdot)=-\frac{\partial lnZ}{\partial\beta}
M_1=\sum_{\begin{subarray}{I}(u_{11},...,u_{1r_1})\in (\mathbb{F}_q^*)^{r_1}, a_{11}u_{11}+...+a_{1r_1}u_{1r_1}=b_1.\end{subarray}}N\Big(\mathrm{x}^{E^{(1)}_i}=u_{1i},\ 1\leq i\leq r_1\{\rm and}\x_{n_1+1}...x_{n_3}=0\Big). \ \ (3.7)
s_n(\mu_n | \nu_{\widehat\theta, n}^N) = \sup_{U\in C_b( \widehat \Sigma_{N,n})}\Big\{\int U(\xi,\omega) d\mu_n(\xi,\omega) - \log \int e^{U(\xi,\omega)}d\nu_{\widehat\theta, n}^N (\eta,\xi)\Big\}.
\epsilon_{i2}\alpha_2 =\gamma_{\sigma_2(i)}- \beta_i=\epsilon_{\sigma_2(i)j}\alpha_j + \beta_{\sigma_j\circ\sigma_2(i)}- \beta_i \epsilon_{\sigma_j(i)2}\alpha_2 =\gamma_{\sigma_j(i)}- \beta_{\sigma_j\circ\sigma_2(i)}=\epsilon_{ij}\alpha_j + \beta_{i}- \beta_{\sigma_j\circ\sigma_2(i)}.
\mathbb{P}\bigg(\Big|\Big|\Lambda^{\big(\mathsf{B}_V^{(n)}(i),m\big)}_{\mathbf{Z}|\mathbf{U}=\mathbf{u}}-Q^n_{Z|U=\mathbf{u}}\Big|\Big|_{\mathsf{TV}}>e^{-n\delta_1}\bigg)\leq e^{-e^{n\delta_2}},
\tilde R^p_{n,H,f}(V):=\sum_{\substack{j_1,\ldots,j_{p}\geq1,j_{p+1}\geq0 j_1+\ldots+j_{p+1}=n-1}}\!\, \Big( T^{H,H_{j_1},H,\ldots,H}_{(fu^p)^{[p]}}(V(H+V-iI)^{-1}\tilde{V}^{j_1-1}, \ldots,\tilde{V}^{j_p})\,\tilde{V}^{j_{p+1}}&-T^{H,\ldots,H}_{(fu^p)^{[p]}}(\tilde{V}^{j_1},\ldots,\tilde{V}^{j_p})\tilde{V}^{j_{p+1}}\Big)
E_{n_{1}, n_{2}}^{d}( w_{1}, w_{2}) = k^{2 ( n_{1}+ n_{2}- 1 ) - d}\int_{0}^{\infty}{d}x x^{\frac{d}{2}}\dot{P}^{2}( P + w_{1})^{- n_{1}}( P + w_{2})^{- n_{2}}