harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_v^{(k+1)}(x,t) = g_1(t)*F_v(\cdot,0)+\int_0^t e^{(1-v_0^2)(t-\tau)}g_1(t-\tau) * \left(((v_0^2-(v_0+V)^2) + \varphi_{1})F_v^{(k)}- F_w^{(k)}+ \varphi_3\right) d\tau F_w^{(k+1)}(x,t) = g_\rho(t) * F_w(\cdot,0) + \varepsilon\int_0^{t}e^{-\varepsilon \gamma ( t-\tau)}g_\rho(t-\tau)*\left(F_v^{(k+1)}+ J_0\right) d\tau,
I_{n o n - p l a n a r}^{T = 0}= \frac{g}{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{d^{2}k}{( 2 \pi )^{2}}\frac{e^{i \tilde{p}k}}{e^{\beta ( \frac{k^{2}}{2}- \mu )}- 1}= \frac{g}{8 \pi^{2}\theta^{2}}\frac{1}{e^{- \frac{\mu}{T}}- 1}\, .
\Big(\frac{1}{|P|}\int_P{\sum_{k=\mu}^{\infty}{2^{skq}\Big|\Pi_k\Big(\sum_{j=3}^{\infty}{T_{[c_j]}f}\Big)(x) \Big|^q}}dx \Big)^{1/q}\lesssim \sup_{R\in\mathcal{D}_{\mu}}{\Big( \dfrac{1}{|R|}\int_R{\sum_{j=\max{(3,\mu-2)}}^{\infty}{2^{sjq}\big|T_{[c_j]}f(x) \big|^q}}dx \Big)^{1/q}}.
\left[ \frac{A_{1}h}{B_{2}}-\left( \frac{h}{B_{2}}\right)^{\prime}\right]f+\left[ -\frac{2h}{B_{2}}\right] f^{\prime}+\left[ \frac{A_{1}B_{1}}{B_{2}}-\left( \frac{B_{1}}{B_{2}}\right)^{\prime}-\frac{B_{1}A_{2}}{B_{2}}\right]f_{c_{2}}=0.
D^2_B(j)= \frac{1}{30}(N^4-10j(j+1)N^2 +10j(2j+1)(j+1)N-10j^2(j+1)^2-1)\,, C_B(j)= \textstyle\frac{30N}{N^4-10j(j+1)N^2+10j(2j+1)(j+1)N-10j^2(j+1)^2-1}\,.
\langle T_{\mu}^{\mu}\rangle_{R}- ( T_{\mu}^{\mu})_{0}= 2 \pi m^{2}\int_{- \infty}^{+ \infty}\frac{d \theta}{2 \pi}\left[ e^{- m R \cosh \theta}+ e^{- 2 m R \cosh \theta}+{\cal O}( e^{- 3 m R \cosh \theta}) \right] \, \, \, .
+\sum_{i<j}\xi_i\xi_j\xi_i^{*}\xi_j^{*}+\cdots+\sum_{i_1<\cdots <i_n}\xi_{i_1}\cdots \xi_{i_n}\xi_{i_1}^{*}\cdots \xi_{i_n}^{*}+\cdots+\xi_1\cdots \xi_p\xi_1^{*}\cdots \xi_p^{*}
\lVert f_{j}\rVert^2 \leqslant \lVert f_{j-1}\rVert^2 - 2c_{j}t_{j}c\lVert f_{j-1}\rVert + c_{j}^2 < &< \cfrac{\varepsilon^2}{c^2}+ \cfrac{\varepsilon^2}{c^2}= = 2\,\cfrac{\varepsilon^2}{c^2}.
4 P'(c_*) \dot{h}= 12 |b|^2 \langle \partial_c u_{c_*}, \psi_*^2 \rangle_{L^2}+ \mathcal{O}(|b|^4)= \frac{48}{5 \sqrt{c_*}}|b|^2 + \mathcal{O}(|b|^4),
Q_{m}(a; b;c):=\sum_{(h^1,\ldots, h^m)\in\mathbb{N}^m\,:\,0\le h^m\le\ldots\le h^1\le c\atop h^1+\ldots+h^m=a}\frac{1}{\prod_{j=1}^{m}w(b+h^j)},
\Phi(t_ww)= J(t_ww)-\lambda_3\frac{t_{w}^{p+1}}{p+1}\int_\Omega \frac{w^{p+1}}{|x|^{s_3}}dx \leq J(w)+|\lambda_3||T|^{p+1}\int_\Omega \frac{w^{p+1}}{|x|^{s_3}}dx = \hat{c}_{\lambda_1,\lambda_2}+|\lambda_3||T|^{p+1}\int_\Omega \frac{w^{p+1}}{|x|^{s_3}}dx < c_{\lambda_1,\lambda_2}.
a^{^{\dag}}(\vec{p})=\int{{d^{\nu}x}\over{(2 \pi)^{\nu\over 2}}}{e^{i \vec{p}\cdot \vec{x}}\over{\sqrt{2 g(\vec{p})}}}\left(\int d^{\nu}y{1\over 2}G(x,y) \eta(y) -{\delta\over{\delta \eta(\vec{x})}}\right).
H(\rho , q) = \kappa(\rho) q \otimes q\, , s(\rho , q) = p (\rho) + A(\rho) |q|^2 \, , \mbox{where $A(\rho) = \tfrac{1}{2}(\rho \kappa'(\rho) + \kappa (\rho) )$}\, , r(\rho , q) = B(\rho) q \, , \mbox{where $B(\rho) = \rho \kappa(\rho) $}\, ,
\iota_{1\leq 3}^{(3)}[6]=(S^{(3)}_{0,3})^{3}\iota_{1\leq 3}^{(3)}(S^{(3)}_{0,1})^{-3}=\Sigma^3\iota_{1\leq 3}^{(3)}\Sigma^{-2}=\Sigma\iota_{1\leq 3}^{(3)}.
h_{\mu \nu}^{\mathrm{s . p .}}[ y ( \xi ) ] = \frac{i m^{3}}{8 \pi g_{m}^{2}}\int d \sigma_{\mu \nu}( y ( \xi^{\prime}) ) \frac{K_{1}\left( m \left| y ( \xi ) - y ( \xi^{\prime}) \right| \right)}{\left| y ( \xi ) - y ( \xi^{\prime}) \right|},
{\cal H}= \int d^{d}x \Bigl [{\frac{1}{2}}\sum_{i}( \partial_{\mu}\Phi_{i})^{2}+{\frac{1}{2}}\sum_{i}r_{i}\Phi_{i}^{2}+{\frac{1}{4 !}}\sum_{i j k l}u_{i j k l}\; \Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\Bigr ] ,
\hat{\cal A}=\hat{\cal A}_1+\hat{\cal A}_2=(-1)^{n+1}{1\over{2^{2n-1}n!}}({g\over \pi})^n \epsilon_{\mu_1\nu_1...\mu_n\nu_n}[\hat F_{\mu_1\nu_1}... \hat F_{\mu_n\nu_n}-g\theta^{\alpha \beta}\partial_\beta (\hat A_\alpha\hat F_{\mu_1\nu_1}...\hat F_{\mu_n\nu_n})].
\mathrm{(1)}e^{- t \mathcal{A}_p}u = \big[A^{\frac{1}{2}}_{p'}\big]^*\Phi^{-1}e^{- t A_p}\Phi \big[ A_{p^{\prime}}^{- \frac{1}{2}}\big]^* u, \mathrm{(2)}\Phi^{-1}e^{- t A_p}f = e^{- t \mathcal{A}_p}\Phi^{-1}f;
G_{\kappa}(t)=\sum_{|\alpha|+|a|\leq\kappa-1}\sum_{1\leq i \leq 2}\int_{\mathbb{R}^2}\frac{|\nabla_iV^{(\alpha,a)}+\nabla_iH^{(\alpha,a)}\cdot\omega|^2+|\nabla_iH^{(\alpha,a)}\cdot\omega^\perp|^2}{\langle t-r \rangle^2}e^qdx.
R_{ij}(0)=0, Sym_{ijk}R_{ij,k}(0)=:\frac{1}{6}\sum_{\sigma}R_{\sigma(i),\sigma(j),\sigma(k)}(0)=0, & R_{,i}(0)=0,\Delta_{\tilde g}R(0)=-\frac{1}{3!}|W_{\tilde g}(0)|^2,
\Delta E_{a1}(\|p\|,\alpha_*,M,\kappa)=\alpha_* \left(\dfrac{\beta^{(1)}(\kappa,\alpha_*=0)}{\|p\|}+O_0(M^{-1/2}\|p\|^{-2})+O(M^{-3/2}\|p\|^{-4})\right),
\lim_{t\to \infty}\frac{{\bf{I}}(x,t)}{A_{1}(b_{1}(t))}= \lim_{t\to \infty}\frac{A_d(b_{d}(t))}{A_1(b_{1}(t))}\times \frac{{\bf{I}}(x,t)}{A_{d}(b_{d}(t))}=0.
\frac{\vec{p^{2}}}{2 m}\psi ( \vec{p}) - \frac{\alpha}{2 \pi^{2}}\int d^{3}\vec{p^{\prime}}\frac{\psi ( \vec{p^{\prime}})}{( \vec{p}- \vec{p^{\prime}})^{2}}= E \psi ( \vec{p}) .
|D^{c}\ell|(J)= | D \ell | (J) = \lim_{k \rightarrow \infty}\sum_{i=1}^{N_{k}-1}| \ell (z_{i+1}^{k}) - \ell(z_{i}^{k}) | = \lim_{k \rightarrow \infty}|D \ell^{k}| (J) =\lim_{k \rightarrow \infty}|D^{c}\ell^{k}|(J).
\delta^2_0(x_i) = x_i+\epsilon_i \delta^2_1(x_i) = x_i \delta^2_2(x_i) = x_i &\delta^2_0(x_i+\epsilon_i) = x_i+\tau_i \delta^2_1(x_i+\epsilon_i) = x_i + \tau_i \delta^2_2(x_i+\epsilon_i) = x_i +\epsilon_i.
h^{\mu}( z ) = a^{\mu}( \theta , \bar{\theta}) + \lambda ( \theta , \bar{\theta}) x^{\mu}+ w_{\nu}^{\mu}( \theta , \bar{\theta}) x^{\nu}+ 2 x{\cdot b}( \theta , \bar{\theta}) x^{\mu}- x^{2}b^{\mu}( \theta , \bar{\theta}) ,
\left\langle \sum\limits_{i=1}^{n}a_{i}\otimes{x}_{i},\sum\limits_{j=1}^{m}a'_{j}\otimes{y}_{j}\right\rangle=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\left\langle{x}_{i},\tau\left( \alpha \left( a^{\ast}_{i}\right) a'_{j}\right){y}_{j}\right\rangle
\mbox{\ensuremath{\langle}Hess}_{\rho}\nabla\varphi,\nabla\varphi\rangle=-K\int_{\mathbb T}\nabla\varphi D^{2}(\sigma\cdot J)\nabla\varphi \rho(\theta) ~ d\theta-K\bigg|\int_{\mathbb T}\nabla\varphi\rho ~ d\theta\bigg|^{2}.
W = - 4 ( \frac{\pi}{2})^{\frac{p}{2}}( 3 - l ) ! \prod_{a = 1}^{p - 1}\left( \frac{n^{\frac{1}{l}}}{L_{a}}\right) \omega_{l}^{3}\frac 1{r^{8 - p}}+ \cdot \cdot \cdot ,
(n-1)!V(k) = \sum_{r,s\in\mathbb{Z}}(-1)^{n+k+s}\sum_{m=1}^{n-1}{k\choose n-s-m+r}{n-k \choose m-r}M(r-sa)^{n-1}= \sum_{0\le s\le n\atop1\le r\le n-1}(-1)^{n+k+s}\sum_{m=1}^{n-1}{k\choose n-s-m+r}{n-k \choose m-r}M(r-sa)^{n-1}
g_0^{j,t}(x)= f(x)-\frac{1}{\omega(B^g(2\sqrt{t}))}\int_{B^g(x^t_j, 2\sqrt{t})}f(y)d\mu_2(y) = \frac{1}{\omega(B^g(2\sqrt{t}))}\int_{B^g(x^t_j, 2\sqrt{t})}(f(x)-f(y))d\mu_2(y).
G_{p,q+2}^{m+1,n}(\begin{matrix}a_{p}\\ \beta,b_{q},\beta^{\prime}\end{matrix}|z)=(-1)^{\beta^{\prime}-\beta}G_{p,q+2}^{m+1,n}(\begin{matrix}a_{p}\\ \beta^{\prime},b_{q},\beta\end{matrix}|z),m\le q,\beta^{\prime}-\beta\in\mathbb{Z},
\log g(n) = \sum_{p\leq \sqrt{n}}{\frac{\xi_{p}(1+\xi_{p})}{2}}\log p + \sum_{\sqrt{n}< p\leq n}\log p &= \sum_{p\leq \sqrt{n}}{\frac{\xi_{p}(1+\xi_{p})}{2}}\log p + \sum_{\sqrt{n}< p\leq \frac{n}{2}}\log p + \sum_{\frac{n}{2}< p\leq n}\log p\,.
\left(z^1(z^2)^2-z^1(z^3)^2\right)\frac{\partial\alpha}{\partial z^1}+\left(z^2(z^3)^2-z^2(z^1)^2\right)\frac{\partial\alpha}{\partial z^2}+\left(z^3(z^1)^2-z^3(z^2)^2\right)\frac{\partial\alpha}{\partial z^3}=0.
K_{l}( u ) = \left( \begin{array}{l l}{{2 x_{0}( q^{- 1}s^{- 1}u - q s u^{- 1}) \;}}&{{x_{+}( q^{- 1}u^{2}- q u^{- 2})}}\\{{x_{-}( q^{- 1}u^{2}- q u^{- 2}) \;}}&{{- 2 x_{0}( s u - s^{- 1}u^{- 1})}}\\ \end{array}\right) \, ,
\Vert \{\lambda_{k}\}\Vert_{l_{q}^{s,\alpha}}=\begin{cases}\left( \sum_{k\in \mathbb{Z}^{n}}^{{}}\langle k\rangle^{\frac{sq}{1-\alpha}}|\lambda_{k}|^{q}\right)^{\frac{1}{q}}~ ~0<q<\infty , \sup_{k\in \mathbb{Z}^{n}}\left( \langle k\rangle^{\frac{s}{1-\alpha}}|\lambda_{k}|\right) ~ ~q=\infty .\end{cases}
\sum_{j=1}^{N_v}\langle g_\mu^A \vert j \rangle \langle j \vert f_\nu^B \rangle \langle j \vert \upsilon \rangle = \left( \tilde{T}_{\rm exp}\right)_{\mu\nu}\ , 1\leq \mu \leq M_A \ , \ \ 1 \leq \nu \leq M_B \ ,
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}\boldsymbol{y}_{t,j}\left(\boldsymbol{x}_{t,j}\right) ={\boldsymbol{W}_{t,j}^{\alpha,out}}\left[{{\boldsymbol{v}_{t,j}^\alpha};{\boldsymbol{x}_{t,j}}}\right],
L_{B}^{r}= q^{1 / 2}p_{\mu}\dot{x}^{\mu}- \frac{e}{1 + q^{2}}( p^{2}- m^{2}) +{\cal B}\dot{e}+ \frac{{\cal B}^{2}}{1 + q^{2}}+ \dot{\bar{\lambda}}\frac{d}{d \tau}( \lambda e ) ,
|\widehat{f}_r(z)-f(z)|^p \leq \frac{1}{|D(z,r)|^p}\int_{D(z,r)}|f(w)-f(z)|^p dV(w) |D(z,r)|^{p(1-1/p)}&=\frac{1}{|D(z,r)|}\int_{D(z,r)}|f(w)-f(z)|^p dV(w).
f_1(q,z_2):=-\frac{[q\xi'(q)-\xi(q)](1-q)(1+z_2)}{\xi'(1)-\xi'(q)}- q^2\log\frac{q[\xi'(1)-\xi'(q)]}{(1+z_2)\xi'(q)(1-q)}+ q^2-\frac{2\xi(q)q}{\xi'(q)}+ \frac{\xi(q) q^2 [\xi'(1)-\xi'(q)]}{(1+z_2)\xi'(q)^2(1-q)}=0, f_2(q,z_2):=(1-q)[\xi'(1)-\xi'(q)]\Big(\frac{1+z_2}{z_2^2}\log(1+z_2) -\frac1{z_2}\Big)+\xi'(q)(1-q)-1+\xi(q) =0.
\frac{(-4096)^{N-1}G(N,1)}{2N^2\binom{2N}{N}^2}=\frac{N^2 (N+1) \binom{2 N}{N}\binom{2 N-2}{N-1}\binom{2 N+2}{N+1}}{64 (2 N+1)}\in \mathbb{Z}.
S_{\left\langle p , \phi \right\rangle}^{\left\langle q , \psi \right\rangle}= \frac{1}{| \Omega_{p}| | \Omega_{q}|}\sum_{z \in \Omega}\sum_{x , y \in \Omega_{p}\cap \Omega_{z q}}\overline{{{\phi}}}( x , y ) \overline{{{\psi}}}^{z}( y , x ) \prod_{\xi \in \mathcal{O}( x , y )}\Lambda_{p_{\xi}}^{( z q )_{\xi}}\left( \frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
\epsilon^{(1)}_{\alpha,\delta_1}= \frac{\frac{1}{2}(R_1-\delta_1) + (\alpha - 1) d_{\alpha}(p_{U,W},p_U p_W)}{\frac{1}{2}+ (\alpha- 1)}- I (U;W), \epsilon^{(2)}_{\alpha,\delta_2}= \frac{\frac{1}{2}(R_1+R_2-\delta_2) + (\alpha - 1) d_{\alpha}(p_{U,V,W},p_{U,V}p_W)}{\frac{1}{2}+ (\alpha- 1)}- I (U,V;W).
\frac{E_2^*(\tau_0)}{\omega_0^2}=\frac{E_2^*(\tau_1)}{\omega_1^2},\frac{E_4(\tau_0)}{\omega_0^4}=\frac{E_4(\tau_1)}{\omega_1^4},\frac{E_6(\tau_0)}{\omega_0^6}=\frac{E_6(\tau_1)}{\omega_1^6}.
\limsup_{\delta \to 0}\limsup_{n \to \infty}\sup_{|t-s|\le\delta}|\psi^n(t) - \psi^n(s)| = 0, & \limsup_{\delta \to 0}\limsup_{n \to \infty}\sup_{|t-s|\le\delta}|B^n_k(t) - B^n_k(s)| = 0, k \in \mathbb{N}_0.
||R^{m_{1},m_{2}}_{l_{1},l_{2}}(n)||\leq\displaystyle{\sum_{l=|l_1-l_2|}^{l_1+l_2}}||D_{m_{1},m_{2}}^{l_{1},l_{2},l}(n)Y^{m}_l||\leq \K\displaystyle{\sum_{l=|l_1-l_2|}^{l_1+l_2}}||Y^{m}_l|| \ ,
\omega_n (T_{\mathfrak u}(x_{i,k}) ,T_{\mathfrak u}(x_{j,l}) &=\omega_n\Big(\big(\frac{2^{k-1}}{(k-1)!}u_i\log^{k-1}|u_i|,\ldots,u_i,\ldots,0\big),\big(\frac{2^{l-1}}{(l-1)!}u_j\log^{l-1}|u_j|,\ldots,u_j,\ldots,0\big)\Big) &=(-1)^k\langle u_i,u_j \rangle=\omega_n(x_{i,k},x_{j,l}).
\Bigg\langle \!\exp \left( - S_{int}[ \Phi^{(I)}]- i e \Bigg[ \frac{1}{\sqrt{\pi\!+\!gN}}\Big( A^{(1)}_\mu, \varepsilon_{\nu \mu}\partial_\nu \Phi^{(1)}\Big) +\sum_{I=2}^N \frac{1}{\sqrt{\pi}}\Big( A^{(I)}_\mu, \varepsilon_{\nu \mu}\partial_\nu \Phi^{(I)}\Big) \Bigg]\right)\!\Bigg\rangle_{\!\{K^{(I)}\}}.
\begin{array}{c}{\left( \sum_{i}r_{i}^{( l )}\right)_{L}\times \left( \sum_{i}r_{i}^{( l )}\right)_{R}}\\{\times \left[ \begin{array}{l}{\{\left( 4 4 + 8 4 \right)_{B}+ 1 2 8_{F}\}_{L}}\\{\times \{\left( 4 4 + 8 4 \right)_{B}+ 1 2 8_{F}\}_{R}}\\ \end{array}\right]}\\ \end{array}
D(k, l, q_-, q_+, m) := \Bigl\{Y_{k-1}(s) < q_- < Y_k(s) = Y_{k+1}(s) = \ldots = Y_l(s) < q_+ < Y_{l+1}(s) \ \ \mbox{for all}\ \s \in [t - m^{-1}, t + m^{-1}]\Bigr\}.
Z_{\nu}= \int dU \sum_{m_1=0}^{+\infty}\sum_{m_2 =0}^{+\infty}\cdots \sum_{m_{N-1}=0}^{+\infty}\sum_{n_N= - \infty}^{+\infty}\det (I_{n_j +i -j}(V \Sigma m)) \left(\det U\right)^{\nu - n_N}\chi_{(\ell_1, \ell_2, \cdots, \ell_N)}(U^{\dagger}),
~\begin{aligned}\Phi(z,s,a) =\frac{1}{a^{s}}+\frac{z}{2(a+1)^{s}}&\quad+\int_{1}^{\infty}\frac{z^{x}}{(x+a)^{s}}dx+\int_{1}^{\infty}\left[\frac{z^{x}\ln z}{(x+a)^{s}}-\frac{sz^{x}}{(x+a)^{s+1}}\right]P_{1}(x)dx.\end{aligned}
\begin{array}{r c l}{{[ ( 1 a ) + ( 1 b ) + ( 1 c ) + ( 2 a ) + ( 2 b ) ]_{\delta Z_{\phi}}}}&{{=}}&{{\displaystyle ( \mathrm{~ o n e ~ l o o p ~ r e s u l t ~}) \times ( - \delta Z_{\phi}) ,}}\\ \end{array}
x = \frac{- b \pm \sqrt{b^{2}- 4 a c}}{2 a}= \frac{- 1 \pm \sqrt{( - 6 )^{2}- 4 \times 1 \times ( - 6 )}}{2 \times 1}= \frac{- 1 \pm \sqrt{2 5}}{2}= \frac{- 1 \pm 5}{2}
\int_{\mathbf{R}^{d+1}}g(t,x)\mathcal{G}f(t,x)dxdt = \int_{\mathbf{R}}\left(\int_{\mathbf{R}^{d}}f(-s,-y)\mathcal{G}\tilde{g}(s,y)dy\right)ds &\leq \int_{\mathbf{R}}\|f(-s,\cdot)\|_{L_{p}}\|\mathcal{G}\tilde{g}(s,\cdot)\|_{L_{p'}}ds &\leq N(d,\alpha,p) \|f\|_{L_{q}(\mathbf{R},L_{p})}\|g\|_{L_{q'}(\mathbf{R},L_{p'})}
\begin{aligned}\rho = \mathbb{E}\Bigg \{\frac{1}{T}\frac{1}{N_{c}}\sum_{t=1}^{T}\sum_{n=1}^{N_{c}}\frac{|\hat{\bf h}_{n}^{H}{\bf h}_{n}|}{\|\hat{\bf h}_{n}\|_{2}\|{\bf h}_{n}\|_{2}}\Bigg \},\end{aligned}
W^{j}_{5}=-\varepsilon\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w^{2})\left(\Theta_{p_{i}}(t,x,\mu^{1},Dw^{1})\partial^{2}_{x_{i}x_{j}}w^{1}-\Theta_{p_{i}}(t,x,\mu^{2},Dw^{2})\partial^{2}_{x_{i}x_{j}}w^{1}\right)\dx,
S_{l , l^{\prime}}^{( 2 )}( E ) = \delta_{l 0}\, \delta_{l^{\prime}0}\; \frac{\ln \left( E / | E_{_\mathrm{( g s )}}| \right) + i \pi}{\ln \left( E / | E_{_\mathrm{( g s )}}| \right) - i \pi}\; .
\left\lbrace \begin{aligned}\delta ( a(x_{0}) \lvert dv_{j}\rvert^{p-2}dv_{j}) ) = 0 \frac{1}{2}B_{j}, \delta v_{j}= 0 \frac{1}{2}B_{j}, \nu\wedge v_{j}= \nu\wedge w_{j}\partial \left( \frac{1}{2}B_{j}\right).\end{aligned}\right.
\sum_{k=N}^\infty \ln2 \left( \frac{\lambda_{N}^{2}+\mu_{k}^{2}}{\lambda_{N}^{2}+\lambda_{k}^{2}}-1 \right) = \sum_{k=N}^\infty -\ln2 \frac{\lambda_{k}^{2}- \mu_{k}^{2}}{\lambda_{N}^{2}+\lambda_{k}^{2}}\ge -\ln2 \sum_{k=N}^{\infty}\frac{\lambda_k^2-\mu_k^2}{\lambda_N^2}\ge -\ln2
\int_{B_{\delta}(\hat{x}^\prime)\times(0,\delta)}J_\varepsilon(|A_\gamma(\hat{x},\hat{x})(\hat{x}-\bar{y})|)\nabla_{x^\prime}u(\hat{x})\cdot(\hat{x}^\prime-\bar{y}^\prime)\big|\det F_\gamma(\hat{x})\big|\;y = 0.
\begin{aligned}\mathcal Q(t) = ~{}-\frac{1}{2L}b \int \varphi' f^2 - \frac{3}{2L}(1+ b) \int \varphi' f_x^2 - \frac{1}{2L}(4+3b) \int \varphi' f_{xx}^2 - \frac{1}{2L}(1+b)\int \varphi' f_{xxx}^2.\end{aligned}
\int_{0}^{\infty}dk \hspace{.2cm}Q_{\ell}({k^2 + q^2) \over2kq}){q^{\ell+d_1}\over{(c^2+k^2)^{d_2}}}B_j(k)=q^{\ell+1}\sum_{j=0}^3a_j(q-p)^j.
(L^m_{p,q}f\ast \Phi_i)(z)=z+\sum\limits_{k=2}^\infty \lambda_k[k]_{p,q}^ma_{k}{z^k}+(-1)^{m+i}\sum\limits_{k=1}^\infty \mu_k[k]_{p,q}^mb_{k}{\overline{z}^k},
p_{\tilde{S}_1}(0)=p_{\bar{S}_2}(0)=H^{-1}_b(\frac{2}{3}), &p_{\tilde{S}_0(\alpha)}(0)=p_{\bar{S}_0(\alpha)}(0)=\alpha, &p_{\bar{S}_0(\alpha)|\tilde{S}_0(\alpha)}(1|0)=p_{\tilde{S}_0(\alpha)|\bar{S}_0(\alpha)}(1|0)=\beta(\alpha), &p_{\bar{S}_0(\alpha)|\tilde{S}_0(\alpha)}(0|1)=p_{\tilde{S}_0(\alpha)|\bar{S}_0(\alpha)}(0|1)=1-\frac{\alpha\beta(\alpha)}{1-\alpha},
E \left ( F(x,y) - \tilde F_{n,l}(x,y) \right )^2 = \ \frac 1{l^2}\frac 1{(n-l+1)^2}\sum_{q=m}^{\infty}\frac{J_q^2(x,y)}{q!}\frac 1{q!}\sum_{i,j \leq n}a_{n,i}a_{n,j}E[H_q(X_i)H_q(X_j)] \leq \ \frac 1{(n-l+1)^2}F(x,y) \sum_{i,j \leq n}\lvert r(i-j) \rvert^m.
c_{\bf m}= 0 \, \mbox{for}\Sigma_{2_1}, \Sigma_{2_2}c_{\bf m \bf n}= 1 \, \mbox{for}\Sigma_3, \Sigma_4, c_{\bf m \bf n}= 0 \, \mbox{for}\Sigma_5, \Sigma_6, \Sigma_7
\begin{cases}(u,t)\in K\times[0,1]\,, \noalign{\medskip}\displaystyle{\int_{\Omega}\bigl[A(x,u)\nabla u\cdot\nabla (v-u) +B(x,u)\,|\nabla u|^2\,(v-u)\bigr]\,dx \geq \int_\Omega g_t(x,u)(v-u)}\noalign{\medskip}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,,\end{cases}
(\alpha_0 - \beta_0) \cdot (\alpha_1 - \beta_1) = \frac{1}{2^2}a_{-1}+ \frac{11}{2^2 \cdot 3}(a_2 + a_{-2}+ a_3) - \frac{1}{2^3 \cdot 3}(v_{(1,2)}+ v_{(1,3)}) - \frac{4}{3}v_{(2,3)}
h_n\le e^{-\frac{3^{n+1}}{\left(n+e_{[k]}^1\right)\ln \left(n+e_{[k]}^1\right)\dots \ln_k \left(n+e_{[k]}^1\right)}}<e^{-3^{\frac n2+1}}\le e^{-3^{2}}=e^{-9}.
\begin{array}{l}{{\xi_{v}= \eta_{u}\hspace{1 c m}\mathrm{~ ( i n t e g r a b l e ~ c o n d i t i o n )}}}\\{{( q u + u^{2}) \xi_{u}+ ( a + 2 u v - p u + q v ) \xi_{v}- ( p v - v^{2}) \eta_{v}= 0}}\\ \end{array}
\Omega_\psi(-m)\tilde x_m^-b =(-1)^{j-1}\Omega_\psi(-m)\tilde x_{i_1-1}^-\tilde x_{i_2-1}^- \cdots \tilde x_{m+j-1}^-\tilde x_{i_j}^-\cdots \tilde x_{i_l}^-v_\lambda +q\mathcal L(\lambda) = \tilde x_{i_1}^-\tilde x_{i_2}^- \cdots \tilde x_{i_j}^-\cdots \tilde x_{i_l}^-v_\lambda +q\mathcal L(\lambda) =b.
|b_+ b_+ b_+ \rangle =\int_0^{\infty}dk_1 dk_2 dk_3\hspace{1mm}\delta (\sum_{i=1}^3 k_i - P^+) f_{b_+ b_+ b_+}(k_1,k_2,k_3) \frac{1}{N^{3/2}}\mbox{Tr}[b_+^{\dagger}(k_1) b_+^{\dagger}(k_2)b_+^{\dagger}(k_3)]|0\rangle
Corr_{P_n}(m_j(X_i,\theta^\prime_n),m_{j+R_1}(X_i,\theta^\prime_n)) =\frac{-\sigma^2_{P_n,j}(\theta^\prime_n)-Cov_{P_n}(m_{j}(X_i,\theta^\prime_n),t_j(X_i,\theta^\prime_n))}{\sigma_{P_n,j}(\theta^\prime_n)\sigma_{P_n,j+R_1}(\theta^\prime_n)}&\to -1,
Z_{total}=\frac{1}{(\frac{a_{1}}{a_{1}^{2}+b_{1}^{2}}-\frac{b_{1}}{a_{1}^{2}+b_{1}^{2}}i)+(\frac{a_{2}}{a_{2}^{2}+b_{2}^{2}}-\frac{b_{2}}{a_{2}^{2}+b_{2}^{2}}i)}.
Z \sim \begin{cases}\mathrm{Uniform}\Big(z \in \{-B,B\}^k \Big|\langle z,\tilde{V}\rangle \geq 0\Big), T = 1, \mathrm{Uniform}\Big(z \in \{-B,B\}^k \Big|\langle z,\tilde{V}\rangle \leq 0\Big), T = 0, \end{cases}
\frac{1}{n}\sum_{i=1}^n \bigl(\|S_i^\ast y_i^\ast+x^\ast\|+\|S_i^\ast y_i^\ast-x^\ast\|\bigr)>(\delta-\varepsilon)\|x^\ast\|+\frac{2}{n}\sum_{i=1}^n\|S_i^\ast y_i^\ast\|.
\left[e^{-t\bar{H}_s^2}\right]_{\mbox{diag}}={1 \over (4\pi t)^{3/2}}~~~,{}~~~\left[ e^{-t\bar{H}_d^2}\right]_{\mbox{diag}}={r_d \over (4\pi t)^{3/2}}\left[1+\frac 12 \kappa^2 t\right]~~~.
\cos d(Q_{1-\eta'}, P_{1-\eta}) +\cos d(Q_{\eta'}, P_{\eta}) &\ge (\alpha_\eta \beta_{\eta'}+ \alpha_{1-\eta}\beta_{1-\eta'}) (\cos d_{PQ}+ \cos d_{SR}) + (\alpha_\eta \beta_{1-\eta'}+\alpha_{1-\eta}\beta_{\eta'}) (\cos d_{PR}+ \cos d_{SQ}).
R^{N}_{\alpha\beta\gamma}{}^\delta=\overline{\Gamma}_{\beta\gamma,\alpha}^\delta-\overline{\Gamma}_{\alpha\gamma,\beta}^\delta +\sum_\eta \overline{\Gamma}_{\beta\gamma}^\eta\overline{\Gamma}_{\alpha\eta}^\delta-\overline{\Gamma}_{\alpha\gamma}^\eta\overline{\Gamma}_{\beta\eta}^\delta.
A_{123}= -\left(\frac{z^*}{|z|}\right)^3 e^{\hat{K}/2}\left[\hat{K}^SF_S + \hat{K}^TF_T \left(1+\sum_{i=1}^3 \frac{n_i}{3}- \frac{T+T^*}{3}\frac{y'_{123}(T)}{y_{123}(T)}\right) \right]
\begin{cases}U_1(s^{*}) = \sup\limits_{(s_1,s_2^*) \in \mathcal C}U_1(s_1,s_2^*), U_2(s^{*}) = \sup\limits_{(s_1^*,s_2) \in \mathcal C}U_2(s_1^*,s_2).\end{cases}
u_{n+1}(t,x) = \int_F p^b_t(x,y) u_0(y) \mu(dy) + \int_0^t \int_F p^b_{t-s}(x,y) f(s,u_n(s,y)) \mu(dy)ds &+ \int_0^t \int_F p^b_{t-s}(x,y) g(s,u_n(s,y)) \xi(s,y) \mu(dy)ds.
u_{n,k}:=\min_{\lfloor \zeta q^{-n}\rfloor \leq i\leq \lfloor \zeta q^{-n}\rfloor + \lfloor n^\alpha q^{-n}\rfloor + k}\frac{L(i)i^{\gamma-1}(1-q^n)^{2i}}{\Gamma(\gamma)v(1-q^n)}, o_{n,k}:=\max_{j\geq \lfloor n^\delta q^{-n}\rfloor + k}\frac{L(j)j^{\gamma-1}(1-q^n)^{2j}}{\Gamma(\gamma)v(1-q^n)}.
\|\tilde{y}^k-x\|^2-\|x^{k+1}-x\|^2\ge 2\alpha_k\left[f(\tilde{y}^k)+g(x^{k+1})-(f+g)(x)+f(x^{k+1})-f(\tilde{y}^k)\right] = 2\alpha_k\left[(f+g)(x^{k+1})-(f+g)(x)\right].
\mathbb{E}[\lambda_{-s}^{\dagger,1}(t)]=\int_{-\infty}^{-s}\gamma_{1}(t-u)\overline{\gamma}_{0}du=\overline{\gamma}_{0}\int_{t+s}^{\infty}\gamma_{1}(u)du=\overline{\gamma}_{0}H_{1}(t+s),
Op_h^w(x\xi + p(\xi))\theta_h(x) v^{\Sigma}_{\Lambda}= Op_h^w\big((x\xi + p(\xi))\theta_h(x) \big) v^{\Sigma}_{\Lambda}+ \frac{h}{2i}Op_h^w\big(\theta_h'(x)(x + p'(\xi))\big)v^{\Sigma}_{\Lambda}+ hR(v) & = Op_h^w\big((x\xi + p(\xi))\theta_h(x) \big) v^{\Sigma}_{\Lambda}+ hR(v)\, ,
|w(t)|_{\alpha + 1/2-\rho,\sigma}\le |w(T_*)|_{\alpha + 1/2-\rho,\sigma}e^{-t+T_*}e^{K^{\alpha+\frac{1}{2}-\rho}\int_{T_*}^t 2D_0/\tau d\tau}\le |w(T_*)|_{\alpha + 1/2-\rho,\sigma}e^{-t+T_*}e^{2K^{\alpha+\frac{1}{2}-\rho}D_0 \ln t}= M_2 t^\beta e^{-t},
[P(f)]_k=\frac{L^2}{4\pi^2}\big(f_k-\frac{(k\cdot f_k)f_k}{\vert k\vert^2}\big),\;\k\in\mathbb{Z}^2\setminus\{0\},\; f=\sum_{n\in\mathbb{Z}^2\setminus\{0\}}f_n e^{\frac{2\pi i n\cdot x}{L}}\in L^2_0(\mathbb{T}^2,\mathbb{R}^2).
\Delta_{x_1,\xi_2,\vartheta}u = \frac{\partial^2 u}{\partial x_1^2}+ \frac{1}{\varepsilon^2}\frac{\partial^2 u}{\partial \xi_2^2}+ \frac{1}{\varepsilon\xi_2+1}\frac{1}{\varepsilon}\frac{\partial u}{\partial \xi_2}+ \frac{1}{(\varepsilon\xi_2+1)^2}\frac{\partial^2 u}{\partial \vartheta^2},
\psi^+_N=\frac{1}{[2r(r+r_3)]^{\frac{1}{2}}}\left( \begin{array}{c}r+r_3 r_1+ir_2 \end{array}\right ) = \left( \begin{array}{c}\cos\frac{\theta}{2}e^{i\phi}\sin\frac{\theta}{2}\end{array}\right )
A_{0}^{3}( \alpha^{\prime}\rightarrow 0 ) = 2 g_{d}\, \, \varepsilon_{\lambda}^{( 1 )}\varepsilon_{\mu}^{( 2 )}\varepsilon_{\nu}^{( 3 )}\left\{\eta^{\lambda \mu}\left( p_{1}^{\nu}- p_{2}^{\nu}\right) + \eta^{\lambda \nu}\left( p_{3}^{\mu}- p_{1}^{\mu}\right) + \eta^{\mu \nu}\left( p_{2}^{\lambda}- p_{3}^{\lambda}\right) \right\}.
\frac{1}{|B_R|}\int_{B_R}||\Theta(v)||^2 \dv \leq \frac{C(m,k)}{|B_R|}\sum_{i_1,i_2,j_1,j_2}\int_{B_R}(k_{i_1 i_2}(x_{j_1}- x_{j_2}+v))^2 \dv &\leq \frac{C(m,k)}{|B_R|}\sum_{1 \leq i_1,i_2 \leq m}\int_{B_{R+T}}(k_{i_1 i_2})^2(v) \dv,
|D^kv(x)|\leq C\int_{\mathbb{R}^n}\frac{e^{nu(y)}}{|x-y|^k}dy \leq C\int_{\mathbb{R}^n}\frac{|y|^{n\beta}e^{n\left(v(y)+p(y)+q\left(\frac{y}{|y|^2}\right)\right)}}{|x-y|^k}dy.
\lambda(\tau)=(\frac{\sqrt{2}\eta(\frac{\tau}{2})\eta^{2}(2\tau)}{\eta^{3}(\tau)})^{8}=\frac{16}{(\frac{\eta(\tau/2)}{\eta(2\tau)})^{8}+16}=\frac{\theta_{2}^{4}(\tau)}{\theta_{3}^{4}(\tau)}
a_n=\left(\left|\begin{array}{cc}\Lambda_{n-1}(\theta_n^n)^\top \theta_{n+1}^n \boxed{m_{2n+1}}\end{array}\right|+\left|\begin{array}{cc}\Lambda_{n-1}e_{n-1}^\top \theta_n^n \boxed{0}\end{array}\right|H_n\right)H_n^{-1},b_n=H_{n}H_{n-1}^{-1}
\#\{p\in Spl_X(f)\mid \left\{\frac{mr}{p}\right\}< \left\{\frac{nr}{p}\right\}^\exists r \in\mathbb{Z}f(r)\equiv0\bmod p \}= \#Spl_X(f)-\#\{p\in Spl_X(f)\mid \left\{\frac{nr}{p}\right\}\le \left\{\frac{mr}{p}\right\}^\forall r f(r)\equiv0\bmod p \},
-\mbox{exp}\left[2\pi i\frac{P}{Q}(h_{n_{2},e_{2}}-h_{e_{1},n_{1}})\right](\delta_{-e_{2},e_{1}+pe}\delta_{1-n_{2},n_{1}+pn}-\delta_{-e_{2},e_{1}+pe}\delta_{1-n_{2},n_{1}+pn-p})
\theta(k,i) =\beta_{k,M_-'}\circ \beta_{k,M_+'}^{-1}(k,i) &=\beta_{k,M_-'}(j) &=\beta_{k,M_-}(j+1) &=\beta_{k,M_-}\circ \beta_{k,M_+}^{-1}\circ \beta_{k,M_+'}(j) &=\beta_{k,M_-}\circ \beta_{k,M_+}^{-1}(k,i) &= \eta(k,i)
m_F(C) =\sum_{i=1}^n \sum_{j=1, j\ne i}^n e(C(z_j) ,F(z_i) )z_i^*z_j^* +\sum_{i=1}^{n}e(C(z_i) ,F(z_i) ) (z_i^*)^2 &=\sum_{i=1}^n \sum_{j=1, j\ne i}^n e(C(z_j) ,\sum_{1\leq k\leq r}f_{ik}e_k )z_i^*z_j^* +\sum_{i=1}^{n}e(C(z_i) ,\sum_{1\leq k\leq r}f_{ik}e_k ) (z_i^*)^2.
+ \frac{R^{3}}{N \kappa^{2}}G_{\bar{A}B}D \bar{z}^{\bar{A}}D z^{B}+ \frac{i}{2 \kappa}D z^{B}( \bar{\lambda}G_{\bar{A}B}\bar{\chi}^{\bar{A}}+ \lambda G_{\bar{A}B}\bar{\phi}^{\bar{A}})
\epsilon^{r_{1}\cdots r_{9}m}\epsilon^{s_{1}\cdots s_{9}}{}_{m}\, W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}W_{r_{5}r_{6}s_{5}s_{6}}\, \Big ( H_{r_{7}r_{8}r_{9}}H_{s_{7}s_{8}s_{9}}- 3 \, H_{r_{7}r_{8}s_{9}}H_{s_{7}s_{8}r_{9}}\Big ) = 0 \, .
\begin{bmatrix}A_{1}^{\left(\ell\right)}\left(\xi,x\right) A_{2}^{\left(\ell\right)}\left(\xi,x\right)\end{bmatrix}=T^{(0)}_{x,0}\left(\xi\right)^{-1}\begin{bmatrix}u^{\left(\ell\right)}\left(\xi,x\right) u^{\left(\ell\right)}\,'\left(\xi,x\right)\end{bmatrix}.
L_{a d d i t i o n a l}= - i l ({\dot{\Theta}}^{\nu}D \Theta_{\mu}E_{\nu}^{\alpha}E_{\alpha}^{\mu}+ D \Theta^{\nu}D \Theta_{\mu}D X^{\underline{{{k}}}}({\tilde{\Omega}}_{\underline{{{k}}}})_{\nu}^{\mu})
N_f\,{\rm Tr}\,\sum_{n=2}^\infty\frac{1}{(p-m+i\epsilon)^n}\,f_n(A)=N_f\sum_{n=2}^\infty\int\frac{dp}{2\pi}\,\frac{1}{(p-m+i\epsilon)^n}\int dt\,f_n(A)
\sum_{\substack{\mathbf{r}\in\mathcal{D}_k' r_m=1}}\frac{(y_{\mathbf{r}}^{(m)})^2}{\phi_\omega(r)}=\Bigl(1+\Bigl(\frac{1}{(\log{x})^{1/10}}\Bigr)\Bigr)(\log{R})^{k+1}\frac{W^{k-1}B^{k-1}\mathfrak{S}_{WB}(\mathcal{L})}{\phi(WB)^{k-1}}J_k(F)\prod_{p|a_m, p\nmid W B}\frac{p-1}{p}.
\mathbb{M}=\mathbb{U}\Lambda\mathbb{\overline{V}}^T=\sum_{m=1}^{N}\sigma_m\mathbf{U}_m\overline{\mathbf{V}}_m^T\approx\sum_{m=1}^{3M}\sigma_m\mathbf{U}_m\overline{\mathbf{V}}_m^T,
\dot{x}^{2}( \tau , 0 ) = A^{2}\frac{[ f - g ]^{2}}{4 \acute{f}\acute{g}}, \; \; \dot{x}^{2}( \tau , l ) = A^{2}\frac{[ f_{l}- g_{l}]^{2}}{4 \acute{f}_{l}\acute{g}_{l}}.
f(\sin^2\theta_{p,p})=\frac{1}{\lambda_p^{-1}+\{\mathrm{Tr}[(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}]-2\lambda_p^{-1}\}\sin^2\theta_{p,p}},