harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{u_{+}-v_{-}+h}{u_{+}-v_{-}}f_{1}^{-}(v)k_{1}^{-}(v)k_{1}^{+}(u)=\frac{h}{u_{-}-v_{+}}f_{1}^{+}(u)k_{1}^{+}(u)k_{1}^{-}(v)+k_{1}^{+}(u)f_{1}^{-}(v)k_{1}^{-}(v)
{\bf E}_v({\rm e}^{-\beta X_t}) = \frac{\psi_{\beta}(v)}{\psi_{\beta}(t)}= \frac{v^{1-\delta/2}\,I_{\delta/2-1}(v\sqrt{2\beta})}{t^{1-\delta/2}\,I_{\delta/2-1}((t+v)\sqrt{2\beta})},
G ( a ) = \left[ \frac{m k \Gamma \left( \frac{1 - x}{h}\right) \Gamma \left( \frac{x}{h}\right) x}{2 \Gamma \left( \frac{1}{h}\right) h}\right]^{- a^{2}}\exp \left( \int_{0}^{\infty}\frac{d t}{t}[ a^{2}e^{- 2 t}- F ( a , t ) ] \right)
\frac{1+\alpha}{\pi}\int_0^1\d r\,r^{2p+1}\frac{r^{2l}}{2^{2l}}(1-r^2)^{\alpha}=\frac{\Gamma(2+\alpha)\Gamma(1+p+l)}{2^{2l+1}\pi\Gamma(2+\alpha+p+l)}.
\langle A(\vec{R}_{n})\rangle=\frac{\int d\vec{R}_{N}d\vec{R}_{n}d\vec{R}_{0}G(\vec{R}_{N},\vec{R}_{n};N-n)G(\vec{R}_{n},\vec{R}_{0};n)A(\vec{R}_{n})}{\int d\vec{R}_{N}\vec{d}R_{0}G(\vec{R}_{N},\vec{R}_{0};N)}
\frac{d^{\ell-t}\bigl[(1-z)^{\ell-1}K_{\ell}\bigr]}{dz^{\ell-t}}=\sum_{\mu=0}^{\ell-t}(-1)^{\mu}\binom{\ell-t}{\mu}(\ell-1)_{\mu}(1-z)^{\ell-1-\mu}\frac{d^{\ell-t-\mu}K_{\ell}}{dz^{\ell-t-\mu}},
\left|s_{i}(t)^{p/2}-s_{i}((1-h)t)^{p/2}\right| \leq\int_{(1-h)t}^{t}\left|\frac{ds_{i}(t')^{p/2}}{dt'}\right|dt' \leq\frac{p^{3}}{p-1}\sqrt{n}t^{\frac{p-2}{2}}\cdot ht=\frac{p^{3}}{p-1}\sqrt{n}t^{\frac{p}{2}}h
H = - \int d{\bf x}\int_{\Gamma_{+}+ \Gamma_{-}}d k \int_{\Gamma_{+}^{\prime}+ \Gamma_{-}^{\prime}}d k^{\prime}e^{i k x}e^{i k^{\prime}x}\frac{a ( k ) a ( k^{\prime})}{f ( - k^{2}) f ( - k^{2})}( k_{0}^{2}- k_{0}k_{0}^{\prime}) \frac{f ( - k^{2}) - f ( - k^{2})}{k^{2}- k^{2}}
\varphi(x^{(t)})-\min \varphi= \left( \varphi(x^{(t)})- \varphi(x^{(t+1)})\right)+ \left(\varphi(x^{(t+1)})-\min \varphi\right) \geq \frac{L_{\min}}{2}\|x^{(t)}-x^{(t+1)}\|^2 +\left(\varphi(x^{(t+1)})-\min \varphi\right) \geq \left(\frac{\eta^2L_{\min}}{4pL^2+4L_{\max}^2}+1\right) \left(\varphi(x^{(t+1)})-\min \varphi\right),
\Big\lVert\|x\|^2\Big\lVert_{L^{\infty}\left(\mathbb{R}_{q,+}^2, d m_{\alpha,q}\right)}=\sup \Big\{\|x\|^2, x \in{supp(m_{\alpha,q})}\Big\}&=\sup \Big\{\|x\|^2,x \in{supp}\operatorname{(F)}\cap \mathbb{R}_{q,+}^{2}\Big\}.
\frac{\partial\Gamma_{\Lambda}}{\partial\Lambda}=-N_{f}\mathrm{tr}\left(\frac{\partial \Delta^{-1}_{IR}}{\partial \Lambda}^{\phantom{\alpha}\beta}_{\alpha}\left(\frac{\delta\Gamma_{\Lambda}}{\delta z}I + \Delta^{-1}_{IR}\right)^{-1\phantom{\beta}\alpha}_{\phantom{-1}\beta}\right).
|\gamma_{\mathbf{r}}| \le 2^{\sum_{i=1}^{t-1}(\sum_{j=1}^i (q_j - 2 r_j) + q_{i+1}+ r_i )}\le 2^{t(q_1 +\ldots + q_t) - \sum_{i=1}^{t-1}r_{i}}.
\mathrm{P}_{\Phi +\mathcal{K}_{\gamma}\left( \Psi ,f\right)}=-\inf_{\rho\in E_{1}}\left\{f_{\Phi}\left( \rho \right) +e_{\mathcal{K}_{\gamma}\left( \Psi ,f\right)}\left( \rho \right) \right\}\ ,\gamma \in\left( 0,1\right) \ ,
{\cal L}_{\mathrm{e f f}}( \vec{E}= 0 , \vec{n}\parallel \vec{B}) = - [ \Pi ( m a ) + F_{2}( m a ) ] \frac{\vert B \vert^{2}}{2}\equiv \frac{- 1}{\mu ( a m )}\frac{\vert B \vert^{2}}{2}.
G_1^1 - \frac{1}{2}u_{(1,1)}+ 6u_1u_{(2,0)}- F_{(3,0),1}= 0 ; G_1^2 - \frac{1}{2}u_{(2,0)}= 0 \, , begin{align*}6pt]G_2^1 - \frac{1}{2}u_{(0,2)}+ 6u_1u_{(1,1)}- F_{(3,0),2}= 0 ; G_2^2 - \frac{1}{2}u_{(1,1)}= 0 \, .
\big\langle \widetilde \Lambda_{v_1}\widetilde P_{v_1, \rm R}\widetilde F (v_1), \widetilde P_{v_1, \rm R}\widetilde F (v_1) \big\rangle = \big\langle \Lambda_{v_1}P_{v_1, \rm R}F (v_1), P_{v_1, \rm R}F (v_1) \big\rangle.
\sigma \left(\frac{L^N(r,\pi \boxtimes \chi, \varrho_{5})}{(2\pi i)^{2k+3r}G(\chi)^3 \langle F, F\rangle}\right) = \frac{L^N(r,{}^\sigma\pi \boxtimes{}^\sigma\!\chi, \varrho_{5})}{(2\pi i)^{2k+3r}G({}^\sigma\!\chi)^3 \langle{}^\sigma\!F,{}^\sigma\!F\rangle}.
\frac{\mathrm{d}}{\mathrm{d}n}\ln{f(n)}=\frac{9}{2n}+\frac{6}{(3n+1)(6n+4)}-0.163 begin{align*}5pt] \leq\frac{9}{2\times168}+\frac{6}{(3\times168+1)(6\times168+4)}-0.163< -0.13<0.
\int \frac{d^{3}p}{( 2 \pi )^{3}}\left( \frac{1}{4 \sqrt{p^{2}+ m^{2}}}+ \frac{1}{2 \sqrt{p^{2}+ m^{2}}}\frac{1}{\exp{\sqrt{p^{2}+ m^{2}}/ T}- 1}\right)
\mathcal{E}_{i}^{*}\mathcal{E}_{i}\sim q^{-2}\mathcal{E}_{i}\mathcal{E}_{i}^{*}-q^{-1}(q - q^{-1}) \sum_{j = i + 1}^{N}q^{i - j}\mathcal{E}_{j}\mathcal{E}_{j}^{*}.
C = \sum_{l=1}^L C_l \leq \sum_{l=1}^L \chi_{l}+ K' \log_2 \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^\mathrm{H}\left(\frac{\mathbf{R}}{K'}\right) \mathbf{W}^{\mathrm{RF}}\right] ,
U_1(t,s)x_1 =\frac{\mu'(s)}{\mu'(t)}\left(\frac{\mu(t)}{\mu(s)}\right)^{- a}e^{\alpha \sin^2 s\log \mu(s)-\alpha \sin^2 t\log \mu(t)}x_1, U_2(t,s)x_2 =\frac{\mu'(s)}{\mu'(t)}\left(\frac{\mu(s)}{\mu(t)}\right)^{- b}e^{\alpha \sin^2 t\log \mu(t)-\alpha \sin^2 s\log \mu(s)}x_2.
Op_h^w(\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}))v^{\Sigma}= Op_h^w(\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}))Op_h^w(\Sigma(\xi)\chi(h^{\beta}\xi))v & + Op_h^w(\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}))Op_h^w(\Sigma(\xi)(1-\chi)(h^{\beta}\xi))v \,,
S[\varphi_R]=\int\,{\mbox d}x\,{\mbox d}y\,\left\{\theta(x)\,\left(\frac{1}{2}(\nabla\varphi_R)^2+\frac{m^2}{2}\varphi_R^2\right) +\frac{c}{2}\delta(x)\varphi_R^2\right\}
x' = f_1(x,y;u):=\frac{y (b + 2 a x - c x^2 + c y^2)}{b + 2 a x - c x^2 - c y^2}y' = f_2(x,y;u):=\frac{2 (a - c x) y^2}{b + 2 a x - c x^2 - c y^2}.
[\delta_1, \delta_2]M ={t_1 \delta (\epsilon_{12}, t_1) -t_2 \delta (\epsilon_{12}, t_2)\over t_1 - t_2}M +{t_1 t_2\over 1 - t_1 t_2}\left( \delta(\epsilon'_{12}, t_2) - \delta(\epsilon'_{21},t_1)\right) M.
c^{-1}|\xi |^2 |x-w|^{\frac{(p-2) (1-n)}{p-1}}\leq \sum_{i,j=1}^n b_{ij}(x) \xi_i \xi_j \leq c |\xi |^2 |x-w|^{\frac{(p-2) (1-n)}{p-1}}
S_{s t a t}\equiv \log d ( N_{L \, 0}, N_{R \, 0}) \approx 4 \pi \sqrt{N_{L \, 0}}= 4 \pi \left( N_{L}-{\frac{1}{2}}\sum_{i = 1}^{[{\frac{D - 1}{2}}]}J_{i}^{2}\right)^{\frac{1}{2}},
{}\sum_{i}B_{ij}(t)\widehat{w}_{i}^{(\epsilon )}(t)\geq\sum_{i\in I_{>}}B_{ij}(t)\widehat{w}_{i}^{(\epsilon )}(t)=\sum_{i\in I_{>}}B_{ij}(t)\rho_{\epsilon}(t)\geq B_{i_{0}j}(t)\rho_{\epsilon}(t)\geq\sigma\rho_{\epsilon}(t).
\mathbf{R}_0(\mathbf{z})= \begin{bmatrix}\lambda_l z_l \bar{\lambda}_{l}\bar{z}_l\end{bmatrix}+\sum_{m=1}^M\begin{bmatrix}\beta_m z^{m+1}_l\bar{z}^{m}_l \overline{\beta}_m z^{m}_l\bar{z}^{m+1}_l\end{bmatrix}+\mathcal{O}(\mathbf{z}^{2M+3}).
\left({}^{c}D_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}t^{\rho-1}\right)(x)=\frac{\Gamma(\rho)\Gamma(\alpha-\beta+\rho-m)\Gamma(\alpha+\alpha'+\beta'-\gamma+\rho-m)}{\Gamma(-\beta+\rho-m)\Gamma(\alpha+\alpha'-\gamma+\rho)\Gamma(\alpha+\beta'-\gamma+\rho-m)}x^{\alpha+\alpha'-\gamma+\rho-1}.
- \mathcal{T}_0 = \frac{1}{4 \pi^2}\sum_{k\neq0}\int \!\!\! \int\!\!\! \int \left(A(k,\xi)-A(k,\xi-\eta)\right)\frac{\eta}{\eta^2}\hat{f}(0,\eta)k\hat{f}(k,\xi-\eta) A(k,\xi)\bar{\hat{f}}(k,\xi)\,d\eta d\xi dt.
H_{\mathrm{v i s , \pm}}^{2}= \frac{\rho_{\mathrm{v i s}}( \rho_{\mathrm{v i s}}+ 2 \Lambda_{2}^{\pm})}{3 6 M^{6}( 1 + 4 \alpha \Lambda_{b}/ 3 M^{5})}= \frac{\pm \rho_{\mathrm{v i s}}}{3 M_{P}^{2}\sqrt{1 + 4 \alpha \Lambda_{b}/ 3 M^{5}}}\left[ 1 + \frac{\rho_{\mathrm{v i s}}}{2 \Lambda_{2}^{\pm}}\right] ~ ~ .
T_k(N) := \sum^{\lfloor \frac{N}{2}\rfloor}_{j=1}l^*(h_{1,k}) \otimes E_{2j-1, 2j}(N) + \sum^{\lfloor \frac{N-1}{2}\rfloor}_{j=1}l(h_{1,k}) \otimes E_{2j, 2j+1}(N)
\tilde h_d(\alpha_d) \leq O(1) + \log\frac{d\cdot \Gamma(\frac{d}{2})}{\alpha_dd^{\frac{d-1}{2}}}+ d\log 2 &= O(1) + \log\frac{\Gamma(\frac{d}{2})}{Ce^{-cd}d^{\frac{d-1}{2}}}+ d\log 2.
2^{N-1}=2b_0+ b_{2^{n-1}}+\sum_{\substack{v_2(k)\neq 0,n,{n-1}b_k\in B_1}}b_k, 2^{N-1}=2b_{2^{n-2}}+\sum_{\substack{v_2(k)\neq{n-2}b_k\in B_2}}b_k.
C_{ij}= \sum_{u\in E}c_u^{(ij)}\nu(u) + \sum_{(u,v)\in E^2}(f_u^{(i)}f_v^{(j)}+f_u^{(j)}f_v^{(i)})\nu_2(u,v) \Gamma_{ij}= \sum_{u\in E}\gamma_u^{(ij)}\nu(u)+ \frac{1}{2}\sum_{(u,v)\in E^2}(f_u^{(i)}f_v^{(j)}-f_u^{(j)}f_v^{(i)})\nu_2(u,v)
\begin{aligned}f^{\prime}(x)=\dfrac{df(x)}{dx}=\dfrac32S-\dfrac32x-\dfrac{\sqrt{(4S-3x)x}}4-\dfrac{x(2S-3x)}{4\sqrt{(4S-3x)x}}&=\dfrac32(S-x)-\dfrac{3x(S-x)}{2\sqrt{(4S-3x)x}}&=\dfrac32(S-x)\bigl(1-\dfrac{x}{\sqrt{(4S-3x)x}}\bigl)>0\end{aligned}
\sum_{k=0}^{\frac{n-1}{2}}[3k+1]\frac{(q;q^2)_k^3 q^{-{k+1\choose 2}}}{(q;q)_k^2 (q^2;q^2)_k}\equiv [n]q^{\frac{1-n}{2}}\pmod{[n]\Phi_n(q)^2}, begin{align*}5pt]\sum_{k=0}^{n-1}[3k+1]\frac{(q;q^2)_k^3 q^{-{k+1\choose 2}}}{(q;q)_k^2 (q^2;q^2)_k}\equiv [n]q^{\frac{1-n}{2}}\pmod{[n]\Phi_n(q)^2}.
\rho_{\alpha\beta}(\omega)=\frac{1}{Z}\sum_{m}2\pi\delta(\xi_{\alpha}-\omega)\delta_{\xi_{\alpha},\xi_{\beta}}\langle m|\psi_{\alpha}\psi_{\beta}^{\dagger}e^{-\beta(H-\mu N)}|m\rangle(1-\zeta e^{-\beta\xi_{\alpha}}),
q_{k}=\sum_{s_{1}=-\infty}^{\infty}\sum_{s_{2}=-\infty}^{\infty}\cdot\cdot\cdot\sum_{s_{N}=-\infty}^{\infty}A_{s_{1},s_{2},...,s_{N}}^{k}e^{i2\pi s_{1}w_{1}}e^{i2\pi s_{2}w_{2}}\cdot\cdot\cdot e^{i2\pi s_{N}w_{N}}
I = \left\|{\tilde \sigma}\right\|_{L^\infty\left({\Gamma_-}\right)}+ (1/A_0)\left\|{g_0}\right\|_{L^\infty \left({\mathbb{R}_+}\right)}N_1\left\| Q \right\|_{L^\infty \left({\mathbb{R}_+\times \Omega}\right)},
Y^{{\varepsilon},u}_x(t) = \sqrt{{\varepsilon}}\int_0^t S(t-s)G(s,\mathcal{M}(Y^{{\varepsilon},u}_x + S(\cdot)x)(s))dw(s) & + \int_0^t S(t-s)G(s,\mathcal{M}(Y^{{\varepsilon},u}_x +S(\cdot)x)(s))u(s)ds.
\frac{Z_n^{(N_f+1)}(M,M_{N_f+1})}{C_n^{(N_f+1)}(M,M_{N_f+1})}\ = \ \frac{Z_n^{(N_f)}(M)}{C_n^{(N_f)}(M)}\, \frac{(n+N_f)!}{(N\Sigma^2)^{n+N_f}}\, \frac{1}{\prod_{f=1}^{N_f}(M_f^2-M_{N_f+1}^2)}
T_{f}T_{g}+{\Gamma^{*}_{\bar{f}}}{\Gamma_{g}}=T_{fg}, T_{f}{\Gamma^{*}_{\bar{g}}}+{\Gamma^{*}_{\bar{f}}}{S_{g}}={\Gamma^{*}_{\overline{fg}}}, \Gamma_{f}T_{g}+S_{f}{\Gamma_{g}}=\Gamma_{fg}, \Gamma_{f}{\Gamma^{*}_{\bar{g}}}+S_{f}S_{g}=S_{fg}.
\lim_{x^{+}\rightarrow 0}\langle 0 | T ( \varphi^{\dagger}( x ) \sigma_{1}\chi ( 0 ) ) | 0 \rangle \neq \langle 0 | \varphi^{\dagger}( x^{+}= 0^{+}, x^{-}, x^{\perp}) \sigma_{1}\chi ( 0 ) | 0 \rangle \ .
\gamma\beta(\gamma)=\frac{\gamma_{t-1}(1-\kappa_t)}{(1-\kappa_{t-1})}\left(1+\frac{2\mu\gamma_{t-1}}{1-\kappa_{t-1}}\right)^{-1}\leq\frac{3}{2}\gamma_0, \alpha(\gamma)=\frac{\kappa_{t-1}\gamma\beta(\gamma)}{\gamma_{t-1}}=\frac{\kappa_{t-1}(1-\kappa_{t})}{1-\kappa_{t-1}}\left(1+\frac{2\mu\gamma_{t-1}}{1-\kappa_{t-1}}\right)^{-1}\leq\frac{1}{2}.
\phi_{n}^{\pm}(x)=\frac{e^{i \left[\left(1 \pm 2n \right) \pi x/a\mp \theta \right]}}{\sqrt{a}}; E_{n}^{\pm}=\frac{\hslash^{2}}{2m}\left(\frac{2n\pi \pm \pi}{a}\right)^{2}+\frac{\beta \hslash^{4}}{3m}\left(\frac{2n\pi \pm \pi}{a}\right)^{4},
a_t+u^0\cdot\nabla a=-v\cdot\nabla\rho,{\rm in}\ \Omega, &\rho(v_t+u\cdot\nabla v)-\nu\Delta v+\nabla q =-\rho v\cdot\nabla u^0+(\nabla p^0/\rho^0)a+\nu\Delta u^0\equiv F,{\rm in}\ \Omega, &\mbox{div}\v=0,{\rm in}\ \Omega,
\langle 0|\Gamma_+(\{x^{\pm}\};z)\Gamma_-(\{y\})|0\rangle=\prod_{1\leq k< l\leq n+1}(1-y_ky_l)^{-1}\sum_{\lambda\subseteq [n+1,\infty)}sp_\lambda(\mathbf{x}^{\pm};z)s_\lambda(y), \langle 0|\Gamma_+(\{x^{\pm}\};z)\Gamma_-(\{y\})|0\rangle=\prod^{n}_{i}\prod^{n+1}_{j}\left[(1-x_iy_j)(1-x^{-1}_iy_j)\right]^{-1}\prod^{n+1}_{j}(1-zy_j)^{-1}.
\|f_{\varepsilon, m}\|_{H_p(\mathbb{T}^m,X)}\leq \int_{\mathbb{T}^{m}}{|g_{m}(\omega)|^{p}\: d \omega}= \| g_m\|_{p}^{p}\leq \| g\|_{p}^{p}= \| T\|_{\Lambda}^{p}.
\int_{\big(\frac{1}{|a|}\Omega\big)\setminus D_{R}}\bigg|(i\nabla+A_{\mathbf e}) \Big(\tilde \varphi_a(x) - e^{\frac i2(\theta_{\mathbf e}-\theta_0)}\tfrac{|a|^{k/2}}{\sqrt{H(\varphi_a,K_\delta|a|)}}W_a\Big)\bigg|^2dx=O(1),\quad|a|\to0^+.
d^2(x'_2) =\phi^*(\bar{d}^2(x_2)) =\phi^*(y_1v_1+y_2v_2+y_1v_2+2y_1u_1 +2y_2u_2+y_1u_2+y_2u_1) &=2y'_1\gamma_1+2y'_2\gamma_2+y'_1\gamma_2+y'_2\gamma_1
d s^{2}= e^{- 2 k \mid y \mid}\left[ - \left( 1 - \frac{U_{T}^{4}}{k^{4}}e^{4 k \mid y \mid}\right) d t^{2}+ \sum_{i = 1}^{3}d x^{i}d x^{i}\right] + \frac{d y^{2}}{1 - \frac{U_{T}^{4}}{k^{4}}e^{4 k \mid y \mid}}
\sum\limits_{p = 1}^n{\left\|{{X_p}{u_i}}\right\|_{{L^2}}^2}+ \sum\limits_{p = 1}^n{\left\|{{Y_p}{u_i}}\right\|_{{L^2}}^2}= \int_\Omega{{\nabla_L}{u_i}{\nabla_L}{u_i}}= \int_\Omega{L{u_i}{u_i}}= \int_\Omega{{\lambda_i}{u_i}{u_i}}={\lambda_i}.
V' := \mathrm{Cov}(Y_i') = \begin{pmatrix}p_{n,x,u}^{(j)}(1-p_{n,x,u}^{(j)}) p_\cap-p_{n,x,u}^{(j)}p_{n,y,v}^{(l)}p_\cap-p_{n,x,u}^{(j)}p_{n,y,v}^{(l)}p_{n,y,v}^{(l)}(1-p_{n,y,v}^{(l)}) \end{pmatrix},
\mathbb{P}\left(Q(t) = M\right) = \frac{\Big(\prod_{i}x_i!\Big)\left(\prod_{j}y_j!\right)}{n!\left(\prod_{i,j}m_{ij}!\right)}\sum_{\substack{M'\in\mathcal{E}': M'\ge M}}\prod_{i,j}\frac{\left(1-e^{-\pi_{ij}t}\right)^{m_{ij}}\left(e^{-\pi_{ij}t}\right)^{m'_{ij}-m_{ij}}}{(m'_{ij}- m_{ij})!}
\Gamma_{0 , \left( 5 \right)}= \sum_{i , k ( i < k )}\varphi^{i k \dagger}\gamma^{0}\left( \gamma^{5}\right) \varphi^{i k}+ \sum_{p = 1}^{N - 1}\psi^{p}\gamma^{0}\left( \gamma^{5}\right) \psi^{p}\ ,
\Phi_{qt}(1,t) =\int_{\Omega}a\left( x\right) [\phi_{1}\log(t\phi_{1})+\phi_{1}]\phi_{1}& =\int_{\Omega}a\left( x\right) \phi^{2}\log(t\phi_{1})+\int_{\Omega}a\left( x\right) \phi_{1}^{2}.
\left\vert E[K_{h}(Y-\hat{\gamma}(X))|X]-E[K_{h}(U)|X]\right\vert =\left\vert \int K_{h}(u-\hat{\Delta}(X))f(u|X)du-\int K_{h}(u)f(u|X)du\right\vert & =\left\vert \int K(v)[f(hv+\hat{\Delta}(X)|X)-f(hv|X)]dv\right\vert & \leq\int\left\vert K(v)\right\vert \left\vert f(hv+\hat{\Delta}(X)|X)-f(hv|X)\right\vert dv\leq C\hat{\Delta}(X).
{\frac{1}{g^{2}( q^{2})}}={\frac{1}{g^{2}( \mu )}}-{\frac{1}{g_{5}^{2}k}}\ln \left({\frac{q}{k}}\right) -{\frac{1}{4 8 \pi^{2}}}\ln \left({\frac{q^{2}}{\mu^{2}}}\right) ,
\aligned\int_{0}^{\infty}\int_{0}^{\infty}\tilde{v}(\tau, s)^{4}s^{2}(\frac{s}{\sinh s})^{2}ds d\tau = \int_{0}^{\infty}\int_{0}^{\infty}\frac{e^{4 \tau}(\sinh s)^{4}}{s^{4}}(\frac{s^{4}}{(\sinh s)^{2}}) v(e^{\tau}\cosh s, e^{\tau}\sinh s)^{4}ds d\tau = \int \int_{t^{2}- r^{2}\geq 1}v(t,r)^{4}r^{2}dr dt.\endaligned
A_{1}= 2^{- 1 / 2}\frac{b_{1}^{- 1 / 8}}{\sqrt{n}}\biggl ( \frac{\pi}{\epsilon}\biggr )^{1 / 4}\exp \biggl [ - i ( b_{1})^{1 / 2}\biggl ( \frac{n^{2}\epsilon}{4 \pi}\biggr ) \vert \eta_{\mathrm{i}}\vert^{2}- i \frac{\pi}{4}- i \frac{\phi_{2}}{2}\biggr ] , \quad A_{2}= 0 .
\dot{u}_{\alpha}= -b_u \, u_{\alpha}+ a_u \, (uv)_{\alpha}+ f \delta_{\alpha \boldsymbol{0}}+ \sigma_u \, E[\dot{W}_u T_{\alpha}] , \dot{v}_{\alpha}= -b_v\, v_{\alpha}+ a_v \, (uv)_{\alpha}+ \sigma_v \,E[\dot{W}_v T_{\alpha}].
\gamma(E) = \begin{cases}\frac 1{\pi^2}\delta_\alpha^2(\sqrt E) \ \ \delta_\alpha(\sqrt E)\le \frac\pi 2 \frac 1{\pi^2}\big(\delta_\alpha(\sqrt E)-\pi\big)^2 \ \ \delta_\alpha(\sqrt E)\ge \frac\pi 2. \end{cases}
\sum_{i=0}^{n}\dfrac{a_{n+1-i}}{\binom{n+1}{i}}\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot A\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right)=0,
\left[ \begin{array}{cccc}I I \dots I A_{i_1}A_{i_2}\dots A_{i_{d+1}}A_{i_1}^2 A_{i_2}^2 \dots A_{i_{d+1}}^2 \vdots \vdots \vdots \vdots A_{i_1}^{s-1}A_{i_2}^{s-1}\dots A_{i_{d+1}}^{s-1}\end{array}\right]\left[ \begin{array}{c}W_1C_{i_1}W_2C_{i_2}\vdots W_{d+1}C_{i_{d+1}}\end{array}\right]=0,
V ( \psi ) = \frac{h}{4}\sum_{j}\left[ \frac{4}{\lambda^{2}h^{2}}\sin^{2}\frac{\lambda}{2}( \psi_{j + 1}- \psi_{j}) + \frac{m^{2}}{\lambda^{2}}\sin^{2}\frac{\lambda}{2}( \psi_{j + 1}+ \psi_{j}) \right]
\begin{aligned}\widetilde{G}^{(n)}(t;\eta,P) =\pm 2\eta^3+P^2\pm \eta\left(\frac{3v_2^2}{2}+\ddot{v}\right)+\frac{v_2^3}{2}+\ddot{v}_2v_2-\frac{\dot{v}_2^2}{4}\ ,\end{aligned}
\mathbb{P}\left(|L_n(z)|\leq e^{-n\epsilon}\right) \leq C_\epsilon\left\{\sum_{i=1}^{l_n}\left(1-Q\bigg(\frac{a_{i_k}}{z-z_{i_k}},e^{-n\epsilon}\bigg)\right)\right\}^{-\frac{1}{2}}.
\varphi^{-1}\left( Y_{[bwa]}(\varphi(P)) \right)n = \varphi^{-1}\rho^{-1}\left( X_{[bwa]^{\ast}}(\rho \varphi(P)) \right)n &=\left(\rho^{-1}\hat{\varphi}^{-1}X_{[bwa]^{\ast}}(\hat{\varphi}(\rho(P))) \right)n
H_2H_0(\varphi_{0,\frac 52})=H_2\left(-\frac{11}5 E_4 \varphi_{-2,1}\varphi_{0,\frac 32}\right)=\frac{44}5 E_6 \varphi_{-2,1}\varphi_{0,\frac 32}+\frac{11}{25}E_4\varphi_{0,\frac 52}.
||\mathcal{J}_I^m||_{\infty}= |\lambda_I|^m +{m\choose 1}|\lambda_I|^{m-1}+ \ldots +{m\choose n_I-1}|\lambda_I|^{m-n_I+1}&= |\lambda_I|^m \left ( 1 + \frac{{m\choose 1}}{|\lambda_I|}+ \ldots + \frac{{m\choose n_I-1}}{|\lambda_I|^{n_I-1}}\right ).
\frac{s\sigma_{s}\sqrt{2\pi n}}{r_{s, f}(v)}e^{- n\Lambda^{*}(q)}I_n(f, v) = \sqrt{2\pi}a_n \int_0^{\infty}e^{- y}\left[ F_n\left( \frac{y}{a_n}\right) - F_n(0) \right] dy.
Y_{n}^{\ell}=\Pi_{n-1}Y_{n-1}^{\ell}=S_{\Delta t}Y_{n-1}^{\ell}+\Delta tS_{\Delta t}G'(X_{n-1}).Y_{n-1}^{\ell}+S_{\Delta t}e^{\tau A}\bigl(\sigma'(X_{n-1}).Y_{n-1}^{\ell}\bigr)\Delta W_{n-1}.
f(x)=\frac{1}{\pi}\int\limits_{\gamma(s)}(\phi(x,\lambda))\frac{1}{p(\lambda)}d\lambda+\frac{1}{\pi}\sum_{k\in\mathbb{Z}\backslash S}\left(\int\limits_{\Gamma_{k}}(\phi(x,\lambda))\frac{1}{p(\lambda)}d\lambda\right),
\sum_{\substack{q\in \mathcal{P}_{X}q\not = p}}\sum_{\eta\in \Gamma_{X,q}\backslash\Gamma_{X}}P_{\mathrm{gen},q}(\eta z)+\sum_{\substack{\eta\in \Gamma_{X,p}\backslash\Gamma_{X}\eta\not = \mathrm{id}}}P_{\mathrm{gen},p}(\eta z)=O\left(\Im(\sigma_{p}^{-1}z)^{-1}\right).
\boldsymbol{n}\left( F_{n}\right) h_{2n+2}^{1+2\alpha_{2},3\alpha_{2}}(\alpha_{1}1)h_{2n+3}^{1+2\alpha_{2},\alpha_{2}}2(\alpha_{1}1)\left( 1\alpha_{2}\right)f_{n}f_{n+1}.\medskip
\frac{\pi}{4}= 1 - \frac{1}{3}+ \frac{1}{5}- \frac{1}{7}+ \ldots + \frac{( - 1 )^{n}}{2 n + 1}+ \ldots = \sum \limits_{k = 0}^{\infty}\frac{( - 1 )^{k}}{2 k + 1}
\frac{d}{dt}\xi^N(k,t)= \i \sum_{\substack{m+n=k, |m|, |n|,|k|\leq N}}m|n| e^{\frac12\mu(|k|^\alpha-|m|^\alpha-|n|^\alpha)t}\xi^N(m) \xi^NB(n) &-\frac12\mu |k|^\alpha\xi^N(k).
\overline{\mu}= \begin{bmatrix}0 0.9 0.3 0.3 0.3 0.3 0 0.3 0.8 0.3 0.3 0.3 0 0.3 0.3 0.7 0.3 0.3 0 0.3 0.3 0.3 0.6 0.3 \end{bmatrix}
2\begin{bmatrix}O \left( \dfrac{z}{\overline{z}}\right)^{\kappa}a_{11}(u)^{\ast}c_{12}a_{22}(u) \left( \dfrac{\overline{z}}{z}\right)^{\kappa}a_{22}(u)^{\ast}c_{12}^{\ast}a_{11}(u) O\end{bmatrix},
\mu = \binom{n-2}{v_t-2}p^{e_t}\geq \left ( \frac{n}{v_t-2}\right )^{v_t-2}n^{-\frac{v_t-2}{e_t}e_t}( \log n )^{e_t}= \frac{(\log n)^{e_t}}{(v_t-2)^{v_t-2}}\geq \left(\frac{\log n}{\tau^t}\right)^{\tau^t},
\varphi'(x) = c_k^{-\frac{1}{2}}\cdot \left( x_1 - x_2 \frac{\i}{u}\right)^k \cdot e^{- \frac{\pi v}{2}( x_1^2 - \frac{1}{u}x_2^2 + x_3^2 - \frac{1}{u}x_4^2 )}
\begin{aligned}10 W_{j - 1}W_j^2 = - 2 W_{j - 2}W_{j - 1}W_j - 6 W_{j - 2}W_{j - 1}W_{j + 1}- 9 W_{j - 2}W_{j}W_{j + 1}+ 27 W_{j - 1}W_j W_{j + 1}& + 10 W_{j - 1}W_{j}+ E_j^{(8)}. \end{aligned}
h_{1, \Delta}(x) := \left((F_{Z,1})'(0) \Delta\right) (x, g(x), g'(x)), h_{2, \Delta}(x) :=\left((F_{Z,2})'(0) \Delta\right) (x, g(x), g'(x)), h_{3, \Delta}(x) : = h_{2, \Delta}'(x).
f_0^{\lambda_i(x,y)}=-f_i^{\lambda_0(x,y)}-\sum_{m+n=i,~m,n>0}f_m^{\lambda_n(x,y)}+\sum_{m+n=i,~m,n>0}[f_m^x,f_n^{y}]+[f_0^x,f_i^{y}]+[f_i^x,f_0^{y}].
\frac{V_{k_0+1}^\rho}{a^{k_0+1}}+\sum_{k=1}^{k_0}\frac{V_{k}}{a^{k}}\leq \sum_{k=0}^{k_0}\frac{V_{k+1}^\rho}{a^{k+1}}\leq \sum_{k=1}^{k_0}\frac{V_k}{a^k}+V_0.
d\mu_z(x) = \frac{1}{Z\,Q(z)}\exp\Big[-\beta\Big(V_0(x) + \frac{1}{\epsilon}V_1(x)\Big)\Big] \Big[\mbox{det}(\nabla \xi\nabla \xi^T)(x)\Big]^{-\frac{1}{2}}\,d\nu_z(x)\,,
\int{\cal D}v \; \delta^{n}\left( \int v d \tau \right) F_{q G}[ v , I ] = \int{\cal D}v \; \delta^{n}\left( \int \phi d \tau \right) F_{q G}[ \phi , I ] \; \mathrm{D e t}\; \frac{\delta \phi_{\tau}( v )}{\delta v ( \tau^{\prime})}\; ,
\begin{aligned}0 =\mathbf{P}_x\left(T_{>x}<\infty, \exists \varepsilon>0X_{T_{>x}+t}\leq x0\leq t\leq \varepsilon \right) &=\mathbf{E}_x\left(1_\Lambda\circ \theta_{T_{>x}}; T_{>x}<\infty\right). \end{aligned}
\bar{\partial}\Lambda=\sum_{\alpha,\beta}\left(\sum_{v=1}^n \frac{\partial \Lambda_{\alpha\beta}}{\partial \bar{z}_v}d\bar{z}_v+\sum_{\mu=1}^m \frac{\partial \Lambda_{\alpha\beta}}{\partial \bar{t}_{\mu}}d\bar{t}_{\mu}\right)\frac{\partial}{\partial z_{\alpha}}\wedge \frac{\partial}{\partial z_{\beta}}
D^{\mathbf i}_\alpha \left[ - 3 i{\bar\sigma}^{0\dot\alpha\gamma}{\bar D}_{\dot\alpha{\mathbf i}}D_{\beta\gamma}T \right] = -24\left[ \sigma^{\mu 0\,\,\gamma}_\alpha \partial_\mu D_{\beta\gamma}T - \frac{1}{2}\partial^0 D_{\alpha\beta}T\right].
\frac{1-|b(r\xi)|^2}{|1-b(r\xi)|^2}=\int_{\partial \mathbb{D}}\frac{1-r^2}{|t-r\xi|^2}d\mu(t)\geq \frac{\mu\left(\{t:\ |t-\xi|<1-r\}\right)}{4(1-r)}\geq C_1\ ,
M_{1 1}^{\beta \gamma}= \frac{T^{\beta \gamma}}{1 - C I^{\beta \gamma \, - 1}T^{\beta \gamma}+ T^{\beta \gamma \, 2}}= \frac{( C I^{\beta \gamma})^{5}- C I^{\beta \gamma}}{( 1 - ( C I^{\beta \gamma})^{2}) ( 1 - 3 ( C I^{\beta \gamma})^{2})}
a_{A}( k ) \, a_{A}^{\dagger}( k^{\prime}) + q^{- 2}\sum_{l , l^{\prime}\in{\cal I}}{\hat{\cal R}}\left( k^{\prime}, l ; k , l^{\prime}\right) a_{A}^{\dagger}( l^{\prime}) \, a_{A}( l ) = \delta_{k \; k^{\prime}}\; ,
X[a_1,a_2] =M[a_1,a_2]+\sum\limits_{(a'_1,a'_2)<(a_1,a_2)}q^{a_1,a_2}_{a'_1,a'_2}M[a'_1,a'_2]; C[a_1,a_2] =X[a_1,a_2]+\sum\limits_{(a'_1,a'_2)<(a_1,a_2)}r^{a_1,a_2}_{a'_1,a'_2}X[a'_1,a'_2],
2\hat q = \left({\begin{array}{*{20}c}0{M_0}0 \cdots 0{M_0}0{M_1}\cdots 0 0{M_1}0 \ddots{}\vdots \vdots \ddots \ddots{M_{2j - 1}}0 0{}{M_{2j - 1}}0 \end{array}}\right) \equiv M^q,
\frac{\partial}{\partial t}\nabla_{i}\nabla_{j}\Phi=\nabla_{i}\nabla_{j}\frac{\partial\Phi}{\partial t}-\frac{1}{2}g^{kp}(\nabla_{i}v_{jp}+\nabla_{j}v_{ip}-\nabla_{p}v_{ij})\frac{\partial\Phi}{\partial x^{k}}
\alpha_1:=\{\{1,2\},\{3,4\},\{5,6\},\{7,8\},\ldots,\{2b-1,2b\}\}, \alpha_2:=\{\{1,4\},\{3,2\},\{5,6\},\{7,8\},\ldots,\{2b-1,2b\}\}, \alpha_3:=\{\{1,6\},\{3,4\},\{5,2\},\{7,8\},\ldots,\{2b-1,2b\}\}, \alpha_4:=\{\{1,8\},\{3,4\},\{5,6\},\{7,8\},\ldots,\{2b-1,2b\}\}.
\lim_{m \rightarrow 0^{+}}\; \lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= 0 \lim_{m \rightarrow 0^{-}}\; \lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= \frac{e B}{2 \pi}\; ,
{\cal D}:= \big\{(u,{\cal P}, T): \u \in SBV(\Omega), \{\cal P}\Omega,\T \in{\cal R}({\cal P}) \big\},{\cal D}_\infty := \big\{(u,{\cal P}, T): \,{\cal P}\Omega,\, T \in{\cal R}({\cal P}), \, \nabla T^T u \in GSBD^2(\Omega) \big\}.
g_{\mu \nu}\equiv{\frac{\rho_{0}}{c_{0}}}\left[ \begin{matrix}{- ( c_{0}^{2}- v_{0}^{2})}&{\vdots}&{- v_{0}^{j}}\\{\cdots \cdots \cdots \cdots}&{\cdot}&{\cdots \cdots}\\{- v_{0}^{i}}&{\vdots}&{\delta_{i j}}\\ \end{matrix}\right] .
f_i(x,m,v_i) = \chi(v_i) + \varphi_{i,\lambda}(m(x)) \, , h_i(x,m) = \psi_i(x) \, , &\chi(v) = \frac{|v|^2}{2}\, , \varphi_{1,\lambda}(\mu_1,\mu_2) = \mu_1 + \lambda \mu_{2}, \varphi_{2,\lambda}(\mu_1,\mu_2) = \mu_2 + \lambda \mu_{1}\, ,
A_{3,\zeta}=Q\rho\tau_{,\zeta}\ \ ,\ \ \ \A_{3,\bar{\zeta}}=-Q\rho\tau_{,\bar{\zeta}}\ \ ,\ \ \ \ \kappa^2=I^3={h^3e^{\tau_0\tau}\over(1-{2m\over r})g_{22}^{\beta}}
U := \frac{1}{X}\sum_{\mathcal{J}\subseteq \{1, \dotsc, J\}}(-1)^{\# \mathcal{J}}\sum_{n \leq X}g_{\mathcal{J}}(n) f(n) n^{-it_1}\ll \exp\left(-\frac{1}{2}M(f; X)\right) + (\log X)^{-1/16}.
\begin{array}{rcl}\mathrm{sign}(\Phi(H)) = \mathrm{sign}(H)+(-2,1,0), \mathrm{sign}(\Psi(H)) = \mathrm{sign}(H)+(-1,-1,1), \mathrm{sign}(\Theta(H)) = \mathrm{sign}(H)+(0,0,-1), \mathrm{sign}(\Lambda(H)) = \mathrm{sign}(H)+(1,-2,0), \end{array}
\Gamma^0 ={\rm i}\epsilon_2 \otimes \epsilon_1 \hspace{6mm}\Gamma^1 ={\rm i}\epsilon_1 \otimes{\boldmath 1}\hspace{6mm}\Gamma^2 ={\rm i}\epsilon_3 \otimes{\boldmath 1}\hspace{6mm}\Gamma^3 ={\rm i}\epsilon_2 \otimes \epsilon_2
\begin{array}{rcl}C_1 = \{(1,1), (2,2), \ldots, (n-1,n-1), (n,n)\}, C_2 = \{(1,2), (2,3), \ldots, (n-1,n), (n,1)\}, & \vdots C_n = \{(1,n), (2,1), \ldots, (n-1,n-2), (n,n-1)\}. \end{array}
\mu_{\alpha}(x) = \frac{\sqrt{x}}{\pi}- \frac{\alpha}{2}, \sigma^{2}(x) = \frac{\log(4 \sqrt{x})}{2 \pi^{2}}, \Sigma(x_{k},x_{j}) = \frac{1}{2\pi^{2}}\log \frac{\sqrt{x_{k}}+\sqrt{x_{j}}}{\sqrt{x_{k}}-\sqrt{x_{j}}}.
(\tilde{\Delta}_E\tilde{h}_{3,i}^{(1)},\tilde{h}_{3,i}^{(1)})_{L^2(\tilde{g})}=\int_{J}\varphi^2f^{n-2}dr\cdot({\Delta_E}{h},{h})_{L^2({g})}+\int_{J}(\varphi')^2f^n dr\cdot \left\|h\right\|^2_{L^2(g)},
I = - \Vert u - \widetilde{u}\Vert_{V}^2 , II \leq \varepsilon \Vert u - \widetilde{u}\Vert_{V}^2 + C_{\varepsilon}\Vert u\Vert_{V}^2 \Vert u - \widetilde{u}\Vert_{H}^2 .
f_{k;i}(t)-f_{k;j}(t) =\sum_{m=i}^{k}(t-\alpha_{m-i+1})^++\sum_{m=k+1}^{M}(\beta_m-\alpha_{m-i+1})^+-\sum_{m=j}^{k}(t-\alpha_{m-j+1})^+-\sum_{m=k+1}^{M}(\beta_m-\alpha_{m-j+1})^+ &=\sum_{m=i}^{j-1}(t-\alpha_{m-i+1})^++\sum_{m=j}^{k}[(t-\alpha_{m-i+1})^+-(t-\alpha_{m-j+1})^+]+\sum_{m=k+1}^{M}[(\beta_m-\alpha_{m-i+1})^+-(\beta_m-\alpha_{m-j+1})^+],