harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|v_{3,1,2}(t,r)| \leq \begin{cases}\frac{C r ||e_{1}-e_{2}||_{X}}{t^{2}\sqrt{\log(\log(t))}\log^{b}(t)}, r \leq \frac{t}{2}\frac{C ||e_{1}-e_{2}||_{X}}{r \log^{b}(t) \sqrt{\log(\log(t))}}, r > \frac{t}{2}\end{cases}
\mu(\mathcal{C}_n)= \prod_{i\in\mathcal{I}_b}p^n_U\big(\mathbf{u}(i)\big) \prod_{\substack{\big(\hat{i},j,m\big) \in\mathcal{I}_n\times\mathcal{J}_n\times\mathcal{M}_m}}p^n_{V|U}\Big(\mathbf{v}\big(\hat{i},j,m\big)\Big|\mathbf{u}(\hat{i})\Big).
\mathcal{L}(\mathbf{x}, \mathbf{y})=\frac{1}{2}\mathbf{x}^TH\mathbf{x}+\mathbf{g}^T\mathbf{x}-\mathbf{y}^T(A\mathbf{x}-\mathbf{b}) + I_D(\mathbf{x}, \mathbf{y}),
y_k= \sqrt{P_0}\sum_{i \in \mathcal{K}_{\mathrm{0}}}{{\sqrt{q_i}\mathbf{h}^H_k \mathbf{v}_i x_i}}+ \sqrt{P^{(c)}}\mathbf{h}_k^H\mathbf{v}^{(c)}x^{(c)}+ \sum_{i \in \mathcal{K}_{\mathrm{\alpha}}}{\sqrt{P_i^{(p)}}}\mathbf{h}^H_k \mathbf{v}_i^{(p)}x_i^{(p)}+{n_k},
{D^{{\rm{opt}}}}= \frac{{{{\left({{B_1}+{B_2}+ 2{C_2}}\right)}^{\frac{1}{{v - 1}}}}}}{{{{\left({{B_1}+{B_2}+ 2{C_1}}\right)}^{\frac{1}{{v - 1}}}}+{{\left({{B_1}+{B_2}+ 2{C_2}}\right)}^{\frac{1}{{v - 1}}}}}}
\zeta_{\xi/t}(|\xi|) = |\xi| \left(\hat{\psi}_{\xi/t}(|\xi|)-\hat{\psi}_{\xi/t}(0)\right) \eta (|\xi|^2) +|\xi| \hat{\psi}_{\xi/t}(0) \eta (|\xi|^2)+ |\xi|\hat{\psi}_{\xi/t}(|\xi|) (1-\eta (|\xi|^2 )) &= R^{(a)}_{\xi/t}(|\xi|) + R^{(b)}_{\xi/t}(|\xi|)+ R^{(c)}_{\xi/t}(|\xi|).
\begin{array}{l}\chi(X_1 \#_{\psi}X_2)=\chi(X_1)+\chi(X_2)+(g-1), c_{1}^{2}(X_1 \#_{\psi}X_2) = c_{1}^{2}(X_1)+ c_{1}^{2}(X_2)+8(g-1)\end{array}
\prod_{(i,j)\in F_k}X_{i,j}=\frac{L_{1,b-k+1}}{L_{1,b-k+2}}\frac{L_{2,b-k+2}}{L_{2,b-k+3}}\cdots \frac{L_{a,n-k}}{L_{a,n-k+1}}=\frac{\prod\limits_{i=1}^aL_{i,b-k+i}}{\prod\limits_{i=1}^aL_{i,b-k+i+1}}.
\frac{\partial F}{\partial\zeta_i}=0,\qquad \zeta_i=\zeta_{i-1},\ 2\leq i\leq N \frac{\partial F}{\partial{t}}=\sum_{n=1}^N\left(\frac{1}{2}\frac{\partial^2}{\partial\zeta_n^2}+\mu_n\frac{\partial}{\partial\zeta_n}\right)F.
S_{\pm}( x , y ) = \langle q_{\alpha}^{a}( x ){\bar{q}}_{\beta}^{b}( y ) \rangle = \int d^{4}z \frac{\left[ \Psi_{z}^{\pm}( x ){\bar{\Psi}}_{z}^{\pm}( y ) \right]_{\alpha \beta}^{a b}}{m^{*}}
{\cal D}_{0}^{- 1}= \left( \begin{array}{c c}{{\left( 2 \beta p^{2}\theta_{\mu \nu}+ 2 p^{2}\omega_{\mu \nu}\right)}}&{{0}}\\{{0}}&{{\left( p^{2}- \mu^{2}\right)}}\\ \end{array}\right) \mathrm{{.}}
\sum_{n=0}^\infty\overline{\textrm{spt}}_\omega{\left(2\times5^{2m}(5n+2)+5^{2m}\right)}q^n =\sum_{i=1}^\infty x_{2m,i}U(q^{-2}\delta\xi^{i-1}) =\sum_{i=1}^\infty x_{2m,i}\gamma\sum_{j=1}^{5i}\beta_{i,j}\xi^{j-1}=\gamma\sum_{j=1}^\infty{\left(\sum_{i=1}^\infty x_{2m,i}\beta_{i,j}\right)}\xi^{j-1}=\gamma\sum_{j=1}^\infty x_{2m+1,i}\xi^{j-1},
\phi(t,q) := (1+t^{a(1)}q+t^{2a(1)}q^2+\ldots)(1+t^{a(2)}q^2+t^{2a(2)}q^4+\ldots) \times\cdots &\times (1+t^{a(n)}q^n+t^{2a(n)}q^{2n}+\ldots)\times \cdots.
\begin{aligned}|\tilde{M}_s|^2|H-2|^2+|\tilde{M}_s|^2|\mathring{A}|^2\leq~ Ce^{-2^{(2-\alpha)}\cdot t}|\tilde{M}_s|^3|\nabla A|^2 \leq~ Ce^{-3\cdot 2^{(1-\alpha)}\cdot t}\end{aligned}
\sqrt{(f') + J(f')}\; = \; \sqrt{\bigl(x_1x_2'^2+x_1x_3'^2,x_3'+2x_1^2x_2',x_2'+2x_1^2x_3',x_2'x_3'\bigr)}= \; \sqrt{\bigl(x_3'+2x_1^2x_2',x_2'+2x_1^2x_3',x_2'x_3',x_1x_2',x_1x_3'\bigr)}\; = \; \bigl(x_2',x_3'\bigr)
\theta_t(v) := \frac{\eta_t v^2}{(1+\delta)^2}+\frac{(1+\delta-v \eta_t)^2}{(1+\delta)^2 (1-\eta_t)}- 1 \ge \theta_t(1) \ge \frac{\eta_t \delta^2}{(1+\delta)^2}\,.
E_{g_{1}}E_{g_{2}}=V_{b_{1}^{'}}^{*}E_{a_{1}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{*}E_{a_{2}^{'}}V_{b_{1}^{'}}= V_{b_{1}^{'}}^{*}E_{a_{2}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{*}E_{a_{1}^{'}}V_{b_{1}^{'}}= E_{g_{2}}E_{g_{1}}.
\Lambda_R(C) =\big\{\rho > 0 :\; \det \big( e^{4iR\rho}C(\rho) - \mathrm{Id}\big) = 0 \big\}, \Lambda^*_R(C) =\big\{\lambda > 0:\; \det \big( e^{4iR\lambda}C(0) - \mathrm{Id}\big) = 0 \big\}.
d\tilde s^2= F ds^2 = F \left ( w_1 w_2 w_3 \, d\mu^2 +\frac{w_2 w_3}{w_1}\sigma_1^2 +\frac{w_3 w_1}{w_2}\sigma_2^2+\frac{w_1 w_2}{w_3}\sigma_3^2 \right ),
\begin{aligned}f_m=0, \qquad f_{m+k}= \frac{\delta y_n}{g_{n+1}}\sum_{l=0}^{k-1}u_{m+l,n}\, \Delta x_{m+l}, &g_n=0, g_{n+k}= \frac{g_{n+1}}{u_{m,n}\delta y_n}\sum_{l=0}^{k-1}u_{m,n+l}\, \delta y_{n+l},\end{aligned}
\overline{R}=\overline{\mathbf{Q}}[Z_i^+,Z_i^- \mid 0 \leqslant i \leqslant n]/\left(\Delta_{jk}Z_i^+Z_i^-+\Delta_{ki}Z_j^+Z_j^-+\Delta_{ij}Z_{k}^+Z_{k}^-\right)_{1 \leqslant i<j<k\leqslant n}
P^{N+1}((\xi^{(N+1)}-\mu_{N})^{\top}\Sigma_{N}^{-1}(\xi^{(N+1)}-\mu_{N})\ge\lambda^{2})\le min\{1,\frac{n_{\xi}(N^{2}-1+N\lambda^{2})}{N^{2}\lambda^{2}}\}.
\tilde{J}= \frac{6 4}{3}\frac{1}{({\cal V}_{2 , 2}^{( 1 )}+{\cal V}_{2 , 2}^{( 2 )})^{2}}\frac{{\cal V}_{2 , 2}^{( 2 )}}{{\cal V}_{2 , 1}^{( 1 )}}.
\frac{\partial(\alpha,\beta)}{\partial(\rho,w)}=\left( \begin{array}{cc}\eta/\rho 1 \eta/ \rho -1 \end{array}\right).\intertext{Therefore,}\frac{\partial(\rho,w)}{\partial(\alpha,\beta)}=\left( \begin{array}{cc}\rho/2\eta \rho/2\eta 1/2 -1/2 \end{array}\right).
\nabla^{2}( \nabla^{2}- s^{2}) A_{0}= - \nabla^{2}\rho - s \overrightarrow{\nabla}\times \overrightarrow{j}- s v_{o}\nabla^{2}\varphi + \nabla^{2}\left( \overrightarrow{v}\times \overrightarrow{\nabla}\varphi \right) .
\Gamma=\frac{1}{2}\int d^3xd^3y\,A_i(x)\left(K(x-y)+\frac{\lambda}{2}\,K^2(x-y)\right)A_i(y)+(A_i\longleftrightarrow A_i+\partial_i\varphi )-\frac{\lambda}{4}\int d^3x\,F_{ij}F_{ij}.
=\pi^{-(1-s)}\Gamma(1-s)\frac{\Gamma(s)^{2}}{\Gamma(s+2q)\Gamma(s-2q)}(ab)^{2s-3/2}c^{-(1-s)}\sum_{j=0}^{\infty}{2j+4q \choose j}\frac{d^{j+2q}\Gamma(j+2q-1+s)}{2^{2q-1}c^{j+2q}\Gamma(-1+s)}(1/2)^{j}.
\widehat{S}_{( 0 )}p^{( 0 )}= p^{( 0 )}, ~ ~ \widehat{S}_{( 0 )}p^{( \widetilde{0})}= p^{( \widetilde{0})}, ~ ~ S_{( 1 )}^{2}p^{( 1 )}= p^{( 1 )}, ~ ~ S_{( 1 )}^{2}p^{( \widetilde{1})}= p^{( \widetilde{1})}.
\frac{\partial n_p}{\partial\rho^\ast}=-\frac{\sigma_b^2\left[2\kappa+n\rho^\ast\lambda_{ab}-2\sqrt{\kappa(\kappa+n\rho^\ast\lambda_{ab})}\right]}{2\lambda_{ab}{\rho^\ast}^2\sqrt{\kappa(\kappa+n\rho^\ast\lambda_{ab})}}.
\sigma^\mu_\partial(T)(x^\prime,\xi^\prime)u= \sum_{j=0}^{d-1}\sigma_\psi^{\mu-j-1/2}(S_j)(x^\prime,\xi^\prime)\frac{d^ju}{dx_n^j}(0) + \sigma^\mu_\partial(T_0)(x^\prime,\xi^\prime)u.
| u \rangle_{\tau}= N ( u ) \cdot \frac{\sqrt{u}}{\sqrt{1 - u \tau}}\cdot e x p \Bigl ( \sum_{m = 1}^{\infty}\left( \frac{m - u}{m + u}\right) a_{m}^{\dagger}\tilde{a}_{m}^{\dagger}e^{2 m \tau}\Bigr ) e x p \Bigl ( - \frac{u \hat{q}^{2}}{4 ( 1 - u \tau )}\Bigr ) | 0 \rangle \, .
\lambda(g_c,0) =\int_M (R_c-\frac{1}{12}|H_c|^2)\omega_c^2+4|\nabla\omega_c|^2 dV_{g_c}&\leq \int_M (R_c-\frac{1}{12}|H_c|^2)\omega^2_{\infty}+4|\nabla\omega_{\infty}|^2dV_{g_c}=\lim_{i\to\infty}\lambda(g_i,b_i) &\leq \lim_{i\to\infty}\mathcal{F}(g_i,b_i,f_c)=\lambda(g_c,0).
L_{n}^{(p,q),\gamma}(f;x)=\frac{1}{\ell_{n}^{p,q}(x)}\sum_{k=0}^{n}f\left(\frac{p^{n-k+1}[k]_{p,q}+\gamma}{b_{n,k}}\right) p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}}\left[\begin{array}{c}n k\end{array}\right]_{p,q}x^{k},~~~~~~(\gamma \in \mathbb{R})
\widetilde{B}=\limsup_{q\rightarrow \infty}\sup_{f\in L^{Q}_{1}(B(x_{0},r))\backslash\{0\}}\frac{\left\|\frac{f}{|\cdot|^{\frac{\beta}{q}}}\right\|_{L^{q}(B(x_{0},r))}}{q^{1-1/Q}\|\nabla_{H}f\|_{L^{Q}(B(x_{0},r))}}.
N_{+}(x) = N_{-}(x) \begin{pmatrix}0 x^\beta 0 -x^{-\beta}0 0 0 0 1\end{pmatrix}, x \in (0,q), N_{+}(x) = N_{-}(x) \begin{pmatrix}1 0 0 0 0 -1 0 1 0\end{pmatrix}, x<0.
(\phi, Z) = - \Big[ \rho_1^{-1}(D\Phi^W_{(g_1, \pi_1)}-D\Phi^W_{(g_2, \pi_2)})(h_2, w_2) + \rho_1^{-1}\Big( Q^W_{(g_1,\pi_1)}(h_1, w_1)-Q^W_{(g_2,\pi_2)}(h_2, w_2)\Big) & + \rho_1^{-1}D\Phi^W_{(g_1, \pi_1)}\Big( \big( \rho_1 (D\Phi^W_{(g_1, \pi_1)})^*-\rho_2 (D\Phi^W_{(g_2, \pi_2)})^*\big)(f_2, X_2)\Big)\Big].
T_{\mu \nu}^{m a t t e r}= \frac{1}{4}\left( H_{\mu \nu}^{2}- \frac{1}{6}\tilde{G}_{\mu \nu}H^{2}\right) e^{- \frac{8}{\sqrt{2 | D - 2 |}}\tilde{\phi}}+ \left( \partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}- \frac{1}{2}( \partial \tilde{\phi})^{2}\tilde{G}_{\mu \nu}\right) ~ .
|I_2| \le C_1 \sum_{a\in\alpha}\mu_Y(a)D_\theta V_\omega(a) d_\theta(y_a,y_a') \le \theta C_1C \sum_{a\in\alpha}\mu_Y(a)|v|_\eta r(a) d_\theta(y,y') =\theta C_1C |v|_\eta |r|_1 d_\theta(y,y').
\begin{aligned}\Phi^0_j(q_j, t) = -(1+o(1))\int_{t_0}^{t}\frac{\dot{\mu}_j(\tilde{s})}{\mu_j(\tilde{s})}\mu_j^{-(n-2s)}(\tilde{s})F\left(\frac{(t-\tilde{s})^{\frac{1}{2s}}}{\mu_j(\tilde{s})}\right)d\tilde{s}&= \frac{2s b_j^{4s-n}}{(n-4s)c_{n,s}^{n-4s}}A + o(1):= B b_j^{4s-n}+ o(1)\end{aligned}
\left\| R_{\alpha}^{i}( A ) \right\|_{A = A_{0}}= \left( \begin{array}{c c c}{{0}}&{{\bar{\partial}- h \partial + h^{\prime}}}&{{B^{\prime}- 2 B \partial}}\\{{0}}&{{0}}&{{\bar{\partial}- h \partial + 2 h^{\prime}}}\\ \end{array}\right) ,
e^{-\xi/D}\le \frac{\Gamma(\xi)}{\Gamma_0}\le e^{-\delta_0 \xi}\ , \xi\in [0,\infty)\ , \;\;\;\;\;\; \delta_0 := \frac{1}{4D+2\Gamma_0 H_* \|\sigma'\|_{L_\infty(0,\Gamma_0)}}>0\ .
\hat{\rho}_{C}=\frac{(\sum_{i=1}^{k}\hat{\lambda}_{i})^{2}}{(\sum_{i=1}^{k}\hat{\lambda}_{i})^{2}+\sum_{i=1}^{k}\hat{\sigma}_{e_{i}}^{2}}=\frac{106.22}{106.22+18.01}=.8550
\langle\chi_{k^{\prime}}|[P_{A\alpha},H_{e}]|\chi_{k}\rangle_{(r)}=iZ_{A}\sum_{i}\langle\chi_{k^{\prime}}|\frac{(r_{iA})_{\alpha}}{r_{iA}^{3}}|\chi_{k}\rangle_{(r)}withr_{iA}\equiv r_{i}-R_{A}.
\|f\|_{B_{p,\infty}^{\sigma}(\mathbb{M})}:= \sup_{t>0}t^{-\sigma/2m}\inf_{g\in W_{p}^{2m}(\mathbb{M})}\bigl(\|f-g\|_{L_p(\mathbb{M})}+ t\|g\|_{W_{p}^{2m}(\mathbb{M})}\bigr)
B(a, b)^{2}= \binom{a+b}{a}^{-2}\cdot \frac{(a+b)^2}{a^2 b^2}= \int_0^{\infty}\int_0^{\infty}\frac{(ts)^{a-1}}{\left[(1+t)(1+s)\right]^{a+b}}dt\ds,
\Theta(z) =\begin{bmatrix}\Theta_{11}(z) \Theta_{12}(z) \Theta_{21}(z) \Theta_{22}(z)\end{bmatrix}&=I+(z-\mu)\begin{bmatrix}E N\end{bmatrix}(I-zT)^{-1}P^{-1}(\mu I-T^*)^{-1}\begin{bmatrix}E^* -N^*\end{bmatrix}.
(\frac{p_{\mu}u^{\mu}}{c})^{2}+p_{\mu}p^{\mu}=\frac{E_{1}^{2}}{c^{2}}-(mc)^{2}=(\gamma_{1}^{2}-1)(mc)^{2}=\gamma_{1}^{2}v_{1}\cdot v_{1}m^{2}=p_{1}\cdot p_{1}.
\begin{aligned}\frac{d}{dt}F(A^{(t)}) =\int_{A}\sum_{i=1}^{n}V_{x_{i}}(\Psi(x,t),t)\Psi_{t}^{(i)}(x,t)J\Psi(x,t)+V(\Psi(x,t),t)\frac{d}{dt}(J\Psi(x,t)) &\qquad+V_{t}(\Psi(x,t),t)J\Psi(x,t).\end{aligned}
C_z(n) = \sum_{0\leq a \leq n}M_z(a)D_z(n-a) &= q^n \left[ \sum_{0\leq a < n}\frac{M_z(a)}{q^a}\frac{(n-a)^{z-1}}{\Gamma(z)}+ O_A\left(\sum_{0\leq a < n}\frac{|M_z(a)|}{q^a}(n-a)^{\Re{z}-2}\right) + \frac{M_z(n)}{q^n}\right].
H_{j}\;=\; \frac{i}{2}[\:\eta\tau_{j}^{y}\tau_{j+1}^{x}\: -\:\eta^{-1}\tau_{j}^{x}\tau_{j+1}^{y}\: +\: q \tau_{j}^{x}\tau_{j}^{y}\: +\: q^{-1}\tau_{j+1}^{x}\tau_{j+1}^{y}].
\tilde{\alpha}_{- m}^{( A )}\rightarrow \frac{1}{\sqrt{m}}\tilde{\alpha}_{- m}^{( A )}\, \, , \, \, \tilde{A}_{- r}\rightarrow \frac{1}{\sqrt{r}\gamma}\tilde{A}_{- r}\, \, , \, \, \tilde{B}_{- s}\rightarrow \frac{1}{\sqrt{s}\gamma}\tilde{B}_{- s}\, .
({\mathrm{D i}}\oplus{\mathrm{R a c}}) \otimes ({\mathrm{D i}}\oplus{\mathrm{R a c}}) = ( D ( 1 , 0 ) \oplus D ( 2 , 0 ) ) \oplus 2 \bigoplus_{s = \frac{1}{2}}^{\infty}D ( s + 1 , s ) .
\alpha_6 = 35+24\sqrt{2}-20\sqrt{3}-14\sqrt{6}, \intertext{and}\widehat{D}_6\!\left(e^{-\pi/\sqrt{6}}\right) =\sqrt{\frac{111}{16}+5\sqrt{2}+\frac{33}{8}\sqrt{3}+\frac{45}{16}\sqrt{6}}.
\mathsf{d}_{cc}(y(\{X_{ni}\}_{n=1,\dots,N,i=1,\dots,j})) =\mathsf{d}_{cc}\Bigg(\prod_{n=1}^N[X_{n1},\dots,X_{nj}]_c\Bigg) \leq \sum_{n=1}^N\mathsf{d}_{cc}([X_{n1},\dots,X_{nj}]_c).
u_{0,1}(x)= \frac1{\sqrt{2}}+\frac{\bf{i}}{\sqrt{2}},u_{0,2}(x)=1,u_{0,3}(x)=2x, u_{0,4}(x)= \left(1-\sqrt{\frac{\pi}{2}(\exp{\frac14}-1)}\right)(1-\exp{(x(1-x))}), u_{0,5}(x)= c_*(\frac{x}{\sqrt{2}})\exp{({\bf{i}}\frac{x}2)}
(e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}w|H_{-\tau}e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}&=(w|e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}H_{-\tau}e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}&=(w|e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}.
\langle l|\psi(t+T)\rangle=exp(-i\frac{l^{2}\hbar T}{2I})\sum_{n,m=-\infty}^{\infty}(-i)^{n}J_{n}(\frac{KT}{\hbar})[\int_{-\pi}^{\pi}\frac{d\theta}{2\pi}e^{i(m-l-n)\theta}]\langle m|\psi(t)\rangle
{}_4\phi_3\!\left[\begin{matrix}q^{-n},a,b,c d,e,abcq^{1-n}/de\end{matrix};q,q\right]=\frac{(e/a,de/bc;q)_n}{(e,de/abc;q)_n}\,{}_4\phi_3\!\left[\begin{matrix}q^{-n},a,d/b,d/c d,aq^{1-n}/e,de/bc\end{matrix};q,q\right].
\sum_{k=1}^{N}\big|a_{k}^1\big|\le C\Delta^{\frac12-5\kappa}(1+|x|_{L^{\max(p,2q)}})^{K+1}\int_{0}^{T}\frac{1}{(T-t)^{1-\kappa}}\bigl(1+\frac{1}{t^{1-\kappa}}\bigr)dt.
\int_H \phi_R(U) d \mu(U) = \int_{B(\rho)}\int_{H}\phi_R(\tilde{U}) P_t(U, d\tilde{U}) d \mu (U) + \int_{B(\rho)^c}\int_{H}\phi_R(\tilde{U}) P_t(U, d\tilde{U}) d \mu (U) &\leq C \exp\left(\eta \left(e^{-\epsilon_1 t}\kappa_1 \rho^2 + C_5\right) \right) + R \mu( B(\rho)^c).
- \frac{1}{8 \pi g^{2}}\int d^{2}x t r \sum_{\mu \neq 2 , 3 , 4 , 5}\left[ ( ( 2 M + 1 ) / B + ( 2 N + 1 ) / B + U^{2}) \left( c_{\mu}c_{\mu}^{\dagger}+ c_{\mu}^{\dagger}c_{\mu}\right) \right]_{\star}.
\nu_{m}= \begin{array}{r l}{\frac{n m}{2},}&{\quad \mathrm{f o r m \ne 0 m o d n + 1}}\\{\frac{n m}{2}+ ( n + 1 ) b_{h_{m}+ 1},}&{\quad \mathrm{f o r m = 0 m o d n + 1}.}\\ \end{array}
D(t) = \int_U a_0(x)^{-1}|\nabla\Phi(x,t)|^2dx+\Big(\int_U a_0(x)^{-1}|\nabla\Phi(x,t)|^2dx\Big)^\frac12 +\Big(\int_U |\Phi_t(x,t)|^2\phi(x) dx\Big)^\frac12, h_1(t) =B_1+ \sum_{i=1}^2\int_U H(x,|\nabla p_i(x,t)|) dx, h_2(t) = 1+\sum_{i=1}^2 \int_U\Big [H(x,|\nabla p_i(x,t)|) +\bar p_i^2(x,t)\phi(x)\Big]dx.
\begin{aligned}x^{1,+}=x^{1}+u^{1}T\cos\left(x^{3}+u^{2}\tfrac{T}{2}\right)\tfrac{\sin\left(u^{2}\tfrac{T}{2}\right)}{u^{2}\tfrac{T}{2}}x^{2,+}=x^{2}+u^{1}T\sin\left(x^{3}+u^{2}\tfrac{T}{2}\right)\tfrac{\sin\left(u^{2}\tfrac{T}{2}\right)}{u^{2}\tfrac{T}{2}}x^{3,+}=x^{3}+u^{2}T\,,\end{aligned}
\frac{1}{2}\frac{d\|X^{x_2}(t)-X^{x_1}(t)\|^2}{dt}=\langle f(X^{x_2}(t))-f(X^{x_1}(t)),X^{x_2}(t)-X^{x_1}(t)\rangle\le -\gamma\|X^{x_2}(t)-X^{x_1}(t)\|^2,
\left[\frac{d^{2}}{d\varrho^{2}}+ \widetilde{E}-{\rm sgn}\left( \widetilde{\beta}\right)\, \widetilde{\lambda}\varrho^{\widetilde{\beta}}- \frac{\left(\widetilde{l}+ \widetilde{\nu}\right)^{2}- 1/4}{\varrho^{2}}\right] \widetilde{u}_{l}(\varrho) = 0\; ,
s = \psi(t,x), y^{i}=\phi^{i}(x), t^{*}= \left(\frac{\partial \psi}{\partial t}(t,x)\right) s^{*}, x^{*}_{a}= \left(\frac{\partial \phi^{i}}{\partial x^{a}}(x) \right) y^{*}_{i}+\left(\frac{\partial \psi}{\partial x^{a}}(t, x) \right)s^{*}.
\frac{i}{4 \pi}\int_{- \infty}^{\infty}\left( \frac{\sinh \left( \frac{\alpha}{2}\right)}{\cosh \left( \frac{\alpha}{2}\right)}\right) e^{- i \alpha x}d \alpha = \int_{- \infty}^{\infty}\frac{\cosh ( \pi u )}{\sinh ( 2 \pi u )}\delta ( x - u ) d u
f(T)=F(T) - \sum_i F(T_i) = \log \left(\frac{I(T)}{\prod_i I(T_i)}\right) = \log \left( \frac{I(T)}{I_0(T)}\right) = \log \left(1 + \prod_i \frac{I_0(T_i)}{I(T_i)}\right).
p^2\cdot\frac{(1)_{p-1}^2}{(\frac32)_{p-1}^2}= \frac{\Gamma_p(1+p)^2\Gamma_p(\frac12)^2}{\Gamma_p(1)^2\Gamma_p(\frac12+p)^2}\equiv 1+p\cdot\frac{\Gamma_p(\frac12)^2}{\Gamma_p(1)^2}\cdot\frac{d}{dx}\bigg(\frac{\Gamma(1+x)^2}{\Gamma(\frac{p+1}{2}+x)^2}\bigg)\bigg|_{x=0}=1-2pH_{p-1}\pmod{p^2},
\mathbf{P}_{p}^{D}[\mu ](x,t) \leq (p-2)D+\frac{1}{2(p-1)^{p-1}}\sum_{i=0}^{\infty}\frac{|\mu |(\tilde{Q}_{\rho_{i}}(x,t))}{\rho_{i}^{N}}& \leq (p-2)D+C_{5}\int_{0}^{2D}\frac{|\mu |(\tilde{Q}_{\rho}(x,t))}{\rho^{N}}\frac{d\rho}{\rho}.
f'_{h-1}+1-f_{h-1}=\frac{1}{2}\left(f_h-(k-2)f_{h-1}-2\right)+1-f_{h-1}begin{align*}5pt] =\frac{1}{2}(f_h-kf_{h-1}) begin{align*}5pt] =\frac{1}{2}\left(\frac{2n-1+(-1)^h}{k+1}-k\frac{2\frac{n-1}{k}-1+(-1)^{h-1}}{k+1}\right) begin{align*}5pt] =1-(-1)^{h-1}\geq 0. begin{align*}5pt]
{{P}}_1={\rm d}z-2 l_xz_x{\rm d}t-2\left(x+\varepsilon\right)^{2\gamma}l_yz_y{\rm d}t-\tau l_{xx}z{\rm d}t,{{P}}_2= z_{xx}+\left(x+\varepsilon\right)^{2\gamma}z_{yy}+ l_x^2z+\left(x+\varepsilon\right)^{2\gamma}l_y^2z+\frac{\sigma}{(x+\varepsilon)^2}z,{{P}}= (\tau-1) l_{xx}z- l_tz-\left(x+\varepsilon\right)^{2\gamma}l_{yy}z.
\lim_{\epsilon \rightarrow 0}\sup_{x_0 \in D}E_{\epsilon,0}^{x_0,\gamma}\Bigl\{\operatorname{dist}\Bigl(X^{\epsilon,\gamma}(t), \exp \Bigl\{\Bigl( A + \sum\nolimits_{i=1}^n B_i \gamma_i\Bigr)t\Bigr\}x_0 \Bigr) > \delta \Bigr\}= 0, \,\, t \ge 0.
\hat{\mathfrak{k}}= \{u \in L(\mathfrak{g}, \sigma) \mid \rho(u(-t)) = u(t) \}, &\hat{\mathfrak{m}}= \{u + \alpha c + \beta d \mid u \in L(\mathfrak{g}, \sigma), \ \rho(u(-t)) = - u(t), \ \alpha , \beta \in \mathbb{R}\}.
|\Phi_{\alpha_{k}}^{[1..k]}\rangle=\sum_{\alpha_{1},\alpha_{2}..\alpha_{k-1}}\Gamma_{\alpha_{1}}^{[1]i_{1}}\lambda_{\alpha_{1}}^{[1]}\cdot\cdot\Gamma_{\alpha_{k-1}\alpha_{k}}^{[k]i_{k}}|i_{1}i_{2}..i_{k}\rangle
s_1(T_v) = \sum_{j,k=1}^n T_v(e_j, \bar e_j, e_k, \bar e_k)= \sum_{j,k=1}^n R(e_j, \bar e_j, v, \bar v) R(e_k, \bar e_k, v, \bar v)= r(v, \bar v)^2, s_2(T_v) = \sum_{j,k=1}^n T_v(e_j, \bar e_k, e_k, \bar e_j)= \sum_{j,k=1}^n |R(e_j, \bar e_k, v, \bar v)|^2.
mRMR=max_{x\in\{0,1\}^{n}}[\frac{\sum_{i=1}^{n}c_{i}x_{i}}{\sum_{i=1}^{n}x_{i}}-\frac{\sum_{i,j=1}^{n}a_{ij}x_{i}x_{j}}{(\sum_{i=1}^{n}x_{i})^{2}}].
{\cal E}_{a s 1}^{s p i n}\ = \ \frac{\beta^{2}m_{e}^{2}}{4 \pi}\left( \frac 1 s + \ln \left( \frac{4 \mu^{2}}{m_{e}^{2}}\right) - 1 \right) \- \ - \frac{\beta^{2}m_{e}}{1 6 R}
d_\lambda=\frac{\prod_{\alpha\in\Phi^+,\alpha|_\mathfrak{a}\neq \gamma_j,\forall j}(\lambda+\rho',\alpha)}{\prod_{\alpha\in\Phi^+, \alpha|_\mathfrak{a}\neq 0}(\rho', \alpha)}\cdot\prod_{1\leq j\leq r}\prod_{\alpha\in\Phi^+,\alpha|_\mathfrak{a}=\gamma_j}(n_j(\gamma_j,\gamma_j)+(\rho',\alpha)).
J^{( 2 )}( 3 ; 1 , 1 ) = \frac{\mathrm{i}\pi^{2}}{\sqrt{k_{1 2}^{2}}}\operatorname{l n}\left( \frac{m_{1}+ m_{2}+ \sqrt{k_{1 2}^{2}}}{m_{1}+ m_{2}- \sqrt{k_{1 2}^{2}}}\right) .
S_{m m^{\prime}, n n^{\prime}}^{N S}= \frac{1}{u}( - 1 )^{m + n}e^{- i \pi ( u + 1 ) ( u - m - 1 ) ( u - n - 1 ) / u}e^{- i \pi ( u - 1 ) ( m - 2 m^{\prime}+ u ) ( n - 2 n^{\prime}+ u ) / u},
{\cal L}= \int d x \frac{d t}{2 \pi R^{\prime}}\frac{1}{2 g}[ \mathrm{~ t r}\dot{Y}^{i}\dot{Y}^{i}+ \mathrm{~ t r ~}\dot{A}^{1}\dot{A}^{1}+ \frac{1}{2}\mathrm{~ t r ~}[ Y^{i}, Y^{j}]^{2}- \mathrm{~ t r ~}[ \partial_{1}Y^{i}- i [ A^{1}, Y^{i}] ]^{2},
\dim ((\mathbb{C}\phi^1)_2 \cap \ker A_2) + \dim ((\mathbb{C}\phi^2)_2 \cap \ker A_2) + \dim ((\mathbb{C}\phi^3)_2 \cap \ker A_2) = 1 + 1 + 1 & = \dim(M_2\cap \ker A_i).
\frac{\partial f(t)}{\partial t}=e_{\lambda}^{1-\lambda}(t)|z|^{2}f(t)\ \Longleftrightarrow\ \frac{f^{\prime}(t)}{f(t)}=e_{\lambda}^{1-\lambda}(t)|z|^{2},\bigg(f^{\prime}(t)=\frac{d}{dt}f(t)\bigg).
k=\left\{\frac{C_1(A_1t+Kt+1)\rho^{\frac{2n}{n+2}}e^{C(\rho+tK)}b^{\beta}}{K^{\frac{n}{n+2}}V_{x_0}\left(\rho/\sqrt{K}\right)^{\frac{2}{n+2}}\min(t,\rho^2/K)}\right\}^{\frac{n+2}{2p}}\left(1+\int_{t/4}^t\int_{B_{2\rho/\sqrt{K}}}v^p\,dv_0\,dt\right)^{\frac{n+4}{2p}}.
S_{n}^{\operatorname{II}}=\sum_{0\leq m\leq\lfloor\tilde{\delta}_{0}n\rfloor}\frac{a_{m,n}}{\tilde{b}_{n}}\left( \frac{\lambda_{0}}{n}\right)^{2m}\leq2C_{1}\sum_{0\leq m\leq\lfloor\tilde{\delta}_{0}n\rfloor}\left(1-\frac{c_{2}}{c_{0}}\lambda_{0}^{-2}\right)^{n}\leq2C_{1}(n+1)\left(1-\frac{c_{2}}{c_{0}}\lambda_{0}^{-2}\right)^{n}.
\vec{\mathbb{I}}^2 = \langle \langle \mathbb{I}^2_{\gamma}\colon \gamma \leq \delta \rangle, \langle \dot{\mathbb{J}}^{2}_{\gamma}\colon \gamma < \delta \rangle \rangle \dot{\vec{\mathbb{I}}}^1 = \langle \langle \dot{\mathbb{I}}^1_{\gamma}\colon \gamma \leq \delta \rangle, \langle \dot{\mathbb{J}}^{1}_{\gamma}\colon \gamma < \delta \rangle \rangle
\Psi\big(e^{(2)}_{11}\big) + \Phi\big(1_{M_{p}}\big) = 1_{\ell^{\infty}(A)/c_{0}(A)}\mbox{and}\Psi\big(e^{(2)}_{22}\big) \Phi\big(e^{(p)}_{11}\big) = \Psi\big(e^{(2)}_{22}\big).
\phi_{\ast}\left(\bar{m}_{K}\times\eta\right) =\int_{G/Q}\int_{K\cap Q}\tau(y)k.\lambda dm_{K\cap Q}(k)d\bar{m}_{K}(y) =\left(\int_{G/Q}\tau(y).m_{K\cap Q}d\bar{m}_{K}(y)\right)\ast\lambda=m_{K}\ast\lambda=\nu.
|\frac{L_{1}(u_{1})(t,R\lambda(t))-L_{1}(u_{2})(t,R\lambda(t))}{R}| \leq \frac{C |\overline{v}_{1}-\overline{v}_{2}|(t,R)}{R^{3}\lambda(t)^{2}}\left(|v_{corr}(t,R\lambda(t))|^{2}+\frac{R |v_{corr}(t,R\lambda(t))|}{(R^{2}+1)}\right)
\partial_{y}w=-C_s\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_{s-1}(\lambda_k^{1/2}y)\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^{s-1/2}}\psi_{k}(x)
( \partial_{x_{\perp}}^{2}+ \partial_{w}^{2}) H_{B}( x_{\perp}, w , z_{a}) + H_{A}( x_{\perp}) \partial_{z_{a}}^{2}H_{B}( x_{\perp}, w , z_{a}) = q_{B}\delta ( x_{\perp}, x_{\perp 0}) \delta ( w , w_{0}) \delta ( z_{a}, z_{a 0}) ,
J^{\iota_1, \iota_2,\iota_3}_{k_1,k_2, k_3}:= \int_{t_1}^{t_2}i e^{i L(t,\xi)}I^{\iota_1, \iota_2,\iota_3}_{k_1,k_2, k_3}(t,\xi) d t, (\iota_1, \iota_2,\iota_3)\in \{(+,+,+),(+,-,-), (-,-,-)\}.
o(0) :=0, o(\varepsilon_s) :=\varepsilon_{\eta+1+c(s)}, o(\Omega^{s_0}\cdot\beta_0+\dots +\Omega^{s_n}\cdot\beta_n) :=(\omega^{1+\alpha})^{o(s_0)}\cdot\beta_0+\dots +(\omega^{1+\alpha})^{o(s_n)}\cdot\beta_n.
\frac{1}{(2\pi )^n\, n!}\int_0^\pi\cdots \int_0^\pi R ( \boldsymbol{\xi}) \rho_\epsilon (\boldsymbol{\xi}) \xi_1\cdots \xi_n = & \sum_{{\lambda}\in\Lambda^{(m,n)}}R\bigl( \boldsymbol{\xi}_{{\lambda}}^{(m,n)}\bigr) \rho_\epsilon \bigl(\boldsymbol{\xi}_{{\lambda}}^{(m,n)}\bigr) \Delta_\lambda^{(m,n)}.
\bigg\|B\int_{-t}^0S_B(-r)\omega_2(r)dr\bigg\|\le \bigg\|B\int_{-1}^0 S_B(-r)\omega_2(r)dr\bigg\| &+\bigg\|B\int_{-t}^{-1}S_B(-r)\omega_2(r)dr\bigg\|.
d_{p-;I}(\mathbf{x},\mathbf{y}) := \max_{k = 1,\ldots,N}\left( \sup_{\mathbb{D}}\sum_{t_i \in \mathbb{D}}|\pi_k(\mathbf{x}_{t_i,t_{i+1}}- \mathbf{y}_{t_i,t_{i+1}})|^\frac{p}{k}\right)^{\frac{1}{p}}
\sum\limits_{k=0}^{n-1}\frac{(-1)^kx^{2k}}{(2k+1)^s}=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}{_{s+1}F_s}\left(\left\{\frac{1}{2}\right\}^{s},1-k;\left\{\frac{3}{2}\right\}^{s};-x^2\right).
{\cal L}^{(2)}_{F}=\frac{{\rm i}}{2}(\theta_i \tilde{{\cal D}}_0\theta^i + \eta_i \tilde{{\cal D}}_0 \eta^i-2{\rm i}\eta_i e_0^i{}_j\eta^j)+ e^{2\phi}\eta^i C_{ij}^\prime \tilde{{\cal D}}_1\theta^j +h.c.\ ,
\log \Pr[d(\hat{M}',M) \leq n^{1-\beta}]^{-1}\geq \log \Pr[\hat{M}' = M]^{-1}- n^{1-\beta}(2 + \max(1, \alpha+\beta) \log n) &\geq \omega(n^{1-\beta}\log n) - O(n^{1-\beta}\log n) &\geq \omega(n^{1-\beta}\log n) \geq \omega(\log n).
S=\frac{N^2}{16\pi G_{10}}\int d^{10}x\sqrt{-g}(Ne^{\phi})^{\lambda}\left [R +\frac{4(p-1)(p-4)}{(7-p)^2}(\partial \phi)^2 - \frac{1}{2N^2(8-p)!}F^2_{8-p}\right],
|\tilde u_k(x)| =|\frac{u((x^k)'+d_kx)-u((x^k)')}{d_k^2}| &\le \frac{|\nabla u(r_{x^k})||x|}{d_k}=\frac{|\nabla u(r_{x^k})-\nabla u(x^k)||x|}{d_k}\le T_1,
\frac{i}{2}{\rm Tr}\,\left(\hat{T}_x^{(1)}\frac{1}{\partial^2+m^2}\sigma\frac{1}{\partial^2+m^2}\right)-\frac{i}{4}{\rm Tr}\,\left(\hat{T}_x^{(0)}\frac{1}{\partial^2+m^2}\sigma\frac{1}{\partial^2+m^2}\sigma\frac{1}{\partial^2+m^2}\right)\,.
\left(\begin{array}{ccccccccc}2 -1 0 . . . . 0 -1 -1 2 -1 0 . . . . 0 0 -1 2 -1 0 . . . 0 . . . . . . . . . 0 . . . . 0 -1 2 -1 -1 0 . . . . 0 -1 2 \end{array}\right).
I_{tt}^{1}= \left.\frac{1 + x}{4 (2 + 2 x + x^2)}\right|_{x=-2}^{x=-1}= \frac{1}{8}I_{tt}^{3}= \left.\frac{x-1}{4 (2 - 2 x + x^2)}\right|_{x=1}^{x=2}= \frac{1}{8},
\frac{1}{|B_{r}|}\int_{B_{r}}|X_{i}X_{j}h\left( x\right) -(X_{i}X_{j}h)_{B_{r}}|dx\leq\frac{c}{k}\sum_{i,j=1}^{q}\frac{1}{|B_{kr}|}\int_{B_{kr}}|X_{i}X_{j}u\left( x\right) |dx,
|g(z)| = R_0^m \left|\sum_{j=0}^{J-1}a_j(z-q)^j \right|\le R_0^m\left(\sum_{j=0}^{J-1}(2R_0)^j\right)\max_{0\le j < J}|a_j| \le JR_0^m\max\{1,(2R_0)^J\}\max_{0\le j < J}|a_j|
\left\Vert V^{\left( 1\right)}-V^{\left( 2\right)}\right\Vert_{H^{1}\left( \Omega_{d+c}\right)}\leq C_{2}\left( 1+\left\Vert V^{\left(1\right)}-V^{\left( 2\right)}\right\Vert_{H^{2}\left( \Omega \right)}\right) \sigma^{\rho},\rho =c/\left( m+c\right) .
D_{\Phi(y)}G(\Phi(x),\Phi(y))D_{y_1}\Phi(y)\le \frac{C}{|x-y|}+\sum_{k=1}\alpha^k\frac{C}{|x-y|}-\sum_{k=1}^\infty\alpha^k\frac{2}{|k-2\Phi_1(x)||y|^2}
\begin{aligned}Q^{(m,n)}r(k) = e_{m,n}^{(1)}(k)I_{m,n}(k) + e_{m,n}^{(2)}(k)K_{m,n}(k) m\neq 0 Q^{(m,n)}r(k) = \sum_{i=0}^k \left(\begin{array}{cc}0 0 0 \frac{1}{a_{n}(i)}\prod_{j=i}^{k-1}c_{2,n}(j)\end{array}\right)r(i) m=0.\end{aligned}
\widehat G_\infty = \varprojlim\{\pi_{n+1}\colon Q_{n+1}\to Q_n\mid n\geq 0\}\phantom{--------}= \left\{(g_0 C_0 ,g_1 C_1,\ldots) \in \prod_{n \geq 0}G/C_n \mid \pi^{n+1}_n (g_{n+1}C_{n+1}) = g_nC_n \right\}
\sum_{n\geq L}\big( p^+_M(t,n)-p^+_M(t,{n+1})\big)|B_n|=\sum_{n\geq L}p^+_M(t,n)|C_n|+p^+_M(t,L)|B_{L-1}| &=\sum_{y\not\in B_{L-1}}p_M(t,y)+\sum_{n\geq L}\sum_{y\in C_n}\big(p^+_M(t,n)-p_M(t,y)\big)+p^+_M(t,L)|B_{L-1}|\,.