harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal D}_{\alpha \beta}= \left( \begin{matrix}{\partial_{t}^{2}+ \beta_{k}- i \alpha_{k}}&{\gamma_{k}\partial_{t}+ \textstyle{\frac{1}{2}}\dot{\gamma}_{k}+ i \alpha_{k}}\\{- \gamma_{k}\partial_{t}- \textstyle{\frac{1}{2}}\dot{\gamma}_{k}+ i \alpha_{k}}&{- \partial_{t}^{2}- \beta_{k}- i \alpha_{k}}\\ \end{matrix}\right) ,
q[\mathbf{u},\mathbf{u}]=\int_{\mathbb{R}^d}\langle d\mu (\mathbf{x})\mathbf{u},\mathbf{u}\rangle =\sum \limits_{j,l=1}^n \int_{\mathbb{R}^d}u_l (\mathbf{x})u_j(\mathbf{x})^*\,d\mu_{jl}(\mathbf{x}),\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\delta x = \theta y = \frac{\partial F_{R}}{\partial y}, \hspace{0 . 5 c m}\delta y = - \theta x = - \frac{\partial F_{R}}{\partial x}, \hspace{0 . 5 c m}\Rightarrow \hspace{0 . 2 c m}F_{R}= \frac{\theta}{2}( x^{2}+ y^{2}) .
\mathbb{E}_{\sigma_{0}}\Vert \hat{S}_{t}\Vert_{2}^{2}= \Big(\,1-\frac{3}{2n}\,\Big)^{t}\,\Vert \hat{S}_{0}\Vert_{2}^{2}+ \frac{2}{3n}\Big[\,1-\Big(\,1-\frac{3}{2n}\,\Big)^{t}\,\Big].
\displaystyle{\sum_{\lambda = - \infty}^{+ \infty}\exp \left[ -{\frac{g_{c}^{2}A_{c}N}{2}}\left( \lambda + \frac{l}{N}\right)^{2}\right] = \sqrt{\frac{2 \pi}{g_{c}^{2}A_{c}N}}\sum_{\lambda = - \infty}^{+ \infty}\exp \left[ - \frac{2 \pi^{2}}{g_{c}^{2}A_{c}N}\lambda^{2}- 2 \pi i \lambda \frac{l}{N}\right]}.
\sinh{d(m,m')}\leq \sinh{d(i(\bar{x}),\bar{x}')}\cosh{d(i(\bar{x}),m)}+\cosh{d(i(\bar{x}),\bar{x}')}\sinh{d(i(\bar{x}),m)}\sin{\angle{(\vec{n}'@\bar{x}',\vec{n}@\bar{x}')}}& \leq \frac{\delta}{100}\cosh{d(i(\bar{x}),m)}+\frac{\delta}{100}\sinh{d(i(\bar{x}),m)}.
{U}:=\sum\nolimits_{i=1}^m U_iU_{i-1}^\dagger\otimes e_ie_i^\dagger\quad{W}:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}\otimes \sum\nolimits_{i=1}^{m}e_{i+1}e_{i}^\dagger
X_{A_{0}\cdots A_{r}}^{\left( p\right)}=C_{A_{0}A_{p}}^{B}\left\langle \boldsymbol{T}_{A_{1}}\cdots\boldsymbol{T}_{A_{p-1}}\boldsymbol{T}_{B}\boldsymbol{T}_{A_{p+1}}\cdots\boldsymbol{T}_{A_{r}}\right\rangle .
\overline{{\not \! \partial}}= \gamma^{a}\delta_{a}^{\mu}\partial_{\mu}, \qquad \not \! \partial = \gamma^{\mu}\partial_{\mu}= a^{- 1}\overline{{\not \! \partial}}, \qquad \overline{{\gamma}}_{\mu}= \delta_{\mu}^{a}\gamma_{a}, \qquad \overline{{\gamma}}^{\mu}= \delta_{a}^{\mu}\gamma^{a}.
2m_{1}\partial(a_{1}e^{2m_{1}(\varphi-\eta)})\wedge\bar\partial(\varphi-\eta) &=2m_{1}(e^{2m_{1}(\varphi-\eta)}\partial a_{1}\wedge\bar\partial(\varphi-\eta)+2m_{1}a_{1}e^{2m_{1}(\varphi-\eta)}\partial(\varphi-\eta)\wedge\bar\partial(\varphi-\eta) )
\dot{W}(x_t) = - \begin{bsmallmatrix}x(t) x(t-h)\end{bsmallmatrix}^\top \begin{bsmallmatrix}\eta_0 \mathcal{H}(A) + \frac{1}{2}I_m \eta_0 A_d \ast \frac{1}{2}I_m\end{bsmallmatrix}\begin{bsmallmatrix}x(t) x(t-h)\end{bsmallmatrix},
V_{P}( B ) = \frac{1}{4 \pi^{2}l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u \, u^{2}\ln \left[ \frac{[ r + c ] [ r + c - c \, \eta^{2}/ 2 ]}{[ r + \pi / 2 u ]^{2}}\right] .
|g(X_1,\dots, X_n)-g(X_1',\dots, X_n')|\leq &12\frac{\|\Sigma\|^{1/2}+\sqrt{2\delta}}{\sqrt{n}}\sup_{\|u\|\leq 1}\biggl(\sum_{j=1}^n \langle X_j-X_j',u\rangle^2\biggr)^{1/2}\bigvee\frac{12}{n}\sup_{\|u\|\leq 1}\sum_{j=1}^n \langle X_j-X_j',u\rangle^2.
\begin{array}{lll}\sum_{i=0}^{k-1}\widetilde{b}_{i+1}\cdot \mu h^{-1}[b_i , b_{i+1}] \leq \sum_{i=0}^{k-1}\widetilde{(b_{i+1}+ \epsilon_i )}\cdot \mu h^{-1}[b_{i}-\epsilon_i , b_{i+1}+ \epsilon_i ] = & \sum_{i=0}^{k-1}\widetilde{a_i}\cdot \mu h_{0}^{-1}[a_i , a_{i+1}] < \widetilde{d'}.\end{array}
{\hat H}=\frac{H}{M}=\frac{2\pi}{l}\left( H_{\mathrm{CFT}}+\kappa(h)\frac{l^{2-h}}{\left( 2\pi \right)^{1-h}}B\right)\, ,\qquad H_{\mathrm{CFT}}=L_{0}+\bar{L}_{0}-\frac{c}{12}{\mathrm{Id}}\: .
\Phi_1(h_1) \ge c \| h_1 \|_2^2, \forall \; h_1 \int \left( \partial_{x_j}Q \right) h_1 = \int (|\cdot|^{-(N-\gamma)}* Q^p) Q^{p-1}h_1 =0, \forall 1 \le j \le N.
\sum\limits_{n\geq 0}a(B^{s}n+m)f(B^{t}n+x)= \sum\limits_{n\geq 0}a(B^{s-1}n+v_B(b_2...b_s))f(B^{t-1}n+x-B^{t-1}b_1) -\sum\limits_{b\in \{0,\ldots,B-1\}\backslash \{b_1\}}\sum\limits_{n\geq 0}a(B^{s-1}n+v_B(bb_2...b_{s-1}))f(B^{t}n+x+B^{t-1}(b-b_1)).
\sup_{\tilde X\in{{B_\epsilon(X)}}}u(\tilde X,\tilde Y-(\tilde t-t)X,\tilde t) =\max_{\tilde X\in\overline{B_\epsilon(X)}}u(\tilde X,\tilde Y-(\tilde t-t)X,\tilde t), \inf_{\tilde X\in{{B_\epsilon(X)}}}u(\tilde X,\tilde Y-(\tilde t-t)X,\tilde t) =\min_{\tilde X\in\overline{B_\epsilon(X)}}u(\tilde X,\tilde Y-(\tilde t-t)X,\tilde t),
\phi_{i,, \beta}(\xi) = \sum_{j=0}^i c_{i,j}(2\beta)^j b^j T_j(\xi) + \tilde \lambda \sum_{j=0}^i b^{j+1}\left(c_{i,j}(2\beta)^{j+1}T_{j+1}(\xi) + S_j (\xi)\right) + bR_i(\xi),
u_{a,b}(\alpha(z))=\prod_{\gamma\in \Gamma}\frac{\alpha(z)-\gamma a}{\alpha(z)-\gamma b}=\prod_{\gamma\in\Gamma}=\prod_{\gamma\in \Gamma}\frac{\frac{pz+q}{rz+s}-\gamma a}{\frac{pz+q}{rz+s}-\gamma b}=\prod_{\gamma\in \Gamma}\frac{pz+q-rz\gamma a-s\gamma a}{pz+q-rz\gamma b- s\gamma b}
e_1=\partial_t,e_2=\partial_\varphi,e_3=\sin(2\sqrt{\epsilon}t)\partial_t+\sqrt{\epsilon}\cos(2\sqrt{\epsilon}t)r\partial_r, e_4=\cos(2\sqrt{\epsilon}t)\partial_t-\sqrt{\epsilon}\sin(2\sqrt{\epsilon}t)r\partial_r.
\gamma_{5}\left( \begin{array}{c}{\lambda_{A}^{i^{\star}}}\\{\zeta^{\alpha}}\\{\psi^{A}}\\ \end{array}\right) = - \left( \begin{array}{c}{\lambda_{A}^{i^{\star}}}\\{\zeta^{\alpha}}\\{\psi^{A}}\\ \end{array}\right)
=\pi\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{(z^{2}+t^{2}+1)^{3}+(z^{2}+t^{2}+1)^{2}-1}{(z^{2}+t^{2}+1)^{3}}exp(-\frac{(z^{2}+t^{2})^{2}}{z^{2}+t^{2}+1})dzdt,
\|e^{s\varphi}Pw\|_{L^2(\mathbb{T}\times(0,T))}^{2}= \|P_pv\|_{L^2(\mathbb{T}\times(0,T))}^{2}+ \|P_nv\|_{L^2(\mathbb{T}\times(0,T))}^{2}+2(P_pv,P_nv)_{L^2(\mathbb{T}\times(0,T))}^{2}.
\varPhi(M_1)-\varPhi(M_0) =\int_0^1\varPhi'(M_{t-})\, dM_t+{1\over2}\int_0^1\varPhi''(M_{t-})\, d \bigl\langle M^c\bigr\rangle_{t}&\quad+\sum_{0< t\leq1}\bigl[\varPhi(M_{t})-\varPhi(M_{t-})-\varPhi '(M_{t-})(M_t-M_{t-}) \bigr].
[\dots,(\alpha_{n-1},\epsilon_{n-1}),(\alpha_n,\epsilon_n),(1,1),(\alpha_{n+2},\epsilon_{n+2}),\dots] &\qquad=[\dots,(\alpha_{n-1},\epsilon_{n-1}),(\alpha_n+\epsilon_n,-\epsilon_n),(\alpha_{n+2}+1,\epsilon_{n+2}),\dots].
\varphi_P(f)=\sum_{I\in P}m(I)\varphi_I(f)=\sum_{I\in P}\sum_{I'\in P'}m(I\cap I')\varphi_{I\cap I'}(f)=\sum_{I'\in\P'}m(I')\varphi_{I'}(f)=\varphi_{P'}(f)\,.
Pr[|\frac{1}{m}(\sum_{i}(w_{\sigma_{i}}^{j}-w_{\sigma_{m+i}}^{j}))|\ge\frac{\epsilon}{2}]=Pr[|\frac{1}{m}(\sum_{i}|w_{i}^{j}-w_{m+i}^{j}|\beta_{i})|\ge\frac{\epsilon}{2}],
\left(\frac{d}{2}\right)^{-\frac{(k-1)p}{2}}\prod_{i=1}^{k-1}\frac{\Gamma \left( (d-i+1)/2\right)}{\Gamma \left((d-p-i+1)/2\right)}\det(S_{k-1})^{-\frac{p}{2}}\left(\prod_{i=1}^{k-1}c_i'\beta_i \right)^{d-p-k}e^{\frac{(k-1)}2 \|x\|^2}
E\left[\int_s^t \partial_r \beta(r,q_r^m) \|z_r^m\|^2dr\right] = E\left[\int_s^t \partial_r \beta(r,q_r) \int \|z\|^2 h(r,q_r,z)dr\right]+O(m^{1/2}) = E\left[n\int_s^t \beta^{-1}(r,q_r)\partial_r \beta(r,q_r)dr \right]+O(m^{1/2}).
\|\theta(t_{1})-\theta(t_{2})\|_{H^{s}}=\Big\{(\sum_{k<N}+ \sum_{k\geq N}) (2^{ks}\|\Delta_{k}\theta(t_{1})-\Delta_{k}\theta(t_{2})\|_{L^{2}})^{2}\Big\}^{\frac{1}{2}}.
\upsilon_{k+1}\leq\Big(1+\frac{2}{n}\Big)\frac{1}{k}\sum_{i=1}^k\upsilon_i+\Big[\Big(\frac{2}{n}\frac{1}{k}\sum_{i=1}^k\upsilon_i\Big)^2-\Big(1+\frac{4}{n}\Big)\frac{1}{k}\sum_{j=1}^k\Big(\upsilon_j-\frac{1}{k}\sum_{i=1}^k\upsilon_i\Big)^2\Big]^\frac{1}{2},
\int_{\mathbb{R}}J_2(y)(-y)\int_{0}^{1}[\psi(\xi_n-\tau y)-\psi(-\tau y)]d\tau dy begin{align*}0.2cm]\leq 2\int_{\mathbb{R}}J_2(y)|y|dy\leq4\int_{\mathbb{R}}J_2(y)e^{y}dy<\infty.
H_{n,k}(z)=\begin{cases}z\int_{a_{k-1}}^{b_{k-1}}\frac{P_{n,k-1}^{2}(\tau)}{z-\tau}\,\frac{H_{n,k-1}(\tau)\,d\sigma_{n,k-1}(\tau)}{P_{n,k-2}(\tau)\,P_{n,k}(\tau)}, \ell<k, begin{align*}1em]\int_{a_{k-1}}^{b_{k-1}}\frac{P_{n,k-1}^{2}(\tau)}{z-\tau}\,\frac{H_{n,k-1}(\tau)\,d\sigma_{n,k-1}(\tau)}{P_{n,k-2}(\tau)\,P_{n,k}(\tau)}, k\leq \ell. \end{cases}
A_k(t,\xi)=\frac{1}{(\xi-kt)^2+\varepsilon_0^{-1}k^2},\,k\neq 0,A_0(t,\xi)=\frac{1}{\xi^2+\varepsilon_0^{-1}};A_{*}(t,\xi)=\sum_{k}e^{-2|k|}A_{k}(t,\xi),
k \cdot \gcd\left( C,C_1,C_2, \dots ,C_m\right) = \gcd\left( k C,k C_1,k C_2, \dots ,k C_m\right) & = \gcd\left( k C, j_1 C, j_2 C, \dots, j_m C\right) & = C\gcd(k,j_1,j_2,\dots,j_m).
k(x)=\int_{\left[\frac{x}{c},\infty\right)}h\left(\frac{x}{r}\right)\nu(dr)=\int_{\left[\frac{x}{c},\infty\right)\setminus \{\frac{x}{r}\in g(T)\}}h\left(\frac{x}{r}\right)\nu(dr)+\int_{\{\frac{x}{r}\in g(T)\}}h\left(\frac{x}{r}\right)\nu(dr).
W(0, \mbox{Id}) = 0, DW(0, \mbox{Id}) = 0, ~~~ \mbox{and:}D^2W(0,\mbox{Id}) :(\tilde\phi, \tilde F)^{\otimes 2}\geq \gamma (|\tilde\phi|^2 + |\mbox{sym~}\tilde F|^2) \mbox{\ \for all \}(\tilde \phi, \tilde F)\in \mathbb{R}\times\mathbb{R}^{3\times 3}.
\sigma_{0}:= \dfrac{2(\mathit{e}-1)}{\mathit{e}(2-2\alpha+\beta)+\sqrt{(\mathit{e}(2-2\alpha+\beta))^2-4(\mathit{e}-1)(1-\mathit{e}(2\alpha-1+\beta))}},\intertext{and}\tilde{\sigma_{0}}:= \dfrac{2(\mathit{e}-1)}{(2-2\alpha+\beta)+\sqrt{(2-2\alpha+\beta)^2-4(\mathit{e}-1)(2\alpha-1+\beta-\mathit{e})}}.
L_{i}(x)= F_{i+1}(x)+F_{i-1}(x), Q_{i}(x)=P_{i+1}(x)+P_{i-1}(x), f_{i}(x)= \mathcal{F}_{i+1}-2\, \mathcal{F}_{i-1}(x), 2\, T_{i}(x)=U_{i}(x)-U_{i-2}(x), D^{\alpha}_{i}(x)= E^{\alpha}_{i}(x)-\alpha\, E^{\alpha}_{i-2}(x).
\Lambda_{M_j}^{r,s}(g_j^{v-1}(z_j,s),h^{v-1}(s))-\Lambda_{M_{0j}}^{r,s}(z_j)-L(g_j^{v-1}(z_j,s)-z_j,h^{v-1}(s)) &\ll \sum_{\mu=2}^\infty{d_0}^\mu(n+m)^\mu A(s)^\mu\ll \sum_{\mu=2}^\infty d_0^{\mu}(m+n)^\mu\left(\frac{b}{c}\right)^{\mu-1}A(s)= \frac{bd_0^2(m+n)^2}{c}\sum_{\mu=2}^\infty\left(\frac{bd_0(m+n)}{c}\right)^{\mu-2}A(s)
\left\vert \phi_{k}^{\pm}\left( I_{\pm}\right) \right\vert =\left\vert\frac{1}{2\pi ki}\int_{0}^{2\pi}e^{-ik\theta_{\pm}}\phi^{\prime}(x)\frac{\partial x}{\partial\theta_{\pm}}d\theta_{\pm}\right\vert \leq\frac{1}{2\pi k}||\phi||_{W^{1,1}}\lesssim\frac{1}{k}||\phi||_{H^{1}}.
\mathcal E^{1D}_{\alpha,b}(f)=\int_{-\infty}^\infty\left(|f'(t)|^2+\left(\frac{t^2}2+\alpha\right)^2|f(t)|^2-b\, |f(t)|^2+\frac{b}2\, |f(t)|^4\right)\,dt\,,
\begin{cases}\displaystyle \sum_{j=-2}^2 \frac{c_j(\frac{3}{5}-b_j)}{(\frac{1}{16}+(\frac{3}{5}-b_j)^2)^2}= 0, \displaystyle \sum_{j=-2}^2 \frac{c_j(\frac{1}{16}-3b_j^2)}{(\frac{1}{16}+b_j^2)^3}= 0.\end{cases}
[f_1, \cdots, f_n]=\hbox{det}\left( \begin{array}{cc}f_1\quad\cdots f_n D_1(f_1)\, \cdots \,D_1(f_n) \cdots\cdots\cdots\cdots\cdots\cdots \cdots D_{n-1}(f_1)\, \cdots \,D_{n-1}(f_n) \end{array}\right)
\mathcal{K}_{n}\left( \mathcal{E},0\right) =\frac{\left( \gamma +\frac{3}{2}-D\right)_{n}}{n!}_{3}F_{2}\left(\begin{array}{ccc}-n \gamma +\frac{1}{2}-i\lambda \left( \mathcal{E}\right) \gamma +\frac{1}{2}+i\lambda \left( \mathcal{E}\right) . \gamma +\frac{3}{2}-D 2\gamma +1\end{array}\mid 1\right)
||f||^2 =(1-|a|^4)^2||h_1-\overline{a}h_0||^2 &=(1-|a|^4)^2||e_1-\overline{a}e_0||^2 &=(1-|a|^4)^2\frac{1+|a|^2}{1-|a|^2}&=(1-|a|^4)(1+|a|^2)^2.
|G|= |\Phi| + |G|\sum_{i=1}^{k_1}\frac{1}{\lambda_{i}}+ |G|\sum_{i=1}^{k_2}\frac{\nu_i}{\mu_i}+|G|\sum_{i=1}^{k_3}\left(\frac{1}{\omega_i}- \frac{1}{q_i}\right),
\rho(f^I) = \sum_{\nu=0}^{\lfloor \frac{k-1}{2}\rfloor}e_{k-1-2\nu}(X_i \mid i\in I)Y^\nu \frac{1}{2}\rho(f^I(y_2-y_1)) = \sum_{\nu=0}^{\lfloor \frac{k}{2}\rfloor}e_{k-2\nu}(X_i \mid i\in I)Y^\nu.
v_b^{w \, \prime}(b+) = \left( \theta \dfrac{{C^{(\theta,r)}(b;w)}+ \rho_b^{(\theta)}(b;w)}{Z^{(\theta)}(b)}- w(0) \right) W^{(\theta)}(b) - \rho_b^{(\theta)}(b;w') = v_b^{w \, \prime}(b-).
G_{T}^{+}( x , y ) = \sum_{m = - \infty}^{\infty}\int_{0}^{\infty}d q \int_{- \infty}^{\infty}d k \, N_{2}^{2}\, e^{- i [ \frac{m}{R_{0}}\sinh \Omega R_{0}+ \omega \cosh \Omega R_{0}] \Delta t}J_{m}^{2}( q R_{0}) .
\begin{array}{r c l}{{S_{3}}}&{{=}}&{{r_{1}( - \gamma ) r_{2}( - 2 \gamma ) r_{1}( - \gamma )}}\\{{\tilde{S}_{3}}}&{{=}}&{{r_{5}( - \gamma ) r_{4}( - 2 \gamma ) r_{5}( - \gamma ) \; .}}\\ \end{array}
\int_{P(z,2^{-(i-1)}\delta_{\Omega}(z))\setminus P(z,2^{-i}\delta_{\Omega}(z))}\frac{d\lambda(\zeta)}{\left|z-\zeta\right|^{1+\eta}}\lesssim\left(2^{-i}\right)^{2}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right)\tau_{n}^{\frac{1-\eta}{m}}\left(z,\delta_{\Omega}(z)\right).
\Delta t_{E}-\Delta t_{B}=\frac{1}{\lambda}ln(\frac{(^{129}I/^{127}I)_{B}}{(^{129}I/^{127}I)_{E}})=\frac{1}{\lambda}ln(\frac{(^{129}I/^{127}I)_{B}}{(^{129}Xe/^{127}I)_{BSE}})
s= -\frac{17}{5},-\frac{16}{5},-3,-\frac{14}{5},-\frac{13}{5},-\frac{12}{5},-\frac{9}{5},-\frac{8}{5},-\frac{6}{5},-\frac{3}{5},0,\frac{2}{5},\frac{12}{5},\frac{18}{5},\frac{27}{5},\frac{42}{5},\frac{72}{5},\frac{162}{5}.
\begin{cases}\partial_t \omega^j+ (u^j \cdot \nabla )\omega^j = (\omega^j \cdot \nabla) u^j, 0<t\le 1, u^j =-\Delta^{-1}\nabla \times \omega^j, \omega^j \Bigr|_{t=0}=f^j \in C_c^{\infty}(\mathbb R^3), j=1,2. \end{cases}
\mathbf{\Pi}_{N}\left( \overline{\mathbf{F}}_{N}(C) \in B( \overline{\mathbf{P}}, \delta) \right) \geq \mathbf{\Pi}_{N}\left( \frac{|C|}{N}\in (1-\epsilon, 1+\epsilon) \right) \min_{\frac{j}{N}\in (1-\epsilon, 1+\epsilon)}\mathbf{\Pi}_{N}\left( \overline{\mathbf{F}}_{N}(C) \in B( \overline{\mathbf{P}}, \delta) \Big| |C| = j \right).
\langle w, [\mathcal{Q}+\,(\mathcal{S}+(1-\tau)\sigma\mathcal{A}\mathcal{A}^*, \mathcal{T}+ (1-\tau)\sigma\mathcal{B}\mathcal{B}^*)]w\rangle = 0 \Rightarrow \|u\|\|v\| = 0,
\langle D \phi_N(\sigma^N(t)),\sigma^N (t) -\varrho(t)\rangle =-\langle\dot u_3^N(t)-\dot w_3(t),\ddot u_3^N(t)\rangle+\langle E\dot w(t),\sigma^N(t)-\varrho(t)\rangle-\langle\mathbb A_r\dot \sigma^N(t),\sigma^N(t)-\varrho(t)\rangle \leq C(\|\ddot u_3^N(t)\|_{L^2}+\|\dot \sigma^N(t)\|_{L^2}+1),
\kappa_1a+\kappa_2=\int \sqrt{h(b)}db \simeq \int\sqrt{\frac{3\pi}{8}-2b^2}\, db ={\frac{4\,b\,{\sqrt{3\,\pi-16\,{b^2}}}+ 3\,\pi \,\arcsin ({\frac{4\,b}{{\sqrt{3\,\pi}}}})}{16\,{\sqrt{2}}}}\quad .
{\bf Q}_f(\mu)(\phi^{M}\,[\lambda/\zeta]^M{\bf Q}_f(q\lambda)+\phi^{- M}\,[\lambda \zeta]^M{\bf Q}_f(q^{-1}\lambda))= = (\phi^{M}\,[\lambda/\zeta]^M{\bf Q}_f(q\lambda)+\phi^{- M}\,[\lambda \zeta]^M{\bf Q}_f(q^{-1}\lambda)){\bf Q}_f(\mu)
R_{j}(r,v,\varphi,\varepsilon) \bigl|_{r_{j}=0}= 0 \forall (r, v, \varphi, \varepsilon) \in \mathcal{S}_{\varrho}\times B_{R^{\ast}}^{m}\times \mathbb{T}^{n}\times (0, \varepsilon_{0}), \; j=1, \ldots, n.
R = \sum_{x = 1}^{N}R_{x}= \sum_{x = 1}^{N}\sum_{a = 1}^{2}R_{x}^{( a )}= \sum_{x = 1}^{N}\sum_{a = 1}^{2}\psi_{a , x + 1}^{\dag}e^{i A}\psi_{a , x}\quad .
\mathcal{W}_q=q\frac{\Delta_{\{b+1,\dots, n-1,1\}}}{\Delta_{\{b+1,\dots, n\}}}+\sum_{i=1}^{n-1}\frac{\Delta_{\{i-a+1,\dots, i-1,i+1\}}}{\Delta_{\{i-a+1,\dots, i\}}}.
\deg_{F_1\cup F_2}(u) ={\deg_{{F_1}}}(u) +{\deg_{{F_2}}}(u) \geq \delta(F)-d_{F_3}(u)\geq v(F) - \sqrt \delta t/2-\left(\frac{1}{2}+\sqrt{\delta}\right)t &\overset{(\ref{F-degree})}{>}\left({1 - \frac{{\frac{1}{2}+\frac{3}{2}\sqrt \delta}}{{1 - \frac{3}{2}\sqrt \delta}}}\right)v(F)> \left( \frac{1}{2}-3\sqrt{\delta}\right) v(F).
\sum_{n=M_-}^{M_+-1}A_2(n)|C_n|\leq c_1\,\frac{(\gamma t)^{-\frac12+\varepsilon(d-1)}}{M^d}\sum_{n=M_-}^{M_+}n^{d-1}\leq c_2\,\frac{(\gamma t)^{\varepsilon d}}{M}\leq c_3\,{(\gamma t)^{-1/2+\varepsilon d}}.
{\mathfrak M}_\psi^+: =\bigl\{a\in A^+:\psi(a)<\infty\bigr\}{\mathfrak N}_\psi: =\bigl\{a\in A:\psi(a^*a)<\infty\bigr\}{\mathfrak M}_\psi: ={\mathfrak N}_\psi^*{\mathfrak N}_\psi=\operatorname{span}\{y^*x:x,y\in{\mathfrak N}_\psi\}
\exp^*(\alpha) = 1 + \alpha + \frac{1}{2!}\alpha*\alpha + \frac{1}{3!}\alpha*\alpha*\alpha + \ldots \exp^{\cdot}(\alpha) = 1 + \alpha + \frac{1}{2!}\alpha \cdot \alpha + \frac{1}{3!}\alpha \cdot \alpha \cdot \alpha + \ldots .
[ g_{I J}] = \frac{R_{2}}{\mathrm{I m}\tau}\left( \begin{matrix}{1}&{\mathrm{R e}\tau}\\{\mathrm{R e}\tau}&{| \tau |^{2}}\\ \end{matrix}\right) , [ \beta_{I J}] = \left( \begin{matrix}{0}&{R_{1}}\\{- R_{1}}&{0}\\ \end{matrix}\right)
p(a_1, \, a_2, \, \ldots, \, a_n,\, b_1, \, b_2, \, \ldots, \, b_n) = \begin{cases}(a_1, \, a_n), \;\; \mbox{if}\;\; n \; \mbox{is even}, (a_1, \, b_n), \;\; \mbox{if}\;\; n \; \mbox{is odd}\end{cases}.
\Pr \left[K_{1,n,0}^c=k^c\right]=g(\mathcal F_{1,-n}^c,k^c-1),k^c=1,\cdots, K_1^c, &\Pr \left[\overline{K}_{1,n,0}^b=k^b\right]=g(\mathcal F_{1}^b,k^b),k^b=0,\cdots, F_1^b, &\Pr \left[\overline{K}_{1,n,0}^c=k^c\right]= g(\mathcal F_{1}^c,k^c),k^c=0,\cdots, K_1^c, &\Pr \left[K_{1,n,0}^b=k^b\right]=g(\mathcal F_{1,-n}^b,k^b-1),k^b=1,\cdots, F_1^b,
Y_s^n \ = \g(\hat X_T^{t,x,a}) + \int_s^T f(r,\hat X_r^{t,x,a},\hat I_r^{t,a}) dr + n\int_s^T \int_\Lambda \big(R_r^n(b)\big)_+ \lambda_\theta(db) dr &\ - \int_s^T Z_r^n d\hat W_r - \int_s^T\int_\Lambda R_r^n(b) \hat\theta(dr,db) - \int_s^T\int_{U\setminus \{0\}}L_r^n(z)\,(\hat\pi(dr\,dz)-\lambda_\pi(dz)\,dr),
\prod_{j=1}^k\frac{z-\cos(y+2\gamma(s-j))}{z-\cos(y-2\gamma(s+j))}=1+\sum_{m=0}^{k-1}\prod_{l=0}^m\frac{2v_nv_s\sin\gamma(2s-l)}{\sin\gamma(l+1)}\frac{\sin\gamma(n-k+l)\sin\gamma(k-l)}{z-v_nv_s\cos\gamma(n+1-2k+2l-2s)}
\|L\|^2 = \sup_{\|x\| = 1}\int_{0}^{2\pi}Tx(t)^2 dt = \sup_{\|x\| = 1}\int_{0}^{2\pi}t^2 dt \left( \int_{0}^{2 \pi}x(s) ds \right)^2 \leq \sup_{\|x\| = 1}\frac{8}{3}\pi^3 \int_{0}^{2 \pi}x(s)^2 ds \cdot 2\pi = \frac{16}{3}\pi^4.
\partial_{\tilde{t}}\ \left( \frac{\lambda}{\rho ( \varphi ) h ( x )}\right) \; = \; \lambda \ \sqrt{\tilde{g}\;}/ \tilde{\rho}\h^{2}( x ) \n^{i}\ \nabla_{i}\ \left( \frac{1}{\rho \, h}\right) \ ,
\varphi (g_1 g_3) \varphi (g_1 g_2) \overset{(\ref{eq:anti})}{=}\varphi (g_3) \varphi (g_1) \varphi (g_1) \varphi (g_2) \overset{{\bf A3}}{=}\varphi (g_3) \varphi (g^{2}_1) \varphi (g_2) \overset{(\ref{eq:comm})}{=}\varphi (g_3 g^{2}_1) \varphi (g_2) \,,
\frac{1}{3v^2}\int_0^vA(v')v'^3dv'=\frac{1}{3v^2}\int_0^v\left(\frac{\rho_0(v')}{v'}+\mathcal{O}(1)\right)v'^3dv'=\frac{1}{3v^2}\int_0^v\rho_0(v')v'^2dv'+\mathcal{O}(v^2).
\int\limits_{1}^{\infty}y \, e^{-q_1 y}\, dy= \left[ - \frac{e^{-q_1 x}}{q_1^2}\, (q_1 \, x + 1)\right]_1^\infty= \frac{e^{-q_1}}{q_1^2}\, (q_1 + 1)
{M_{c}= \left( \begin{matrix}{{0}}&{{1}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{\cdots}}&{{0}}\\{{\cdots}}&{{\cdots}}&{{\cdots}}&{{\cdots}}&{{\cdots}}&{{\cdots}}\\{{1}}&{{0}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\ \end{matrix}\right) ,}
\begin{array}{c}{{A_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) , \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) \right) ,}}\\{{B_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) , - \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) \right) ,}}\\ \end{array}
F(x,\sigma^{2}|\mu,\lambda,\alpha,\beta)=\frac{e^{-\frac{\beta}{\sigma^{2}}}(\frac{\beta}{\sigma^{2}})^{\alpha}(erf(\frac{\sqrt{\lambda}(x-\mu)}{\sqrt{2}\sigma})+1)}{2\sigma^{2}\Gamma(\alpha)}
\frac{1}{2\pi}\int_{0}^{T}\left[\log \left| -\frac{1}{2}-\epsilon+it\right|-\log \left| -\frac{1}{2}+\epsilon+it\right|\right]\dt+ \frac{1}{2\pi}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\left[\arg \left(\sigma-1+iT\right)-\arg (\sigma-1)\right]\d\sigma=0
R_{f}^{-}( \hat{k}) = K_{f}^{-}( \hat{k})_{c l a s s}\left[ 1 - \frac{i \beta^{2}}{1 6 \pi}\sinh \theta \left( \frac{1}{\cosh \theta + 1}- \frac{1}{\cosh \theta}\right) + \frac{i \beta^{2}}{8 \pi}\frac{\theta}{\cosh \theta}\right] .
\eta^{k(\ell)-1}= f_{j_{k(\ell)-1}}(x^{\ell}) + \sum_{i\neq j_{k(\ell)-1}}\tilde{f}^{k(\ell)-1}_{i}(x^{(\ell)}) + \frac{1}{2}\big\|x^{(\ell)}- x^{(\ell-1)}\big\|_D^2.
\begin{aligned}L_1(f) :=4f_1^{0.8}+ 2^{2.8}f_2^{0.8}- 2^{2.8}, L_2(f) := 2^{2.8}f_1^{0.8}+ 4f_2^{0.8}- 2^{2.8}, R_1(f) := - \bigl((2^{0.6}- 1)f_1 -2^{0.6}+2\bigr) - \bigl((2^{0.6}- 1)f_2 -2^{0.6}+2\bigr) +1, R_2(f) := -2^{0.6}\bigl((2^{0.6}- 1)f_1 -2^{0.6}+2\bigr) -2^{0.6}\bigl((2^{0.6}- 1)f_2 -2^{0.6}+2\bigr) +2^{1.2},\end{aligned}
V_n := \bigcup_{| \boldsymbol{i}| = n}\psi_{\boldsymbol{i}}(V_0) = \bigcup_{| \boldsymbol{i}| = n}\Delta_{\boldsymbol{i}}, E_n := \bigcup_{| \boldsymbol{i}| = n}\psi_{\boldsymbol{i}}(E_0).
\frac{\mathsf{a}_\ell^k}{\mathsf{S}_\ell^k}\le\frac{\theta\sqrt{k(\ln k)^{1+\lambda}}}{\ell(\ln\ell)^{1+\lambda}}\le\frac{\theta r^{-1}}{\sqrt{k}}\cdot\frac{\sqrt{(\ln k)^{1+\lambda}}}{(\ln(rk))^{1+\lambda}}=\theta r^{-1}\frac{(\ln k)^{\frac{1+\lambda}{2}}}{\sqrt{k}\left[\ln k -\ln(r^{-1})\right]^{1+\lambda}},
W_{0,0}(f,g)(n)= a_{n+1}f(n)g(n+1)-a_{n+1}f(n+1)g(n), W_{a',a'}(f,g)(n)= (a_{n+1}+a_{n+1}') f(n)g(n+1)-(a_{n+1}+a_{n+1}')f(n+1)g(n), W_{0,a'}(f,g)(n)= (a_{n+1}+a_{n+1}') f(n)g(n+1)-a_{n+1}f(n+1)g(n).
\mathbb{E}[N_a^{\hat{x}}(G') - N_a^{\hat{x}}(H) \mid \hat{x}] = \frac{(a-b)}{n}{\Big(}\left((n^{++}-n_C^{++}) - (n^{+-}- n_C^{+-})\right) (n_C^{--}- n_C^{-+}) \quad+ \left((n^{--}-n_C^{--}) - (n^{-+}- n_C^{-+})\right) (n_C^{++}- n_C^{+-}){\Big)}.
|g(z)| \leq \left(\frac{1-\alpha}{1+\alpha}\right)^2\log{\frac{1+\alpha |z|}{1-|z|}}+\frac{|z|}{(1-|z|)^2}-\left(\frac{1-\alpha}{1+\alpha}\right)\frac{2|z|}{1-|z|}&\leq \frac{|z|(\alpha+|z|)}{(1+\alpha |z|)(1-|z|)^2}, z \in \Delta.
\left(\left\lceil \frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{\sum_{j=1}^i \mathcal E_j}\right\rceil,\left\lceil \frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{\sum_{j=1}^i \mathcal E'_j}\right\rceil\right)_{1\le i\le k},c:=\frac{\pi}{\sqrt{6}},
\tilde{r}_{i}( \nu ) \equiv \int d t e^{- i \nu t}r_{i}( t ) \, , \kern 0 . 3 5 i n \tilde{\rho}_{\theta}^{i}( \nu ) \equiv \int d t e^{- i \nu t}\rho_{\theta}^{i}( t ) \, , \kern 0 . 3 5 i n \tilde{\rho}_{\phi}^{i}( \nu ) \equiv \int d t e^{- i \nu t - i \delta_{i}( t )}\rho_{\phi}^{i}( t ) \, .
\begin{array}{rccccl}dp = e_j\omega^j + e_{n+j}\omega^{n+j}+ T\omega^{2n+1}, de_j= e_k\omega^k_j + e_{n+k}\omega^{n+k}_j + T\omega^{n+j}, de_{n+j}= e_k\omega^k_{n+j}+ e_{n+k}\omega^{n+k}_{n+j}- T\omega^j, dT= 0. \end{array}
| D ( \varphi_0) \rangle = \frac{1}{\sqrt{2 r}}\exp{[ - \sum_{n=1}^{\infty}a^{\dagger}_{Ln}a^{\dagger}_{Rn}]}\sum_{k= - \infty}^{\infty}e^{- i k \varphi_0/r}| ( 0, k ) \rangle .
Z(H) =Z_{00}(H)+Z_{11}(H)+Z_{01}(H)+Z_{10}(H) &\leq Z_{00}(H)+Z_{11}(H)+2\sqrt{Z_{00}(H)Z_{11}(H)}&\leq Z_{00}(G\cup G)+Z_{11}(G\cup G)+2\sqrt{Z_{00}(G\cup G)Z_{11}(G\cup G)}&=Z_{00}(G\cup G)+Z_{11}(G\cup G)+Z_{01}(G\cup G)+Z_{10}(G\cup G) &=Z(G\cup G).
\begin{aligned}\mathbb{E}_j\tau(1+\gamma_1+\cdots+\gamma_{k-i-1})= (\kappa_{j-i}+\cdots+\kappa_{k-i-1}) \qquad+\sum_{\begin{subarray}{c}1\le \ell_1\le j-i-1 j-i\le\ell_2\le k-i-1 \end{subarray}}(\gamma_{\ell_1}\kappa_{\ell_2}-\kappa_{\ell_1}\gamma_{\ell_2})\end{aligned}
\left( x_{N}^{0}\right)^{\gamma+m-j}\frac{|\Delta_{\overline{h}^{\prime},x^{\prime}}^{s}D_{x}^{\alpha}v(x_{0},t_{0})|}{|\overline{h}^{\prime}|^{\gamma}}\leq C\left( \left( x_{N}^{0}\right)^{m+\gamma}{\displaystyle\sum\limits_{i=1}^{N-1}}\left\langle D_{x_{i}}^{m}u\right\rangle_{x_{i},\overline{Q}_{(3/4)x_{N}^{0}}}^{(\gamma)}+\right.
\begin{aligned}x^2 = (x_0+2x_1+4x_2+ 8x_3+ 16x_4 + \ldots)^2{}= x_0^2+4(x_0x_1+x_1^2)+8(x_0x_2)+16(x_2^2+x_1x_2+x_0x_3){}+2^5(x_0x_4+x_1x_3)+2^6(x_3^2+x_0x_5+x_1x_4+x_2x_3)+ \ldots.\end{aligned}
p_{i,s;j,t}:= \pi_{u^{i,s}}\pi_{\theta_i^{s+1}}+ \pi_{u^{j,t}}\pi_{\theta_j^{t+1}}- \pi_{u^{i,s}}\pi_{\theta_i^{s+1}}\pi_{u^{j,t}}\pi_{\theta_j^{t+1}}.
\begin{aligned}\sum_{(n,m)\in\mathbb{Z}^{2}}\frac{1}{(a+n\omega_{1}+m\omega_{2})^{k}}=\sum_{y_{0}+\epsilon<m<\infty}\sum_{-\infty <n<\infty}\frac1{(a+nw_1+mw_2)^k}&\quad+\sum_{-\infty <m\leq y_{0}-\epsilon}\sum_{-\infty <n<\infty}\frac1{(a+nw_1+mw_2)^k},\end{aligned}
{\bf ( p_{L}^{D})}_{i}={\cal R}_{i}^{j}{\bf ( p_{L})}_{j}\mathrm{w i t h}{\cal R}_{i}^{j}= -{\bf e^{a}}_{i}( e^{t}. e^{- 1})_{a}^{b}{\bf e_{b}}^{j}.
a = a_{k(n+1)+l}= a_{kn+l}\prod_{i=1}^k f(kn+l+i) + \sum_{j=1}^k h_1(kn+l+j)h_2(kn+l+j)^{kn+l+j}\prod_{i=j+1}^k f(kn+l+i) = a\prod_{i=1}^k f(kn+l+i) + h_2(kn+l)^{kn+l}\sum_{j=1}^k h_1(kn+l+j)\times & \times h_2(kn+l+j)^j\left(\frac{h_2(kn+l+j)}{h_2(kn+l)}\right)^{kn+l}\prod_{i=j+1}^k f(kn+l+i).
\frac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}=\sum_{i=m}^{l}(\begin{matrix}l+m\\i\end{matrix})\frac{d^{l+m-i}}{dx^{l+m-i}}(x+1)^{l}\frac{d^{i}}{dx^{i}}(x-1)^{l}
c_{i,n-l}= \frac{(\alpha+2l+1)_{i-k-l}}{(i-k-l)!}\binom{n}{l}^{-1}, &c_{i,n-l-1}= c_{i,n-l}\frac{(n-k-l)(l+1)}{n-l}\left[1-\frac{(k+l-i)(i+k+l+\sigma)}{(k+l-n)(\alpha+2l+1)}\right], &c_{ih}= F_{i}(h)c_{i,h+1}+ G(h)c_{i,h+2}(h = n-l-2,n-l-3,\ldots,k),
f_D(d)=\frac{1}{\pi R_0^2}\Big[ h_2'(d) \frac{\partial \mathbb{A}(R_0, h_2(d))}{\partial h}+ \frac{\partial \mathbb{A}(d, h_1(d))}{\partial R}+ h_1'(d) \frac{\partial \mathbb{A}(d, h_1(d))}{\partial h}\Big]
\begin{aligned}\left\|\mathbf{R}_{xs}-\mathbf{W_{xs}}\right\| \le 2\gamma\dot{\nu}, &\left\|\mathbf{W}_{xs}-\dot{\nu}\mathbf{I}\right\| \le \gamma\dot{\nu}, &\left\|\mathbf{R}_{xs}-\dot{\nu}\mathbf{I}\right\| \le 3\gamma\dot{\nu},\end{aligned}