harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{dN}{dt}=(N_r-N)D-I_{NR}\frac{N}{\kappa_{NR}+N}R, \frac{dR}{dt}=\bigg(\mu_{NR}\frac{N}{\kappa_{NR}+N}-(D+m_R(R))\bigg)R-I_{RP}\frac{R}{\kappa_{RP}+R}P, \frac{dP}{dt}=\bigg(\mu_{RP}\frac{R}{\kappa_{RP}+R}-(D+m_{P0})\bigg)P,
\dim_P(\mathcal{S}_K^{\ast}(w))\geq \dim_P(\mathcal{S}_K^{(b)\ast}(w))\geq \left(1-\frac{1}{m}\right)\dim(K)=\left(1-\frac{1}{m}\right)\frac{\sum_{i=1}^{m}\log |W_i|}{\log b}.
U ( f , h , u ) = \frac{( u^{2}- 1 )^{2}}{2 x^{4}}+ \frac{u^{2}f^{2}}{x^{2}}+ \frac{f^{2}}{x^{2}}+ \frac \beta 4 ( h^{2}- 1 )^{2}+ \frac \beta 4 ( f^{2}- 1 )^{2}\ .
g({\rm grad}_E\,f,X)=X^hf, f\in C^{\infty}(P),X\in TL, &{\rm Hess}\,f(X,Y)=g(\nabla^E_{X^h}{\rm grad}_E\,f,Y), f\in C^{\infty}(P),X,Y\in TL, &{\rm div}_E(W)=\sum_i g(\nabla^E_{e_i^h}W,e_i\circ\pi_P), W\in \Gamma(E),
E ={\frac{M ( \rho ) \operatorname{c o s h}^{2}\! \rho}{\sqrt{\operatorname{c o s h}^{2}\! \rho -{\dot{\rho}}^{2}}}}-{2 \pi q L T_{\mathrm{F}}}\operatorname{s i n h}^{2}\! \rho \, ,
K^{n,3}_\tau =n^{2\alpha+1}\int^{\tau}_{(\delta+\frac 1n) \wedge \tau}\gamma_s [ (\sigma'(X_s) \sigma( X_{\eta_n(s) -\delta}))^2 - ( \sigma'\sigma)^2(X_{\eta_n(s)})] \left( \int_{\eta_n(s)-\delta}^{\eta_n(s)}\psi_{n,2}(u,s) dW_u \right)^2 ds
\frac{v(x,t)}{u(x,t)}= \frac{\int_{0}^t \int_{\partial B}K(x,t,y,s) s^{\beta}d\sigma(y) ds}{\int_{0}^t \int_{\partial B}K(x,t,y,s) s^{\alpha}d\sigma(y) ds}\geq \frac{1}{t^{\alpha - \beta}}\frac{\int_0^t \int_{\partial B}K(x,t, y, s) s^{\alpha}dy ds}{\int_0^t \int_{\partial B}K(x,t, y, s) s^{\alpha}dy ds}= \frac{1}{t^{\alpha - \beta}}.
\frac{\partial}{\partial ( \partial_{0}^{2})}D_{0 0}^{c o l}( \partial_{\mu}) = 0 , \quad D_{0 i}^{c o l}( \partial_{\mu}) = D_{i 0}^{c o l}( \partial_{\mu}) = 0 , \quad i = 1 , 2 , 3
G(x,N)=\frac{1}{N}\sum_{n=0}^{N-1}(\sum_{k=-n}^{n}G_{k}e^{-2\pi ikx})=\int_{0}^{1}G(t)\frac{1}{N}\sum_{n=0}^{N-1}\sum_{k=-n}^{n}e^{2\pi ik(t-x)}dt,
\frac{d^{2}\beta}{dt^{2}}+(\frac{4k}{MV}+\frac{2k(a^{2}+b^{2})}{VI})\frac{d\beta}{dt}+(\frac{4k^{2}(a+b)^{2}}{MV^{2}I}+\frac{2k(b-a)}{I})\beta=0
K_0 = K_0 K_1 = 2^1 K_0 K_2 = K_0 K_3 = 2^2 K_0 K_4 = K_0 K_5 = 2^1 K_0 K_6 = K_0 K_7 = 2^3 K_0 &K_8 = K_0 K_9 = 2^1 K_0 K_{10}= K_0 K_{11}= 2^2 K_0 &K_{12}= K_0 K_{13}= 2^1 K_0 K_{14}= K_0 K_{15}= 2^4 K_0
\lim_{b\to 0}\frac{(1-\alpha^2)b}{g(z_-)}= 1 \lim_{b\to 0}z_-(b) = -\infty \,, \lim_{b\to 0}\frac{b g(z_+)}{1-\alpha^2}= \frac1{1-\alpha^2}>1 \lim_{b\to 0}z_+(b) = \alpha \,.
|| D \mathrm{u}||_{q( z ), \mu ( z )}^{q^{+}}= - || D \mathrm{u}||_{p( z )}^{p^{+}}+ \int_{M}g( z ) | \mathrm{u}( z ) |^{1 - \gamma ( z )}\,\, dv_{g}( z ) + \lambda || \mathrm{u}||_{r( z )}^{r^{-}},
\beta_{n,q}^{\left(h,k\right)}\left(x\right)=\frac{1}{\left(1-q\right)^{n}}\sum_{j=0}^{n}\dbinom{n}{j}\left(-1\right)^{j}q^{jx}\frac{\left(j+h\right)_{k}}{\left[j+h\right]_{q,k}},
\sum_{\substack{j=1 2 \notin I_j}}^h \left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{P}^{Cont}_j}+ \frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{P}^{Cont}_j}\right) +\sum_{j=1}^h \mathbf{1}_{2\in I_j}-\frac{1+\sum_{i=0}^hk_1(i)}{r+k_1(0)}a^{\widehat{P}^{Cont, +r}}-\frac{1+\sum_{i=0}^hk_2(i)}{s+k_2(0)}b^{\widehat{P}^{Cont, +s}}.
\lambda_{a}\equiv l_{j_{a}}\, - \sum_{q = 1}^{a - 1}k_{j_{q}j_{a}}; \; \theta_{a}= \sum_{j}( l_{j}\, - \sum_{q = 1}^{a}k_{j_{q}j}) x_{j}, \; \rho_{j_{a}}= \sum_{j}k_{j_{a}j}x_{j}.
\mathcal I_1:=-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}\right)_x\varphi_{t,\gamma}\,dx\,dt\,, &\mathcal I_2:=-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\varphi_{t,\gamma}\,dx\,dt\,.
\chi e^{-ith^{-1}\tilde{P}_h(0)}q_1 e^{-(t_0-t)X}\tilde{R}_h(z)\chi=ih^{-1}\int_0^\infty \chi e^{-ith^{-1}\tilde{P}_h(0)}q_1 e^{-(t_0-t)X}e^{-ish^{-1}\tilde{P}_h(z)}\chi ds.
\beta(J) = \sum_{K\, :\, \emptyset \not=K,\J^c\subset K}(-1)^{|J\cap K|+1}\theta(K), \beta(J) = \sum_{L\, :\, J \subset L \subset [p]}(-1)^{|L\setminus J|}\lambda(L) &\theta(K) = \sum_{L\, :\, \emptyset\not=L\subset K}(-1)^{|L|-1}\lambda(L), \lambda(L) = \sum_{K\, :\, \emptyset \not=K\subset L}(-1)^{|K|-1}\theta(K).
\mathcal{A}_P^{(1)}(UF,UG) = max_{i=1}^N \frac{q_i\bigg\{| \langle Uf_i,Ug_i \rangle | + \|Uf_i\|\; \|Ug_i\| \bigg\}}{2}&= max_{i=1}^N \frac{q_i \bigg\{| \langle f_i,g_i \rangle | + \|f_i\|\; \|g_i\| \bigg\}}{2}&=\mathcal{A}_{P}(F,G).
\Delta U_T= 2 A \int_0^{\infty}dz\, \Delta w_T(z)={\pi\over 4}A{(k_B T)^3\over \hbar^2}\int_{-\infty}^{\infty}d{\cal Z}\left[{2\over{\cal Z}^4}-{\coth{\cal Z}\over{\cal Z}^3}-{1\over{\cal Z}^2\sinh^2{\cal Z}}\right].
W ( \phi ) = \int_{\phi_{0}}^{\phi}e^{\alpha \eta}\left( \frac{\beta}{2}\eta^{2}- \Lambda \right) d \eta = \left. e^{\alpha \eta}\left[ \frac{\beta}{2}\left( \frac{2}{\alpha^{3}}- \frac{2 \eta}{\alpha^{2}}+ \frac{\eta^{2}}{\alpha}\right) - \frac{\Lambda}{\alpha}\right] \right|_{\phi_{0}}^{\phi}.
Q=[\begin{matrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\ 2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\ 2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{matrix}].
J_{X,j,h(j)}=\left\langle d_{R,C}:R\subset \left\lbrace 1, 2, \ldots n\right\rbrace C\subset \left\lbrace u_1, \ldots u_{h(j)}, Xu_1,\ldots, Xu_j\right\rbrace\right. &\qquad\qquad\left.\; |R|=|C|=h(j)+1\right\rangle, J'_{X,j,h(j)+1}=\left\langle d_{R,C}: R=\{1,\ldots,h(j)+1\}C=\{u_1,\ldots u_{h(j)},Xu_j\}\right\rangle,
\begin{aligned}\{x_i,x_{i+l}\}=-x_ix_{i+l},\; 1\le l \le k-2; \quad \{y_i,y_{i+l}\}=-y_iy_{i+l},\; 1\le l\le k-1; \{y_i,x_{i+l}\}=-y_ix_{i+l},\; 1\le l\le k-1;\quad \{y_i,x_{i-l}\}=y_ix_{i-l},\; 0\le l\le k-2,\end{aligned}
\langle \, \partial_{-}f ( \sigma ) \, \partial_{+}f ( \sigma^{\prime}) \, \rangle \,_{\mathrm{{i n}}}= - \, \frac{1}{4 \pi}\, \frac{p^{\prime}( \sigma^{-})}{[ \: p ( \sigma^{-}) - \sigma^{+}- i \epsilon \: ]^{2}}\, ,
\Omega^V_1:=\Bigg\{\omega: H^V_t=\lim_{n_m\to\infty}\sum_{0<s\leq t}K_s\Big(V(X_s)-V(X_{s-})-(\Delta X_s\cdot\nabla V)(X_{s-}) \Big)1_{|\Delta L^0_s|>\frac{1}{n_m}}\mbox{and}&G^V_t=\lim_{n_m\to\infty}\sum_{s\in(0,t]}K_s(A_2\Delta L^0_s\cdot \nabla) V(X_{s-})1_{|\Delta L^0_s|>\frac{1}{n_m}}\mbox{uniformly in $t\in[0,1]$.}\Bigg\}.
\limsup_{j \to +\infty}\big\| \exp(A)^j \exp(B)^{-\lfloor \varepsilon j \rfloor}\big\| = \infty, \mbox{or}\limsup_{j\to -\infty}\big\| \exp(A)^j \exp(B)^{-\lfloor \varepsilon j \rfloor}\big\| =\infty,
(V \pi_X (a) - \pi^\infty_Y (a) V)^*(V \pi_X (a) - \pi^\infty_Y (a) V) =\pi_X (a^*) (\pi_X (a)- V^* \pi_Y^\infty (a) V) &+ (\pi_X (a^*) - V^*\pi_Y^\infty (a^*)V )\pi_X (a) &+ V^* \pi_Y^\infty (a^*a ) V - \pi_X (a^*a ).
\begin{matrix}\overline{X}_{w}&=&\frac{1+p}{1-p}\\ \overline{M}_{n}&=&M_{o}\frac{1}{1-p}\\ \overline{M}_{w}&=&M_{o}\frac{1+p}{1-p}\\PDI&=&\frac{\overline{M}_{w}}{\overline{M}_{n}}=1+p\\ \end{matrix}
0 =(p-1) |v'|^{p-2}f''(r)Y(\theta) + \frac{(p-1)(n-1)}{r}|v'|^{p-2}f'(r)Y(\theta) +\frac{1}{r^2}|v'|^{p-2}f(r)\Delta_{\mathbb{S}^{n-1}}Y(\theta)+(p-1)(p-2)|v'|^{p-4}v' v'' f'(r)Y(\theta) +\alpha v^{p^*-2}f(r)Y(\theta),
\|[b_0,T](f)\|_{L^2(\mathbb R^3)}= \|b_0T(f) - T(b_0 f)\|_{L^2(\mathbb R^3)}&\leq \|b_0T_1(f) - T_1(b_0 f)\|_{L^2(\mathbb R^3)}+ \|b_0T_2(f) - T_2(b_0 f)\|_{L^2(\mathbb R^3)},
T^{\mu\nu}=\Sigma_n\frac{m_n}{\sqrt{g}}\int(1-\alpha\phi)U^{\mu}_nU^{\nu}_n\delta^4(x-x_n(\tau_n))d\tau_n+\frac{1}{\beta}\left[\nabla^{\mu}\phi\nabla^{\nu}\phi-\frac{1}{2}g^{\mu\nu}\left(\nabla_{\alpha}\phi\nabla^{\alpha}\phi-\Lambda^2 e^{2\phi}\right)\right] .
[ \: \frac{1}{4}( \frac{\ddot{z}^{+}}{\dot{z}^{+}}- \frac{\ddot{z}^{-}}{\dot{z}^{-}}) l + \frac{1}{2}( \dot{z}^{+}\partial_{+}l - \dot{z}^{-}\partial_{-}l ) \: ] \sqrt{\frac{\dot{z}^{+}}{\dot{z}^{-}}}= F_{-},
<q|n>=\left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}}\,\frac{1}{\sqrt{2^n n!}}\,e^{-\frac{m\omega}{2\hbar}q^2}\,H_n\left(q\sqrt{\frac{m\omega}{\hbar}}\right)\ \ \ ,\ \ \n=0,1,2,\dots\ \ \ ,
\int_{0}^{T}\,\int_{\Omega}u\,\frac{\partial \phi}{\partial t}\,dx\,dt +\displaystyle\sum_{j=1}^{d}\int_{0}^{T}\,\int_{\Omega}f_{j}(u)\,\frac{\partial \phi}{\partial x_{j}}\,dx \,dt =\int_{\Omega}u_{0}(x)\,\phi(x,0)\,dx\,dt.
\int_{|y|<\frac{1}{2|x_i-x_k|}}Q'(\tau_i^{-1}\tau_kQ)^{p-1}A_{ij,0}=\int_{|y|<\frac{1}{2|x_i-x_k|}}Q'A_{ij}\bigg(\frac{\kappa_0}{[y-(x_k-x_i)]^2}\bigg)^{p-1}+O\bigg(\frac{1}{d^3}\bigg).
\left(x \frac{d}{dx}\right)^3 \log \left( \theta_3(x) \right) = x \, \psi(x) + 3 x^2 \, \psi'(x) + x^3 \, \psi''(x) & < \left( 6.35 \, x - 118.2 \, x^2 + 255.5 \, x^3 \right) e^{-2 \pi x}+ \left( -6 \, x + 60.3 \, x^2 - 55.1 \, x^3 \right) e^{-\pi x}.
\begin{array}{c}e_{i}= A_{i}^{-}A_{i+1}^{+},~~~ 1 \le i \le n+1 f_{i}= A_{i}^{+}A_{i+1}^{-},~~~ 1 \le i \le n+1 k_{i}= q^{-N_{A_{i}}+ N_{A_{i+1}}},~~~ 1\le i \le n+1 e_{0}= A_{n+1}^{-}A_{1}^{+}, f_{0}= A_{n+1}^{+}A_{1}^{-}, k_{0}= q^{N_{A_{n+1}}+ N_{A_{1}}}\end{array}
\widehat{Q}_{k+1}=(\sum\limits_{j=0}^{q}{r \choose 2j}\widehat{Q}_k^{r-2j}S^j)(\sum\limits_{j=0}^{q-1}{r \choose 2j+1}\widehat{Q}_k^{r-2j-1}S^{j})^{-1}.
\Delta^{I\cup\{i\},J\cup\{j\}}\Delta^{I\cup\{k\},J\cup\{l\}}=\Delta^{I\cup \{i\},J\cup\{l\}}\Delta^{I\cup\{k\},J\cup\{j\}}+\Delta^{I,J}\Delta^{I\cup\{i,k\},J\cup\{j,l\}}.
\dot{x}= \lambda-dx+ rx \left( 1- \frac{x}{K}\right)- \frac{\beta xv}{(1+ y)(1+ v)}, \dot{y}= \frac{\beta x(t- \tau_1)v(t- \tau_1)}{(1+ y(t- \tau_1))(1+v(t- \tau_1))}e^{- \alpha_1 \tau_1}- a y - pyz, \dot{v}= k e^{- \alpha_2 \tau_2}y(t- \tau_2)-uv, \dot{z}= cyz-bz,
\sum_{n\le x}\omega^*(n)^2 =x\sum_{m\le x}\varphi(m)\bigg(\sum_{\substack{p\le x p\equiv 1\!\!\!\pmod{m}}}\frac{1}{p}+O\left(\frac{1}{m^2}\right)\bigg)^2+O(x) &=x\sum_{m\le x}\varphi(m)\bigg(\sum_{\substack{p\le x p\equiv 1\!\!\!\pmod{m}}}\frac{1}{p}\bigg)^2+O(x),
\begin{aligned}\frac{|x_iu_{x_i}(P_1) - x_iu_{x_i}(P_2)|}{\rho^{\alpha}(P_1,P_2)}\leq 2 \eta^{-\frac{\alpha}{2}}\|x_iu_{x_i}\|_{C([0,T]\times\bar S_{n,m})},\end{aligned}
F(u+d)-F(u) & =\left\langle \left( \gamma;d_{1},...,d_{p}\right) ,\mathbf{d}\int_{u}^{u+d}\omega+\int_{u}^{u+d}\mathbf{d}\omega+\varphi_{u}^{\ast}\omega-\varphi_{u+d}^{\ast}\omega\right\rangle & =0
{\cal A}={\frac{V}{2 \pi}}\, ( 8 \pi^{2}\alpha^{\prime})^{- 1 / 2}\int_{0}^{\infty}d t \left({\frac{\pi}{t}}\right)^{\frac{9}{2}}\left[{\frac{f_{3}^{8}( q )}{f_{1}^{8}( q )}}-{\frac{f_{4}^{8}( q )}{f_{1}^{8}( q )}}\right]
\psi_k(z,\bar z)={1\over \sqrt 2}\left[ \left ({f_{n, m}(z)\over\overline f_{n, m}(z)}\right)^{k}+\left ({\overline{f}_{n, m}(z)\over \f_{n, m}(z)}\right)^{k}\right],\qquad k=0,1,2,\ldots,
V_{\mathrm{\footnotesize ~ 1 - l o o p}}( H ) = \frac{3 N}{2 \pi^{6}R^{4}}\sum_{k = 0}^{\infty}\frac{\cos \left[ ( 2 k + 1 ) \pi R m_{t}( H ) \right]}{( 2 k + 1 )^{5}}.
\lim_{\epsilon \to + 0}\left[ K_{\mu \nu}\right]_{x^{\perp}= - \epsilon}^{x^{\perp}= \epsilon}= \left. \frac{1}{m_{d}^{d - 1}}\left( T_{\mu \nu}- \frac{1}{d - 1}g_{\mu \nu}g^{\alpha \beta}T_{\alpha \beta}\right) \right|_{x^{\perp}= 0}
\mathcal{Q}^u(s)=2|u_x(0,s)|^2 + 2\big(u(0,s)\bar{u}_s(0,s)\big)-\beta |u(0,s)|^4, &\mathcal{Q}^v(s)=\frac{\alpha}{\gamma}v_x^2(0,s)-\frac{2\alpha}{\gamma}v_{xx}(0,s)v(0,s)-\frac{2\alpha}{3\gamma}v^3(0,s).
\lim_{n\to\infty}\,\sup_{0<\tau<\frac{t_n}{16}}\frac{1}{\tau}\int_{|t_n-t|<\tau}\,\int_{|x|<4t}\,\left(\partial_tu+\frac{x}{t}\cdot\nabla u+\left(\frac{d}{2}-1\right)\frac{u}{t}\right)^2(x,t)\,dx\,dt = 0.
e_{\mu}^{a}\; \rightarrow \; e^{a}= e_{\mu}^{a}d x^{\mu}, \hspace{0 . 5 c m}\omega_{\mu}{}^{a}{}_{b}\; \rightarrow \; \omega^{a}{}_{b}= \omega_{\mu}{}^{a}{}_{b}\, d x^{\mu}
E_8 := K E \int_{t_0}^{u}|x_s^n|^{p-2}|\sigma_s^{n}(x_{\kappa(n,s)}^n)|^2ds \leq K+K \int_{t_0}^{u}E\sup_{t_0 \leq r \leq s}|x_r^n|^p ds.
w_{k+1}=\frac{1}{\alpha_{k+1}}\left[\begin{array}{ccccc}\left(-1\right)^{k}\prod_{i=1}^{k}\frac{\beta_{i}}{\alpha_{i}}, \dots, \frac{\beta_{k-1}}{\alpha_{k-1}}\frac{\beta_{k}}{\alpha_{k}}, -\frac{\beta_{k}}{\alpha_{k}}, 1\end{array}\right]^{T}.
\psi ( z ) ={\frac{{2 \kappa^{\frac{1}{2}}| f^{\prime}( z ) |}}{{( 1 + | f ( z ) |^{2})}}}e^{i \omega ( z )}, \qquad f ( z ) = \sum_{n = 1}^{N}{\frac{{c_{n}}}{{z - z_{n}}}},
\Gamma_{k}[g,\overline{g},\xi,\overline{\xi}]=\frac{1}{16\pi G_{k}}\int d^{d}x\sqrt{g}(-R(g)+2\Lambda_{k})+\Gamma_{k}^{gf}[g,\overline{g}]+\Gamma_{k}^{gh}[g,\overline{g},\xi,\overline{\xi}].
\eta(-\frac{5\xi_{1}^{2}-\xi_{2}^{2}}{2\xi_{1}^{2}(\xi_{1}^{2}-\xi_{2}^{2})}c_{\eta}(0,\xi_{1})-\frac{n_{\eta}^{\prime}(\xi_{1})}{2\xi_{1}^{2}})+(1\leftrightarrow2)
0>(K_Y+E+T_Y)\cdot \widetilde{B_k^+}= (-1+a_k)\Big(\widetilde{B_k^+}\Big)^2+T'_Y\cdot\widetilde{B_k^+}\geq \geq (-1+a_k)\Big(\widetilde{B_k^+}\Big)^2\geq 0.
\int_{\Omega}u^{1-\gamma}dx \leq\int_{\Omega}u_n^{1-\gamma}dx+\int_{\Omega}|u_n-u|^{1-\gamma}dx \leq\int_{\Omega}u_n^{1-\gamma}dx+C\|u_n-u\|_{L^2(\Omega)}^{1-\gamma}=\int_{\Omega}u_n^{1-\gamma}dx +o(1)
\partial_{n+1}u =e^{\tau\phi}\bigg(x_{n+1}^{\frac{1-2s}{2}}\partial_{n+1}w-\frac{2s-1}{2}x_{n+1}^{-\frac{1+2s}{2}}w\bigg)+x_{n+1}^{\frac{3-2s}{2}}R \nabla'u =e^{\tau\phi}x_{n+1}^{\frac{1-2s}{2}}\nabla'w+x_{n+1}^{s+\frac{1}{2}}R',
\mathrm{w i t h}\;{\cal{X}}^{\alpha}({\bf{r}}) ={\textstyle \frac{\partial_{i}}{\partial^{2}}}A_{i}^{\alpha}({\bf{r}}) \; \mathrm{a n d}\; \overline{{{{\cal Y}^{\alpha}}}}({\bf r}) ={\textstyle \frac{\partial_{j}}{\partial^{2}}\overline{{{{\cal A}_{j}^{\alpha}}}}({\bf r})}.
\footnotesize\mathbf{C}=\frac{1}{\alpha}\left[\begin{array}{rrrrrrrrr}1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 \end{array}\right].
R^{( 1 )}( \theta ) = \frac{\left( \frac{1}{2}\right)_{\theta}\left( \frac{1}{2 \lambda}+ 1 \right)_{\theta}}{\left( \frac{1}{2 \lambda}+ \frac{3}{2}\right)_{\theta}}\frac{\left( \frac{\eta}{\pi \lambda}- \frac{1}{2}\right)_{\theta}}{\left( \frac{\eta}{\pi \lambda}+ \frac{1}{2}\right)_{\theta}}\ .
0\le \Delta_1:=\prod_{j=\ell+2}^{\infty}\left(1-x^j\right)^2-\left( 1-2\sum_{j=\ell+2}^{\infty}x^j\right) \le 4\sum_{j,j'\ge \ell+2}x^{j+j'}=\frac{4x^{2\ell+4}}{(1-x)^2}\le x^{2\ell}\,,
|\Gamma(t)f|_q\leq \mathcal{B}e^{\sum_{i=1}^N\left[\beta_i(t)\theta_i-\frac{t}{2}\theta_i^2\right]}\left|f\right|_q |\Gamma^{-1}(t)f|_q\leq e^{\sum_{i=1}^N\left[-\beta_i(t)\theta_i+\frac{t}{2}\theta_i^2\right]}\left|f\right|_q
\frac{C}{|t-r|}\int_{0}^{\infty}d\xi |\partial_{\xi}\left(\left(1-\chi_{\leq 1}(r \xi)\right) r \xi \phi(r \xi) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}\right)| \leq \frac{C}{|t-r|\log^{b-1}(r)}
\delta \psi^{i}(x) = \zeta^{i}+ i \kappa (\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x)) \partial_{\rho}\psi^{i}(x),\quad\delta{e^{a}}_{\mu}(x) = i \kappa (\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x))\partial_{[\rho}{e^{a}}_{\mu]}(x),
\varepsilon(\{a,b\}) =(-1)^{|b_{(1)}||a_{(2)}|}\varepsilon(\varepsilon(\{a_{(1)},b_{(1)}\})a_{(2)}b_{(2)}+\{a, b\}) &=(-1)^{|b_{(1)}||a_{(2)}|}(\varepsilon(\{a_{(1)},b_{(1)}\})\varepsilon(a_{(2)}b_{(2)})+\varepsilon(\{a, b\})) &=(-1)^{|b_{(1)}||a_{(2)}|}(\varepsilon(\{a, b\})+\varepsilon(\{a,b\})),
e_{(k)}H_{2k+1}H_{2k}H_{2k+1}^{-1}e_{(k)}= H_{2k+1}e_{(k)}H_{2k}e_{(k)}H_{2k+1}^{-1}= z\Big(\frac{z-z^{-1}}{q-q^{-1}}\Big)^{k-1}H_{2k+1}e_{(k)}H_{2k+1}^{-1}= e_{(k)}H_{2k}e_{(k)}.
\left(\frac{2}{c(d_x)c(d_{x^*})}\right)=(-1)^{\frac{(c(d_x)c(d_{x^*}))^2-1}{8}}=(-1)^{\frac{(d_xd_{x^*})^2-1}8}=\left(\frac{2}{d_xd_{x^*}}\right).
\partial_{\pm 2}\alpha_{A B}= - \frac 1 4 \alpha_{A C}\Omega_{\pm 2}^{( i ) ( j )}\gamma_{C B}^{( i ) ( j )}, \ \, p a r t i a l_{\pm 2}\beta_{\dot{A}\dot{B}}= - \frac 1 4 \beta_{\dot{A}\dot{C}}\Omega_{\pm 2}^{( i ) ( j )}\tilde{\gamma}_{\dot{C}\dot{B}}^{( i ) ( j )}.
[\begin{matrix}2&-1\\ 4&-2\end{matrix}][\begin{matrix}2&2&-2\\ 5&1&-3\\ 1&5&-3\end{matrix}][\begin{matrix}2&2&2&-3\\ 6&1&1&-4\\ 1&6&1&-4\\ 1&1&6&-4\end{matrix}][\begin{matrix}2&2&2&2&-4\\ 7&1&1&1&-5\\ 1&7&1&1&-5\\ 1&1&7&1&-5\\ 1&1&1&7&-5\end{matrix}]...
\widetilde Y \equiv \tilde{e}_{\mu}{}^{r}e_{r}{}^{\rho}g ( \tilde{e})_{\rho \nu}\, \mathrm{d}x^{\mu}\wedge \mathrm{d}x^{\nu}= - 2 m \, \mathrm{d}r \wedge \Sigma^{3}+ \frac{1}{m}( r + 2 m ) ( r + m ) \, \Sigma^{1}\wedge \Sigma^{2}\, ,
\mathcal{Z}(\hat{\mu}_{M,D}) =\bigcup_{j=1}^{\infty}M^{*j}(\mathcal{Z}(m_D))=\bigcup_{j=1}^{\infty}M^{*j}(\mathcal{Z}_{D}^2+\mathbb{Z}^2) &\supset\bigcup_{j=1}^{\infty}L^{\varphi(p)j}(M^{*j}\mathcal{Z}_{D}^2+\mathbb{Z}^2)\supset\bigcup_{j=1}^{p^2-1}L^{\varphi(p)j}(M^{*j}\mathcal{Z}_{D}^2+\mathbb{Z}^2).
\lambda^{2}( - \frac{1}{2}\frac{\partial^{2}}{\partial X^{2}}+ \frac{i Q_{B}}{2}\frac{\partial}{\partial Y}+ \frac{1}{2}\frac{\partial^{2}}{\partial Y^{2}}+ e^{\sqrt{\frac{2}{| \kappa |}}( X + Y )}){\cal T}= \frac{N}{2 4 ( x^{+})^{2}}{\cal T}
I_{i}= \frac{h_{1}^{i}+ h_{2}^{i}}{\alpha^{\prime}\pi}\left[ \sum_{n = 1}^{\infty}\left( \overline{{b}}_{n}^{i}b_{n}^{i}- \overline{{a}}_{n}^{i}a_{n}^{i}\right) - \frac{\left| \epsilon^{i}\right|}{\epsilon^{i}}\overline{{a}}_{0}^{i}a_{0}^{i}\right] ,
\psi_0(x)=\log(1+\epsilon\alpha e^x),\psi_i(x)=\log\left(1+\frac{\epsilon}{\lambda}(e^x-1)\right),\phi_{i0}(x)=\log\frac{1-\lambda(\alpha-\epsilon)e^x}{1-\alpha(\lambda-\epsilon)e^x}, \intertext{and}\psi_k(x)=\log(1+\epsilon(\lambda+\alpha)e^x),\phi_{k0}(x)=\log\frac{1-\lambda^{-1}(\alpha-\epsilon)(e^x-1)}{1-\lambda^{-1}\alpha(e^x-1)},
\begin{aligned}3\Phi_{2,w}(X,Y) = -3(D(a,a,X) + D(b,b,X) + D(c,c,X))Y -3(D(a,a,Y) + D(b,b,Y) + D(c,c,Y))X & = (s_3+s_1+s_2)Y + (t_3+t_1+t_2)X & = (Y)X+(X)Y.\end{aligned}
V_{\cal N}^{-}( q ) = \frac{C_{1}^{2}}{2}q^{4}+ C_{1}C_{2}q^{3}+ \left( \frac{C_{2}^{2}}{2}+ C_{1}C_{3}\right) q^{2}+ ( C_{2}C_{3}-{\cal N}C_{1}) q .
d_g(x,p_i)^{\frac{n-2}{2}}u_i(x) \leq C u_i(p_i)^{-1}r_i^{-\frac{n-2}{2}}= C i^{-\frac{n-2}{2}}\{ir_i \leq d_g(x,p_i) \leq r_0\}.
\Lambda^{( 1 , 1 )}: \quad \quad \quad ({\frac{\alpha^{\prime}}{2}})^{1 / 2}( p_{L}, p_{R}) \equiv ( l_{L}, l_{R}) = ({\frac{n}{x}}+{\frac{w x}{2}},{\frac{n}{x}}-{\frac{w x}{2}}) \quad ,
\displaystyle\sum_{i}b_{i}c_{i}= \tfrac{1}{2}, &\displaystyle\sum_{i,j}b_{i}a_{ij}c_{j}+\displaystyle\sum_{i,j}b_{j}c_{j}a_{ji}= \tfrac{1}{2}, &\displaystyle\sum_{i,j}b_{i}c_{i}a_{ij}+\displaystyle\sum_{i,j}b_{j}a_{ji}c_{i}= \tfrac{1}{2}, &\displaystyle\sum_{i,j}b_{i}c_{i}a_{ij}c_{j}= \tfrac{1}{8}.
4 \pi \sqrt n \, ( n - 2 ) \,{\frac{| \xi |}{\sqrt \lambda}}\, \left({\frac{6 | \xi |}{1 + 6 | \xi |}}\right)^{1 / 2}< \left({\frac{\phi}{M_{\mathrm{p}}}}\right)^{n - 2}< 8 \pi \sqrt{2 n}\,{\frac{| \xi |}{\sqrt \lambda}}\ ,
\begin{aligned}\liminf\limits_{t\rightarrow+\infty}\frac{cap_p(B^{g^+}_q(t))}{e^{nt}}\geq\liminf\limits_{t\rightarrow+\infty}\frac{n^p}{(p-1)^{p-1}}\frac{Vol(B^{g^+}_q(t))}{e^{nt}}&=\frac{n^p}{(p-1)^{p-1}}\lim\limits_{t\rightarrow+\infty}\frac{\frac{d}{dt}[Vol(B^{g^+}_q(t))]}{ne^{nt}}&=\frac{1}{2^n}(\frac{n}{p-1})^{p-1}\mathcal{A}(q)\end{aligned}
\begin{array}{c}\displaystyle \displaystyle\sum^{M}_{j=1}\phi (\Theta -\vartheta_{j},1/2)=\displaystyle\int_{asymp}^{\Theta}dx\, W(x) \displaystyle\sum^{M}_{j=1}\phi (\Theta +\vartheta_{j},1/2)=-\displaystyle\sum^{M}_{j=1}\phi (-\Theta -\vartheta_{j},1/2)=-\displaystyle\int_{asymp}^{-\Theta}dx\, W(x).\end{array}
\hat{\mathbf{V}}_i = \mathbf{F}_i^{-1}\boldsymbol{\Gamma}_i -d^{\max}_i\nu_i\mathbf{F}_i^{-1}\mathbf{T}_i\mathbf{F}_i^{-1}\boldsymbol{\Gamma}_i +\sum_{k=2}^{\infty}\left(d^{\max}_i\right)^k\mathbf{A}_k\boldsymbol{\Gamma}_i
||x^*||_1\,\varphi(t^*,x^*) - \mathbb{E}\left[L(\nu_{\ell})X(\nu_{\ell})\varphi \big(t^*-\nu_{\ell},\mathfrak{X}(\nu_{\ell})\big)\right]=\mathbb{E}\left[\int_0^{\nu_{\ell}}L(s)X(s)g(t^*-s,s,\mathfrak{X})\, \mathrm{d}s \right]> 0\,.
I = \frac1{A_1}[\xi'(q)-\xi'(b)]\frac1{A_1\xi'(s)+\frac1{|\nu|}}\Big|_{s=0}^q = \frac{[\xi'(b)-\xi'(q)]|\nu| \xi'(q)}{A_1 \xi'(q)+\frac1{|\nu|}},
\beta_{nk}^{l}=-2\varepsilon \exp \left[ -iT\left( \omega_{ln}^{(0)}+\omega_{lk}^{(0)}\right) /2\right] \sum_{\sigma =+,-}v_{k-,n-}^{\sigma}e^{i\sigma \Omega T/2}\frac{\sin \left[ T\left( \omega_{ln}^{(0)}+\omega_{lk}^{(0)}-\sigma \Omega \right) /2\right]}{\omega_{ln}^{(0)}+\omega_{lk}^{(0)}-\sigma \Omega}
\min_{u_V\in `R_+^V}\varphi(u_V) =\min_{u_V\in `R_+^V}\max_{e\in E}\frac{u( `x(e))}{u(V)}=\min_{u_V\in `R_+^V: u(\xi(e))\leq 1, \forall e\in E}\frac{1}{u(V)}=\frac{1}{`t}
P\lim_{\delta\rightarrow 0}\frac{(Q_{t,\gamma(t+\delta)}-Q_{t,\gamma(t)})\frac{d}{dt}\gamma(t+\delta)}{\delta}=\frac{d}{ds}\left(\frac{\frac{d\gamma}{dt}(t)}{\partial_x\Phi_{t,\gamma(s)}}\circ \Phi_{t,\gamma(s)}^{-1}\right)_{s=t}
g_{ij,kl}=\frac{-q^{-1}}{\left[2\right]}D_{si}\hat{R}_{js,kl}^{\epsilon}= \frac{-q^{-1}}{\left[2\right]}\epsilon_{im}\hat{R}_{jm,kt}\epsilon^{-1}_{lt}\quad, \quad(g=\frac{-q^{-1}}{\left[2\right]}D_1{\cal P}\hat{R}_{12}^{\epsilon}) \quad,
u^i_s(q_i) = 0; u^i_s\in\mathcal{O}_{\frac{3}{2}}(M'_i); \int_{B(q_i, 1)}u^i_s\overline{u^i_t}= c^i_{st}\delta_{st}; \max\limits_{B(q_i, 1)}|u^i_s| = 1.
{\cal L}= \frac{m^{2}}{2}f_{\mu}f^{\mu}- m^{2}f_{\mu}F^{\mu}+ \frac{m^{2}}{2}F^{\mu}A_{\mu}- e J_{\mu}f^{\mu}- e A^{\mu}g_{\mu}+ \frac{\lambda}{2}( \partial_{\mu}A^{\mu})^{2}+{\cal L}_{D},
\lambda_{a|a|c|d}^{c|c|a|a}+ \lambda_{a|a|c|d}^{c|d|a|a}\lambda_{c|d|e|e}^{e|e|c|c}= \lambda_{c|d|e|e}^{e|e|c|c}+ \lambda_{c|d|e|e}^{e|e|c|d}\lambda_{a|a|c|d}^{c|c|a|a},
\Omega_{a_1 a_2 \ldots a_{2m+1}}\psi^{a_1}_+ \psi^{a_2}_+ \ldots\psi^{a_{2m+1}}_+ \; , \qquad \Omega_{a_1 \ldots a_{2m}a_{2m+1}}\psi^{a_1}_+ \ldots \psi^{a_{2m}}_+ j^{a_{2m+1}}_+ \,,
F_{( d ) 0}^{[ 1 , 0 ]}\left( p_{1}^{2}, p_{2}^{2}; p_{3}^{2}\right) = x_{1 , 8}\left( \Pi_{1}- \Pi_{2}\right) + \left[ \left( x_{1 , 9}+ x_{1 , 9}^{\prime}\frac{1}{\Pi_{2}}\right) \Pi_{1}- \left( x_{1 , 9}+ x_{1 , 9}^{\prime}\frac{1}{\Pi_{1}}\right) \Pi_{2}\right] \Pi_{3}\, ,
(y-z)^{2}-np\rho_{1}y-np\frac{\sqrt{c_{2}}}{\rho}y^{\frac{1}{2}}(y-\alpha z) &=\frac{1}{\alpha^{2}}(y-\alpha z)^{2}+\left(\frac{\alpha-1}{\alpha^{2}}y^{2}-np\rho_{1}y\right) &+\left(\frac{2(\alpha-1)}{\alpha^{2}}y-np\frac{\sqrt{c_{2}}}{\rho}y^{\frac{1}{2}}\right)(y-\alpha z),
\lim_{p\to 0}\hat{R}^*_\lambda\big(\hat{z};a,p^{1/2}b;q,t,T;p\big)=t^{-2n(\lambda)}q^{n(\lambda')}(-aT/t)^{|\lambda|}C^-_\lambda(t;q,t)\bar{P}^*_\lambda(\hat{z};q,t,T;a).
-\int_\Omega (\mu_1 - \mu_2) e^{i\xi\cdot x}a_1 b_2\,dx + \int_\Omega (\mu_1 - \mu_2) e^{i\xi\cdot x}a_1 \rho\cdot r_2\,dx - \int_\Omega (\mu_1 - \mu_2) e^{i\xi\cdot x}r_1\cdot \Big(a_2\rho+\frac{1}{2}b_2\overline\rho\Big)\,dx + \int_\Omega (\mu_1 - \mu_2) e^{i\xi\cdot x}r_1\cdot r_2\,dx= 0.
\left\{\left({\mathcal{R}}^{},{\mathcal{R}}^{}\right)|0\leq{\mathcal{R}}^{}\leq{L}^{-1}\sum\nolimits_{i=1}^{L}\bar{\mathcal{R}}_i,0\leq{\mathcal{R}}^{}\leq{\mathcal{R}}_{}\right\},
H_{0}= \frac{1}{l_{0}}\sum_{i}\mathrm{T r}( \dot{\bf x}^{i}\dot{\bf x}^{i}) + \frac{1}{l_{0}^{5}}\sum_{i , j}\mathrm{T r}( [{\bf x}^{i},{\bf x}^{j}]^{\dagger}[{\bf x}^{i},{\bf x}^{j}] ) \ .
\Bigl| \int u_{1}\overline{v}_{2}w_{3,\pm}dt dx \Bigr| & \lesssim (L_1L_2 L_3)^{\frac12}C(N_1,N_2,N_3) \|u_{1}\|_{L_{t,x}^2}\|v_{2}\|_{L_{t,x}^2}\|w_{3,\pm}\|_{L_{t,x}^2},
F \equiv ( F_{\mu \nu}) = \left( \begin{array}{c c c c}{{0}}&{{E_{1}}}&{{E_{2}}}&{{E_{3}}}\\{{- E_{1}}}&{{0}}&{{B_{3}}}&{{- B_{2}}}\\{{- E_{2}}}&{{- B_{3}}}&{{0}}&{{B_{1}}}\\{{- E_{3}}}&{{B_{2}}}&{{- B_{1}}}&{{0}}\\ \end{array}\right) .
\sum_{j=1}^\infty \langle j \rangle^{2\rho}\vert a_j(p) \vert^2 \leq 2^\rho a_1(p)^2+c(p) \sum_{\substack{j = 3 j\equiv_2 1}}^\infty \frac{1}{j^{1+\epsilon(p)}}< \infty
\Lambda_{M_j}^{r,s}(z_j+\xi,t)-\Lambda_{M_{0j}}^{r,s}(z_j)-L(\xi,t)\ll \sum_{\mu=2}^{\infty}d_0^\mu(\xi_1+\cdots+\xi_n+t_1+\cdots +t_m)^\mu\,\,\,\,\,
U_{1}(x)=\left(\begin{array}{c}x-1 2 \lfloor \frac{x-1}{2}\rfloor\end{array}\right),U_{2}(x)=\left(\begin{array}{c}2 \lceil \frac{x-1}{2}\rceil 2 \lceil \frac{x-1}{2}\rceil\end{array}\right)\mbox{and}U_{3}(x)=\left(\begin{array}{c}2 \lfloor \frac{x-1}{2}\rfloor 2 \lceil \frac{x-1}{2}\rceil\end{array}\right).
{\cal L}= \frac{1}{2 g l_{s}}\mathrm{T r}\left[ \dot{X}^{a}\dot{X}^{a}+ \frac{1}{2}[ X^{a}, X^{b}]^{2}+ \theta^{T}( i \dot{\theta}- \Gamma_{a}[ X^{a}, \theta ] ) \right] .
U^M_{k,(\alpha,\beta)}(t) = C_{k,(\alpha,\beta)}(t) - U^M_{k,(\alpha,M)}(t) \kern+1pt C_{k,(M+1,\beta)}(t) = C_{k,(\alpha,\beta)}(t) - \frac{C_{k,(\alpha,M)}(t) \kern+1pt C_{k,(M+1,\beta)}(t)}{1+C_{k,(M+1,M)}(t)}.
{\cal L}= \frac{3}{4}\Omega^{\mu , \nu \alpha \beta}\Omega_{\nu , \mu \alpha \beta}- \frac{3}{4}\Omega^{\alpha \beta}\Omega_{\alpha \beta}- \frac{1}{2}\Omega^{\mu , \nu \alpha \beta}T_{\nu \alpha \beta , \mu}+ \frac{3}{2}\Omega^{\alpha \beta}T_{\alpha \beta}
E^{T, \Upsilon, Y}\subseteq \bigg( \bigcup_{i=1}^n E^{T_i, \Upsilon_i, Y_i}\bigg) \cup \Big\{\tilde{\phi}^{T, \Upsilon, Y}\big[ \tilde{h}_j^{Y}\big] : \tilde{h}_j^{Y}\in Z^a\big[ Y_1, ..., Y_n \big], \tilde{h}_j^{\Upsilon}\in Z^a\big[ \Upsilon_1, ..., \Upsilon_n \big] \Big\}