harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\Lambda}= \int d^{2}x d u \left({\frac{i}{2}}\lambda_{+}^{a}\partial_{--}\lambda_{+}^{a}+ i \sigma_{-}^{-- a}\partial^{+ +}\lambda_{+}^{a}+{\frac{1}{4}}g^{- a A^{\prime}}\partial^{+ +}g_{A^{\prime}}^{- a}\right) .
X = \begin{pmatrix}0 q_1 &r_1 q_2 & r_2 \ddots & \ddots q_{n-1}& r_{n-1}\end{pmatrix},Y = \begin{pmatrix}0 r_1 q_1 &r_2 q_2 & \ddots \ddots & r_{n-1}q_{n-1}\end{pmatrix}.
q^u_{n+1}=\min\{e^{\epsilon}q^u_n,e^{-\epsilon/3}Q_\epsilon(f^{n+1}(x))\}&\geq \min\{e^{\epsilon}e^{-\epsilon}p^u_n,e^{-2\epsilon/3}Q_\epsilon(x_{n+1})\}, \textrm{by (b$_{n+1}$) with $i=0$}&\geq e^{-\epsilon}\min\{e^{\epsilon}p^u_n,Q_\epsilon(x_{n+1})\}=e^{-\epsilon}p^u_{n+1}.
\Gamma_3 =\{(z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3):\,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3 \mathbb G_3 =\{(z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|< 1,i=1,2,3 \}\subseteq \Gamma_3.
d^n(f)(g_1,\ldots, g_{n+1})= g_1f(g_2,\ldots, g_{n+1})+ \sum_{i=1}^{n}(-1)^{i}f(g_1,\ldots, g_{i-1}, g_ig_{i+1}, \ldots, g_{n+1})+ (-1)^{n+1}f(g_1,\ldots, g_n).
\frac{\partial}{\partial q^i}\, \bigg( X^j \; \frac{\partial H}{\partial q^j}\; + \; X_j^\nu \; \frac{\partial H}{\partial p\>\!_j^\nu}\, \bigg)~ = \, - \, \frac{\partial}{\partial x^\mu}\left( X_i^\mu \, + \, X^\mu \; \frac{\partial H}{\partial q^i}\right) .
s(f,\Psi,{\mathcal P}_n^1)= \sum_{k}f(x_k^{n,1})\,|I_k^{n,1}|_\Psi =\sum_{k}f(x_k^{n,1})\int_{I_k^{n,1}}\big(\psi-\psi(x_k^{n,1})\big) +\sum_{k}f(x_k^{n,1}) \psi(x_k^{n,1})\,|I_k^{n,1}| &=A_n+B_n,
\begin{bmatrix}2x_0^2 bx_3^2 a x_1^2 a x_4^2 b x_2^2 b x_3^2 2x_1^2 b x_4^2 a x_2^2 a x_0^2 a x_1^2 b x_4^2 2 x_2^2 b x_0^2 a x_3^2 a x_4^2 a x_2^2 b x_0^2 2 x_3^2 b x_1^2 b x_2^2 a x_0^2 a x_3^2 b x_1^2 2 x_4^2\end{bmatrix}
\int\limits_{\mathbb{R}^{N}}\left\vert V_{0}\right\vert^{q}\frac{dx}{\left\Vert x\right\Vert^{\theta}}=d^{1+\frac{p-1}{p}q}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert U_{0}\right\vert^{q}}{\left\Vert x\right\Vert^{N+\theta d-Nd}}dx
V_{3,i}= (2\pi)^3 r_{m_i}r_{m'_i}r_{m''_i}\ ,\quad\mathbf{\omega}_{{\mathbf 3},i}= \frac{1}{V_{3,i}}\mathbf{dy}^{m_i}\wedge \mathbf{dy}^{m'_i}\wedge \mathbf{dy}^{m''_i}\ .
\sum_{j=1}^{p-1}\omega_q^{\left(a^{t_1}_{\sigma^j, r}-a^{t_2}_{\sigma^j, s}\right)-\left(a^{t_1}_{\sigma, r}-a^{t_2}_{\sigma, s}\right)}=\sum_{j=1}^{p-1}\omega_p^{j\left({r}_{\pi_\beta(1)}-{s}_{\pi_\beta(1)}\right)}=-1.
\Gamma_{e.f.}= \int \{Y_1 (-{e}z_e \sqrt{\frac{z_2}{z_1}}\sqrt{z_A}\bar{\varphi}_2 c + x_1 q_1) \; + \;Y_2 ({e}z_e \sqrt{\frac{z_1}{z_2}}\sqrt{z_A}(\bar{\varphi}_1 + \frac{\sqrt{z_m}m}{\sqrt{z_1}{z_e}e}) c + x_2 q_2) \}
\frac{1}{2}\frac{d}{dt}| [\theta - M_0]^+|_{L^2(\Omega)}^2 + \frac{\kappa}{2}| \nabla [\theta - M_0]^+|_{L^2(\Omega)}^2 \leq \ \frac{\tau^2 \sqrt{N}}{2 \kappa}\sum_{i=1}^N | \nabla^{\delta_0}\hat{u}_i|_{L^{\infty}(Q(T))}| [\theta - M_0]^+|_{L^2(\Omega)}^2 \mbox{a.e. on}[0,T].
\bar{\textbf{u}}= \bar{B}\left(\begin{array}{c}1 \dfrac{k_{E}k_{D}}{\epsilon k_{B}}\dfrac{\alpha k_{D}}{k_{N}}0\end{array}\right) = \frac{a}{1+\frac{\alpha k_{D}}{k_{N}}+ \frac{k_{D}k_{E}}{\epsilon k_{B}}}\left(\begin{array}{c}1 \dfrac{k_{E}k_{D}}{\epsilon k_{B}}\dfrac{\alpha k_{D}}{k_{N}}0\end{array}\right),
\Phi_{8}( \psi )^{\ddagger}= \Phi_{8}( \psi ) \quad \quad \Phi_{\alpha}( \psi )^{\ddagger}= \sum_{\beta = 6 , 7}\tilde{C}_{\alpha \beta}\, \Phi_{\beta}( \psi ) \, , \quad \quad \alpha = 6 , 7
\sum_{n \le X^{\theta}}\frac{d_k^2(n)}{n^{1+\frac1k}}\sin^2\Big(\pi \frac{\sqrt[k]{n}}{L}\Big) =\int_1^{X^{\theta}}\frac{Q_{k^2-1}(\log x)+Q_{k^2-1}'(\log x)}{x^{1+\frac1k}}\sin^2\Big(\pi \frac{\sqrt[k]{x}}{L}\Big) \, dx+O\Big(\frac{1}{L}\Big).
\Phi_3(b;\,c;\,x,\,y)= \exp \left( x+\frac{y}{x}\right)\,\sum_{k=0}^\infty\,\frac{1}{k!}\, \left(-\frac{y}{x}\right)^k{}_2F_1\left[ \begin{array}{c}-k,\,-k-b+1 c \end{array};\,\frac{x^2}{y}\right] \, .
\Delta_\Sigma r = (4 - 2 \, |\nabla_\Sigma r|^2) e^{- 2 \, r}- (H - 2) + (H - 2) \, (1 - \langle \nu, \nabla r \rangle) + (1 - \langle \nu, \nabla r \rangle)^2 + O (e^{- 3 \, r}).
P^{(\mathcal{C}_n)}_{\mathbf{Y}_2|M_p=1,M_1=1,M_{22}=1,\mathbf{U}_0=\mathbf{u}_0,\mathbf{U}_2=\mathbf{u}_2}=Q^n_{Y_2|U_0,U_2}\big(\mspace{-1.5mu}\cdot\mspace{-1.5mu}\big|\mathbf{u}_0,\mathbf{u}_2\big).
\mathcal{D}_{\theta}\overset{}{=}{\displaystyle\bigotimes\limits_{k=1}^{n}}\mathcal{I}_{\theta^{k}}=\left( \mathcal{I}_{\theta^{1}}\otimes\mathcal{I}_{\theta^{2}}\otimes\mathcal{I}_{\theta^{n}}\right)\subseteq\mathbb{R}^{n}
\sup_{f \in \mathcal{F}_{d,\theta}}\bigl| \mathbb{E}_f(\hat{H}_n^w) - H(f)\bigr| = O \biggl( \max \biggl\{\frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\frac{2( \lfloor d/4 \rfloor +1)}{d}}}{n^{\frac{2( \lfloor d/4 \rfloor +1)}{d}}}\, , \, \frac{k^{\frac{\beta}{d}}}{n^{\frac{\beta}{d}}}\biggr\}\biggr),
\bigl\|f(e^{i Z}U) -f(U)-\frac{d}{dt}\bigl(f(e^{it Z}U)\bigr)_{\vert t=0}\bigr\|_1 & =\sum_{n=1}^\infty\, \bigl\|f(e^{i \tilde Z_n}\tilde U_n) -f(\tilde U_n)-\frac{d}{dt}\bigl(f(e^{it \tilde Z_n}\tilde U_n)\bigr)_{\vert t=0}\bigr\|_1\,.
\int_{-\infty}^\infty \!\widehat{h}(k)\widehat{h}(k)^\dag dk\!+\!\int_{-\infty}^\infty \!\widehat{h}(-k) [S(k)\!-U_0]\widehat{h}(k)^\dag dk &+2\pi\sum_{j=1}^N [\widehat{h}(-ik_j)C_j][\widehat{h}(-ik_j)C_j]^\dag\!=0.
E_{6}( n ) = - ( 1 + 2 n ) ( 6 5 5 1 8 4 0 1 + 1 4 6 3 3 8 8 9 5 n + 2 1 3 1 7 2 4 3 0 n^{2}+ 1 3 9 9 3 1 8 6 8 n^{3}+ 8 5 6 2 7 9 2 9 n^{4}
\beta(t, u^{1}, \dotsc, u^{m-d-1}) ={}\gamma(t) + u^{1}X_{1}(t) + \dotsb + u^{m-d-1}X_{m-d-1}(t) &+ u^{m-d}(t, u^{1}, \dotsc, u^{m-d-1}) X_{m-d}(t) + \dotsb &+ u^{m-1}(t, u^{1}, \dotsc, u^{m-d-1}) X_{m-1}(t).
\bar D_{3,a,2}= \frac{|X_1-X_2|}{|z|}| ( D_t Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,z+t_0(X_\tau))) &\qquad- D_t Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,t_0(X_\tau))) ) D_X t_0(X_\tau)|
h\beta_1\nu_{\delta}\varpi_p = \left(\begin{array}{cc}t^{-1}\det(h) p^{-1}\ast 0 -t \end{array}\right)\left(\begin{array}{cc}\delta 1 1 - \delta t^{-1}p^{-1}z -t^{-1}p^{-1}z \end{array}\right).
\partial^+\mathcal{O}_{(x,0), \rho}:= \{(x_1, x_2) \in \partial\mathcal{O}_{(x,0), \rho}: x_2= (|x_1|, \mathcal{O}_{(x,0), \rho}), 2/3 \le x_1 \le 1\}\cap \{(x_1, x_2) \in \partial\mathcal{O}_{(x,0), \rho}: x_2 \ge 3 (1, \mathcal{O}_{(x,0), \rho})x_1 - 2(1, \mathcal{O}_{(x,0), \rho}) \}.
L_1\,=\, L_0 \,+\,\dot\lambda_1^i \lbrace{( X^i_1 - X^i_0 ) \over \epsilon}\, - \, 2 \pi \alpha^\prime\, M^{-1}_{ij}{\cal B}^{jk}P_0^k \,\rbrace\,+\,\dot\lambda_3^i \lbrace{( X^i_N - X^i_{N-1}) \over \epsilon}\, - \, 2 \pi \alpha^\prime\, M^{-1}_{ij}{\cal B}^{jk}P_{N-1}^k \,\rbrace
\frac{\partial t^*_{\alpha'}(\theta)}{\partial\theta}= \biggl\{\frac{\beta(({\partial F_0( t, \theta )})/({\partial\theta}))-(({\partial F_1( t, \theta )})/({\partial\theta}))}{(({\partial F_1 ( t, \theta )})/({\partial t}))- \beta(({\partial F_0( t, \theta )})/({\partial t}))}\biggr\}\bigg|_{t=t^*_{\alpha'}(\theta)}.
\begin{aligned}\frac{1}{2C}\int |\nabla u^k|^2 dx-\int |\nabla d^k|^2 |\nabla u^k|dx+\frac{1}{4}\int_0^t\int \rho^k |u_t^k|^2dxd\tau\le C+C\int_0^t \Phi^{N_{5}}_K d\tau\end{aligned}
- \int d^{1 0}\sigma \sqrt{- \mathrm{d e t}\big ( \eta_{\mu \nu}+ F_{\mu \nu}- 2 \bar{\lambda}\Gamma_{\mu}\partial_{\nu}\lambda + \bar{\lambda}\Gamma^{\rho}\partial_{\mu}\lambda \bar{\lambda}\Gamma_{\rho}\partial_{\nu}\lambda \big )}.
\begin{aligned}\|d(\nabla\varphi,SO(n))\|_{L^{p(\cdot)}(Q')}\le \|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(Q')}+\|\nabla \psi\|_{L^{p(\cdot)}(Q')}&\le C\|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(Q)}\end{aligned}
X^{(d)}_{M,h}(n;z)=2(d-1)\delta_{h,0}+\frac{2n\bigl(-(2d-1)\bigr)^{\frac{n-h}{2}}}{n+h}\binom{h}{z_1,\ldots,z_d}P^{(z,-1)}_{d,\frac{n-h}{2}}\Bigl(\frac{2d-3}{2d-1}\Bigr),
G(\psi^{k}(z_0,t_0)) = \big(w(\psi^{k}(z_0,t_0)), s(\psi^{k}(z_0,t_0))\big) = \big(w(z_k), s(\psi^{k}(z_0,t_0))\big) \;, \psi'^k(G(z_0,t_0) = \psi'^k(w_0,s_0) = (\mu'^{-k/2}U'^k(w_0),\mu'^{-k}s_0) \;.
|I_2(D^{\beta_\perp}\sigma_\rho \chi_{\cup_{i=0}^{j-1}{\bf C}_i^\rho})(x)| \le C \rho^{-2-k_\perp}\|\sigma\|_{H^1(\Lambda)}\sum_{i=j+1}^{J}(2^i\rho)^{-1}(2^i\rho) \le C \rho^{-2-k_\perp}\|\sigma\|_{H^1(\Lambda)}J \le C \rho^{-2-k_\perp}|\log(\rho)|\|\sigma\|_{H^1(\Lambda)},
|S_{f}(x';Q,A) - S_{f}(x;Q,A)|= O\Bigg( \frac{\varphi_C (x)}{\log x}\frac{Q}{\phi(Q)}\prod_{\substack{p \leq x, \,p\nmid Q}}\left(1 + \frac{|f(p)|}{p}\right) \Bigg)
\begin{array}{llllll}p_i = (S^{-1})^j_i\pi_j p{_i}^{ab}= (S^{-1})^j_i\pi{_j}^{ab}p{_i}^{ak}= (S^{-1})^j_iS^k_\ell \pi{_j}^{a\ell}p{_i}^{jk}= (S^{-1})^{\ell}_iS^j_{m}S^k_{n}\pi{_\ell}^{mn}\end{array}
\P(R_n> x_n) =\P\left(M_n R_{n-1}>(1-\delta_n)x_{n-1}\right) &\geq \P(M_n> 1-\delta_n)\P(R_{n-1}>x_{n-1}) &\geq \prod_{i=1}^{n}\P(M>1-\delta_i).
N\leq C\sum\limits_{k=1}^{\infty}\frac{e^{-C2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|(b(z)-b_B)_{\sigma}||f(z)|dz &=C\sum\limits_{k=1}^{\infty}\frac{e^{-C2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|b_{\sigma_1}(z)-(b_{\sigma_1})_B|\cdots|b_{\sigma_j}(z)-(b_{\sigma_j})_B||f(z)|dz
G_2(x;d) =\frac{x^{2+d}(F_T(x)-1)C(x)}{(1-x)^d\big(1-xC(x)\big)}+\left(\sum_{m=2}^d\frac{x^{3+d}C(x)}{(1-x)^m\big(1-xC(x)\big)}\right)+\frac{x^{2+d}C(x)}{(1-x)\big(1-xC(x)\big)^d}&= \frac{x^{2+d}(F_T(x)-1)C(x)^2}{(1-x)^d}+\left(\sum_{m=2}^d\frac{x^{3+d}C(x)^2}{(1-x)^m)}\right)+\frac{x^{2+d}C(x)}{(1-x)\big(1-xC(x)\big)^d}
U_{2n,N-x}= U_{2n,N+x}= \frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x);\gamma,\delta,N), x=1,\ldots,N, U_{2n+1,N-x}= -U_{2n+1,N+x}= -\frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x-1);\gamma+1,\delta+1,N-1), x=1,\ldots,N, U_{2n,N}= (-1)^n \tilde R_n(\lambda(0);\gamma,\delta,N),U_{2n+1,N}=0,
\begin{matrix}{{\zeta^{+}\wedge E^{z}= 0 , \, \,}}&{{\quad \zeta^{-}\wedge E^{z}= 0 , \, \,}}&{{\quad \tilde{\zeta}_{+}\wedge E^{\bar{z}}= 0 , \, \,}}&{{\quad \tilde{\zeta}_{-}\wedge E^{\bar{z}}= 0 ,}}\\ \end{matrix}
a^{i+1}=\widetilde{a^{i}}\vDash n-i-1, l_{i+1}=\begin{cases}l(\mu^{i},a^{i},1) i\in A, l(\mu^{i},a^{i},l_{i}) \end{cases}\mu^{i+1}=(\mu^{i})^{(l_{i+1})}\vdash n-i-1,
u^0 = \left( \left( y+ \alpha (y^2-y)\cos (\pi x)/\pi \right)^2 \ln \left( y+ \alpha (y^2-y)\cos (\pi x)/\pi \right) - 1.5 \right) +7.5 \left( y+ \alpha (y^2-y)\cos (\pi x)/\pi \right)\,,
\left\langle\lambda_0a-(Bc)*a+B\left(c*a\right),b\right\rangle_{X,X^\ast}=0\iff\left\langle a,\lambda_0b-(Bc)*b\right\rangle_{X,X^\ast}=-\left\langle B\left(c*a\right),b\right\rangle_{X,X^\ast}.
\left<\mathcal W_\Psi f,\mathcal W_\Psi h\right>=\int_0^\infty\int_{SO(n+1)}\overline{\mathcal W_\Psi f(\rho,g)}\,\mathcal W_\Psi h(\rho,g)\,d\nu(g)\,\alpha(\rho)\,d\rho &=\int_0^\infty\!\!\int_{SO(n+1)}\int_{\mathcal S^n}\!\Psi(g^{-1}x)\,\overline{f(x)}d\sigma(x)\, \int_{\mathcal S^n}\!\overline{\Psi(g^{-1}y)}\,h(y)\,d\sigma(y)\,d\nu(g)\,\alpha(\rho)\,d\rho.
{\cal J}= -i\int{d{\bf x}\,\psi^\dagger\epsilon_{ln}x_l(\partial_n - ieA_n)\psi}-\int{d{\bf x}\,\psi^\dagger\frac{\gamma_0}{2}\psi}-\frac{1}{2}\int{d{\bf x}\,x_nF_{n0}\epsilon_{ij}F_{ij}},
r^{S_0 \oplus S_1}_{(1, i)}= R_i + \beta \sum_{j \in N}p_{ij}f^{S_0 \oplus S_1}_{(1, j)}- \beta f^{S_0 \oplus S_1}_{(0, i)}= f_i^{S_1}= \frac{r^{S_1}_i}{1-\beta},
1 = 2cd+2d^2+c^2+b^2+db+a^2+ca+2da = \left(\frac{a}{2}+ d + c\right)^2 + \left(\frac{d}{2}+ b\right)^2 + \left(\frac{d}{\sqrt{2}}+ \frac{a}{\sqrt{2}}\right)^2 + \left(\frac{d}{2}\right)^2 + \left(\frac{a}{2}\right)^2 .
{\rm tr}'e^{-(\Delta +m^2)t}= \sum_{\delta \neq{\bf 1}}\sum_{p=1}^{\infty}\int_{H/\Gamma^{(\delta)}}{{d x d y}\over{y^2}}{\hat G}(z, e^{p l(\delta[ z])};t ) + A \cdot{\hat G}(z,z;t) \quad ,
\int_0^tu(x)\, \mu(dx) =\int_0^tu(x)\frac{\phi_0(x,t)}{e(x,t)}\frac{e(x,t)}{\phi_0(x,t)}\, \mu(dx) &=\int_0^t\sum_{n=1}^{+\infty}c_n\sin(\frac{x}{t}n\pi)\frac{e(x,t)}{\phi_0(x,t)}\, \mu(dx)
{\cal L}_{Weyl}^{nm}= \partial_{+}{A}_i \left(\partial_{-}{A}_i - \partial_i A_{-}\right) + \frac 1 2 \left(\partial_{+}{A}_{-}\right)^2 - \frac 1 4\left( \partial_i{A}_j - \partial_j{A}_i\right)^2 +{A}_{-}{J}^{-}+{A}_i{J}^i \;,
\dfrac{\partial p}{\partial t}+\triangle p\, =\,y_d-y\,\,\, in\,\, Q:=\Omega\times (0,T), p\, =\,0\,\,\, on\,\, \Sigma :=\Gamma\times (0,T), p\, =\,0\,\,\, at\,\, t=T.
\frac{\partial L^{\prime}}{\partial r_{k}}-\frac{d}{dt}\frac{\partial L^{\prime}}{\partial\dot{r}_{k}}=0\Rightarrow\frac{\partial L}{\partial r_{k}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{r}_{k}}+\sum_{i=1}^{C}\lambda_{i}\frac{\partial f_{i}}{\partial r_{k}}=0,
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}\dot{u}_{j})_{,K}\big)_{,J}a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}\dot{\tau}_{,IJ}+ M_{jLKI}u_{j,LKI}+ K_{IJ}\tau_{,IJ}
\Pi_{k}^{o u t}= \sqrt{\pi k_{s}}2^{1 - n_{s}^{\pi}}\Gamma \left( - n_{s}^{\pi}\right) a_{s}^{m}g_{S}^{l}\left( \frac k{k_{s}}\right)^{n_{s}}\mathrm{f o r}n_{1}> 0 ,
\varphi^* \left(\begin{array}{c c}a b 0 d \end{array}\right) := \left(\begin{array}{c c c}a^{\ell_1}0 0 0 a^{\ell_2}0 0 0 a^{\ell_3}\end{array}\right) = \left(\begin{array}{c c c}1 0 0 0 1 0 0 0 1\end{array}\right) .
\begin{array}{ccc}R_{BCD}^A = \tilde{R}_{BCD}^A, R_{iBj}^A = -a_kD_B\partial^Aa_l\eta_{ij}^{kl}, R_{AjB}^i = -a^kD_B\partial_Aa_l\eta_{kj}^{li}, R_{jkl}^i = -a^m\partial^Aa_na_r\partial_Aa_s(\eta_{mk}^{ni}\eta_{jl}^{rs}-\eta_{ml}^{ni}\eta_{jk}^{rs}).\end{array}
\gamma_{2k-1}=\prod_{j=1}^{k-1}\frac{w_{2j}}{w_{2j-1}}\,,\mbox{}\gamma_{2k}=w_{1}\prod_{j=1}^{k-1}\frac{w_{2j+1}}{w_{2j}}\,,\mbox{}k=1,2,3,\ldots.
\gamma^{\mu}\gamma^{\lambda}= g^{\mu \lambda}+ \sigma^{\mu \lambda}, \ \ \ \left( g^{\mu \lambda}= \frac{1}{2}\left\{\gamma^{\mu}, \gamma^{\lambda}\right\}, \ \sigma^{\mu \lambda}= \frac{1}{2}\left[ \gamma^{\mu}, \gamma^{\lambda}\right] \right) ,
\Phi(e^{i\varphi},s,a)=L(\frac{\varphi}{2\pi},s,a)=\frac{1}{a^{s}}+\frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}(e^{i\varphi}-e^{-t})}{cosht-cos\varphi}dt,
{\cal U}= \left( \begin{matrix}{{2 \delta_{x}{}^{y}}}&{{- \frac 1{W^{2}}( \partial_{x}K^{Z}){\cal J}_{Z}{}^{Y}}}\\{{\frac 1{W^{2}}{\cal J}_{X Z}\partial^{y}K^{Z}}}&{{\frac 3 2 \delta_{X}{}^{Y}- \frac{1}{W^{2}}{\cal J}_{X}{}^{Z}{\cal L}_{Z}{}^{Y}}}\\ \end{matrix}\right) \, ,
\left( \begin{pmatrix}w 0 0 z \end{pmatrix}+ \begin{pmatrix}a b c d \end{pmatrix}p \right)^{p-1}= I + \begin{pmatrix}-aw^{p-2}0 0 -dz^{p-2}\end{pmatrix}p &= I + \begin{pmatrix}- \frac{a}{w}0 0 -\frac{d}{z}\end{pmatrix}p.
((\begin{matrix}1&1\\ 0&1\end{matrix})-1(\begin{matrix}1&0\\ 0&1\end{matrix}))(\begin{matrix}v_{21}\\v_{22}\end{matrix})=(\begin{matrix}0&1\\ 0&0\end{matrix})(\begin{matrix}v_{21}\\v_{22}\end{matrix})=(\begin{matrix}1\\ 0\end{matrix}).
K_{1,2}\le 2C_\alpha L\phi'(|e|)\int_{\bar{x}}^{\bar{y}}\frac{dz}{z^{\alpha+1}}=\frac{2C_\alpha L\phi'(|e|)}{\alpha}\left(\frac{1}{\bar{x}^\alpha}-\frac{1}{\bar{y}^\alpha}\right).
[f, \tilde{p}_{i, j}] = -\frac{1}{2 \cdot i!}\big[\tilde{X}_{2, 0}(C), \tilde{X}_{0, i}(\eta^{(j - i + 2)/2})\big] & = \frac{1}{(i - 1)!}\tilde{X}_{1, i - 1}(\eta^{(j - i + 2)/2}) - \frac{1}{(i - 2)!}\tilde{X}_{0, i - 2}(\eta^{(j - i + 4)/2}).
\Lambda(x)=\frac{L({\sigma_{0}}^{2}|x)}{L({\sigma_{1}}^{2}|x)}=(\frac{\sigma_{0}^{2}}{\sigma_{1}^{2}})^{-n/2}exp\{-\frac{1}{2}(\sigma_{0}^{-2}-\sigma_{1}^{-2})\sum_{i=1}^{n}(x_{i}-\mu)^{2}\}.
r_{n}(\lambda,b,\mathbf{m}):=\det\left(M_{n\mathbf{m}}(\lambda,b,\Omega_{\mathbf{m}}^{\pm}(\lambda,b))\right)-d_{\infty}(\lambda,b,\mathbf{m})\underset{n\rightarrow\infty}{=}\frac{\widetilde{d}_{\infty}(\lambda,b,\mathbf{m})}{n}+O_{\lambda,b,\mathbf{m}}\left(\frac{1}{n^{3}}\right).
\widetilde{N}_3 = \left( \begin{array}{llll}1 4 6 4 1 5 8 5 1 5 9 6 \end{array}\right)\quad\rightsquigarrow{N}_3 = \left(\begin{array}{llll}1 4 6 4 0 1 2 1 0 0 1 1 \end{array}\right).
\int \eta_i(x) \eta_j(y) f(\xi^{x,y}, \omega^{x,y}) d\nu_{\widehat \theta}^N (\xi,\omega) = \int \eta_j(x) \eta_i(y) (R_{i,j}^{x,y}(\widehat \theta)+1) f(\xi, \omega) d\nu_{\widehat \theta}^N (\xi,\omega)
|F_{A,0}^{(1)}(z_0,z_1)|<|F_{A,1}^{(1)}(z_0,z_1)|\quad \begin{aligned}\|F_A(z_0,z_1)\|= |F_{A,1}^{(1)}(z_0,z_1)|=p^{1/2}|cz_0+dz_1|^p = p^{1/2}|dz_1|^p=p^{1/2}|z_1|^p =p^{1/2}\|(z_0,z_1)\|^p.\end{aligned}
a_n = \sum_{j=0}^n h_1(j) \prod_{i=j+1}^n f(i) \equiv \sum_{j=n-d+1}^n h_1(j) \prod_{i=j+1}^n f(i) = \sum_{j=0}^{d-1}h_1(n-j) \prod_{i=0}^{j-1}f(n-i) \pmod{d}.
\left\vert G^{(2)}_{\mathcal R,Res}(x,y)\right\vert =\left\vert \left(I_1+I_2\right)/(4\pi)\right\vert\leq C_{R_0}\left((\sigma\vert x \vert)^{-3/2}+e^{-\vert x \vert\sigma}\right) &\leq\frac{C_{R_0}}{{{\left\vert \sin\left({(\theta_c-\theta_{\hat x})}/2\right)\right\vert}^{\frac 32}}\vert x \vert^{\frac 32}}
a_3(3^1) = 728 = \;\; 2^3\cdot 7 \cdot 13, a_3(3^2) = 2028730080 = \;\; 2^5\cdot 3 \cdot 5 \cdot 13^2 \cdot 89 \cdot 281, a_3(3^3) = 1747100845087920 = \;\;2^4\cdot 3^3 \cdot 5 \cdot 7 \cdot 617 \cdot 187275523, a_3(3^4) = 1249380857829754167840 = \;\; 2^5\cdot 3^5 \cdot 5 \cdot 7^2 \cdot 794953 \cdot 824956519.
S^{(1)}_{\ell,N}=\sum_{n=1}^{N_\ell-1}F_\ell(\xi_n,...,\xi_{\ell n})\,\,\, \mbox{and}\,\,\, S^{(2)}_{\ell,N}=S_{\ell,N}-S_{\ell,N}^{(1)}.
{E}^{0,i}\big[e^{-\alpha\mathbf{e}_q}, J_{\mathbf{e}_q}=k\big] = \Big[\int_0^\infty qe^{-(\alpha+q) t}e^{Qt}dt\Big]_{i,k}&= q\big[\big((q+\alpha)I -Q\big)^{-1}\big]_{i,k}.
{\mathrm{negative}}\hspace{0.25 cm}{\mathrm{energy:}}\hspace{0.3 cm}E \leq -m \hspace{0.3 cm}{\mathrm{and}}\hspace{0.3 cm}\theta \hspace{0.15 cm}{\mathrm{decreasing}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}t_{2}< 0 \hspace{0.3 cm}{\mathrm{and}}\hspace{0.3 cm}t_{2}\hspace{0.15 cm}{\mathrm{decreasing}}.
\begin{aligned}(\lfloor\frac{a-1}{2}\rfloor+\lfloor\frac{j-l}{2}\rfloor+1 -\lceil\frac{p-1}{2}\rceil)+\frac{a}{2}+l-a+p &\geq \lfloor\frac{j-l}{2}\rfloor+l+\lceil\frac{p}{2}\rceil &\geq \lfloor\frac{j-l}{2}\rfloor+l+\lceil\frac{j-l+1}{2}\rceil &\geq j, \end{aligned}
\Delta_\phi(\alpha)=\alpha \otimes \alpha - q e^{-4i\phi}\gamma^*\otimes \gamma &\Delta_\phi(\alpha^*)=\alpha^*\otimes \alpha^* - q e^{-4i \phi}\gamma \otimes \gamma^* &\Delta_\phi(\gamma)=\gamma\otimes\alpha +\alpha^*\otimes \gamma &\Delta_\phi(\gamma^*)=\gamma^*\otimes \alpha^*+\alpha\otimes \gamma^*
\begin{array}{ll}\lambda_1 = \frac{5x(y_1-y_2)(4x+4-4y_1-5y_2)}{(32x-24) (9y_2-36x+28)}, \lambda_2 = 9\lambda_1 + \lambda_3 + 9\lambda_4, \lambda_3 = \frac{4(x-\lambda_4)(5y_1-4y_2+4x-4)}{(41y_2+164x-164)}, \lambda_4 = \frac{x(y_2-y_1)(12-5y_1-4y_2-4x)}{(10x-8) (9y_2-36x+28)}. \end{array}
\sum_{k=0}^{n-1}(-1)^k(2k+1)^{2r-1}= 2^{2r-2}\sum_{k=0}^{n-1}\left((-1)^kE_{2r-1}\left(k+\frac12\right)-(-1)^{k+1}E_{2r-1}\left(k+\frac32\right)\right) = 2^{2r-2}\left(E_{2r-1}\left(\frac12\right)-(-1)^nE_{2r-1}\left(n+\frac12\right)\right) = \frac{E_{2r-1}}2-(-1)^n\frac{E_{2r-1}}2-(-1)^n2^{2r-2}\sum_{k=0}^{2r-2}\binom{2r-1}k\frac{E_k}{2^k}n^{2r-1-k}.
S^{\mp ,a,p}(x,x^{\prime}) =\int_{-\infty}^{+\infty}\,dz\,\int_{-\infty}^{+\infty}\,\frac{dp_3}{2\pi}\,e^{-ip_3y^3}S_Q^{\mp ,a,p}\;,\quad y_\mu=x_\mu -x_\mu^{\prime}\;,
A_0\setminus\{1\}=\{3, 5, 7\}, q(0,3) =d_3^{-1}d_{16-3}=d_3^{-1}d_{13}, q(0,5) =d_5^{-1}d_{11}, q(0,7) =d_7^{-1}d_9, F_0 =\langle q(0,3)\rangle\times\langle q(0,5)\rangle\times\langle q(0,7)\rangle.
H_1(y,q) =\widetilde{DG}_2(y,q),H_2(y,q)=\frac{F_1(y,q)}{(y^{1/2}-y^{-1/2})^4}+\frac{F_2(y,q)}{(y^{1/2}-y^{-1/2})^2(y-y^{-1})},
d s^{2}= - \frac{r^{2}}{l^{2}}d t^{2}+ \left( \frac{r^{2}}{l^{2}}- r_{0}^{2}\right) d \theta^{2}+ \left( \frac{r^{2}}{l^{2}}- r_{0}^{2}\right)^{- 1}d r^{2}+ d x_{1}^{2}+ \sinh^{2}\left( \frac{\sqrt{2}x_{1}}{l}\right) d x_{2}^{2}
f|_{-\frac{1}{2}}\rho=q^{-\frac{(c,24)^2}{576}}\left(a_{f}\left(\rho : -\frac{(c,24)^2}{576}\right)+a_{f}\left(\rho : \frac{23(c,24)^2}{576}\right)q^{\frac{(c,24)^2}{24}}+\cdots\right).
\mathcal B_\Omega f(x)=\int_{\Omega}f(y)\bigg(\frac{x-y}{|x-y|^{n}}\int_{|x-y|}^\infty\omega\Big(y+ r\frac{x-y}{|x-y|}\Big)\zeta^{n-1}\,dr \bigg)\,dy \hbox{for $x\in \Omega$,}
B = \frac{1}{2}\int_{\partial M}{\left\langle{\nabla ({{\left|{\nabla u}\right|}^2}),A\overrightarrow{n}}\right\rangle dvo{l_{\bar g}}}- \int_{\partial M}{({L_A}u)\left\langle{\nabla u,\overrightarrow{n}}\right\rangle dvo{l_{\bar g}}},
\vert \zeta +1\vert^{1/2}\Vert \mathbf{D}(B_\varepsilon -\zeta^*Q_0^\varepsilon )^{-1}\Vert_{L_2\rightarrow L_2}&\leqslant c_*^{-1/2}\Vert f\Vert_{L_\infty}(c_\flat +2)^{1/2}(c_\flat +1)^{1/2}\varrho (\zeta )^{1/2}.
\Delta^{\mathrm{c l}}= \int d^{3}x \left( [ A_{\mu}^{*}, A^{\mu}] - \left[ c^{*}, c \right] + \left( \overline{{{\Psi}}}^{*}T^{a}\, \Psi + \overline{{{\Psi}}}\, T^{a}\, \Psi^{*}\right) \tau_{a}\right) .
\sum_{k}h^{p^{\ast}}_{ijk}\omega_k=dh^{p^{\ast}}_{ij}+\sum_kh^{p^{\ast}}_{ik}\omega_{kj}+\sum_k h^{p^{\ast}}_{kj}\omega_{ki}+\sum_{q}h^{q^{\ast}}_{ij}\omega_{p^{\ast}q^{\ast}},
{\sf s}_{[i,j]}(X\cup Y) =\{c:\exists a\in(X\cup Y), a\equiv_{ij}c\}&=\{c:(\exists a\in X\vee\exists a\in Y), a\equiv_{ij}c\}&=\{c:\exists a\in X, a\equiv_{ij}c\}\cup\{c:\exists a\in Y,a\equiv_{ij}c\}&={\sf s}_{[i,j]}X\cup{\sf s}_{ij}Y.
r_{s,p}=\frac{\overline{\beta'(b_s)}(\overline{a}_1-\overline{b}_s)r_{s,1}+\overline{\beta'(b_1)}(\overline{a}_p-\overline{b}_1)r_{1,p}}{\overline{\beta'(b_s)}(\overline{a}_p-\overline{b}_s)}
\hat{L}_{\delta}= \hat{L}_{\delta,\mathcal{F}_{RDT,p}(C)}= \frac{\sum_{i=1}^{n}g^{*}_{+,\delta,C}(x_{i})y_{i}/\sigma^{2}(x_{i})}{\sum_{i=1}^{n}g^{*}_{+,\delta,C}(x_{i})/\sigma^{2}(x_{i})}-\frac{\sum_{i=1}^{n}g^{*}_{-,\delta,C}(x_{i})y_{i}/\sigma^{2}(x_{i})}{\sum_{i=1}^{n}g^{*}_{-,\delta,C}(x_{i})/\sigma^{2}(x_{i})}.
A \equiv \frac{( - K^{\prime}( 0 ) ) ( -{\textstyle \int}p^{2}K^{\prime})}{( -{\textstyle \int}K^{\prime}) K ( 0 )}, \ \ \B \equiv \frac{K^{\prime \prime}( 0 ) K ( 0 )}{( - K^{\prime}( 0 ) )^{2}}.
u_i u_j =( 3c_i+4 s_{i}+4z_{i})( 3c_j+4 s_{j}+4z_{j}) &=9 c_i c_j+12\left(c_i(s_{j}+ z_{j})+ c_j(s_{i}+ z_{i})\right) &\phantom{{}={}}+ 16(s_{i}s_{j}+s_i z_{j}+s_j z_{i}+z_{i}z_{j}) &= 9c_{i,j}-6 t_{i,j}+ 9c_ic_j + 16(s_i z_{j}+s_j z_{i}+z_{i}z_{j})
{\cal A}_{\rm eff}^{\rm mf}\,=\,\frac{R\beta M^2}{2}\,\sqrt{\rho_{0}\rho_{1}}\,\left[\left(\frac{\lambda_0\,+\,1}{\rho_0}\,+\,\frac{\lambda_1\,+\,1}{\rho_1}\right)\,-\,\left(\lambda_0\,+\lambda_1\right)\right],
C(m,n,D)=C_{m,n,\mathrm{diam}D}\left[\frac{2}{2-\eta}(2\rho)^{2-\eta}\sqrt{1+\mathcal{L}^{2}}\sigma(S^{n-1})+\left(\frac{2}{d}\right)^{n-2+\eta}\sigma(\partial D)\right],
\int_0^T ||\mathcal{\mathbf{B^{\ast}_{n+j}}}U_{n+j}||_{G_{n+j}}^2 \,dt & \le D(T)\Big(e_1(U_{n+j})(0) + \int_0^T (|u_{n-1}|^2 + |u_n|^2 + \sum_{k=1}^{j-1}|u_{n+k}|^2)\,dt\Big)\,.
T_{\i}(x)-x\in E_n\iff A_{\i}^{-1}(x)-x\in \sum_{k=1}^{n}A_{i_1\ldots i_{k}}^{-1}t_{i_{k}}+E_n \iff x\in (A_{\i}^{-1}-I)^{-1}\left(\sum_{k=1}^{n}A_{i_1\ldots i_{k}}^{-1}t_{i_{k}}+E_n\right).
\langle \Psi\varphi, w \rangle_{V^{*}, V}:= (L\varphi, w)_{H}+ h\langle B_{1}^{*}\varphi, w \rangle_{V^{*}, V}+ h^2\langle A_{2}^{*}\varphi, w \rangle_{V^{*}, V}+ h^2(\Phi_{\lambda}\varphi, w)_{H}\,\quad+ h^2({\cal L}\varphi, w)_{H}+ \eta h^2 (B_2 (I+hA_1)^{-1}\varphi, w)_{H}\mbox{for}\ \varphi, w \in V.
\int_{\varphi_n(\mathsf{U}_n)}\left|w_{ij}^{(m,n)}(x)\right|^p \,d\mu(x)=\int_{\varphi_n(\mathsf{U}_n)}\Bigl|\sum_{1\leq k\leq d}u_{ik}^{(n)}(\phi_{m,n}(x))v_{kj}^{(m,n)}(x) \, \Bigr|^p \,d\mu(x) &=\int_{\varphi_n(\mathsf{U}_n)}\Bigl|\sum_{1\leq k\leq d}u_{ik}^{(n)}(y)v_{kj}^{(m,n)}(\phi_{m,n}^{-1}(y)) \, \Bigr|^p J_{\phi_{m,n}^{-1}}(y) \,d\mu(y) \ ,
\Phi_0(x) :=[Q,\iota_{\nu_0(x)}]\in\mathfrak{X}_{0}(E)[1],\; \; x\in \mathfrak g[1]. \Phi_1(x,y) :=[Q,\iota_{\nu_1(x,y)}]^{(-1)}-\sum_{k\geq 0}[[Q,\iota_{\nu_0(x)}],\iota_{\nu_0(y)}]^{(k)}\in\mathfrak{X}_{-1}(E)[1],\; \; x,y\in \mathfrak g[1].
\left(D_{-}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(-\frac{d}{dx}\right)^{[\operatorname{Re}(\gamma)]+1}\left(I_{-}^{-\alpha',-\alpha,-\beta',-\beta+[\operatorname{Re}(\gamma)]+1,-\gamma+[\operatorname{Re}(\gamma)]+1}f\right)(x).
Q_{13}=-r_{11}\ell_1'-r_{12}\ell_2'+c_{12}\ell_2=c_{13}Q_{14}=r_{11}\ell_2'-r_{12}\ell_1'+c_{12}\ell_1=c_{14}Q_{23}=-r_{12}\ell_1'-r_{22}\ell_2'-c_{12}\ell_1=c_{23}Q_{24}=r_{12}\ell_2'-r_{22}\ell_1'+c_{12}\ell_2=c_{24}\ .
\bar D_{3,a,6}= \frac{|X_1-X_2|}{|z|}\big|D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,t_0(X_\tau))) &\qquad\big[D^2_{tt}\tilde h(X_\tau,z+t_0(X_\tau)) -D^2_{tt}\tilde h(X_\tau,t_0(X_\tau)) \big] D_X t_0(X_\tau) .
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{K}_{3}(t)g\right\|_{2}\le Ct^{1- \ell}(1+t)^{-\frac{n}{2}-k}\| g \|_{1}+ C e^{-ct}\| \nabla^{(k+\ell-1)_{+}}_{x}g\|_{2},
b_{2}=\frac{b_{22}m_{02}}{m_{01}+m_{02}}=\frac{b_{22}}{1+\frac{m_{01}}{m_{02}}}=\frac{b_{11}}{1+\frac{m_{s1}}{m_{s2}}\frac{1+b_{22}M_{2}}{1+b_{11}M_{1}}}