harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{2}^{\ast\ast}= \frac{\Gamma\left((m+1)/2\right)m^{3/2}\left[\xi_{p}\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-3)/2}-\xi_{q}\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-3)/2}\right]}{\Gamma(m/2)(m-1)(m-3)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})}+\frac{m^{2}}{(m-2)(m-4)}\frac{F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},~m>4,
\tilde{P}^{\epsilon,\Delta t}(t)-\tilde{P}^{\epsilon,\Delta t}(t_n)=\bigl(e^{-\frac{t-t_n}{\epsilon^2}}-1\bigr)\tilde{P}^{\epsilon,\Delta t}(t_n)+\int_{t_n}^{t}e^{-\frac{t-s}{\epsilon^2}}f(q_n^{\epsilon,\Delta t})ds+\int_{t_n}^{t}e^{-\frac{t-s}{\epsilon^2}}\sigma(q_n^{\epsilon,\Delta t})d\beta(s).
1 - \frac 1 4 \omega_{0}^{2}r_{0}^{2}+ \frac{1}{6 4}\omega_{0}^{4}r_{0}^{4}- \frac{1}{3 8 4}\omega_{0}^{6}r_{0}^{6}- \frac{7}{6 4}\kappa^{2}r_{0}^{2}- \frac{1 5}{1 2 8}\omega_{0}^{2}\kappa^{2}r_{0}^{4}= \frac{c_{1}}{\alpha_{0}}K_{0}( k_{0}r_{0}) ,
B_{m,m-n}(b_1, b_2, b_3, b_4, \ldots) = \frac{m!}{(m-n-1)!n!}\sum_{k=1}^n\binom{m-1+k}{k-1}(k-1)!B_{n,k}(x_1, x_2, \ldots) & = \sum_{k=1}^n \frac{(m-1+k)!}{(m-n-1)!n!}B_{n,k}(x_1, x_2, \ldots)
D^{ij}_{rs}(ab) =[D(ab\otimes E_{rs})]_{ij}&=[D((a\otimes E_{rm})(b\otimes E_{ms}))]_{ij}&=[D(a\otimes E_{rm})(b\otimes E_{ms})]_{ij}+[(a\otimes E_{rm})D(b\otimes E_{ms})]_{ij}&=\delta_{js}[D(a\otimes E_{rm})]_{im}b+\delta_{ir}a[D(b\otimes E_{ms})]_{mj},
\lambda_k[f(A_{\mathcal{S'}})] =\min_{h\in{\mathcal{F}};\ \Vert h\Vert=1}\langle h,f(A_{\mathcal{F}})h \rangle &= \min\{f(\lambda_1(A_{\mathcal{F}}))\,;\,f(\lambda_k(A_{\mathcal{F}}))\}&= \min_{h\in{\mathcal{F}};\ \Vert h\Vert=1}f(\langle h,A_{\mathcal{F}}h \rangle) &= \min_{h\in{\mathcal{F}};\ \Vert h\Vert=1}f(\langle h,Ah \rangle)
S_1 = \{x \in \mathbb{R}^2 : \dfrac{x_1}{a_1}+ \dfrac{x_2}{a_2}\leq 1 \}S_2 = \{x \in \mathbb{R}^2 : \dfrac{-x_1}{a_1}+ \dfrac{x_2}{a_2}\leq 1 \}S_3 = \{x \in \mathbb{R}^2 : \dfrac{x_1}{a_1}+ \dfrac{-x_2}{a_2}\leq 1 \}S_4 = \{x \in \mathbb{R}^2 : \dfrac{-x_1}{a_1}+ \dfrac{-x_2}{a_2}\leq 1 \}.
P^\Sigma(U,V):= ( A_S(U,\nu) \nu_S^\flat(V) + A_S(V,\nu) \nu_S^\flat(U)) \; g_S(\nu,\nu) & -( g_S(U,\nu) \nu_S^\flat(V) + g_S(V,\nu) \nu_S^\flat(U) ) \; A_S(\nu,\nu)
B_{\alpha}^{(i,j)}\tau((i,j))-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\tau((i,j)) =\widehat{\mu}_{\alpha}\tau((i,j)) , \tau((i,j)) B_{(i,j) \alpha}^{(j,i)}-\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}=\tau((i,j)) \widehat{\mu}_{(i,j)\alpha};
\frac{H_{1}^{\prime}( \rho_{b})}{H_{1}( \rho_{b})}+ \frac{2 ( d - 1 ) ( d - 2 ) \rho_{b}^{2 d - 6}- 2 ( d^{2}- d - 2 ) \omega M \rho_{b}^{d - 3}+ ( d - 1 )^{2}\omega^{2}M^{2}}{\rho_{b}( \rho_{b}^{d - 3}- \omega M ) [ 2 ( d - 2 ) \rho_{b}^{d - 3}- ( d - 1 ) \omega M ]}= 0 .
(x_{j+1},y_{j+1})=(x_{j},y_{j})-\frac{F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F_{y}(x_{j},y_{j})^{2}}(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j}))
\phi ( x ) = \langle I N | \tilde{T}( e^{- i \int^{t}d^{4}x^{\prime}~ J ( x^{\prime}) \Phi ( x^{\prime})}) \Phi ( x ) T ( e^{i \int d^{4}x^{\prime \prime}~ J ( x^{\prime \prime}) \Phi ( x^{\prime \prime})}) | I N \rangle |
\frac{1}{2\pi i}\int_{(5)}L^s\tilde{w}(s)\frac{L'(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}{L(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}ds = \frac{1}{2\pi i}\int_{\mathcal{C}_1}L^s\tilde{w}(s)\frac{L'(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}{L(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}ds.
\mathcal{T}_{V_{N,\beta,a}}\big(n_N(t),m_N(t)\big) \leq T_{V_{N,\beta,a}}\big(\omega_N(t)\big) &\leq e^{-\gamma t}T_{V_{N,\beta,a}}\big(\omega_N(0)\big) &=e^{-\gamma t}\iint V_{N,\beta,a}([s]_N,[s']_N)\omega_N(0,d[s]_N,d[s']_N).
{\psi}_{o d d}^{+}= e^{- \frac{1}{2}z_{+}^{2}}\left( \begin{array}{c}{{\sqrt{\frac{\Delta}{2}}e^{i \frac{\pi}{4}}F ( - \frac{\nu}{2}; \frac{1}{2}; z_{+}^{2}) + \bar{G}z_{+}F ( \frac{1 - \nu}{2}; \frac{3}{2}; z_{+}^{2})}}\\{{z_{+}F ( 1 - \frac{\nu}{2}; \frac{3}{2}; z_{+}^{2}) - \bar{G}\sqrt{\frac{\Delta}{2}}e^{i \frac{\pi}{4}}F ( \frac{1 - \nu}{2}; \frac{3}{2}; z_{+}^{2})}}\\ \end{array}\right)
\theta_1(v,u;q;z):=\sum_{k\geq0}\theta_{1,k}(v,u;q) z^k= -2e^{zv}\sum_{m\geq0}\Bigl(\gamma-\frac12 u+\psi(m+1)\Bigr)q^m e^{m u}\frac{z^{2m}}{m!^2}, &\theta_2(v,u;q;z):=\sum_{k\geq0}\theta_{2,k}(v,u;q) z^k = z^{-1}\biggl(\sum_{m\geq0}q^m e^{m u+z v}\frac{z^{2m}}{m!^2}-1\biggr),
\begin{cases}\partial_{t}V_{1}= D \partial_{xx}^{2}V_{1}+ b - V_{1}(\beta_{1}2m^{\ast}+ \beta_{2}+ d) \triangleq D \partial_{xx}^{2}V_{1}+ \tilde{F}_{1}, \partial_{x}V_{1}(x,t) \rvert_{x=0,1}= 0.\end{cases}
\Gamma_{3}^{U V}= \frac{\lambda^{3}}{6 4 \pi^{3}}\frac{M^{2 \epsilon}}{2 \epsilon}\simeq \frac{1}{6 4 \pi^{3}}\left[ \frac{1}{2 \epsilon}+ \mathrm{l n}\left( M \right) \right] ,
\frac{p}{\tau}\widetilde{B}(-\tau) = \begin{pmatrix}0 0 0 0 \frac{p}{\tau}\widetilde{\Psi_0}(-\tau,5/2) 0 0 0 0 \end{pmatrix}= \begin{pmatrix}0 0 0 0 - (\frac{5}{2})^{\frac{1}{p}-1}~e^{-\tau (5/2)^{1/p}}0 0 0 0 \end{pmatrix}.
T_2 \le \Gamma(2m-2)\frac{\Gamma(m+1/2)\Gamma(m-5/2)}{\Gamma(2m-2)}\int_{0}^1y^{2}(1-y)^{m-3/2}\,dy &=\frac{\Gamma(m+1/2)\Gamma(m-5/2)\Gamma(3)\Gamma(m-1/2)}{\Gamma(m+5/2)}.
\int_{B_1^c}\frac1{\, |z|^{N+2s}\,}\int_{B^c_{R_t}(\mp z)}\frac{\, 1+|y|^{2s-\sigma}\,}{\, |y\pm z|^{N+2s}\,}\,dy\,dz = \int_{B_1^c}\frac1{\, |z|^{N+2s}\,}\int_{B^c_{R_t}}\frac{\, 1+|w\mp z|^{2s-\sigma}\,}{|w|^{N+2s}}\,dw\,dz &\le \int_{B_1^c}C \, \frac{\, 1+|z|^{2s-\sigma}\,}{|z|^{N+2s}}\,dz < +\infty.
\max_{1 \le j \le d_n}| ( \overline{B}_n'(1)_0 - M_{n,0}^{(n)}( 0 ), M_{n,0}^{(n)}( j ) )_{L_2}| \le \sup_{1 \le j}\| \overline{B}_n'(1)_0 - M_{n,0}^{(n)}( 0 ) \|_{L_2}\| M_{n,0}^{(n)}( j ) \|_{L_2}= o(1),
\tau^{e_2}(x)=\begin{cases}\tau_{d_0}^{-1}(x)\quad\;\; x\in R_{d_0}\tau_{d_1}^{-1}(x)\quad\;\; x\in R_{d_1}\tau_{b_0}^{-1}(x)\quad\;\; x\in R_{b_0}\tau_{b_1}^{-1}(x)\quad\;\; x\in R_{b_1}\end{cases}
\L u(a + \tau k) = \tau^{\alpha}c_{\alpha}\sum_{-\infty<i <k}\frac{E_{\alpha,\alpha}\big[c \tau^{\alpha}(k-i)^{\alpha}\big] \big[ u(a+ \tau k) - u(a+ \tau i) \big]}{\big(k-i\big)^{1-\alpha}}
\left\{\frac{1}{z_1^{\alpha_{1}^1+\cdots+\alpha_{k}^1}}(d^{1}z)^{\alpha_1}(d^{2}z)^{\alpha_2}\cdots (d^{k}z)^{\alpha_k}\right\}_{|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=N},
f^{n, k}(t, y, z): = \inf_{y^\prime, z^\prime}\big\{f^+ (t, y^\prime, z^\prime) + n|y-y^\prime| + n|\lambda_t(z-z^\prime)|\big \}&- \inf_{y^\prime, z^\prime}\big\{f^- (t, y^\prime, z^\prime) + k|y-y^\prime| + k|\lambda_t(z-z^\prime)|\big\}.
(q)_{\infty}\sum_{n=1}^{\infty}N_{\textup{SC}}(n)q^n+\frac{1}{2}\frac{(q)_{\infty}}{(-q)_{\infty}}\sum_{n=1}^{\infty}\frac{q^{\frac{n(n+1)}{2}}}{(1-q^n)(q)_n}\left(\frac{(-q)_n}{(q)_n}-1\right) &=\frac{1}{4}-\frac{1}{4}\frac{(q)_{\infty}}{(-q)_{\infty}}+\frac{1}{2}\frac{(q)_{\infty}}{(-q)_{\infty}}\sum_{n=1}^{\infty}\frac{(-q)_n}{(q)_n}\frac{q^n}{1-q^n}.
A_1 =\big(\sum^{2m+1}_{i=0}E^*_i\big)A_1\big(\sum^{2m+1}_{i=0}E^*_i\big)=\sum^{2m}_{i=0}(E^*_{i}A_1E^*_{i+1}+E^*_{i+1}A_1E^*_{i}),
\frac{\partial M_{r}}{\partial c}=\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2}}\sum_{k=0}^{r-1}(\frac{1}{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{2}{c+1+k}).
R^{(1)}_2 :=\big[\delta^hd+(\tfrac{\beta}{2}-1)\tfrac{\beta}{2}\tfrac{2J}{n}\big] \big[\delta^hd+(\tfrac{\beta}{2}-2)(\tfrac{\beta}{2}+1)\tfrac{2J}{n}\big], R^{(2)}_2 := \tfrac{\beta}{2}(\tfrac{\beta}{2}+1)(\tfrac{2J}{n}) d\delta^h +d\delta^h\big[d\delta^h+ (\tfrac{\beta}{2}-1)(\tfrac{\beta}{2}+2)\tfrac{2J}{n}\big].
u_{n}=C\lambda^{n}+O\left( 1\right) \quad\quad\left\vert\nu\right\vert <1, u_{n}=C\lambda^{n}+O\left( n\right) \quad\quad\left\vert\nu\right\vert =1, u_{n}=C\lambda^{n}+D\nu^{n}+O\left( 1\right) \quad\quad\left\vert \nu\right\vert >1,
\Phi_{conv}:=\left\{\phi:X\to\mathbb{R}\ |\ \phi\left(x\right) =\left\langle u,x\right\rangle_X +c,u\in X^*, c\in \mathbb{R}\right\}\Psi_{conv}:=\left\{\psi:Y\to\mathbb{R}\ |\ \psi\left(y\right) =\left\langle v,y\right\rangle_Y +d,v\in Y^*, d\in \mathbb{R}\right\}
\tilde{a}= \frac{\frac{p+\gamma-1}{2}-\frac{(p+\gamma-1)(q-1)}{2q(p-\gamma+1)}}{\frac{p+\gamma-1}{2}+\frac{1}{2}}=\frac{(q(p-\gamma)+1)(p+\gamma-1)}{q(p+\gamma)(p-\gamma+1)}.
\sum_{n=0}^{\infty}\frac{\|\big(A^{(k, i)}\big)^{n}e^{(k, i)}_{p}\| t^{n}}{n!}\geq \sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sqrt{\frac{[i+(p+n)k]!}{(i+pk)!}}.
f_{1/n}^{\#}(x) :=\sup_{Q\backepsilon x}\frac{1}{\left\vert Q\right\vert^{1+1/n}}\int_{Q}\left\vert f(x)-f_{Q}\right\vert dx & \leq c(f)\sup_{Q\backepsilon x}\frac{1}{\left\vert Q\right\vert}\int_{Q}g(x)dx=c(fMg(x),
\det(\xi^{\mu\pm i\nu,\varkappa\pm i\vartheta})= K^2\big((\varkappa-\mu)^2+(\vartheta-\nu)^2\big)\big(\varkappa-\mu)^2+(\vartheta+\nu)^2\big)=K^2(\delta-\lambda)(\delta-\bar\lambda)(\bar\delta-\lambda)(\bar\delta-\bar\lambda).
\frac{\theta'(t)}{\theta_{k_0-1}'(t)}=\frac{\theta'(t)}{\theta'(t_1)}\cdot \frac{\theta'(t_1)}{\theta_{k_0-1}'(t_1)}\cdot \frac{\theta_{k_0-1}'(t_1)}{\theta_{k_0-1}'(t)}< M'^2\alpha=M'\sqrt{d}.
A(r,U,p_r/\rho) = \int_0^r \frac{dy}{\sqrt{U^2 + \tfrac{2}{n}(1-(y/r)^n)r p_r/\rho}}= \int_0^1 \frac{r\,dy}{\sqrt{U^2 +\tfrac{2}{n}(1-y^n) r p_r/\rho}}.
< \! \mu ( z , \bar{z}) \mu ( 0 ) \! >^{2}= < \! \cos ({\frac{\Phi ( z , \bar{z})}{2}}) \cos ({\frac{\Phi ( 0 )}{2}}) \! > ={\frac{1}{2}}\left( G ({\frac{1}{2}}, -{\frac{1}{2}}) + G ({\frac{1}{2}},{\frac{1}{2}}) \right) ,
\sum_{i\, =\, 1}^{m}\overline{D}_{t}E_{i}=\sum_{i \, =\, 1}^{m}D_{t}E_{i}+ E_{m+1}\sum_{i \,=\, 1}^{m}\tau_{i}&= \sum_{i \, =\, 1}^{m}\left\{(D_{t}E_{i}\cdot E_{1}) E_{1}+ \dotsb + (D_{t}E_{i}\cdot E_{m}) E_{m}\right\}+ E_{m+1}\sum_{i \,=\, 1}^{m}\tau_{i}\,.
\begin{array}{r c c c c c}{{\mathrm{\underline{{{s e c t o r}}}}}}&{{{}}}&{{\mathrm{\underline{{{s t a t e}}}}}}&{{{}}}&{{{}}}&{{\mathrm{\underline{{{~{\cal ~ R}~}}}}}}\\{{\mathrm{N S}}}&{{{}}}&{{\psi_{-{\frac{1}{2}}}^{i}| 0 \rangle}}&{{i = 2 , \cdots , 9}}&{{{}}}&{{-}}\\{{\mathrm{R}}}&{{{}}}&{{| + , 2 s_{1}, \cdots , 2 s_{4}\rangle}}&{{8 s_{1}s_{2}s_{3}s_{4}= + 1}}&{{{}}}&{{+}}\\ \end{array}
W(y_{1},y_{2})(x)=|\begin{matrix}y_{1}(x)&y_{2}(x)\\y_{1}^{\prime}(x)&y_{2}^{\prime}(x)\end{matrix}|=y_{1}(x)y_{2}^{\prime}(x)-y_{1}^{\prime}(x)y_{2}(x),x\in I,
f_{( \eta )}^{\vec{\alpha}\beta \gamma}= f^{\alpha [ 1 ] \beta b [ 1 ]}\, \, f^{b [ 1 ] \alpha [ 2 ] b [ 2 ]}\, f^{b [ 2 ] \alpha [ 3 ] b [ 3 ]}\, \cdots \, \, f^{b [ \eta - 2 ] \alpha [ \eta - 1 ] b [ \eta - 1 ]}f^{b [ \eta - 1 ] \alpha [ \eta ] \gamma}\; ,
^* \vec{F}_{\mu \nu}( x ) = - \frac{e_{0}}{4 \pi \varepsilon_{0}c}\vec{R}_{\mu \nu}( x ) = - \frac{e_{0}}{4 \pi \varepsilon_{0}c}\partial_{\mu}\vec{n}( x ) \wedge \partial_{\nu}\vec{n}( x ) .
\Pi_{2,2}^{0,\beta}(m_0,i \Omega, \vec k=0)= -\frac{\lambda^2 ~ I_{\beta}^{\epsilon}(m_0)}{2(4\pi)^2}\left[\frac{1}{\tilde \epsilon}- \ln \left( \frac{m_0^2}{\mu^2}\right) +2 \right] + F(\Omega^2),
\mathrm{Ad}_Z\circ\Phi(X) =Z^*V^*\pi(X)(UU^*)VZ =(U^*VZ)^*\pi(X)(U^*VZ) =(T^{\frac{1}{2}}V)^*\pi(X)(T^{\frac{1}{2}}V) =V^*T\pi(X)V =\Psi(X)
E[h(U_i+\xi_iV_i)] =\frac{1}{2}\sum_{p,q=1}^dE\left[\frac{\partial^2h}{\partial x_p\partial x_q}(U_i)V_{p,i}V_{q_i}\right] &\quad+\frac{1}{2}\sum_{p,q,r=1}^d\int_0^1(1-t)^2E\left[\frac{\partial^3h}{\partial x_p\partial x_q\partial x_r}(U_i+t\xi_iV_i)V_{p,i}V_{q,i}V_{r,i}\xi_i^{3}\right]dt
{\cal{F}}\left\{\left( P -{\mu}^{2}\right)_{A d}^{- 1}\right\}( x ) = \pi{\mu}^{\frac{\nu}{2}- 1}Q_{-}^{\frac{1}{2}\left( 1 - \frac{\nu}{2}\right)}{\cal{I}}_{1 - \frac{\nu}{2}}\left( \mu Q_{-}^{\frac{1}{2}}\right)
\|W_tf-f\|_{\ell^2(\mathbb{N})}^2=\|c_n((e^{(\cdot-s^{+})t}-1)F)\|_{\ell^2(\mathbb{N})}^2=\int_{X}(e^{(x-s^{+})t}-1)^2F^2(x)\, d\mu(x)
\begin{aligned}\zeta_B'(0;a,1,1)-\left(\frac 1{12}-\zeta'_R(-1)\right)\frac 1 a +\frac 1 4 \log (2\pi)-\frac{\gamma a}{12}=\int_0^\infty\left( \frac t{e^t-1}-1+\frac 1 2 t-\frac 1{12}t^2 \right)\frac 1{t^2(e^{t/a}-1)}\,dt =\int_0^\infty\frac{F(at)}{t(e^{t}-1)}\,dt,\end{aligned}
A = \ell_{b}(m_{a}^2-u_{a}^2) + \ell_{a}(m_{b}^2-u_{b}^2) + \ell_{ab}\big((m_{a}+u_{a})(m_{b}+u_{b}) + (m_{a}-u_{a})(m_{b}-u_{b})\big) &= \ell_{b}m_{a}^2+\ell_{a}m_{b}^2+2 \ell_{ab}m_{a}m_{b}+ 2 \ell_{ab}u_{a}u_{b}- \ell_{b}u_{a}^2-\ell_{a}u_{b}^2.
\left\vert P_\ell\Bigr(a_s(t,x)-\bigl(\sum\limits^{p_r}_{s=1}e^{\frac{2\pi(s-1)}{p_r}mi}\bigr)\alpha_{s}(x)\Bigr)\right\vert\leq C_0t^{-\frac{\ell}{2}}e^{-\frac{\varepsilon_0\hat d(x,X^{r}_{\mathrm{sing\,}})^2}{t}},\ \forall t\in\mathbb{R}_+,\ \ \forall x\in X_{p_r}
\begin{array}{c}a X^{0}_{\pm}= \frac{\sqrt{\alpha'}(D-1)}{2 Q}\log \frac{(D-1) \sqrt{\alpha'}}{2 \tilde{Q}\sqrt{\eta}\, ( t_0 \pm t )}\ \hfill (\tilde{Q}= Q e^{\frac{2 \phi_0}{D-1}}\ , \t_0 = \mbox{const.}\ ,\ \eta = \pm 1\ \mbox{for\Minkowskean\or\Euclidean\signature}\ ) \ . \end{array}
\begin{aligned}\dot{\bar{x}}_{1,1}=\bar{x}_{1,2}\dot{\bar{x}}_{1,2}=\bar{x}_{1,3}&\vdotswithin{=}\dot{\bar{x}}_{1,\rho_1}=\bar{u}_1\end{aligned}\begin{aligned}\dot{\bar{x}}_{2,1}=\bar{x}_{2,2}\dot{\bar{x}}_{2,2}=\bar{x}_{2,3}&\vdotswithin{=}\dot{\bar{x}}_{2,\rho_2}=\bar{u}_2\end{aligned}
d \overline{{{s}}}^{2}= \overline{{{g}}}_{\mu \nu}d x^{\mu}d x^{\nu}= d t^{2}- \overline{{{a}}}^{2}( t ) d l^{2}, d l^{2}= [ d r^{2}/ ( 1 - k r^{2}) + r^{2}d \Omega^{2}] ,
\frac{\left( \widehat{P}_\hbar +\widehat{\chi}_{R}-z\right)^{-1}-\left( \widehat{P}_{\hbar'}+\widehat{\chi}_{R}-z\right)^{-1}}{\hbar-\hbar'}=-\left( \widehat{P}_\hbar +\widehat{\chi}_{R}-z\right)^{-1}\widehat{Q}_2\left( \widehat{P}_{\hbar'}+\widehat{\chi}_{R}-z\right)^{-1},
\mu_{N - 1}^{( N )}= -{\frac{1}{2}}\int_{- \infty}^{\infty}d x \; w^{2}( x ) \left({\cal P}_{N - 1}^{\prime}\right)^{2}+{\frac{1}{2}}\sum_{m = 0}^{N - 1}K_{N - 1 , m}.
{\rm ex}(tn,H_k^r) \:\geq\: t^r{\rm ex}(n,H_k^r) + \, \frac{1}{n}\binom{tn}{r}- \, \frac{1}{n}\, t^r \binom{n}{r}- \frac{1}{2}\, t^{r-1}(t-1) \binom{n-1}{r-2}.
\begin{array}{l}\displaystyle|| U^{t}g||_{\mathbf{B}_{\tau}}=|| g||_{\mathbf{B}_{\tau}},\forall g\in \mathbf{B}_{\tau}, t\in \mathbb{R}, \displaystyle || U^{t}g||_{\mathbb{L}^{1}}=|| g||_{\mathbb{L}^{1}}, \forall g\in \mathbb{L}^{1},t\in \mathbb{R}. \end{array}
F ( z ) = F ( 0 ) + \sum_{m = 1}^{\infty}\left\{\frac{b_{m}}{z - a_{m}}+ \frac{b_{m}}{a_{m}}\right\}+ \sum_{m = 1}^{\infty}\left\{\frac{b_{- m}}{z - a_{- m}}+ \frac{b_{- m}}{a_{- m}}\right\}.
\Delta S=\int\Delta L\approx\int\frac{\partial L}{\partial\dot{q}}\Delta\dot{q}\approx\int\frac{\partial L}{\partial\dot{q}}(\pm\frac{\delta q}{\tau})\approx\pm\frac{\partial L}{\partial\dot{q}}\delta q=\pm\frac{\partial L}{\partial\dot{q}}\varphi.
\Psi(g) = -\frac{1}{2\tau}\int_0^M\int_0^M(M-\max\{\eta,\eta'\})\sum_{i=0}^{k-1}a_i(g(\eta)-g^{n+i}(\eta))(g(\eta')-g^{n+i}(\eta')) \, d\eta\,d\eta' &\phantom{xx}{}+ \frac{1}{\alpha(1-\alpha)}\int_0^M g(\omega)^{1-\alpha}\, d\omega.
K_{j+1}^{i+1}(z) = i \frac{\wp'(z)}2\cdot \frac 1 j K_j^{i-1}(z) - i\frac{g_2}{12}\cdot \frac 1 j K_{j-1}^{i-1}(z) - \frac 1 j DK_i^j(z)
\exp \, i \int d^{4}x \,{\cal L}_{c o r r}( A_{\mu}) = \int{\cal D}\Phi \, \int{\cal D}\Phi^{*}\, \exp - i \int d^{4}x \; \Phi^{*}{\hat{D}}\Phi = \left( \mathrm{d e t}\,{\hat{D}}\right)^{- 1},
\left[ \left( \frac{1}{2}p_{1 2}p_{2 1}+ \frac{2 9 8 - N}{3 6 0}\right) G + \left( \frac{1}{2}p_{1 2}p_{2 1}+ \frac{N + 4 2}{1 2 0}\right) C_{\mu \nu \alpha \beta}C^{\mu \nu \alpha \beta}\right.
\begin{aligned}\psi^k_v \circ F_k = \psi^k_v \circ (\psi^k_v)^{-1}\circ F_{k-1}^2 \circ \psi^k_v begin{align*}0.2em] = F_{k-1}^2 \circ \psi^k_v = F_{k-1}\circ \big[\,F_{k-1}\circ \psi^k_v \,\big] begin{align*}0.2em] = F_{k-1}\circ \psi^k_c\end{aligned}
\begin{cases}-\Delta P_{\tilde{\Omega}_{\varepsilon}}w + \tilde{\lambda}_0 \underline{m}P_{\tilde{\Omega}_{\varepsilon}}w = f_0(w) \tilde{\Omega}_{\varepsilon}\; , P_{\tilde{\Omega}_{\varepsilon}}w = 0 \partial\tilde{\Omega}_{\varepsilon}\; .\end{cases}
(1+t)^{\tfrac{x}{2}}= \sum_{m=0}^\infty \Big( \frac{x}{2}\Big)^m \frac{\big(\log(1+t)\big)^m}{m!}= \sum_{m=0}^\infty \Big( \frac{x}{2}\Big)^m \sum_{n=m}^\infty S_1(n,m) \frac{t^n}{n!}&= \sum_{n=0}^\infty \left( \sum_{m=0}^n \Big( \frac{x}{2}\Big)^m S_1(n,m) \right) \frac{t^n}{n!}.
|S| =|(S_1\sqcup S_1^{-1})||(S_3\sqcup S_3^{-1})^{\gamma}|+|(S_3\sqcup S_3^{-1})||(S_1\sqcup S_1^{-1})^\gamma| &\phantom{=}+|S_1||S_2^\gamma|+|S_2||S_1^\gamma|+|S_1^{-1}||S_4^\gamma|+|S_4||S_1^{-\gamma}| &=(4+4)\cdot(4+4)+(4+4)\cdot(4+4)+4\cdot2+4\cdot2+4\cdot2+4\cdot2 &=160,
w_a=\Big(\begin{array}{ccccccccc}1\! \!2\! \!\cdots\! \!a\! \!a\!+\!1\! \!a\!+\!2\! \!\cdots\! r r\!-\!a\!+\!1\! \!r\!-\!a\!+\!3\! \!\cdots\! \! r \! 1 \! \!\! 2\!\! \!\cdots\! \!r\!-\!a\!\end{array}\Big).
{\rm H}_{31}^{F}=-\frac{1}{4}{\cal F}_{31}\left( 9\alpha_{31}+6\beta_{31}\right) \int d^{3}{\vec{x}}\;i\pi ({\vec{x}})\tau_{a}\partial_{a}{\xi}(\vec{x})\left( \frac{\ell_{P}}{{\cal L}}\right)^{2\left( 1+\Upsilon\right)}
\forall m\exists p T_1(e,m,p) \rightarrow \neg\forall n\left[\begin{gathered}\neg\exists m\exists p (T_1(e,m,p) \wedge U(p)=n) \rightarrow n \not\in X_0 \wedge \exists m\exists p (T_1(e,m,p) \wedge U(p)=n)\rightarrow n \in X_0\end{gathered}\right]
\mathrm{\boldmath ~ \Psi ~ \u n b o l d m a t h}= \left( \begin{array}{c}{{\Psi_{+ 1}( x )}}\\{{\Psi_{- 1}( x )}}\\ \end{array}\right) \; , \; \; \Psi_{\zeta}( x ) = \left( \begin{array}{c}{{\chi_{\zeta}( x )}}\\{{\varphi_{\zeta}( x )}}\\ \end{array}\right) \; ,
- \frac{v(x,z)}{u(x,z)}= - k - \int_0^x V(t) e^{-2kt}\,dt + \frac 1k \int_0^x \int_0^{t_1}e^{-2kt_1}(1 -e^{-2kt_2}) V(t_1) V(t_2)\,dt_2\,dt_1 + O(\lvert k\rvert^{-2})
x_i(a) y_j(\pm b) = y_j(\pm b) x_i(a), x_i(a) y_i(b+c) = y_i(\frac{b+c}{a(b+c)+1}) \alpha_i^\vee (a(b+c)+1) x_i(\frac{a}{a(b+c)+1}), x_i(\frac{a}{a(b+c)+1}) y_i(-c) = y_i(\frac{-c(a(b+c)+1)}{ab+1}) \alpha_i^\vee (\frac{a(b)+1}{a(b+c)+1}) x_i(\frac{a}{ab+1}).
\hat{\rho}_{\mathrm{eq}}(\|p\|) =\|p\|\hat{\rho}_{\mathrm{eq}}(\|p\|^2/2) =\widetilde{\mathcal{N}}(\xi_+)\|p\|^{d}\|p\|^{2C/k_{scatt}^2}\exp\left(-\dfrac{1}{2 k_{scatt}^2}\|p\|^2\right).
\left[ \begin{array}{c}{}begin{align*}-3mm]{\sum\sum}begin{align*}-3mm]{}_{j<l}\end{array}\alpha_j \alpha_l k_{jl}^2 - \sum \alpha_i m_i^2 \right]\Rightarrow - \left[ \sum \alpha_i^2 m_i^2 \!+\!2\!\begin{array}{c}{}begin{align*}-3mm]{\sum\sum}begin{align*}-3mm]{}_{j<l}\end{array}\alpha_j \alpha_l m_j m_l c_{jl}\right] ,
\mathbb{E}\left[\cos(W_{(k-1)L+1}+\Phi_{S,k-1})\right] &= \mathbb{E}\left[\cos(W_{(k-1)L+1})\right] \mathbb{E}\left[\cos(\Phi_{S,k-1})\right] - \mathbb{E}\left[\sin(W_{(k-1)L+1})\right] \mathbb{E}\left[\sin(\Phi_{S,k-1})\right]
\nu_j = \frac{h}{\mu}\, \left[ \left( \frac{2j+1}{(\rho -1)^2}-\frac{2}{\left( \rho -1\right)^{3}}\right) \rho^{j+2}- \frac{j\left(j+2\right)}{\rho-1}+\frac{3}{(\rho -1)^2}+\frac{2}{\left( \rho -1\right)^{3}}\right]
p(Z|\theta,\eta)= p(Z|\eta_{11},\eta_{22}) &=\prod_{t=1}^{T}{p(z_{t+1}|z_t,\eta_{11},\eta_{22})}&=\eta_{11}^{n_{11}}(1-\eta_{11})^{n_{12}}\eta_{22}^{n_{22}}(1-\eta_{22})^{n_{21}},
T_{23}= \left(\begin{array}{rrr}1 0 0 0 1 1 0 0 1\end{array}\right),T_{31}= \left(\begin{array}{rrr}1 0 0 0 1 0 1 0 1\end{array}\right),T_{32}= \left(\begin{array}{rrr}1 0 0 0 1 0 0 1 1\end{array}\right).
\lim_{x \to 0}\frac{x\left(\frac{1}{2}+\frac{1}{3}\sin\left(\frac{1}{x}\right) \right)-0}{x-0}=\lim_{x \to 0}\left(\frac{1}{2}+\frac{1}{3}\sin\left(\frac{1}{x}\right) \right)=D.N.E,
\sum_{n=1}^{\left\lfloor Tr\right\rfloor}\prod_{i=1}^{m}(\mathsf{D}_{d}^{i-1}R_{T,\left\lfloor Tr\right\rfloor -n}^{(1)}(d))^{j_{i}}T^{1/2-d}\pi_{\left\lfloor Tr\right\rfloor -n}(d)\xi_{n}\Rightarrow\int_{0}^{r}\prod_{i=1}^{m}(\mathsf{D}_{d}^{i-1}R^{(1)}(d))^{j_{i}}\mathsf{d}W.
\begin{array}{lcllcl}G_1 = \langle (3,4), (2,4,3) \rangle, G_3 = \langle (1,2)(3,4), (3,4) \rangle G_2 = \langle (1,3), (1,3,2) \rangle, G_4 = \langle (1,3)(2,4), (2,4) \rangle G_5 = \langle (1,4)(2,3), (1,3)(2,4) \rangle\end{array}
F \; \; = \; \;{\cal{Y}}_{[ p_{1}+ 1 , . . . , p_{N}+ 1 ]}\circ \left( \, \prod_{i = 1}^{N}\; d^{( i )}A \, \right) \; \; \equiv \; \; \prod_{i = 1}^{N}\; d^{( i )}A
V = \frac{N T_{0}}{L}\sqrt{( 1 + \frac{r^{2}}{{\tilde{L}}^{2}}) ( 1 + \frac{4 L^{2}r^{4}}{N^{2}- 1})}- \frac{2 N T_{0}}{\sqrt{N^{2}- 1}}\frac{r^{3}}{{\tilde{L}}}.
\mathbb{E}\left [\left( X_{\, \alpha}^{\, \left ( n \right )}\right)^{r}\right ]= \frac{n\alpha\,B(n\alpha,r+1)}{\mathbb{E}[X^{n\alpha}]}\, \mathbb{E}[X^{n\alpha+r}],r\in\mathbb{R}^{+},
( Tu, u + B(Tu)) \cdot_H ( Tv , v + B (Tv)) &= ( T(u)T(v), (Tu) \cdot v + (Tu) \cdot B(Tv) + u \cdot (Tv) + B (Tu) \cdot Tv + H (Tu, Tv)) &= ( T(u )T(v), (Tu) \cdot v + u \cdot (Tv) + B (Tu, Tv) + H (Tu, Tv)) \in \tau_B (\mathrm{Gr}(T)).
H_t^{d,p}(k)=\{x\in T^d(r):\sum_{y:y\sim x}\eta_t^{d,p}(y)=k\}, &I_t^{d,p}(k)=\{x\in T^d(r):\sum_{y:y\sim x}\eta_t^{d,p}(y)\geq k\}=\bigcup_{l\geq k}H_t^{d,p}(l), &J_t^{d,p}(k)=\{x\in T^d(r):\sum_{y:y\sim x}\eta_t^{d,p}(y)\leq k\}=\bigcup_{l\leq k}H_t^{d,p}(l).
M_{1}^{s}=\left( \begin{array}{ccc}1 0 0 0 1 0 0 0 1\end{array}\right) ,M_{2}^{s}=\left( \begin{array}{ccc}1 0 0 0 1 0 0 -s_{1}1\end{array}\right) ,M_{3}^{s}=\left( \begin{array}{ccc}1 0 0 0 e^{s_{2}}0 0 0 1\end{array}\right) .
\Psi ( x ) = \left( f ( x ) +{\cal F}^{\mu}( x ) \widehat{b}_{\mu}^{+}+ \frac{1}{2 !}\varepsilon_{\mu \nu \lambda}{\tilde{\cal F}}^{\mu}( x ) \widehat{b}^{+ \nu}\widehat{b}^{+ \lambda}+ \frac{1}{3 !}\tilde{f}( x ) \varepsilon_{\mu \nu \lambda}\widehat{b}^{+ \mu}\widehat{b}^{+ \nu}\widehat{b}^{+ \lambda}\right) | 0 \rangle .
\hat{V}^{( B = 0 )}( x , t ) = \mathrm{d i a g .}\, ( \overline{{{V}}}_{1}, \cdots , \overline{{{V}}}_{4}) = \mathrm{d i a g .}\, ( \scriptstyle{{\sqrt{-{\frac{t_{3}}{t_{1}}}}{\frac{t - t_{1}}{t - t_{3}}}, \sqrt{-{\frac{t_{4}}{t_{2}}}}{\frac{t - t_{2}}{t - t_{4}}}, \sqrt{-{\ \frac{t_{1}}{t_{3}}}}{\frac{t - t_{3}}{t - t_{1}}}, \sqrt{-{\frac{t_{2}}{t_{4}}}}{\ \frac{t - t_{4}}{t - t_{2}}}}})
\left|(1-\chi(4r/r')-\chi(4r'/r) )(\mathcal{L}_V+1)^{-1}(z,z')\right|\lesssim\begin{cases}d(z,z')^{2-n}, d(z,z')\leq 1; d(z,z')^{-N}, d(z,z')\geq 1.\end{cases}
p_{\mathcal{S}}(\psi_i,\Phi) = \prod_{j=1}^n p_{\mathcal{S}_j}(\psi_{ij},\Phi_j) = \prod_{j=1}^n p_{K_j}(_{\mathcal{S}_j/K_j}(\psi_{ij}), \Phi'_j)=p_Y(_{\mathcal{S}/Y}(\psi_i), \Phi'),
a^{ij}= [D^2w-A(\cdot,w,Dw)]^{ij}, b^i = -a^{ij}A_{ij,p_k}(\cdot,w_\tau,Dw_\tau)+\tilde{B}_{p_k}(\cdot,w_\tau,Dw_\tau), c = -a^{ij}A_{ij,u}(\cdot,w_\tau,Dw_\tau)+\tilde{B}_{u}(\cdot,w_\tau,Dw_\tau).
\Omega'\star d\Phi_\mathcal{E}(0,E) :=\,\frac{d{\Omega}(u)}{d u}\large|_{u=\Phi_\mathcal{E}(0,E)}\star \,\frac{d\Phi_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}}& =\,\frac{d{\Omega}(u)}{d u}\large|_{u=E}\star \,\frac{d\Phi_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}},
\mathcal{A}^2 = \frac{1}{16}\begin{vmatrix}0 1 1 1 1 0 L_{1,2}^2 L_{1,3}^2 1 L_{1,2}^2 0 L_{2,3}^2 1 L_{1,3}^2 L_{2,3}^2 0 \end{vmatrix}.
\alpha (-\Delta)^{\alpha/2}u(x)+ \langle x, \nabla (-\Delta)^{\alpha/2}u(x)\rangle =\partial_t \big(t^{\alpha}[(-\Delta)^{\alpha/2}u](tx)\big) |_{t=1}&=\partial_t \big((-\Delta)^{\alpha/2}u(tx)\big)|_{t=1}.
\textstyle C=C_0\bigcup\Big( \bigcup_{\ell=\lceil\frac{1}{2}\mu_q(n)\rceil}^{\frac{n-1}{2}}\bigcup_{L\in\Omega_{2\ell}}\tilde L^*\Big);~~~~~|\Omega_{2\ell}|\le n^{2\ell/\mu_q(n)}.
T_3 = \frac1k \sum_{j=0}^{k-2}\left( \frac{j-k+1}{Q(T) - j}\right) Q'(T) T' - \left( d_2 (Q(T) - k+1) + \frac{k-1}2 \right) T' - \left( \frac{k-1}2 \right) \frac{T''}{T'}.
S[A_\mu, h]= - \frac{1}{2 g^2}\int dx^3 \,{\rm tr}[F_{\mu \nu}\, F^{\mu \nu}] + \int dx^3 \left({\rm tr}(D_\mu h)^2 + \frac{\lambda}{4}(2{\rm tr}h^\dagger h -v^2)^2 \right)\,\,\, ,
\frac{d^{2}r}{d\theta^{2}}\cdot(\frac{H}{r^{2}})^{2}+\frac{dr}{d\theta}\cdot(-\frac{2\cdot H\cdot\dot{r}}{r^{3}})-r(\frac{H}{r^{2}})^{2}=-\frac{\alpha}{r^{2}}
\sum_{e \in \beta^{-1}(1)}|x_{v_1(e), n}- x_{v_1(e), n}|^2/4t_e = \sum_{e \in \beta^{-1}(1)}\left|\sum_{i = 1}^{l(e)}y_{f^e_i, n}\right|^2\bigg/4t_e,
\frac{\partial}{\partial t}|C_2|^2_g =\Delta_g|C_2|^2_g-2|\nabla C_2|^2_g-16\langle\mathrm{Ric}, C^2_2\rangle_g+8R|C_2|_g^2 &-4\langle\mathrm{Ric},\mathrm{div}_gD\rangle_g+4\langle\mathrm{Ric}^2,\mathrm{div}_gC_3\rangle_g-2\langle\nabla R,\mathrm{div}_g(\mathrm{div}_gC_3)\rangle_g.
X^+_q\bigl(g(U,V)\bigr) =\left.\frac{d\}{dt}\right|_{t=0}g(U,V)_{\exp(tX)\cdot q}=\left.\frac{d\}{dt}\right|_{t=0}g\bigl(\tau(\exp(-tX))_*(U),\tau(\exp(-tX))_*(V)\bigr)_q =g\bigl([X^+,U],V\bigr)_q+g\bigl(U,[X^+,V]\bigr)_q\quad\textrm{(see \eqref{eq:XYPhi}).}
\overline{\Gamma}_{ij}^k= 0\ \ \ \ \ \ \ \ \ \overline{\Gamma}_{ij}^i= -\frac{\varphi_{,x_{j}}}{\varphi}\ \ \ \ \ \ \ \overline{\Gamma}_{ii}^k= \varepsilon_{i}\varepsilon_{k}\frac{\varphi_{,x_{k}}}{\varphi}\ \ \ \ \ \ \ \overline{\Gamma}_{ii}^i= -\frac{\varphi_{,x_{j}}}{\varphi}.