harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{v}p=[\begin{matrix}v_{x}&0&0&0\\ 0&v_{y}&0&0\\ 0&0&v_{z}&0\\ 0&0&0&1\end{matrix}][\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ 1\end{matrix}]=[\begin{matrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\ 1\end{matrix}].
\begin{aligned}\alpha_n^{(M)}(x_1,x_2,\dots x_M) =\left\langle\prod_{k=1}^M\det(x_k-X)\right\rangle_{n\times n}\ \beta_n^{(M)}(y_1,y_2,\dots y_M) =\left\langle\prod_{k=1}^M\det(y_k-Y)\right\rangle_{n\times n}\ \end{aligned}
\{{Q_{1 1}^{\alpha}},{Q_{1 1}^{\beta}}\}= ({\gamma}^{p})^{\alpha \beta}P_{p}+ ({\gamma}^{p_{1}p_{2}})^{\alpha \beta}Z_{p_{1}p_{2}}+ ({\gamma}^{p_{1}\dots p_{5}})^{\alpha \beta}Z_{p_{1}\dots p_{5}},
Q\left({a,b,n}\right)=\frac{{{d^n}}}{{d{a^n}}}\left({\frac{{\Gamma \left({a + 1}\right)}}{{{b^{a + 1}}}}}\right) & =\sum\limits_{k = 0}^n{\left({\begin{array}{*{20}{c}}n k\end{array}}\right)}\frac{{{{\left({- 1}\right)}^{n - k}}{{\ln}^{n - k}}\left( b \right)}}{{{b^{a + 1}}}}\frac{{{d^k}\Gamma \left({a + 1}\right)}}{{d{a^k}}}.
\sup_{Q}\Big(\frac{1}{|Q|}\int_{Q}\big(v_{\vec{w}}(x)\big)^{q}dx \Big)^{1/q}\prod_{j=1}^{m}\Big(\frac{1}{|Q|}\int_{Q}\big( w_{j}(x) \big)^{-p'_{j}}dx \Big)^{1/p'_{j}}<\infty,
[T^{p^v}\textbf{a}]_i \equiv\sum_{j=0}^{p^v}\binom{p^v}{j}\textbf{a}_{i+j}\equiv\sum_{j=0}^{p^2}\binom{p^v}{jp^{v-2}}\textbf{a}_{i+jp^{v-2}}\equiv \sum_{j=0}^{p^2}\binom{p^2}{j}\textbf{a}_{i+jp^{v-2}}\pmod{p^3},
\begin{array}[c]{rl}-d{p}(t) = \{b_x(\bar{X}{(t)},\bar{u}(t))^{}p(t) -f_x(\bar{X}{(t)},\bar{u}(t))\}dt,\t\in(t_{i-1},t_{i}), p(t_{i}) =-\psi_{x_i}(\bar{X}(t_1),\bar{X}(t_2),\cdots,\bar{X}(t_n))+p(t_{i}^{+}),i=1,2,\cdots,n,\end{array}
\alpha_j = \frac{e_j}{q_j}+ z_j (e_j, q_j) =1, 0 < q_j \leq \tau_j := X^{j/2}|z_j| \leq \frac{1}{q_j \tau_j}, z_j = \delta_j X^{-j}(j=1,2, \dots, t).
\sum_{n=1}^{\infty}A_{\pi}(r,n)\,e\left(\frac{(a+bq)n}{p_1q}\right) \,v_1(n) = p_1q\, \sum_{\pm}\sum_{n_{1}|p_1 qr}\,\, \sum_{n_{2}=1}^{\infty}\frac{A_{\pi}(n_{2},n_{1})}{n_{1}n_{2}}S\left( r\overline{(a+bq)}, \pm n_{2}; p_1qr/n_{1}\right) \, \mathcal{V}^{\pm}_1 \left(\frac{n_{1}^2 n_{2}}{(p_1q)^3r}\right).
\begin{minipage}[c]{35pc}If $s\in S$ satisfies $d(x^n(s)) = n$ for all $n\geq 0,$then we take $y(s)$ to be any $\!y\!$-stable element(any element satisfying~\eqref{d.stable}), subject only tothe condition that if $s\approx s',$ then $y(s)=y(s').$\end{minipage}
\frac{Q_{i,j}\left(u+1\right)}{Q_{i,j}\left(u\right)}=\frac{u-\left(a-1\right)}{u-a}\frac{u-\left(b-1\right)}{u-b}&= \left(1+u^{-1}+au^{-2}+a^2u^{-3}+\ldots\right)\left(1+u^{-1}+bu^{-2}+b^2u^{-3}+\ldots\right) &= 1+2u^{-1}+\left(a+b+1\right)u^{-2}+\left(a^2+b^2+a+b\right)u^{-3}+\ldots.
\delta{\widetilde A}_{01}=-\frac{{\rm e}^{\varphi_0}\,(m-\chi_0\,n)}{{\widetilde\Delta}_{m,n}^{1/2}}\,\frac{{\widetilde Q}_1}{r^6}~~~,~~~\delta{\widetilde C}_{01}= -\frac{{\rm e}^{\varphi_0}\,(|\lambda_0|^2\,n-\chi_0\,m)}{{\widetilde\Delta}_{m,n}^{1/2}}\,\frac{{\widetilde Q}_1}{r^6}
\gamma^\prime = rn + 12r - 14n - 6m - 8\Theta - 4\Delta \, ; t = \frac{1}{6}\left(r^3 - 3r^2 - 58r - 3r \left(n + 3m + 3\Theta \right) +42n + 78m + 78\Theta \right) \, ; t^\prime = \frac{1}{6}\left(r^3 - 3r^2 - 58r - 3r \left(m + 3n + 3\Theta \right)+42m + 78n + 78\Theta\right) \, ,
e_1 = - \frac{i}{2}\widetilde F\sigma_1\widetilde F^{-1}, \;\; e_2 = -\frac{i}{2}\widetilde F\sigma_2\widetilde F^{-1}\;\; \mbox{and}\;\; e_3 = - \frac{i}{2}\widetilde F\sigma_3\widetilde F^{-1}.
|z^n-w^n|_p =|z-w|_p|z^{n-1}+z^{n-2}w+\cdots+w^{n-1}|_p &\leq|z-w|_p\max_{0\leq j\leq n-1}|z^{n-1-j}w^j|_p &=|z-w|_p|z|^{n-1}_p
\sum_{k=-1,-2,\dots}\sum_{\ell = 0, -1, \dots , k+1}\int_{s=1}^{+\infty}\int_{t=s}^{s+1}A_1 ((s,s+1],(t,t+1]) f_{(X_\ell , X_k)}(s,t) \mathrm{d}t \mathrm{d}s &= \int_{s=1}^{+\infty}\int_{t=s}^{s+1}\frac{\theta^2}{st}\mathrm{d}t \mathrm{d}s &\leq \theta^2 \int_{s=1}^{+\infty}\frac{1}{s^2}\mathrm{d}s = \theta^2 <+\infty
-S-\frac{S}{\alpha^{1/p}-1}-\alpha^{-\frac{N+1}{p}}\left|\partial_\theta\phi^1(\theta)\right|\ \ \leq partial_\theta f_{\tau,\theta}^{N+1}(\phi^1(\theta))\ \ \leq \-s+\frac{S}{\alpha^{1/p}-1}+\alpha^{-\frac{N+1}{p}}\left|\partial_\theta\phi^1(\theta)\right|\ ;
C_{1,k}:= \left[ (3k+1) a - 6 k, (3k+1) a - 4 \right]_2 \cup [(3k+1) a - 2, (3k+1) a], C_{2,k}:= \left[ (3k+2) a - 6 k, (3k+2) a - 4 \right]_2 \cup [(3k+2) a - 2, (3k+2) a],
R_{( 1 )}^{\mathrm{n e a r}}( r ) = C_{\mathrm{i n}}e^{- i \frac{\omega \ell^{2}}{2 r_{H}}\ln \left( \frac{r^{2}- r_{H}^{2}}{r^{2}}\right)}+ C_{\mathrm{o u t}}e^{i \frac{\omega \ell^{2}}{2 r_{H}}\ln \left( \frac{r^{2}- r_{H}^{2}}{r^{2}}\right)}.
\nabla^{2}\varphi_{x}- \int_{- \infty}^{\infty}{\frac{d \alpha}{2 \sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V^{( 1 )}({\frac{\alpha}{2}}\sqrt{J_{1}( g ) - I_{1}( Q^{2})}+ \varphi_{x}) = 0 ,
\begin{array}{l}{f_{( + )}( \theta_{+ k}) = C e^{\chi_{R}( \theta_{+ k}) - i \chi_{J}( \theta_{+ k})},}\\{f_{( - )}( \theta_{- k}) = \left. f_{( + )}^{*}( \theta_{+ k}) \right|_{\theta_{+ k}= \theta_{- k}}= C e^{\chi_{R}( \theta_{- k}) + i \chi_{J}( \theta_{- k})}.}\\ \end{array}
\psi: = Q\phi = (H-E-i0)^{-1}P\phi= (H-E-i\epsilon)^{-1}P\phi + \psi_\epsilon, & ~~where~ \|\psi_\epsilon \| = o(1), (H_\lambda-E)\psi = (H - E - i\epsilon)\psi + \lambda P \psi + i\epsilon \psi &= P\phi + (H - E - i\epsilon)\psi_\epsilon + \lambda P \psi + i\epsilon \psi &= P\phi + (H - E - i\epsilon)\psi_\epsilon + \lambda G(E+i0)\phi + o(1).
\begin{cases}c_{\alpha + n +1}- c_{n+1}= b_{\alpha + n}- b_{\alpha + n +1}d_n = c_{\alpha + 1}+ (b_{\alpha + 1}- b_\alpha) = d b_{\alpha + n}= b_\alpha + (d_1 - c_{\alpha+1}) + \sum_{j=2}^n (c_j - c_{\alpha + j}) b = b_\alpha + d - c_{\alpha + 1}+ \sum_{j=2}^\infty (c_j - c_{\alpha+j})\end{cases}
T(z) = \begin{pmatrix}1 0 0 0 1 i n m_{\frac{1}{2}}0 0 1\end{pmatrix}L^{-1}X(z) \begin{pmatrix}e^{-n g_{1}(z)}0 0 0 e^{n(g_{1}(z)-g_{2}(z))}0 0 0 e^{n g_{2}(z)}\end{pmatrix}L,
\|\Pi_{\partial K}v\|_{\partial K}^2\lesssim\sum_{j=1}^6|\alpha_j^K\!(v)|^2\|\psi_j^K\|_{\partial K}^2\lesssim\frac{|\partial K|}{\xi_K}\|v\|_{\partial K}^2=|\partial\widehat{K}|\|v\|_{\partial K}^2\lesssim\|v\|_{\partial K}^2.
{\cal L}= \frac{1}{2}\partial_{\mu}\varphi \partial^{\mu}\varphi - \frac{1}{2}m^{2}\varphi^{2}- \frac{\lambda}{4 !}\varphi^{4}+ \frac{1}{2}\delta Z ~ \partial_{\mu}\varphi \partial^{\mu}\varphi - \frac{1}{2}\delta m^{2}\varphi^{2}- \frac{1}{4 !}\delta \lambda ~ \varphi^{4}\,
\begin{aligned}d X_i(t) = \left(b_i( Z(t))+ \sqrt{X_i(t)}\sum_{j=1}^{n}f_{ij}( Z(t))h_{ij}(X_j(t))\right)\, dt &\quad+\sqrt{X_i(t)}\sum_{k=1}^{n+m}\sigma_{ik}( Z(t))\, d W_k(t), d Y_l(t) = \left(e_l( Z(t))+\sum_{j=1}^{n}f_{n+l,j}( Z(t)) h_{n+l,j}(X_j(t))\right)\, dt+\sum_{k=1}^{n+m}\sigma_{n+l,k}( Z(t))\, d W_k(t),\end{aligned}
\alpha^1=\left(\begin{array}{ccc}1 0 0 0 1 0 0 0 1\end{array}\right),\ \ \alpha^2=\left(\begin{array}{ccc}0 1 0 0 0 1 1 0 0\end{array}\right),\ \ \alpha^3=\left(\begin{array}{ccc}0 0 1 1 0 0 0 1 0\end{array}\right)
\begin{array}{l}\displaystyle \# \{\sum_{\substack{j\in J\setminus J_{k}}}\Phi(\frac{B_{t_{j}}+B_{t_{j+1}}}{2},t_{j})[B_{t_{j+1}}-B_{t_{j}}]|\forall J_{k}=\{i_1,\cdots,i_k\}\subset J,1\leq k\leq K \}=C_{n}^{K}(2^{K}-1).\end{array}
{J(f,x,\hat{h},\hat{\gamma};T)}=\sup_{h \in{\mathcal{H}}(T)}\inf_{\gamma \in \Gamma(T)}{\frac{-2}{\theta}\log{E[(\frac{V^{h}_{T}}{L^{\gamma}_{T}})^{-\frac{\theta}{2}}]}}=\inf_{\gamma \in \Gamma(T)}\sup_{h \in{\mathcal{H}}(T)}{\frac{-2}{\theta}\log{E[(\frac{V^{h}_{T}}{L^{\gamma}_{T}})^{-\frac{\theta}{2}}]}}
\phi_m(1) = \sum_{\lambda \in E^*}e^{-\beta |\lambda|}\int_{x \in \varprojlim E^{<n_k}}\chi_{Z(s(\lambda), 1)}(x)\,d\varepsilon(x) = \sum_{\lambda \in E^*}e^{-\beta|\lambda|}\varepsilon(Z(s(\lambda), 1)) = \sum_{w \in E^0}m(Z(w,1)) = 1.
Q(t)=\iint u(r,t)dA=Re\{\pi R^{2}\sum_{n=1}^{N}\frac{iP_{n}^{\prime}}{\rho n\omega}[1-\frac{2}{\Lambda_{n}}\frac{J_{1}(\Lambda_{n})}{J_{0}(\Lambda_{n})}]e^{in\omega t}\}.
\|d\nu\|_{{\frak B_{\lambda}}}=\sup\lim_{r\rightarrow1-}c_{\lambda}\left|\int_{-\pi}^{\pi}P(h,re^{i\theta})|\sin\theta|^{2\lambda}d\nu(\theta)\right| &=\sup\lim_{r\rightarrow1-}\left|\int_{-\pi}^{\pi}h(\varphi)P(d\nu,re^{i\varphi})dm_{\lambda}(\varphi)\right|,
H^{\prime}\left( \begin{array}{c}\psi_{+}\psi_{-}\end{array}\right) =\left( \begin{array}{c}P_{-}P_{+}+p_3^2\qquad 0 0\qquad P_{+}P_{-}+p_3^2 \end{array}\right) \left( \begin{array}{c}\psi_{+}\psi_{-}\end{array}\right) .
\det\big(A(e^{\chi})A(e^{-\chi})\big)=\mathbf c(e^{\chi_-},e^{\chi_0})\mathbf c(e^{\chi_0},e^{\chi_+})\det\big(\big\{e^{-A(\chi_+)},e^{A(\chi_-)}\big\}\big)\det\big(A(e^{\chi_0})A(e^{-\chi_0})\big).
\langle \:{\hat \phi}_{\bar t}\prod_{i \in S}\phi_i \:\rangle_{g, \:{\Delta_{1}}}\; = \; \frac{1}{2}\:B_{\bar t}^{' \alpha \beta}\langle \: \phi_{\alpha}\phi_{\beta}\prod_{i \in S}\phi_i \: \rangle_{g-1}
\varphi(a_1,\ldots,a_m)=\left(\prod_{0<i\le l_1}a_{i},\prod_{l_1<i\le l_2}a_i,\ldots,\prod_{l_{n-1}<i\le l_n}a_{i}\right)\textup{with $0<l_1<\cdots<l_n\le m$,}
L_{ij}(v,v)\,dx^i\otimes dx^j:=\nabla^l(\nabla_iv\,dx^i)\cdot\nabla_l(\nabla_jv\,dx^j)+\{\frac{1}{2}e\nabla_iv \cdot\nabla_jv+\frac{1}{2}(R_i{}^k\nabla_jv+R_j{}^k\nabla_iv)\cdot \nabla_kv\}dx^i\otimes dx^j-\Delta v\cdot\nabla_i(\nabla_jv\,dx^j)dx^i .
h_{in}(n)=\frac{\mathbb{E}[k_{in}]}{n-1}\tilde{u}_{in}^{*n}(n-2),n>1,\tilde{u}_{in}=\frac{k_{in}+1}{\mathbb{E}[k_{in}]}\sum_{k_{out}\ge0}u(k_{in}+1,k_{out}),
\left\langle \frac{\partial^{2}V}{\partial \left( \mathrm{R e}F \right)^{2}}\right\rangle = - \left\langle \frac{\partial^{2}V}{\partial \left( \mathrm{I m F}\right)^{2}}\right\rangle = \frac{2}{\Lambda_{Y M}^{4}}\left( \frac{1 1 N_{c}}{1 2}\right)^{2}\ .
\int_{\Lambda^0}g^{m,J}\overline{g^{m',J'}}\, d\tilde{\nu}= \delta_{J, J'}\sum_{w: \lambda_w\in D_{J}}c^{m,J}_{\lambda_w}\overline{c^{m',J}_{\lambda_w}}\rho(\Lambda)^{-J}x^\Lambda_{w}&= \delta_{J, J'}\delta_{m,m'}
A_\varphi \varphi_x(t, x) = \int \left(\frac{1}{|y|}- \frac{1}{\sqrt{y^2+(\varphi(t,x+y)-\varphi(t,x))^2}}\right)\cdot (\varphi_x(t,x + y) - \varphi_x(t,x)) \,dy.
\Vert \delta h\Vert_{\dot{B}^{\frac{N}{2}-1}_{2,1}}= \|h(\sigma_1) -h(\sigma_2)\|_{\dot{B}^{\frac{N}{2}-1}_{2,1}}\lesssim \big(1+ \|(\sigma_1,\sigma_2)\|_{L^\infty_t(\dot{B}^{\frac{N}{2}}_{2,1})}\big)^{[\frac{N}{2}+1]}\Vert \delta \sigma\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}\le C \Vert \delta \sigma\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}.
S = Tr[\frac{m_{1}^{2}}{2}(\phi^{1})^{2}+ \frac{m_{2}^{2}}{2}(\phi^{2})^{2}+ \frac{\lambda_{1}}{4N}(\phi^{1})^{4}+\frac{\lambda_{2}}{4N}(\phi^{2})^{4}+ g \phi^{1}\phi^{2}]
Q_{\pm}=-\frac{|2\alpha\widetilde{N_{\pm}}|^4}{\langle n\rangle_{\pm}}e^{-2|\alpha|^2\sin^2\varphi}\sin^2\varphi\left(\cos^2\varphi\, e^{-2| \alpha|^2\sin^2\varphi}\pm\cos(|\alpha|^2\sin2\varphi+2\varphi)\right).
\int_{B_r(x_0)}\left|\nabla u(x)-\nabla v(x)\right|^p\,dx &\qquad\le C\left(\int_{B_r(x_0)}\left(\left|\nabla u(x)\right|^p-\left|\nabla v(x)\right|^p\right)\,dx\right)^{\frac{p}{2}}\left(\int_{B_r(x_0)}\big(\left|\nabla u(x)\right|+\left|\nabla v(x)\right|\big)^p\,dx\right)^{1-\frac{p}{2}},
\frac{1}{\varepsilon_{k}^{n}}\int_{M}f^{\prime}(W_{\varepsilon_{k},\xi_{k}}(x))u_{k}^{2}(x)d\mu_{g}\leq\frac{1}{\varepsilon_{k}^{n}}\int_{B_{g}(0,r)}f^{\prime}\left( V_{\varepsilon}^{\xi}\left( \exp_{\xi_{k}}^{-1}(x)\right) \right) u_{k}^{2}(x)d\mu_{g}& =C\int_{B(0,r/\varepsilon_{k})}f^{\prime}(V^{\xi}(z))\tilde{u}_{k}^{2}(z)=o(1),
q_0 = 1, q_1 = v, q_2 = 2-u^2-2(u^2-v)p^{-1}+p^{-2}, q_3 = v-2(u+v)(u-v)p^{-1}+vp^{-2}, q_4 = 1-2(u^2-v)p^{-1}+(2-u^2)p^{-2}, q_5 = vp^{-2}, q_6 = p^{-2}.
\Big\{\tilde{\phi}^{T, \Upsilon, Y}[ \tilde{h}_j^Y]: \tilde{h}_j^Y \in Z^a[Y_1, ..., Y_n], \tilde{h}_j^\Upsilon \in Z^a[\Upsilon_1, ..., \Upsilon_n] \Big\}= Z^a[T_1, ..., T_n].
{\cal L}_J =\frac{N}{G}\Biggl[(\partial_\mu z)^\dagger (\partial^\mu z)-J_\mu J^\mu +\theta G\,\epsilon^{\mu\nu\rho}J_\mu \partial_\nu J_\rho+\frac{1}{4}\theta^2 G^2 J_\mu (\partial^2 g^{\mu\nu}-\partial^\mu\partial^\nu) J_\nu \Biggr],
{\cal U}= \left( \begin{matrix}{{\frac 3 2 \delta_{X}{}^{Y}- \frac{1}{W^{2}}{\cal J}_{X}{}^{Z}{\cal L}_{Z}{}^{Y}}}&{{\frac 1{W^{2}}{\cal J}_{X Z}\partial^{y}K^{Z}}}\\{{- \frac 1{W^{2}}( \partial_{x}K^{Z}){\cal J}_{Z}{}^{Y}}}&{{2 \delta_{x}{}^{y}}}\\ \end{matrix}\right) \, ,
e^{{\cal{G}}_{\mathrm{a c}}\{{\phi}\}}= \frac{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_{0}\{\phi^{\prime}\}- S_{\mathrm{i n t}}\{\phi^{\prime}+{\phi}\}}}{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_{0}\{\phi^{\prime}\}}}\; .
\left(e_{r}(\widehat{X}_{\tau(j_{1})}, q\widehat{X}_{\tau(j_{2})}, \ldots , q^{m-1}\widehat{X}_{\tau(j_{m})})f\right)(s_{i}x)=\left(e_{r}(\widehat{X}_{j_{1}}, q\widehat{X}_{j_{2}}, \ldots , q^{m-1}\widehat{X}_{j_{m}})f\right)(x)
v_{max}=(\sqrt{\frac{f^{3}}{27k^{3}}+\frac{H^{2}}{4k^{2}}}+\frac{H}{2k})^{\frac{1}{3}}+(\frac{H}{2k}-\sqrt{\frac{f^{3}}{27k^{3}}+\frac{H^{2}}{4k^{2}}})^{\frac{1}{3}}
\sum_{n \leq N}\pi_{X}(n) R_{X}^{n}= \log N + \gamma+ \log C_{X}- H_{X}& - \sum_{s=1}^{k}\left( \frac{a^{s}}{s}+ \sum_{m=0}^{s-1}\binom{s-1}{m}\frac{a^{m}B_{s-m}\Delta_{X}^{s-m}}{s-m}\right) \frac{1}{N^{s}}+ O \left( \frac{1}{N^{k+1}}\right)
f_{\alpha(i)}(x) - f_{\alpha(i+1)}(x)= xq^{\alpha(i)}f_{v(\alpha(i))}\left(xq^{Nw(\alpha(i))}\right) + dxq^{\alpha(i)}f_{v(\alpha(i))}\left(xq^{N\big(w(\alpha(i))-1 \big)}\right),
(\begin{matrix}\frac{1}{15}&0&0&0&0\\ 0&\frac{1}{15}&0&0&0\\ 0&0&\frac{1}{15}&0&0\\ 0&0&0&-\frac{1}{10}&0\\ 0&0&0&0&-\frac{1}{10}\end{matrix})\in SU(5),\chi/5.
\begin{cases}-\sigma_t - \sigma_x - v_{x}=f (0,T)\times (0,1), -v_{t}- v_{xx}- v_x - \sigma_x = g (0,T)\times (0,1), \sigma(t,0)=\sigma(t,1) t\in(0,T), v(t,0)=v(t,1)=0 t\in(0,T), \sigma(T,x)=0,\ \v(T,x)=0 x\in(0,1),\end{cases}
\Psi (th) :=\int_{\mathbb R^N}\frac{1}{p(x)}(\left\vert \nabla th\right\vert^{p(x)}+V(x)\left\vert th\right\vert^{p(x)})\,dx-\int_{\mathbb R^N}C_{1}\left\vert th\right\vert^{p(x)}[\ln(e+\left\vert th\right\vert )]^{a(x)}dx &\rightarrow -\infty t\rightarrow +\infty .
x^{3}+sx+t = (x-u)(x-ru)(x+(r+1)u)=x^{3}-u^{2}(r^{2}+r+1)x+u^{3}r(r+1) &=:x^{3}+u^{2}s_{2,2}(r)x+u^{3}t_{2,2}(r),
\lim_{i\rightarrow\infty}I_3^{\varepsilon_i}(g)=\lambda\int_{\Omega}u\,{\rm div}\,g\,dx=-2\lambda \int_{M}\Big\langle g,\frac{\nabla u}{|\nabla u|}\Big\rangle\, d\mathcal H^{n-1}+\lambda\int_{\partial\Omega}u\,\langle g,\nu\rangle\, d\mathcal H^{n-1},
Pr(f,\sigma):=\lim_{X\to\infty}\frac{\#Spl_X(f,\sigma)}{\#Spl_X(f)}, Pr(f,\sigma,\{k_j\}):=\lim_{X\to\infty}\frac{\#Spl_X(f,\sigma,\{k_j\})}{\#Spl_X(f,\sigma)}, Pr(f,\sigma,\{k_j\},L,\{R_i\}):=\lim_{X\to\infty}\frac{\#Spl_X(f,\sigma,\{k_j\},L,\{R_i\})}{\#Spl_X(f,\sigma,\{k_j\})},
d s^{2}= \exp \left[ - \frac{4 \sigma}{\sqrt{d - 2}}\right] ( d \tau + \omega_{i}d x^{i})^{2}+ \exp \left[ \frac{4 \sigma}{( d - 3 ) \sqrt{d - 2}}\right] \gamma_{i j}d x^{i}d x^{j},
Z_{0}( k_{s}; l_{1 s}, l_{2 s}) = \frac{( - 1 )^{2 l_{1 s}+ 2 l_{2 s}- 1}( 1 - k_{s})^{2}}{4^{2 l_{1 s}}k_{s}^{3 / 2}[ 1 + ( - 1 )^{2 l_{2 s}}\sqrt k_{s}k_{s}^{l_{1 s}}]^{2 - 2 l_{1 s}}}.
\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}(u_{n}-u_{m})dx \le C\left\|\phi_{u_{n}}\right\|_{6}\left\|u_{n}\right\|_{2}\left\|u_{n}-u_{m}\right\|_{3}&\le C \left\|u_{n}\right\|^{2}\left\|u_{n}\right\|_{2}\left\|u_{n}-u_{m}\right\|_{3}&=o(1).
\Psi_{M-1}^{I,\alpha}(t) =\int_{t_{M-1}}^t \Big[\int_{t_{M-1}}^t\int_\tau^s \lambda_\alpha e^{-\lambda_\alpha(t-u)}dud\tau+\int_t^{t_M}e^{-\lambda_\alpha(t-s)}d\tau\Big]^2ds &+ \int_t^{t_M}\Big[\int_{t_{M-1}}^t e^{-\lambda_\alpha(t-\tau)}d\tau \Big]^2ds :=\Psi_{M-1,1}^{I,\alpha}(t)+\Psi_{M-1,2}^{I,\alpha}(t).
\sup_{Q^{(4)}\cap 2Q_i}\left|\frac{\partial\widetilde{v}}{\partial t}\right|\leq c\mu\lambda_1^{(1-p_0^i)/p_0^i}r_{i,\lambda_1}^{-1}=c\mu\lambda_1^{\frac{2}{p_0^i}-\frac{3}{2}}r_{i}^{-1},
\Gamma_{N P}= \int{\frac{d^{d}k}{( 2 \pi )^{d}}}A_{a}( k ) A_{b}( - k ) \left( -{\frac{g^{2}}{2 \pi^{\frac{d}{2}}}}\Gamma ({\frac{d}{2}}){\frac{\tilde{k}^{a}\tilde{k}^{b}}{| \tilde{k}|^{d}}}\right)
Z_{s}(K,\mu )=\int_{K^{m_{s}(n)}}{\lvert \V(\zeta^{(1)},\ldots ,\zeta^{(m_{s}(n))})\rvert}^{2}d\mu (\zeta^{(1)})d\mu\ldots d\mu (\zeta^{(m_{s}(n))}).
M_{F 4}^{g r a v - m e d .}= 4 \left( \frac{\kappa}{1 2}\right)^{2}\left( T_{F}^{\alpha \beta}( k_{1}, k_{2}) \frac{1}{s}\: G_{\alpha \beta ; \mu \nu}T_{F}^{\mu \nu}( k_{3}, k_{4}) + \mathrm{p e r m .}\right) .
\frac{1}{2}i \psi_{\pm}^{\dagger}\left( \partial_{1}- i e A^{1}\right) \psi_{\pm}- \frac{1}{2}i \left( \partial_{1}+ i e A^{1}\right) \psi_{\pm}^{\dagger}\psi_{\pm}= \pm \frac{1}{4 \pi}\left( \partial_{1}\phi_{\pm}\mp e \lambda A^{1}\right)^{2}\mp \frac{\pi}{1 2 L^{2}}\ .
\lambda\sigma^{1-1/p}\leq\lambda x_{k}^{p-1}=\sum_{i\in\Gamma\left(k\right)}x_{i}\leq\delta^{1-1/p}\left( \sum_{i\in\Gamma\left( k\right)}x_{i}^{p}\right)^{1/p}=\delta^{1-1/p}\left( 1-\sum_{i\notin\Gamma\left(k\right)}x_{i}^{p}\right)^{1/p}& \leq\delta^{1-1/p}\left( 1-\left( n-\delta\right) \sigma\right)^{1/p}.
a . e_0 = e_1 \,,a . e_1 = e_0 \,, \phi(a, e_0) = 1_G \,, \phi(a, e_1) = 1_G \,,\! begin{align*}-1mm]b . e_0 = e_0 \,,b . e_1 = e_1 \,, \phi(b, e_0) = a \,, ~\quad\ \phi(b, e_1) = c \,,~ begin{align*}-1mm]c . e_0 = e_0 \,,c . e_1 = e_1 \,, \phi(c, e_0) = a \,, ~\quad\ \phi(c, e_1) = d \,,~ begin{align*}-1mm]d . e_0 = e_0 \,,d . e_1 = e_1 \,, \phi(d, e_0) = 1_G \,, \phi(d, e_1) = b \,.~
P_{\vert\boldsymbol{Y}_{\mathcal{I}}\vert=n}(\boldsymbol{Y}_{\mathcal{I}}=\boldsymbol{y}_{\mathcal{I}}) = \frac{1}{a_{\vert\boldsymbol{\theta}_{\mathcal{I}}\vert}(n)}\prod_{j\in\mathcal{I}}a_{\theta_j}(y_j) \mathbb{1}_{\Delta_n}(\boldsymbol{y}_{\mathcal{I}}),
\begin{aligned}F_1F_2(\psi)=F_1(\psi) F_2(\psi), \Delta(F)(\psi_1,\psi_2)=F(\psi_1 \psi_2) , &1(\psi)=1, \varepsilon (F)= F(e), S (F)(\psi)= F(\psi^{-1}).\end{aligned}
-{\cal L}_{\rm mass}= \sum_{i=1}^{N_f}{\big[}\mu_{Li}^2|\phi_i|^2 + \mu_{Ri}^2|{\bar \phi}_i|^2 +\mu_i^2(\phi_i{\bar \phi}_i+\phi_i^\ast{\bar \phi}_i^\ast ){\big]}+ m_{\lambda}(\lambda\lambda+\bar{\lambda}\bar{\lambda}).
\begin{array}{lll}\{y_1\}=\{x_{41}x_{42}\}, \{z_1\}=\{x_{41}, x_{42}\}, \{y_2 \}= \emptyset, \{z_2\}=\{x_{41}^2x_{42}+x_{41}x_{42}^2 \}, \{e_2\}=\{x_{41}^2, x_{42}^2 \}, \{y_3\}=\emptyset, \{z_3\}=\emptyset, \{e_3\}=\{x_{41}^2x_{42}^4+x_{41}^4x_{42}^2 \}.\end{array}
\frac{1}{N}\sum_{i=1}^{N}{\mathbf{1}}_{\{X_{t-}^{i}\geq1-\alpha\lambda-4h^{1/16}\}}\geq\frac{1}{N}\sum_{i=1}^{N}{\mathbf{1}}_{B^{i,1}(k)}-\frac{1}{N}\sum_{i=1}^{N}{\mathbf{1}}_{B^{i,2}}\geq\frac{k}{N}- h \geq \lambda,
\pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-s/2}\sum_{n=1}^\infty{n}^{-s}\int_0^\infty\frac{\d{t}}{t}t^{s/2}e^{-{t}}=\int_0^\infty\frac{\d{t}}{t}t^{s/2}\Psi(t)
U_{e}=\frac{1}{\beta}\left[ \frac{e^2}{8}\left(|\sigma_1|^2 - |\sigma_2|^2\right)^2 +\left(\beta \tilde m^2- \frac{1}{2}\alpha e\tilde k\right)|\sigma_1|^2 + \left(\beta \tilde m^2+\frac{1}{2}\alpha e\tilde k\right)|\sigma_2|^2 \right],
U_n(x,p;\lambda)=\begin{pmatrix}1+\lambda\left( e^{p_n}+e^{x_n-x_{n-1}}\right) -\lambda(\lambda-\alpha)e^{x_n-x_{n-1}+p_{n-1}}1 0\end{pmatrix}, \intertext{and further}T_{N}(x,p;\lambda)=U_N(x,p;\lambda)\ldots U_2(x,p;\lambda)U_1(x,p;\lambda).
\sup_{f\in M}\|f\|_{Y_q(\tilde \Omega_{[\hat t(n)]})}\stackrel{(\ref{nu2}), (\ref{emb_nu})}{\underset{\mathfrak{Z}_0}{\lesssim}}2^{((1‐\lambda)\mu_*‐\lambda \alpha_*)k_*\hat t(n)}\stackrel{(\ref{hat_mt})}{=}n^{((1‐\lambda)\mu_*‐\lambda \alpha_*)/\gamma_*}= n^{‐\theta_3}
\sum_{\lambda \in \sigma_{disc}(H)}\frac{\textup{dist}\big( \lambda,\cup_{j=0}^{\infty}\lbrace \Lambda_{j}\rbrace \big)^{p}}{\big( 1 + \vert \lambda \vert \big)^{2p}}\leq C_{1}\Vert F \Vert_{L^{p}}^{p},
U_{24}:=\{A_{1}^{\pm1},A_{2}^{\pm1},A_{3}^{\pm1},B_{1}^{\pm1},B_{2}^{\pm1},B_{3}^{\pm1},A_{1}'^{\pm1},A_{2}'^{\pm1},A_{3}'^{\pm1},B_{1}'^{\pm1},B_{2}'^{\pm1},B_{3}'^{\pm1}\}.
\begin{aligned}\textbf{I}_S\left(\hat{\boldsymbol{\psi}}_{k-1}, \textbf{W}_k\right)^{-1}= \frac{{{\sigma_z^2}}}{{2{{\lvert\textbf{s}\rvert}^2}}}\left\{{{{\widetilde{\bf{I}}}_{i{p_1}}}+{{\widetilde{\bf{I}}}_{i{p_2}}}\left(\hat{\beta}_{k-1}\right)}\right\},\end{aligned}
w = \frac{p_{\Phi}}{\rho_{\Phi}}= \frac{\dot{R}^{2}+ \frac{\Omega^{2}}{a^{6}R^{2}}+ 2 V ( R )}{\dot{R}^{2}+ \frac{\Omega^{2}}{a^{6}R^{2}}- 2 V ( R )}
NL =t\left(\partial_tK(0)+\frac{M^2}t+sc_1+tc_2\right)=t\bar a_{22}, N^2 =N^2(0)+(N^2-N^2(0)), A_{12}=3K_t(0)+(N^2-N^2(s,0))c_3+sc_4+Mc_5+tc_6,
\Delta^{( + )}( x ) = \frac{1}{( 2 \pi )^{3}}\int_{0}^{\infty}\frac{d p^{+}}{2 p^{+}}e^{- i p^{-}x^{+}- i p^{+}x^{-}+ i p^{\bot}x^{\bot}}= \frac{m}{4 \pi^{2}\sqrt{- x^{2}}}K_{1}( m \sqrt{- x^{2}}) ,
\hat{X}_\mu\hat{W}\equiv -\frac{\partial}{\partial p_\nu}\left(\int_0^1 (1-\frac{1}{2}s)e^{-\frac{i(1-s)}{2}\bigtriangleup}F_{\mu\nu}\hat{W}ds+ \hat{W}\int_0^1\frac{1-s}{2}e^{\frac{1}{2}is\bigtriangleup}F_{\mu\nu}ds\right)
U ( 1 6 ) \stackrel{k i n}{\rightarrow}U ( 8 ) \times U ( 8 ) \stackrel{k i n k , 2}{\rightarrow}U ( 8 ) \stackrel{k i n k}{\rightarrow}U ( 4 ) \times U ( 4 ) \stackrel{k i n k , 4}{\rightarrow}U ( 4 ) \stackrel{k i n k}{\rightarrow}U ( 4 )
G_{\alpha}^{( n )}\equiv \sum_{m = 0}^{n}\{\tau_{\alpha}^{( n - m )}, \tilde{H}^{( m )}\}+ \sum_{m = 0}^{n - 2}\{\tau_{\alpha}^{( n - m )}, \tilde{H}^{( m + 2 )}\}_{( \eta )}+ \{\tau_{\alpha}^{( n + 1 )}, \tilde{H}^{( 1 )}\}_{( \eta )}; \hspace{0 . 5 c m}n \geq 2 .
d_{CE}(x_{1}Y_{1}\wedge \dots \wedge x_{k}Y_{k}) = \sum_{i < j}\pm x_{1}Y_{1}\wedge \dots \wedge x_{i}x_{j}[Y_{i}, Y_{j}] \wedge \dots \wedge \widehat{x_{j}Y_{j}}\wedge \dots \wedge x_{k}Y_{k}.
\frac{\partial}{\partial t}\widetilde R(p_{\infty},0)=\lim_{i\rightarrow\infty}\frac{1}{R^{2}(p_{i},0)}\frac{\partial}{\partial t}R(p_{i},0)=\lim_{i_{k}\rightarrow\infty}\frac{1}{R^{2}(p,-t_{i_{k}})}\frac{\partial}{\partial t}R(p,-t_{i}).
|I_1| \le \frac{\Delta_n^2 \, n}{T_n^2}\sum_{j = 0}^{j_{\delta_1}}\frac{c}{\prod_{l = 1}^d h_l}= c \frac{\Delta_n^2 \, n}{T_n^2}\frac{1}{\prod_{l = 1}^d h_l}(j_{\delta_1}+ 1).
\nabla_x = \left(\frac{\partial}{\partial x_1}, \, \frac{\partial}{\partial x_2}\right) = \left( \cos \theta \frac{\partial}{\partial r}- \frac{1}{r}\sin \theta \frac{\partial}{\partial \theta}, \, \sin \theta \frac{\partial}{\partial r}+ \frac{1}{r}\cos \theta \frac{\partial}{\partial \theta}\right),
\begin{array}{lllll}\partial\phi([\mathbf{a}, \mathbf{a'}]) = \partial\left([\mathbf{a}]\otimes (s^0)^{k+1}[\mathbf{a'}]\right) &= \partial[\mathbf{a}]\otimes (s^0)^{k+1}[\mathbf{a'}] + (-1)^{|\mathbf a|+1}[\mathbf{a}]\otimes \partial (s^0)^{k+1}[\mathbf{a'}] = \phi(\partial([\mathbf{a}, \mathbf{a'}])) \end{array}
\tilde{\lambda}(\xi) =\begin{cases}e^{-1/\xi}, \xi>0, 0, \xi\leq 0, \end{cases}\lambda(\xi) =\frac{\tilde{\lambda}(\xi)}{\tilde{\lambda}(\xi)+\tilde{\lambda}(1-\xi)}, \lambda_\epsilon(\xi) =\lambda\left(\frac{\xi}{\epsilon}\right)
T=\int_{(r_{1},\theta_{1})}^{(r_{2},\theta_{2})}\frac{n(r)}{c}\sqrt{(rd\theta)^{2}+dr^{2}}=\frac{1}{c}\int_{\theta_{1}}^{\theta_{2}}n(r)\sqrt{r^{2}+(\frac{dr}{d\theta})^{2}}d\theta,
\int_s^t \|H(\tau) U(\tau,s) x\|^p d\tau &=\int_s^t \|H(\tau) U(\tau,s)P(s) x + H(\tau) U(\tau,s)Q(s) x\|^p d\tau &\leq 2^{p-1}\int_s^{t}\|H(\tau) U(\tau,s) P(s) x\|^p d\tau+2^{p-1}\int_s^{t}\|H(\tau) U(\tau,s) Q(s) x\|^p d\tau.
|E|\le\sum_{i,i'= k+2}^m\sum_{j,j'\ge 1}\frac{n^{i+i'}jj'(C_{i,j}r_n)^{d(i+i'-1)}}{i!i'!}2\|f\|_\infty^{i+i'-1}\theta_d^{i+i'-1}\le C(f,d,k)(nr_n^d)^{k+2}(n^{k+2}r_n^{d(k+1)}),
S_{m m^{\prime}, n n^{\prime}}^{R}= \frac{1}{u}( - 1 )^{G^{\prime}+ m + n + u ( u - n - 1 )}e^{- i \pi ( u + 1 ) m ( u - n - 1 ) / u}e^{- i \pi ( u - 1 ) ( m - 2 m^{\prime}+ u ) ( n - 2 n^{\prime}+ u ) / u};
\begin{array}[c]{l}Ps_{n}^{m}\left({x,\gamma^{2}}\right) =c_{n}^{m}\left( \gamma\right)\left\{{\dfrac{\eta}{\left({x^{2}-1}\right) \left({x^{2}-\sigma^{2}}\right)}}\right\}^{1/4}\times\left[{J_{m}\left({\gamma\eta^{1/2}}\right) +{O}\left({\gamma^{-1}}\right) \operatorname{env}J_{m}\left({\gamma\eta^{1/2}}\right)}\right] ,\end{array}
\mathcal{V}^{(n,c)}_{\mu}(\xi ) = \sum_{1\le j < k \le n}\int_0^{\xi_j-\xi_k}v() + \sum_{1\leq j\leq n}\left({\textstyle \frac{c}{2}}\xi_j^2-2\pi ( \rho_j+\mu_j)\xi_j \right) ,
{\cal W}\,(\phi,\widetilde \phi) = \frac{m^2}{4\lambda}\,\phi-\frac{\lambda}{3}\,\phi^3 +\lambda\,\phi\,\widetilde\phi^{\,2}+\frac{p\,m}{2}\,\widetilde\phi^{\,2}+\frac{q\lambda}{3}\,\widetilde\phi^{\,3}\,,
\mathbf{v}_{m}^{\left(i\right)}=\left[\sum_{l\in\mathcal{S}^{\left(i\right)}}a_{ml}\left(\mathbf{t}_{l}-Q_{\Lambda_{s,l}}\left(\mathbf{t}_{l}-\mathbf{d}_{l}\right)\right)\right]\bmod\Lambda_{e,m}
C^2 2\pi\frac{2^{2\beta}}{(z_0(1+2\beta))^{2\beta}}\int_0^\infty dz z^{2\beta+1}e^{-\frac{2z}{z_0 (1+2\beta)}}=1{\rm and}C=\sqrt{\frac{2}{\pi\Gamma(2\beta+2)z_0^2(1+2\beta)^2}}.