harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_u(\mathcal{S}) = \sum_{t=1}^{M-1}\left( M - W_u^1(t) + 1 + \kappa(t) - \kappa(t)\right) &= \sum_{t=1}^{M-1}\left(M - \kappa(t) + 1\right) + \sum_{t=1}^{M-1}\left(\kappa(t) - W_u^1(t) \right).
i \Pi_{\mu \nu}^{( d )}( p ) = \frac{i e^{2}N_{f}}{6 \pi^{2}\epsilon}( p_{\mu}p_{\nu}- \eta_{\mu \nu}p^{2}) = \frac{i e^{2}}{( 4 \pi )^{2}\epsilon}\frac{- 8 N_{f}}{3}( \eta_{\mu \nu}p^{2}- p_{\mu}p_{\nu})
\Phi_{k}( \sigma_{1}, \sigma_{2}) = \sum_{p \in{\bf Z}}~ e^{2 \pi i ( \sigma_{2}/ 2 \pi + k ) p / n}~ e^{2 \pi i ( \sigma_{1}/ 2 \pi - p / m ) m / n}~ \widehat{\psi}_{p}( \sigma_{1}/ 2 \pi - p / m ) ,
G ({\bf x}_{n + 1},{\bf x}_{n}; \varepsilon \vert{\hat{\bf A}}) \simeq \biggl ({\frac{m}{2 \pi i \varepsilon}}\biggr )^{N}e x p \biggl [ i \sum_{a = 1}^{N}\biggl ({\frac{m ( \Delta{\bf x}_{n}^{a})^{2}}{2 \varepsilon}}+ e \Delta{\bf x}_{n}^{a}\cdot{\hat{\bf A}}({\bar{\bf x}}^{a}) \biggr ) \biggr ] \, .
I_{1}= \lim_{t\rightarrow\infty}\frac{t}{a(t)}\theta\left[\frac{1}{t}\int_{0}^{t}\int_{0}^{t-r}h(s)ds\gamma_{0}(r)dr-m\right] &=\lim_{t\rightarrow\infty}\frac{t}{a(t)}\theta\left[\frac{1}{t}\int_{0}^{t}h(s)ds\int_{0}^{t}\gamma_{0}(r)dr -\frac{1}{t}\int_{0}^{t}h(s)ds\int_{t-r}^{t}\gamma_{0}(r)dr-m\right]
R_c(\frac{1}{K})= \frac{\min \left(\frac{K(K-1)}{2},N(K-1) \right)-KN}{N}\cdot \frac{1}{K}+N &=N-1 + \min(\frac{K-1}{2N},1-\frac{1}{K}) &\geq N-1 + \min(\frac{N-1}{2N},1-\frac{1}{N}) &= N - \max \left( \frac{N+1}{2N}, 1/N \right) &\ge N-\frac{K}{N}=R(\frac{1}{K}).
\beta(-p) = \mathring{P}^\prime(\mathring{y}) \mathring{y}_p+ \frac{1}{2}\rho^\prime c^2 + g \rho^\prime \mathring{y}+ g \rho \mathring{y}_p & = \rho^\prime \left[ \frac{1}{2}c^2 + g\mathring{y}\right].
\vec{\nabla}\cdot[\epsilon(\vec{r})\vec{\nabla}\Psi(\vec{r})]=-4\pi\rho^{f}(\vec{r})-4\pi\sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi(\vec{r})}{kT}}
\frac{1}{z}=\frac{\overline{z}}{z\overline{z}}=\frac{\overline{z}}{|z|^{2}}=\frac{x-yi}{x^{2}+y^{2}}=\frac{x}{x^{2}+y^{2}}-\frac{y}{x^{2}+y^{2}}i.
\mathcal{O}_N(x-y)=\mathcal{O}_N\bigl(x-\sigma^\ell(x)\bigr)=\mathcal{O}_N\Bigl(\textstyle\sum\limits_{i=0}^{\ell-1}\sigma^i(x)-\sigma\bigl(\sigma^i(x)\bigr)\Bigr)\in P_\ell,
\lim_{r\to r_0}\frac{r^{n+1-2s}\mu(rz)U^2(rz)-r_0^{n+1-2s}\mu(r_0z)U^2(r_0z)}{r-r_0}= (n+1-2s)r_0^{n-2s}\mu(r_0z)U^2(r_0z) +r_0^{n+1-2s}\frac{\partial}{\partial r}|_{r=r_0}\mu(rz)U^2(r_0z) +2r_0^{n+1-2s}\mu(r_0z)U(r_0z)U_{\nu}(r_0z),
= i \int \frac{d^{4}x_{1}d^{4}p^{\prime}}{( 2 \pi )^{4}}e^{- i p_{\mu}^{\prime}x_{2}^{\mu}}\frac{p_{\nu}^{\prime}\gamma^{\nu}+ m}{p_{\tau}^{\prime}p^{\tau \prime}- m + i \epsilon}e^{i p_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi ( 1 )
\mathcal{H}^{\rm LB}(D)=\big\{[W^{\rm LB}(D, C)] \in H^{\rm LB}_3(X) \ \big| \C \in{\rm Col}_{X^2}^{\rm LB}(D) \big\},\mbox{and}\Phi_{\theta}^{\rm LB}(D)=\big\{\theta\big(W^{\rm LB}(D, C)\big) \in A \ \big| \C \in{\rm Col}_{X^2}^{\rm LB}(D) \big\}
\frac 12 - \frac{\alpha - d+ 2+[ d-\alpha]}{2 \beta_{[d-\alpha]+ 2}(\alpha-d+ 2 +[d-\alpha])}= \frac12 - \frac{2-\langle d-\alpha\rangle}{2 ([d-\alpha]+1)(2-\langle d-\alpha\rangle) +2}
\varphi(x) =T_{\lambda+0i}\varphi(x) = i\int_{-\infty}^x e^{i\lambda(x-y)}u\varphi(y)~dy - \widetilde{G}_{\lambda}*(u\varphi) (x) &= -i\int_x^{\infty}e^{i\lambda(x-y)}u\varphi(y)~dy - \widetilde{G}_{\lambda}*(u\varphi) (x)
\frac{\delta_{\mathsf{P,Ach}}(\mu,r)}{\delta_{\mathsf{P}}^*(\mu,r)}\leq \frac{M+K-1}{M}\times \frac{\min\{M,K\}}{K}\leq \frac{M+K}{\max\{M,K\}}~= 1 + \frac{\min\{M,K\}}{\max\{M,K\}}\leq 2.
\mathrm{R}\left(f(x)-\frac{c}{\Phi_0(\nu;x)}\right)=\nu+\mathrm{R}\left(E_0(x)\right), &\mathrm{R}\left(f(x)-\frac{c}{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}\right)=\nu+\mathrm{R}\left(E_k(x)\right),k\ge 1.
\frac{\P (S_\delta^b (\tau_U) \in J_\alpha \ |\S_\delta^b (\tau_D) = a_\delta)}{\P (S_\delta^b (\tau_U) \not\in J_\alpha \ |\S_\delta^b (\tau_D)=a_\delta )}\leq C\frac{\P (S_\delta^b (\tau_U) \in J_\alpha \ |\S_\delta^b (\tau_D) \in A)}{\P (S_\delta^b (\tau_U) \not\in J_\alpha \ |\S_\delta^b (\tau_D) \in A )}
\aligned0 =(-d\sigma^{-1}(I-\Delta^{tr}\Delta)+\theta\Delta)\Delta^{tr}\mu^{tr}-2(d\Delta-\theta\sigma)\sigma\mu^{tr}&=d(-\sigma^{-1}(I-\Delta^{tr}\Delta)\Delta^{tr}-2\Delta\sigma)u^{tr}+\theta(\Delta\Delta^{tr}+2\sigma^2)u^{tr}&=d\; (\sqrt{2},0,0)\mu^{tr}+\theta\mu^{tr}&=(\sqrt{2}p,0,0)\mu^{tr},\endaligned
D_1 \sum_{i=1}^{r_1}l(e_1^i) = D_1 \sum_{i=1}^{r_2}l(e_2^i) = \ldots = D_1 \sum_{i=1}^{r_m}l(e_m^i) = = D_2 \sum_{j=1}^{s_1}l'(f_1^j) = D_2 \sum_{j=1}^{s_2}l'(f_2^j) = \ldots = D_2 \sum_{j=1}^{s_n}l'(f_n^j).
{\cal{F}}\left\{\left(P-{\mu}^2\right)_{Ad}^{-1}\right\}(x) =\pi{\mu}^{\frac{\nu}{2}-1}Q_{-}^{\frac{1}{2}\left(1-\frac{\nu}{2}\right)}{\cal{I}}_{1-\frac{\nu}{2}}\left(\mu Q_{-}^{\frac{1}{2}}\right)
f_{e+f+g+h}^{(2)}= \frac{g^2_1}{8 \pi^2}\frac{g^2_2}{8 \pi^2}\int_{\lambda_1^2}^1\frac{d\alpha_1}{\alpha_1}\int_{\lambda_1^2/\alpha_1}^1\frac{d\beta_1}{\beta_1}\int_{\lambda_2^2}^1\frac{d\alpha_2}{\alpha_2}\int_{\lambda_2^2/\alpha_2}^1\frac{d\beta_2}{\beta_2}.
{}~\times \frac{w(\omega x_3x_4z_1z_2/x_1x_2z_3z_4|c+d-a-b)}{w\bigg(\frac{\displaystyle x_4z_1}{\displaystyle x_1z_4}|d-a\bigg)w\bigg(\frac{\displaystyle x_3z_2}{\displaystyle x_2z_3}|c-b\bigg)w\bigg(\frac{\displaystyle x_3z_1}{\displaystyle x_1z_3}|c-a\bigg)w\bigg(\frac{\displaystyle x_4z_2}{\displaystyle x_2z_4}|d-b\bigg)},
H_{\tilde{u}}(e_1) =\frac{|p|^2}{(1-|p|^2)^2}e_1+\frac{1}{1-|p|^2}e_2, H_{\tilde{u}}(e_2) =\frac{2|p|^2}{(1-|p|^2)^3}e_1+\frac{1}{(1-|p|^2)^2}e_2.
k^{2}a_{0,n}+{\displaystyle\sum\limits_{m=0}^{n-1}}\left( a_{m+1,n}+C_{n}\beta a_{m,n}\right) \dfrac{1}{k^{2m}}+C_{n}\beta\dfrac{a_{n,n}}{k^{2n}}& \leq k^{2}a_{0,n}+{\displaystyle\sum\limits_{m=0}^{n-1}}\left( \left( m+2\right) a_{m+1,n}+\beta\left( m+1\right) a_{m,n}\right)\dfrac{1}{k^{2m}}+\beta\left( n+1\right) \dfrac{a_{n,n}}{k^{2n}}.
L_{b}={\frac{1}{g^{2}}}\int_{{\cal B}\times R}d t d^{d}\sigma \t r \left\{-{\frac{1}{4}}F^{2}+{\frac{1}{2}}( D_{\alpha}X^{i})^{2}-{\frac{1}{4}}( [ X^{i}, X^{j}] )^{2}\right\}\quad ,
H_c = \int d^3x \left[ \frac{1}{2}(\pi_i^{a})^2 + \frac{1}{4}F_{ij}^{a}F^{a, ij}+ \frac{1}{2}m^2 \{(A^{a,0})^2 + (A^{a,i})^2 \}- A_0^{a}\Theta_2^{a}\right],
\sum_{j=1}^m \left[ \left(e^{ib_j\left(\frac{\alpha-\beta}{2}\right)}+e^{-ib_j\left(\frac{\alpha-\beta}{2}\right)}\right) \delta_0\, -\left(e^{ib_j\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{b_j}+e^{-ib_j\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{- b_j}\right) \right]^s \ast f_j.
2[{}^{\ell}Q_n,{}^{\ell}Q_m]= +2\omega\tilde{g}\sum_{t=0}^{n+m-1}\sum_{j,k}\left((\ell_k^+\ell_k^-)^{t}m_{jk}(\ell_k^+\ell_k^-)^{n+m-t-1}-(\ell_k^+\ell_k^-)^{t}m_{jk}(\ell_k^+\ell_k^-)^{n+m-t-1}\right)=0.
\left({2\pi}{\zeta_1^{-1}}\right)^{\frac 12}e^{\frac{(n\zeta_0)^2}{2\zeta_1}}e^{\zeta_0 nc+ \frac12 \zeta_1 c^2}\,=\, \int_{\mathbb{R}}e^{x_i n\zeta_0 +(cx_i- \frac12 x^2_i)\zeta_1}dx_i ,
\tilde{A}_{\mu \nu}( k ) ~ \sim ~ \frac{\tilde{B}_{A}}{( k^{2})^{3 / 2 - \beta_{A}}}\left[ \eta_{\mu \nu}~ - ~ ( 1 - \rho ) \frac{k_{\mu}k_{\nu}}{k^{2}}~ + ~ \vartheta \frac{\epsilon_{\mu \nu \sigma}k^{\sigma}}{( k^{2})^{1 / 2}}\right]
SO_2(G) = \sum_{uv\in E(G)}\Big| \frac{d_u^2-d_v^2}{d_u^2+d_v^2}\Big|\geq \sum_{uv\in E(G)}\frac{|d_u^2-d_v^2|}{2\Delta^2}&\geq \frac{1}{\Delta^2}\left( \frac{1}{2}\sum_{uv\in E(G)}|d_u^2-d_v^2| \right)= \frac{SO_1(G)}{\Delta^2}.
s_{d,w}(t) =\sum_{i\in [n]}d_i^2+M_n''(t)-\int_0^t \sum_{i\neq j}\frac{w_i(u)w_j(u)(d_i+d_j)}{s_1(u)-1}\mathrm{d}u-\int_0^t \sum_{i\in [n]}\frac{w_i(u)(w_i(u)-1)2d_i}{s_1(u)-1}\mathrm{d}u = n\mu_n(1+\nu_n)+M_n''(t)- 2 \int_0^ts_{d,w}(u)\mathrm{d}u,
Z_j (\varphi, \Psi_j ; (\Psi_k)_{k<j}) = \sum_{X\in \mathcal{P}_{j}}e^{U_j (\Lambda \backslash X)}(K_j (X ; (\Psi_k)_{k<j}) + \Psi_j (X) ) = \sum_{X\in \mathcal{P}_{j+1}}e^{U_j (\Lambda \backslash X)}\bar{K}^{\Psi}_j (X)
o(1) = I_\lambda' (w_{k_1},w_{k_2})(w_{k_1},0) &= \int_{-1}^1 |\nabla w_{k_1}|^2 dx-\lambda \frac{p}{p+q}\int_{-1}^1f(x)\vert w_{k_1}\vert^p\vert w_{k_2}\vert^q dx-\int_{-1}^1 g_1(w_{k_1},w_{k_2}) w_{k_1}dx &= I_\lambda' (\tilde w_{1},\tilde w_{2})(\tilde w_{1},0)+ o(1)
c_{A_{1}, A_{2}, . . . A_{N}}( t ) \, = \, < \, \Theta_{A_{1}}^{( 0 )}, . . . . , \, \Theta_{A_{N}}^{( 0 )}\, \exp \left[ \, \sum_{A}\, t_{A}\, \int \, \Theta_{A}^{( 2 )}\, \right] \, >
\mathcal{D}_\alpha (L^\Delta_{0,{\rm ren}}) = \mathcal{D}_\alpha(V)= \mathcal{D}_\alpha (A_0) \cap \mathcal{D}_\alpha (B) \supset\mathcal{D}_\alpha (L^\Delta_{\varepsilon,{\rm ren}}), \varepsilon \in (0,1].
\left\vert E , p_{1}^{\mathrm{m i n}}, \ldots , p_{N - 1}^{\mathrm{m i n}}\right> = \left\vert \begin{array}{c}{E , \left[ 0 \right]}\\{\left[ q \right]}\\ \end{array}\right> = \left\vert \begin{array}{c}{E , 0 , \ldots , 0}\\{q_{1}, \ldots , q_{k}, \ldots , q_{N}}\\ \end{array}\right> .
\begin{array}{c}{{A_{2 \rightarrow 2 + n}= s \frac{( s_{1}/ m^{2})^{\alpha ( q_{1}^{2})}}{( q_{1}^{2}- m^{2})}\cdots \frac{( s_{n + 1}/ m^{2})^{\alpha ( q_{n + 1}^{2})}}{( q_{n + 1}^{2}- m^{2})}}}\\{{\Gamma_{A D_{0}}^{i_{1}}\gamma_{i_{1}i_{2}}^{D_{1}}\cdots \gamma_{i_{n}i_{n + 1}}^{D_{n}}\Gamma_{B D_{n + 1}}^{i_{n + 1}}}}\\ \end{array}
g_n(X)=\sum_{i\in\mathcal{I}^-}L(A_i)\nabla\psi_{i,T}(X)\cdot\bar{\mathbf{n}}, ~~~~ g_t(X)=\sum_{i\in\mathcal{I}^-}L(A_i)\nabla\psi_{i,T}(X)\cdot\bar{\mathbf{t}}.
\Phi(\Psi(x\otimes X)) = x\otimes T_nE_{o_+}(\lambda n^{-1})X & = x\otimes j_{P'}^+(E_{o_{+,P'}}^{P'}(n\lambda n^{-1}))X & = x(n\sigma)(E_{o_{+,P'}}^{P'}(n\lambda n^{-1}))\otimes X & = x\sigma(E^P_{o_{+,P}}(\lambda))\otimes X.
{\cal C}\left[ \begin{array}{c}{{{\alpha}_{1}\ \cdots \ \alpha_{g}}}\\{{{\beta}_{1}\ \cdots \ \beta_{g}}}\\ \end{array}\right] ={\cal C}\left[ \begin{array}{c}{{{\alpha}_{1}}}\\{{{\beta}_{1}}}\\ \end{array}\right]{\cal C}\left[ \begin{array}{c}{{{\alpha}_{2}}}\\{{{\beta}_{2}}}\\ \end{array}\right] \cdots{\cal C}\left[ \begin{array}{c}{{{\alpha}_{g}}}\\{{{\beta}_{g}}}\\ \end{array}\right]
\int M d{\sigma}{\Delta_{1}}^{- 1}{\Delta_{2}}^{- 1}= 2{\pi}i D^{- 1}({\hat{q}}) ; \quad D ({\hat{q}}) = 4{\hat{\omega}}({\hat{\omega}}^{2}- M^{2}/ 4 ) ; \quad{\hat{\omega}}^{2}={m_{q}}^{2}+{\hat{q}}^{2}
c'=\begin{cases}N/c \: u_2\equiv\frac{cu_2+2}{N/c}\pmod* 2, N/(2c) \: 2\mid u_22\nmid\frac{cu_2+2}{N/c}, 2N/c \: 2\nmid u_22\mid\frac{cu_2+2}{N/c}.\end{cases}
S^{stair}(\ell) \le S(3;0,0) S(3;2,2)^{\ell-1}\le S(3;0,0) S_U(3;2,2)^{\ell-1}=: S_U^{stair}(\ell) \approx 10^{21.8241 + 11.2034 (\ell-1)}.
\widehat{c}_* = \alpha_0 \|g^{-1}\|_{L_\infty}^{-1}, &\widehat{\delta}= \frac{1}{4}\alpha_0 \|g^{-1}\|_{L_\infty}^{-1}r^2_0, &\widehat{t}^{\,0}= \frac{r_0}{2}\alpha_0^{1/2}\alpha_1^{-1/2}\left( \| h \|_{L_{\infty}}\| h^{-1}\|_{L_{\infty}}\right)^{-1}.
I\leq |\ddot{z}_1(t)|\|\sum_{j=1}^2\frac{\lambda_j i}{2\pi}\frac{1}{(\zeta(\alpha,t)-z_j(t))^2}\|_{H^s}\leq (10\epsilon+\frac{6|\lambda|}{x(t)}\epsilon) |\lambda x(0)|((s+6)!)^2 d_I(t)^{-5/2}\leq K_s^{-1}\epsilon^2 d_I(t)^{-5/2}.
C_+(z)=C_-(z)\begin{pmatrix}1 0 e^{\pi i \alpha}1\end{pmatrix}, z\in\Sigma_1, &C_+(z)=C_-(z)\begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \Sigma_2, &C_+(z)=C_-(z)\begin{pmatrix}1 0 e^{-\pi i \alpha}1\end{pmatrix}, z\in\Sigma_3, &C_+(z)=C_-(z) e^{\pi i \alpha \sigma_3}, z\in\Sigma_4.
\frac{\partial \ell_i^{(k)}}{\partial \zeta_q^{(p)}}= 2\delta_{k,p}\delta_{i,q}e^{t(2\zeta_q^{(p)}-\ell_i^{(k)})}+ O(e^{-t\delta}),\,\frac{\partial \ell_i^{(k)}}{\partial \varphi_q^{(p)}}= O(e^{-t\delta}),
S=\int d\tau\left\{p_a \dot{x}^a -\frac{\varsigma}{2}q_\alpha\dot{q}^\alpha- \frac{e(\tau )}{2}(p^2+m^2)+ \lambda (\tau)\left(\frac{\varsigma}{4}(p^a\sigma_a)_{\alpha\beta}q^\alpha q^\beta + ms\right)\right\}
\sum_{r=1}^\infty \sum_{s=1}^\infty x^r x^s x^{\max\{r,s\}}= 2 \sum_{r=1}^\infty \sum_{s=r}^\infty x^r x^s x^s - \sum_{r=1}^\infty x^r x^r x^r &= 2 \sum_{r=1}^\infty \frac{x^{3r}}{1 - x^2}- \sum_{r=1}^\infty x^{3r}= \frac{1+x^2}{1-x^2}\frac{x^3}{1-x^3},
\sqrt{e^{-2\pi t}\sin^2{\pi\sigma}+ (1 \pm e^{-\pi t}\cos{\pi\sigma})^2}= \sqrt{1 + e^{-2\pi t}\pm 2 e^{-\pi t}\cos{\pi\sigma}}\geq \sqrt{1 + e^{-2\pi t}- 2 e^{-\pi t}}= \sqrt{(1 - e^{-\pi t})^2}= 1 - e^{-\pi t}.
\mathbf{\Phi}_{{\nu}}(z)=\frac{z^{2{\nu}+1}}{2^{2{\nu}}\Gamma \left({\nu}+1\right) \Gamma \left({\nu}+2\right)}\prod_{n\geq1}\left( 1-\frac{z^{4}}{\gamma_{{\nu},n}^{4}}\right)
\cosh\Theta=\frac{1}{\sqrt{2}}\left[\frac{S'(p)}{\sqrt{{S'}^2(p)-R^2(p)}}+1 \right]^{1/2},\ \ \sinh\Theta=\frac{1}{\sqrt{2}}\left[\frac{S'(p)}{\sqrt{{S'}^2(p)-R^2(p)}}-1 \right]^{1/2}.
\left|\frac{d}{dz}\psi(z) \right| \leq 2\pi \left(e^{2\pi \cdot .865}+ \sum_{n=1}^N n|s(n)|e^{-2\pi n \cdot .865}\right) + 2\pi \sum_{n=N+1}^\infty n|s(n)| e^{-2\pi n \cdot .865}.
\nabla_uu =\nabla_{u_re_1+u_\theta e_2}(u_re_1+u_\theta e_2) &=u_r\nabla_{e_1}(u_re_1+u_\theta e_2)+u_\theta\nabla_{e_2}(u_re_1+u_\theta e_2) &=u_r(\partial_r u_r e_1+\partial_r u_\theta e_2)+u_\theta(\frac 1s\partial_\theta u_r e_1+u_r\frac cs e_2+\frac 1s\partial_\theta u_\theta e_2-u_\theta\frac cs e_1).
\sum_{t=a+m}^{n-2}\sum_{s=t+1}^{n-1}\binom{n-t+a-1}{a-1}=\sum_{t=a+m}^{n-1}\binom{n-t+a-1}{a-1}(n-t-1) &=\sum_{t=a+m}^{n}\binom{n-t+a-1}{a-1}(n-t-1)+1 &=\sum_{t=a+m}^n\binom{n-t+a-1}{a-1}(n-t+a)-(a+1)\binom{n-m}{a}+1 &= a\binom{n-m+1}{a+1}-(a+1)\binom{n-m}{a}+1=f(a,m)
\zeta_{c y l}^{D}( s ) = \frac{a^{s - 1}}{2 \sqrt{\pi}\Gamma \left( \frac{\displaystyle s}{\displaystyle 2}\right) \Gamma \left( \frac{\displaystyle 3 - s}{\displaystyle 2}\right)}\sum_{n = - \infty}^{\infty}\int_{0}^{\infty}d y y^{1 - s}\frac{d}{d y}\ln \left[ 2 y I_{n}( y ) K_{n}( y ) \right] .
(1-r_L)^{mL-n+j-1}r_L^{n-j}=L^{-\frac{n+1}L (mL-n+j-1)}\left(\frac{n+1}L \log L+ \textrm{o}(L^{-n})\right)^{n-j}&=\textrm{O}\left(\frac{\log^{n+j}L}{L^{(n+1)m+n+j}}\right) \ .
\begin{array}{c c c}\hline{{\hline}}&{{}}&{{}}\\{{M 2^{e}\mapsto M 5^{m}}}&{{\qquad d * F_{4}= F_{4}\wedge F_{4}}}&{{\qquad d \star{\cal G}_{3}= d{\cal G}_{3}= - \hat{F}_{4}^{( 6 )}}}\\{{}}&{{}}&{{}}\\ \hline{{\hline}}\\ \end{array}
-\frac{1}{N}\int{(y - \eta)\cdot \nabla_Y \bar{f}(y) \bar{\mu}(dy)}&= \frac{1}{N}\int{(\nabla_Y \cdot y)\bar{f}(y)\bar{\mu}(dy)}- \int{(y - \eta) \cdot \nabla_Y \bar{H}(y) \bar{\mu}(dy)}&= \frac{M-1}{N}- \int{(y - \eta) \cdot \nabla_Y \bar{H}(y) \bar{\mu}(dy)}.
\sum_\pi q^{\vert\pi\vert}=\prod_{i=1}^{a}\frac{1-q^{3i-1}}{1-q^{3i-2}}\prod_{1\le i<j\le a}^{}\frac{1-q^{3(2i+j-1)}}{1-q^{3(2i+j-2)}}\prod_{1\le i<j,k\le a}^{}\frac{1-q^{3(i+j+k-1)}}{1-q^{3(i+j+k-2)}},
\mu(A) =\mu\left(\bigcap_{n=1}^r\{\omega\in\Omega:\, \omega_{k_n}\in(\bar\omega_{p_n}-\epsilon\, b_n^{-1}, \bar\omega_{p_n}+\epsilon\, b_n^{-1})\}\right) &=\prod_{n=1}^r\nu (\bar\omega_{p_n}-\epsilon\, b_n^{-1}, \bar\omega_{p_n}+\epsilon\, b_n^{-1}) &>0.
\widehat{(\mathcal{P}_k Q_k)}(\xi) =(|\xi+k|^{2\sigma}-|k|^{2\sigma}-2\sigma |k|^{2\sigma-2}k\cdot\xi) |k|^{\sigma-1}\hat{Q}_1(\tfrac{\xi}{k}) &=|k|^{3\sigma-1}(|\tfrac{\xi}{k}+1|^{2\sigma}-1-2\sigma\tfrac{\xi}{k})\hat{Q}_1(\tfrac{\xi}{k}) &=|k|^{3\sigma-1}(\mathcal{P}_1Q_1)^\wedge(\tfrac{\xi}{k}).
E_{-}(a_{i}^{*},z) =\exp\left(\sum_{r=1}^{\infty}\frac{q^{r/2}}{[r]}a_{i}^{*}(-r)z^{r}\right), E_{+}(a_{i}^{*},z) =\exp\left(-\sum_{r=1}^{\infty}\frac{q^{r/2}}{[r]}a_{i}^{*}(r)z^{-r}\right).
c ( 1 ) = 4 [ ( p_{1}\! \cdot \! p_{2}) ( \epsilon_{1}\! \cdot \! \epsilon_{2}) ( p_{3}^{+}p_{4}^{-}p_{5}^{+}p_{6}^{-}) ( \epsilon_{3}^{-}\epsilon_{4}^{+}\epsilon_{5}^{-}\epsilon_{6}^{+}) ] .
Q^{k_{1}}Q^{k_{2}}Q^{k_{3}}= \varrho_{k_{1}k_{2}k_{3}}( \sigma ) \, Q^{k_{\sigma ( 1 )}}Q^{k_{\sigma ( 2 )}}Q^{k_{\sigma ( 3 )}}+ 3 \,{\eta}^{k_{1}k_{2}k_{3}}\, 1 ,
d_{n,t}+d_{n,x}-b_1d_{n,txx}+a_1d_{n,xxx}+bd_{txxxx}+ad_{xxxxx}= -\sigma(I-b_1\partial_{x}^{2}+b\partial_{x}^{4})[\sigma d_n]-\frac{3}{2}\alpha_nv_nv_{n,x}-\gamma\alpha_n(v_{n}^{2})_{xxx}+\frac{7}{48}\alpha_n(v_{n,x}^{2})_x+\frac{1}{8}\alpha_{n}^{2}(v_{n}^{3})_x d_n(0)=0.
\begin{array}{ll}\Delta(\widehat{x}^\mu) = 1\otimes\widehat{x}^\mu \Delta(\widehat{p}_k) = \widehat{p}_k\otimes e^{\-\frac{\widehat{p}_0}\kappa}+ I\otimes\widehat{p}_k, \Delta(\widehat{p}_0) = \widehat{p}_0\otimes I + I\otimes\widehat{p}_0.\end{array}
\sum_{j=1}^n\begin{pmatrix}b_j^\dagger b_{\kappa(j)}\end{pmatrix}\begin{pmatrix}\epsilon\mu_j-1 -\epsilon\mu_j \epsilon\mu_{\kappa(j)}-\epsilon\mu_{\kappa(j)}+1\end{pmatrix}\begin{pmatrix}b_j b_{\kappa(j)}^\dagger\end{pmatrix}
\beta_{i,j}= \begin{cases}\pm N([c(g_i)_{\pm}]) \; j = j_{\pm}(i) \; \; j_{-}(i) \not= j_{+}(i); N([c(g_i)_{+}]) - N([c(g_i)_{-}]) \; j = j_{-}(i) = j_{+}(i); 0 \; j \notin \{j_{-}(i), j_{+}(i)\}. \end{cases}
\widehat{I}\left(\alpha^{(0)},\alpha^{(1)}\right)=\dfrac{\sum\limits_{l_0=2}^N\sum\limits_{l_1=2}^N \widehat{A}(l_0,l_1)\alpha_{l_0}^{(0)}\alpha_{l_1}^{(1)}}{\sum\limits_{l_0=2}^N\sum\limits_{l_1=2}^N \widehat{B}(l_0,l_1)\alpha_{l_0}^{(0)}\alpha_{l_1}^{(1)}},
\limsup_{n\to\infty}\angle^k_{\gamma(t_n)}(\gamma(0),s_n) \leq \limsup_{n\to\infty}\angle_{\gamma(t_n)}(\gamma|_{[t_n,0]}, \sigma_{t_n}) = \pi - \liminf_{n\to\infty}\angle_{\gamma(t_n)}(\gamma|_{[t_n,T]},\sigma_{t_n}) \leq \pi - \angle (\gamma,\sigma_0)
\|b\|_{L^1}=\|\Lambda_1(b)\|=\max_{x\in X}\sum_{t\in G}|b(t)(x)|, \|b\|_{L^\infty}=\|\Lambda_{\infty}(b)\|=\max_{x\in X}\sum_{t\in G}|b(t)(\varphi_{t}(x))|,
\frac{1}{|{\bf x - x^{\prime}}|}= 4 \pi \sum_{l = 0}^{\infty}\sum_{m = - l}^{l}\frac{1}{( 2 l + 1 )}\frac{r^{\prime l}}{r^{l + 1}}Y_{l m}^{*}( \theta^{\prime}, \phi^{\prime}) Y_{l m}( \theta , \phi ) ,
G_{M N}^{( 1 1 )}= \left( \begin{matrix}{{e^{-{\frac{\hat{\Phi}}{3}}}G_{\hat{\mu}\hat{\nu}}^{( 1 0 )}}}&{{e^{{\frac{2}{3}}\hat{\Phi}}B_{M}}}\\{{e^{{\frac{2}{3}}\hat{\Phi}}B_{N}}}&{{e^{{\frac{2}{3}}\hat{\Phi}}}}\\ \end{matrix}\right) ,
+ \; \frac{1}{2}\frac{1}{( 2 \pi )^{2}}\Big ( m c \Big )^{2}\int_{\Lambda}d^{2}x \; d^{2}y \bigg [ 2 e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) - \delta ( y ) , C_{m_{d}}\alpha \big )}
\begin{matrix}A^{\prime}=&0&:&g(b,c,a)&:&-g(c,b,a)\\B^{\prime}=&-g(a,c,b)&:&0&:&g(c,a,b)\\C^{\prime}=&g(a,b,c)&:&-g(b,a,c)&:&0\end{matrix}
P_{(\hat{e},r)}(A) = \frac{1}{2}( \delta_{\hat{e}}\otimes\mu + \delta_{-\hat{e}}\otimes\delta_{0})(A) , P_{(-\hat{e},r)}(A) = \frac{1}{2}( \delta_{-\hat{e}}\otimes\mu + \delta_{\hat{e}}\otimes\delta_{0})(A).
\mathcal{L}_6:=\mathcal{S}^{-1}\mathcal{L}_4 \mathcal{S}=\Pi_S^{\perp}(\mathcal{D}_{\omega}+m_3 \partial_{xxx}+m_1 \partial_x)\Pi_S^{\perp}+R_6, R_6:=\mathcal{S}^{-1}\tilde{R}_6.
\tilde{\Psi}_{y}= \tilde{\Psi}_{x x}- 2 ( \Psi \Psi^{*}) \tilde{\Psi}_{x}+ \alpha ( -{\theta}_{y}+ \alpha{\theta}_{x}^{2}+{\theta}_{x x}- 2 \theta_{x}( \tilde{\Psi}\tilde{\Psi}^{*}) ) \tilde{\Psi}+ 2 \alpha{\theta}_{x}\tilde{\Psi}_{x},
\begin{aligned}\mathbb{E}\bigg[\sup\limits_{t\in[0,T]}|Y(t)|^\gamma+\Big(\int_0^T|Z(t)|^2dt\Big)^\frac{\gamma}{2}\bigg] \leq C\mathbb{E}\Big[ |\xi|^\gamma+\Big(\int_0^T|F(t,0,0,\alpha(t-))| dt\Big)^\gamma\Big]. \end{aligned}
\epsilon_{\mu}^{2}(n)=\frac{1}{\sqrt{2}}(\frac{in_{1}n_{2}+1+n_{3}-n_{1}^{2}}{1+n_{3}},\frac{-n_{1}n_{2}-in_{1}^{2}-in_{3}^{2}-in_{3}}{1+n_{3}},-n_{1}+in_{2},0).
\begin{array}{c}a \bar{a}+ \bar{a}a = 1 + c_{1}N , \quad c_{1}\in{\bf C}, begin{align*}4mm]\varphi (n) = (-)^{n-1}\{(1- c_{1}q\frac{d}{dq})\frac{q^{-n}- 1}{q^{-1}-1}\}_{q=-1}begin{align*}4mm]= (-)^{n-1}\{S_{0}(n,-1) + c_{1}S_{1}(n,-1)\}= \frac{1}{2}c_{1}n +[n]_{-1}(1 - \frac{1}{2}c_{1}) . \end{array}
\zeta^{\prime}\left( 0 | A \right) = \frac{1}{4 \pi}\int \left[ d^{2}z \right] \, t r^{\prime}[ k_{1}\left( z \right) ] \qquad , \qquad \left[ \frac{d}{d s}\zeta^{\prime}\left( s | A \right) \right]_{s = 0}= \gamma \zeta^{\prime}\left( 0 | A \right)
\displaystyle\widetilde{\cal H}_{\scriptscriptstyle K}=\frac{1}{(k+1)!}\Biggl(\lambda_jK^j_{j_1j_2\cdots j_{k+1}}+i\bar{c}_j(\partial_dK^{j}_{j_1j_2\cdots j_{k+1}}c^d)\Biggr)c^{j_1}\cdots c^{j_{k+1}}
\sum_{a=0}^c\binom{p-1-a}{c-a}^2+\binom{2c}c \equiv_{p^2}\sum_{a=0}^c\binom{c}a^2(1-2p(H_c-H_a))+\binom{2c}{c}&=2\binom{2c}c(1-pH_c)+2p\sum_{a=0}^c\binom{c}a^2H_a &=2\binom{2c}c\left(1-pH_c+p(2H_c-H_{2c})\right) &\equiv_{p^2}2\binom{2c}c(1-pH_{2c})\equiv_{p^2}2\binom{2c}c(1-1)=0,
{n;b;\omega_n^{g'}\brack k}=\sum_{\substack{0 \le i \le \lfloor \frac{n-k}{2}\rfloor n/(g',n) \mid i}}\binom{(g',n)}{i/(n/(g',n))}{n-i \brack i+k}_{\omega_{n-i}^{g'(n-i)/n}}.
\displaystyle g(\lambda |\lambda_{j})=\sum^{N_{H}}_{k=1}\chi (\lambda -h_{k})-2\sum^{N_{S}}_{k=1}\chi (\lambda -y_{k})-\sum^{M_{C}}_{k=1}\chi (\lambda -c_{k})-\sum^{M_{W}}_{k=1}\chi (\lambda -w_{k})_{\mathrm{II}}\: ,
\sum_{i=1}^{\infty}w_i\leq \sum_{i=1}^{\infty}2^{2(\alpha- \alpha_i)}w_i \ , \sum_{i=1}^{\infty}2^{-2\alpha_i}w_i \geq \sum_{i=1}^{\infty}2^{2(\alpha- 2\alpha_i)}w_i
\vec{X}_{11}= \Bigg( 0, 1, - \frac{\lambda}{m-n}\bigg(\frac{1}{1+ B^{-1}}\bigg)\Bigg), \vec{X}_{12}'= \Bigg(1 \;,\; -\lambda c - 2\lambda n \frac{e}{1+B^{-1}}\;,\; 0\Bigg).
\xi^{m *}( x ) = \pi_{m - 6}^{\perp}( x ) + 2 i e \int d^{\, 3}y \, \partial_{m - 6}^{x}\nabla^{- 2}\pi_{\psi}( y ) \, \psi ( y ) \; , m = 7 , 8 , 9 \; .
\alpha_M\circ \beta_M=\beta_M\circ \alpha_M, \alpha_M(a\cdot m)=\alpha_A(a)\cdot \alpha_M(m), \beta_M(a\cdot m)=\beta_A(a)\cdot \beta_M(m), \alpha_A(a)\cdot (a^{\prime}\cdot m)=(aa^{\prime})\cdot \beta_M(m).
\gamma_{\mathbf{n};j}^{-}\left(\mbox{\boldmath$ \psi$}^{-}\right) =\left\langle \left.\mbox{\boldmath$ \psi$}^{-}\right\vert_{\Gamma_{j}},\mathbf{n}_{j}\right\rangle \quad\gamma_{\mathbf{n};j}^{+}\left(\mbox{\boldmath$ \psi$}^{+}\right) =\left\langle \left.\mbox{\boldmath$ \psi$}^{+}\right\vert_{\Gamma_{j}},-\mathbf{n}_{j}\right\rangle ,
L_{2,n}^{2,k}\cong_{ht}\begin{cases}S^2 \textnormal{if}0< k < n \bigvee_{i=1}^{e-1}S^1 \textnormal{if}k=n e \textnormal{points}\textnormal{if}k = n+1 \emptyset \textnormal{otherwise,}\end{cases}
p(\bar{H}_{(m,n)}) =\int_{-\infty}^{\infty}p(\bar{H}_{(m,n)}|\gamma_{(m,n)})p(\gamma_{(m,n)})d\gamma_{(m,n)}=\frac{a}{\pi b}\left(\frac{||\bar{H}_{(m,n)}||^{2}}{b}+1\right)^{-(a+1)}.
\prod_{j=2}^{2i-2}d_j=d_2\prod_{j=3}^{2i-2}d_j \leq \frac{2}{5}n\prod_{j=1}^{i-2}(d_{2j+1}d_{2j+2}) \leq \frac{2}{5}n\prod_{j=1}^{i-2}\left(\left\lfloor \frac{n-2j-1}{2}\right\rfloor \cdot \left\lceil \frac{n-2j-1}{2}\right\rceil\right) & =\frac{2}{5}n\prod_{j=1}^{i-2}\left(\frac{n-2j-1}{2}\right)^2.
\Gamma\left(-N-\frac12,w\right)=\frac{(-1)^N}{\Gamma\left(N+\frac32\right)}\left[-\pi\,\mathrm{erfc}\left(\sqrt{w}\,\right)+e^{-w}\sum_{l=0}^N (-1)^l\Gamma\left(l+\frac12\right)w^{-l-\frac12}\right]\,,
\Bigg\langle x_1,x_2,x_3,y_1,y_2,y_3~\mid~x_1x_2x_1x_2=x_2x_1x_2x_1, x_1x_3=x_3x_1, x_2x_3x_2=x_3x_2x_3, &\hskip3mm y_1y_2y_1=y_2y_1y_2, y_1y_3=y_3y_1, y_2y_3y_2=y_3y_2y_3, (x_1x_2x_3)^6=(y_1y_2y_3y_1y_2y_1)^3\Bigg\rangle
\begin{array}{rclrclrclrcl}A = T_a A^a, \quad \hat{\psi}= T_a \hat{\psi}^a, \quad \psi = \Sigma_\alpha \psi^\alpha, \quad \hat{A}= \Sigma_\alpha \hat{A}^\alpha, \hat{a}= T_a \hat{a}^a, \quad \alpha = T_a \alpha^a, \quad \hat{\alpha}= \Sigma_\alpha \hat{\alpha}^\alpha, \quad a = \Sigma_\alpha a^\alpha. \end{array}
1 = \sum_{l \in \bf{N}_{0}}\tilde{\epsilon}_{l}\frac{( ( l + 3 ) ! )^{2}}{( 2 l + 7 ) !}( v - 1 )^{l}\,_{2}F_{1}( l + 4 , l + 4 ; 2 l + 8 ; 1 - v )
\begin{aligned}\sum\limits_{1\leq i,j\leq n}a^{ij}\Big(\sum\limits_{t\in I}k_{it}\sigma_{t}-2\alpha_i\Big)\Big(\sum\limits_{s\in I}k_{js}\sigma_s-2\alpha_{j}\Big)-4\sum\limits_{1\leq i,j\leq n}a^{ij}\alpha_i\alpha_j-4\sum\limits_{i\in I}\sigma_{i}=0.\end{aligned}
Z [ \bar{\eta}, \eta ] = \int D A ~ e^{- S^{\prime}[ A ]}~ e x p{[ - \int d^{2}x \bar{\Psi}( i \raise . 1 5 e x \mathrm{~ / ~}\kern - . 5 7 e m \mathrm{~ \partial ~}+ \not \! \! A ) \Psi ]}~ e x p{[ - \int d x ( \bar{\eta}\Psi + \bar{\Psi}\eta ) ]}
\langle 1 \cdots M|\mathcal{B}^\prime(z_1) \cdots\mathcal{B}^\prime(z_N)| \overline{x_1}\cdots \overline{x_N}\rangle=(-1)^{N(N-1)/2}\mathrm{det}_N(z_j^{\lambda_k+N-k+1}-z_j^{-\lambda_k-N+k-1}).
\sup\limits_{I^\theta_{r_0^{\,k+1}}\times B_{r_0^{\,k+1}}}{v}(t,x) = \sup\limits_{I^\theta_{r_0^{\,k}}\times B_{r_0^{\,k}}}\tilde{v}(t,x) \, r_0^\frac{2}{m}\leq r_0^{\,(k+1)\frac{2}{m}}
\beta_n(z) = \sum_{n \geq m_{2k-1}\geq \ldots \geq m_1 \geq 0}\frac{q^{\sum_{\nu=1}^{k-1}(m_{k+\nu}^2+m_{k+\nu}) + \binom{m_k+1}{2}- \sum_{\nu=1}^{k-1}m_\nu m_{\nu+1}- \sum_{\nu=1}^{k-\ell}m_\nu}(-z)^{m_k}}{(q)_{m_{2k}-m_{2k-1}}(q)_{m_{2k-1}-m_{2k-2}}\cdot\ldots\cdot (q)_{m_2-m_1}(q)_{m_1}},
g(x_i) =a_i+ \sum_{s=1}^n b_{is}x_s+\sum_{t=1}^n c_{it}y_t=a_i+X_i, g(y_i) =d_i+ \sum_{s=1}^n e_{is}x_s+\sum_{t=1}^n f_{it}y_t=d_i+Y_i,
W_n=\bigcup_{w\in \Sigma_{\beta}^n}\bigcup_{0\leq k\leq \left\lfloor\frac{\beta^n}{\varphi(n)}\right\rfloor}\left\{(x,y)\in I_{n,\beta}(w)\times J_{n,\beta}(k): |T_{\beta}^nx-f(x,y)|<\varphi(n)\right\}.