harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_n - n^p \ \ = \ \ \mathcal{C}_n^{(p)}\ - \ 2\sum_{j=0}^{p/2}\binom{p}{2j+1}\mathcal{C}_n^{(p-2j-1)}\ + 2 \binom{p}{2 \lfloor p/2 \rfloor + 1}F_n \mbox{for $p$ odd}.
\textup{StoC}:= \min\Big\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta}(\kappa- r), 1-\varepsilon \Big\}= \min\Big\{\frac{\alpha r}{2\beta}+ (\gamma-\frac{1}{2})^{+}, 1-\varepsilon \Big\}.
\int_M \sum_{ij}g_{ij}(t) V^i(t) W^j(t) = \int_M \sum_{ij}g_{ij}(0) V^i(0) W^j(0) \: + & \int_0^t \int_M \sum_{i,j}\left[g_{ij}(\partial_s V^i) W^j + g_{ij}V^i (\partial_s W^j) -2 \left( \nabla_i V^i \right) \left( \nabla_j W^j \right) + 2 \left( \nabla_i V^j \right) \left( \nabla_j W^i \right) \right](s) \:ds.
\check{e}'_1 := \check{e}_1 + (\sum_{i=1}^n a_{1i})x_{n+1}\xi_{n+1}= in_{(0,\ldots,0,-e,1,\ldots,1,1+e)}(\check{E}'_1 +\beta_1) \check{e}'_k := \check{e}_k = in_{(0,\ldots,0,-e,1,\ldots,1,1+e)}(\check{E}'_k + \beta_k) k= 2, \ldots ,d.
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int |\nabla^3 B|^2 dx+\int \!|\nabla^4 B|^2 dx &=\int \nabla^3\left[-u\cdot \nabla B+B\cdot \nabla u-B{\rm div}u-{\rm curl}\left(g(\varrho)\!\left(B\cdot \nabla B-\frac{1}{2}\nabla(|B|^2)\right)\right)\right]\nabla^3 B dx &=I\!I\!I_1+I\!I\!I_2+I\!I\!I_3+I\!I\!I_4.\end{aligned}
\begin{array}{c c c}{\phi_{1}= 3 D A_{1}= 3 D A_{2}= 3 D 0}&{,}&{\phi_{2}( x ) = 3 D \phi_{2}( x_{3}, x_{4}= )}\\{A_{3}( x ) = 3 D A_{3}( x_{3}, x_{4})}&{;}&{A_{4}( x ) = 3 D A_{4}( x_{3}, x_{4})}\\ \end{array}
L_{D}^{\ast}=\int_{X}d^{3}x\varepsilon^{ijk}\bigg \{-{\frac{3}{4a}}(a\partial_{i}B_{jAB}+bG_{ijAB})[{\bf R}^{-1}]_{kn}^{ABCD}\varepsilon^{lmn}(a\partial_{l}B_{mCD}+bG_{lmCD})+c\alpha_{i}^{AB}G_{jkAB}\bigg\},
f_{1}( s ) \equiv \int_{0}^{s}d x \frac{s i n h x}{c o s h^{5}x}\phi_{2}^{( 2 )}( x ) , f_{2}( s ) \equiv \int_{0}^{s}d x \frac{x}{c o s h^{2}x}\phi_{2}^{( 2 )}( x ) ,
\langle Q_{2n}(x),Q_{2m+1}(x)\rangle=0,\langle Q_{2n}(x),Q_{2m}(x)\rangle=H_n^{(0)}\delta_{n,m}, \langle Q_{2n+1}(x),Q_{2m+1}(x)\rangle=H_n^{(1)}\delta_{n,m},
\frac{1}{n}\left . V(K)^{\frac{p}{n}-1}\frac{d}{dt}\right |_{t=0+}V(K +_p t \cdot L) \geq \lim_{t\rightarrow \infty}\frac{\frac{1}{p}V(K +_p t \cdot L)^{\frac{p}{n}}- \frac{1}{p}V(K)^{\frac{p}{n}}}{t}= \frac{1}{p}V(L)^{\frac{p}{n}}.
r(t) = \mathcal{I}_{L}(\mathbf{x}_t;\mathbf{y}_t|\widehat{\mathbf{H}}_{t}) = \mathbb{E}_{\boldsymbol{\mathrm{x}}_{t},\boldsymbol{\mathrm{y}}_{t}}\bigg \{\log_2 \frac{f_{\boldsymbol{\mathrm{y}}_{t}|\boldsymbol{\mathrm{x}}_{t}}(\boldsymbol{\mathrm{y}}_{t}|\boldsymbol{\mathrm{x}}_{t})}{f_{\boldsymbol{\mathrm{y}}_{t}}(\boldsymbol{\mathrm{y}}_{t})}\bigg \},
a = \frac{1}{1 + \mu_{1}\cdots \mu_{n-1}}b = \frac{\mu_{1}\cdots\mu_{n-2}}{1 + \mu_{1}\cdots \mu_{n-1}}c = \frac{\mu_{1}\cdots\mu_{n-2}(\mu_{n-1}- 1)}{1 + \mu_{1}\cdots\mu_{n-1}}
g\tau^{r}=\left(\sigma\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}\right)\tau^{r}=\rho^{ra_{3}}\sigma\tau^{r}\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}=\rho^{r\left(a_{3}-1\right)}\tau^{r}g,
i_1<...<i_{k-1}, \; \; j_1<...<t<...<j_{b-k-1}, \{i_1,...,i_{k-1}\}\cap \{j_1,...,j_{b-k-1}\}= \emptyset, \; \{i_1,...,i_{k-1}\}\cup \{j_1,...,j_{b-k-1}\}= \{2,...,\widehat{t},...,b\}
\alpha_{n+2}^2 \leq \frac{\sum \limits_{k=1}^{n}\cos^2 \frac{\pi(2k-1)}{n}}{4n^2 \sin^2 \frac{\pi}{n}}\sum \limits_{k=1}^{n}\alpha_k^2, \alpha_{n+3}^2 \leq \frac{\sum \limits_{k=1}^{n}\sin^2 \frac{\pi(2k-1)}{n}}{4n^2 \sin^2 \frac{\pi}{n}}\sum \limits_{k=1}^{n}\alpha_k^2;
[\bar{x}^{\pm}_{i,r+1}, \bar{x}^{\pm}_{j,s}]-[\bar{x}^{\pm}_{i,r}, \bar{x}^{\pm}_{j,s+1}]= -m_{i,j}\beta [\bar{x}^{\pm}_{i,r}, \bar{x}^{\pm}_{j,s}],
\binom{n + 1}{2}- t \geq \frac{n(n + 1) - (s + 1)(2n - s)}{2}= \frac{n^2 + n - 2ns + s^2 - 2n + s}{2}&= \frac{(n - s)^2 - (n - s)}{2}= \binom{n - s}{2}
< (O^{\dagger}O)_{cl}>_{PCP}= \langle \Psi|O^{\dagger}O|\Psi \rangle = \langle \Psi|({\cal P}^{\psi}_o(1,\xi))^{\dagger}{\cal P}^{\psi}_o(1,\xi)|\Psi \rangle = < O^*_{cl}O_{cl}>_{PCP}.
l(\sigma_{t_i}(s) ) = \int_0^1 |\dot{\sigma}_{t_i}(s)| \, ds = \int_0^1 \sqrt{|\dot{\tilde{\sigma}}_{t_i}(s)|^2 + O(t_i^2)}\, ds =\int_0^1 |\dot{\tilde{\sigma}}_{t_i}(s)| + O(t_i^2) \, ds = l(\tilde{\sigma}_{t_i}(s) ) + O(t_i^2) = |t_i v - t_i v_i | +O(t_i^2)
k=\sum_{j\geq 1}e_{j,i}(k) p_i^{j-1},\; e_{j,i}(k) \in \{0,1, \ldots ,p_i-1\}, \;{\rm and}\; \phi_i(k)= \sum_{j\geq 1}e_{j,i}(k) p_i^{-j}.
\hspace{1 5 m m}\times \prod_{i = 1}^{p}v_{i}^{a}\prod_{i = 1}^{p - 1}( v_{i}- 1 )^{b - 1}( v_{i}- z )^{c}\prod_{i = p - k + 1}^{p}( 1 - v_{i})^{b - 1}( z - v_{i})^{c}
\quad\quad\quad\quad0 \leq \phi \leq 1, 0 \leq -\dot{\phi}\leq \frac{4}{t_{2}-t_{1}}, & \phi = 1 [0,t_{1}-t_{0}], \phi = 0 [t_{1}-t_{0}+(t_{2}-t_{1})/2,t_{2}-t_{0}].
\begin{aligned}-\iint g^\epsilon \partial_y h_s u^\epsilon \leq L \Vert y\partial_y h_s \Vert_{L^\infty}\Vert u^\epsilon_x \Vert_{L^2}\Vert g^\epsilon_y\Vert_{L^2}&=L \Vert y\partial_y h_s \Vert_{L^\infty}\Vert v^\epsilon_y \Vert_{L^2}\Vert g^\epsilon_y\Vert_{L^2},\end{aligned}
\begin{array}{c}\displaystyle P(\cdot) \in C^1([0,\infty)) \cap C^2(0,\infty), \displaystyle P(0) = 0, P'(0) = p_0 >0, P'(Z) >0, Z>0, \displaystyle \lim_{Z \to \infty}\frac{P(Z)}{Z^{\gamma}}= p_\infty >0, begin{align*}10pt]\displaystyle 0 < \frac{1}{\gamma-1}\frac{\gamma P(Z) - Z P'(Z)}{Z}\leq c_7 <\infty, Z>0.\end{array}
G:=\langle u,v,x,y\mid\, u^{p^{n_1}}=v^{p^{n_2}}=[u,v]=x^{dl_1}=y^{l_2}=[u,y]=[v,x]=1, &xux^{-1}=u^{\gamma_1},\yvy^{-1}=v^{\gamma_2},\yxy^{-1}=x^{1+l_1}\rangle
{\mathcal O}={\mathcal O}_{k_j}, I^l=I^l_{k_j}, \B_i=B_{r_{n_j}}(x_{k_{n_j},i}) \mbox{and}\tB_i=B_{tr_{n_j}}(x_{k_{n_j},i}) \mbox{for}t \in \mathbb{R}^+.
\begin{cases}\partial_\tau\phi = -(-\Delta)^s\phi + pU^{p-1}(y)\phi + h(y,\tau), y\in \mathbb{R}^n,~ \tau\geq \tau_0, \phi(y,\tau_0) = e_{0}Z_0(y), y\in \mathbb{R}^n.\end{cases}
\int_{pr_{g+1}}\delta_{\lbrace P_g\in\sigma(P_1+\dots+P_{g-1})\rbrace}\gamma^*\omega_0^{g}=\sum_{j=1}^{g-1}\int_{pr_{g+1}}\delta_{\lbrace P_j=P_g\rbrace}\gamma_{\sigma}^*\omega_0^g.
= \sum_{\mu, \nu, \kappa\in\{+,-\}}\frac{1}{8}\textup{Re}\Big(\int \Phi_{\mu}^{\alpha}Q^4_{\nu, \kappa}(\Phi_{\nu}^{\alpha}, \Phi_{\kappa})\Big)= \sum_{\nu, \kappa\in\{+,-\}}\frac{1}{4}\textup{Re}\Big(\int \overline{\Phi^{\alpha}}Q^4_{\nu, \kappa}(\Phi_{\nu}^{\alpha}, \Phi_{\kappa})\Big),
G_2(\varepsilon ;\zeta ):=\widetilde{g}^\varepsilon S_\varepsilon b(\mathbf{D})(B^0-\zeta \overline{Q_0})^{-1}+g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon (B^0-\zeta \overline{Q_0})^{-1}.
I{}=\int\limits_0^{T}(T-u) \bigl(e^{-2au^2}+e^{-a| u+\tau|^2}e^{-a|u-\tau|^2}\bigr)du{}=\int\limits_0^{T}Te^{-2au^2}du +T\int\limits_0^{T}e^{-a| u+\tau|^2}e^{-a| u-\tau|^2}du-\int\limits_0^{T}ue^{-2au^2}du \quad{}-\int\limits_0^{T}ue^{-a| u+\tau|^2}e^{-a| u-\tau|^2}du{}=I_1+I_2+I_3+I_4.
\frac{n^2 (n+1) \binom{2 n}{n}\binom{2 n-2}{n-1}\binom{2 n+2}{n+1}}{64(2n+1)}=\frac{(2n-1)^2(2n-3)!^3}{(n-2)!^3(n-1)!^3}&=(2n-1)^2\binom{2n-3}{n-1}^3.
[\,x_{i-1}^-(0),\,y_{1\, i+1}^-(1)\,]=0, 1<i<n [\,x_{i}^-(0),\,y_{1\, i}^-(1)\,]_{s^{-1}}=0, \qquad1\leqslant i \leqslant n-1 [\,x_{n-1}^-(0),\,x_{1\, n}^-(1)\,]=0, [\,x_{n-1}^-(0),\,x_{1\, n-1}^-(1)\,]_r=0 [\,x_{n}^-(0),\,y_{1\, n}^-(1)\,]_r=0.
2^{\lceil \log_g{\frac{m}{g-1}}\rceil}\leq 2^{\log_g{\frac{m}{g-1}}+ 1}= 2\left({\frac{m}{g-1}}\right)^{\frac{1}{\log_2 g}}< o_g(m)=(o_g(p_1),\dots,o_g(p_k))
\tilde{\Phi}^{\mu V^{(l)}}_{\lambda m}(z) = g^{\mu V^{(l)}}_{\lambda}(z) \int_{0}^{s_{1}\infty}d_{p}t_{1}J^{S}(t_{1}) \cdots \int_{0}^{s_{l-m}\infty}d_{p}t_{l-m}J^{S}(t_{l-m}) \phi^{(l)}_{m}(z),
\inf_{x \in \mathcal{X}}\ \sum_{i \in [m]}\max\{0, x_i - \overline{b}_i, x_i - \overline{b}_i + \Delta b_i - \Delta b_k \}+ \Gamma \Delta b_k \mathrm{s.t.}\x_k \in [\overline{b}_k - \Delta b_k, \overline{b}_k]
\left \{-\frac{\beta_{D \cup \{w\}}^{i}(u)}{\beta_{D \cup \{w\}}^{i}(v)}: i \in \{1, \ldots, t\}\right \}\cup \left \{\frac{x_{2}}{x_{1}}: x \in S \setminus B \right \}= \mathbb{F}_{q}\setminus \{0\}.
\sin^2{\psi}= \frac{1}{2}\pm \frac{1}{4}\frac{r_{1}^2 + r_{5}^2 -2 r_{K}^2}{\sqrt{r_{1}^4 + r_{5}^4 + r_{K}^4 - r_{1}^2 r_{5}^2 - r_{1}^2 r_{K}^2 - r_{5}^2 r_{K}^2}}.
S_{n}(\kappa_1^{-1}, \ldots,\kappa_n^{-1}) =\tilde\psi^k S_{n-k}(\kappa_1^{-1}, \ldots,\kappa_n^{-1}) & \geq \tilde\psi^k S_{n}(\kappa_1^{-1}, \ldots,\kappa_n^{-1})^{\frac{n-k}{n}}.
G_d:=-2[\bar{q}, \mathcal{H}]\frac{\partial_{\alpha}\bar{\mathfrak{F}}}{\zeta_{\alpha}}-2[\bar{\mathfrak{F}},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-2[\bar{q},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-4D_t q,
f(\widetilde{\varphi_t\;}(z)) - f(\widetilde{\varphi_{t_0}}(z)) = f\big(z_{[j]}, t\widetilde{G}z'_{[j]}\big) - f\big(z_{[j]}, t_0\widetilde{G}z'_{[j]}\big) & = (t - t_0) \sum_{i = j+1}^n \widetilde{g}_{ii}z_i \dfrac{\partial f}{\partial z_i}\big(z_{[j]}, \tau \widetilde{G}z'_{[j]}\big),
\left( \begin{array}{c}{\psi_{+}( z , t )}\\{\psi_{-}( z , t )}\\ \end{array}\right) = \mathcal{F}\left( \begin{array}{c}{\psi_{+}( z - \Delta t , t - \Delta t )}\\{\psi_{-}( z + \Delta t , t - \Delta t )}\\ \end{array}\right) ,
\int_{z\in\partial D_{\lambda}}\phi(z)d^c_zV\wedge(dd^c_zV)^{n-1}& = \ \int_{t\in\lambda^{-1}(\partial D_{\lambda})}\!\! \phi(t\lambda)d^c_t(V(t\lambda)-\log|\lambda|)\wedge(dd^c_t(V(t\lambda)-\log|\lambda|))^{n-1}.
A_n = \left\{\sum_{k = 0}^\infty\left|\sum_{t = 0}^k f_n(t) - f(t)\right| \le n^{-\eta/(2 + 2\eta)}\right\}B_n = \left\{\sum_{k = 0}^\infty\left|f_n^\ast(k) - f^\ast(k)\right| \le n^{-\varepsilon}\right\}
\mathbf{sing}( \epsilon ) \, = \, \bigcap_{R > 0}\{z_0 \, \in \, Q(T) \, ; \, \, \overline{\mathcal{M}}\! \bigl(P_R (z_0)\bigr) \, \ge \, \epsilon, \, P_R (z_0) \, \subset\!\subset \, Q(T)\}, &\mathbf{reg}( \epsilon ) \; = \; Q(T) \setminus \mathbf{sing}.
- \nabla \cdot \big(\nabla \chi_i + e_i\big) =0 \mbox{in}Y^{\ast}, -\big(\nabla \chi_i + e_i\big) \cdot \nu = 0 \mbox{on}\Gamma, \chi_i \mbox{is}Y\mbox{-periodic,}\int_{Y^{\ast}}\chi_i dy = 0.
R(z) = \begin{cases}S(z) N^{-1}(z), z \in \mathbb{C}\setminus \left(\Sigma_S \cup \overline{D(0,R)}\cup \overline{D(q,r_q)}\right), S(z) P^{-1}(z), z \in D(0,R) \setminus \left(\Sigma_S \cup \partial D(0,r_n) \right), S(z) Q^{-1}(z), z\in D(q,r_q) \setminus \Sigma_S.\end{cases}
W^{( 1 )}= - i g^{2}\, t^{a}t^{a}\int \frac{d^{d}q}{( 2 \pi )^{d}}\frac{v_{\mu}{\mit \Pi}_{R}^{\mu \nu}( q ) v_{\nu}^{\prime}}{q^{4}( q \cdot v ) ( q \cdot v^{\prime})}\, .
=\frac{\Delta_{h}^{k}u(y)}{h^{l}}\eta(y_{k}^{(\eta)})+{\displaystyle\sum\limits_{i=0}^{k-1}}C_{i}\frac{\Delta_{h}^{i}u(y_{i}^{(u)})\Delta_{h}^{k-i}\eta(y_{i}^{(\eta)})}{h^{l}}\equiv I_{k}+{\displaystyle\sum\limits_{i=0}^{k-1}}I_{i},
(-\frac{a^{2}}{b^{2}c}+\frac{1}{c})y^{2}+(\frac{a^{2}}{c^{3}}+\frac{1}{c})z^{2}=0\leftrightarrow z=\pm\frac{c}{b}\sqrt{\frac{a^{2}-b^{2}}{a^{2}+c^{2}}}y.
b[h_1,k_1][h_2,h_3]b^{-1}= b[h_1,k_1]b^{-1}b[h_2,h_3]b^{-1}&=[bh_1b^{-1},bk_1b^{-1}][bh_2b^{-1},bh_3b^{-1}] &\in [R,L]\cdot [R,R];
\frac{1}{Z_n}\det \left[I_{\kappa}(2\alpha_i\sqrt{x_j})\right]_{i,j=1}^n\det\left[x_j^{\frac{\nu+i-1}{2}}K_{\nu-\kappa+i-1}(2\beta\sqrt{x_j})\right]_{i,j=1}^n,
0 =L\left([e_{ii},e_{kl}]\right)-\delta_{ik}L(e_{il})+\delta_{il}L(e_{ki}) &=L(e_{ii})e_{kl}-e_{kl}L(e_{ii})+e_{ii}L(e_{kl})-L(e_{kl})e_{ii}&-\delta_{ik}L(e_{il})+\delta_{il}L(e_{ki}).
K_{C}^{-1}=[\begin{matrix}\frac{2}{\sqrt{6}}&0&\frac{1}{\sqrt{3}}\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{3}}\\ -\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{3}}\end{matrix}]
\int_0^{+\infty}u_{xx}\bar{u}_xdx =\frac12\int_0^{+\infty}(u_{xx}\bar{u}_x+\bar{u}_{xx}u_x)dx &=\frac12 \int_0^{+\infty}\frac{d}{dx}|u_x|^2=-\frac12 |u_x(0,t)|^2.
\frac{P_{t}}{P_{t-1}}=\prod_{i=1}^{n}(\frac{p_{it}}{p_{i,t-1}})^{\frac{1}{2}[\frac{p_{i,t-1}q_{i,t-1}}{p_{t-1}\cdot q_{t-1}}+\frac{p_{i,t}q_{i,t}}{p_{t}\cdot q_{t}}]}
( \delta h_{i j}, \delta h_{i j}) = ( P \xi^{0}+ \alpha_{k}P \xi^{k}+ \bar{\alpha}_{k}P \bar{\xi}^{k}, P \xi^{0}+ \alpha_{k}P \xi^{k}+ \bar{\alpha}_{k}P \bar{\xi}^{k}) + 2 C ( \delta \sigma_{R}, \delta \sigma_{R}) .
( \Delta S )_{r e g .}={\frac{i}{4 \pi}}\int d^{2}x ( c - \tilde{c}) \Big ( ( 1 - a ) \partial_{\mu}( A^{\mu}- \tilde{A}^{\mu}) - \epsilon^{\mu \nu}\partial_{\mu}( A_{\nu}- \tilde{A}_{\nu}) \Big )
\frac{1}{a-1}(V_a(x)-V(x)) = \frac{1}{a-1}\frac{-C}{|x|^\gamma}\left(\frac{1}{a^\gamma}-1\right) = V(x) a^{-\gamma}\frac{1-a^\gamma}{a-1}\; \stackrel{a \uparrow 1}\longrightarrow \; -\gamma V(x) =: W(x).
W_{I \times S_{4}}^{D , N}= \pm \frac 1 4 \zeta_{S_{4}}^{\prime}( 0 ) \pm \frac 1 4 \zeta_{S_{4}}( 0 ) \log \mu^{2}- \frac{1}{1 6}\sum_{n = 1}^{\infty}\frac{\cosh n L}{n ( \sinh n L )^{4}}
V_s'(v) =(s-1)\frac{v^{s-2}}{(v+1)^{\frac{N}{2}}}-\frac{N}{2}\frac{v^{s-1}}{(v+1)^{\frac{N}{2}+1}}=V_s(v)\frac{(s-1)(v+1)-\frac{N}{2}v}{v(v+1)}
G =\begin{bmatrix}-(Z-I) q(T)^{-1}(Z-I)^{-1}G_q (Z-I) q(T)^{-1}(Z-I)^{-1}G_p e_1,\end{bmatrix}B =\begin{bmatrix}(Z-I) p(T)^* q(T)^{-*}(Z-I)^{-1}B_q B_p (Z-I) p(T)^* q(T)^{-*}(Z-I)^{-1}e_1\end{bmatrix}.
\int |\nabla \psi_{R_{n}}|^{2}= \int_{B_{4R_{n}}(0) \smallsetminus B_{2R_{n}}(0)}|\nabla \psi_{R_{n}}|^{2}\leq C \int_{B_{4R_{n}}(0) \smallsetminus B_{2R_{n}}(0)}R_{n}^{-2}\leq C_{1}R_{n}.
\lim_{s\to\overline\rho_-}\frac{p(s)}{(b(s))^\gamma}=\lim_{s\to\overline\rho_-}\frac{P(s)}{(b(s))^\gamma}=\lim_{s\to\overline\rho_-}\frac{p(s)}{(b'(s))^\beta}=\lim_{s\to\overline\rho_-}\frac{P(s)}{(b(s))^{\beta-1}}=+\infty.
\Phi_{\alpha}(A_{\alpha}^{1},\cdot\cdot\cdot,A_{\alpha}^{n})_{-}=min_{W_{\alpha-}^{i}\le w_{i}\le W_{\alpha+}^{i}A_{\alpha-}^{i}\le a_{i}\le A_{\alpha+}^{i}}\sum_{i=1}^{n}w_{i}a_{\sigma(i)}/\sum_{i=1}^{n}w_{i}
d \Pi^{- 2}= - \Omega^{( 0 )}\Pi^{- 2}- \Omega^{- 2 i}\Pi^{i}- 2 i \pi_{\dot{q}}^{1 -}\pi_{\dot{q}}^{1 -}- 2 i \pi_{q}^{2 -}\pi_{q}^{2 -},
\langle \delta j^{\mu}( X ) \rangle \simeq \frac{1}{2}\int \frac{d^{3}p}{( 2 \pi )^{3}}\left[ \hat{\cal P}_{+}^{\mu}Z_{+}\frac{- i \Sigma_{o f f}^{( + )}}{| I m \Sigma_{R}^{( + )}|}- \hat{\cal P}_{-}^{\mu}Z_{-}\frac{- i \Sigma_{o f f}^{( - )}}{| I m \Sigma_{R}^{( - )}|}\right] ,
-d(\phi^a\cdot \iota_a\Theta ) = -\omega^a\wedge \iota_a\Theta \;+\;\frac 12f^c_{ab}\,\phi^a\wedge \phi^b\cdot \iota_c\Theta \;-\;\phi^a\wedge \iota_a\Omega \;+\; \phi^a\wedge [\iota_a\Theta , \Theta ].
A_{N}= 2 \pi \sqrt{l}\left( \frac{1}{\pi}\right)^{{\textstyle \frac{k + 1 - D}{2}}}\Gamma \left( \frac{D - 3 - k}{2}\right) \left( \frac{1}{Y^{2}}\right)^{{\textstyle \frac{D - 3 - k}{2}}}
( \exp{(-tX)}\vec{x}_j, \exp{(-tX)}\vec{x}_j)_{z_0}= \sum_{i=1}^n |(\tilde{v}_i,(\exp{-(tX)}\vec{x}_j)_{z_0}|^2 = \sum_{i=1}^n |\big(\tilde{v}_i,\exp{(-tX)}(\sum_{k=1}^n a_j^k \tilde{v}_k) \big)_{z_0}|^2 = \sum_{i=1}^n |(\tilde{v}_i, \sum_{k=1}^n a_j^k \exp{(-\lambda_k t)}\tilde{v}_k)_{z_0}|^2 = \sum_{i=1}^n |a_j^i|^2 \exp{( - 2 \lambda_i t)}.
[ ( - D_{1}D_{2}D_{3}D_{4}\partial_{t}^{2}+ f{\frac{1}{r^{2}}}({\frac{1}{\sin \theta}}\partial_{\theta}\sin \theta \partial_{\theta}+ \partial_{\phi}^{2}) +{\frac{1}{r^{2}}}f \partial_{r}( r^{2}f \partial_{r}) ] \chi = 0
\tilde b({\mathrm y}) = \sigma^{-1}({\mathrm y})\Big(b({\mathrm y}) -{1\over 2}(\nabla_{{\mathrm y}}\sigma^T({\mathrm y}))^T\Big ), ~~~~~~\nabla_{\mathrm y}= \Big (\partial_{{\mathrm y}_1}, \ldots, \partial_{{\mathrm y}_M}\Big )
\left. \begin{array}{l l}\frac{\partial V(\textbf{x})}{\partial \textbf{x}}\cdot \textbf{f}(\textbf{x},\textbf{u}_E^*,\textbf{u}_i^**) = \frac{d_E}{\Lambda}[ \nu_1 +\nu_2 +\nu_3 -\nu_E] .\end{array}\right.
\| A^{poly}_{N}f\|_{L^{2}(X,\mu)}^{2}\leq \frac{\| f\|^{2}_{L^{2}(X,\mu)}}{N}+\frac{1}{N^2}\sum_{n,m=0,\n>m}^{N}\left<T^{p(n)-p(m)}f,f\right>.
c(ab,b\bar{a}) \, = \, c(ab\bar{a}\bar{a},\bar{a}\bar{a}\bar{b}a) \, = \, \frac{(1+\lambda)\, c(b\bar{a},\bar{a}b)-2\lambda}{3\lambda+1}\, = \, \lambda^{-1}\bigl(c(b\bar{a},\bar{a}b)-\sqrt{2}\,\bigr),
\left(D_{b-}^\alpha y\right)(x):=\frac{1}{\Gamma(1-\alpha)}\left(-\frac{d}{dx}\right)^n\underset{x}{\overset{b}{\int}}\frac{y(t)dt}{(t-x)^{\alpha-n+1}},\,(n=\left[Re(\alpha)\right]+1,\,x<b),
W=-\int_{t_{1}}^{t_{2}}\frac{GmM}{r^{3}}(re_{r})\cdot(\dot{r}e_{r}+r\dot{\theta}e_{t})dt=-\int_{t_{1}}^{t_{2}}\frac{GmM}{r^{3}}r\dot{r}dt=\frac{GMm}{r(t_{2})}-\frac{GMm}{r(t_{1})}.
\frac{d}{dt}\Bigl(\dot q_i+\lambda^2 q_i\Bigr)=-\biggl[\lambda^2+\frac{2\tilde T_2(2\tilde T_1+1)}{(q^2+2\eta)^2}\biggr](\dot q_i+\lambda^2 q_i)-\frac{1}{q^2}(\dot q_k+\lambda^2q_k)^2\,q_i\,.
\int_X f(\pi(x,y))\> \overline{g(\pi(z,y))}\> d\omega_X(y) = T^{f\circ\pi}(\overline{g\circ \pi_z})(x)= T_f(\overline{g\circ \pi_z})(x) &= ((f^-*\bar g)\circ \pi_z)(x)= (f^-*\bar g)(\pi(z,x))= \int_D f(h) \> \overline{g(\pi(z,x)*h)}\>d\omega_D(h)
\hat{\alpha}_{L\ell}(x)=\hat{\rho}_{L\ell}^{\prime}b(x),\hat{\rho}_{L\ell}=\arg\min_{\rho}\left\{-2\hat{M}_{\ell}^{\prime}\rho+\rho^{\prime}\hat{Q}_{\ell}\rho+2r\sum_{j=1}^{p}\left\vert \rho_{j}\right\vert \right\}.
\tau=f(l,\sigma) =f(l,\alpha\beta)=f(l,\alpha)\beta=l\alpha\beta=l\sigma f(k,\tau) =f(k,l\alpha\beta)=f(k,l\alpha)\beta=f(k,\alpha)l\beta f(k,\sigma) =f(k,\alpha\beta)=f(k,\alpha)\beta.
\int_{M}s^{*}( \Phi_{\Gamma}( E ) ) \wedge{\cal O}_{1}\wedge \dots \wedge{\cal O}_{k}= \int_{\mathrm{s u p e r s p a c e}}e^{- S_{\mathrm{q}}}\tilde{{\cal O}}_{1}\dots \tilde{{\cal O}}_{k},
f_1(x^*)=a_1\|x^*\|^2+b_1^T{x^*}+c_1=0 \|x^*\|^2=-\frac{b_1^T{x^*}}{a_1}-\frac{c_1}{a_1}\|x^*\|^2=\frac{1}{a_1w}b_1^T(\gamma_1 b_1+\gamma_2 b_2)-\frac{c_1}{a_1}
|V_k|= \mathrm{O}(1)\int^{\tilde{x}_{k+1}}_{\tilde{x}_k}\frac{v-\tilde{x}_k}{b_n}\,\mathrm{d}v = \frac{\mathrm{O}[(\tilde{x}_{k+1}-\tilde{x}_{k})^2]}{b_n}= \frac{\mathrm{O}(1)}{N^2_n b_n}.
0 =[D(1\otimes E_{ii})]_{ii}=[D((1\otimes E_{im})(1\otimes E_{mi}))]_{ii}&=[D((1\otimes E_{im}))(1\otimes E_{mi})]_{ii}+[(1\otimes E_{im})D((1\otimes E_{mi}))]_{ii}&=D^{im}_{im}(1)+D^{mi}_{mi}(1),
\|\sum a_{ij}u_{ij}\| = \|\sum_i (\sum_j |a_{ij}|^p)^{1/p}x_i\|_X =\|\sum_i \big( \sup_{\sum_j |\alpha_j|^{p^\prime}\leq 1}|\sum_j \alpha_{ij}a_{ij}| \big) x_i\|_X .
\Delta_M u=\frac{2}{\sin^{2}\theta}\frac{\partial^{2}\overline{u}}{\partial\phi^{2}}+\frac{2}{\sin^{2}\theta}\frac{\partial^{2}\overline{u}}{\partial\psi^{2}}+\frac{\partial^{2}\overline{u}}{\partial\theta^{2}}+(2\cot\theta)\frac{\partial \overline{u}}{\partial\theta}
q^n-1 + \gamma - r = \sum_{i \in [0, n-1]}(q-1)q^i + \sum_{i \in G}(1 - r_i)q^i - \sum_{i \in E \setminus G}r_i q^i = \sum_{i \in G}(q - r_i)q^i + \sum_{i \in E \setminus G}(q-1 - r_i)q^i + \sum_{i \in [0,n-1] \setminus E}(q-1)q^i.
{\frac{\overline{\alpha}( Q )}{Q^{2}}}\Longrightarrow{\frac{1}{2}}\sqrt{\overline{\alpha}( Q_{q}) \overline{\alpha}( Q_{\bar{q}})}\left({\frac{1}{Q_{q}^{2}}}+{\frac{1}{Q_{\bar{q}}^{2}}}\right) ,
\begin{array}{ccll}\langle1\rangle E = -(A+B)^2,~~~~ \qquad \sigma=\{A,B\}\langle2\rangle E = -(1+A-B)^2, \qquad \sigma=\{-A-1,B\}\langle3\rangle E = -(1-A+B)^2, \qquad \sigma=\{A,-B-1\}\langle4\rangle E = -(2+A+B)^2, \qquad \sigma=\{-A-1,-B-1\}\end{array}
\Phi_{A}^{* ( a )}\equiv \left( \Phi_{\alpha_{0}}^{* ( a )}, \; \eta_{b \alpha_{1}}^{* ( a )}, \; \pi_{b \alpha_{1}}^{* ( a )}, \; \lambda_{\alpha_{1}}^{* ( a )}\right) , \; a , b = 1 , 2 , 3 .
\frac{d}{dt}E^2_q(g) - \sum_{i,j}(A_{ij}(\Delta_q g_2 )+2\Delta_qc\delta_{ij}, \partial_j \Delta_q\partial_tb_i + \partial_i \Delta_q\partial_tb_j) &+ 2\Vert \nabla_x \Delta_q b\Vert^2_{L^2_x}+ 2\Vert \nabla_x \cdot \Delta_q b\Vert^2_{L^2_x}= ( A_{ij}(\Delta_qR_1+\Delta_qR_2), \partial_j \Delta_qb_i + \partial_i \Delta_qb_j).
R_\gamma : = (A_{x}^{-1}P_\alpha + Q_\alpha)\Big ( C_{z_\gamma}( AP_\alpha + Q_\alpha) C_{-z_\gamma}- (A_{x}P_\alpha + Q_\alpha) \Big ) M_{\chi_{B(0,R)}}&= (A_{x}^{-1}P_\alpha + Q_\alpha)C_{z_\gamma}(AP_\alpha + Q_\alpha)C_{-z_\gamma}M_{\chi_{B(0,R)}}- M_{\chi_{B(0,R)}}
(-\Delta)^{\alpha/2}u(x):=C_{d,\alpha}{\rm p.v.}\int_{\mathbb{R}^d}\dfrac{u(x)-u(y)}{\lvert x-y\rvert^{d+\alpha}}dy \;\; \textmd{with}\;\;C_{d,\alpha}:=\dfrac{{\alpha}2^{\alpha-1}\Gamma\Big(\dfrac{\alpha+d}{2}\Big)}{\pi^{d/2}\Gamma\Big(\dfrac{2-\alpha}{2}\Big)}.
b_k = \frac{1}{\sqrt{1-\alpha_k^2}}\Big( a_k + \alpha_k a_{-k}^\dagger \Big), \alpha_k = \frac{k^2 + \rho_z \widehat g(k) - \sqrt{k^4 + 2k^2 \rho_z \widehat g(k)}}{\rho_z \widehat g(k)},
A_{\sigma l}= (l+1)(l+2+2\sigma_n)p_{\sigma,l+1}+ 2s(\sigma_n + 1)p_{\sigma+\bar{n},l}+ (\sigma_i + 1)(\sigma_i+2)p_{\sigma + 2\bar{\imath},l-1}+ c_{\sigma l}^{\mu m}p_{\mu m},
\varphi_{\pi_1}^{U_1,\psi_{U_1}}(ug)=\sum_{t\in T(F)}\int\limits_{U(B)(F)\backslash U(B)({\bf A})}\varphi_{\pi_1}^{U_1,\psi_{U_1}}(u_1tug)\psi_{U(B)}(u_1)du_1du
\mathbb{P}(Y_n > y) = \mathbb{P}(Y_n > y, \, X_n - Y_n \ge \varepsilon) + \mathbb{P}(Y_n > y, \, |X_n - Y_n| < \varepsilon) & \le \mathbb{P}(X_n > y + \varepsilon, \, X_n - Y_n \ge \varepsilon) + \mathbb{P}(X_n > y, \, |X_n - Y_n| < \varepsilon) &= \mathbb{P}(X_n > y + \varepsilon) + \mathbb{P}(X_n \in (y, y+\varepsilon], \, |X_n - Y_n| < \varepsilon),
\mathcal{Z}_{m,n}^{\gamma}(z,\overline{z})= \dfrac{-(\gamma+m+1)_{n}m!}{2\pi i}\left(1- |z|^{2}\right)^{-\gamma}\oint_{\mid t\mid=1}t^{n}\dfrac{\left(1-{t\overline{z}}\right)^{\gamma+m}}{(z-t)^{m+1}}dt
\|j(x)\| = \sup_{x^* \in X^*, \|k(x^*)\|=1}\langle k(x^*), j(x)\rangle = \sup_{x^*\in X^*, \|k(x^*)\|=1}\langle x^*, x\rangle \geq \sup_{x^* \in X^*, \|x^*\|=\frac 1{\|k\|}}\langle x^*, x\rangle = \frac 1{\|k\|}\|x\|,
C_{ik~pr}^{~jn~ms}\theta_{m}^{~p}t_{s}^{~r}\otimes t_{n}^{~l}+D_{ir~k}^{~pj~n}\otimes \theta_{p}^{~r}t_{n}^{~l}+C_{pn~bd}^{~rl~ms}t_{i}^{~p}S(t_{r}^{~j})t_{k}^{~n}\otimes\theta_{m}^{~b}t_{s}^{~d}=
\psi(q,x)=1+h(q,x),\,\,\,\,\,\,\,\,h(q,x)=O\left(\frac{Q}{q}\left(\frac{q}{Q}+|x|\right)^A\right) &\psi(q,x)\ll |x|^{-A}&x^j \frac{\partial^j}{\partial x^j}\psi(q, x) \ll \ \min \left\lbrace \frac{Q}{q}, \frac{1}{|x|}\right\rbrace \ \log Q
\Delta A_{-}= 8 \pi G \int d x^{2}\ldots d x^{D - 1}\int_{x_{+}^{1}}^{x_{-}^{1}}d \hat{x}^{1}\int_{\hat{x}^{1}}^{x_{-}^{1}}d \hat{\hat{x}}^{1}T_{a b}k^{a}k^{b}.
\mu\left(\{C_{r_n}(\sigma^i\underline{x})=C_{r_n}(\sigma^{i+k}\underline{x})\}\right)= \mu\left(\sigma^{-i}S_k(r_n)\right)=\sum_{C\in\mathcal{C}_{r_n}}\mu(C\cap\sigma^{-k}C) \geq (1-\psi(k-r_n))\sum_{C\in\mathcal{C}_{r_n}}\mu(C)^2\geq(1-\psi(r_n))Z_{r_n}(1),
\mathcal{G}\triangleq \left\{\Theta \mathbf{G}: \mathbf{G}=\left({{\mathbf{G}}^{1}},\cdots ,{{\mathbf{G}}^{k}}\right), \mathbf{G}^i \in \mathbf{R}^{n_i\times n_i}, (\mathbf{G}^i)^T \mathbf{Q}^i \mathbf{G}^i = \mathbf{Q}^i\right\}
\begin{aligned}[z^n]\sum_{h\ge 0}(T-T_h)(z) = \frac{5^n}{n}+\frac{3\sqrt{5}}{2}\frac{5^n}{\Gamma(-1/2)n^{3/2}}+O\biggl(5^n\frac{\log n}{n^2}\biggr) = \frac{5^n}{n}-\frac{3\cdot 5^{n+1/2}}{4\sqrt{\pi n^3}}+O\biggl(5^n\frac{\log n}{n^2}\biggr). \end{aligned}
F^C(\beta) \sim -{\pi^2 \over 90}{c_0 \over \beta^4}-{1 \over24}{c_1 \over \beta^2}-{1 \over 16 \pi^2}\ln(\beta\rho)c_2-{1 \over 16\pi^{5/2}}\sum_{n=3}^{\infty}\Gamma\left(n-\frac 32\right)\zeta(2n-3)c_n\left({\beta \over 2\pi}\right)^{2n-4},
g^{\mu \rho}M^{\nu \sigma}-g^{\nu \rho}M^{\mu \sigma}+g^{\mu \sigma}M^{\rho \nu}-g^{\nu \sigma}M^{\rho \mu}=q(F^{\nu \sigma}x^{\mu}x^{\rho}-F^{\mu \sigma}x^{\upsilon}x^{\rho}+F^{\rho \mu}x^{\mu}x^{\sigma}-F^{\rho \mu}x^{\nu}x^{\sigma}),
I_1^p:=\sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}v_1^{p}(x)\frac{\Bigl|\bigl[V_1^-(x)\bigr]'\Bigr|^p}{\bigl[V_1^-(x)\bigr]^{2p}}\biggl(\int_{\eta_{k-1}}^x v_1^{-p'}(t) \biggl(\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta}[V_1(t)]^{\delta}\bigl|G^{(\delta)}_{1,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx
\int_{\mathbb{R}^m}|u|_g^2dV_g = \int_{\mathbb{R}^m}{}|u|_{g_0}^2 e^{-\frac{m}{4(m-2)}|x|^2}dV_{g_0}\leq \liminf \int_{\mathbb{R}^m}{}|u_k|_{g_0}^2 e^{-\frac{m}{4(m-2)}|x|^2}dV_{g_0}&=\liminf \int_{\mathbb{R}^m}|u_k|_g^2 dV_g.
\begin{array}{r@{}c@{}ll}\partial_t u_n + A_\varepsilon u_\varepsilon{}={}0 \mbox{in}(0,\infty ) \times \Omega , begin{align*}5pt] u_\varepsilon = 0 \mbox{on}(0,\infty ) \times \partial\Omega , begin{align*}5pt] u_n (0,\cdot ) = u_{0,\varepsilon}\mbox{in}\Omega ,\end{array}
\sum_{k \leq j < m}\delta_{0, j}+ \sum_{0 < i < k}\delta_{i, k}- \sum_{k < j \leq m}\delta_{k, j}- \sum_{0< i < k}\delta_{i, m}= 1 + \sum_{0 \leq i < m}(\delta_{0, i}+ \delta_{i, k}- 1),
X_{\pm}(x) :=a\otimes \left[-2\pi\,\mathsf{M}^{\mathbb{R}}\mp\frac{1}{b}\frac{\partial\;}{\partial r}\pm \frac{\partial\;}{\partial \Theta}\right](\Phi)+\left[2\pi\, \mathsf{M}^{\mathbb{Z}}\pm\frac{\partial\;}{\partial \Theta}\right](a)\otimes \Phi \ \in \C_{c}^{\infty}(\mathcal{A}_{\theta}) \odot C_{c}^{\infty}(\mathcal{Z}_{b}),
\rho_R^{k}\frac{\partial p_{k-1}}{\partial \rho_V}= \sum\limits_{i=1}^{k-1}i \rho_V^{i-1}\frac{\rho_R^{2k-i-1}}{(k-i-1)!}= \sum\limits_{i=0}^{k-2}(i+1) \rho_V^{i}\frac{\rho_R^{2k-i-2}}{(k-i-2)!}.