harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma \lbrack A \rbrack = \mathrm{t r}\int_{0}^{\infty}{\frac{d T}{T}}\mathrm{e}^{- m^{2}T}\int{\cal D}x{\cal P}\mathrm{e x p}\left[ - \int_{0}^{T}\! \! \! d \tau \left({\frac{1}{4}}{\dot{x}}^{2}+ i g A_{\mu}\dot{x}^{\mu}\right) \right]
\varphi ({\overline{\alpha V}},{\overline{\beta V}})= ({\overline{\alpha V}}-W ({\overline{\alpha V}}))\cdot ({\overline{\beta V}}-W({\overline{\beta V}}))+ f({\overline{\alpha V}},{\overline{\beta V}}) + \chi ({\overline{\alpha V}},{\overline{\beta V}}) ~,
\Gamma_{N P}^{( 1 )}= \frac{g^{2}}{4}\frac{1}{1 6 \pi^{2}}\int_{0}^{1}d x \int \frac{d \alpha}{\alpha}e^{i \alpha [ x ( 1 - x ) p^{2}- m^{2}] + i \frac{p \circ p}{4 \alpha}- \alpha \epsilon}\Phi ( - p , \theta ) \bar{\Phi}( p , \theta )
\times \prod_{b = 1}^{N}\prod_{j = 1}^{m_{b}}\exp \left( - 2 e \sum_{I = 1}^{N}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}U_{I b}\delta ( y_{j}^{( b )}) \Big ) \right)
\hat{H}=-\frac{1}{2}\sum_{j=1}^{N}(J_{x}\sigma_{j}^{x}\sigma_{j+1}^{x}+J_{y}\sigma_{j}^{y}\sigma_{j+1}^{y}+J_{z}\sigma_{j}^{z}\sigma_{j+1}^{z}+h\sigma_{j}^{z})
\begin{array}{ll}\displaystyle E|\sum_{\substack{j\in J_{K}}}X(u_{j}) (t_{j+1}-t_{j})|^2 =\displaystyle \| \sum_{\substack{j\in J_{K}}}X(u_{j}) (t_{j+1}-t_{j}) \|^2 \leq \displaystyle [\sum_{\substack{j\in J_{K}}}\| X(u_{j})\| (t_{j+1}-t_{j}) ]^2 \end{array}
d\big(\tilde J_\rho\rho\big)= \, \, q_1e^{1234}+q_2e^{1235}+q_3e^{1245}+q_4(e^{1246}-e^{1356})+q_5e^{1256}+q_6e^{1345}+q_7(e^{1346}+e^{2356}) begin{align*}2pt] + q_8e^{2345}+q_9e^{2346},
\sum^{\infty}_{n=-\infty}(-1)^nq^{2n^2+3n/2}=q^{-11/96}\eta(q^4)\sqrt[48]{\frac{4\left(1-k_r\right)^4\left(2+k_r-2\sqrt{1+k_r}\right)^{12}}{k_r^{13}\left(1+k_r\right)^2}}
\hat{S}_{\lambda \lambda^{\prime}}= \left[ \nabla_{\lambda}, \nabla_{\lambda^{\prime}}\right] \hat{1}+ \frac{1}{2}\left( \nabla_{\lambda}\hat{E}_{\lambda^{\prime}}- \nabla_{\lambda^{\prime}}\hat{E}_{\lambda}\right) + \frac{1}{4}\left( \hat{E}_{\lambda}\hat{E}_{\lambda^{\prime}}- \hat{E}_{\lambda^{\prime}}\hat{E}_{\lambda}\right)
af'+\frac{1}{2}bf'' =\lambda g \frac{\Pi}{\pi}\frac{s'}{s}-\lambda\bigg(\frac{\Pi}{\pi}g'+g+\frac{\Pi}{\pi}g \frac{s'}{s}\bigg) &=-\lambda\bigg(\frac{\Pi}{\pi}g'+g\bigg)=-\lambda f.
[\ker(\Theta^1_S)]=\frac{[\mathrm{cok}(\Theta^1_S)][\mathbb{III}^1(T)][\mathbb{III}^2(T)]\prod_{v\in S}[H^1(K_v,T)]}{[H^1(K,\hat{T})]}\prod_{v\notin S}\frac{[H^0(\hat{\mathbb{Z}},H^1(I_v,\hat{T}))^{D}]}{[S_v]}.
{\frac{d^{2}x^{i}}{d \lambda^{2}}}+ 2{\frac{\dot{a}}{a}}{\frac{d t}{d \lambda}}{\frac{d x^{i}}{d \lambda}}+ 2{\frac{a^{\prime}}{a}}{\frac{d y}{d \lambda}}{\frac{d x^{i}}{d \lambda}}= 0 ,
\pi_{n + 1}( x ) - ( x - \check{S}_{n^{n}( n + 1 )}^{*}\check{S}_{n^{n + 1}}^{- 1}+ \check{S}_{( n - 1 )^{n - 1}n}^{*}\, \check{S}_{( n - 1 )^{n}}^{- 1}) \pi_{n}( x ) + \check{S}_{n^{n + 1}}\, \check{S}_{( n - 1 )^{n}}^{- 1}\, \pi_{n - 1}( x ) = 0
\Theta(a) =\Psi(v(1_B\otimes a\otimes 1_V)v^*)=\Pi(v(1_B\otimes a\otimes 1_V)v^*)\Psi(1_{(B\otimes A\otimes V)_\omega}) &=\Pi^{1/3}(v)\Psi(1)^{1/2}\Pi^{1/3}(1_B\otimes a\otimes 1_V)\Psi(1)^{1/2}\Pi^{1/3}(v^*),a\in A.
\widetilde{H}_{b}^{p}(M)\cong\left\{\varphi\in\mathcal{H}_{b}^{p}(M)\,|\,\varphi\wedge d\eta\in d_b\left(F_b^{p+1}(M)\right)\right\}\cong\left\{\varphi\in\mathcal{H}^{p}_{b}(M)\,|\,\varphi\wedge d\eta=0\right\}.
Bis(v,w)=\displaystyle\frac{\displaystyle \sum_{1\leq i,j,k,l \leq 2}R_{i \bar{j}k \bar{l}}v_i \bar{v_j}w_k \bar{w_l}}{\left(\sum_{1\leq i,j\leq 2}g_{i\bar{j}}v_i\overline{v_j}\right)\left(\sum_{1\leq i,j\leq 2}g_{i\bar{j}}w_i\overline{w_j}\right)},
B_n\Big(\frac{\alpha^j}{L_j}\Big) = \sum_{k = 0}^n{\binom nk\frac{{B_k\alpha^{j(n - k)}}}{{L_j^{n - k}}}}&= \alpha \sum_{k = 0}^n \binom nk\frac{B_k F_{j(n - k)}}{L_j^{n - k}}+ \sum_{k = 0}^n \binom nk\frac{B_k F_{j(n-k) - 1}}{L_j^{n - k}},
S_{b o u n d a r y}= - \frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{( 1 )}}{\sqrt{- g}V^{- 1}\alpha^{( 1 )}}- \frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{( 2 )}}{\sqrt{- g}V^{- 1}\alpha^{( 2 )}}
\begin{gathered}\begin{bmatrix}-h 1\end{bmatrix}\begin{bmatrix}a b c d \end{bmatrix}V(x=0) = 0, \begin{bmatrix}H \, \, 1 \end{bmatrix}\begin{bmatrix}a b c d \end{bmatrix}V(x=1) = 0,\end{gathered}
m\lfloor\frac{m-1}{k-1}\rfloor-\frac{k-1}{2}\lfloor\frac{m-1}{k-1}\rfloor(\lfloor\frac{m-1}{k-1}\rfloor+1)<m\frac{m-1}{k-1}-\frac{k-1}{2}(\frac{m-1}{k-1}-1)((\frac{m-1}{k-1}-1)+1)
T_{5}( k , k^{\prime}; a , b ) = \frac{T_{0}^{5}}{g}\sqrt{\left[ ( k - \chi_{B 0}\, k^{\prime})^{2}g^{2}+ k^{2}\right] \left[ ( a - \chi_{0}\, b )^{2}+ b^{2}g^{- 2}\right]}
I= \langle ((x_2)_1^m-m!(x_2)_m)j, \; (x_4)_1^m-m!(x_4)_m, \; (x_2)_a(\gamma_1^2+\gamma_2^2+\gamma_1\gamma_2),\;(x_2)_a\gamma_1^3,\; (x_2)_a(y_2(\gamma_1+2\gamma_2)+y_1(2\gamma_1+\gamma_2)) \rangle
\begin{array}{ll}K_{N}(x,y) = \int \int \int \psi \left( \frac{\xi}{640 N \mu_{2}^{-1}}\right) \chi_{\mu_4}(z)\phi \left( \frac{\eta}{10 \mu_{2}^{-1}}\right) e^{i(\xi \cdot x - \eta \cdot y - (\xi- \eta) \cdot z )}\, d \xi \, d \eta \, dz \cdot\end{array}
min_{\beta\in\mathbb{R}^{p}}\{\frac{(y-X\beta)^{\prime}(y-X\beta)}{(y-X\beta_{0})^{\prime}(y-X\beta_{0})}+2\lambda\sum_{i=1}^{p}\frac{|\beta_{i}-\beta_{0,i}|}{q_{i}}\}
P\left(\sup_{x\in \mathbb{R},t>0}\frac{L_B(x,t)}{\sqrt{t\log t}\vee 1}> R \right)\leq \sum_{k=0}^{\infty}P\left(\sup_{x\in \mathbb{R}}\frac{L_B(x,k +1)}{\sqrt{\lfloor k \log k\rfloor}\vee 1}> R \right).
P_{\chi_f}\varphi := \frac{\langle \chi_f, \varphi\rangle_{H_1}}{\|\chi_f\|_{H_1}^2}\, \chi_f, \varphi\in H_1, P_{\chi_g}\psi := \frac{\langle \chi_g, \psi \rangle_{H_2}}{\|\chi_g\|_{H_2}^2}\, \chi_g, \psi\in H_2.
- i \Sigma_{ab}(q,p) = - (i e)^2 \epsilon_a \epsilon_b \int{d^4 k \over (2 \pi)^4}{1 \over i}\left[D^{(0)}\right]^{\rho\sigma}_{ba}(k) \gamma_\sigma i S^{(0)}_{ba}({p}{\mkern -9.0mu /}- k) \gamma_\rho \, .
\| h_\perp(t) \|_{H_x^1}^2 \simeq B(h_\perp, h_\perp) = \Phi(h) - 2 \alpha_+ \alpha_- \lesssim e^{-(c_0 + c_1) t}+ e^{-2c_1^- t}\lesssim e^{-(c_0 + c_1) t}
\delta_{j_{1}, 0}\delta_{j_{2}, 0}= \int_{0}^{\frac{2 \pi}{l}}d k \int_{0}^{l}d q e^{- i j_{1}l k + i j_{2}l \tau_{2}q}| \tilde{c}( k , q ) |^{2}| C_{0}( k , q ) |^{2}.
y=\frac{3(1+x^{2})^{1.5}}{2+3x^{2}+(1+x^{2})^{1.5}}=3-9x^{-1}\cdot\frac{2x^{-2}+3}{6x^{-3}+9x^{-1}+3(1+x^{-2})^{1.5}}\approx3-9x^{-1}
{\cal H}= \frac{1}{2 \mathcal{C_{R}}}\sum_{\mu \in \mathcal{R}}\ell_{\mu}^{+}\ell_{\mu}^{-}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}\left( \omega + \frac{1}{2}{\frac{| \rho |^{2}}{( \rho \cdot q )^{2}}}\right) ( \check{s}_{\rho}- 1 ) ,
m{\frac{d^{2}x^{\beta}}{d \tau^{2}}}( \delta^{\alpha}{}_{\beta}+ \mu^{2}\mathrm{D}_{\beta \mu}g^{\mu \alpha}) +{\frac{d x^{\nu}}{d \tau}}{\frac{d x^{\lambda}}{d \tau}}\left( \Gamma_{\nu \lambda}{}^{\alpha}+ \frac{1}{2}\mu^{2}g^{\alpha \gamma}( \partial_{\gamma}\mathrm{D}_{\nu \lambda}- \partial_{\nu}\mathrm{D}_{\gamma \lambda}- \partial_{\lambda}\mathrm{D}_{\gamma \nu}) \right) = 0 ,
\frac{\partial^\gamma u}{\partial t^\gamma}+ u \frac{\partial u}{\partial x}-\nu \frac{\partial^2 u}{\partial x^2}=0,\,\,\,\,\,\,a\leq x \leq b,\,\,\,\,\,0 < \gamma < 1,\,\,\,\,t>0.
\omega_{\sigma}^{2}( q ) = q^{2}\frac{1 + \frac{2}{\pi}V_{( 0 )}^{1 1}+ \frac{1}{2 \pi^{2}}\{( U_{( 0 )}^{1 1})^{2}- 2 U_{( 0 )}^{1 1}U_{( 1 )}^{1 1}\}}{1 + \frac{2}{\pi}V_{( 1 )}^{1 1}- \frac{1}{2 \pi^{2}}( U_{( 1 )}^{1 1})^{2}},
\left[ \Psi_{M}^{-}( x ) , \overline{{\Psi}}_{N}^{+}( y ) \right] = - ( 2 \pi )^{- 3}\int \frac{\left( m + i \widehat{p}\right)_{M N}}{2 p_{0}}e^{- i p ( x - y )}d^{3}p
Z_4(q,\zeta) = \; \zeta^2 (q^d-1)^2(q^{i-1}-1)(q^{d-i}-1) \; - \zeta (q-1) (q^d-1) \big( (q^{d-i}-1)(q^{d+i-2}-1) - q^{d-i}(q^{i-1}-1)^2 \big) \; - (q^{d-1}-1)(q^{i-1}-1) \big( (q^{d-1}+1)(q^{d-i+2}+1) - 2 q^{d-i+1}(q^{i-1}+1) \big).
(1-\eta^{2})\frac{d^{2}S_{mn}(-ic,\eta)}{d\eta^{2}}-2\eta\frac{dS_{mn}(-ic,\eta)}{d\eta}+(\lambda_{mn}(c)+c^{2}\eta^{2}-\frac{m^{2}}{1-\eta^{2}})S_{mn}(-ic,\eta)=0
\sup_{\psi\in}\inf_{x\in X}f\left(x\right)+\psi\left(x\right)-g^{*}_\Psi\left(\psi\right) =\sup_{\psi\in}\inf_{x\in X}x^{2}-x-10+b-b+1 =\sup_{\psi\in}\inf_{x\in X}x^{2}-x-9 =\inf_{x\in X}x^{2}-x-9=\inf_{x\in\mathbb{R}}f\left(x\right)+g\left(x\right),
d\mathcal L[\varphi_0](\varphi^\prime)=\mathbb L^\prime[\varphi^a+\varphi_0]\varphi^\prime\,, d^2\mathcal L[\varphi_0](\varphi^\prime, \psi^\prime)=\mathbb L^{\prime\prime}[\varphi^a+\varphi_0](\varphi^\prime, \psi^\prime)\,.
|D|= \left|(1-i)(\delta_1)\frac{K_{\psi}(-3,n,4N)}{4N}\left(I_{\frac 12}\left(\frac{\pi\sqrt{3n}}{N}\right)-\sqrt{\frac{2\sqrt{3n}}{N}}\right)\right| \leq \sqrt{2}\frac{\delta_1}{2}\frac{\sqrt{N}}{\pi\sqrt{2}(3n)^\frac 14}e^{\frac{\pi\sqrt{3n}}{N}}\leq \frac{2\sqrt{N}}{\pi(3n)^\frac 14}e^{\frac{\pi\sqrt{3n}}{N}}.
\left\lfloor \frac{n}{b}\right\rfloor+ \left\lceil \epsilon' \left\lfloor \frac{n}{b}\right\rfloor \right\rceil \le \frac{n}{b}+\epsilon' \frac{n}{b}+1 &= (b^{-1}+\epsilon)n + \left( 1 - \frac{n\epsilon}{2}\right) \le (b^{-1}+ \epsilon)n.
\begin{aligned}u(t) = \zeta E_{\alpha}\big[-\gamma t^{\alpha}\big]u(0) &+ \frac{\alpha c_{0}\zeta}{B(\alpha)}\int_{0}^{t}\bigg( E_{\alpha,\alpha}\big[-\gamma(t-s)^{\alpha}\big] + \frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big[-\gamma^{-2\alpha}(t-s)^{\alpha}\big]\bigg)(t-s)^{\alpha-1}h(s)ds,\end{aligned}
V_{m}\bar{\Phi}_{A}^{(n)}=\left( -\right)^{\varepsilon_{A}}\varepsilon_{mnp}\Phi_{A}^{*(p)}\,, V_{m}\tilde{\Phi}_{A}=\left( -\right)^{\varepsilon_{A}+1}\bar{\Phi}_{A}^{(m)}\,,
\nu(x_{2m,2}) =\nu{\left(\sum_{k=1}^\infty x_{2m-1,k}\alpha_{k,2}\right)}=\min_{k\geq4}{\left\{\big(\nu(x_{2m-1,2})+\nu(\alpha_{2,2})\big),\big(\nu(x_{2m-1,3})+\nu(\alpha_{3,2})\big)\right.}, \qquad\qquad\left.\big(\nu(x_{2m-1,k})+\nu(\alpha_{k,2})\big)\right\}\geq2m.
a^{\rm M}_{1,2,3,5,8,11,14,15}= 0 \, ,a^{\rm M}_{4}= \mu \, L_c^2 \frac{2a_3-a_1}{3}\, ,a^{\rm M}_{10}= \mu \, L_c^2 \frac{a_1 + a_2}{2}\, ,a^{\rm M}_{13}= \mu \, L_c^2 \frac{a_1 - a_2}{2}\, .
\begin{aligned}\zeta*\mathcal{T}_{\alpha\beta}(t)=\ \mathcal{\zeta*\hat{T}}_{\alpha\beta}(t)-\xi*\theta^{(\alpha)}*\rho\hat{\eta}^{(\beta)}(t) -\ \xi*\mathcal{L}_{ij}^{(\alpha)}*e_{ij}^{(\beta)}(t)-\xi*\mathcal{M}_{ij}^{(\alpha)}*V_{ij}^{(\beta)}(t).\end{aligned}
\left[\hat{x}^\mu,\hat{\pi}_\nu\right]_{-}=i\left\{x^\mu,\pi_\nu\right\}_{D(\Phi^{(1)}_3)}=i\delta^\mu_\nu\;,\;\;\left[\hat{\psi}^\mu,\hat{\psi}^\nu\right]_{+}=i\left\{\psi^\mu,\psi^\nu\right\}_{D(\Phi^{(1)}_3)}=-\frac{1}{2}g^{\mu \nu}\;.
\mathcal{W}_{p}(\mu_{n_{k}},\nu_{n_{k}})=W_{p}((\bar{\mu}_{n_{k}})_{u_{n_{k}}},(\bar{\nu}_{n_{k}})_{u_{n_{k}}}) \mathcal{W}_{p}(\mu,\nu)=W_{p}(\bar{\mu}_{u},\bar{\nu}_{u}),
\tilde{g}_{k}( \eta ) = \frac{i}{2 \Omega_{k 0}}\int_{\eta_{0}}^{\eta}d \eta^{\prime}\left( e^{2 i \Omega_{k 0}( \eta - \eta^{\prime})}- 1 \right) \Delta \Omega^{2}( \eta^{\prime}) \left[ 1 + \tilde{g}_{k}( \eta^{\prime}) \right] .
0\leq v(X_{T_n}^Z) = v(x) + \int_0^{T_n}(\mathcal{A}v)(X_s^Z)ds + \int_0^{T_n}\sigma(X_s^Z)v'(X_s^Z)dB_s -\int_0^{T_n}v'(X_s^Z)dZ_s^c + \sum_{s\leq T_n}(v(X_s^Z)-v(X_{s-}^Z)),
\hat{g}_{\ell}(z,\zeta) \:=\: e_\ell(\zeta) + e_{\ell-1}(\zeta) e_2(z),\ell = 1, 2, 3, \hat{g}_4(z,\zeta) \:=\: e_4(\zeta) + e_3(\zeta) e_2 (z) - e_5(\zeta) - q \sum_{s=0}^5 (-1)^s e_s(\zeta), \hat{g}_5(z,\zeta) \:=\: e_5(\zeta) + e_2(z)e_4(\zeta) - e_1(z) e_5(\zeta) - q \sum_{s=0}^5 (-1)^s e_s(\zeta).
\nabla_Lf(x) :=L\,\Big(f(x+L^{-1})-f(x)\Big) \qquad\qquad x\in e^n, \Delta_Lf(x) := L^2\,\Big(f(x+L^{-1})+f(x-L^{-1})-2f(x)\Big) x\in e^n\setminus\{x^e_1\}.
\hat{f}(u,v)= \frac{1}{2}[f(u)+f(v)-a_1(v-u)],a_1=\max_u|f'(u)|, \hat{g}(u,v)= \frac{1}{2}[g(u)+g(v)-a_2(v-u)],a_2=\max_u|g'(u)|.
\widehat{w}(\xi)= \widehat{v}(\xi)\left((2\pi i \xi)^{-s}+(-2\pi i\xi)^s \right) = \dfrac{1}{\xi}\widehat{\psi}(\dfrac{\xi}{\epsilon})\left((2\pi i \xi)^{-s}+(-2\pi i\xi)^s \right).
X^m_{\nu(k)}=\bigl(\bigcup_{I\subset [m]\,,\, |I|=k}\Delta_I\bigr)\setminus \bigl(\bigcup_{I\subset [m]\,,\, |I|=k+1}\Delta_I\bigr)\quad,\bar X^m_{\nu(k)}=X^m\setminus \bigl(\bigcup_{I\subset [m]\,,\, |I|=k}\Delta_I\bigr)\,.
g_{f}(n+1)=\sum_{b=\pm1}\sum_{k=1}^{\lfloor\frac{\sqrt{24n+1}-b}{6}\rfloor}(-1)^{k+1}g_{f}(n+1-\frac{k(3k+b)}{2})+\sum_{k=1}^{n+1}s_{n+1,k}f(k),
(1-\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{I^-}+r_2\delta_{\alpha_2 y}^{I^+}\}) I_{i,j}^{n+1}+(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{I^-}+r_2\delta_{\alpha_2 y}^{I^+}\}) I_{i,j}^{n}=2 I_{i,j}^*
\sup\limits_{\lambda\in \left[0, 1\right]}\left|\frac{\lfloor n\lambda\rfloor}{n}-\lambda\right|=\sup\limits_{\lambda\in \left[0, 1\right]}\left(\lambda-\frac{\lfloor n\lambda\rfloor}{n}\right)\leq \sup\limits_{\lambda\in \left[0, 1\right]}\left(\lambda-\frac{n\lambda-1}{n}\right)=\frac{1}{n}\longrightarrow 0,
{\cal Z}_{\rm mon.}=\int D\rho D\phi\exp\left\{-\frac{\pi}{2e^2}\int d^3xd^3y\rho\left(\vec x{\,}\right)\frac{1}{\left|\vec x-\vec y{\,}\right|}\rho\left(\vec y{\,}\right)+\int d^3x\left(2\zeta\cos\phi-i\phi\rho\right)\right\}.
\begin{array}{l}{\displaystyle{\frac{A_{n}}{B_{n}}}\sim r_{0}^{2 \left( k + \alpha - n \right)}\qquad n - k + \frac{1}{2}- \alpha \leq 0}\\{\displaystyle{\frac{B_{n}}{A_{n}}}\sim r_{0}^{2 \left( n + 1 - k - \alpha \right)}\qquad n - k + \frac{1}{2}- \alpha > 0}\\ \end{array}
A_{i}^{\prime}\, \left( A + \delta_{\lambda}A \right) - A_{i}^{\prime}\, \left( A \right) - \partial_{i}\lambda^{\prime}- i \left[ \lambda^{\prime}, A_{i}\right] - i \left[ \lambda , A_{i}^{\prime}\right] = - \frac{1}{2}\theta^{k l}\left( \partial_{k}\lambda \partial_{l}A_{i}+ \partial_{l}A_{i}\partial_{k}\lambda \right) + o \left( \theta^{2}\right) ,
+ ( 1 - \mu^{2}) \stackrel{1}{J_{1}}( x ){]}\; \delta ( x - y ) = \,{[}\; \frac{P}{2 ( \lambda - \mu )}\; , \; \stackrel{1}{U^{\prime}}( x , \lambda ) \; - \stackrel{1}{U^{\prime}}( x , \mu ) \;{]}\; \delta ( x - y ) =
\left( w^{-\frac{p^{\prime}}{p}}\left( B\right) \right)^{\frac{1}{p^{\prime}}}=\left \Vert w^{-\frac{1}{p}}\right \Vert_{L_{p^{\prime}}\left( B\right)}\leq C\left \vert B\right \vert w\left( B\right)^{-\frac{1}{p}}
f(X,Y,Z) = \sum_k \int_{\eta \in \mathbb{R}}\sum_\ell \widehat{f}(k,\eta,\ell) e^{2\pi i(kX + \eta Y + \ell Z)}\,d\eta & \widehat{f}(k,\eta,\ell) = \int_{X \in \mathbb{T}}\int_{Y \in \mathbb{R}}\int_{Z \in \mathbb{T}}f(X,Y,Z) e^{-i 2\pi (kX + \eta Y + \ell Z)}\,dX\,dY\,dZ.
\int_{\mathcal{Y}^d}\Big(\big |\nabla_y \widetilde{m}^{\frac{1}{2}}\big|^2+ \big(\widetilde{m}^{\frac{1}{2}}\big)^2 \Big)dy= \int_{\mathcal{Y}^d}\Big(\frac{1}{4}\widetilde{m}|\nabla_yh |^2+\widetilde{m}\Big)dy\leq C.
\underline{m}^*\circ\mathsf{cof}^{\underline{1}}= \infty^*\circ (0_{\underline{m}^{\vee}})^*\circ\mathsf{cof}^{\underline{1}}&\cong \infty^* \circ\mathsf{cof}^{\underline{1}}\circ (1_{\underline{m}^{\vee}})^* &= C^{|\underline{m}|}\circ (1_{\underline{m}^{\vee}})^*.
f_{+}(z,t,s) :=\exp \left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\{S_n >0\}}e^{itS_n}e^{i{s}\cdot{C}_n}d \mathbb{P}\right), f_{-}(z,t, s) :=\exp \left(+\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\{S_n \le 0 \}}e^{itS_n}e^{i{s}\cdot{C}_n}d \mathbb{P}\right).
{\psi}_{o d d}^{-}= e^{- \frac{1}{2}z_{-}^{2}}\left( \begin{array}{c}{{z_{-}F ( 1 - \frac{\nu}{2}; \frac{3}{2}; z_{-}^{2}) + \bar{G}\sqrt{\frac{\Delta}{2}}e^{i \frac{\pi}{4}}F ( \frac{1 - \nu}{2}; \frac{1}{2}; z_{-}^{2})}}\\{{- \sqrt{\frac{\Delta}{2}}e^{i \frac{\pi}{4}}F ( - \frac{\nu}{2}; \frac{1}{2}; z_{-}^{2}) + \bar{G}z_{-}F ( \frac{1 - \nu}{2}; \frac{3}{2}; z_{-}^{2})}}\\ \end{array}\right)
H^{2a}(B^\vee X Z) = H^{a}( H^{a}(B^\vee X) Z) = H^{a}( B^\vee H^{a}(XZ)), H^{-2a}(B^\vee X Z) = H^{-a}( H^{-a}(B^\vee X) Z) = H^{-a}( B^\vee H^{-a}(XZ))
L_{1 1}^{( \pm )}L_{1 2}^{( + )}= q^{\mp 1}L_{1 2}^{( + )}L_{1 1}^{( \pm )}, L_{2 2}^{( \pm )}L_{1 2}^{( + )}= - q^{\mp 1}L_{1 2}^{( + )}L_{2 2}^{( \pm )}
\varphi_1 = \tfrac{1}{\gamma(P_1)}\left[ \tau_1+\tau_2 -(\gamma(P_1+P_2)-\gamma(P_1))\tfrac{\tau_2}{\gamma(P_2)}\right], \varphi_2 = \tfrac{1}{\gamma(P_2)}\left[ \tau_1+\tau_2 -(\gamma(P_1+P_2)-\gamma(P_2))\tfrac{\tau_1}{\gamma(P_1)}\right].
{\bf D}_\mu (\sigma )F={\frac{\delta F}{\delta z^\mu (\sigma )}}=D_\mu\,f_{\rho_1 \rho_2}{\cal D}z^{\prime \rho_1}{\cal D}z^{\prime \rho_2}+2{\cal D}^2(f_{\mu \rho_2}{\cal D}z^{\prime \rho_2})-2f_{\rho_1 \rho_2}z^{\prime \kappa}z^{\prime \lambda}R_{\mu \kappa \lambda}^{{\,\,\,\,\,\,\,\,}\rho_1}{\cal D}z^{\prime \rho_2}
\Phi_k(z,s,a)=\frac1{a^s}+\sum_{r=0}^{k-1}\binom{k}r\sum_{m_{r+1},\ldots,m_k=1}^\infty\frac{z^{m_{r+1}+\dots+m_k}}{(m_{r+1}+\dots+m_k+a)^{s}}.
\aligned\Phi(u_j,-p_0,B_{3r,{r}/{4}}) \le \Phi(u_j,1+\tilde \beta_0,B_{\tilde \rho_0+\tilde \delta_0, \tilde \sigma_0-\tilde \delta_0}) &\leq\prod_{k=0}^m (C'/\tilde \delta_k^2)^{1/|1+\tilde \beta_k|}\Phi(u_j,-\bar p,B_{2r, r}) & \le (C')^{\frac 2{p_0}}2^{\frac c{p_0}}r^{-4(\frac{1}{p_0}-\frac{1}{\bar p})}\Phi(u_j,-\bar p,B_{2r, r}).\endaligned
\alpha = (\underbrace{1,\ldots,1}_n,\underbrace{r_n}_{\neq 1},r_{n+1},r_{n+2}\ldots) \mapsto \textstyle{\int}\alpha = (\underbrace{0,\ldots,0}_{n},r_n-1,r_{n+1},r_{n+2},\ldots):\Gamma_{\log}\to \Gamma_{\log}^{\neq}
- c ( l_{0}, l_{1}; l_{0}, l_{1}+ 1 ) d ( l_{0}, l_{1}+ 1 ; l_{0}, l_{1}) - d ( l_{0}, l_{1}; l_{0}, l_{1}+ 1 ) c ( l_{0}, l_{1}+ 1 ; l_{0}, l_{1}) ] = 0 ,
E\left[T^{(2)}\right] = (\nu_1 + \lambda \nu_2)^2 M^2 \left(H_{\nu_1 M}^2 + \sum_{j=1}^{\nu_1 M}\frac{1}{j^2}\right)+ O\left(M^{3 - \lambda + \varepsilon}\right),M \to \infty.
[\begin{matrix}t_{i}\\A_{i}\end{matrix}]=[\begin{matrix}0,&(2\pi/3)/\omega_{d},&(4\pi/3)/\omega_{d},&2\pi/\omega_{d}\\I/(1+m),&I/K^{2/3},&I/K^{4/3},&mI/[(1+m)K^{2}]\end{matrix}]
\lim_{n \to +\infty}J_{n, j}= \lim_{n \to +\infty}\int_{A_{j,K_{n}, \sigma_{n}}}(( v_{j}-K_{n})_{+})^{\gamma^*}dx =\int_{A_{j,\frac{K}{2}, \frac{R_1}{2}}}(( v_{j}-\frac{K}{2})_{+})^{\gamma^*}dx.
\left| \frac1{2l_n}\int\limits^{l_n}_{-l_n}\widetilde{{\bf D}}(\tau,0)\,{\rm d}\tau-2\right|\le \frac{2\cdot 2^{(2n-1)^2}}{2\cdot2^{(2n)^2}}=2^{-4n+1}\to 0 n\to\infty,
u= 3( \lambda_1 - \lambda_2 - \lambda_3) -l, v= \sqrt{-3( \lambda_1 +3 \lambda_2 - \lambda_3 +l)( \lambda_1 - \lambda_2 +3 \lambda_3 +l)}, l= \sqrt{-3(\lambda_1 +2\lambda_2)(\lambda_1 +2 \lambda_3)}.
\left. + \left( 1 - \frac{r_{0}^{4}}{r^{4}}\right)^{a_{\theta}}d \theta^{2}\right] + \frac{l^{2}}{r^{2}}\left( 1 - \frac{r_{0}^{4}}{r^{4}}\right)^{- 1}d r^{2}
\mathbf{R}_{\mathbf{\tilde{\mathbf{G}}\boldsymbol{\psi}}}=\left(m_0m_1\rho_p\right)^{-2}\mathbb{E}\left\{\mathbf{Z}\boldsymbol{\Psi}_p^{\dagger}\boldsymbol{\psi}\boldsymbol{\psi}^{\dagger}\boldsymbol{\Psi}_p\mathbf{Z}^{\dagger}\right\}.
h(x,u)=E_{\xi_1^\theta(u)}\left[\varphi\Bigl(x+u+\sum_{i=2}^{n}Z_i^{\theta[u]}\Bigr)\right]=E_{\xi_1^\theta(u)}\left[\varphi\Bigl(x+u+\sum_{i=1}^{n-1}Z_i^{\rho}\Bigr)\right].
\begin{aligned}\mbox{there are positive constants}\,\,\, M_0\,\, \mbox{and}\,\, \sigma\,\, \mbox{such that}& \,\,\,\,\,\,\,\displaystyle\max_{(x,s)\in \mathop D\limits^\circ\times[-M_0,M_0]}f_+(x,s)\leq \mu_0^2\frac{M_0}{(\sigma+1)}\frac{\lambda_1}{2}.\end{aligned}
\sqrt{| \hat{\hat{g}}|}{\textstyle \frac{- 1}{2 \cdot 4 !}}\hat{\hat{G}}{}^{2}= \sqrt{| g_{E}|}\left\{{\textstyle \frac{- 1}{2 \cdot 4 !}}K^{6 / 7}F_{( 4 )}^{2}+{\textstyle \frac{1}{2 \cdot 3 !}}K^{- 3 / 7}\vec{F}_{( 3 )}^{\T}{\cal M}^{- 1}\vec{F}_{( 3 )}-{\textstyle \frac{1}{4}}K^{- 1 2 / 7}F_{( 2 )}^{2}\right\}\, .
S_{3}= - \frac{m^{2}}{2}\int d^{4}\theta d^{d}x e^{\sigma + \bar{\sigma}}\int \frac{d^{d}k}{{( 2 \pi )}^{d}}\frac{- A^{2}k^{2}+ m^{2}}{k^{2}{( - A^{2}k^{2}+ m^{2})}^{2}- 8 1 \Lambda \bar{\Lambda}}\frac{\bar{D}_{1}^{2}D_{1}^{2}}{1 6}\delta_{1 1}
g_-=\left(\begin{array}{c c}1 0 \frac{x_-}{\sqrt{2}}1 \end{array}\right) \;\;\;\;\;\;g_0 =\left(\begin{array}{c c}e^{\frac{x}{2}}0 0 e^{-\frac{x}{2}}\end{array}\right) \;\;\;\;\;\;g_+ = \left( \begin{array}{c c}1 \frac{x_+}{\sqrt{2}}0 1\end{array}\right).
\Vert f\Vert_{\mu}^2=\int_{\mathbb R^{n-1}}|\mathcal F f(\xi)|^2 |\xi|^{n-1-2\mu}d\xi=\Vert \mathcal F f(\cdot) |\cdot |^{\frac{\widetilde\mu}{2}}\Vert^2_{L^2(\mathbb R^{n-1})},
G^{+}( x , 0 ) = \frac{1}{2^{\beta + 2}\pi ( 1 + \sigma / l^{2})^{1 + \beta}}{}_{2}F_{1}\left( \frac{1 + \beta}{2}, \frac{\beta + 2}{2}, 1 + \beta , \frac{1}{( 1 + \sigma / l^{2})^{2}}\right)
\begin{aligned}\frac{1}{\tau^k}[\theta(u)-\theta(\breve{u}^k)]-\frac{1}{\tau^{k-1}}[\theta(u)-\theta(\breve{u}^{k-1})]+(w-\widetilde{w}^k)^TF(\widetilde{w}^k) \geq \frac{1}{2}(\|v^{k+1}-v\|^2_{H}-\|v^k-v\|^2_{H})+\frac{1}{2}\|v^k-\widetilde{v}^k\|^2_{G},~\forall w\in\Omega.\end{aligned}
\frac{1}{8}[(1+x)^{n^{2}}+2(1+x)(1+x^{4})^{\frac{n^{2}-1}{4}}+(1+x)(1+x^{2})^{n^{2}-1}+4(1+x)^{n}(1+x^{2})^{\frac{(n-1)(2n-1)}{2}}]
\zeta_r(s) ={{1}\over{3^{s}}}-{{3}\over{\sqrt{\pi}}}{{1}\over{2^{2s+1}}}{{\Gamma \left(s +{{1}\over{2}}\right)}\over{\Gamma(s + 1)}}-{{3}\over{\sqrt{\pi}}}{{1}\over{2^{2s+3}}}{{\Gamma \left(s +{{3}\over{2}}\right)}\over{\Gamma(s + 2)}}\F\left(1, s +{{3}\over{2}}, s + 2,{{3}\over{4}}\right)
\frac{\partial A}{\partial q_{j}}=0 &\frac{\partial X}{\partial q_{j}}=\frac{1-\alpha}{\alpha}\left(\sum_{i}p^{\alpha}_{i}q^{1-\alpha}_{i}\right)^{\frac{1}{\alpha}-1}p^{\alpha}_{j}q^{-\alpha}_{j}&\frac{\partial Y}{\partial q_{j}}=\frac{\alpha-1}{\alpha}\left(\sum_{i}q_{i}\right)^{-\frac{1}{\alpha}}.\ 1
\textbf{FI}(T)-\textbf{FI}(T^{\ast})=\textbf{FI}_{\boldsymbol{\beta}}(T)-\textbf{FI}_{-\boldsymbol{\beta}-1}(T)=\sum_{\underline{\zeta}\in\mathcal{D}_{T_{\infty}}, -\boldsymbol{\beta}-1\leq\underline{\zeta}\leq\boldsymbol{\beta}}\textbf{m}_{T_{\infty}}(\underline{\zeta})
\mathbb{E}_{x}\left[(D_{t})^{2}\right]=\sigma^{2}\int_{0}^{t}\mathbb{E}_{x}\left[X_{s}\left(1+X_{s}\right)^{-2}\right]\mathrm{d}s\leqslant\sigma^{2}\int_{0}^{t}\mathbb{E}_{x}\left[\left(1+X_{s}\right)^{-1}\right]\mathrm{d}s\leqslant t\sigma^{2}<\infty,
\partial_z q^i(z)+\langle Z, \alpha_i\rangle q^i(z)=\Big[L_i(z){p^i(z)}\Big]^{2}\prod_{j\neq i, j\in I_w\cup I_g}p^j(z)^{2a_{ji}}\prod_{j\neq i, j\in I_b}p^j(z)^{a_{ji}}.
g_{\mathrm{t o t a l}}= \frac{4 \pi^{2}R}{e^{2}}\, \left( M_{i j}d \vec{x}_{i}\cdot d \vec{x}_{j}+ ( M^{- 1})_{i j}( d \xi_{i}+ \vec{v}_{i k}\cdot d \vec{x}_{k}) ( d \xi_{j}+ \vec{v}_{j m}\cdot d \vec{x}_{m}) \right) .
\phi_1(v_1,v_1v_2) = v_1 \left(a_{1,1}+ a_{1,2}v_2 + v_1\Phi_1 \right), \phi_2(v_1,v_1v_2) = v_1 \left(a_{2,1}+ a_{2,2}v_2 + v_1\Phi_2 \right)
P = \int_{0}^{t}\bigg\{\int_{0}^{1}\chi_{\{x+z\in[0,1]\}}\bigg|\int_{0}^{1}\chi_{\{|x-y|\leq |z|\}}|G(t-s,x+z,y)-G(t-s,x,y)| & \times v^{2\delta+1}(s,y)\d y\bigg|^p\d x\bigg\}^{\frac{1}{p}}\d s,
\left. \frac{\partial}{\partial a_{\ell}}\langle \lambda_{\mathrm{s}}\rangle \right|_{g_{j}= 0}= \left. \frac{\partial}{\partial a_{\ell}}\left\langle \frac 1 N \, \mathrm{t r}\, \frac{1}{z - \phi}\right\rangle \right|_{g_{j}= 0}= 0
\sigma \sum_{n = 0}^{\infty}\int d^{4}z_{1}\dots \int d^{4}z_{n}\left( W_{x z_{1}z_{2}\dots z_{n}y}\right)_{A B}\frac{e^{- \frac{( x - z_{1})^{2}}{4 \sigma}}}{( 4 \pi \sigma )^{2}}\frac{e^{- \frac{( z_{1}- z_{2})^{2}}{4 \sigma}}}{( 4 \pi \sigma )^{2}}\dots \frac{e^{- \frac{( z_{n}- y )^{2}}{4 \sigma}}}{( 4 \pi \sigma )^{2}}
|\sum_{i=1}^n x_i c_{n-i,n}\frac{r^{n-i}}{(n-i)!}| \geq|x_1|\frac{|r|^{n-1}}{(n-1)!}|c_{n-1,n}| -\sum_{i=2}^n |x_i| \frac{|r|^{n-i}}{(n-i)!}|c_{n-i,n}| \geq |x_1| (\frac{|r|^{n-1}}{(n-1)!}|c_{n-1,n}| -\sum_{i=2}^n \frac{|r|^{n-i}}{(n-i)!}|c_{n-i,n}|).
\;\;\;\;\;(s_i+\sum_{j\leq i-1}(\pi_{i j}, \tilde\pi_{i j})^\ast \tau_{j})|_{\overline{W''_{ik}}}&=(s_i+\sum_{j\leq k}(\pi_{i j}, \tilde\pi_{i j})^\ast \tau_{j})|_{\overline{W''_{ik}}}&=(\pi_{i k}, \tilde\pi_{i k})^\ast(s_k+\sum_{j\leq k-1}(\pi_{i j}, \tilde\pi_{i j})^\ast \tau_{j}+\tau_k)|_{\overline{W''_{ik}}}
H=\frac{g'}{\delta}, \, K=\frac{g'^{2}}{\delta^{2}}, \, S = \frac{-g\left[\kappa v^{2}+\delta^{2}\left(\kappa-\lambda\right)\right]}{2\delta^{2}v}, \, J= \frac{3\left[\kappa g v^{2}+2\delta g' v+ \delta^{2}g \left(\kappa-\lambda\right)\right]}{2\delta^{2}v}.
\!\!f(-k,x)+f(k,x)(S(k)-{U_0})= e^{-ikx}I_n+\int_{-\infty}^\infty K(x,t)e^{-ikt}dt &+ e^{ikx}\int_{-\infty}^\infty F_S(s)e^{-iks}ds &+\int_{-\infty}^\infty K(x,t)e^{ikt}dt\int_{-\infty}^\infty F_S(s)e^{-iks}ds.
\begin{aligned}G_{ij}= \frac{u_1 u_{1ij}}{w^2 \ln w}+ \frac{\sum_k u_{ki}u_{kj}}{w^2 \ln w}- \frac{2 u^2_1 u_{1i}u_{1j}}{w^4 \ln w}- \frac{2 u^2_1 u_{1i}u_{1j}}{w^4 \ln^2 w}&+ \varphi' u_{ij}+ \frac{\rho_{ij}}{\rho}+ \frac{\varphi'}{\rho}(u_i \rho_j + u_j \rho_i) .\end{aligned}
- 2 \Bigl ( e^{\psi}\partial_{1}\partial_{\bar{1}}+ e^{\psi}\partial_{2}\partial_{\bar{2}}+ \partial_{3}\partial_{\bar{3}}\Bigr ) Z = ( 4 \pi )^{1 / 2}\kappa e^{\psi}\rho_{3}\ , \quad e^{\psi}\rho_{3}= \sum_{j = 1}^{N}\delta^{6}( x^{m}- x_{j}^{m}) \ .