harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Sigma_{0, \, N}^{\alpha_0}= \{f \in \Sigma_{0, \, N}\ | \ \|f\|_{\Sigma_{0, \, N}}\leq \alpha_0\}, D\Sigma_{i-1, \, N}^{\alpha_i}= \{f \in D\Sigma_{i-1, \, N}\ | \ \|f\|_{D\Sigma_{i-1, \, N}}\leq \alpha_i\}, i\in \{1, \, \dots , r\},
\begin{gathered}\int_0^\infty u^{2\alpha}C(u) J_\nu (u\rho; q^2) \,d_q u- \int_0^\infty u^{2\alpha}C(u)\left[\frac{w(u)}{1+w(u)}\right] J_\nu (u\rho; q^2) \,d_q u = f_1 (\rho), \quad \rho\in A_{q,a}\end{gathered}
L_xp_t^0(x,y) =\Big[a(x)\frac{1}{a^{d/\alpha}(y)t^{d/\alpha}}(L^{(\alpha)}g^{(\alpha)})\left(\frac{w}{a(y)t^{1/\alpha}}\right) &-\Big(b(x), \frac{1}{a^{d/\alpha+1}(y)t^{d/\alpha+1}}(\nabla g^{(\alpha)})\left(\frac{w}{a(y)t^{1/\alpha}}\right)\Big)\Big]\Big|_{w=\kappa_t(y)-x}.
\sup_{\genfrac{}{}{0pt}{}{\scriptstyle{x,\bar x\in \overline{B},\, t,\bar t\in[s,s+1],}}{{\scriptstyle{x \not= \bar x,\, t \not= \bar t,\, s\in[-1,1]}}}}\frac{|\nabla v_n(x,t_n+t)-\nabla v_n(\bar x,t_n+\bar t)|}{|x-\bar x|^\gamma+|t-\bar t|^{\frac{\gamma}{2}}}< K
X_j f(x,u) = \frac{d}{dt}f((x,u)(tX_j,0))\big|_{t=0}& = \partial_{x_j}f(x,u) + \frac 1 2 \sum_{k=1}^{d_2}\langle U_k,[x,X_j]\rangle \partial_{u_k}f(x,u), U_k f(x,u) = \partial_{u_k}f(x,u).
\sum_{n= 1}^{N}\frac{1}{n}\sum_{\lambda \in (W)}(\lambda R)^{n}= S_{1}(N)+ S_{2}(N) &= \sum_{n=1}^{[{N}/{\Delta}]}\frac{1}{n}+ \log C_{X}+ \log \Delta+ O \left( (\rho R)^{N}\right).
\sum_{a = 1}^{2}\overline{{{\psi_{a}}}}( x ) \psi_{a}( x ) \longrightarrow \frac{( - 1 )^{x}}{2 a}( \psi_{1 , x}^{\dagger}\psi_{1 , x}+ \psi_{2 , x}^{\dagger}\psi_{2 , x}- 1 ) \quad ;
\begin{array}{c}{{( B_{\rho}- ( l + \frac{d - 3}{2})^{2}) R_{l}= - ( l + 1 ) ( l + 2 ) R_{l + 2}}}\\{{( B_{\rho^{\prime}}- ( l + \frac{d - 3}{2})^{2}) R_{l}= - ( l + 1 ) ( l + 2 ) R_{l + 2},}}\\ \end{array}
\left\langle \alpha_{1},\alpha_{2},...,\alpha_{2N},t\right| U_{+}\left| \alpha_{1}',\alpha_{2}',...,\alpha_{2N}',t\right\rangle =R^{\alpha_{1}'\alpha_{2}'}_{\alpha_{1}\alpha_{2}}\: R^{\alpha_{3}'\alpha_{4}'}_{\alpha_{3}\alpha_{4}}\: ...\: R^{\alpha_{2N-1}'\alpha_{2N}'}_{\alpha_{2N-1}\alpha_{2N}}
\frac{b_{p,q}(x_n)-b_{p,q}(x)}{x_n-x}=\frac{\pi_q(0^m1b_{n,m+2}b_{n,m+3}\cdots)}{\pi_p(0^m1b_{n,m+2}b_{n,m+3}\cdots)}\ge \left(\frac{p}{q}\right)^m\frac{\pi_q(10^{\infty})}{\pi_p(1^{\infty})}
\begin{aligned}\frac{|x^1_iu_{x_i}(P_1) - x^2_iu_{x_i}(P_2)|}{\rho^{\alpha}(P_1,P_2)}\leq C\eta^{\frac{1-\alpha}{2}}\|u\|_{C^{2+\alpha}_{WF}([0,T]\times\bar S_{n,m})}+ 2 \eta^{-\frac{\alpha}{2}}\|u_{x_i}\|_{C([0,T]\times\bar S_{n,m})}.\end{aligned}
\sum_{i, \ell \colon \mu_{i,\ell}^{(n)}> k'}\check f_n(i,\ell)^2 \leq \left(\sqrt{\frac{k}{k'}\cdot{\sum_{i, \ell \colon 0<\lambda_{i,\ell}^{(n)}\leq k}\hat f_n(i,\ell)^2}}+ \sqrt{\sum_{i, \ell \colon \lambda_{i,\ell}^{(n)}> k}\hat f_n(i,\ell)^2}\right)^2 & \leq \left(\sqrt{\delta/4}+ \sqrt{\delta/4}\right)^2=\delta.
\sum_{i=0}^{2\ell}\binom{2\ell}{i}\, i^2\, K_k(i)K_j(i)=2^{2\ell}\,\binom{2\ell}{k}^{-1}\begin{cases}\frac{(k+1)(k+2)}{4}, j=k+2, -\ell(k+1), j=k+1, (\ell(\ell+1/2)+k(\ell-k/2)), j=k, - \ell(2\ell-k+1), j=k-1, \frac{(2\ell-k+1)(2\ell-k+2)}{4}, j=k-2, 0, \end{cases}
G^-_c(\gamma_1,\gamma_2) = \frac{2\sqrt{J_1J_2}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty\frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\cos(\gamma_1 q^{2[n_1]_q}-\gamma_2q^{2[n_2]_q})\,;
T = \left( \begin{array}{c c c c c}{{\mathrm{r o t}\theta_{1}}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\mathrm{r o t}\theta_{2}}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{\mathrm{r o t}\theta_{N}}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{{\large 1}}}\\ \end{array}\right)
(\psi_2^{i+1 i}\circ \frak n^i_k-\frak n^{i+1}_k\circ \psi_1^{i+1 i})_{\alpha'_2\alpha_1}=(\hat d_{2}^{i+1}\circ \frak h^{i+1 i}_k+ \frak h^{i+1 i}_k \circ \hat d_{1}^{i+1})_{\alpha'_2\alpha_1}
- \frac{i}{2 \mu}\left[ \epsilon^{i a b}k_{a}( - k^{2}\chi_{b}^{j}+ k_{e}k^{j}\chi_{b}^{e}) ~ + ~ \epsilon^{j a b}k_{a}( - k^{2}\chi_{b}^{i}+ k_{e}k^{i}\chi_{b}^{e}) \right] = 0 .
2(n+1)(n + \beta + 1)(2n + \beta) P_{n+1}^{(0,\beta)}(t) = (2n + \beta +1)[(2n + \beta+2)(2n + \beta )t - \beta^2]P_n^{(0,\beta)}(t) - 2n(n + \beta)(2n + \beta + 2)P_{n-1}^{(0,\beta)}(t)
\varphi_{A_1...A_na_1...a_m}(y)=\int_{{\cal F}\times S^2}\omega(p,\zeta )\ \widetilde{\varphi}(p,\zeta )\Phi (p,\zeta|y)_{A_1...A_na_1...a_m}\ \in{\cal H}_{E_o,s}
I_{\alpha}= -\sum_{i=1}^{N-1}\int_{1}^{x_{i}(\Lambda_{0}^{2})}\! dx_{i}\,{\displaystyle{(x_{i}-1)^{3}\over x_{i}^{2}(x_{i}+1)}+{2\over x_{i}}\, \left({x_{i}-1\over x_{i}+1}\right)^{2}\, \ln x_{i}\over\displaystyle\sum_{j=1}^{N-1}{x_{j}(x_{i})-1\over x_{j}(x_{i}) + 1}\, \ln x_{j}(x_{i})}\ \raise 2pt\hbox{,}
S = \int d s \, p_{0}\, \frac{d}{d s}\left( x_{0}+ \frac 1 \kappa \, e^{p_{0}/ \kappa}\, \vec{x}\cdot \vec{p}\right) - e^{p_{0}/ \kappa}\, \vec{p}\, \frac{d}{d s}\vec{x}- H
S^{(n+2)}(0)=-K^{(n+2)}(0)-5(n+2)K^{(n+1)}(0)-15(n+2)(n+1)K^{(n)}(0) -(n+2)!\dbinom{n+6}{n+2}K(0)=3(n+3)!n\left( n^{2}+9n+2\right) -15(n+2)(n+3)!n\left( n+5\right) +15(n+3)!(n+2)^{2}(n+1)+ 12(n-1)(n+2)!\dbinom{n+6}{n+2}
\int_{\partial B_n}f(x(u))^{1-b-\frac{2}{n-1}}\int_0^{\phi_f(u,1)}s^{N+b-2+\frac{2}{n-1}}\,ds\,d\mu_{\partial B_n}(u) &=\int_{\partial B_n}f(x(u))^{1-b-\frac{2}{n-1}}\int_{1-\phi_f(u,1)}^{1}(1-w)^{N+b-2+\frac{2}{n-1}}\,dw\,d\mu_{\partial B_n}(u).
= \frac{1}{1 6 R}\ \Big [ \ \frac{3}{r^{5}}+ \sum_{m = 1}^{\infty}e^{- m r / R}( \frac{m^{2}}{R^{2}}\frac{1}{r^{3}}+ \frac{m}{R}\frac{3}{r^{4}}+ \frac{3}{r^{5}}) 2 \cos ( m x_{1 1}/ R ) \Big ]
\int dP_{\mathsf{B}_n,2}=2^{-n(R_1+R_2)}\sum_{(i,j)\in\mathcal{I}_n\times\mathcal{J}_n}\mathbb{P}_{p_{W|U,V}^n}\Big(\big(\mathbf{U}(i),\mathbf{V}(i,j),\mathbf{W}\big)\notin\mathcal{A}_{\epsilon_1,\epsilon_2}\Big|\mathbf{U}=\mathbf{U}(i),\mathbf{V}=\mathbf{V}(i,j)\Big),
I_\lambda(\zeta,\bar{\Lambda}) = ka_0+ a_1 \Big[ -\sigma_1(\varepsilon,\zeta)^2 \bar\Lambda_1^2 + |\sigma_1(\varepsilon,\zeta)| (\bar\Lambda')^T Q(\varepsilon,\zeta) \bar\Lambda' \Bigr] & +\sigma_1(\varepsilon,\zeta)^2\,\mathcal Poly_4(\varepsilon,\zeta,\bar \Lambda)+ \theta_\lambda(\zeta,\bar \Lambda),
\mathcal{G}_0 (U) := \bigg\{f \in \mathcal{O}_{X\times \mathbb{B}^1}(U \times \mathbb{B}^1) = \mathcal{O}_X (U) \langle T \rangle \, \bigg| f = 1+ \sum_{i=1}^\infty f_iT^i \vert f_i(u) \vert < 1 \ \forall i \geq 1 \ \forall u \in U \bigg\}.
\ast (\overline{\zeta_j}\wedge \beta ) = \ast \sum_{m=1}^n \sum_I (-1)^p \overline{f_{j,m}}\beta_I dz_I \wedge d \overline{z_m}&=\sum_{m=1}^n \sum_I (-1)^{\bullet}\overline{f_{j,m}}\beta_{I}dz_{\{1, \dots , n \}\setminus \{m \}}\wedge d \overline{z_{\{1 , \dots , n\}\setminus I}},
\begin{gathered}\forall \zeta\in E_s(z), ||d\varphi_t(x).\zeta||_{G}\leq Ce^{-\nu t}||\zeta||_{G}, \quad\forall\, t\geq 0, \forall \zeta \in E_u(z), ||d\varphi_t(x).\zeta||_{G}\leq Ce^{-\nu |t|}||\zeta||_{G}, \forall\, t\leq 0,\end{gathered}
\xi_i\le \min_{\substack{p\in S p(\xi_1)=\ldots=p(\xi_{i-1})=0}}\frac{\int_{-1}^1tp(t)w(t)dt}{\int_{-1}^1p(t)w(t)dt}\le\max_{-1<z_1<\ldots<z_{i-1}<1}\min_{\substack{p\in S p(z_1)=\ldots=p(z_{i-1})=0}}\frac{\int_{-1}^1tp(t)w(t)dt}{\int_{-1}^1p(t)w(t)dt}.
\exp \left\{\int_{0}^{T}d t \rho^{\mu}( t ) \gamma_{\mu}\right\}= \exp \left\{\gamma_{\mu}\frac{\partial}{\partial \Gamma_{\mu}}\right\}\exp \left\{\int_{0}^{T}d t \rho^{\mu}( t ) \Gamma_{\mu}\right\}\Biggr |_{\Gamma_{\mu}= 0}
(\int_{\mathbb{R}^{n}}|\hat{f}(\xi)|^{p^{\prime}}d\xi)^{1/p^{\prime}}\le(\frac{p^{1/p}}{(p^{\prime})^{1/p^{\prime}}})^{n/2}(\int_{\mathbb{R}^{n}}|f(x)|^{p}dx)^{1/p}
F(a,b):=\tilde{\kappa}_k^b\left(\sum_{i=1}^{t_1\wedge r}g_1(a,\tilde{R}_{i,0}^1),\sum_{i=1}^{t_1\wedge r}g_2(a,\tilde{R}_{i,0}^1),\cdots,\sum_{i=1}^{t_1\wedge r}g_k(a,\tilde{R}_{i,0}^1) \right).
{\cal L}= \bar{\psi}_\alpha (i \partial \! \! \! /- e A \! \! \! /)\psi_\alpha+ \frac{1}{2}( \phi^* \bar{\psi}^c_\alpha \psi_\alpha - \phi\bar{\psi}_\alpha \psi^c_\alpha ) - \lambda N \phi^* \phi \, \, \, \, .
S [ x ( \sigma ) , p ( \sigma ) , \xi ( \sigma ) ] ={\frac{1}{2}}\int_{X ( \sigma )}p_{\mu \nu}d x^{\mu}\wedge d x^{\nu}-{\frac{1}{2}}\epsilon_{a b}\int_{\Sigma ( \sigma )}d \xi^{a}\wedge d \xi^{b}H ( p )
{\bf J}^{\prime}\frac{\partial}{\partial x_{1}}= \frac{\partial}{\partial p_{m}}, \qquad{\bf J}^{\prime}\frac{\partial}{\partial x_{m}}= \frac{\partial}{\partial p_{1}}, \qquad{\bf J}^{\prime}\frac{\partial}{\partial p_{1}}= - \frac{\partial}{\partial x_{m}}, \qquad{\bf J}^{\prime}\frac{\partial}{\partial p_{m}}= - \frac{\partial}{\partial x_{1}}, \qquad
\| \nabla_{X,z}^{\mu}\tilde u \|_{H^{k,0}(\mathcal{S})}\leq M(k+1) ( |g|_{H^k}+\|\tilde{f}\|_{H^{k,0}}) \leq M(k+1) (|g|_{H^k}+ \sum\limits_{j=0}^k\|\partial_z^j f\|_{H^{k-j,0}}).
||(S\widetilde{T}- \widetilde{T}S)v|| \leq ||(S\widetilde{T}- \widetilde{S}\widetilde{T})v|| + ||(\widetilde{S}\widetilde{T}- \widetilde{T}S)v|| = ||(S - \widetilde{S})\widetilde{T}v|| + ||(\widetilde{T}(\widetilde{S}- S)v|| \leq \epsilon + ||\widetilde{T}||\epsilon
\nabla_{j}= \nabla_{j}^{0}+ i \left( \begin{array}{c c}{( A_{j})_{i}^{k}}&{\langle ( b_{j})_{i}^{\beta}|}\\{| ( b_{j})_{\alpha}^{k}\rangle}&{( \hat{D}_{j})_{\alpha}^{\beta}}\\ \end{array}\right) .
\left[ 0 \right]_q ! = 1, \left[ n \right]_q ! = \left[ n\right]_q \left[{n - 1}\right]_q \cdots \left[ 2 \right]_q\cdot \left[ 1 \right]_q \left[{\begin{array}{*{20}c}n j \end{array}}\right]_q = \frac{{\left[ n \right]_q !}}{{\left[ j \right]_q !\left[{n - j}\right]_q !}}
d s^{2}= ( \frac{r^{2}}{b^{2}}- \frac{b^{n - 2}}{r^{n - 2}}) d \tau^{2}+ \frac{d r^{2}}{( \frac{r^{2}}{b^{2}}- \frac{b^{n - 2}}{r^{n - 2}})}+ r^{2}\sum_{i = 1}^{n - 1}d x_{i}^{2}.
(x-u_1)(L_1-\zeta_1)+(x-u_2)(L_2-\zeta_2)+(x-u_3)(L_3-\zeta_3)+(x-u_4)(L_4-\zeta_4)=0 (y-v_1)(L_1-\zeta_1)+(y-v_2)(L_2-\zeta_2)+(y-v_3)(L_3-\zeta_3)+(y-v_4)(L_4-\zeta_4)=0 (z-w_1)(L_1-\zeta_1)+(z-w_2)(L_2-\zeta_2)+(z-w_3)(L_3-\zeta_3)+(z-w_4)(L_4-\zeta_4)=0
e_n^j(z_{1, 1}, \dots, z_{1, n}) = e_n^j(z_{i, 1}, \dots, z_{i, n}), 3 \leq i \leq m,\ 1 \leq j \leq n - h - 1; z_{1, j}= z_{2, j}, 1 \leq j \leq n; z_{1, n - h - 2}= z_{1, j}, n - h - 1 \leq j \leq n.
\sum_{\ell=1}^{\infty}\ell^{2s/m}|{\widehat f}_{\phi}(q_{\phi(\ell),\ell}+\alpha)|^2=\sum_{\ell=1}^{\infty}\frac{\ell^{2s/m}}{\ell^{1+2s/m}}=\sum_{\ell=1}^{\infty}\frac{1}{\ell}=\infty.
C_{hk\ell}(N)=\frac{(-1)^{s}\rho^\ell}{k^{2s}s!}\left( \prod_{d=1}^{N-k s}\frac{1}{1-\rho^d}\right)\sum_{m=1}^{s-\ell}\sum_{\substack{i_1, i_2, \dots ,i_m \geqslant 1 i_1+i_2+ \cdots +i_m=s-\ell 1\leqslant r_1 \leqslant r_2 \leqslant \cdots \leqslant r_m \leqslant N}}G_\rho(r_1,i_1) G_\rho(r_2,i_2) \cdots G_\rho(r_m,i_m).
\Delta ~ \partial_{1}J_{1 1 1}^{( d )}= \frac 1 2 ( d - 3 ) ( \partial_{1}\Delta ) J_{1 1 1}^{( d )}+ ( d - 2 ) J_{0 1 1}^{( d )}+ \frac{( d - 2 )}{4 m_{1}^{2}}\left[ ( \partial_{3}\Delta ) J_{1 0 1}^{( d )}+ ( \partial_{2}\Delta ) J_{1 1 0}^{( d )}\right] ,
R_{12}^{\prime \prime}(x_{i}, x_{i^{\prime}})M_{1}(x_{i}|t)M_{2}^{\prime}(x_{i^{\prime}}|t)=M_{2}^{\prime}(x_{i^{\prime}}|t)M_{1}(x_{i}|t)R_{12}^{\prime \prime}(x_{i},x_{i^{\prime}})
\begin{array}{| r c l |}\hline{\mathrm{m o m e n t u m m o d e s}}&{\leftrightarrow}&{\mathrm{w r a p p e d N S 5 - b r a n e s}}\\ \hline{\mathrm{w r a p p e d s t r i n g s}}&{\leftrightarrow}&{\mathrm{K K - m o n o p o l e s}}\\ \hline{}\\ \end{array}
1=\|\psi^{D}\|^2_{L^2(0,\delta)}=C^2\Big(\dfrac{e^{2\delta k}-e^{-2\delta k}}{2k}-2\delta\Big),C^2=\dfrac{2ke^{-2\delta k}}{1-4\delta k e^{-2\delta k}-e^{-4\delta k}},
\langle \rho, y(\varepsilon,\cdot) \rangle = \left\langle \frac{1}\varepsilon \tilde{y}(1,\cdot), \rho \right\rangle \geq k_2 \varepsilon^{-1/2}|\tilde{U}|_{H^{-1/4}(0,1)}^2 \geq k_2 |U|_{H^{-1/4}(0,\varepsilon)}^2,
\limsup_{[0,\infty)\ni s \to t}\|(e^{At}-e^{As})B\|^2_{HS(U,H_1)}= \limsup_{[0,\infty)\ni s \to t}\Bigg[ \sum_{u\in\mathbb{U}}\|(e^{At}-e^{As})Bu\|^2_{H_1}\Bigg] =0.
2T=c^{2}=(\frac{ds}{d\tau})^{2}=g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=(1-\frac{r_{s}}{r})c^{2}(\frac{dt}{d\tau})^{2}-\frac{1}{1-\frac{r_{s}}{r}}(\frac{dr}{d\tau})^{2}-r^{2}(\frac{d\varphi}{d\tau})^{2}
\big(a_k\big)_{k = 1}^\infty = \left(\pi \left(k-\tfrac{1}{2}\right)^2 e^{-\pi \left(k-\tfrac{1}{2}\right)^2 x/2}\right)_{k = 1}^\infty \big(b_k\big)_{k = 1}^\infty = \left(e^{-\pi \left(k-\tfrac{1}{2}\right)^2 x/2}\right)_{k = 1}^\infty \, .
\sum_{k=0}^{m}\frac{(1+aq^{4k+1})\,(a^2q^2;q^4)_k (bq^2;q^4)_k (cq^2;q^4)_k}{(1+aq)\,(a^2q^4/b;q^4)_k (a^2q^4/c;q^4)_k (q^4;q^4)_k}\biggl(\frac{aq}{bc}\biggr)^k\equiv 0\pmod{(1-a^2q^{2n})};
(Q_1 Q_2^{*})(k+1:n, 1:k) = \sigma e_1 e_k^{*}Y_1^{*}+ \underbrace{X_2 V_2}_{\widetilde{V}_2}U_2^{*}&= \sigma e_1 Y_1(:,k)^{*}+ \widetilde{V}_2 U_2^{*}
C\int_{|y| \le t^{\beta/\alpha}}|y|^{d-1}\left( t^{-\beta d/\alpha}- |y|^{\alpha}t^{-\beta}t^{-\beta d/\alpha}\right)dy \le C t^{-\beta d/\alpha}t^{\beta d/\alpha}- C t^{\beta + d\beta/\alpha - \beta - \beta d/\alpha}= C t^{0}.
T(f_j)(x) =T(g)(x)+T(h_j)(x)=T(g)(x)+\int_{2R\leqslant d(0,y)\le j}K(x,y)f(y)d\omega(y) &= T(g)(x)+\int_{2R\leqslant d(0,y)\le j}[K(x,y)-K(0,y)]f(y)d\omega(y)+c_j-C(R),
\frac{\partial \eta(u^{\varepsilon})}{\partial t}+ \displaystyle\sum_{j=1}^{d}\frac{\partial q_{j}(u^{\varepsilon})}{\partial x_{j}}= \varepsilon\, \eta^{'}(u^{\varepsilon}) \frac{\partial}{\partial x_{j}}\left(B(u^{\varepsilon})\frac{\partial u^{\varepsilon}}{\partial x_{j}}\right).
B_{\Delta 1}^{j}= \frac{i}{2}\int [ 1 2 3 ] \left[ \sum_{l = 1}^{2}\left( p_{l}^{-}\frac{\partial}{\partial p_{l}^{j}}+ 2 p_{l}^{j}\frac{\partial}{\partial p_{l}^{+}}\right) r_{\Delta}( 1 2 ) \right] \tilde{\delta}a_{1}^{\dagger}a_{2}^{\dagger}a_{3}- h . c . ,
R_7:=R_6[Y_{33};\alpha_7,\delta_7]; \alpha_7: Y_{23}\mapsto t^{-1}Y_{23}, \Y_{13}\mapsto t^{-1}Y_{13}, \Y_{32}\mapsto t^{-1}Y_{32}, \Y_{31}\mapsto t^{-1}Y_{31}; \delta_7: Y_{11}\mapsto (t^{-1}-t)Y_{13}Y_{31}, \Y_{12}\mapsto (t^{-1}-t)Y_{13}Y_{32}, \Y_{21}\mapsto (t^{-1}-t)Y_{23}Y_{31}, Y_{22}\mapsto (t^{-1}-t)Y_{23}Y_{32}.
\left|\frac{\Re G_{\nu}(z)-G_{\nu}^{*}(0)}{\Im G_{\nu}(z)}\right| =\frac{|z|}{\Im z}\left|\frac{\Re G_{\nu}(z)-G_{\nu}^{*}(0)}{z}\right|\frac{\Im z}{-\Im G_{\nu}(z)}&\leq \sqrt{1+\delta^2}\left|\frac{G_{\nu}(z)-G_{\nu}^{*}(0)}{z}\right|\frac{2}{c}& \rightarrow 2\sqrt{1+\delta^2}.
\begin{cases}\mathcal{H}_{q}e_{0}=\sum_{k=0}^{p}a_{q}^{(k)}e_{k}, \mathcal{H}_{q}e_{n}=e_{n-1}+\sum_{k=0}^{p}a_{n+q}^{(k)}\,e_{n+k},n\geq 1,\end{cases}
C_{\omega,\mathbf{c}}:=\left\{U\in \mathcal{H}^1\backslash \{(\mathbf{0},\mathbf{0},\mathbf{0})\}\ |\S_{\omega,\mathbf{c}}(U)< \widetilde{\mu}_{\omega,\mathbf{c}},\I_{\omega,\mathbf{c}}(U)<0,\ \mathbf{c}\cdot \mathbf{P}(U)\ge 0\right\}.
\int_0^\infty [(u-\psi_n)']^2\,dt = \int_0^\infty [(1-\phi_n)u' - \phi_n' u]^2\,dt &\le 2 \int_0^\infty \Bigl((1-\phi_n)^2[u']^2 + [\phi_n']^2 u^2\Bigr)\,dt,
\left\| \nabla^{k}_{x}\left( K_{1}(t)g-\mathcal{F}^{-1}\left[e^{-\frac{\nu t |\xi|^{2 \sigma}}{2}}\cos (t |\xi|) \right] \ast g\right)\right\|_{2}& \le C (1+t)^{-\frac{n}{4 \sigma}-\frac{k}{2 \sigma}-2+\frac{1}{2 \sigma}}\| g \|_{1}+ C e^{-ct}\| \nabla^{k}_{x}g\|_{2},
S^\wedge_a(x,y) =-\frac{1}{2 \pi}\: \delta(\xi^2)\: \Theta(-\xi^0) + 2\, \: \Theta(\xi^2)\: \Theta(-\xi^0) \:H_a(x,y) &= -\frac{1}{2 \pi}\: \delta(\xi^2)\: \Theta(-\xi^0) + \frac{a}{8 \pi}\: \Theta(\xi^2)\: \Theta(-\xi^0) + \O\big( a^2 \big) \:,
\frac{1}{r_1^2}\int_{\{\Psi_1\le2\log r_1\}}|f|^2 \ge \lim_{j\rightarrow+\infty}\frac{1}{r_1^2}\int_{\{\Psi_1\le2\log r_1\}\cap D_j}|f|^2 \ge \lim_{j\rightarrow+\infty}C_j \ge G(0;\Psi_1,I_+(\Psi_1)_o,f).
\frac{1}{n}e_{2}\otimes([\alpha_{2},\alpha_{1}]_{A^{-}}\alpha_{3}-\langle\alpha_{2},\alpha_{1}\rangle\alpha_{3})=\frac{1}{2}e_{2}\otimes([\alpha_{3},\alpha_{2}]_{A^{-}}\alpha_{1}+(\alpha_{3}\circ\alpha_{2})_{A^{+}}\alpha_{1}-[\alpha_{3},\alpha_{1}]_{C}\alpha_{2}-(\alpha_{3}\circ\alpha_{1})_{E}\alpha_{2}),
S_{\mathrm{c l}}= \mathrm{t r}\left( \hat{\bar{\sigma}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}D_{\alpha}\hat{\sigma}^{\alpha}\right) + \mathrm{t r}\left({\hat{\bar{\sigma}}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}\left( e^{- V}- 1 \right) D_{\alpha}{\hat{\sigma}}^{\alpha}\right) \, .
D_{\vec{m}\vec{n}}=D^{\theta_{(1)}}_{m_1n_1}(z_1,\bar{z}_1) D^{\theta_{(2)}}_{m_2n_2}(z_2,\bar{z}_2) \dots D^{\theta_{(p/2)}}_{m_{p/2}n_{p/2}}(z_{p/2},\bar{z}_{p/2}),
(p_1)_\# ( a ) = ( \beta_{1,1}, \beta_{1,2}) \; (p_1)_\# ( b l_\sigma (b)) = (r + 1 , s - \beta_{2,1}) + ( -1 , 1 + 2 \beta_{2,1}) + ( r + 1 , s - \beta_{2,1}) = ( \beta_{1,2}, \beta_{2,2}),
c_2(\ell,k)=\frac 1{(2\pi i)^2}\int_{\Gamma_{D}}dw\int_{\gamma_{\tau}}d\omega\frac{G_{n_2-k,\Delta\mu,\Delta\xi}(w)}{G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(w-\omega)},
K_\alpha^{\beta,\,\rho}:= \beta^{-1}([0,\rho))\cap K_\alpha =\{u\in\mathcal{C}(I)\ :\ \alpha (u)\ge0,\ 0\le\beta(u)<\rho\}, K_\alpha^{\gamma,\,\rho}:= \gamma^{-1}([0,\rho))\cap K_\alpha =\{u\in\mathcal{C}(I)\ :\ \alpha (u)\ge0,\ 0\le\gamma(u)<\rho\}.
\begin{array}{c}{{\left( q^{S_{3}}\otimes{\frac{E}{z}}+{\frac{E}{w}}\otimes q^{- S_{3}}\right) \left| \left({\frac{n}{2}},{\frac{n}{2}}\right)_{p}, p \right\rangle =}}\\{{= \left({\frac{q^{n / 2}}{z}}-{\frac{q^{- 2 p - 2 +{\frac{n}{2}}}}{w}}\right) \left|{\frac{n}{2}},{\frac{n}{2}}\right\rangle \otimes \left|{\frac{n}{2}}, p -{\frac{n}{2}}+ 1 \right\rangle + \ldots}}\\ \end{array}
[K_1, K_3 ] = L_2, [K_4, L_2 ] = K_4^2+\{K_1, K_4\}+d_2K_4+e_{21}, [L_2, K_1 ] = K_1^2+\{K_1, K_4\}+d_2K_1+e_{22}, d_2=\frac{1}{2}\big(2K_5-Q^{(1)}-Q^{(2)}\big), e_{21}=-\frac{1}{4}\big(Q-Q^{(1)}\big)\big(2K_5+Q^{(2)}\big), e_{22}=\frac{1}{4}\big(Q^{(1)}-Q^{(2)}\big)(2K_5+Q) .
Y_m = y_{\beta^{j(1)}_{k(1)}}^{m_1}\cdots y_{\beta^{j(r)}_{k(r)}}^{m_r}= \Gamma\cdot\left( y_{\beta^1_1}^{m^1_1}\cdots y_{\beta^1_{r_1}}^{m^1_{r_1}}\right) \cdots \left( y_{\beta^\ell_1}^{m^\ell_1}\cdots y_{\beta^\ell_{r_\ell}}^{m^\ell_{r_\ell}}\right).
Q_{C}= i \int d^{3}x \left[ \overline{{c}}^{a}\left( x \right) \nabla_{0}c^{a}\left( x \right) + g \overline{{c}}^{a}\left( x \right) f^{a b c}A_{0}^{b}\left( x \right) c^{c}\left( x \right) \right] ,
\overline w_{2p}'(t) \,>\, -\frac{a}{p}\min_{\beta = 2,4}\biggl\{\Bigl( \frac{\overline w_{2p}(t)}{\overline w_p(t)}\Bigr)^{\beta p}\biggr\}\overline w_{2p}(t) + pb\alpha^2 \overline w_{2p}(t)~, t > 0~.
\sqrt{1-\tfrac{4\gamma\sigma}{1+2\gamma\sigma+(\gamma\beta)^2}}= \sqrt{1-\tfrac{2\sigma/\beta}{1+\sigma/\beta}}=\sqrt{\tfrac{1-\sigma/\beta}{1+\sigma/\beta}}=\sqrt{\tfrac{\beta/\sigma-1}{\beta/\sigma+1}}.
\begin{array}{r}{{\overline{{{M}}}^{n}: ~ ~ 1 + \displaystyle \sum_{i , j = 0}^{n}\eta_{j k}x^{j}x^{k}= 0 ~ ,}}\\{{1 + \displaystyle \sum_{i , j = 0}^{n}\eta_{j k}y^{j}y^{k}= 0 ~ .}}\\ \end{array}
W(G_{2}(n,d,x-1,s))-W(G_{2}(n,d,x,s))\overset{}{\leq}& \overset{}{\leq}W(G_{2}(n,d,x_{2}^{\max}-1,s))-W(G_{2}(n,d,x_{2}^{\max},s))\overset{}{=}& \overset{}{=}2(n-5)+2.
\begin{pmatrix}d\Theta_1 d\bar\Theta_1 d\Theta_2 d\bar\Theta_2\end{pmatrix}=\begin{pmatrix}l k 0 -1 \bar k l -1 0 \bar k l 1 0 l k 0 1\end{pmatrix}\begin{pmatrix}dz d\bar z dw d\bar w \end{pmatrix},
V_{k_1,\dots, k_\ell}(G_N,s_1,\dots, s_\ell,D) &= \sum_{n=1}^{N-\lfloor \frac{N}{\log N}\rfloor}e_{n+d_1^1}(s_1)\dots e_{n+d_{k_1}^1}(s_1) \dots e_{n+d_1^\ell}(s_\ell)\dots e_{n+d_{k_\ell}^\ell}(s_\ell).
\iota(\upsilon) = \frac{-1}{2\pi}\sin(2\pi \upsilon)+\frac{-\sin(2\pi \upsilon) + 4\pi \upsilon - \sin(4\pi \upsilon) + \frac{1}{3}\sin(6\pi \upsilon)}{8\left(-\sin(2\pi \upsilon) + \frac 13 \sin(6\pi \upsilon) - (2 \pi \upsilon + \frac 12 \sin(4\pi \upsilon) - 2\sin(2\pi \upsilon))\right)}\left(-4\upsilon + \frac{1}{\pi}\sin(4\pi \upsilon)\right)
\langle 0 \mid e^{H[\boldsymbol{t}]}\psi^\ast_{k+1/2}\mid -1 \rangle = \langle 0 \mid e^{H[\boldsymbol{t}]}\psi^\ast(z) \mid -1 \rangle \mid_{z^{k}}& = \left[ e^{\sum_{q \geq 1}t_q z^q}\langle 0 \mid \psi^\ast(z) \mid -1 \rangle \right] \mid_{z^{k}}= \left[ e^{\sum_{q \geq 1}t_q z^q}\right] \mid_{z^{k+1}}=: h_{k+1}[\boldsymbol{t}].
|\mathcal K_{\kappa,2}| = \left |\mathcal T_{\kappa,2}\right | + \frac12\left |\mathcal S_{\kappa,2}\right|- \frac12\left |\mathcal T_{\kappa,2}\right | = \frac12\left( \left|\mathcal S_{\kappa,2}\right |+ \left |\mathcal T_{\kappa,2}\right |\right)= \frac12\left( \sum_{i=0}^{\lfloor\kappa/2\rfloor}(\kappa-2i +1)+\binom{\kappa+3}{3}\right ).
T_H(N; y, \alpha) = e(N \alpha) \sum_{n = -H}^y (H - |n|) e(n \alpha) = \frac{e( (N + y + 1) \alpha)}{1 - e(\alpha)}\cdot (y - H) + \frac{e( (N + 1) \alpha)}{(1 - e(\alpha))^2}\cdot \bigl( e(y \alpha) - 2 + e(- H \alpha) \bigr).
\frac{(\beta(T)+3)!}{(\beta(T)+2-n-\alpha_i)!}u^{\beta(T)+2-n-\alpha_i}\sum_{k=n}^{\beta(T)+2-\alpha_i}\frac{k!(\beta(T)+2-k)!}{(k-n)!(\beta(T)+2-k-\alpha_i)!}u^{\beta(T)+2-n-\alpha_i},
\dot \Delta(\dot \lambda_n) = -\dot\theta_N2 \sin \theta_N\big|_{\lambda = \dot \lambda_n}+ 2i \frac{(-1)^{n+1}}{(2i \dot \lambda_n)^{N+1}}\int_0^1\varphi_1 r_{N+1}\,dt + \frac{\ell^2_n}{n^{N+1}}\, .
\log \hat{\rho}_s(A) =\max_{\mu \in \mathcal{M}_f}\left\lbrace \gamma_1(A,\mu)+\gamma_2(A,\mu)+\cdots +\gamma_s(A,\mu)\right\rbrace &= \max_{\mu \in \mathcal{M}_f(Per)}\left\lbrace \gamma_1(A,\mu)+\gamma_2(A,\mu)+\cdots +\gamma_s(A,\mu)\right\rbrace,
D_{1}= \mathrm{diag}\big(\varrho_1 / \eta_{-}, \ldots, \varrho_N / \eta_{-}, 1, \eta_{-}, \ldots, \eta_{-}^{n-1}\big), D_{2}= \mathrm{diag}\big((1 - 2\varrho_{1})^{-1/2}, \ldots, (1 - 2\varrho_{N+n})^{-1/2}\big),
D^+_kf(x) = \sum_{j=1}^n D^+_k a_j(x)\prod_{i\neq j}a_i(x) + \sum_{j=2}^N h^{j-1}\sum_{|I|=j}\prod_{i \in I}D_k^+ a_i(x) \prod_{i\not\in I}a_i(x).
0<\Phi_{I,\lambda,P}-\Phi_{I,\lambda,Q}(f) =f(t_{1})-f(a)-\left[|f(t_{1})-f(x)|+|f(x)-f(a)|\right]+\lambda \le f(t_{1})-f(a)-\left[f(t_{1})-f(x)+f(a)-f(x)\right]+\lambda =2[f(x)-f(a)]+\lambda,
\zeta_{2n-1}(-l,k+n+1-q)=\sum_{i=0}^{2n-2}b_{2n-1,i}\sum_{p=0}^i(q-n-1-k)^{i-p}\binom{i}{p}\frac{B_{l+p+1}(k+n+1-q)}{p+l+1},
Q(\xi) =\sum_{x}(-\alpha-\beta\nu(x))\xi^2_x +\beta\sum\limits_{x\sim y}(\xi_x-\xi_y)^2 &=\sum\limits_{x\in \L}\bigl(-\alpha\xi_x^2-\beta\xi_x\phi(x,\xi)\bigr)=-\sum\limits_{x\in \L}\xi_xU(x, \xi).
\lim_{n\to \infty}\frac{(\rho_p+1)^{n-j_p}h_{j_p}}{h_{n+1}}= \lim_{n\to \infty}\frac{(\rho_p+1)^{n-j_p}h_{j_p}}{\rho_p^{n-j_p+1}h_{j_p}+ \frac{\rho_p(\rho_p^{n-j_p+1}-1)}{2}}&\geq \lim_{n\to \infty}\frac{(\rho_p+1)^{n-j_p}h_{j_p}}{\rho_p^{n-j_p+1}h_{j_p}+ \rho_p^{n-j+2}}= \infty .
-\int_{\mathcal{B}^+(0, R)}(y^{1-2s_1}\nabla w+y^{1-2s_2}\nabla w)\cdot \nabla ((x, y) \cdot \nabla w)\,dxdy &= k_{s_1, s_2}\int_{B(0, R)}(|u|^{p-2}u-\lambda u)(x \cdot \nabla u) \, dx -R \int_{\partial^+\mathcal{B}^+(0, R)}y^{1-2s_1}|\nabla w|^2 + y^{1-2s_2}|\nabla w|^2\,dS,
\partial_{n}, \; n \geq 0 , \, \; \mathrm{\boldmath ~ \g a m m a ~}= \sum_{n \geq 0}( n + 1 ) t_{n + 1}\partial_{n}, \, \; \mathrm{\boldmath ~ \v a r s i g m a ~}= \sum_{n \geq 1}n t_{n}\partial_{n},
2 \pi^{3}x{\binom{-{\frac{3}{2}}}{0}}{\frac{1}{( 2 \pi )^{3}}}\int_{1}^{\infty}{\frac{B_{2}( t - [ t ] )}{t^{3}}}\, d t ={\frac{x}{4}}\left({\frac{7}{1 2}}- \gamma \right) \ ,
\delta_{\epsilon}q^{n}= \epsilon \, \left\{q^{n}, \phi \right\}= \epsilon \, \frac{\partial H}{\partial p_{n}}\ \ , \ \ \delta_{\epsilon}p_{n}= \epsilon \, \left\{p^{n}, \phi \right\}= - \epsilon \, \frac{\partial H}{\partial q^{n}}\ \ , \ \ \delta_{\epsilon}\lambda = \dot{\epsilon}\ ,
E_0^{(\lambda)}= E_0 + \lambda\langle \psi_0|\mathtt{H}_1\psi_0 \rangle + \lambda^2\langle \psi_0|\mathtt{H}_2\psi_0 \rangle - \lambda^2\sum_{k=1}^{\mathtt{w}}\frac{|\langle \psi_0|\mathtt{H}_1\psi_k \rangle|^2}{E_k-E_0}+ \lambda^3E^{(\lambda)}
f_{m}^{\pm}( i k ) \ = \k R \left[ i^{\beta}\left( I_{m}^{\prime}K_{m - \beta}+ I_{m}K_{m - \beta}^{\prime}\right) \ + \i^{- \beta}\left( I_{m}^{\prime}K_{m + \beta}+ I_{m}K_{m + \beta}^{\prime}\right) \right] \ ,
g(z,-m,r_1,r_2) :=\lim_{s\to-m}\frac{\Gamma(s)}{(2\pi i)^s}\biggl\{-\lambda(r_1)(cz+d)^{-s}(e^{\pi is}\zeta(s,r_2)+\zeta(s,-r_2)) &+\lambda(R_1)(\zeta(s,R_2)+e^{-\pi is}\zeta(s,-R_2))\biggr\}.
\frac{\alpha^2}{4}-b_0(x){}^2 u(x)-\frac{1}{4}b_0'(x){}^2+\frac{1}{2}b_0(x) b_0''(x)=0, &b_1(x){}^2 (1-u(x))-\frac{1}{4}b_1'(x){}^2+\frac{1}{2}b_1(x) b_1''(x)+1=0, &b_2(x){}^2 (a-u(x))-\frac{1}{4}b_2'(x){}^2+\frac{1}{2}b_2(x) b_2''(x)=0, &\frac{x^2}{4}-\left(b_0(x)+b_1(x)+b_2(x)\right){}^2=0.
\dot W_i(z_i; y_i) = e^{L_i \tau_i}\dot V_i(x_i; y_{i1}) + L_i e^{L_i \tau_i}V_i(x_i)\, y_{i2}\leq -c_i e^{L_i \tau_i}V_i (x_i) + L_i \delta_i e^{L_i \tau_i}V_i(x_i) = -\tilde c_i W_i(z_i).
\langle f,g\rangle_{L^2_{p,\mathcal{I}}}=\int_{\mathbb{R}^3}\int_0^\infty f({p,\mathcal{I}})g({p,\mathcal{I}})\phi(\mathcal{I})\, d\mathcal{I}dp,\cr\langle f,g\rangle_{L^2_{x,p,\mathcal{I}}}=\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\int_0^\infty f(x,{p,\mathcal{I}})g(x,{p,\mathcal{I}})\phi(\mathcal{I})\, d\mathcal{I}dpdx