harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2}\|w(t)\|^2_{L^2(\Omega)}+\int_s^t\|\nabla w\|^2_{L^2(\Omega)}d\tau &\leq\frac{1}{2}\|w(s)\|^2_{L^2(\Omega)}+\int_s^t\langle (hu_s+\widetilde U)\otimes w, \nabla w \rangle d\tau+\int_s^t\langle f, w\rangle d\tau
M_1 = M_{(a, x)}\; \; a=(1,\tau), \; x=\bigl(1+\tau(1+\lambda_1), \, 1+\tau(1+\lambda_2)\bigr), M_2 = M_{(a, x)}\; \; a=(1,\tau), \; x=(-\tau\lambda_1, \,-\tau\lambda_2).
h(\omega,k.)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}w(k.,0,\omega)dk.=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W(.,t)exp(\omega,t,k.,x.)dtdxdydk.
\left| \left| P_{(S_i^j)^\perp}D(m)^{l}z_i(m) \right| \right|_2 = \min_{p \in{\cal P}_j}\left| \left| \left( D(m)^l - p(D(m)) \right) z_i(m) \right| \right|_2 & \leq \Phi_{k,j}||z_i(m)||_2,
(D_t - \Delta) (P\widetilde \nabla^k \widetilde A - \nabla^k A) = ((D_t-\Delta )P) \widetilde\nabla^k \widetilde A - 2g^{ij}(\nabla_i P)(\nabla_j \widetilde\nabla^k \widetilde A) &+ P( (D_t - \Delta)\widetilde \nabla^k \widetilde A) - (D_t - \Delta)\nabla^k A.
\big( \mu_{(\alpha ,\beta )}\circ ( \operatorname{Id}\otimes S ) \circ\Delta_{(\psi ,\omega )}\big) (X) =\alpha (X)S ( 1 ) +\alpha ( 1 ) S(X) =X^{2}S (1 ) +S(X), ( \mu_{(\alpha ,\beta )}\circ ( S\otimes \operatorname{Id}) \circ\Delta_{(\psi ,\omega )}) (X) =\alpha ( S (X ) ) 1+\alpha ( S ( 1 ) ) X,
L_{2}(\mathbb{R}) = \overline{V_{p,0}\oplus \bigoplus_{j = 0}^{\infty}W_{p,j}^{\star}}= \overline{V_{p,0}\oplus \bigoplus_{j = 0}^{\infty}\hat{\bigoplus}_{q = 0}^{p}W_{q,j}^{\star \star}}.
\rho(x)=\frac{x\sinh(2\pi x)}{\pi^{2}\Gamma(a)\Gamma(b)\Gamma(c)}\left|\Gamma\!\left(\frac{b+c-a}{2}+ix\right)\!\Gamma\!\left(\frac{a+c-b}{2}+ix\right)\!\Gamma\!\left(\frac{a+b-c}{2}+ix\right)\right|^{2}\!.
\frac{\partial^n\alpha}{\partial u^n}(u,v) =\frac{\partial^n\alpha}{\partial u^n}(u,0)+\int_0^v\left(\frac{\partial^n}{\partial u^n}(\nu F)\right)(u,v')dv' =F_1(u,v)+\int_0^v\left(F_2\frac{\partial^{n+1}t}{\partial u^n\partial v}+F_3\frac{\partial^nF}{\partial u^n}+F_4\frac{\partial^nt}{\partial u^{n-1}\partial v}\right)(u,v')dv'.
\mu_{-}= \left( \begin{array}{c c}{I_{n}}&{0}\\{\mu_{- 2 1}}&{I_{k}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{I_{n}}&{\mu_{+ 1 2}}\\{0}&{I_{k}}\\ \end{array}\right) ,
F(r) =I(U_r)+O(\|l\|\|\omega\|+\|\omega\|^2) &=k\Big(A-\frac{B_0k^{N+2s}}{r^{N+2s}}+\frac{B_1}{r^m}+O\Big(\frac{1}{r^{m+\sigma}}\Big)+O\Bigl(\Bigl(\frac{k}{r}\Bigr)^{N+2s+2\tau}+\frac{1}{r^{m+2\tau}}\Bigr)\Big) &=k\Big(A-\frac{B_0k^{N+2s}}{r^{N+2s}}+\frac{B_1}{r^m}+O\Big(\frac{1}{r^{m+\sigma}}\Big)\Big).
-\cos\psi\cdot\left(\frac{F[p,q](\frac{i}{\mu_{0}})w_{1}[p,q](\frac{i}{\mu_{0}})w_{3}[p,q](\frac{i}{\mu_{0}})}{w_{2}[p,q](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\lim_{r \to \infty}\frac{1}{r^n}\sum_{B \in P : B \subseteq rC}\lambda(B) \geq \lim_{r \to \infty}\frac{\lambda(|P| \cap (r-r_0)C)}{r^n}= \lim_{r \to \infty}\frac{\lambda(|P| \cap rC)}{(r+r_0)^n}= \rho(|P|)
s^{(j)}_{n}:=\lim_{q\to p_j}\dfrac{q^{-n}-r^{-n}}{r-q}= \begin{cases}\dfrac{p_j^{-n}-r^{-n}}{r-p_j}, p_j\ne r, \smallskip \dfrac{n}{r^{n+1}}, p_j=r. \end{cases}
\phi_1=\frac{\mu+\frac{1}{2}\partial(\varphi)}{\varphi}=\frac{(z^2 +1)^3\mu+z^4-z^2}{(z^2+1) ((z^2+1)^2\lambda+z^4+z^2+1)}, \mbox{where}\varphi=\lambda+1-\frac{1}{\eta^2},
0= (x_0 x_1 x_2-1) \cdot (x_0^{-1}x_1 x_2-1) \cdot (x_0 x_1^{-1}x_2-1) \cdot (x_0 x_1 x_2^{-1}-1) = (-x_0 x_1 x_2)^{-1}\cdot (x_1 x_2-x_0) (x_0x_2 -x_1) (-x_0x_1 + x_2) (x_0x_1x_2 - 1).
\kappa_\pi\left(T_1 \cdots T_{k(1)}, T_{k(1)+1}\cdots T_{k(2)}, \ldots, T_{k(m-1)+1}\cdots T_{k(m)}\right) = \sum_{\substack{\sigma \in BNC(\widehat{\chi}) \sigma \vee \widehat{0_\chi}= \widehat{\pi}}}\kappa_\sigma(T_1, \ldots, T_n).
\lim_{|\xi|_{p,q,r}+\theta^{\ell}\to\infty}\frac{|\partial_F\hat{\psi}|+|\partial_{\zeta}\hat{\psi}|^{\frac{p}{p-1}}+|\partial_w\hat{\psi}|^{\frac{p}{p-2}}}{|\xi|_{p,q,r}+\theta^{\ell}}=0\;,
I_{\epsilon_n}(u)=\frac{1}{2}\int_{\Omega}\big(|\nabla u|^{2}-a(x)u^{2}\big)-\frac{\mu}{2^{*}(t)-\epsilon_n}\int_{\Omega}\frac{|u|^{2^{*}(t)-\epsilon_n}}{|y|^{t}}-\frac{1}{2^{*}(s)-\epsilon_n}\int_{\Omega}\frac{|u|^{2^{*}(s)-\epsilon_n}}{|y|^{s}}.
\frac{1}{2}\int_{0}^{1}x(\rho^2(T,x)-\rho_0^2(x))dx+\frac{1}{2}\int_{0}^{T}\rho^2(t,1)dt-\frac{1}{2}\int_{0}^{T}\int_{0}^{1}\rho^2dxdt+\int_{0}^{T}\int_{0}^{1}x\rho u_xdxdt=0.
a^{(r)}\theta_i - \theta_i = \theta_io(r^{J})=o(r^{J}\theta_i), (a^{(r)})^3 \sigma^2_{(r)}\theta^2_i - c^2_{e,i}\theta^2_i = \theta^2_io(r^{J-1})=o(r^{J-1}\theta^2_i)=o(\theta_i^2).
\pi^{\mu}\,\rho(\tilde{\eta}_{o})\,U({\textstyle{\frac{1}{2}}}\tilde{\eta}_{o},{\textstyle{\frac{1}{2}}}\tilde{\eta}_{o},d-\tilde{\eta}_{o})=\frac{\Gamma(\tilde{\eta}_{o})\Gamma(\mu-\frac{1}{2}\tilde{\eta}_{o})}{\Gamma(\mu-\tilde{\eta}_{o})\Gamma(\frac{1}{2}\tilde{\eta}_{o})}.
\begin{bmatrix}x_1 x_2 \dots x_{n-1}x_n fx_n x_1 \dots x_{n-2}x_{n-1}\vdots \vdots \ddots \vdots \vdots fx_3 fx_4 \dots x_1 x_2 fx_2 fx_3 \dots fx_n x_1\end{bmatrix}\in \mathbb{C}^{n \times n}.
H^0(X,\, E^*\otimes K^{1/2}_X)\,=\,H^1(X,\, E\otimes K^{1/2}_X)^*\,=\,0\,=\, H^0(X,\, E\otimes K^{1/2}_X)^*\,=\, H^1(X,\, E^*\otimes K^{1/2}_X)\, .
|N_4| =\frac{1}{2\sqrt{n}}\bigg|\sum_{j=1}^m\bigg\{\frac{3}{\sqrt{p_j}}(1-p_j)(1-2p_j)\mathbb{E}S_jf''(W) &\quad+2\bigg[\frac{1}{\sqrt{p_j}}(1-p_j)(1-2p_j)+\sqrt{p_j}\bigg]\mathbb{E}S_j^3f^{(3)}(W)\bigg\}\bigg| &\leq\frac{1}{\sqrt{n}}\sum_{j=1}^m\frac{1}{\sqrt{p_j}}\bigg\{\frac{3}{2}|\mathbb{E}S_jf''(W)|+|\mathbb{E}S_j^3f^{(3)}(W)|\bigg\}.
|w_{d_{1}-1}|\ge \frac{c_{d_{1}, 1}}{(d_1-1)!}, \ \w_{d_{1}-2}=0, \ \ |w_{d_{1}-3}|\leq \frac{(s_{d_{1}})^2}{(d_1-3)!}, |w_{d_{1}-4}|\leq \frac{(s_{d_{1}})^3}{(d_1-4)!}, \dots, |w_0|\leq (s_{d_{1}})^{d_{1}-1}.
\bigl|\triangle_2^{\ell}R_n (\cos\theta) \bigr|\le C \begin{cases}\theta^\ell(1+n\theta)^{-\frac{d-1}2}, \theta\in [0, \pi/2], (\pi-\theta)^{\ell}(1+n(\pi-\theta))^{-\frac{d-1}2}, \theta\in [\pi/2, \pi].\end{cases}
\mathcal{U}[h] = (-1)^{h}(L_0-h)! \sigma_{L_0-h}\! \left(\overline l,\overline{\alpha}\right)\cdot (-1)^{L}F_{m+1}\cdot \alpha_1^{l_0l_1}\cdots \alpha_m^{l_0l_m}\prod_{1 \le i < j \le m}(\alpha_i - \alpha_j)^{l_i l_j}
P = (x+4)I_{20}+ \frac{10(x+6)}{(x-22)(x+8)}J_{20}+ X; Q = B + \frac{9x+82}{(x-22)(x+8)}J_{20,s}; R = (x+4)I_{s}+ \frac{10(x+6)}{(x-22)(x+8)}J_{s}.
\frac{(n-2-a)^2}{4}\left\|\frac{g}{|x|^{1+\frac{a}{2}}}\right\|_2^2=\left\|\frac{\partial_rg}{|x|^{\frac{a}{2}}}\right\|_2^2-\left\|\frac{n-2-a}{2}\frac{g}{|x|^{1+\frac{a}{2}}}+\frac{\partial_rg}{|x|^{\frac{a}{2}}}\right\|_2^2
\frac{(2\kappa\beta)^{-m/\alpha}}{1+\kappa\frac{m}{\alpha}}\frac{\Gamma(\frac{1}{2\kappa}-\frac{m}{2\alpha})}{\Gamma(\frac{1}{2\kappa}+\frac{m}{2\alpha})}\Gamma(1+\frac{m}{\alpha})
\Big[ \int_0^1\int_{\Omega}\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_{R}\sum_{k=\mu}^{\infty}{2^{skq}\big| \Pi_k f_L^{\omega,t,\mu}(x)\big|^q}dx\Big) d\lambda dt\Big]^{1/q}\lesssim 1
\left( \begin{array}{cccc}0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 \end{array}\right)^{2n}=\left( \begin{array}{cccc}1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 \end{array}\right)
{\cal L}= - \frac{1}{4}F_{\mu \nu}\left({1 + e^{2}a \left[{1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right]}\right) F^{\mu \nu}+ \frac{\chi}{2}\varepsilon_{\mu \nu \lambda}A^{\mu}\partial^{\nu}A^{\lambda}.
T_N(T_N(X,Y),Z) = [N T_N(X,Y),NZ]-N[N T_N(X,Y),Z]-N[T_N(X,Y),NZ]+N^2[T_N(X,Y),Z] = N[N T_N(X,Y),NZ]-N[T_N(X,Y),Z]-N[T_N(X,Y),NZ]+N[T_N(X,Y),Z] = N[T_N(X,Y),NZ]-N[T_N(X,Y),Z]-N[T_N(X,Y),NZ]+N[T_N(X,Y),Z]=0,
\frac{d}{dt}|\rho|^2_{D^2}\leq C|\nabla u|_\infty|\rho|^2_{D^2}+C|\rho|_{\infty}|u|_{D^3}|\rho|_{D^2}+|\nabla \rho|_3|\nabla^2 \rho|_2\|\nabla^2 u\|_1,
\textit{H}(e,m,|\widetilde{q}|^{2},\widetilde{q\hspace{-0.2cm}/})=-\frac{4e^{2}\widetilde{q\hspace{-0.2cm}/}}{(4\pi)^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\biggl\{\frac{(M^{2}_{a})}{Z_{a}}K_{-1}(Z_{a})+\frac{(M^{2}_{b})}{Z_{b}}K_{-1}(Z_{b})\biggr\}
\sum_{i\in I}\mu(A_i)^q = \sum_{i\in I}\left( \sum_{j\in J}\mu(A_i\cap B_j) \right)^q &\le \sum_{i\in I}M^{q-1}\sum_{j\in J}\mu(A_i\cap B_j)^q &= M^{q-1}\sum_{j\in J}\sum_{i\in I}\mu(A_i\cap B_j)^q &\le M^{q-1}\sum_{j\in J}\mu(B_j)^q.
d(T^mx, T^m y) \leq C_0d(Tx,Ty)\prod_{l=1}^{N}(1+\lambda_l)\cdot\frac{\cos\varphi_l}{\cos\varphi_{l+1}}\cdot\sqrt{\frac{1+(d\varphi_{l+1}/dr_{l+1})^2}{1+(d\varphi_{l}/dr_{l})^2}}\leq C_1 N^{\frac{\beta-1}{2\beta-1}}N^{\frac{\beta}{2\beta-1}}d(Tx,Ty) \cos\varphi_1=C_1 N d(Tx,Ty) \cos\varphi_1,
H_{0}= \frac{1}{2}\int d^{3}x \left( B_{a}^{i j}F_{i j}^{a}- A_{i}^{a}A_{a}^{i}+ \left( \partial^{i}B_{0 i}^{a}\right) \left( \partial_{j}B_{a}^{0 j}\right) \right) \equiv \int d^{3}x \, h .
\Psi(E_k(x, d-1)) + \sum_{j = 0}^{k-2}(d-1)^{k-1-j}\Psi(E_{j+1}(x, d-1)) = = \sum_{j = 0}^{k-1}(d-1)^{k-1-j}\Psi(E_{j+1}(x, d-1)) ,
\mathbb{E}\|\partial^2 \mu^n\|_{L^2}^2 = \frac{16\pi^4\sigma^2}{n}\sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}}\frac{l^4}{(1+16\pi^4\lambda n^p l^4)^2}=: S(n)
1-\mu(r,0,(AB)^s B^{r-2}) &\qquad{}=1-\bigg(1-\frac{2(s-1)}{r+2(s-1)}\bigg)\bigg(1-\frac{1}{2^s}\bigg) &\qquad{}=\frac{2s}{r+2(s-1)}-\frac{2}{r+2(s-1)}+\frac{1}{2^s}-\frac{2(s-1)}{r+2(s-1)}\cdot\frac{1}{2^s},
R_{A}^{B}= \kappa \tau_{A}^{B}\, - \,\epsilon \,\biggl[ \delta_{A}^{B}\frac{1}{d}\,{\cal R}^2_{\rm EGB}\, - 2 \, R\, R_{A}^{B}+ 4\, R_{A\,C}\, R^{C\,B}+ 4 \, R_{C\, D}R_{A}^{~~C\,B\, D}- 2\, R_{A\,C\,D\,E}\, R^{B\,C\,D\,E}\biggr],
{}B = \tilde{O}_1 \bigl( Q^{-2}I_n - \Sigma^2 \bigr)^{\frac 12}\tilde{O}_1^t \,, C = \tilde{O}_2^t \bigl( Q^{-2}I_n - \Sigma^2 \bigr)^{\frac 12}\tilde{O}_2 \,,
\operatorname{tr}X \mathcal{L}(\rho) =-\operatorname{tr}(X\frac{1}{2}\sum\limits_{j}[C_{j},[C_{j},\rho]]) = -\frac{1}{2}\sum\limits_{j}\operatorname{tr}(X(C_{j}^{2}\rho - 2C_{j}\rho C_{j}+ \rho C_{j}^{2}) = -\operatorname{tr}(\rho\frac{1}{2}\sum\limits_{j}[C_{j},[C_{j},X]]) = \operatorname{tr}\mathcal{L}(X)\rho.
G_{l + \nu}\left( \left. r^{\prime \prime}, r^{\prime}; E \right| \lambda ; a \right) = - \frac{2 M}{\hbar^{2}}\, \frac{\sqrt{r^{\prime}r^{\prime \prime}}}{K_{s_{l}}( \kappa a )}\, \left[ K_{s_{l}}( \kappa a ) I_{s_{l}}( \kappa r_{<}) - I_{s_{l}}( \kappa a ) K_{s_{l}}( \kappa r_{<}) \right] K_{s_{l}}( \kappa r_{>}) \; ,
\mathcal{P}^1=r,\ \mathcal{P}^2 =\int \left(\frac{\nu}{m}+\mu\cdot m\right)\sqrt{\frac{1}{m^2-\nu^2}}dr+C_1 &=\theta(r)+\mu\int \cdot\frac{m}{\sqrt{m^2-\nu^2}}dr+C_1,
\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}\left(cx\right)^{n}}{\left(q;q\right)_{\infty}}A_{q}\left(cq^{n-1}\right)=\frac{\left(c;q\right)_{\infty}}{\left(cx;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}-n}\left(-c^{2}\right)^{n}\prod_{k=0}^{2n-1}\left(x-q^{k}\right)}{\left(q^{2},c,cq;q^{2}\right)_{n}}.
\sum_{k=0}^m c_e(k,m-k;q^l s|-a,b,-q^{1/2}a,q^{1/2}b,q)={(a^2;q)_m(q^l s;q)_m\over (q;q)_m(q^{l+1}s/a^2;q)_m}(q /a^2)^m.
\mathcal{J}_{3}(y; \mu) = \Bigl| \frac{y_1}{y_2}\Bigr|^{\frac{1}{2}\mu_2}\int_0^\infty \tilde{K}_{3\nu_3}\left(2\pi |y_1|^{1/2}\sqrt{1+u^2}\right) J^{-}_{3\nu_3}\left(2\pi |y_2|^{1/2}\sqrt{1+u^{-2}}\right) u^{3\mu_2}\frac{du}{u},
\sum_{i=k_0+1}^{N_0}\tilde{\theta}(2^{-i}) \geq \sum_{i=k_0+1}^{N_0}\theta(2^{-i}) \geq \sum_{i=k_0+1}^{N_0}\int_{2^{-i-1}}^{2^{-i}}\frac{\theta(r)}{r}dr = \int_{2^{-N_0-1}}^{2^{-k_0-1}}\frac{\theta(r)}{r}dr \geq \int_{|\epsilon|/2}^{x_0/4}\frac{\theta(r)}{r}dr \to +\infty,
M^{}:=\{(Z^0,...,Z^n)=Z^0\cdot(1,z) \in \mathbb{C}^{n+1}\; | \; Z^0\in \mathbb{C}^{\times}, z\in \overline{M}^{}\},\, \mathfrak{F}^{}:=-\frac{1}{6}k_{abc}\frac{Z^aZ^bZ^c}{Z^0}\,.
P_n^+\lbrace \mathcal{T}(t) > \beta \rbrace + P_n^-\lbrace \mathcal{T}(t)> \beta\rbrace = 2P\lbrace \mathcal{T}(t)> \beta\ |\N(t) = n\rbrace = P_n^+\lbrace M(t) > \beta \rbrace & = P_n^+\lbrace M(t) > \beta,\ \mathcal{T}(t) > \beta \rbrace + P_n^+\lbrace M(t) > \beta, \mathcal{T}(t)\le \beta \rbrace,
\mathcal E[u] =\|\sqrt{\mathcal K_\lambda}u\|^2-\frac1p\mathcal P[u] &\geq\Big(1-\frac{1}p\Big(\frac{\mathcal P[u]}{\mathcal P[\varphi]}\Big)^\frac{p-1}p\Big)\|\sqrt{\mathcal K_\lambda}u\|^2 &\geq\Big(1-\frac{c^{(p-1)/p}}p\Big)\|\sqrt{\mathcal K_\lambda}u\|^2.
\begin{array}{rl}M=\left\{(0,0,0),(2,0,1),(4,0,2),(6,0,3),(0,0,4),(2,0,5),(4,0,6),(6,0,7),\right. \left. (0,4,0),(2,4,1),(4,4,2),(6,4,3),(0,4,4),(2,4,5),(4,4,6),(6,4,7)\right\}\end{array}
[J^{\chi_A}]_\mathcal{M}:=\lambda\alpha\in\mathbb{N}^\mathbb{N},x\in X.\begin{cases}J^A_{r_\alpha}x r_\alpha> 0,\rho>-r_\alpha/2x\in\mathrm{dom}(J^A_{r_\alpha}), 0 ,\end{cases}
f_{\gamma}(y \, | \, \sigma_{2}, \rho) \pi_{\delta}( \sigma_{2}) =2^{\tfrac{-n-\gamma-1}{2}}\tfrac{\pi^{1-n}}{n}(n s_{1}^{2})^{\tfrac{1+\gamma-n}{2}}(1-\rho^{2})^{\tfrac{-\gamma}{2}}\big [ \breve{A}_{\gamma}(y \, | \, \sigma_{2}, \rho) + \breve{B}_{\gamma}(y \, | \, \sigma_{2}, \rho) \big ],
\beta(P_0;\kappa) = -P_0^2\, \frac{e^{P_0/\kappa}}{2f(P_0)}\left(\frac{n \Phi(P_0)}{\kappa}+ \Phi'(P_0) -\frac{e^{-P_0/\kappa}\Phi(P_0)}{f(P_0)}\right).
\setlength{\arraycolsep}{3pt}\renewcommand{\arraystretch}{0.8}\begin{pmatrix}a q u -a \sqrt{-q}u -a u 1 -a \sqrt{-q}u -a u 1 a q -a u 1 a q -a \sqrt{-q}1 a q -a \sqrt{-q}-a \end{pmatrix}
v = \frac{1}{detD}\sqrt{(e^{\phi}\vec{\pi}.\vec{\pi}+1)(e^{-\phi}\vec{B}.\vec{B}+ 1) - [(\vec{\pi}.\vec{B})^2 + 2\psi e^{\phi}(\vec{\pi}.\vec{B}) - \psi^2 e^{\phi}\vec{B}.\vec{B}]}.
\sum_{n=-N}^{N}\log | \tanh(C_{d}(x - a(n))) | \leq \int_{a(-N-1)}^{a(N+1)}\log | \tanh(C_{d}(x - z)) | \, b'(z)\, \d z & + (2c+1) (\lambda \log N - \log (C_{d}/2)) + 2c,
F_{\beta_{t}}F_{\beta_{l+1}}-q^{-(\beta_t,\beta_{l+1})}F_{\beta_{l+1}}F_{\beta_{t}}=\sum_{n_{l+2},\ldots,n_{t-1}\geq 0}c(n_{l+2},\ldots,n_{t-1})F_{\beta_{l+2}}^{n_{l+2}}\ldots F_{\beta_{t-1}}^{n_{t-1}}.
{\bf E}\Bigl[\Bigl(\int\limits_0^1 f(t)\,M_\mu(dt)\Bigr)^n\Bigr] = n!\int\limits_{0<t_1<\cdots<t_n<1}\prod\limits_{i=1}^n f(t_i)\prod\limits_{k<p}^n |t_p-t_k|^{-\mu}\,dt.
\begin{array}{cc}\dot \pi_g = 0 \Rightarrow \pi^2 = \frac{b^2}{a^2}\dot \pi_\varphi = 0 \Rightarrow v^2 = a^2 \pi \dot \pi = 0 \Rightarrow v \pi = 0 (v \pi)\dot{}= 0 \Rightarrow \frac{2}{\varphi}a^2 + g \frac{b^2}{a^2}=0. \end{array}
C=k\frac{2\pi \kappa_{B}}{G_{k}}\frac{\Omega_{d-2}}{\Sigma_{d-2}}r_{+}^{d-2k}\left( \gamma +\frac{r_{+}^{2}}{l^{2}}\right)^{k-1}\left[ \frac{r_{+}^{2}+\gamma r_{c}^{2}}{r_{+}^{2}-\gamma r_{c}^{2}}\right] ,
|x^k\partial^n_x(\frac{1}{x}\varphi(x))| \le \sum_{j=0}^n\frac{n!}{(n-j)!}x^{k-j-1}|\varphi^{(n-j)}(x)| &\le \sum_{j=0}^n\frac{n!}{(n-j)!}x^{-n-1}|\varphi^{(n-j)}(x)| &\le \sum_{j=0}^n\frac{1}{(n-j)!(n+1)}\sup_y|\varphi^{(2n+1-j)}(y)|,
\mathrm{H}^i_\mathrm{cont}(G) =\mathrm{H}^i_\mathrm{cont}(G,\mathbb{F}_p) \mathrm{H}^i_\mathrm{disc}(G) =\mathrm{H}^i_\mathrm{disc}(G,\mathbb{F}_p) \text{for $i \in \mathbb{N}_0$}.
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_3,[e_2,e_{n+1}]=e_2,[e_{i},e_{n+1}]=e_{i}+\epsilon e_{i+2}+\sum_{k=i+3}^n{b_{k-i-2}e_k},[e_{n+1},e_1]=-e_3, \displaystyle [e_{n+1},e_i]=-e_i-\epsilon e_{i+2}-\sum_{k=i+3}^n{b_{k-i-2}e_k},(\epsilon=0,\pm1,3\leq i\leq n,n\geq4).\end{array}
\begin{aligned}{\rm div}(J_{M_{g,k+1}}^{*}\chi_{g}^{8}) ={\rm div}(i^{*}J_{M_{g,k}}^{*}\chi_{g}^{8})=i^{*}{\rm div}(J_{M_{g,k}}^{*}\chi_{g}^{8})=i^{*}(a_{g,k}{\mathcal D}_{\Lambda_{g,k}}^{-}+b_{g,k}{\mathcal H}_{\Lambda_{g,k}}) &=a_{g,k}{\mathcal D}_{\Lambda_{g,k+1}}^{-}+2b_{g,k}{\mathcal H}_{\Lambda_{g,k+1}}.\end{aligned}
\delta \phi^{i}\left( \frac{\partial}{\partial \phi^{i}}f^{( p )}\right) \left[ \psi^{p}\right]_{\mathrm{m a x}}= \delta \phi^{i}\phi^{i}\left( \frac{d}{\phi d \phi}f^{( p )}\right) \left[ \psi^{p}\right]_{\mathrm{m a x}}= - f^{( p + 2 )}\delta_{\psi}\left[ \psi^{p + 2}\right]_{\mathrm{m a x}},
\int_{0}^{\frac{\pi}{2}}sin^{2m+1}(x)dx=\int_{0}^{\frac{\pi}{2}}cos^{2m+1}(x)dx=\frac{2\times4\times6\times\cdot\cdot\cdot\times2m}{1\times3\times5\times\cdot\cdot\cdot\times(2m+1)}form=1,2,3...
\left ( \prod_{i=1}^mY_i^{- \frac{2n_i}{n}}\right )^{-1}\frac{d \bar{\lambda}(Y(u))}{du}= \frac{2}{n}\left (2 \sum_{i=1}^m n_i p_i Y_i - \frac{1}{2}E(Y)\right)^2 -\frac{1}{2}E(Y)^2 - \sum_{i=1}^m n_iY_i^2 (2p_i -q_i^2Y_i )^2 .
Z_2 (\{b_k\},\{a_k\}) = \prod_{k=1}^\infty\exp\left(-2k\left( \frac{1+q^{2k}}{1-q^{2k}}(a_ka_{-k}+ b_kb_{-k}) - \frac{2q^k}{1-q^{2k}}(a_kb_{-k}+ b_ka_{-k}) \right)\right).
[\begin{matrix}E_{x}\\E_{y}\\E_{z}\end{matrix}]=[\begin{matrix}\rho_{xx}&\rho_{xy}&\rho_{xz}\\ \rho_{yx}&\rho_{yy}&\rho_{yz}\\ \rho_{zx}&\rho_{zy}&\rho_{zz}\end{matrix}][\begin{matrix}J_{x}\\J_{y}\\J_{z}\end{matrix}],
\mathcal{B}_{t}(f,g) :=e^{t^{\frac{1}{\alpha}}\Lambda_{1}}(e^{-t^{\frac{1}{\alpha}}\Lambda_{1}}fe^{-t^{\frac{1}{\alpha}}\Lambda_{1}}g) =\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot(\xi+\eta)}e^{t^{\frac{1}{\alpha}}(|\xi+\eta|_{1}-|\xi|_{1}-|\eta|_{1})}\hat{f}(\xi)\hat{g}(\xi)d\xi d\eta.
\begin{cases}\begin{array}{ll}\dot{u}_m=\log\det (u_m)_{\alpha\bar{\beta}}-Au_m+f(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), u_m=\varphi_m \mbox{on}\;\partial\Omega\times [0,T), u_m=u_{0,m}\mbox{on}\;\bar{\Omega}\times\{0\}. \end{array}\end{cases}
[A_1, A_2]= \left(\begin{array}{cc}0 -1 1 0 \end{array}\right), [A_1, A_2]^{(t_2,s_1)}=\left(\begin{array}{cc}-\cosh (s_1) \sinh (2 t_2) \sinh (s_1)-\cosh (s_1) \cosh (2 t_2) \cosh (s_1) \cosh (2 t_2)+\sinh (s_1) \cosh (s_1) \sinh (2 t_2) \end{array}\right).
\mathcal{E}^{( q )}( B_{r}) = \frac{m \cos ( \frac{\pi}{4}- \frac{\pi}{2 h})}{4 \sin ( \frac{\pi x}{2 h}) \cos ( \frac{\pi ( 1 - x )}{2 h})}+ \frac{m ( \cos ( \frac{\pi w_{r}}{h}) - \cos ( \frac{\pi x}{2 h}) )}{2 \sqrt{2}\sin ( \frac{\pi x}{2 h}) \, \sin ( \frac{\pi ( 1 - x )}{h})};
z_{s,t}^\epsilon :=\begin{cases}y^\ell_{s,t}q_1+ y^\ell_{s,t+1}(p_1 - q_1) \epsilon=\ell q_1y_{s,r}^r + (p_1-q_1)y_{s,t+1}^r \epsilon=r q_1y^c_{s,t}q_1 + \left[ y_{s,t+1}^c - q_1 y_{s,t+1}^c q_1\right] \epsilon=c\end{cases}.
{\rm dim}_q{\tilde{\rho}}_{\ell}={([N][N+1] \ldots [N+2\ell-1])\prod_{a=1}^{N-2}([N-a] [N-a+1] \ldots [N-a+\ell-1])\over [\ell]! ([N+2 \ell-2] [N+2 \ell-3] \ldots [N+\ell-1]) \prod_{a=1}^{N-2}([a][a+1] \ldots [a + \ell-1])}
t([\varphi_1\iota_{Q_1}^{Q_2}]) =t([\iota_{Q_2}^{Q_1}])^{\varphi_1^{-1}}t([\varphi_1])=t([\varphi_1]), t([\iota_{P_1}^{P_2}\varphi_2]) =t([\varphi_2]t([\iota_{P_1}^{P_2}])=([\varphi_2]),
\xi^* \left( \left[\overline{\mathcal{W}}_2 \right]\right) = 3 \left( \pi_1^*(\psi) + \pi_2^*(\psi) \right) - \frac{1}{10}\left(\pi_1^*(\delta_0) + \pi_2^*(\delta_{0})\right) - \frac{6}{5}\left(\pi_1^*(\delta_{1}) + \pi_2^*(\delta_{1})\right).
\begin{matrix}{\tilde{h}^{0}= h^{0}}\\{\tilde{h}^{1}= - h^{2}\sin ( 2 t ) + h^{1}\cos ( 2 t )}\\{\tilde{h}^{2}= h^{2}\cos ( 2 t ) + h^{1}\sin ( 2 t )}\\ \end{matrix}
\langle ab, c \rangle = \langle \nabla(a \otimes b), c \rangle = \langle a \otimes b, \Delta(c) \rangle, \langle a, bc \rangle = \langle a, \nabla(b \otimes c) \rangle = \langle \Delta(a), b \otimes c \rangle, \langle a, 1 \rangle = \langle 1, a \rangle = \varepsilon(a).
\frac{A^{(m-1)}(\omega^{-nM/2}E_{k}^{(m)},s)}{A^{(m-1)}(\omega^{nM/2}E_{k}^{(m)},s)}\, \frac{A^{(m)}(\omega^{nM}E_{k}^{(m)},s)}{A^{(m)}(\omega^{-nM}E_{k}^{(m)},s)}\, \frac{A^{(m+1)}(\omega^{-nM/2}E_{k}^{(m)},s)}{A^{(m+1)}(\omega^{nM/2}E_{k}^{(m)},s)}= -\omega^{-2 \beta_{m}+ \beta_{m-1}+ \beta_{m+1}}\,.
\| \partial_\nu u\|_{L^2(\Sigma)}=\| (\Lambda_{q_1,\lambda}^1-\Lambda_{q_2,\lambda}^1)(u_2{_{|\partial M}})\|_{L^2(\Sigma)}&\le \| \Lambda_{q_1,\lambda}^1-\Lambda_{q_2,\lambda}^1\| \|u_2\|_{H^{3/2}(\partial \mathrm{M})}
\mathrm{T r}( B P_{[{\cal H \/}_{k}^{\perp}]}e^{- t \Delta_{k}}) = \sum_{0{\leq}l{\leq}l_{0}}a_{l}( B P_{[{\cal H \/}_{k}^{\perp}]}| \Delta_{k}) t^{- l}+ O ( t^{\epsilon})
\frac 1 2 \Sigma_{\mu \nu}^{( + )}\widetilde{F}_{\mu \nu}= \mathbf{S}\widetilde{\mathbf{X}}, \frac 1 2 \Sigma_{\mu \nu}^{( - )}\widetilde{F}_{\mu \nu}= \mathbf{S}\widetilde{\mathbf{X}}^{*},
\int_T\|k_t\|_{p,v\nu}h(t)\,d\lambda(t) =\int_T\bigg(\int_Xk(x,t)^pv(x)\,d\nu(x)\bigg)^{1/p}h(t)\,d\lambda(t) &\le\bigg(\int_X\bigg(\int_Tk(x,t)h(t)\,d\lambda(t)\bigg)^pv(x)\,d\nu(x)\bigg)^{1/p}&=\|Kh\|_{p,v\nu}.
C_{s,r}^{+}:=\bigg\{(x',x_{n+1})\in\mathbb{R}_{+}^{n+1}:x_{n+1}\le\bigg[(1-s)\bigg(r-\frac{|x'|^{2}}{4}\bigg)\bigg]^{\frac{1}{2-2s}}\bigg\}C_{s,r}' := \bigg\{(x',0)\in \mathbb{R}^{n}\times \{0\}: 0\le\bigg[(1-s)\bigg(r-\frac{|x'|^{2}}{4}\bigg)\bigg]^{\frac{1}{2-2s}}\bigg\}.
f(m,n)=(\begin{matrix}f_{1,1}&f_{2,1}&\cdot\cdot\cdot&f_{M,1}\\f_{1,2}&f_{2,2}&\cdot\cdot\cdot&f_{M,2}\\ \cdot\cdot\cdot&\cdot\cdot\cdot&\cdot\cdot\cdot&\cdot\cdot\cdot\\f_{1,n}&f_{2,N}&\cdot\cdot\cdot&f_{M,N}\end{matrix}),
\mathbf{E}[ f(Z^x_t) ] =W^{+}_{t,x}f (t,x) & = W^{-}_{t,x}(W^{-}_{t,x})^{-1}W^{+}_{t,x}f (t,x) & = W^{-}_{t,x}\mathcal{N}_{t,x}f (t,x).
X_1= \begin{bmatrix}a+\frac{1}{a}\frac{a}{a^2-2a+1}-a+2-\frac{1}{a}0 \end{bmatrix}, X_2=\begin{bmatrix}0 -\frac{a}{a^2-2a+1}a-2+\frac{1}{a}2\end{bmatrix}, X_3=\begin{bmatrix}1 1 0 1\end{bmatrix}.
II_1 \leq c_n\left(\int_{\{t<r_1<\cdots<r_n<\widetilde{t}\}}\prod_{j=1}^n(r_j-r_{j-1})^{-\frac{H_1H_2}{H_1+H_2}}dr\right)^2 &\leq c_n\left((\widetilde{t}-t)^{\sum_{j=1}^n\left(1-\frac{H_1H_2}{H_1+H_2}\right)}\right)^2 &\leq c_n(\widetilde{t}-t)^{2n\left(1-\frac{H_1H_2}{H_1+H_2}\right)}
\frac{1}{8\pi}\left( \int_{\Omega}|\nabla \phi_{a}|^{2}+ a^{2}\int_{\Omega}\phi_{a}^{2}\right) = \frac{1}{8\pi}\int_{\Omega}(- \Delta \phi_{a}+ a^{2}\phi_{a}) \phi_{a}+ \int_{\partial \Omega}\phi_{a}\nabla \phi_{a}\cdot n = \frac{1}{2}\int_{\Omega}\phi_{a}(m - u^{2}_{a}),
\partial_\tau \Phi -\L_{\rm ss}\Phi + \Phi \cdot \nabla (\Phi N) + \div ( \tilde{N}\Phi \otimes \Psi + \tilde{N}\Psi \otimes \Phi) + \bar{U}\cdot \nabla \Psi + \Psi \cdot \nabla \bar{U}+ \nabla \Pi =0\, ,
lim_{x\rightarrow\infty}\frac{x^{\frac{1}{2}}+x^{-\frac{1}{2}}}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}=lim_{y\rightarrow\infty}\frac{y+y^{-1}}{y-y^{-1}}=lim_{y\rightarrow\infty}\frac{1-y^{-2}}{1+y^{-2}}=\frac{1}{1}=1.
\begin{aligned}\int_{|z|<|w|}z^m d\mathrm{arg}(z)d\mathrm{arg}(z-1) =-i\pi\ \frac{w^m \overline w^m}{2m},\ \ \int_{|w|<|z|<1}\frac{1}{z^m}d\mathrm{arg}(z)d\mathrm{arg}(z-1) =i\pi\ \log(|w|), \int_{|z|<1}z^m d\mathrm{arg}(z)d\mathrm{arg}(z-1) =-i\pi\ \frac{1}{2m},\m\geq 1,\end{aligned}
Q_{2,k}\leq C_{H}\int_{0}^{1}\left|\sum_{l=1}^{k-1}\int_{0}^{1}\frac{\left|\dot{\bar{h}}_{k}(u)\right|+\left(\frac{\sum_{j_1=1}^k|I_{j_1}|}{\sum_{j_2=1}^{l-1}|I_{j_2}|}\right)^{H-\frac{1}{2}}\cdot\left|\dot{\bar{h}}_{l}(v)\right|}{|q_{k,l}(u,v)|^{H+\frac{1}{2}}}|I_{l}|dv\right|^{2}|I_{k}|du.
(\{y_j=S_{x,\tau}^{-1}x_j+Ve_j-V\theta_\tau S_{x,\tau}^{-1}x_j\}_{j\in \mathbb{J}}, \{\omega_j=S_{x,\tau}^{-1}\tau_j+Ue_j-U\theta_xS_{x,\tau}^{-1}\tau_j\}_{j\in \mathbb{J}})
\begin{aligned}y\left[ n \right] = \Big({\sum\nolimits_{l = 1}^L{{\bf{h}}_l^H{{\bf{f}}_l}}}\Big)s\left[{n -{n_{\max}}}\right] + \sum\nolimits_{l = 1}^L{\sum\nolimits_{l' \ne l}^L{{\bf{h}}_l^H{{\bf{f}}_{l'}}s\left[{n -{n_{\max}}+{\Delta_{l',l}}}\right]}}+ z\left[ n \right]. \end{aligned}
e_{r}\big(\widehat{\boldsymbol{X}}\big)=\sum_{I,A}x_{a_1}^{(i_1)}\cdots x_{a_s}^{(i_s)}, e_{0}\big(\widehat{\boldsymbol{X}}\big)=1,e_{-r}\big(\widehat{\boldsymbol{X}}\big)=0, r > 0,
\frac{\delta F \left[ f \right]}{\delta f ( k )}= \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon}\left( F \left[ f ( p ) + \varepsilon \delta_{D}^{( 4 )}( p \, \dot{+}( \dot{-}k ) ) \right] - F \left[ f ( p ) \right] \right) ~ ,
\bar{S}\kappa_{e}\bar{R}= \kappa_{o}R \bar{R}= \kappa_{o}+ ( 1 + \bar{w}w ) \kappa_{o}v \bar{v}\, , \quad \sum_{e > 0}S_{e o}\kappa_{e}w_{e}= \kappa_{o}R w_{e}= \kappa_{o}v ( 1 + \bar{w}w ) \, .
\left(z^{2}q;q^{2}\right)_{\infty}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}(-1)^{n}z^{2n}}{(q^{2};q^{2})_{n}}=\sum_{k=0}^{\infty}\left(-z\right)^{k}q^{\binom{k}{2}}S_{k}\left(q^{-k};q\right).
q_{2}^{*}=\frac{a-b\cdot\frac{a+\frac{\partial C_{2}(q_{2})}{\partial q_{2}}-2\cdot\frac{\partial C_{1}(q_{1})}{\partial q_{1}}}{2b}-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}}{2b},
x \otimes (y, \pm y) \cdot \iota(h^{-1},1) = (x \otimes y, \pm x \otimes y) \cdot \iota(h^{-1},1) = (h x \otimes y, \pm x \otimes y) = \frac{1}{2}((h \pm 1) x \otimes y, (h \pm 1) x \otimes y) + \frac{1}{2}((h \mp 1) x \otimes y, - (h \mp 1) x \otimes y) = \frac{1}{2}(h \pm 1) x \otimes (y, y) + \frac{1}{2}(h \mp 1) x \otimes (y, -y)
Z_{D}( x , x , s ) ={\frac{1}{( s - 1 ) ( s -{\frac{1}{2}}) s \Gamma ( s )}}\int_{0}^{\infty}d t t^{s}{\frac{\partial}{\partial t}}[ t^{\frac{1}{2}}{\frac{\partial}{\partial t}}[ t^{\frac{1}{2}}{\frac{\partial}{\partial t}}t K_{D}( x , x , t ) ] ]
a(\Gamma) = a_{n}^{(1)}(\{p_{ij}\}) \bigg\vert_{p_{ij}=0,\, (ij) \in \Gamma,\atop p_{ij}=1, \, (ij) \notin \Gamma}(\ell(\lambda)-2) !{\cal{N}}_{\lambda}(\Gamma)\beta^{\ell(\lambda)-2},