harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M^{-(a_1+a_2+a_4)(l+1)+a_3(l+1)}\sum_{m=4}^{N(\beta_2)}c_0^{\frac{m}{2}}M^{\frac{a_1(l+1)}{2}m}\alpha^m|J_m^{l+1}(\beta_1)-J_m^{l+1}(\beta_2)|_{l+1}\le \beta_1^{-\frac{1}{2}}.
\int_{C_1}\rho^* \omega = \lim_{\varepsilon \rightarrow 0}\int_{-\pi}^{\pi}\int_{\epsilon}^{1}\rho^*\left(\frac{1}{h}dh\wedge d\theta\right) = \lim_{\varepsilon \rightarrow 0}\int_{-\pi}^{\pi}\int_{\epsilon}^{1}\frac{1}{-h}d(-h)\wedge d\theta = \int_{C_1}\omega.
(\alpha_{opt}^{2})^{(ev)}=\sqrt{\frac{2\mid \beta \mid^{2}tanh \mid \beta \mid^{2}+2\mid \beta \mid^{2}cos~2\theta+1}{2\mid \beta \mid^{2}tanh \mid \beta \mid^{2}-2\mid \beta \mid^{2}cos~2\theta+1}}(\alpha_{opt}^{2})^{(o)}
Y_O (P, G) := \sup \{\| \phi_1 * \phi_2 \|_{q (P)}\mid ( \phi_1 , \phi_2 ) \in \mathcal{B}\}, Y_R (P, G) := \inf \{\| \phi_1 * \phi_2 \|_{q (P)}\mid ( \phi_1 , \phi_2 ) \in \mathcal{B}\}.
a_2=2a_1-a_0,\; a_3=\nu a_0 (2a_1-a_0)-a_1, \; a_4=\nu a_0 (2a_1-a_0)-a_0,\; d_1=\nu a_0(2a_1-a_0), d_2=(\nu a_0+1) (2a_1-a_0)-a_1,\; \mbox{where}\; \gcd(\nu a_0-1,a_1-a_0)=1.
{{\mathbf{\Omega}}_{pF}}= \left\{{\left.{\left({{{\sin}^2}{\theta_1}, \cdots ,{{\sin}^2}{\theta_k}}\right)}\right|\sum\limits_{i = 1}^k{{{\sin}^2}{\theta_i}}\leq{\delta^2},{{\sin}^2}{\theta_k}\leq \cdots \leq{{\sin}^2}{\theta_1}\leq 1}\right\}.
K_{\lambda \nu}^{\hspace{. 3 t r u e c m}\alpha}= - \frac{1}{2}( S_{\lambda \hspace{. 2 t r u e c m}\nu}^{\hspace{. 1 t r u e c m}\alpha}+ S_{\nu \hspace{. 2 t r u e c m}\lambda}^{\hspace{. 1 t r u e c m}\alpha}- S_{\lambda \nu}^{\hspace{. 3 t r u e c m}\alpha}) \; .
\left(\begin{array}{c}y_1 y_2\end{array}\right)\to\left(\begin{array}{rc}d c -b a\end{array}\right)\left(\begin{array}{c}y_1 y_2\end{array}\right),\quad\left(\begin{array}{c}\bar{y}_1 \bar{y}_2\end{array}\right)\to\left(\begin{array}{rc}a b -c d\end{array}\right)\left(\begin{array}{c}\bar{y}_1 \bar{y}_2\end{array}\right).
\sup_{x\in U}f :=\inf\{a\in \mathbb{R}:f^{-1}(a,\infty)=\varnothing\}\,, \mathrm{ess}\,\sup_{x\in U}f :=\inf\{a\in \mathbb{R}:\mu(f^{-1}(a,\infty))=0\}\,.
\left(\left\{\mathbf{\bar{G}}_{ij,kl}\mathbf{\bar{v}}_{kp,l}^{[\sf bs](\mathbf{s}^{[\sf d]})}\right\}_{k\in[1:K], l\in[1:M], \mathbf{s}^{[\sf d]}\in \mathcal{S}_{T_{\sf d}}^{[\sf d]}~}\right)
u(t,x)=\int_{\mathbb{R}}G^{\varepsilon}(t,x-y)\,u_0 (y)\, dy - \int_0^t\int_{\mathbb{R}}G_{x}^{\varepsilon}(t-s,\, x-y) f\bigl(s,y, u(s,y)\bigr)\, dy \,ds,
a_{t}= \frac{- \alpha_{1}+ 3 \alpha_{2}+ \sqrt{D}}{4 \sqrt{D}}\ , \a_{1}= a_{2}= \frac{- \alpha_{1}- \alpha_{2}+ \sqrt{D}}{4 \sqrt{D}}\ , \a_{\theta}= \frac{3 \alpha_{1}- \alpha_{2}+ \sqrt{D}}{4 \sqrt{D}},
{\cal A}^{*}({\cal E}) = \sum_{i}\sum_{j}\int_{- \infty}^{\infty}d \tau e^{- i{\cal E}\tau}u_{i}[{\tilde{x}}( \tau ) ] v_{j}^{*}[{\tilde{x}}( \tau ) ] \langle 0 \vert{\hat{a}}_{i}{\hat{b}}_{j}\vert \Psi \rangle .
\delta_{\Phi}^* (\eta_1, \ldots,\eta_{n-1}) = &(\phi_1^*\eta_1^*,\mbox{}\phi_2^*\eta_2^* - \eta_1^*\phi_1^*,\ldots,\phi_{n-1}^*\eta_{n-1}^* - \eta_{n-2}^*\phi_{n-2}^*,-\eta_{n-1}^*\phi_{n-1}^*)
\|(I-M_{2r})\l_{p_N}M_r\| \leq \sum_{n \geq N_r}\|\l_{g_{2n}f}\| &\leq \sum_{n \geq N_r}C_1 R^{-2n}J_D(f) = 2 C_1 R^{-2N_r}J_D(f) .
[A\xi\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u},c_0\frac{\partial}{\partial u}+\left(c_1-c_0t_0\wp(u-p)\right)\xi\frac{\partial}{\partial \xi}]=0,\,\,\,\,\,c_0,c_1\in \mathbb{C}.
Equ= p^s\!-\!(p\!-\!\tau)p^{s-t-1}\!+\!1\!-\!(\left\lceil\frac{(p\!-\!\tau)p^{s-t-1}}{(p\!-\!1\!-\!\tau)\lceil \frac{\delta}{\tau+1}\rceil \!+\! 1}\right\rceil \!-\!1)(\delta\!-\!1) \!-\!(\tau\!+\!1)p^{t}.
\begin{cases}\mathcal{L}_\lambda \psi = \lambda (p-1) a u_{2,\lambda}^{p-2}\psi + \gamma \psi \ \ \Omega, \frac{\partial \psi}{\partial \mathbf{n}}= \lambda (q-1) b u_{2,\lambda}^{q-2}\psi + \gamma \psi \ \ \partial \Omega, \end{cases}
\begin{array}{l}S_1 = xI_n - A(F) - \left({\frac{{I_k \otimes \textbf{1}_{m - 1}^T}}{0}}\right)\cdot \left({I_k \otimes (xI_{m - 1}- A(H_1 ))}\right)^{- 1}\cdot(I_k \otimes \textbf{1}_{m - 1}|0) \;\;\;\; = xI_n - A(F) - \Gamma_{A(H_1 )}(x)\left({\begin{array}{*{20}c}{I_k}{0^T}0 0 \end{array}}\right) \end{array}
e^{- S_{\mathrm{d u a l}}( A ) \;}= \int D B_{\mu}D B_{\mu}^{\dagger}D \eta_{\mu}e^{- \left( S ( B , ( \partial + i \eta ) B ) + \int d^{3}x A_{\mu}\epsilon^{\mu \nu \rho}\partial_{\nu}\eta_{\rho}\right) \;}\; ,
g(\tilde R(X,Y)Z,W) = g(R(X,Y)Z,W)+g(h(Y,W),h(X,Z)) -g(h(X,W),h(Y,Z)), g(\tilde R(X,Y)\xi,\eta) = g(R(X,Y)\xi,\eta)-g([A_\xi,A_\eta]X,Y),
\tilde v(x,t) = v_i(k),x\in[x_i,x_{i+1}],t\in[t_{k-1},t_k],i =\overline{0,m-1},k =\overline{0,n}, \hat v(x;k) = v_i(k)+v_{ix}(k)(x-x_i),x\in[x_i,x_{i+1}],i=\overline{0,m-1}, v^{\tau}(x,t) = \hat v(x;k),t\in[t_{k-1},t_k], \hat v^{\tau}(x,t) = \hat v(x;k-1) +\hat v_{\bar t}(x;k)(t-t_{k-1}),t\in[t_{k-1},t_k],k = \overline{1,n}.
s(u_0\vert\cdots\vert u_n) \quad= \sum_{\substack{0\leq i\leq n+1 n+1-i}}d_i(u_0\vert\cdots\vert u_n) \quad+s'(u_0\vert\cdots\vert u_n) t(u_0\vert\cdots\vert u_n) \quad= \sum_{\substack{0\leq i\leq n+1 n+1-i}}d_i(u_0\vert\cdots\vert u_n) \quad+t'(u_0\vert\cdots\vert u_n),
\int_{s}^t \beta(r)(\partial_t\psi(r,q_r)+\nabla_qV(r,q_r))\circ dq_r = \int_{s}^t \beta(r)\partial_t\psi(r,q_r) dq_r+\frac{1}{2}[\beta(r)\partial_t\psi(r,q_r),q_r]_{s,t}+(\beta V)(t,q_t)-(\beta V)(s,q_s)-\int_s^t \partial_t(\beta V)(r,q_r)dr
Z^{v_1,v_2}(t)=\exp\big\{\sum_{j=1}^2\int_0^th_j(s,x(s),v_1(s),v_2(s))dY_j(s)-\frac{1}{2}\sum_{j=1}^2\int_0^th^2_j(s,x(s),v_1(s),v_2(s))ds\big\}.
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-q^2;q^2)_k}{(q^2;q^2)_k}q^{n^2-(k+1)^2}begin{align*}7pt] =\sum_{k=0}^{n-1}\frac{(q;q^2)_n q^{k^2+k}}{(q^2;q^2)_k (q^2;q^2)_{n-k-1}(1-q^{2n-2k-1})}begin{align*}7pt] =\sum_{k=0}^{n-1}\frac{(q;q^2)_n q^{k^2+k}}{(q^2;q^2)_{n-1}(1-q^{2n-2k-1})}{n-1\brack k}_{q^2}.
\begin{aligned}\frac{d(r_{s/N}(x),x)^2}{2}= 1 - \cos d(r_{s/N}(x),x) + O\left(\frac{s^4}{N^4}\right) = c_0^2 \frac{s^2}{2N^2}+ O\left(\frac{s^4}{N^4}\right)\end{aligned}
(I-\tilde{T})^{2n}= 2^n (-\tilde T)^n \begin{pmatrix}(I-T)^n 0 0 (I-T)^n \end{pmatrix}, (I+\tilde{T})^{2n}= 2^n \tilde T^n \begin{pmatrix}(I+T)^n 0 0 (I+T)^n \end{pmatrix}.
\frac{d}{dt}P_{\lambda,d,p}(\eta_t(x)=1)= -P_{\lambda,d,p}(\eta_t(x)=1) &+\lambda\sum_{i=1}^dP_{\lambda,d,p}\big(\eta_t(x)=0,\eta_t(x-e_i)=1,x-e_i\rightarrow x\big).
[(\alpha, g), (\beta\gamma, 1_G), (\beta, k)] = [(\alpha, g)u,v(\beta\gamma, 1_G)u, v(\beta, k)] &= [(\alpha, g(\left.a\right|_{\beta\gamma})^{-1}), (a\cdot(\beta\gamma), 1_G), (a\cdot\beta, 1_G)],
a_i \! = -\gamma\left( 1- \epsilon_i \sum_{j \in \mathcal{I}}{\frac{1}{\epsilon_j}}\right) \quad\forall i \in \mathcal{I}b_i\! = -\frac{a_i}{2\epsilon_i\left( 1+ \gamma \sum_{j \in \mathcal{I}}{\frac{1}{\epsilon_j}}\right)}\forall i \in \mathcal{I}.
\sum_{k=1}^n\binom{m-n-1}{k-1}(k-1)!B_{n,k}(y_1, y_2, \ldots) = \sum_{k=1}^n\frac{(m-n-1)!}{(m-n-k)!}B_{n,k}(b_2/2, b_3/3, \ldots) & = \frac{(m-n-1)!n!}{m!}B_{m,m-n}(b_1, b_2, b_3, b_4, \ldots)
{}B_{ij}^{(u)}=\sup_{t\in (t_{u-1},t_{u})}B_{ij}(t)<+\infty\mbox{and}K_{ij}^{(u,v)}=\inf_{(t,s)\in (t_{u-1},t_{u})\times (t_{v-1},t_{v})}K_{ij}(t,s)>-\infty .
- 1 2 b^{\prime}\frac{a^{\prime \prime}a^{2}}{a^{3}}+ 1 2 b^{\prime}\frac{a^{4}}{a^{4}}- \frac{6}{\kappa}a a^{\prime \prime}- \frac{C}{2}\phi \phi^{\prime \prime \prime \prime}= 0 ,
\Omega=(-1)^{\frac{n}{2}}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\frac{\det(h)}{\det(g)}dv=(-1)^{\frac{n}{2}}\frac{2}{vol(S^{n})}\frac{\det(h)}{\det(g)}dv,
\delta^{-\frac{n-4}{2}}\int_{{B}\backslash\omega_+}\varphi_0\Big(\frac{e_n-x}{\delta}\Big)U_{\delta,e_n}^p \leq \left(C2^{n-3}\delta^{n-3}\right)\left(2^{n+2}\delta^\frac{n+2}{2}\right)=O(\delta^{n+\frac{n-4}{2}})=o(\delta).
T(t)f(x) = \begin{cases}f(e^{-it}x) x \in \mathbb{T}f(x-t) x \in [t,\infty] e^{-(t-x)}f(0) + e^{-(t-x)}\int_0^{t-x}e^s \, \langle \mu, R(s)f|_\mathbb{T}\rangle \, ds x \in [0,t) \end{cases}
\left| \int_{M}^x \frac{\sqrt{y}\log^2 y}{(y-1)^2}\,dy\right| < \frac{2.2024}{2}\int_{M}^x \frac{\log^2 y}{y^{1.5}}\,dy =2.2024\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}-\frac{\log^2 x+4\log x+8}{\sqrt{x}}\right).
t\mapsto (2\pi i)^{-\frac{1}{2g}}\begin{bmatrix}(2\pi i)^{\frac{1}{2g}}\int_{\delta_j}\alpha_0 \int_{\delta_j}\alpha_1 \cdots \int_{\delta_j}\alpha_{g}-\int_{\delta_j}\alpha_{g+1}\cdots -\int_{\delta_j}\alpha_{2g}\end{bmatrix},
{\cal R}\equiv \left( \begin{array}{c c c c}{R^{( 1 )}}&{0}&{0}&{0}\\{0}&{R^{( 3 )}}&{0}&{0}\\{0}&{0}&{R^{( 2 ) - 1}}&{0}\\{0}&{0}&{0}&{{\cal P}R^{( 4 )}{\cal P}}\\ \end{array}\right) \quad . \qquad
\mathcal{T}_{K}^{i,2}(y,h):=\Big\{w\in \mathbb{Y}\;|\;{\rm dist}(y+th+\frac{1}{2}t^2w,K)=o(t^2), t\geq 0\Big\},\qquad \mathcal{T}_{K}^{2}(y,h):=\Big\{w\in \mathbb{Y}\;|\; \exists\;t_n\downarrow 0{\;\rm such\;that\;}{\rm dist}(y+t_nh+\frac{1}{2}t_n^2 w,K)=o(t_n^2)\Big\}.
K(x,y,z) :=\left( \frac{2 x (x^2+y^2) - 8 yz}{((x^2+y^2)^2+16 z^2)^{3/2}}, \frac{2 y(x^2+y^2)+ 8xz}{((x^2+y^2)^2+16 z^2)^{3/2}}\right) = r^{-6}\left(2x (x^2+y^2)-8yz, 2 y(x^2+y^2) + 8 xz \right),
\omega (t) = \lambda_1(t)f^{12}+\lambda_2(t)f^{34}+\lambda_3(t)f^{56}, begin{align*}3pt]\widehat \rho(t)=\sqrt{\lambda_1(t)\lambda_2(t)\lambda_3(t)}\left(-f^{246}+f^{136}+f^{145}+f^{235}\right),
\begin{bmatrix}y_1 y_2^*\end{bmatrix}=\begin{bmatrix}h_1 h_2 h_2^* -h_1^*\end{bmatrix}\begin{bmatrix}s_1 s_2\end{bmatrix}+\begin{bmatrix}w_1 w_2^*\end{bmatrix}.
S_i(x_1,..,x_n)= R_{i,i-1}(x_i/x_{i-1})..R_{i,1}(x_i/x_{1}) K_0(x_i) R_{1,i}(x_i x_1)..R_{i-1,i}(x_i x_{i-1}) \times & R_{i+1,i}(x_i x_{i+1})..R_{L,i}(x_i x_L) K_n(1/x_i) R_{i,L}(x_i/x_{L})..R_{i,i+1}(x_i/x_{i+1}).
\partial^3_\mu Q_\mu=\Big[\Big(\phi_{\partial^2_\mu Q_\mu}+\Big(\phi_{\partial_\mu Q_\mu}-\frac{r^2}{2}\Big)^2+2\phi_{\partial^2_\mu Q_\mu}\Big)\Big(\phi_{\partial_\mu Q_\mu}-\frac{r^2}{2}\Big)+\phi_{\partial^3_\mu Q_\mu}\Big]Q_\mu.
\sum_{n_{1},...,n_{S}=0}^{\infty}\varphi(n_{1}\cdot\cdot\cdot n_{S})\phi(n_{1}\cdot\cdot\cdot n_{S})\prod_{j=1}^{B}\langle-c_{j}+\sum_{k=1}^{S}a_{jk}\cdot n_{k}\rangle
\lim_{t_1,t_2\rightarrow0^+}\mathcal C_tg=\mathcal C_\Gamma g, \lim_{t_1,t_2\rightarrow0^+}\mathcal C_t^{p1}g=\mathcal C_\Gamma^{p1}g, \lim_{t_1,t_2\rightarrow0^+}\mathcal C_t^{p2}g=\mathcal C_\Gamma^{p2}g
\tilde{\varepsilon}_T^{-1}\sum_j E^{\Pi^{D_T}}[\|b_j-P_{V_J}b_{0,j}\|_\infty |X^T] =O_{P_{b_0}}\Big( \frac{J^{\frac{3}{2}}2^{J(d-2)}(\log T)^\eta}{\tilde{\varepsilon}_T \sqrt{T}}\Big) = o_{P_{b_0}}(1).
\Gamma=\int dt \Bigl[\sum_a{N_a\over 2g_s}v_a^2 +{1\over 2}\sum_{a,b}{15N_aN_b \over 16}{v_{ab}^4 \over r_{ab}^7}-{1\over 2}\sum_{a,b}{15 N_aN_b \over 8}{\dot{g}_s \over g_s}{v_{ab}^2 v_{ab}\cdot x_{ab}\over r_{ab}^7}\Bigr] .
\begin{aligned}w(x,y) = \frac{2^{1-s}}{\Gamma(s)}\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_s(\lambda_k^{1/2}y)\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^s}\psi_{k}(x) &= \frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-y^{2}/(4t)}e^{-t\mathcal{L}_{\Omega}}f(x)\,\frac{dt}{t^{1-s}}.\end{aligned}
\sum_{q\leq \frac{N^{\vartheta}}{(\log N)^{B}}}\max_{M\leq N}\max_{\substack{a (a,q)= 1}}\Bigg| \sum_{\substack{n\leq M n \equiv a \pmod q}}f(n) -\frac{1}{\varphi(q)}\sum_{\substack{n\leq M (n, q)=1}}f(n)\Bigg| \ll_{A}\, \frac{N}{(\log N)^A}.
f_{p}\left( x\right) =x^{-4}F_{p}\left( x\right) =\frac{3-5p^{2}}{360}+\sum_{n=3}^{\infty}\left( -1\right)^{n}\frac{a_{n}\left( p^{2}\right)}{3\left( 2n+1\right) !}x^{2n-4},
\begin{array}{ll}f(\psi)=1+\frac{1}{R^{2}}\sum_{r=1}^{N}m_{r}^{2}(\psi_{r})^2 +O(\psi^3) \mbox{with $ \frac{m_{r}^{2}}{R^{2}}\equiv \frac{1}{2}\partial_{r}\partial_{r}f |_{\psi =0}$}.\end{array}
\left[ 1 + \frac{\, g^{2}}{2 m^{2}}\, V^{( 2 )}( \phi ) \right] \frac{d p^{\mu}}{d \tau}= - \frac{\, g^{2}}{2 m}\left[ \partial^{\mu}V^{( 2 )}( \phi ) - \frac{p^{\mu}}{p^{2}}\, p \cdot \partial \, V^{( 2 )}( \phi ) \right] \, ,
S_{E}={\frac{1}{2 \kappa_{D}^{2}}}\int d^{D}x \sqrt{- G^{E}}\left[{\cal R}_{G^{E}}-{\frac{4}{D - 2}}( \partial \phi )^{2}-{\frac{1}{4}}e^{2 a \phi}F_{2}^{2}\right] ,
\bar{a}\bar{\gamma}_2+(1-\bar{a})\bar{\gamma}_3-\bar{\gamma}_1 =\frac{s}{(1-a)\bar{s}+as}\left(a\gamma_2+(1-a)\gamma_3+a\gamma_1\Big(\frac{s}{\bar{s}}-1\Big)\right)- \frac{s}{\bar{s}}\gamma_1 =\frac{s}{(1-a)\bar{s}+as}(a\gamma_2+(1-a)\gamma_3-\gamma_1).
\frac{\mu X_{j-1}(t^*)}{s X_j(t^*)}\leq \frac{\mu (1 + \delta)}{s (1 - \delta)}\cdot \frac{x_{j-1}(t^*)}{x_j(t^*)}= \frac{1 + \delta}{1 - \delta}\cdot \frac{j}{e^{st^*}- 1}\leq \frac{1 + \delta}{1 - \delta}\cdot \frac{k^*}{k_N^2 - 1}.
\mathcal{M}_{|\Delta_n|^2}(s) = \frac{\left({\alpha_n}/{b_{0,n}}\right)^{\alpha_n}}{\Gamma(\alpha_n)}& \times \int_0^{\infty}\frac{b^{\alpha_n-1}}{2bs+1}\exp\left(-\frac{\tilde{A}_ns}{2bs+1}-\frac{\alpha_n b}{b_{0,n}}\right)\mathrm{d}b.
(X^{k}\frac{\partial}{\partial x^{k}}|_{v}+Y^{l}\frac{\partial}{\partial v^{l}}|_{v})+_{*}(X^{k}\frac{\partial}{\partial x^{k}}|_{w}+Z^{l}\frac{\partial}{\partial v^{l}}|_{w})=X^{k}\frac{\partial}{\partial x^{k}}|_{v+w}+(Y^{l}+Z^{l})\frac{\partial}{\partial v^{l}}|_{v+w}
\int_{\mathbb{R}^{3}\times \mathbb{R}^{3}}d\mathbf{x}d\mathbf{v}\varphi_{i}(\mathbf{v})f_{\Delta t,\Delta \mathbf{x}}(t,\mathbf{x},\mathbf{v})=\int_{\mathbb{R}^{3}\times \mathbb{R}^{3}}d\mathbf{x}d\mathbf{v}\varphi_{i}(\mathbf{v})f_{0}(\mathbf{x},\mathbf{v}),
\dbinom{k+r}{i}\dbinom{k-i+r}{j-i}= \dfrac{(k+r)!}{i!(k-i+r)!}\cdot\dfrac{(k-i+r)!}{(j-i)!(k+r-j)!}= \dfrac{(k+r)!}{(k+r-j)!}\cdot\dfrac{1}{i!(j-i)!}= \dbinom{k+r}{j}\cdot\dfrac{j!}{i!(j-i)!}= \dbinom{k+r}{j}\dbinom{j}{i}.
q_{k+1}\left( \| q_{k-1}\theta \| + \| q_k \theta \| \right) = q_{k+1}\left( 2 \| q_k \theta \| + \| q_{k+1}\theta \| \right) &= 2 q_{k+1}\| q_k \theta \| + \left(q_k + q_{k-1}\right) \| q_{k+1}\theta \| &= 2 - q_k \| q_{k+1}\theta \| + q_{k-1}\| q_{k+1}\theta \| < 2.
\langle 0 | W_{\mathrm{e f f}}| 0 \rangle = \sum_{p = 2}^{N}\, g_{p}\langle 0 | u_{p}| 0 \rangle = \sum_{p = 2}^{N}\, \sum_{j = 1}^{N}\, g_{p}\tilde{\lambda}_{j}^{p}
\tilde{{F}}_{s+1}\left( p\right) =\frac{1}{2}p^{2}\left({1-p^{2}}\right) \frac{d\tilde{{F}}_{s}\left( p\right)}{dp}-\frac{1}{2}\sum\limits_{j=1}^{s-1}{\tilde{{F}}_{j}\left( p\right) \tilde{{F}}_{s-j}\left(p\right)}\ \left({s=2,3,4,\cdots}\right) .
\begin{array}{c}{W = X_{A C}X_{C D}X_{D A}- X_{A C}X_{C B}X_{B A}+ X_{C A}X_{A B}X_{B C}- X_{C A}X_{A D}X_{D C}}\\{+ X_{B D}X_{D C}X_{C B}- X_{B D}X_{D A}X_{A B}- X_{D B}X_{B C}X_{C D},}\\ \end{array}
\boldsymbol{v}(x) =\arg\min_{\boldsymbol{v}}\sum_{l=0}^{{T}-1}\ell(z_l,v_l) \mbox{s.\,t.}\ \{z}_{l+1}=A{z}_{l}+B v_{l}, \l=0,\dots,T-1\, , & z_{T}=0, \z_0=x begin{align*}-0.4cm]
H_j(t,x,y)=H_{j, +}(t, x, y)=\chi_j(x)e^{-m\varphi_j(z)-im\theta}A_{B_j,+}(t,z,w)e^{m\varphi_j(w)+im\eta}\tau_j(w)\sigma_j(\eta) &G_j(t,x,y)=G_{j, +}(t, x, y)=\chi_j(x)e^{-m\varphi_j(z)-im\theta}A_{B_j,+}(t,z,w)e^{m\varphi_j(w)+im\eta}\tau_j(w).
\frac{1}{d_{i}}\sum_{j=1}^{N}a_{ij}\sum_{l=1}^{N}a_{il}\left( 1-\delta_{lj}\right) =\frac{1}{d_{i}}\sum_{j=1}^{N}a_{ij}\left\{\sum_{l=1}^{N}a_{il}-a_{ij}\right\}=\frac{1}{d_{i}}\sum_{j=1}^{N}a_{ij}d_{i}-\frac{1}{d_{i}}\sum_{j=1}^{N}a_{ij}^{2}& =d_{i}-1
\left( \begin{array}{c}{{{\hat{\xi}}^{t}}}\\{{{\hat{\xi}}^{r}}}\\{{{\hat{\xi}}^{\varphi}}}\\ \end{array}\right) ={\cal G}\left( \begin{array}{c}{{{\hat{\Sigma}}^{t}}}\\{{{\hat{\Sigma}}^{r}}}\\{{{\hat{\Sigma}}^{\varphi}}}\\ \end{array}\right) ;{\cal G}= \mathrm{E x p}\left( - \int^{\tau}{\cal D}( \tau^{\prime}) \; d \tau^{\prime}\right) ,
\zeta :\! = \alpha_T \!+\! \frac{\beta_T-\alpha_T}{\sqrt{a^2\!+\!b^2\alpha_T^2}\!+\!\sqrt{a^2\!+\!b^2\beta_T^2}}\sqrt{a^2\!+\!b^2\alpha_T^2}= \beta_T \!-\! \frac{\beta_T-\alpha_T}{\sqrt{a^2\!+\!b^2\alpha_T^2}\!+\!\sqrt{a^2\!+\!b^2\beta_T^2}}\sqrt{a^2\!+\!b^2\beta_T^2},
\gamma_{\scriptscriptstyle{l}}=\Phi/2 \pi - l + \sqrt 2 \pi \,b_{\scriptscriptstyle{l-1}}\,a^l / \psi_{\scriptscriptstyle{l}}^<(k,a)+ k^2 [ (\Phi/2 \pi - l - \gamma_{\scriptscriptstyle{l}})\,\chi_{\scriptscriptstyle{l}}(a)-(r \partial_r \chi_{\scriptscriptstyle{l}})_a ] +\mbox{O}(k^4)\,.
\mu \int_\Omega \varepsilon (w_\delta - z_\delta ):\varepsilon v +\nu \int_\Omega (\mathsf{m}_\delta (\varepsilon w_\delta) - \mathsf{m}_\delta (\varepsilon z_\delta) ):\varepsilon v=\langle u_1 -u_2, v\rangle, \forall v \in Y.
\mathbb{E}[x^{V};D_1]=\frac{1}{3}\Big{(}\sum_{k=0}^{\infty}\mathbb{P}(V_b=k)\sum_{j=0}^k\Big{(}\frac{1}{2}\Big{)}^kx^j\binom{k}{j}\Big{)}^2=\frac{1}{3}\Big{(}\mathcal{L}[f]\Big{(}\frac{x+1}{2}\Big{)}\Big{)}^2
a_k\sim b_k\,\,\,\,\mbox{as $k\to\infty$}\quad\Leftrightarrow\quad\lim_{k\to\infty}\frac{a_k}{b_k}=1 &f(t)\sim g(t)\,\,\,\,\mbox{as $t\to\infty$}\quad\Leftrightarrow\quad\lim_{t\to\infty}\frac{f(t)}{g(t)}=1.
\mbox{Tr}\left( P \exp \int_x^{gx}A dx \right) \,\left( P \exp \int_{x}^{hx}(g^* A) dx \right) \,\left( P \exp \int_{gx}^{x}(h^* A) dx \right) \,\left( P \exp \int_{hx}^x A dx \right)
b'_0 = b_0c_0 b'_1 = b_0c_1+b_1c_0+(a_1-d_1)a_2 b'_2 = b_0c_2+b_1c_1+b_2c_0+(a_1-d_1)a_3-a_2d_2 b'_3 = b_0c_3+b_1c_2+b_2c_1+b_3c_0+(a_1-d_1)a_4-a_2d_3-a_3d_2.
U'(x(T,y))= p(0,y) \exp \big( - \int_0^T \{\Phi(s,y) +\frac{b_0(s,y)}{\sigma_0(s,y)}\}dB(s) &+\frac{1}{2}\int_0^T\{\Phi^2(s,y) -\frac{b_0^2(s,y)}{\sigma_0^2(s,y)}\}ds \big)=:p(0,y)\Gamma(T,y) .
W_{\mathrm{c}}^{(1)}=\left\{W_{\mathrm{c},1}^{(1)},W_{\mathrm{c},2}^{(1)}\right\}\rightarrow s_{\mathrm{c}}^{(1)}, W_{\mathrm{c}}^{(2)}=\left\{W_{\mathrm{c},1}^{(2)},W_{\mathrm{c},2}^{(2)}\right\}\rightarrow s_{\mathrm{c}}^{(2)},
\int_{t_{0}}^{\infty}r_{1}^{-\frac{1}{\alpha}}(v)\left(\int_{t_{0}}^{v}r_{2}^{-\frac{1}{\beta}}(u)\left(\int_{t_{0}}^{u}\pi_{1}^{\gamma}\left(\sigma(s)\right)q(s)s\right)^{\frac{1}{\beta}}u\right)^{\frac{1}{\alpha}}v = \infty.
s \ell (a_N t - \tau_{j(t) + 1}) = s \ell (\gamma_{j(t) + 1}- \tau_{j(t) + 1}) - s \ell (\gamma_{j(t) + 1}- a_N t) &\leq \ell \log \bigg( \frac{s}{\mu}\bigg) - \frac{\ell (1/2 - d(t))(1 - \eta/3)}{q(t-1) k_N}\log \bigg( \frac{s}{\mu}\bigg).
L_{s e l f i n t}^{f e r m}= 3 \left[ \frac{1}{b}\left( \frac{m ( h - b )}{4 M ( k - b )}\overline{{{\Psi}}}^{\prime}\Psi^{\prime}\right)^{4}e^{2 \alpha \bar{\phi}/ M_{p}}\right]^{1 / 3}.
\tilde{\cal L}= \frac{1}{2}\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{A}+ \frac{1}{2}\lambda \bigl ( \phi^{A}\phi^{A}- \mathrm{e}^{\eta^{1}}\bigr ) - \frac{1}{8}\mathrm{e}^{\eta^{1}}\partial_{\mu}\eta^{1}\partial^{\mu}\eta^{1}
2\alpha_t^{-1}(Q_t-R_t)=\ -\beta_t\|y^t-z^{t+1}\|^2-(1-\beta_t)\|z^t-z^{t+1}\|^2+\alpha_t^{-2}\|x^{t+1}-y^t\|^2-\mu\gamma_t\alpha_t^{-2}(1-\alpha_t)\|x^t-y^t\|^2 \le \ -\beta_t\|y^t-z^{t+1}\|^2-(1-\beta_t)\|z^t-z^{t+1}\|^2+\alpha_t^{-2}\|x^{t+1}-y^t\|^2 \leq0,
\tilde W_{2,0}(z_i,z_j) =W_{2,0}(z_i,z_j), i\in M,~j\in K \tilde W_{2,0}(z_i,z_j) =W_{0,2}(z_i,z_j)-\frac{x_ix_j}{(x_i-x_j)^2}, i\in K,~j\in K.
s^c_{(3,0;4,1)}=\sum\limits_{\Omega^s\leq (4,1)}K_{(4,1),\Omega^s}m_{(3,0;\Omega^s)}+\sum\limits_{\Omega^s\leq(3,1)}K_{(3,1),\Omega^s}m_{(3,1;\Omega^s)}\phantom{s^c_{(3,0;4,1)}=}{}+\sum\limits_{\Omega^s\leq (2,1)}K_{(2,1),\Omega^s}m_{(3,2;\Omega^s)},
\alpha_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left(\begin{array}{c}1,e^{m\int \kappa (s)\,ds}+\frac{1}{m}\sinh \left( m\int \kappa(s)\,ds\right) , \frac{1}{m}(1+m)\cosh \left( m\int \kappa (s)\,ds\right) +\sinh \left( m\int\kappa (s)\,ds\right) ,\end{array}\right).
\begin{aligned}f(x)-f(x^{k+1})+(x-x^{k+1})^T[-A^T\hat{\lambda}^k+\beta^kA^T(A\hat{x}^k-b) &+\beta^kD(x^{k+1}-\hat{x}^k)+\frac{\sigma(1-\tau^k)}{\tau^k}(x^{k+1}-x^k)]\geq\frac{\sigma}{2}\|x^{k+1}-x\|^2,~\forall x.\end{aligned}
\dfrac{M_{2}(\mathcal{NC}(D_{2m}))}{|e( \mathcal{NC}(D_{2m}) )|}- \dfrac{M_{1}(\mathcal{NC}(D_{2m}))}{|v( \mathcal{NC}(D_{2m}) )|}= \dfrac{m^{2}(m-5)+4(2m-1)}{3m(m-1)(2m-1)}:= \dfrac{f(m)}{g(m)}.
\frac{\tilde{u}^{[n]}(x)-\tilde{u}^{[n-1]}(x)}{\Delta t}= \theta \left[\mathcal{L}(\tilde{u}^{[n]}) + f(t_n,x)\right] + (1-\theta)\left[ \mathcal{L}(\tilde{u}^{[n-1]}) + f(t_{n-1},x) \right],
n^{1-\beta_{3}}m^{\alpha_{3}}\Big( \sum_{i}D_{i}E_{i}\Big)^{1-\alpha_{3}}= O\big( n^{1-\beta_{3}}m^{\alpha_{3}}({n^{-\beta_2}}m^{\alpha_2}(m^{\alpha_1}n^{-\beta_1})^{1-\alpha_2})^{1-\alpha_{3}}\big) & =O\big( m^{1 - \frac{k-1}{4 k -1}}n^{1 - \frac{3}{4 k-1}}\big).
(|\lambda_1|+|\lambda_2|+|\lambda_3|)^2 =(\lambda_1+\lambda_2-\lambda_3)^2 begin{align*}1mm] =(\lambda_1+\lambda_2+\lambda_3)^2-4\,(\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1)+4\,\lambda_1\lambda_2 begin{align*}1mm] =(n-3)^2+4\,(2n-3)+2\,[n-4+\sqrt{n^2+4n-12}] begin{align*}1mm] >n^2+4n-11+2\,\sqrt{n^2+4n-12}=(\sqrt{n^2+4n-12}+1)^2,
\begin{aligned}\int_{\Omega}e_{\eta}^{i}\wedge[\eta,dN_{\phi}(e_{\phi}^{j})]_{1}=-\int_{\Omega}e_{\eta}^{i}\wedge\ast\Big((\ast e_{\eta}^{j}+dN_{\phi}(e_{\phi}^{j}))\wedge(\ast d\eta)\Big). \end{aligned}
U \phi_{\ell}= 2 \alpha \left({\tt \frac{1}{\bar{z}}}\partial_{z}\phi_{\ell}-{\tt \frac{1}{z}}\partial_{\bar{z}}\phi_{\ell}\right) + \frac{\alpha^{2}}{2}\left({\tt \frac{1}{z}{\tt \frac{1}{\bar{z}}}}+{\tt \frac{1}{\bar{z}}}{\tt \frac{1}{z}}\right) \phi_{\ell}\, ,
|\lim_{\mathcal{T}\rightarrow\infty}\frac{1}{\mathcal{T}}\int_0^\mathcal{T}\tilde h(\mu^{N,M}(t),z^{N,M}(t))dt- \int_{F}\tilde{h}(\mu,z)\lambda^{N,M}(d\mu,dz)|\ \leq \ \phi_{\tilde h}(\mathcal{T}), \ \ \ \ \ \ \ \lim_{\mathcal{T}\rightarrow\infty}\phi_{\tilde h}(\mathcal{T}) = 0.
p^+_{\alpha,\rho}(x)= \dfrac{\sin(\pi\rho \alpha)}{\pi}\cdot \displaystyle \dfrac{x^{\alpha-1}}{x^{2\alpha}+2x^\alpha \cos(\pi \rho \alpha)+1}, &p^-_{\alpha,\rho}(x)= \dfrac{\sin(\pi(1-\rho)\alpha)}{\pi}\cdot\displaystyle \dfrac{|x|^{\alpha-1}}{|x|^{2\alpha}+2|x|^\alpha \cos(\pi (1-\rho) \alpha)+1}.
\sum_{|d|\leq \widehat{K}}\sum_{d=e_1f_1f_2^2d_3}\sum_{d_3=d_3'g_1g_2}|e_1f_1|^{-1}|f_2|^{-1/2}|g_2||m(g_2)|^{-1}|d_3'|^{-1/2}\leq \sum_{|d|\leq \widehat{K}}|h_1\cdots h_k|^{-1}\sum_{d=ef_1f_2^2d_2}\sum_{d_3=d_3'g_1g_2}1 \ll \sum_{|d|\leq \widehat{K}}|d|^\varepsilon |h_1\cdots h_k|^{-1}\ll \widehat{K}^{1/k+\varepsilon}.
\sum_{i=1}^{N_n}E\left[\max_{1\leq k\leq d_n}\left|\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)W^{(i)}_j\right|^3\right]\leq5^{3/2}a^3\log^{3/2}(2d_n-1+\sqrt{e})\sum_{i=1}^{N_n}\max_{1\leq k\leq d_n}\left(\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)^2\right)^{3/2}
-\frac{\partial L_{d}FI\left(p\|q\right)}{\partial q_{j}}=-\sum_{j}p_{j}\left[\frac{a-1}{a-b}\sum_{i}\overline{p}_{i}\left(\overline{Z}_{i}\right)^{a-2}-\frac{b-1}{a-b}\sum_{i}\overline{p}_{i}\left(\overline{Z}_{i}\right)^{b-2}\right]\frac{\partial \overline{Z}_{i}}{\partial q_{j}}
G \varphi (\xi) \ = \ \varphi (\xi) \ \int\limits_{\mathbb R^d}a (\xi-q) \mu (\xi, q) \dq \ = \ \varphi (\xi) \int\limits_{\mathbb T^d}\hat a (\xi-\eta) \mu (\xi, \eta) \d\eta, \varphi \in L^2 (\mathbb T^d),
i\hbar\sum_{n}(\dot{c}_{n}|\psi_{n}\rangle+c_{n}|\dot{\psi}_{n}\rangle+ic_{n}|\psi_{n}\rangle T\dot{\theta}_{n})e^{iT\theta_{n}}=\sum_{n}c_{n}TE_{n}|\psi_{n}\rangle e^{iT\theta_{n}}.
\Pr \left( \max_{2 \le R < r}\left| \sum_{i=2}^R E_i \right| \ge t n 2^{(1/2-c(2-\gamma )/8)n}\right) \ll \left( \frac{\log H}{t H}\right)^{1/\gamma}2^{(1-1/(2 \gamma ) +c(2-\gamma )/(8 \gamma ))n}.
\big|R_j(f_2)(g)-R_j(f_2)(\tilde g)\big| \leq\|f\|_{L^\infty(\mathcal G)}\sum_{l=\gamma+1}^{\infty}\int_{2^lB\setminus 2^{l-1}B}\left|K_j(g,g')-K_j(\tilde g, g')\right|dg' &\leq C\|f\|_{L^\infty(\mathcal G)}\sum_{l=\gamma+1}^{\infty}2^{-l}{1\over |2^lB|}\int_{2^lB}dg' &\leq C\|f\|_{L^\infty(\mathcal G)}.
C_{4,1}=[\begin{matrix}c_{1}&c_{2}&c_{3}&c_{4}\\ -c_{2}^{*}&c_{1}^{*}&-c_{4}^{*}&c_{3}^{*}\\ -c_{3}^{*}&-c_{4}^{*}&c_{1}^{*}&c_{2}^{*}\\c_{4}&-c_{3}&-c_{2}&c_{1}\end{matrix}].
{\textit{(i)}}~||\xi_1||_{\mathfrak{Y}}= \sup \bigg\{||S(\xi_1, \widehat{D})||~:~\widehat{D}\in \mathfrak{Y}\bigg\}& =\sup \bigg\{||\sum_{i=1}^{n}\xi_1(d_i)\overline{\mu}(D_i)||~:~\widehat{D}=(D_i, d_i),~i=1,2,..\bigg\}& \geq 0
\hat{\cal H}=-{1\over2}{d^2\over{dq^2}}-ag\rho\cot(a\rho q){d\over{dq}}= -{1\over2}a^2|\rho|^2\left\{(1-z^2){d^2\over{dz^2}}- (1+2g)z{d\over{dz}}\right\}.
I_{f i e l d}= \langle I \rangle = \int d^{4}x \, \left[ - m^{2}\sqrt{-{\frac{1}{2}}W^{\mu \nu}( x ) W_{\mu \nu}( x )}-{\frac{1}{2}}W^{\mu \nu}( x ) \partial_{[ \, \mu}A_{\nu \, ]}\right] \ .
\{Q_\alpha,Q_\beta\}= (C\Gamma^\mu)_{\alpha\beta}P_\mu + (C\Gamma_\mu\Gamma_{11})_{\alpha\beta}{\hat Z}^\mu + (C\Gamma_{\mu\nu})_{\alpha\beta}{\hat Z}^{\mu\nu}+ (C\Gamma_{11})_{\alpha\beta}{\hat Z}\quad
B^{\prime}(k)=\left(\begin{array}{c|c}\begin{matrix}1+e^{2\pi ik}0 -e^{2\pi ik}0\end{matrix}\boldsymbol{O}\hline\boldsymbol{O}\begin{matrix}1 1 e^{2\pi ik}-e^{2\pi ik}\end{matrix}\end{array}\right),
\left[ g^{\mu \nu}( \vec{\sigma}) \right] = \left( \begin{array}{c | c}{{\gamma_{0}^{- 2}}}&{{\displaystyle \frac{\vec{\sigma}^{\mathrm{T}}}{c}\gamma_{0}^{- 2}}}\\ \hline{{\displaystyle \frac{\vec{\sigma}}{c}\gamma_{0}^{- 2}}}&{{- I}}\\ \end{array}\right) .