harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathfrak{d}_2< \mathfrak{d}_1, \ \ \mathfrak{s}_2\in \begin{cases}\{1\}, \mathfrak{d}_2=2; [2^{-\frac{\mathfrak{k}_1}{\mathfrak{d}_2}}, \frac{1}{2}] \mathfrak{d}_2>2, \end{cases}\ \ 2^{\mathfrak{k}_2}=2^{\mathfrak{k}_1}(\mathfrak{s}_2)^{\mathfrak{d}_2}.
\sum_k \hat{\rho}_{k,\infty}= \sum_k \frac{\hat{\lambda}_{k,\infty}}{\mu_{k,\infty}}= \frac{1}{\rho_\infty}\sum_k\frac{\lambda_{k,\infty}}{\mu_{k,\infty}}= \frac{1}{\rho_\infty}\sum_k \rho_{k,\infty}= 1.
I(su^++tu^-) <\left(I(u^+)-\frac{1}{4}I'(u^+)[u^+]\right)+\left(I(u^-)-\frac{1}{4}I'(u^-)[u^-]\right) =\left(I(u^+)-\frac{1}{4}I'(u)[u^+]\right)+\left(I(u^-)-\frac{1}{4}I'(u)[u^-]\right)+\frac{\lambda}{2}\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx =I(u^+)+I(u^-)+\frac{\lambda}{2}\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx =I(u),
\alpha_{1}= e_{1}- e_{2}, \quad \cdots , \ \alpha_{2 n - 2}= e_{2 n - 2}- e_{2 n - 1}, \quad \alpha_{2 n - 1}= e_{2 n - 1}- e_{2 n}\quad \alpha_{2 n}= e_{2 n - 1}+ e_{2 n}
(\begin{matrix}\alpha\\ \beta\\ \gamma\\ \delta\\ \epsilon\end{matrix})=\frac{2}{3}(\begin{matrix}-1&2&-1&-1&2\\ 2&-1&2&-1&-1\\ -1&2&-1&2&-1\\ -1&-1&2&-1&2\\ 2&-1&-1&2&-1\end{matrix})(\begin{matrix}A\\B\\C\\D\\E\end{matrix})
C_{n}^{(d), \sigma}(\mathbf{P}_{\!\sigma})\coloneqq \bigg(\! 1-\!\sum_{m=n+1}^{d}C_m^{(d),\sigma}(\mathbf{P}_{\!\sigma})\cdot \chi_n^{\sigma}(\mathbf{p}_{\sigma_m})\! \bigg) \frac{1}{\chi_n^{\sigma}(\mathbf{p}_{\sigma_n})}.
\begin{aligned}d\big((\pi^{*}f)(\alpha,\beta)\big)=d\big((\pi^{*}f)\alpha,(\pi^{*}_{\partial}f)\beta\big) = \Big((\pi^{*}d^{S}f)\wedge\alpha+(\pi^{*}f) d^{M}\alpha,i^{*}((\pi^{*}f)\alpha)-(\pi^{*}_{\partial}d^{S}f)\wedge\beta-(\pi^{*}_{\partial}f) d^{X}\beta\Big) = \pi^{*}(d^{S}f)\wedge(\alpha,\beta)+(\pi^{*}f)\cdot d(\alpha,\beta),\end{aligned}
C ( \omega , \omega^{\prime}) = \delta_{{\omega + \omega^{\prime}}, 0}f ( \omega , \omega^{\prime}, \theta ) - 4 \sum_{\omega_{1}, \omega_{2}> 0}^{N}{\omega_{1}}^{2}{\omega_{2}}^{2}C_{\omega_{1}, \omega_{2}}^{- \omega^{\prime}}\: C_{\omega_{1}, \omega_{2}}^{\omega}+ \sum_{\omega_{1}> 0}^{N}{\omega_{1}}^{2}\: ( \omega - \omega_{1})^{2}\: C_{\omega_{1}, \omega - \omega_{1}}^{- \omega^{\prime}}
G_{0}^{( 2 )}( \vec{r}_{1}- \vec{r}_{2}, z_{1}, z_{2}) = \frac{2}{L}\frac{1}{( 2 \pi )^{d - 1}}\sum_{n = 1}^{\infty}\sin ( \frac{n \pi z_{1}}{L}) \sin ( \frac{n \pi z_{2}}{L}) \int d^{d - 1}p \frac{e^{i \vec{p}. ( \vec{r}_{1}- \vec{r}_{2})}}{( \vec{p}^{\, 2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}.
\frac{d}{dt}\left[\frac{e^{xt}}{(e^t-A)^p}\right] = e^{xt}\frac{d}{dt}\left[\frac{1}{(e^t-A)^p}\right] +x\frac{e^{xt}}{(e^t-A)^p}& =-pe^{xt}\frac{e^t-A'}{(e^t-A)^{p+1}}+x\frac{e^{xt}}{(e^t-A)^p}& =p(A'-A)\frac{e^{xt}}{(e^t-A)^{p+1}}+(x-p)\frac{e^{xt}}{(e^t-A)^p}
{\mathcal C}={\mathcal C}_{\mathbf{x}_0,\theta_0}:= \left\{\mathbf{y}\in \mathbb R^3 ~|~0\leqslant \angle(\mathbf y-\mathbf{x}_0,\mathbf{v}_0)\leqslant \theta_0\right \}\ (\theta_0 \in(0,\pi/2))
[FS(\langle a_{n}\rangle_{n=0}^{\infty})]_{<}^{k}=\{\{\sum_{t\in\alpha_{1}}a_{t},...,\sum_{t\in\alpha_{k}}a_{t}\}:\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\in P_{f}(\mathbb{N})and\alpha_{1}<\cdot\cdot\cdot<\alpha_{k}\}.
\Upsilon(h):=-\sigma\big(\overline{\lambda},\mathcal{T}_{\mathcal{K}}^2(\overline{y},h)\big) =\langle \overline{\mu},\Xi''(\overline{y})(h,h)\rangle \forall h\in\mathcal{C}_{\mathcal{K}}(\overline{y},\overline{\lambda}).
\bigoplus_{d\geq 0}K_T(BGL(d))=\bigoplus_{d\geq 0}\mathbb{K}\left[z_1^{\pm 1}, \ldots, z_d^{\pm 1}\right]^{\mathfrak{S}_d}\stackrel{\cong}{\to}\bigoplus_{d\geq 0}K_T(\mathcal{Y}(d)).
\begin{array}{rcl}L_{-1}L_{0}L_{1}\quad L_{2}\quad \ldots J_{-2}\quad J_{-1}J_{0}J_{1}\quad J_{2}\quad \ldots \Lambda_{-3}\quad \Lambda_{-2}\quad \Lambda_{-1}\Lambda_{0}\Lambda_{1}\quad \Lambda_{2}\quad \ldots \ldots\ldots\ldots\ldots \ldots\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \quad .\end{array}
\lambda_{k,n}(\alpha_\Sigma) = \min_W \max_{w \in W}\left\{\frac{\int_0^{L_\Sigma}|w'|^2 F_\Sigma + \frac{k^2 |w|^2}{F_\Sigma}\,dt}{\int_0^{L_\Sigma}|w|^2 F_\Sigma \,dt}\right\}
\left\langle \left[ \sigma_+[Q^W] \right]^q\left[ \sigma_-[Q^W] \right]^{n-q}\right\rangle_{Q^\mu}\; = \; \left\langle \left[ \sigma_+[Q^W] \right]^{n-q}\left[ \sigma_-[Q^W] \right]^q \right\rangle_{Q^\mu}\; .
\lambda_m(\alpha)=\sum_{l=0}^m\left({i\alpha\over2}\right)^l{(m+l)!\over l! (m-l)!}\times\left\{\begin{array}{cl}-\lambda_{2N}(\alpha) \hbox{~for odd$m+l$}\lambda_0(\alpha) \hbox{~for even $m+l$}\end{array}\right\}
\|uu_h^{\kappa_1}\|_{p^*}\leq M_5(\kappa_1, u) \left [\|uu_h^{\kappa_1}\|_{p}^p+1\right]^{\frac{1}{p}}\leq M_6(\kappa_1, u) \left [\|u^{\kappa_1+1}\|_{p}^p+1\right]^{\frac{1}{p}}= M_6(\kappa_1, u) \left [\|u\|_{p^*}^{p^*}+1\right]^{\frac{1}{p}}<\infty,
\alpha_{n}^{(N)}(f) =\sum_{j=1}^{m_{N}}P(A_{j}^{(N)})\alpha_{n,j}^{(N-1)}(f), \alpha_{n,j}^{(N-1)}(f) =\sqrt{n}\left( \mathbb{E}_{n}^{(N-1)}(f|A_{j}^{(N)})-\mathbb{E}(f|A_{j}^{(N)})\right).
g_v(x) = \sum_{i=1}^r a_{i}h_{i, v}(x) & = \lim_{n\to\infty}\theta_{n,v}(x) + \sum_{i=1}^r \left( \lim_{n\to\infty}a_{ni}\right) h_{i,v}(x) & = \lim_{n\to\infty}\left( \theta_{n,v}(x) + \sum_{i=1}^r a_{ni}h_{i,v}(x) \right) \geqslant 0.
\mu(x) =\frac{1}{2\pi \i}(R(x-\i 0)-R(x +\i 0))=|\alpha_x)(\beta_x| |\alpha_x) = (x^2+\pi^2)^{-1/2}\left(\binom{1}{\frac{\mathcal{P}}{x-\Omega}|E)}+ x\binom{0}{|\delta_x)}\right) (\beta_x| = (x^2+\pi^2)^{-1/2}\bigg (-(1,(E|\frac{\mathcal{P}}{x-\Omega})+ x(0,(\delta_x|)\bigg)
\| \sum_{i=1}^d \left( \widehat \lambda_i - \lambda_i \right)\langle \theta, q_i\rangle q_i \|^2 = \sum_{i=1}^d \left( \widehat \lambda_i - \lambda_i \right)^2\langle \theta, q_i\rangle^2 \leq B\left(\lambda_1 \right)^2.
\Delta_ku(x)=\Delta u(x)+\sum_{\alpha\in R}k(\alpha)\left(\frac{\langle\nabla u (x),\alpha\rangle}{\langle\alpha,x\rangle}-\frac{|\alpha|^2}{2}\frac{u(x)-u(\sigma_\alpha(x))}{\langle\alpha,x\rangle^2}\right).
\begin{array}{c c}{{Z_{-}( z ) = C^{- 1}\bar{Z}_{+}^{t}( z ) \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{\bar{\cal E}}}\\ \end{array}\right) ~ ~}}&{{~ ~ \bar{Z}_{-}( z ) = \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- \bar{\cal E}}}\\ \end{array}\right) Z_{+}( z ){}^{t}C}}\\ \end{array}
2k-1 > \lim\limits_{n \to +\infty}\mu_k^{1/{p_n}}(p_n) = \lim\limits_{n \to +\infty}\lambda_1^{1/{p_n}}(B_{r_{i+1}^n}\setminus \overline{B_{r_{i}^n}};p_n) =\frac{2}{r_{i+1}- r_i}
\hat{K}(N): =\prod_{p\nmid N}\left( 1-\frac{(\frac{N-1}{p})^2p+1}{(p-1)^2(p+1)}\right)\prod_{\underset{2\nmid \nu_p(N)}{p\mid N}}\left(1-\frac{1}{p^{\nu_p(N)}(p-1)}\right) \prod_{\underset{2\mid \nu_p(N)}{p\mid N}}\left(1-\frac{p-\left(\frac{-N_p}{p}\right)}{p^{\nu_p(N)+1}(p-1)}\right)
\left \|\left(\sum\limits_{j=1}^{\infty}\left(\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right)^{\eta}\right)^{1/\eta}\right \|_{L_{\omega}^{p}}+\left \|\left(\sum\limits_{j=1}^{\infty}\left(\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right)^{\eta}\right)^{1/\eta}\right \|_{L_{\omega}^{p}}\leq C_{\eta}\left \|f\right \|_{h_{\omega}^{p}}
\widetilde\nabla_i (A_{g_u})^l{}_j - \widetilde\nabla_j (A_{g_u})^l{}_i= \frac{1}{2}u^{-\frac{n+2}{n-2}}\,W_{ij}{}^l{}_{s}\,\nabla^s u + \frac{1}{n-2}u^{-\frac{4}{n-2}}\,C^l{}_{ji}.
\int_{T}^{2T}{\left(\tfrac{t}{2\pi}\right)}^{{1}/{4}+{\delta}/{2}}e^{\frac{it}{2}\log ({t}/{2\pi en^{2})}}dt = O \left( T^{{1}/{4}+{\delta}/{2}}T^{{1}/{2}}\right) = O \left( T^{{3}/{4}+{\delta}/{2}}\right).
\begin{array}{lll}l_q = q^2[(x^0)^2-q^{-1}x^+x^- -qx^-x^+ -(x^3)^2] \, = q^2[(x_0)^2-qx_+x_- -q^{-1}x_-x_+ -(x_3)^2] \;; \, \, \, l_q = CD-q^2AB \quad.\end{array}
\sum_{i \neq j}\left| \left[{\rm emp}_{N}(K_{i}) - \rho(K_{i}) \right] \max_{y \in K_{i}}g^{\epsilon}(x-y) \right| \leq{\rm Disc}(\eta, \delta, N) \sum_{i \neq j}\left| \max_{y \in K_{i}}g^{\epsilon}(x-y) \right| \leq \frac{c_{g, \epsilon}}{\eta^{d}}{\rm Disc}(\eta, \delta, N) ,
\hat{A}_p = \hat{A}_pQ_p(Q_p^*Q_p)^{-1}Q_p^* + \hat{A}_p(\mathsf{I}- Q_p(Q_p^*Q_p)^{-1}Q_p^*), \hat{A}_p = M_p(M_p^*M_p)^{-1}M_p^*\hat{A}_p + (\mathsf{I}- M_p(M_p^*M_p)^{-1}M_p^*)\hat{A}_p.
\sum_{k=1}^{n}x^{p_{k}}\widetilde{f_{k}}(x^{q_{k}})= \sum_{k=1}^{n}x^{p_{k}}\left[f_{k}(x^{q_{k}})-f_{k}(1) x^{q_{k}}\right]= \sum_{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})-x^{l}\sum_{k=1}^{n}f_{k}(1)=0
C_{I J K}\tilde{h}^{I}\tilde{h}^{J}\tilde{h}^{K}= a^{3}( y ) C_{I J K}h^{I}h^{J}h^{K}= a^{3}( y ) \qquad \mathrm{o r}\qquad a^{2}( y ) = h^{I}H_{I}\, .
\Vert\zeta\Vert_{p_1\wedge p_2\wedge p_3\wedge\ldots}:= \inf_{\zeta=\zeta_{p_1}+\zeta_{p_2}+\zeta_{p_3}+\ldots}\left( \Vert\zeta_{p_1}\Vert_{p_1}+\Vert\zeta_{p_2}\Vert_{p_2}+\Vert\zeta_{p_3}\Vert_{p_3}+\ldots \right).
x_k = \frac{\tau_k}{1+\tau_k}z_k + \frac{1}{1+\tau_k}y_k \nabla h(z_{k+1}) - \nabla h(z_k) = \tau_k \left( \nabla h(x_k) - \nabla h(z_k) - \frac{1}{\mu}\nabla f(x_k)\right) y_{k+1}= \mathcal{G}(x),
\sum_{\Omega_{n,\mathrm{k}}}\prod_{i\in[n]_{\neq\mathrm{k}}}\frac{c_i^{N_i}}{N_i!}\cdot \prod_{1\le i\le \ell}\frac{c_{k_i}^{N_{k_i}-1}}{N_i!}\le \sum_{\Omega_{n-\Sigma_{\mathrm{k}}}}\prod_{i=1}^{n-\Sigma_{\mathrm{k}}}\frac{c_i^{N_i}}{N_i!}= [x^{n-\Sigma_{\mathrm{k}}}]S(x).
K^{n,4}_\tau :=n^{2\alpha+1}\int^{\tau}_{(\delta+\frac 1n) \wedge \tau}\gamma_s ( \sigma'\sigma)^2(X_{\eta_n(s)}) \times \left[ \left( \int_{\eta_n(s)-\delta}^{\eta_n(s)}\psi_{n,2}(u,s) dW_u \right)^2 -\left( \int_0^{\eta_n(s)}\psi_{n,2}(u,s) dW_u \right)^2 \right]ds
\chi(\xi\otimes\alpha\delta\eta) =|\xi|(2+|\eta|)\varepsilon(\xi\alpha\delta\eta) =|\xi|(2+|\eta|)\varepsilon(\xi\eta) =|\xi|(2+|\eta|)\varepsilon(\xi\delta\alpha\eta)+(q-q^{-1})|\xi|(2+|\eta|)\underbrace{\varepsilon(\xi\beta\gamma)}_{=0}=\chi(\xi\otimes(\delta\alpha+(q-q^{-1})\beta\gamma)\eta)
S = \int \frac{1}{2}\epsilon^{\mu \nu \lambda \rho}\left( B_{\mu \nu}^{a}( x ) R_{\lambda \rho}^{a}( x ) + \underbrace{\phi^{a b}( x ) B_{\mu \nu}^{a}( x ) B_{\lambda \rho}^{b}( x )}_{c o n s t r a i n t \: t e r m}\right) .
\sum_i\frac{i}{2}\nabla_a({\bf V}\cdot{\bf K}_a)\psi^a\psi^b = \frac{2i\pi^2}{\sqrt{3}e^2}\frac{(\hat{\bf v}^i\cdot\hat{\bf v}^i)^{\frac{1}{2}}}{(1+2\mu r)^2}\left(\cos\theta\,\psi^r\psi^\phi+\psi^r\psi^\psi-r\sin\theta(1+2\mu r)\psi^\theta\psi^\phi\right)
\tilde{h}= \tilde{h}_1+ \tilde{h}_2+\tilde{h}_3=\varphi f^2(n\nabla^2 v_i+\Delta v_i\cdot g)+\psi \cdot dr\odot \nabla v_i+\chi\cdot v_i(f^2g-ndr\otimes dr)\in V_{4,i}
\widehat{y}_{0}(x) = \int_0^{\xi \infty}s^{\mu -1}\frac{\big(q^{\lambda+1}s/x, \alpha s ; q\big)_{\infty}}{(qs/x, \beta s ; q)_{\infty}}\,{\rm d}_{q}s = (1-q)\sum_{n=-\infty}^{\infty}(q^n\xi)^{\mu}\frac{\big(q^{\lambda+n+1}\xi / x, q^n \xi\alpha ; q\big)_{\infty}}{\big(q^{n+1}\xi / x, q^n \xi\beta ; q\big)_{\infty}}.
\begin{aligned}\sum_{\alpha_1<n\leq \beta_1}f(a+nw_1+\beta_2w_2) =\int_{\alpha_1}^{\beta_1}f(a+xw_1+\beta_2w_2) dx &\quad+\int_{\alpha_1}^{\beta_1}\frac{\partial f(a+xw_1+\beta_2w_2)}{\partial x}P_1(x) dx &\quad+P_1(\alpha_1)f(a+\alpha_1w_1+\beta_2w_2) &\quad-P_1(\beta_1)f(a+\beta_1w_1+\beta_2w_2).\end{aligned}
{\cal L}^{(0)}=\pi_{ij}\dot A^{ij}+{1\over2}\,\pi_{ij}\pi^{ij}+2\partial_iA_{j0}\pi^{ij}-{1\over2}\,\partial_iA_{jk}\partial^iA^{jk}+\partial_iA_{jk}\partial^jA^{ik}\,,
\begin{aligned}R_{{N}}\triangleq \max_{{P}_i}\sum_{i=1}^K r_i \mathrm{s.t.}r_i=\ln \Big(1+\frac{P_i|h_i|^2}{N_0+|h_i|^2\sum_{j=1}^{i-1}P_j}\Big), &\sum_{i=1}^K P_i\leq P, \min_i r_i\geq r^*.\end{aligned}
\hat{X}^{\Lambda}={\frac{1}{\sqrt 2}}\left( \begin{matrix}{1 - T U}\\{- ( T + U )}\\{- ( 1 + T U )}\\{T - U}\\ \end{matrix}\right) , \qquad \hat{F}_{\Lambda}= \left( \begin{matrix}{S \hat{X}^{0}}\\{S \hat{X}^{1}}\\{- S \hat{X}^{2}}\\{- S \hat{X}^{3}}\\ \end{matrix}\right) = S \eta_{\Lambda \Sigma}\hat{X}^{\Sigma}\ ,
\phi(f,g) :=f(1)\overline{g}(1)+\frac{2}{\beta}(f^{\prime}(1)\overline{g}(1)+f(1)\overline{g}^{\prime}(1)) & +\left( 1+\frac{4}{\beta^{2}}\right) f^{\prime}(1)\overline{g}^{\prime}(1)+\int_{-1}^{1}f^{\prime\prime}(x)\overline{g}^{\prime\prime}(x)(1+x)^{\beta+2}\,dx\,.
{\displaystyle V^{(1)i}_\alpha\frac{i\tilde{\cal P}^{\dot\beta\alpha}}{(m+ie\bar W/2)}\sqrt{\frac{m^{*2}}{-{\cal P}^2}}=\bar V^{(1)\dot\beta i}\mbox{,}V^{(2)i}_\alpha\frac{-i\tilde{\cal P}^{\dot\beta\alpha}}{(m+ie\bar W/2)}\sqrt{\frac{m^{*2}}{-{\cal P}^2}}=\bar V^{(2)\dot\beta i}}.
G^*_2(N) =\frac{N}{\phi(N)}\prod_{\underset{p>2}{p\mid N}}\left(1-\frac{1}{(p-1)^2}\right)^{-1}\prod_{\underset{2\nmid \alpha}{p^\alpha \parallel N}}\left(1-\frac{1}{p^\alpha(p-1)}\right) \intertext{and}G^*_3(N) =\prod_{p^{2\alpha}\parallel N}\left(1-\frac{p-\left(\frac{-N_p}{p}\right)}{p^{2\alpha+1}(p-1)}\right).
\rho_{1}[ \beta ; \phi , \phi ] = - \frac{\lambda}{4 !}\rho_{0}\, \left[ \int_{1}\widehat \varphi^{4}( 1 ) + 6 \int_{1}G_{\eta}( 1 ; 1 ) \widehat \varphi^{2}( 1 ) + \int_{1}\langle \eta^{4}( 1 ) \rangle \right] \; .
\|1 + \rho d X\|_{q}^{q}\le \left( \rho \left( 1 + \frac{1}{\omega}\right) \right)^q \| d X\|_{q}^{q}+ \|\left(1 + \rho d X\right) 1_{\rho |d X| <\omega}\|_{q}^{q}= \beta^q \| d X\|_{q}^{q}+ \|\left(1 + \rho d X\right) 1_{\rho |d X| <\omega}\|_{q}^{q}.
C\Vert\varphi\Vert_{g}^{2}=C\int_{M}\left( |\nabla_{g}\varphi|^{2}+\varphi^{2}\right) d\mu_{g}& \leq\int_{M}\left( c(x)|\nabla_{g}\varphi|^{2}+\frac{d(x)}{\varepsilon^{2}}\varphi^{2}\right) d\mu_{g}=\Vert\varphi\Vert_{\varepsilon}^{2}.
\Pi_{00}^{\rm thermal}(k_{0},\vec{k}) ={4e^{2}\over (2\pi)^{4}i}\int d^{3}p\int_{-i\infty+\epsilon}^{i\infty+\epsilon}dp_{0}\,n_{F}(p_{0})\left[{m^{2}+ p^{2}+ p\cdot k - 2p_{0}k_{0}- 2p_{0}^{2}\over(p^{2}+m^{2})((p+k)^{2}+m^{2})}+ (k_{0}\leftrightarrow -k_{0})\right]
\left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a b c d \end{pmatrix}p \right) \left( I + \begin{pmatrix}0 \beta 0 0 \end{pmatrix}p \right) = \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a b + \beta c d \end{pmatrix}p.
\widetilde\chi^{(3)}\left(e_i,e_j\right)=\chi_F^{(3)}\left(e_i,e_j\right)+\sum_{k=1}^{3h-3}a_k\oint_{\cal C}\Omega^{k+1-h_0}\left[e_i,e_j\right],\qquad h_0\equiv{3\over 2}h.
\begin{array}{rll}{\cal E}(A,B)=\int_{M_3}<(H^A\otimes I_E-I_E\otimes K^B) \wedge (D^A\Phi_A\otimes I_E-I_E\otimes D^B\Phi_B)> -\int_{M_3}<(D^A\Phi_A\otimes I_E) \wedge (H^A\otimes I_E)>\end{array}
\sqrt{- h}e^{- 2 \varphi}h^{\bar{\mu}\bar{\mu}^{\prime}}h^{\bar{\nu}\bar{\nu}^{\prime}}( M L )_{i j}{\cal F}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}}^{j}= \epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\partial_{\bar{\rho}}\psi^{i},
|N^+(x_1)\cap V(C_1)|+|N^+(x_2)\cap V(C_1)| \leq \frac{|C_1|}{2}\,, \mathrm{\and}|N^-(x_1)\cap V(C_1)|+|N^-(x_2)\cap V(C_1)| \leq \frac{|C_1|}{2}\,.
\begin{aligned}\|x(t)-x_a(t)\| \,\le \|x(t)-z(t)\| + \|z(t)-x_a(t)\| &\, \le \|x(t)-z(t)\|+\gamma \|\epsilon(x_a(t),t)\| &\, \le \|x(t)-z(t)\|+\gamma 2M_\mathcal{D}T.\end{aligned}
f^{(n-k)}(z) =\sum_{j=0}^{n-k}h^{(j)}\big(l_{\pi_{\lambda,\beta}}(z)\big)B_{n-k,j}\big(l'_{\pi_{\lambda,\beta}}(z),l''_{\pi_{\lambda,\beta}}(z),\dots,l_{\pi_{\lambda,\beta}}^{(n-k-j+1)}(z)\big). =\sum_{j=0}^{n-k}\frac{(-1)^{j}j!}{(l_{\pi_{\lambda,\beta}}(z))^{j+1}}B_{n-k,j}\big((l_{\pi_{\lambda,\beta}}^{(i)}(z))_{i=1}^{n-k-j+1}\big).
\lvert \lambda\rvert^{1/2}\nabla_H(\lambda-\Delta)^{-1}\left(Q\overline{f}\otimes 1\right)=\lvert \lambda\rvert^{1/2}\int_0^\infty e^{-\lambda t}\left(\nabla_H e^{t\Delta_H}Q\overline{f}\otimes e^{t\Delta_z}1\right)\,dt,
I_{2}\leq C \|u\|_{L^{p}(X)}=C\frac{\mu^{\frac{n-(n-\sigma)p'}{p'}}}{\mu^{\frac{n-(n-\sigma)p'}{p'}}}\|u\|_{L^{p}(X)}\leq C \mu^{\frac{n-(n-\sigma)p'}{p'}}\|u\|_{L^{p}(X)}.
g_7 = 2^{\frac{4}{3}}|\Omega|_\omega^{-\frac{4}{3}}(dr^1)^2 + 2^{-\frac{2}{3}}|\Omega|_\omega^\frac{2}{3}(dr^2)^2 + 2^{-\frac{2}{3}}|\Omega|_\omega^\frac{2}{3}(dr^3)^3 + 2^{-\frac{2}{3}}|\Omega|_\omega^\frac{2}{3}g_4,
F^1 \circ \widetilde{Q}(v_1 \vee \cdots \vee v_n) =F^1 ( Q_0 \vee v_1 \vee \cdots \vee v_n + Q(v_1\vee \cdots \vee v_n)) & =F_{n+1}^1( Q_0 \vee v_1 \vee \cdots \vee v_n) + (Q')^1\circ F (v_1\vee \cdots \vee v_n) & \stackrel{!}{=}\widetilde{Q'}^1 \circ F(v_1\vee \cdots \vee v_n).
\begin{pmatrix}C_p E_1^T \cdot E_p^T \end{pmatrix}U_n= \begin{pmatrix}C_pU_n E_1^TU_n \cdot E_p^TU_n \end{pmatrix}\rightarrow \begin{pmatrix}0 u_1 \cdot u_p \end{pmatrix}
H_{c}={\displaystyle \int d^{3}x ~{\cal H}_{c}}={\displaystyle \int d^{3}x \left( \frac{1}{2}{\pi_{i}}{\pi_{i}}+{\frac{1}{4}}F_{i j}F_{i j}- \frac{m^{2}}{2}( A_{0}^{2}- A_{i}^{2}) + A_{0}({\partial_{i}}\pi_{i}) \right) ,}
P^{\mu}({\cal A})=\frac{4}{3}\left[{\cal A}_\nu\partial_\rho{\cal A}_\sigma+\partial_\rho{\cal A}_\sigma{\cal A}_\nu+\frac{3}{2}{\cal A}_\nu{\cal A}_\rho{\cal A}_\sigma\right]\epsilon^{\mu\nu\rho\sigma}
\Sigma_2 \ll p^{(\frac{3}{2}-\epsilon) h_p}\sum_{m= \max\{0,k_p-h_p\}}^{\infty}\frac{p^{\frac{1}{2}-\epsilon}}{p^{m( \frac{1}{2}-\varepsilon+\epsilon)}p^{( h_p-k_p+1)(1-\varepsilon)}}.
\Psi_{z}( q_{r}) = \frac{1}{\pi^{1 / 4}}e x p \left( - \frac{1}{2}( | z |^{2}- z^{2}) \right) e x p \left( - \frac{1}{2}( q_{R}- \sqrt{2}z )^{2}\right) , \hspace{0 . 7 i n}q_{R}\in R .
W^l_1(x,t)=\sum_{i=1}^{\lceil l/2\rceil}{w}_{1,i}+\sum_{i=1}^{\lfloor l/2\rfloor}\bar{w}_{2,i},\ \W_2^l(x,t)=\sum_{i=1}^{\lfloor l/2\rfloor}\bar{w}_{1,i}+\sum_{i=1}^{\lceil l/2\rceil}{w}_{2,i},
S_1^{CS}(\beta,\alpha=1,y)=-\left[\mbox{Tr}\left(\hat{\rho}^{H}(\beta)\ln\hat{\rho}^{H}(\beta)\right)-\mbox{Tr}\left(\hat{\rho}^R(2\pi\mu)\ln\hat{\rho}^R(2\pi\mu)\right)\right]~~~.
\mathbb{E}\left [ Z_{\alpha}^{r}\right ]= \frac{\alpha B\left ( \alpha ,r+1 \right )}{\mathbb{E}\left [ Y^{\alpha}\right ]-\mathbb{E}[X^{\alpha}]}\left \{\mathbb{E}\left [ Y^{\alpha+r}\right ]-\mathbb{E}[X^{\alpha+r}] \right \},r\in\mathbb{R}^{+}.
d(F_1dx_1+F_2dx_2)= \frac{\partial F_1}{\partial x_2}dx_2\wedge dx_1+\frac{\partial F_2}{\partial x_1}dx_1\wedge d x_2 = (|\nabla u|^{-2}A_{12}-|\nabla u|^{-2}A_{12})dx_2\wedge dx_1=0.
\sum_{k=0}^{M}[4k-1]\frac{(q^{-1};q^2)_k^5(q^{-3};q^2)_k}{(q^2;q^2)_k^5(q^{4};q^2)_k}q^{9k}\equiv\frac{\Omega_q(n)[5][7]}{[2]^3[4]^2[6]}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3(q^9;q^2)_k}{(q^2,q^6,q^6,q^8;q^2)_k}q^{2k}.
{\mathbf q}(x) ={\rm Tr}\left( x^* \begin{pmatrix}0 1 -1 0 \end{pmatrix}W\right) (W= \begin{pmatrix}X_1\otimes X_2 X_1\otimes Y_2 Y_1\otimes X_2 Y_1\otimes Y_2 \end{pmatrix}).
0 = x^3w x^4 + 2 wx^3w x^4 + \frac{1}{2}xwx^4w x^4 - \frac{1}{24}x^3wx^4w x^4 & = x^3w x^4 + 2 wx^3w x^4 + 12 xwx^4 - x^3wx^4
\int_\Omega |\nabla u|^2\, dx+\int_\Omega V\, u^2\, dx=\langle f,u\rangle\le \frac{1}{2}\, \|f\|_{W^{-1,2}(\Omega)}^2+\frac{1}{2}\,\int_\Omega |\nabla u|^2\, dx.
g_{\tilde{r}\tilde{r}}( G_{\tilde{r}}^{\tilde{r}}+ G_{\theta}^{\theta}) = 1 6 \pi G \frac{m}{f}( L_{M}- g^{\tilde{r}\tilde{r}}\frac{\partial L_{M}}{\partial g^{\tilde{r}\tilde{r}}}- g^{\theta \theta}\frac{\partial L_{M}}{\partial g^{\theta \theta}})
\begin{cases}\partial_{t}p+\partial_{v}(hp)-a\partial_{v v}p=0,v\in(-\infty,V_F]/\{V_R\}, p(v,0)=p^0(v),p(-\infty,t)=p(V_F,t)=0, p(V^-_R,t)=p(V^+_R,t),\partial_vp(V^-_R,t)=\partial_vp(V^+_R,t)+\frac{N(t)}{a}, \end{cases}
\begin{aligned}S^N(m)=\{R\in C^\infty(T^*{\bf R}^d): \exists M,\forall\alpha,\beta\in{\bf N}^{2}\cr \exists C_{\alpha\beta}>0, \ |\partial^\alpha_{x}\partial^\beta_pH(x,p;h)|\leq C_\alpha h^N m^M\}\cr\end{aligned}
\frac{1}{b_{1}}\Lambda=\bigcup_{n=0}^{q-1}\left(\frac{n}{q}+ \Lambda_{n}\right)=\bigcup_{i=0}^{\tau_{1}-1}\bigcup_{j=0}^{p_{1}-1}\bigcup_{l=0}^{t_{1}-1}\left(\frac{i+\tau_{1}j+\tau_{1}p_{1}l}{q}+\Lambda_{i+\tau_{1}j+\tau_{1}p_{1}l}\right).
X_{j,1}=1\;(j\geq1), & X_{1,2}=1,\;X_{j,2}=0.5X_{j-1,2}+\epsilon_{j,2},\;(j\geq2), & X_{1,3}=0,\;X_{j,3}=-0.7X_{j-1,3}+\epsilon_{j,3},\;(j\geq2), & X_{1,4}=-1,\;X_{j,4}=0.8X_{j-1,4}+\epsilon_{j,4},\;(j\geq2),
\begin{aligned}u =K^h_{(\Lambda,d\mu)}\bigl[{(1-\rho)A\over H}\bigr]+\cr \bigl({2i\pi\over h}\bigr)^{1/2}K^h_{(\Lambda_+,d\mu_+)}\bigl[(1-\theta(t_0))A(t_0)\bigr]\cr +{i\over h}\int_0^{t_0}\exp\bigl[i\int_{\alpha^*}^{g_H^t(\alpha^*)}p\,dx\bigr]\cr K^h_{(\Lambda_t,d\mu_t)}\rho\theta(t) A(t)\,dt+{\cal O}(h^{1/2})\cr\end{aligned}
\left| n - \sum_{i=1}^m k_i \right| = n \left| \sum_{i=1}^m \lambda_i - \sum_{i=1}^m \frac{k_i}{n}\right| \le n \sum_{i=1}^m \frac{1}{n N^{1/m}}= \frac{m}{N^{1/m}}.
\lim_{z\rightarrow p}\mathcal{C}_{\Omega}^2\Big(f(z), \big(J_f(z)\tau_j\big)_T\big(f(z)\big)\Big)\delta\big(f(z)\big) = \frac12\mathcal{L}_{\Omega}\Big(f(p), \big(J_f(p)\tau_j\big)_T\big(f(p)\big)\Big) = \frac12\mathcal{L}_{\Omega}\big(p, J_f(p)\tau_j\big).
F_{ts}\left({t,s}\right) = \frac{1}{{ts}}f_{st}\left({\frac{1}{t},\frac{1}{s}}\right) - \frac{1}{t}f_t \left({\frac{1}{t},\frac{1}{s}}\right) - \frac{1}{s}f_s \left({\frac{1}{t},\frac{1}{s}}\right) + f\left({\frac{1}{t},\frac{1}{s}}\right) &= F_{st}\left({t,s}\right).
\P_\pi[T_{A}>t]=\langle P_{t,B}1_{B},1_{B}\rangle_{\pi}=\sum_{i=1}^m c_{i}^{2}e^{-t(1-\gamma_i)}= \frac{1}{\|\alpha/\pi\|_2^2}e^{-t/\mathbb{E}_{\alpha}[T_A]}+\sum_{i=2}^m c_{i}^{2}e^{-t(1-\gamma_i)}
s_{\alpha}(x) = y_j+y'_j(x-x_j) &\quad+\left( \dfrac{3(y_{j+1}-y_j)-h_{j+1}(2y'_j+y'_{j+1})}{h_{j+1}^2}+O(\alpha^2)\right)(x-x_j)^2 &\qquad+\left(\dfrac{h_{j+1}(y'_j+y'_{j+1})-2(y_{j+1}-y_j)}{h_{j+1}^3}+O(\alpha^2)\right)(x-x_j)^3+O(\alpha^2) &=\sigma(x)+O(\alpha^2 h_{\max}^2).
\frac{\partial \Gamma_{\Lambda}[ \phi ]}{\partial \Lambda}= - \frac{\Omega}{2}\mathrm{t r}\int_{0}^{\infty}\! \! \! \! \! d q \, \frac{q^{D - 1}}{C ( q , \Lambda )}\frac{\partial C ( q , \Lambda )}{\partial \Lambda}\left\langle \left[ 1 + C . \frac{\delta^{2}\Gamma_{\Lambda}}{\delta \phi \delta \phi}\right]^{- 1}\! \! \! ( \mathbf{q}, \mathbf{- q}) \right\rangle
T_n \sum_\mu f_\mu = \sum_{\mu:\ \mu_n < 0}\left( t_n f_{s_n\mu}+(t_n-1)f_\mu \right) + \sum_{\mu:\ \mu_n = 0}t_n f_\mu + \sum_{\mu:\ \mu_n > 0}f_{s_n \mu}&= \sum_{\mu:\ \mu_n < 0}t_nf_{s_n\mu}+ \sum_{\mu:\ \mu_n \le 0}t_n f_\mu = t_n \sum_\mu f_\mu.
+\frac{b_{0}}{(-2)_{2}}\sum_{r=3}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{(r-3)!r!}[H_{2}+\sum_{k=0}^{k=r-1}\frac{1}{\alpha+k}+\sum_{k=0}^{k=r-1}\frac{1}{\beta+k}-H_{r}-H_{r-3}]x^{r}.
\frac{dn_i}{d\rho_i}=\frac{1}{16\pi G_{11}T_2}\int F^{(7)}_{\rho_i \phi_i \alpha_1\alpha_2 \alpha_3 \alpha_4 \alpha_5}d\phi_i d^5 \alpha =8N^{2}\frac{q_i}{L_7^6}\rho_i \,.
i_L\tau (X_1,\ldots,X_{q+r}) := &= \frac{1}{(q+1)!(r-1)!}\sum\limits_{\sigma\in S_{q+r}}sign(\sigma) \ \tau\left(L\left(X_{\sigma(1)},\ldots,X_{\sigma(q+1)}\right),X_{\sigma(q+2)},\ldots,X_{\sigma(q+r)}\right)
{\cal L}= \int d^4\theta \left(Q^{i \dagger}e^{2V}Q^{i}+ \frac{1}{2}\mbox{Tr}S^{\dagger}S\right) + \int d^2 \theta \left(\frac{1}{4 g^2_h}W^a W^a + \frac{1}{2}\lambda S_{ij}Q^i Q^j\right) + h.c.
k:=k(n,p)=\begin{cases}\left\lceil\log \left( \frac{4n}{(15)^2}\right) /\left(2\log(np) \right) \right\rceil +1 np=c\log n c>0 \left\lceil\log \left( \frac{400n}{81}\right) /\left(2\log(np)\right) \right\rceil np=\omega\left(\log n \right).\end{cases}
E_2(f,p)=\sum_{z\leq p}\frac{1}{p}\sum_{\substack{0< a< p gcd(n,p-1)=1}}e^{\frac{i2\pi a(\tau^{n}-u)}{p}}\cdot \frac{1}{p}\sum_{\substack{0< b< p \gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{em}-v)}{p}}\ll p^{1-2\varepsilon}.
\inf_\theta X_2(\theta) =\min\Big\{X_2(0),X_2\Big(\frac{p-2-\gamma}{(p-2)\gamma}\Big), X_2(1)\Big\}&=\min\Big\{4, \frac{p-2}{\gamma}+\frac{\gamma}{p-2}+2,\big(\sqrt{p-1}+\sqrt{1-\gamma}\big)^2 \Big\}&=\min\Big\{4, \big(\sqrt{p-1}+\sqrt{1-\gamma}\big)^2 \Big\}.
U_k\rightharpoonup U \hbox{in}\dot{H}^s_a(\mathbb{R}^{n+1}_+){\mbox{as}}k\to+\infty{\mbox{and}}U_k\rightarrow U \hbox{a.e. in}\mathbb{R}^{n+1}_+{\mbox{as}}k\to+\infty.
\widetilde{\psi}(d_{1}(^{\ast}b)d_{2}\cdot n) =\widetilde{\psi}(d_{1}\overline{N}(^{\ast}b)(d_{2}n)) &=\widetilde{\psi}\begin{pmatrix}-d_{1}\pi_{1}p \xi_{be_{k}}(d_{2}n) -d_{1}\gamma' \xi_{be_{k}}(d_{2}n) 0 0\end{pmatrix}
\Phi^{\l}( w ) = \left\{\prod_{i = 0}^{J - 1}\prod_{j = 1 + m_{i}}^{m_{i + 1}}\prod_{k = i + 1}^{J}( d_{j}( \beta ) - \zeta_{1 + m_{k}}^{\l}) \right\}J_{\l , N}^{( \beta )}( w ) ;
\frac{2\epsilon_f + 4/a + r}{h(\gamma \bar\Delta)}T + \left[ \sum_{k=0}^{T-1}\Theta_k(1 - \Lambda_k') + (1-\Theta_k)\Lambda_k \right] < \left( \frac{2\epsilon_f + 4/a + r}{h(\gamma \bar\Delta)}+ \frac{1}{2}\right) T + \frac{1}{2}\left|\log_\gamma \frac{\bar\Delta}{\delta_0}\right| + \frac{1}{2}.
\nabla_X J_{\varphi}Y = \frac{2}{\sqrt 5}\nabla_X \varphi Y - \frac{1}{\sqrt 5}\nabla_X Y = \frac{2}{\sqrt 5}\varphi(\nabla_X Y)- \frac{1}{\sqrt 5}\nabla_X Y = J_{\varphi}(\nabla_X Y), \forall X, Y \in{\mathfrak X}(M),
\Delta_{\varepsilon ,\overline{\varepsilon}}^{(1)}=\frac{\sigma^2}2\cdot \frac{m-i\varepsilon \widehat{p}}{2m}\cdot\frac{m-i\overline{\varepsilon}\overline{p}}{2m}\cdot \left(1-\sigma_p^2\right) =\Psi_{\varepsilon , \overline{\varepsilon}}\cdot \overline{\Psi}_{\varepsilon ,\overline{\varepsilon}}