harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_2f_3 = s s_2 x (x-1) g_2 f_1f_3 = s s_1 x (x-1) g_1 f_1f_2/ (x-1)^2 = s_1 s_2 g_1 g_2 f_1f_2f_3/(x-1)^2 = s s_1 s_2 x g_1 g_2.
Q = \alpha^{2}x \left[ \left( r + \frac{\pi}{2}\frac{1}{\sqrt{x}}\right)^{2}+ \frac{\bar{c}^{2}}{x}\right] , \quad R = \frac{\alpha^{2}}{x}\, F \! \left( \frac{1}{\sqrt{x}}; r \right) , \quad{\bf S}^{2}= \frac{\alpha^{4}}{x}\, G \! \left( \frac{1}{\sqrt{x}}; r \right) ,
\chi^m(a)\int_{\mathcal{O}_K^{\times}}{\chi(ac (x^m))|dx|}=\chi(ac (a^m))\int_{\mathcal{O}_K^{\times}}{\chi(ac (x^m))|dx|}&=\int_{\mathcal{O}_K^{\times}}{\chi(ac ((ax)^m))|dx|}&=\int_{\mathcal{O}_K^{\times}}{\chi(ac (x^m))|dx|}.
p_{w}'(\omega) =- \frac{1}{\pi}\int_{-\alpha}^{\alpha}x\, v(x) \cos(\omega\, x)\, \mathrm{d}x\geq 0 (0 \leq \omega \leq \pi/(2\alpha)), p_{w}''(\omega) =+ \frac{1}{\pi}\int_{-\alpha}^{\alpha}x^{2}v(x) \sin(\omega\, x)\, \mathrm{d}x\leq 0 (0 \leq \omega \leq \pi/\alpha).
\||x'|^{\alpha}v(\cdot, x_n)\|_{L^s(B_1')}\le C\||x'|^{\alpha}v(\cdot, x_n)\|_{L^s(B_2'\setminus B_1')}+C\||x'|^{\mu}\nabla'v(\cdot, x_n)\|_{L^1(B_2')}^{ab}\||x'|^{\hat{\gamma}_3}v(\cdot, x_n)\|_{L^{\hat{q}}(B_1')}^{1-ab}.
\begin{cases}\overset \cdot b_0 = b_0 (b_1+b_2), \overset \cdot b_1 = b_1 (b_2 + b_3 - b_1), \overset \cdot b_2 = b_2 (b_3 - b_1 - b_0), \overset \cdot b_3 = b_3 (-b_2 - b_1).\end{cases}
G(t) := 4\pi^2 \, \varepsilon(t)^3 \, \frac{d\varepsilon}{dt}(t) - 32\pi^2 \, \omega \, \varepsilon(t)^8 + \int_{[-\frac{1}{2},\frac{1}{2}]^4 \setminus \{0\}}\langle \psi(t),\bar{o}_{1,\varepsilon(t),\delta(t)}\rangle_{\bar{g}_{\varepsilon(t),\delta(t)}}\, d_{\bar{g}_{\varepsilon(t),\delta(t)}}
f_y(x)-f_y(a)-T(a)(x-a) =\varphi(\delta(x)\otimes y)-\varphi(\delta(a)\otimes y)-\varphi(Dg_U(a)(x-a)\otimes y) &=\varphi\left((\delta(x)-\delta(a)-Dg_U(a)(x-a))\otimes y\right),
u^{(s)}(\vec{p})=\frac{p/+m}{\sqrt{2m(E+m)}}u^{(s)}(\vec{0})=\sqrt{\frac{E+m}{2m}}[\begin{matrix}\phi^{(s)}\\ \frac{\vec{\sigma}\cdot\vec{p}}{E+m}\phi^{(s)}\end{matrix}]
{\frac{1}{2}}R_{a b , c d}^{( + )}\lambda_{+}{}^{a}\lambda_{+}{}^{b}\lambda_{-}{}^{c}\lambda_{-}{}^{d}={\frac{1}{4}}R_{a b , c d}^{( + )}\lambda_{+}{}^{a}\lambda_{+}{}^{b}\lambda_{-}{}^{c}\lambda_{-}{}^{d}+{\frac{1}{4}}R_{a b , c d}^{( - )}\lambda_{-}{}^{a}\lambda_{-}{}^{b}\lambda_{+}{}^{c}\lambda_{+}{}^{d}\ .
\omega_\mathbb{A}\left[\begin{pmatrix}I T 0 -I\end{pmatrix}\right]=\frac{1}{2}\left(\left\|\begin{pmatrix}I T 0 -I\end{pmatrix}\right\|_\mathbb{A}+\left\|\begin{pmatrix}I T 0 -I\end{pmatrix}\right\|_\mathbb{A}^{-1}\right).
J^w(\textbf{X}_{minRSSU}^{(n)}) = -\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}w(F^{-1}(u)) (1-u)^{2i-2}f(F^{-1}(u))du\right) & \geq -\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}w(F^{-1}(u)) f(F^{-1}(u))du\right) &= (n!)^2 J^w(\textbf{X}_{SRS}^{(n)}).
\varsigma_i^2 =-1 \ \ \ (1\leq i\leq n-1), (\varsigma_i\varsigma_j)^2 =-1\ \ \ (|i-j|\geq 2), (\varsigma_i\varsigma_{i+1})^3 =-1\ \ \ ( 1\leq i\leq n-2).
\mathbf{P}\left( \mathbf{X}_{t_1}<\mathbf{x}_1,\ldots,\mathbf{X}_{t_p}<\mathbf{x}_p \right) = C^{t_1,\ldots,t_p}\left( \mathbf{F}_{t_1}(\mathbf{x}_1),\ldots,\mathbf{F}_{t_p}(\mathbf{x}_p) \right).
a_j = (-1)^j \frac{\ell! (\ell+2n)!}{(\ell+j)! (\ell+2n-j)!},b_j = (-1)^j \frac{n-\frac{1}{2}-j}{n+\frac{1}{2}}\frac{p \ell! (\ell+2n+1)!}{(\ell+j+1)! (\ell+2n-j)!}.
( U_h \mapsto U_j = U^{(1)}\mapsto \ldots \mapsto U^{(K)}, &((a_h, a_j), (a_h^{(1)}, a_j^{(1)}), \ldots, (a_h^{(K - 1)}, a_j^{(K - 1)})), &(b, b^{\prime}= b^{(1)}, \ldots, b^{(K)}))
\left( P^{2}- m^{2}- \frac{e}{2}\sigma^{\mu \nu}F_{\mu \nu}\right) \phi ( x ) = 0 \; , F_{\mu \nu}= \partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}\; , \; \sigma^{\mu \nu}= \frac{i}{2}[ \gamma^{\mu}, \gamma^{\nu}] \; .
\mathrm{KL}(A,B) =\sum_{j=1}^N q'(z_i=1) \sum_{i=1}^M (A_0B_0)_{ij}\log \frac{(A_0B_0)_{ij}}{(AB)_{ij}}&\sim \sum_{j=1}^N q'(z_i=1) \sum_{i=1}^M ((AB)_{ij}- (A_0B_0)_{ij})^2 &\sim \sum_{j=1}^N \sum_{i=1}^M ((AB)_{ij}- (A_0B_0)_{ij})^2 = \|AB-A_0B_0\|^2.
X_{i}^{-}V_{\vec{\lambda}, \vec{m}}( z ) = \frac{1}{1 - q_{i}^{- 1}}\int_{G}d t J_{i}( t ) V_{\vec{\lambda}, \vec{m}}( z ) \sim Q_{i}( \infty , z ) V_{\vec{\lambda}, \vec{m}}( z ) ,
d_{\rm Config}(\mathcal{C}_{1}, C_{2}) = \sum_{k=1}^{\infty}\frac{1}{2^{k}}\sup_{f \in{\rm Lip}_{1}(M)}\frac{\int_{\square_{K}}f d \left( \mathcal{C}_{1}- \mathcal{C}_{2}\right)}{|\mathcal{C}_{1}|(\square_{k}) + |\mathcal{C}_{2}|(\square_{k})},
\begin{array}{lll}\tilde{A}^{(r)}:= [\tilde a_{ij}]_{i,j=1,\ldots,n}= (A^{(r)}+\hat{C}\Tilde{K}^{(r)}) =\begin{bmatrix}\hat{c}_{11}\Tilde{k}_{11}\ldots a_{1n}+\hat{c}_{11}\Tilde{k}_{1n}\vdots \ddots \vdots a_{n1}+\hat{c}_{nn}\Tilde{k}_{n1}\ldots \hat{c}_{nn}\Tilde{k}_{nn}\end{bmatrix}.\end{array}
\langle f, g\circ\Phi^n\rangle = V_0\int_{B_{1}}f\circ \varphi_{\lambda^{-n}_0s}(p)\cdot g \circ \varphi_{s}(p)\, ds + \sum_{i=2}^{d^+}\bar{c}_{i,n}(f,g)\left(\frac{\lambda_i}{\lambda_1}\right)^n + \mathcal{O}(\lambda^{-n}_0)
[\begin{matrix}e+i\overline{e}+jv+k\overline{v}\\u_{r}+i\overline{u}_{\overline{r}}+jd_{r}+k\overline{d}_{\overline{r}}\\u_{g}+i\overline{u}_{\overline{g}}+jd_{g}+k\overline{d}_{\overline{g}}\\u_{b}+i\overline{u}_{\overline{b}}+jd_{b}+k\overline{d}_{\overline{b}}\\ \end{matrix}]_{L}
A_{n}(x,\overline{h})=\left( \frac{x_{N}}{|\overline{h}|}\right)^{\omega\gamma^{\prime}}\frac{|v_{n}(x+\overline{h})-v_{n}(x)|}{|\overline{h}|^{(1-\omega)\gamma^{\prime}}}\leq\varepsilon^{-\omega\gamma^{\prime}}\left\langle u_{n}-u_{0}\right\rangle_{K}^{((1-\omega)\gamma^{\prime})}.
\begin{cases}m_{d+1,1^{k-1}}(x_1,x_2,\ldots,x_i) x_{i+1}+ m_{d,1^k}(x_1,x_2,\ldots,x_i) x_{i+1}(d\ge2), m_{2,1^{k-1}}(x_1,x_2,\ldots,x_i) x_{i+1}+ (k+1)m_{1^{k+1}}(x_1,x_2,\ldots,x_i) x_{i+1}(d=1). \end{cases}
u_{\alpha}(\theta, x) \ = \ \inf_{v(\cdot) \in{\mathcal A}}E^v_x \Big[ e^{\theta \int^{\infty}_0 e^{- \alpha s}r(X_s, v_s) ds}\Big]\ = \E^v_x \Big[ e^{\theta \int^{\infty}_0 e^{- \alpha s}r(X_s, v_{\alpha}(X_s)) ds}\Big] \, .
\frac{q^{2j\{n,n'\}}-1}{q^{2ja}-1}= \frac{q^{2\alpha j/sd(n)}-1}{q^{2j/sd(n)}-1}= \begin{cases}\displaystyle\sum_{p=0}^{\alpha-1}q^{2jp/sd(n)}\alpha> 0, 0 \alpha=0, \displaystyle-\sum_{p=1}^{\alpha}q^{-2jp/sd(n)}\alpha< 0.\end{cases}
(z(\varphi)\Psi)^{\pmb{b_k}}_n (\pmb{\theta}) = \sqrt{n+1}\sum_{l=1,2}\int d\theta' \overline{\varphi^{b_l}(\theta')}\Psi^{b_l \pmb{b_k}}_{n+1}(\theta',\pmb{\theta}), z^\dagger(\varphi) = (z(\varphi))^*
\| \nabla_{X,z}^{\mu}u \|_{H^{k,0}(\mathcal{S}_b)}\leq M(k+1)(\| \nabla_{X,z}^{\mu}\tilde u \|_{H^{k,0}(\mathcal{S})}+ \sum \limits_{j=0}^{k}\| \partial_z^j \tilde u\|_{H^{k,0}(\mathcal{S})}) \leq M(k+1)(|g|_{H^k}+ \sum \limits_{j=0}^{k}\|\partial_z^jf\|_{H^{k-j,0}(\mathcal{S}_b)}).
\|Df - Dg\|_{L^p(\Omega)}\; =\; \|D(f|_{\mathcal{O}}) -D\phi\|_{L^p(\mathcal{O})}\leq \|D(f|_{\mathcal{O}})\|_{L^p(\mathcal{O})}+ \|D\phi\|_{L^p(\mathcal{O})}\leq \; c_p^{\frac{1}{p}}\left[\mathcal{E}_p(f|_{\mathcal{O}})^{\frac{1}{p}}+ \mathcal{E}_p(\phi)^{\frac{1}{p}}\right] \ .
\det \begin{pmatrix}\langle e_\xi, d\Phi_0(\partial_x)\rangle \langle e_\xi, d\Phi_0(\partial_y)\rangle \langle e_\zeta, d\Phi_0(\partial_x)\rangle \langle e_\zeta, d\Phi_0(\partial_y)\rangle \end{pmatrix}(p)=0.
PAP^{-1}=(\begin{matrix}O&A_{1}&O&O&...&O\\O&O&A_{2}&O&...&O\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\O&O&O&O&...&A_{h-1}\\A_{h}&O&O&O&...&O\end{matrix}),
f(x,y,\theta):=E_{LJ}(L) &=\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(x^2m^2+y^2n^2+2xymn\cos\theta)^6}-\frac{2}{(x^2m^2+y^2n^2+2xymn\cos\theta)^3}\right).
\frac{\partial F ( m, n )}{\partial n}= \sum_{j=1}^m \left\{\frac{| \mathcal{A}_j |}{n}- \frac{|s_j| - | \mathcal{A}_j |}{N - n}\right\}+ & + \log \alpha - \sum_{h=1}^n \frac{1}{h}+ \gamma = 0,
u \left( x \right) = \left( \begin{array}{c}{{e^{- i \varphi}u_{1}\left( r , t \right)}}\\{{u_{- 1}( r , t )}}\\ \end{array}\right) \exp i \left[ \left( l - l_{0}\right) \varphi - \frac{m^{2}}{2}\int \frac{d u^{0}}{\lambda + g \left( u^{0}\right)}- \frac{\lambda u^{3}}{2}\right] \, ,
a(u, v) = \int_{\cal O}\Big\{\sum^n_{i,\,j=1}a_{ij}(x)\frac{\partial u}{\partial x_i}\frac{{\partial v}}{\partial x_j}+ \sum^n_{i=1}b_i(x)\frac{\partial u}{\partial x_i}v + c(x)uv\Big\}dx,\,\,\,\,x\in{\cal O}.
{\rm d}\left(e^{-\int_0^t \psi(s)\,{\rm d}s}\|V(t)\|^2_{L^2}\right) =-\psi(t)e^{-\int_0^t \psi(s)\,{\rm d}s}\|V(t)\|^2_{L^2}{\rm d}t &+e^{-\int_0^t \psi(s)\,{\rm d}s}{\rm d}\|V(t)\|^2_{L^2},
\MoveEqLeft \langle A(t)e^{-tA(t)}x - A(0)e^{-tA(0)}x, x' \rangle & = \frac{1}{2\pi i}\int_{\Gamma}ze^{-tz}\langle R(z,A_{-1}(t)) [A_{-1}(t) - A(0)] R(z,A(0))x, x' \rangle \d z & = \frac{1}{2\pi i}\int_{\Gamma}ze^{-tz}\langle (A_{-1}(t) - A(0)) R(z,A(0))x, R(z,A(t)')x' \rangle \d z.
e_n\left(x+\dfrac{f_n}{e_n}\right)\mathcal{L}_{n+1}(x) =(p_nx^2+q_nx+r_n)\mathcal{L}_n(x) +t_n(x^2+\omega^2)\left(x+\dfrac{u_n}{t_n}\right)\mathcal{L}_{n-1}(x), n\geq 1.
\|\Psi([\eta])\|^{2}=\langle\Im z,\Im z\rangle^{w}|\Psi(z)|^{2},\|\Psi([\bar{\eta}])\|^{2}=\langle\Im(-\bar{z}),\Im(-\bar{z})\rangle^{w}|\Psi(-\bar{z})|^{2}.
\int_{\mathbb{R}\backslash\{0\}}(e^{i\theta x}-1-i\theta x1_{|x|<1})\Pi(dx)=\Pi(\mathbb{R}\backslash(-1,1))\int_{\mathbb{R}}(e^{i\theta x}-1)\nu(dx)+\int_{\mathbb{R}}(e^{i\theta x}-1-i\theta x)\mu(dx).
\mathrm{R e}\, \chi_{T}^{\mathrm{D}}( \omega ) = - \frac{\hbar A}{\pi^{2}}\int \int_{R_{4}}d \omega_{\mathrm{i n}}d k_{\|}k_{\|}\sqrt{k_{\|}^{2}- ( \omega + \omega_{\mathrm{i n}})^{2}}\sqrt{\omega_{\mathrm{i n}}^{2}- k_{\|}^{2}}\;{\overline{{n}}}( - \omega_{\mathrm{i n}})
h_1^\ell(X^\ell)=\frac{1}{m^\ell}\sum_{k=1}^{m^\ell}S_0\exp\left(\left(r-\frac{\sigma^2}{2}\right)k\frac{T}{m^\ell}+\sigma\sqrt{\frac{T}{m^\ell}}\sum_{i=1}^{k}X_i^\ell\right)
X(t) = \begin{cases}x (-r \le t \le 0) \displaystyle x + \int_0^t A_0 \big( \Pi( X_s) \big) \, X(s) \, \mbox{d}s + \int_0^t A_1 \big( \Pi (X_s) \big) \, X(s) \, \mbox{d}W(s) (0 < t \le T), \end{cases}
S_{b o u n d}= - \frac{1}{\kappa_{5}^{2}}\int d^{5}x ( \delta_{x_{5}}- \delta_{x_{5}- R}) ( \sqrt{- g_{4}}\frac{3}{2}W + \frac{2 g}{4 !}\epsilon^{\mu \nu \rho \sigma}A_{\mu \nu \rho \sigma}) .
\sum_{j=1}^l \left\Vert a_j\right\Vert^n<\left\Vert a\right\Vert_{\mathcal{P}_n}+\varepsilon/2, \sum_{j=1}^m \left\Vert b_j\right\Vert^n<\left\Vert b\right\Vert_{\mathcal{P}_n}+\varepsilon/2.
u_1 = \ \frac{1}{2}( d_{112}-1,d_{212}), u_2 = \u_1 \ + \ \left(1,0\right), u_3 = \ \frac{1}{2}( -1,0 ), u_4 = \u_3 \ + \ \left(1,0\right), u_5 = \ \frac{1}{3}(-d_{112}-1,-d_{212}), u_6 = \u_5 \ + \ \left(\frac{2}{3},0\right).
\gamma_1= \frac{1}{2}\sum_{i=0}^{n_1}\sum_{j=0}^{n_2}|a_{i,j}| i(i-1), \gamma_2 = \frac{1}{2}\sum_{i=0}^{n_1}\sum_{j=0}^{n_2}|a_{i,j}| j(j-1).
\mathcal{M}_{3 / 2}^{\mathcal{N}= 3}= \frac{\mathrm{S U}( 1 , 1 )}{\mathrm{U}( 1 )}\times \frac{\mathrm{U S p}( 3 , 3 )}{\mathrm{S U}( 3 ) \times \mathrm{U}( 1 )}.
M_{\mathbf{X}}(t,\mathbf{z}) :=M_{\mathbf{X}(t)}(\mathbf{z}), \mathfrak{M}_{\mathbf{X}}(t, \mathbf{z}) := \mathbf{z}M_{\mathbf{X}}(t,\mathbf{z})= (z_1 M_{\mathbf{X}}(t,\mathbf{z}),\ldots,z_r M_{\mathbf{X}}(t,\mathbf{z})).
\frac{\partial}{\partial s}G(\alpha,\beta)= G_1(\alpha,\beta)+O\left(\frac{X^4|m^{\ast}\tilde{m}|}{(hd)^{1/2}|u|^{1/4}s^{5/2}}+\frac{X^8|u|^{1/4}|m^{\ast}\tilde{m}|}{(hd)^{3/2}s^{7/2}}\right).
\|Tf\|_{\mathcal{Y}}=\|g\|_{\mathcal{Y}}\leq R = \|T\|_{\overline{\mathcal{P}}^{\mathcal{X}}\to \mathcal{Y}}\sup_{0<r<1}\|T_r\|_{\mathcal{X}\to \mathcal{X}}\|f\|_{\mathcal{X}},
P(d_1,d_2,\dots,d_n) =\sum_{i=2}^n\min\{d_1,d_i\}+\sum_{i=2}^{n-1}\min\{d_i,d_n\}+P(d_2,\dots,d_{n-1}) &=(n-1)d_1+\sum_{i=2}^{n-1}d_i+P(d_2,\dots,d_{n-1}).
D^{2}[ \bar{\rho}] = - \delta_{\mu \nu}\nabla_{\alpha}( A ( x , [ \bar{\rho}] ) ) \nabla^{\alpha}( A ( x , [ \bar{\rho}] ) ) + i \left[{\bar{F}}_{\mu \nu}^{( 3 )}( x , [ \bar{\rho}] ) , ~ ~ \right]
\begin{array}{lll}\alpha_{ik}= 1\otimes \cdots \otimes 1\otimes \alpha_{i}\otimes1 \otimes \cdots\otimes 1\in H^{1}((M)^{N},{\bf{Z}}) \beta_{k}= 1\otimes \cdots \otimes 1\otimes \beta \otimes 1\otimes\cdots \otimes 1\in H^{2}((M)^{N},{\bf{Z}}),\end{array}
\varrho^j := h f(V^j) + \frac{3}{2}g(V^{j-1}) \Delta W^j - \frac{1}{2}g(V^{j-2}) \Delta W^{j-1}- \frac{3}{2}V^{j}+ 2 V^{j-1}- \frac{1}{2}V^{j-2},
\| \mathcal{R}(\mathbf{k}, \varepsilon)^{s/2}(I - \widehat{P}) \|_{L_2(\Omega) \to L_2 (\Omega)}= \sup_{0 \ne \mathbf{b}\in \widetilde{\Gamma}}\varepsilon^s (|\mathbf{b}+ \mathbf{k}|^2 + \varepsilon^2)^{-s/2}\le r_0^{-s}\varepsilon^s,\varepsilon > 0,\ \mathbf{k}\in \widetilde{\Omega}.
\sum_{(i,j)\in B}m_G(i, j)^2\geq \frac{(\sum_{(i,j)\in B}m_G(i,j))^2}{|B|}=\frac{(\sum_{i<j}m_G(i,j))^2}{|B|}\geq\frac{(\sum_{i<j}m_G(i,j))^2}{8b^2n/a}.
\sum_{k=0}^{\infty}P(k)x^{k}=\frac{1}{(x;x)_{\infty}}=\theta_{3}(x)^{-1/6}\theta_{4}(x)^{-2/3}[\frac{\theta_{3}(x)^{4}-\theta_{4}(x)^{4}}{16x}]^{-1/24}
\frac{\tilde{\beta}_{t+\tau}}{\tilde{\beta}_{t'+\tau}}-\frac{\tilde{\beta}_{t+\tau'}}{\tilde{\beta}_{t'+\tau'}}=\frac{\tilde{\beta}_{t+\tau'}}{\tilde{\beta}_{t'+\tau}}\left(\frac{\tilde{\beta}_{t+\tau}}{\tilde{\beta}_{t+\tau'}}-\frac{\tilde{\beta}_{t'+\tau}}{\tilde{\beta}_{t'+\tau'}}\right)=0,
\begin{cases}-\partial_{tt}v_{s}+ \partial_{rr}v_{s}+ \frac{1}{r}\partial_{r}v_{s}-\frac{v_{s}}{r^{2}}= 0 v_{s}(s,r)=0 \partial_{t}v_{s}(s,r) = -2 \lambda''(s) \frac{r}{1+r^{2}}\end{cases}
T_xM^G =f(U^G)=f(M^G\cap U)=f(M^{H_1}\cap M^{H_2}\cap U) &=f((M^{H_1}\cap U)\cap(M^{H_2}\cap U))=f(U^{H_1}\cap U^{H_2}) &=f(U^{H_1})\cap f(U^{H_2})=T_xM^{H_1}\cap T_xM^{H_2}.
\lim_{j\to\infty}\frac{1}{j}\log\big(\|S(j)({\xi}_{[(\sigma,k,l)]})\|_{L^1(I^{(j)})}/|I^{(j)}|\big)=-\lambda_1(n_{\sigma,k}-a_{\sigma,k})=-\lambda_1\mathfrak{o}(\sigma,k) &\lim_{j\to\infty}\frac{1}{j}\log\|S(j)({\xi}_{[(\sigma,k,l)]})\|_{\sup}=-\lambda_1\mathfrak{o}(\sigma,k)\mathfrak{o}(\sigma,k)>0.
\frac{d^2}{dt^2}\lambda(h,K) =\int_M \Big\langle \frac{1}{2}\triangle h+\mathring{R}(h),h \Big\rangle-\frac{1}{6}|dK|^2 dV_g=\int_M \Big\langle \frac{1}{2}\triangle_L h,h \Big\rangle-\frac{1}{6}|dK|^2 dV_g.
\frac{\tilde M_n(2)}{\delta_n(2)}+\frac{\tilde M_n(3)}{\delta_n(3)}= \delta^{-1}_n(3)\cdot \sum_{\ell=1}^{d}\sum_{q=1}^{\ell-1}\left[\frac{\delta_n(3)}{\delta_n(2)}\cdot \tilde M_{n,\{q,\ell \}}+\sum_{p=1}^{q-1}\tilde M_{n,\{p,q,\ell \}}\right].
\gamma_-(su^++u^-) =I'(u^-)[u^-]+s^2\lambda\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx \leq I'(u^-)[u^-]+\lambda\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx=\gamma_-(u)\leq0.
\frac{1}{4!}\int_{{\Lambda}}\frac{d^4\vec{p}_1}{(2{\pi})^4}\frac{d^4\vec{p}_2}{(2{\pi})^4}\frac{d^4\vec{p}_3}{(2{\pi})^4}\frac{d^4\vec{p}_5}{(2{\pi})^4}{\delta}{\lambda}_4(1235){\delta}^4(\vec{p}_1 + \vec{p}_2+\vec{p}_3+\vec{p}_5) \hat{\phi}_2(\vec{p}_1) \hat{\phi}_2(\vec{p}_2) \hat{\phi}_2(\vec{p}_3) \hat{\phi}_2(\vec{p}_5).
C_6(\gamma,\beta) =(1+\beta)\{48[1\vee \gamma(1+\beta)]+ 80[1\vee \gamma^2(1+\beta)^2]+ 70[1\vee \gamma^3(1+\beta)^3]\}&+C_4(\gamma,\beta) (1+\beta) \{6+12 [1\vee \gamma(1+\beta)]+10 [1\vee \gamma^2(1+\beta)^2] \}&+C_5(\gamma,\beta)(1+\beta) \{2+2[1\vee \gamma (1+\beta)]\}.
\overline{\nabla}_a \eta^I e_I = \overline{\nabla}_{\xi^J e_J}\eta^I e_I + \dot{\eta}^I e_I= \big[ \xi^J e_J , \eta^I e_I \big] + \dot{\eta}^I e_I = - C_{I J}^K \xi^J \eta^I e_K + \dot{\eta}^I e_I ,
h''(x) = -x^{-2}\log(e^3/nx^2)^{-3/2}+ x^{-1}\cdot 3x^{-1}\log(e^3/nx^2)^{-5/2}&= x^{-2}\log(e^3/nx^2)^{-3/2}\left(-1 + 3 \log(e^3/nx^2)^{-1}\right) < 0,
F_{k,a}(x,y) =(2\pi)^{-d_1}|a|^{d_1}\int_{\mathbb{R}^{d_1}}e^{i\frac{a}{2}\xi\cdot(x+y)}\varphi_{k,a}^{d_1-1}(\xi,x-y)\,d\xi =(2\pi)^{-d_1}|a|^{d_1}\int_{\mathbb{R}^{d_1}}e^{i\frac{a}{2}\xi\cdot(x+y)}L_k^{d_1-1}\left(\frac{|a|}{2}(|\xi|^2+|x-y|^2)\right)e^{-\frac{|a|}{4}(|\xi|^2+|x-y|^2)}\,d\xi
\langle J^{\mu}_f \rangle_{reg}=- \lim_{x \rightarrow y}Tr\left\{\gamma^\mu(x)\frac{1}{2}\left(f({{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax}{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax}^\dagger/\Lambda^2)-\gamma^5\right)\frac{1}{i{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax}}\frac{1}{2}(1-\sigma^3)\delta(x-y)\right\}.
\int_{M_4}\,dvol_{{\eta}_4}\,\langle\psi|\psi\rangle=\,\int_{M_4}\,dvol_{{\eta}_4}\,\frac{1}{N}\,\sum^N_{k=1}\int_{\varphi^{-1}_{k*}(T_xM_4)}\,d^4z_k\,C_\psi\,n^2_k(\varphi^{-1}_{k}(x),z_k)=\,1,
\bar{N}^K_{kG}(Q) =\bar{N}_{kN}^{K}(Q)\bigoplus\left(\bigoplus_{\substack{z\in [N_G^K(Q)/C_N(Q)] z\notin N}}\bar{N}_{\mathcal{O}Nz}^{\varphi_x}(Q)\right) &=kN_N(Q)\bigoplus\left(\bigoplus_{\substack{z\in [N_G^K(Q)/C_N(Q)] z\notin N}}\bar{N}_{\mathcal{O}Nz}^{\varphi_x}(Q)\right).
H^{2}={\frac{2 g \bar{V}_{e f f}}{1 - N g ( q^{\prime})^{2}/ 2}}, \ \ \{\frac{\ddot{a}}{a}}= - 2 H^{2}+ 3 g [ 2 \bar{V}_{e f f}- ( \rho_{f}+ p_{f}) ] ,
\mathcal F_{p\mu\rho}(\delta p,\delta\mu,\delta\rho,\delta \b u):= \nabla \delta p + 2\nabla\cdot(\delta\mu\;\varepsilon(\b u))+ 2\nabla\cdot(\mu\;\varepsilon(\delta\b u))-\delta\rho\;\b u_{tt}-\rho(\delta\b u)_{tt},
\left|\sum_{n\ge 1}\frac{(-1)^n q^{\frac{n(3n+1)}{2}}}{1+q^n}\right|\le \frac{1}{1-|q|}\sum_{n\ge 1}|q|^{\frac{n(3n+1)}{2}}\ll \frac{1}{y}\cdot y^{-\frac12}=y^{-\frac32}.
D_{\mathcal{K}_{\alpha}^{2}}(\omega )=\mbox{det}(I-\omega \mathcal{K}_{\alpha}^{2})=\exp\left(-\sum_{k=1}^{\infty}\frac{\mbox{Tr}\mathcal{K}_{\alpha}^{2k}}{k}\omega^{k}\right)=\exp\left(-\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}[\lambda_{l}(\mathcal{K}_{\alpha}^{2})]^{k}\frac{\omega^{k}}{k}\right),
\sum_{I,J,a}(1-q_{a1}q_{12})\alpha_{I,J,a}d_J \rho_{I+e_1}\delta_J \sigma_{a+e_1}= \sum_{I,J,a}q_{a1}q_{12}\alpha_{I,J,a}c_J j_1\rho_I \delta_{J-e_1}\sigma_{a+e_1}.
k_{25^nr_0}=\sqrt{1/2-1/2\sqrt{1-4\left(k_{r_0}k'_{r_0}\right)^2\prod^{n}_{j=1}P^{(j)}\left[\sqrt[12]{\frac{k_{r_0}k'_{r_0}}{k_{r_0/25}k'_{r_0/25}}}\right]^{24}}},
\int_{(\infty)}\frac{d^Dp^\mu}{(2\pi)^D}\,e^{i\Delta x\cdot p}\int_0^\infty\frac{d\gamma}{2\pi}\,e^{-\frac{1}{2}i\gamma(p^2+m^2)}=\frac{1}{\pi}\int_{(\infty)}\frac{d^Dp^\mu}{(2\pi)^D}\,e^{i\Delta x\cdot p}\,\frac{i}{p^2+m^2-i\epsilon}\ ,
\Sigma_+\ :\ -1+\tilde\epsilon\left( \tilde{x}^3 - \frac{3}{4}\tilde u_0^4 \tilde{x}- \frac{1}{4}(\tilde u_0^6 + \tilde u_1)- \frac{1}{4}(\zeta + \frac{1}{\zeta})\right) +{\cal O}(\tilde \epsilon^2) = 0 \ ,
W_\ell^\infty(\eta,x)=\sum_{k\in\mathbb{Z}^3}\frac{e^{i\ell\,k\cdot\eta}}{|x+\ell\,k|}=\frac{4\pi}{\ell^3}\sum_{k\in \mathbb{Z}^3}\frac{1}{\left|\frac{2\pi k}{\ell}-\eta\right|^2}\,e^{i\left(\frac{2\pi k}{\ell}-\eta\right)\cdot x}
\int_{\mathfrak{D}_{R\mu^k_\alpha}(x^k_\alpha)\setminus\cup^s_{j=1}\mathfrak{D}_{R'\lambda^j_\alpha}(y^j_\alpha)}|\hat{u}_\alpha-U^k_\alpha|^{2^*}\,d\sigma_{\hat{h}}=o(1)+\epsilon_\alpha(R')
S_{N, N}^{(r)}=\sum_{m = 1}^N\left(S_{N, m}^{(r+1)}- S_{N, m-1}^{(r+1)}\right) \log m &=S_{N, N}^{(r+1)}(\log N) + \sum_{m = 1}^{N-1}S_{N, m}^{(r+1)}\big(\log(m) - \log(m+1)\big).
\langle H_N\rangle_{\psi_\mathrm{gs}}\ge \; E_{2-\mathrm{mode}}+\frac{c \lambda}{N}\big\langle(\mathcal{N}_1-\mathcal{N}_2)^2\big\rangle_{\psi_\mathrm{gs}}+E^\mathrm{Bog}&-C_\Lambda o_N(1)-\frac{C}{(\mu_{2M_\Lambda+2}-\mu_+)}^{1/2},
\|u^n-U_h^{n}\|+\gamma^{\frac12}\Big{(}\Delta t\sum_{k=1}^n\|(\sigma-\Sigma_h)^{k-\theta}\|^2\Big{)}^{\frac12}+\Big{(}\Delta t\sum_{k=1}^n\|\nabla (u-U_{h})^{k-\theta}\|^2\Big{)}^{\frac12}\leq C[h^{m+1}+H^{2m+2}].
\langle L_a^{-1}R_{log,a}L_a^{-1}\tilde{u}_j, \tilde{u}_je^{\tilde{\phi}}\rangle = \langle R_{log,a}L_a^{-1}\tilde{u}_j, L_a^{-1}[\tilde{u}_je^{\tilde{\phi}}] \rangle &= |u_j(x^*)|^2e^{\phi(x^*)}\langle R_{log,a}L_a^{-1}[1], L_a^{-1}[1] \rangle + O(\varepsilon).
I = \frac{1}{2}\mu \int d^{3}x [ - \epsilon^{m n p}A_{m}^{i}F_{n p ( a + v )}^{i}- \mu a_{m}^{i}a^{m i}+ \epsilon^{m n p}v_{m}^{i}( \partial_{n}v_{p}^{i}+ \frac{1}{3}\epsilon^{i j k}v_{n}^{j}v_{p}^{k}) ] ,
N={\cal N}|_{\pi^{*}{\cal S}}={\cal O}_{\pi^*{\cal S}}((n(\frac{1}{2}+\lambda)\sigma+(\frac{1}{2}-\lambda)\pi^{*}\eta+(\frac{1}{2}+n\lambda)\pi^{*}c_{1}(B)) \cdot \pi^{*}{\cal S}).
\begin{array}{l}{n - 2 n_{1}- 1 - 2 ( n_{2}+ \cdots + n_{j}) - \alpha_{1 j - 1}^{( k - 1 )}\geq a_{1}^{( j )}\geq \cdots \geq a_{n_{j}- n_{j + 1}}^{( j )}\geq 0 ,}\\ \end{array}
\beta \delta = - b_{+}\Phi_{1}( Z^{- 1}) , \quad \beta^{2}= q^{- 1}( 1 - b_{3}) \Phi_{1}( Z^{- 1}) , \quad \delta^{2}= b_{+}( 1 - b_{3})^{- 1}b_{+}\Phi_{1}( Z^{- 1})
\left(\mathcal{R}\cdot\mathrm{P}_{0}\psi_{0}\right)\left(-\left|\eta\right|\right)=\mathrm{P}_{0}\mathcal{R}\cdot\psi_{0}\left(-\left|\eta\right|\right)=\mathrm{P}_{0}\psi_{1}\left(\left|\eta\right|\right).
\begin{aligned}x_1^{a_1}\cdots x_{i-1}^{a_{i-1}}x_i^{r_i-1}x_{i+1}^{a_{i+1}}\cdots x_n^{a_n}\Omega = 0 x_i^r x_1^{a_1}\cdots x_n^{a_n}\Omega = - \hbar \tfrac{a_i+1}{r_i}x_1^{a_1}\cdots x_n^{a_n}\Omega.\end{aligned}
\overline{F}(k,t+1,\mathbf{p}_{n+1}) = \begin{dcases}p_{k}^{}\overline{F}(k-1,t,\mathbf{p}_{n+1}) + (1 - p_{k}^{}) \overline{F}(k,t,\mathbf{p}_{n+1}) \textnormal{if}\; k \le t; 0 \textnormal{if}\; k > t; \end{dcases}
T_1=\sum\limits_{p=0}^{n-2}\frac{\tau_{p}^{(\nu ,n)}p !}{(n-1)!}t^{n-p-1}=\sum\limits_{d=1}^{n-1}\frac{\tau_{n-d-1}^{(\nu ,n)}\left(n-d-1\right) !}{(n-1)!}t^{d}.
N(r,t,\eta,\xi) =\left\{2(r+t\eta-2t\eta\xi)(1-2\xi)-2t\eta\right\}(r+t\eta-2t\eta\xi) -\left\{(r+t\eta-2t\eta\xi)^2+r^2-t^2\eta^2\right\}(1-2\xi).
R_jg(x) := \, c_n\, \mathrm{P. V.}\int_{\mathbb R^n}\frac{x_j - y_j}{|x-y|^{n+1}}g(y)\, dy = \, c_n\lim_{\varepsilon \searrow 0}\int_{|x-y|\ge \varepsilon}\frac{x_j - y_j}{|x-y|^{n+1}}g(y)\, dy
\begin{aligned}\varphi_k = 1\quad\hbox{on}P\cap\{p\in P: \hbox{dist}(p, H_i\cap H_j) \leq 1/k\}, &\varphi_k = 0\quad\hbox{on}P \backslash \{p\in P: \hbox{dist}(p, H_i\cap H_j) \geq 2/k\}, &\psi_k(t) = 1\quad\hbox{for all}t\geq 2/k,\quad\hbox{and}\psi_k(t) = 0\quad\hbox{for all}t\leq 1/k,\end{aligned}
\underset{\left\{\pmb{F}_k\right\}_{k=1}^K,\left\{\pmb{W}_k\right\}_{k=1}^{K+1}}{}\sum\limits_{k=1}^K ||\pmb{J}_k||_{*}+ ||\pmb{S}_e||_{*}\,\,\,\,\,\qquad \pmb{S}_k \succeq \pmb{0},\,\,\, \sigma_{min}(\pmb{S}_k) \geq \epsilon.
\MoveEqLeft T^2(\gamma)-T(\gamma)=(P^+_{T(\gamma)}-P^+_\gamma)T(\gamma)\left(P^+_{T(\gamma)}-P^+_{\gamma}+P^+_{\gamma}\right)+P^+_\gamma T(\gamma)(P^+_{T(\gamma)}-P^+_\gamma) \quad=(P^+_{T(\gamma)}-P^+_\gamma)T(\gamma)+ T(\gamma)(P^+_{T(\gamma)}-P^+_\gamma)+(P^+_{T(\gamma)}-P^+_\gamma)T(\gamma)(P^+_{T(\gamma)}-P^+_\gamma).
q_{(i-j)m}= \omega(W_{\Theta_{i}\xi- \Theta_{j}\xi}) = \omega(W_{((i - j)\mathtt{N}\xi_2 , 0 )}) (i-j) \phi_{m,n}= h \sigma(\Theta_{j}\xi , \Theta_{i}\xi) = (i - j) l h \mathtt{N}\xi_2^2 = (i - j) l \frac{2\pi}{d}+ (i - j) l\frac{\epsilon}{4d^2}\,.
\mathbb{P}\Bigl\{\zeta^{\epsilon}(t+\triangle) = m \, \bigl\vert \, \zeta^{\epsilon}(t) = k, X_t^{\epsilon,\hat{v}}= x \Bigr\}= \gamma_{km}(x) \triangle + o(\triangle) \,\,\, \,\,\, \triangle \downarrow 0,
g_{2a}^j=\frac{(-1)^{\frac{j}{2}+a+\frac{k}{2}}j!}{\prod\limits_{d=0,\;d\neq a}^{\lceil l/2\rceil-1}((k+2a)^2-(k+2d)^2)}\sum_{b=0}^{j/2}\frac{\prod\limits_{d=0,\;d\neq a}^{\lceil l/2\rceil-1}((2b)^2-(k+2d)^2)}{\prod\limits_{d=0,\;d\neq b}^{j/2}((2b)^2-(2d)^2)}.
\begin{pmatrix}H_{k,\{X;Y_{l,1}\}}\vdots \vdots H_{k,\{X;Y_{l,l}\}}\end{pmatrix}=B^{(l)}\begin{pmatrix}\vdots \sum_{h=1}^{j-1}(-1)^h T_{h}H_{k,\{X;Y_{l-h,j-h}\}}\vdots T_{l}H_{k,X}\end{pmatrix}_{j=2,..,l}.
\left.\frac{\partial^{2}}{\partial t^{2}}\rho_{\phi}[\iota] \right|_{t=0}=L_{X}L_{X}{\rm vol}_{\phi}[\iota] &=d (i(X) d (i(X) \rho_{\phi}[\iota]{\rm vol}_{\iota^{*}g}))={\rm div}({\rm div}(\rho_{\phi}[\iota] X)X){\rm vol}_{\iota^{*}g},
\delta_{\hat{\chi}}\hat{L}_{\hat{\mu}_{1}\ldots \hat{\mu}_{r}}= m \hat{\lambda}_{\hat{\mu}_{1}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\nu}\hat{\mu}_{2}\ldots \hat{\mu}_{r}}+ \ldots + m \hat{\lambda}_{\hat{\mu}_{r}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\mu}_{1}\ldots \hat{\mu}_{r - 1}\hat{\nu}}\, .
|(1+s) x - (1-s) y|^2 =|(1+s) (x - y) +2sy|^2= (1+s)^2 \Big|x - y + 2\frac{s}{1+s}y\Big|^2 \ge |x-y|^2 - 4s|\langle x-y, y \rangle| \ge |x - y|^2 - 4s |y| m(x) \ge |x - y|^2 - 8s,
\sqrt{\frac{\pi}{16}}\left[ \frac{z(s)}{s}+ \frac{z(1-s)}{1-s}\right] = \frac{1}{s}+ \frac{1}{1-s}+ \int_{0}^{1}y^{\frac{s}{2}- 1}\Psi(y) dy + \int_{0}^{1}y^{\frac{1-s}{2}- 1}\Psi(y) dy
I_{3}( p ) \sim \frac{1}{( \sigma \omega )^{( 3 - D ) / 2}}\Gamma \left( \frac{3 - D}{2}\right) F \left( \frac{3 - D}{4}, \frac{5 - D}{4}; \frac{D}{2}; \frac{p^{2}}{\sigma^{2}}\right) ,
\Sigma_i(t) \leq C \int_0^t \Psi(\tau) \d\tau \ , \ \ \i = 1,2 \ , \Sigma_3(t) \leq C\varepsilon t + C \int_0^t \Psi(\tau) \d\tau \ , \Sigma_4(t) \leq C\delta t \ , \Sigma_j(t) \leq C\delta t + C \int_0^t \Psi(\tau) \d\tau \ , \ \ \j = 5,6 \ .
v_{2n-1}(P_C \otimes 1) = (P_C \otimes s_{n_0,n})(1 \otimes f_n - v_{2n-2}^* v_{2n-2}), v_{2n-1}(P_D \otimes 1) = \sum_{k=1}^{N_D}(S_{c(k)}S_{d_k}S_{d_k}^*\otimes s_{n_k,n})(1 \otimes f_n - v_{2n-2}^* v_{2n-2}),