harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{w\in V(G)\setminus V}|\psi'_w(K)| \leq r\cdot \frac{6n^{2}r}{|V|k_{[r]}}\cdot 3n+n\cdot\frac{12n^{2}r^{2}}{|V|^{2}k_{[r]}}\cdot 2|V|= \frac{42n^3r^2}{|V|k_{[r]}}.
|B(\lambda)\rangle_\Lambda=\exp\left(\sum_{m>0}\frac{1}{m}a^T_{-m}\cdot\Lambda\cdot\bar{a}_{-m}\right)\lim_{z,\bar{z}\to 0}V^M_{Q_0\ \rho-\frac{1}{\sqrt{pp'}}\lambda}(z)\bar{V}^M_{Q_0 \rho+\frac{1}{\sqrt{pp'}}\lambda}(\bar{z})|0\rangle_{M}\;.
u=\sum_{F}\frac{\langle f,F\rangle \cdot F}{\lambda_{s_F}-\lambda_w}+\frac{\langle f,1\rangle\cdot 1}{(\lambda_1 - \lambda_w)\cdot \langle 1,1\rangle}+\frac{1}{4\pi i}\int_{(\frac{1}{2})}\frac{\langle f, E_s\rangle\cdot E_s}{\lambda_s-\lambda_w}ds,
\chi(\widehat{\mathrm{CFG}}(\Sigma, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{w}, \boldsymbol{z})) \doteq \sum_{x\in \mathbb{T}_{\boldsymbol{\alpha}}\cap \mathbb{T}_{\boldsymbol{\beta}}}\mathrm{sign}(\sigma)\prod_{i=1}^{n}\mathrm{sign}(x_{i})\cdot A(x),
\frac{dx^{(j)}}{dt}=F^{(j)}(x^{(j)},O_t^{(j)}(\omega )) & :=e^{-O_t^{(j)}(\omega )}f^{(j)}(e^{O_t^{(j)}(\omega)}x^{(j)})+O_t^{(j)}(\omega )x^{(j)},j=1,\dots ,N
\Delta V_{PF}={\hbar^2\over8M}\Big[g^{ab}\Gamma_a\Gamma_b+2(g^{ab}\Gamma_b)_{,b}+{g^{ab}}_{,ab}\Big] +{\hbar^2\over8M}\Big(2h^{ac}{h^{bc}}_{,ab}-{h^{ac}}_{,a}{h^{bc}}_{,b}-{h^{ac}}_{,b}{h^{bc}}_{,a}\Big)\enspace.
K_\varepsilon = \left\lceil C_0 \cdot \log_2 \left(\frac{1}{\varepsilon}\right) \right\rceil \leq 1 + C_0 \log_2 \left(\frac{1}{\varepsilon}\right) \leq (1 + C_0) \cdot \log_2 \left(\frac{1}{\varepsilon}\right).
0\longrightarrow H^i({{{{\mathfrak N}}}},V)\longrightarrow\bigoplus_{\lambda\in \nabla}H^i({{{{\mathfrak N}}}},{\rm ind}_{{{{{\mathfrak N}}}}[\lambda]}^{{\mathfrak N}}V)\longrightarrow\ldots\longrightarrow \bigoplus_{E\subset \nabla\atop|E|=|\nabla|-i}H^i({{{{\mathfrak N}}}},{\rm ind}_{{{{{\mathfrak N}}}}[E]}^{{\mathfrak N}}V)
\frac{1}{8}\,W^{+}_{ijkl}W^{+}_{ijpq,t}W^{+}_{klpq,t}= \,\, \lambda |d\lambda|^{2}+\mu|d\mu|^{2}+\nu|d\nu|^{2}&-\lambda(\mu-\nu)^{2}|a|^{2}-\mu(\nu-\lambda)^{2}|b|^{2}-\nu(\lambda-\mu)^{2}|c|^{2}\,,
\frac{1}{2}\Delta|\nabla W|^{2}= |\nabla^{2}W|^{2}+ W_{ijkl,s}W_{ijkl,tst}+ 4 W_{ijkl,s}W_{rjkl,t}R_{rist}&= |\nabla^{2}W|^{2}+ W_{ijkl,s}W_{ijkl,tst}+ 4 W_{ijkl,s}W_{rjkl,t}W_{rist}+\frac{R}{3}W_{ijkl,s}W_{sjkl,i}&=|\nabla^{2}W|^{2}+ W_{ijkl,s}W_{ijkl,tst}+ 4 W_{ijkl,s}W_{rjkl,t}W_{rist}+ \frac{R}{6}|\nabla W|^{2}\,,
\sum_{i=0}^{l_1(x)-1}\{\sum_{i'=0}^{l_2 (\sigma_B^i(h(x)))-1}f(\sigma_C^{i'}(g(\sigma_B^i(h(x)))))-\sum_{j'=0}^{k_2 (\sigma_B^i(h(x)))-1}f(\sigma_C^{j'}(g(\sigma_B^{i}(h(x))))) \}= \sum_{i'=0}^{l_2^{l_1(x)}(h(x))-1}f(\sigma_C^{i'}(gh(x)))-\sum_{j'=0}^{k_2^{l_1(x)}(h(x))-1}f(\sigma_C^{j'}(g(\sigma_B^{l_1(x)}(h(x)))))
\mathcal{E}(\boldsymbol{\theta}) = \mathbf{y}^H\mathrm{Diag}\left(e^{j\boldsymbol{\theta}}\right)\mathbf{F}^{H}\left(\mathbf{I}_{N_c}- \mathbf{B}\right)\mathbf{F}\mathrm{Diag}\left(e^{j\boldsymbol{\theta}}\right)^H\mathbf{y},
\begin{aligned}\int_{\mathbb{R}}\frac{d\xi_1}{\left (1 + |\tau-\xi_1^3- (\xi-\xi_1)^3 | \right )^{2b}}\le \frac{1}{\sqrt{3 |\xi|}}\int_{\mathbb{R}}\frac{d\mu}{\left ( 4\tau-\xi^3-4\mu \right )^{1/2}(1+|\mu|)^{2b}}&\lesssim \frac{1}{\sqrt{|\xi|}\left ( 1+ |4 \tau- \xi^3 | \right )^{1/2}}\end{aligned}
\lambda_1 = a-\frac{1}{4\sin^2 \frac{\pi}{2(2N-1)}}= \frac{N(MN-1)}{K}-\frac{1}{4\sin^2 \frac{\pi}{2(2N-1)}}=\frac{N(MN-1)}{K}\left [1- \frac{K}{4N(MN-1)\sin^2 \frac{\pi}{2(2N-1)}}\right] \leq \frac{N(MN-1)}{K}\left ( 1- \frac{n+1}{n+\epsilon}\right ) <0,
( Z , Z ) \equiv \frac{1}{2}t r ( \bar{Z}Z ) = z^{1}z^{3}- q^{2}z^{2}z^{4}\qquad ( \, \bar{Z}= \left( \begin{array}{c c}{{z^{3}}}&{{- q^{2}z^{4}}}\\{{- q^{2}z^{2}}}&{{z^{1}}}\\ \end{array}\right) \, ) \; .
\tilde{L}_{T(K)}(T^{(0)}_* z)(T^{(0)}\theta) = \frac{\SS(T^{(0)}_* z \cdot h_{T(K)}, h_{T(K)}, \ldots,h_{T(K)})(T^{(0)}\theta)}{\SS(h_{T(K)},\ldots,h_{T(K)})(T^{(0)}\theta)}.
F^{\vee}_{{P}_{(Q^{\ell},S^{\ell})}(\ell)}(X_1(t)^{-1},\ldots X(t)_m^{-1})\circ \check{\mu}_k^*=F^{\vee}_{{P}_{(Q'^{\ell},S'^{\ell})}(\ell)}(X_1(t)^{-1},\ldots X_m(t)^{-1}),
\left( \begin{array}{c}\dot{y}_1 \dot{y}_2 \end{array}\right) = \left( \begin{array}{cc}m(l) 1 0 -[n-2-m(l)] \end{array}\right) \left( \begin{array}{c}y_1 y_2 \end{array}\right) +\left( \begin{array}{c}0 - (y_1)^{q-1}\end{array}\right)
A^+_{r,2r}(x_0) :=\{(x'',x_n,x_{n+1})\in B_1^+| \ |x''-x''_0|\leq r, &r\leq \sqrt{(x_n-(x_0)_n)^2+(x_{n+1}-(x_0)_{n+1})^2}\leq 2r\}.
\begin{cases}\displaystyle\frac{d}{dt}\Psi(t)+C|\phi_1\nabla \overline{u}(t)|^2_2\leq F(t)\Psi (t), begin{align*}10pt]\displaystyle\int_{0}^{t}F(s)ds\leq C 0\leq t\leq T_*.\end{cases}
p_{h_1X_1 \zeta_1}(z) * \cdots *p_{h_{\xi-1}X_{\xi-1}\zeta_{\xi-1}}(z)*p_{h_{\xi+1}X_{\xi+1}\zeta_{\xi+1}}(z)*\cdots *p_{h_{N_\mathrm{B}}X_{N_\mathrm{B}}\zeta_{N_\mathrm{B}}}(z) &=\sum\limits_{k=1}^{Q-1}\tilde{\omega}_k\mathcal{CN}(z;0,\tilde{\sigma}_k^2)+\tilde{\omega}_{Q}\delta(z),
dI-\sum_{i=1}^{2}||\nabla w_i||^2_2= \dfrac{2k_{11}}{|K|}\bigg(\sqrt{|K|}w_1,e^{u_0'+\sqrt{|K|}w_1}-1+\dfrac{\sqrt{|K|}}{2k_{11}}h_1-\frac{k_{21}}{2k_{11}}h_2\bigg)_2 &+\dfrac{2k_{11}}{|K|}\bigg(\frac{1}{k_{11}}(k_{21}\sqrt{|K|}w_1+|K|w_2),e^{u_0''+(k_{21}\sqrt{|K|}w_1+|K|w_2)/k_{11}}-1+ \dfrac{1}{2}h_2\bigg)_2.
\mathcal{B}(\mathfrak{E})=\{f:\mathbb{C}\to\mathfrak{X}~\mbox{entire function}~ | E_+^{-1}f\in H^2_\mathfrak{X}(\mathbb{C}_+),~E_-^{-1}f\in H^2_\mathfrak{X}(\mathbb{C}_+)^\perp\}.
{\Lambda}_{i j}= \frac{1}{2}\sqrt{- G}G_{i j}\left({\Phi}^{( p - 1 ) / 2}G^{k l}h_{k l}- ( p - 1 ){\Phi}^{( p + 1 ) / 2}\right) - \sqrt{- G}{\Phi}^{( p - 1 ) / 2}h_{i j},
P_{\lambda,d}(\eta_{t+s}^{\delta_1}(O)=1) \geq P_{\lambda,d}(A_t^O\neq \emptyset)P_{\lambda,d}(A_s^O\neq \emptyset) &=P_{\lambda,d}(\eta_{t}^{\delta_1}(O)=1)P_{\lambda,d}(\eta_{s}^{\delta_1}(O)=1)
\begin{array}{l}{\tilde{Z}( \tilde{z}^{\prime}) = G ( g )^{- 1}\tilde{Z}( \tilde{z}) \tilde{K}( \tilde{z}; g )}\\{{}}\\{\bar{\tilde{Z}}( \tilde{z}^{\prime}) = \bar{\tilde{K}}( \tilde{z}; g ) \bar{\tilde{Z}}( \tilde{z}) G ( g )}\\ \end{array}
(Gu)(t)= \, \,\underbrace{T(t)u_{0}}_{I_{0}(t)}+\chi\underbrace{\int_{0}^{t}T(t-s)\nabla\cdot (u(s)\nabla (\Delta-I)^{-1}u(s))ds}_{I_{1}(t)}\,\,\,\, +\underbrace{\int_{0}^{t}T(t-s)((a+1)u(s)-bu^{2}(s))ds}_{I_{2}(t)}.
\det(zI_{k+1}-A_{k+1}^{(s,t)})=(z-q_{k+1}^{(s,t)}-e_k^{(s,t)})\det(zI_k-A_{k}^{(s,t)})-q_k^{(s,t)}e_k^{(s,t)}\det(zI_{k-1}-A_{k-1}^{(s,t)}).
M_{t}(i,t,n) =\dfrac{p_{t}(n)}{t}; \overline{M}_{t}(i,t,n) =\dfrac{\overline{p}_{t}(n)}{t}\equiv 0 \pmod 2; \hat{M}_{t}(i,t,n) =\dfrac{\hat{p}_{t}(n)}{t}.
\sum_{k=1}^{\overline{r}(n;\theta) + 1}\mathfrak{J}_k^{(1)}(n;\theta) \leq \frac{\lambda_{12}}{n\theta(1-\theta)}+ \frac{\lambda_{13}}{[n\theta(1-\theta)]^{1+\varepsilon(\gamma)}}\log\left(\frac{n}{\theta(1-\theta)}\right)\ .
\lim_{T \to \infty}\frac{1}{T}\int_{0}^{T}\frac{e \left(\mathcal{H}, e^r ; z\right)}{e^{r/2}}d r = \frac{|\Gamma(3/4)|^2}{\pi^{3/2}}\sum_{\mathfrak{a}}\hat{E}_{\mathfrak{a}}(1/2) E_{\mathfrak{a}}(z, 1/2).
\varphi_{\zeta}(x) = 0 \, \, x \in (-\infty,-2\zeta] , \varphi_{\zeta}(x) = 1 \, \, x \in [-\zeta,0], \max \left\{\zeta |\varphi_{\zeta}'(x)|, \zeta^{2}|\varphi_{\zeta}''(x)| \right\}\leq C_{0}\, \, x \in (-\infty,0],
f({\sf s}_{[\kappa, l]}x) =f({\sf s}^{l}_{\mu}{\sf s}^{\kappa}_{l}{\sf s}^{\mu}_{\kappa}{\sf c}_{\mu}x) &={\sf s}^{l}_{\mu}{\sf s}^{\kappa}_{l}{\sf s}^{\mu}_{\kappa}{\sf c}_{\mu}f(x) &={\sf s}_{[\kappa, l]}f(x).
S(u_{\mu, \xi}) = -\sum_{i=1}^k\partial_tu_i + \left(\sum_{i=1}^ku_i\right)^p - \sum_{i=1}^kU^p_{\mu_i,\xi_i}- \sum_{i=1}^k\mu_i^{\frac{n-2s}{2}}(-\Delta)^s\Phi_i^*(x, t).
f(0)=0,f_1(0)=0,f_2(0)=1-\frac{4}{3}(1-2p)^2, f(1^-)=\frac{4}{\pi}-1-\frac{1}{3}(1-2p)^2, f_1(1^-)=\frac{2}{\pi}-\frac{2}{3}(1-2p)^2, f_2(1^-)=+\infty, f_1(r)=\frac{2}{\pi}\bigl[\mathcal{E}-\bigl(1-r^2\bigr)\mathcal{K}\bigr]-\frac{2}{3}(1-2p)^2r^2, f_2(r)=\frac{2}{\pi}\mathcal{K}-\frac{4}{3}(1-2p)^2,
|f_{1}(z)|\le C_2\int_{0}^{R_0}\left|e_{V}\left(\frac{u}{z}\right)\right|\sum_{k=N}^{\infty}(D_2u)^{k}\frac{du}{u}\le \epsilon C_2D_2^{N}\int_{0}^{R_0}\left|e_{V}\left(\frac{u}{z}\right)\right|u^{N-1}du.
\begin{aligned}\nabla\cdot \mathbf{u}= q,\mathbf{u}=\frac{-k^{\ast}}{\mu(c)}\nabla p,x\in \Omega,0 \le t \leq T, &\phi\frac{\partial c}{\partial t}+\nabla\cdot(\mathbf{u}c-D(\mathbf{u})\nabla c)=qc^{\ast},x\in \Omega,0 \le t \leq T,\end{aligned}
T_{\psi}^{A,A,A}(Y,X) =\sum_{i,j,k=1}^n \psi(\lambda_i, \lambda_k,\lambda_j)P_{\xi_i}YP_{\xi_k}XP_{\xi_j}&=\sum_{i,k=1}^n \psi(\lambda_i, \lambda_k,\lambda_k)P_{\xi_i}YP_{\xi_k}\times X &=\sum_{i,k=1}^n \phi_1(\lambda_i, \lambda_k)P_{\xi_i}YP_{\xi_k}\times X=T^{A,A}_{\phi_1}(Y)\times X.
\tilde{E}^{(i)}_{n\delta}=\sum_{p_1+2p_2+\cdots+np_n=n}\frac{\left ( q^{(\alpha_i,\alpha_i)}-q^{-(\alpha_i,\alpha_i)}\right )^{\sum_ip_i-1}}{p_1!\;...\;p_n!}(E^{(i)}_{\delta})^{p_1}(E^{(i)}_{2\delta})^{p_2}... (E^{(i)}_{n\delta})^{p_n}
|\partial_{t}^{2}N_{2}(f)(t,r)| \leq C \begin{cases}\frac{1}{t^{5}\log^{b}(t)(r^{2}+\lambda(t)^{2})}, r \leq \frac{t}{2}\frac{\log(r)}{r^{3}t^{2}|t-r| \log^{3N+2b}(t)}+ \frac{\log^{2}(r)}{r^{2}(t-r)^{2}t^{2}\log^{3N+b}(t)}+ \frac{\log^{3}(r)}{t^{2}r^{2}\log(t) |t-r|^{3}}+ \frac{\log^{5/2}(r)}{r^{9/4}(t-r)^{4}}, \frac{t}{2}\leq r <t\end{cases}
\frac{\alpha_{n}}{m^{2 n + 1}}\int d^{3}x F_{\mu \nu}( \partial^{2})^{n}F^{\mu \nu}\; \; , \; \frac{\beta_{n}}{m^{2 n}}\int d^{3}x \varepsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}( \partial^{2})^{n}A_{\rho}\; , n \geq 1 \; ,
\mathbb{E}[\hat{\tilde{\mathbf{x}}}^{\dag}\tilde{\mathbf{x}}]=\mathbf{w}^{\dag}\mathbf{g},\ mathbb{E}[\hat{\tilde{\mathbf{x}}}^{\dag}\hat{\tilde{\mathbf{x}}}]=\mathbf{w}^{\dag}\mathbf{D}\mathbf{w}.
R_-(\kappa_x) = \frac{1-\alpha}{1+\alpha}= \frac{\mu_2 \kappa_{1z}- \mu_1 \kappa_{2z}}{\mu_2 \kappa_{1z}+ \mu_1 \kappa_{2z}}T_+(\kappa_x) = \frac{2}{1+\alpha}= \frac{2 \mu_2 \kappa_{1z}}{\mu_2 \kappa_{1z}+ \mu_1 \kappa_{2z}}.
v^{\prime}=(yq_{2})^{\prime}=y^{\prime}q_{2}+yq_{2}^{\prime}=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+v\frac{q_{2}^{\prime}}{q_{2}}=q_{0}q_{2}+(q_{1}+\frac{q_{2}^{\prime}}{q_{2}})v+v^{2}.
\begin{array}{rll}\|f\|_{L^p(\Omega)}\le C_p \|\nabla f\|_{L^p(\Omega)}, f\in W^{1,p}(\Omega), f=0 , \|f\|_{L^p(G)}\le C_p \|\nabla_Hf\|_{L^p(G)}, f\in W^{1,p}(G)\cap L^p_0(G).\end{array}
\tilde{C}_{[\alpha_1],[\alpha_2],(\alpha_3)}^\epsilon =i\epsilon\left[\left( \beta_1^2+\beta_2^2\right) C_{[\alpha_1],[\alpha_2],(\alpha_3)}^\epsilon -2\beta_1\beta_2C_{[\alpha_1],[\alpha_2],(\alpha_3)}^{-\epsilon}\right],
p_1(t) = \rho_1 [(\rho_2 + bt)(\rho_3 + ct) + a(\rho_3 + ct) + \delta^2 t^{2\gamma}], p_2(t) = \rho_1 [(\rho_2 + a + bt)(\rho_3 + ct) + \delta^2 t^{2\gamma}], p_3(t) = -at^\frac12[\rho_2(\rho_3 + ct) +\delta^2 t^{2\gamma}], p_4(t) = -at^\frac12\rho_2(\rho_3 + ct), p_5(t) = -\rho_3\delta at^{\gamma + \frac12}.
A_{h,k}(x,w) = \sum_{k\nmid \ell}\frac{x^\ell}{\ell}\frac{e^{-\ell w+ 2 \pi i \ell h/k}}{(1- e^{-\ell w+ 2 \pi i \ell h/k})^2}, B_{h,k}(x,w)=\sum_{\ell=1}^\infty \frac{x^{k \ell}}{k \ell}\frac{e^{- \ell k w}}{( 1- e^{- \ell k w})^2}.
F_{12}=(\sum_{1 \leq i,j \leq 2}H_{ij11}x_{i}y_{j})(\sum_{1 \leq i,j \leq 2}H_{ij22}x_{i}y_{j})-(\sum_{1 \leq i,j \leq 2}H_{ij12}x_{i}y_{j})(\sum_{1 \leq i,j \leq 2}H_{ij21}x_{i}y_{j}).
\xi^\epsilon := \inf \{r > t~:~[\![ X_r^{t,\bar{A}_t^\epsilon; Z_t \otimes \alpha(v), W_t \otimes v}]\!]_{\kappa}> \mu \}&~~~ \wedge \inf\{r > t~:~ \| X_r^{t,\bar{A}_t^\epsilon; Z_t \otimes \alpha(v), W_t \otimes v}\|_\infty > \mu_0 \}.
\frac{1}{2 \pi}\mathcal{P}\! \! \int_{- \infty}^{\infty}\! d p_{\perp}^{\prime}\, \frac{1}{p^{2}+ p_{\perp}^{2}+ M^{2}}= \frac{\Theta ( p^{2}+ M^{2})}{2 \sqrt{p^{2}+ M^{2}}}
f_{y_{11},\cdots, y_{ii}}(y_1,\cdots, y_i) = \underset{y_{ij}, \forall i\neq j}{\int\cdots\int}\frac{\left(\det \mathbf{W}^{-1}\right)^{2N}}{\Gamma}e^{-\left(\mathbf{W}^{-1}\right)}dy_{12}\cdots dy_{N(N-1)}
g(\vec{v}) = \left\vert g(\vec{v})\right\vert \le \sum\limits_{\alpha\in\Gamma}\left\vert c_{\alpha}\right\vert \left\vert v_1^{\alpha_1}\cdots v_n^{\alpha_n}\right\vert \le\sum\limits_{\alpha\in\Gamma}\left\vert c_{\alpha}\right\vert \,\left\vert\, r(\vec{v})^{\frac{\alpha_1}{2m_1}+\cdots+\frac{\alpha_n}{2m_n}}\right\vert.
\begin{array}{lll}p_k^S(n_1,m) = p_k^S(n_2,m) = p_k^S(\infty,m) \quad{\rm if}n_1,n_2\geq 2k, p_k^S(n,m_1) = p_k^S(n,m_2) = p_k^S(n,\infty) \quad{\rm if}m_1,m_2\geq 2k.\end{array}
\left \vert \left\Vert A_{\Omega}\right\Vert_{2}^{2}-\Vert A\Vert_{L_2(\Pi)}^{2}\right \vert > \dfrac{1}{2}\Vert A\Vert_{L_2(\Pi)}^{2}+ \mathcal{E}&> \dfrac{1}{2}\alpha^{l-1}\nu+ \mathcal{E}&= \dfrac{5}{12}\alpha^{l}\nu+ \mathcal{E}.
CA \oplus IX^\infty \longrightarrow CA \oplus H_{IA}\oplus IX^\infty= (C[0,1 ) \oplus H^\circ_{C[0,1]}) \otimes A \oplus IA \oplus IX^\infty &\longrightarrow (H_{C[0,1]}^\circ) \otimes A \oplus IA \oplus IX^\infty =H_{IA}\oplus IX^\infty \longrightarrow IX^\infty.
\hat X_{2,q}= \frac{i\hbar\sqrt{K_1}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_1 -\tilde A_1^\dagger\right) +\frac{i\hbar\sqrt{K_2}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_2 -\tilde A^\dagger_2\right);
X=(\xi^t(x_0,x'_0),\ldots,\xi^t(x_0,x'_{d'}),\xi^t(x_1,x'_0),\ldots,\xi^t(x_1,x'_{d'}),\ldots, \xi^t(x_d,x'_0),\ldots,\xi^t(x_d,x'_{d'})).
\begin{array}{l l}{R_{\mathrm{c r}}= 0 ;}&{D = 2 ,}\\{\displaystyle R_{\mathrm{c r}}= \frac{9}{2}m_{0}^{2};}&{D = 3 ,}\\{R_{\mathrm{c r}}= 1 2 m_{0}^{2};}&{D = 4 .}\\ \end{array}
\sum_{i=1}^{d}u_{i1}(x)\mu_i(x)=\sum_{i=1}^{d}\mu_i(x)\sum_{k \neq i}\frac{1}{x_i-x_k}=\sum_{k,l;k>l}\frac{\mu_k(x)-\mu_k(x)}{x_k-x_l},
\|g\cdot\xi_n-\xi_n\|_{2,B_1}\leqslant\sum_{h\in K}\|g\cdot\xi_n-\xi_n\|_{2,hB_2}=\sum_{h\in K}\|(h^{-1}g)\cdot\xi_n-h^{-1}\cdot\xi_n\|_{2,B_2}&\leqslant\sum_{h\in K}\Big(\|(h^{-1}g)\cdot\xi_n-\xi_n\|_{2,B_2}+\|h^{-1}\cdot\xi_n-\xi_n\|_{2,B_2}\Big).
\begin{matrix}{{\tilde{\psi}^{a}={\frac{e}{2}}( \psi_{\alpha})_{\dot{A}}( \bar{\sigma}^{a})^{\dot{A}\alpha},}}&{{\tilde{\psi}^{a b}= - e{( \bar{\sigma}^{a b})^{\dot{A}}}_{\dot{\alpha}}( \psi^{\dot{\alpha}})_{\dot{A}},}}&{{\tilde{\psi}= - e ( \psi_{\dot{\alpha}})^{\dot{A}}\delta_{\dot{A}}^{\dot{\alpha}},}}\\ \end{matrix}
A \, = \, \left( \begin{array}{c c c}{{0}}&{{{\, \,}}}&{{\Theta_{[ 1 ]}}}\\{{\Theta_{[ 1 ]}}}&{{{\, \,}}}&{{0}}\\ \end{array}\right) \, , \quad \Theta_{[ 1 ]}\, = \, \left( \begin{array}{c c c c c}{{0}}&{{{\, \,}}}&{{0}}&{{{\, \,}}}&{{1}}\\{{0}}&{{{\, \,}}}&{{- 1}}&{{{\, \,}}}&{{0}}\\{{1}}&{{{\, \,}}}&{{0}}&{{{\, \,}}}&{{0}}\\ \end{array}\right) \quad ,
\frac{1}{4\pi}\iint_{[-C, C]^2}\begin{vmatrix}e^{-\frac{1}{2}\xi^2}e^{-\frac{1}{2}\eta^2}e^{-\frac{1}{2}\xi^2}e^{-\frac{1}{2}\eta^2}\end{vmatrix}d\xi d\eta = 0.
\Phi_{1}^{G}= x_{0}- \kappa \tau , \quad \Phi_{2}^{G}= \chi , \quad \Phi_{3}^{G}= e - \frac{\kappa}{p_{0}}, \quad \Phi_{4 \nu}^{G}= b_{\nu}, \quad \Phi_{5}^{G}= \xi_{0},
\left\Vert \beta \right\Vert_{\theta}^{\omega}:=\sup_{t\in \left[ 0,T\right]}\left\Vert \beta_{t}\left( g_{t},\cdot \right) \right\Vert+\max_{k=1,\dots ,\left[ p\right]}\sup_{0\leq s\leq t\leq T}\frac{\left\Vert \left( \beta_{t}-\beta_{s}\right) \left( g_{t},\cdot \right)\right\Vert}{\omega \left( s,t\right)^{\theta -\frac{k}{p}}}
(A+UV^{\ast})^{\dagger}=\big(I+A^{\dagger}UF_{S_{A}}U^{\ast}(A^{\dagger})^{\ast}\big)^{-1}\big(A^{\dagger}-A^{\dagger}US_{A}^{\dagger}V^{\ast}A^{\dagger}\big)\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)^{-1},
\begin{array}{l}{\partial_{-}\partial_{0}= q^{- 1}\partial_{0}\partial_{-}, \partial_{0}\partial_{+}= q^{- 1}\partial_{+}\partial_{0},}\\{\partial_{-}\partial_{+}- \partial_{+}\partial_{-}= ( q^{1 / 2}- q^{- 1 / 2}) \partial_{0}\partial_{0}.}\\ \end{array}
ad(\Box \Psi_{h}^{(\theta)}) Ad(O_{<h}) = ad(\Box \Psi_{h}^{(\theta)}) Ad(O_{<h_\theta-C})+\int_{h_{\theta}-C}^had(\Box \Psi_{h}^{(\theta)}) ad(\Psi_{h_1}) Ad(O_{<h_1}) dh_1
\widehat{N}(m) = \mathbb{E}_sN(s)\prod_{q | m}s_q = 2^{-\pi(y)}\sum_{p\leq x}\prod_{q | m}\mathbb{E}_{s_q}(s_q+ \textstyle{(\frac{q}{p})}) \displaystyle{\prod_{q\nmid m}}\mathbb{E}_{s_q}(1+\textstyle{(\frac{q}{p})}s_q) &= 2^{-\pi(y)}\sum_{p\leq x}\prod_{q | m}\textstyle{(\frac{q}{p})}.
|\alpha|^2+(|\beta|+|\gamma|)^2 \leq |\alpha' |^2+|\beta' z_1+\gamma' z_2|^2+|\delta' |^2 &\leq |\alpha'|^2+(|\beta'|+|\gamma'|)^2+|\delta'|^2 &=|\alpha'|^2+|\beta'|^2+2|\beta'| |\gamma'|+|\gamma'|^2+|\delta'|^2 &=|\alpha|^2+|\beta|^2+|\gamma|^2+2|\beta'| |\gamma'|
\sum_{\substack{d_n\le y p_{n+1}\le X}}f(d_n)=\frac{X}{\log^2 X}\left(1+o(1)\right)\sum_{d\le y}f(d)\mathfrak{S}(\{0, d\})+O\left(\frac{X}{\log^3 X}\int_{2}^{y}f(x)x\,dx\right).
L_{\tau_{i+1}-1}-L_{\eta_i-1}=0, L_{\eta_i-1}-L_{\tau_i-1}=Z_{\tau_i-1}(1+\xi_{\tau_i}+\xi_{\tau_i}\xi_{\tau_i+1}+\dots+\xi_{\tau_i}\xi_{\tau_i+1}\cdots\xi_{\eta_i-2}) \le 1+\xi_{\tau_i}+\xi_{\tau_i}\xi_{\tau_i+1}+\dots+\xi_{\tau_i}\xi_{\tau_i+1}\cdots\xi_{\eta_i-2}\le \tilde V_i,
\sup_{t< T^*}\left(\|z_{tt}\|_{H_\alpha^{4}}+\|z_t\|_{H_\alpha^\frac{9}{2}}\right)+\sup_{\substack{t< T^* \alpha\neq\beta}}\left|\frac{e^{i\alpha}-e^{i\beta}}{z(t,\alpha)-z(t,\beta)}\right|=\infty.
A_{\beta\alpha}\equiv \left[ \frac{\partial U_{\beta}}{\partial z_\alpha},\,\,\,\,\frac{\partial U_{\beta}}{\partial z_{\alpha,j}},\,\,\,\,\left(\frac{\partial U_{\beta}}{\partial z_{\alpha,k}}v_j + \frac{\partial \Phi^\beta_j}{\partial z_{\alpha,k}}\right)\right]\,\,\,\,\,(j,\,k=1,2,3),
\mathcal{E}(t)e^{-rt}+\int^{t}_{0}e^{-rt'}\left[\int_{\mathbb{R/Z}}\frac{1}{2}(\widetilde{W}^{{\rm T}}Q_{r}\widetilde{W})(t',s)ds - \mathcal{R}(t')\right]{\rm d}t'=\mathcal{E}(0)
L^{({\frac{1}{2}}, \sigma )}\left( \theta \right) = \sinh{\left( \theta +{\frac{1}{2}}\gamma \left( 1 + 2 \sigma^{3}\otimes S^{3}\right) \right)}+ \sinh{\gamma}\left( \sigma^{+}\otimes S^{-}+ \sigma^{-}\otimes S^{+}\right)
\int_{B_T}f\circ \varphi_t(x)\, dt = \mathrm{Vol}(B_T)\mu(f) + \sum_{i=2}^{d^+}\mathcal{D}_i^+(f)G_i(T)T^{d\frac{\log \lambda_i}{\log \lambda_1}}+ \mathcal{O}(|\partial B_T|),
\varphi_{{}_{\nu}}(g) ={\textstyle\frac{1}{4\pi}}\int \limits_{\mathbb{S}^2}\, d^2 p \,\overline{(p\cdot u)^{i\nu-1}}(p\cdot v)^{i\nu-1}, \,\,\, v=(1,0,0,0), \, u = gv = \Lambda(g^{-1})v
S^{(b)}_{b+1,0}(b(n))=\sum_{\substack{b_0-b_1+\cdots+(-1)^{k-1}b_{k-1}\equiv 0\mod b b_{k-1}b_{k-2}\cdots b_0<n_{k-1}n_{k-2}\cdots n_0}}(-1)^{b_0+\cdots+b_{k-1}}
\bar{\eta}_{k,L_n}:= \sum_{j=0}^{k-1}\eta_{j+1,k+L_n-j-1}, \tilde{\eta}_{k,L_n}:= \sum_{j=0}^{k-1}\sum_{i=0}^{k+L_n-j-1}\varrho_0^{k+L_n-j-i-1}\eta_{j+1,i}.
p_{\dot{x}}=\frac{\partial L}{\partial \ddot{x}}=2\frac{\ddot{x}}{\left\vert \dot{x}\right\vert^{3}}-2\frac{\left\langle \dot{x},\ddot{x}\right\rangle \dot{x}}{\left\vert \dot{x}\right\vert^{5}},
W^{(2)}=W^{(1)}\cdot W^{(1)}&=W^0_3(p,p_1,p_2)\ast W^0_3(p',p^{'}_1,p^{'}_2) &+h\left(W^1_1(p)\ast_h W^0_3(p',p^{'}_1,p^{'}_2)+W^0_3(p,p_1,p_2)\ast_h W^1_1(p')\right).
U_{i,j}=\begin{cases}\left\{g+3i+5,\frac{1}{2}(2g-3j+6i+7),\frac{1}{2}(2g+3j-12i-23)\right\}_{g-1}\mbox{if $j\equiv 1\pmod{2}$,}\left\{g+3i+5,\frac{1}{2}(2g-3j+6i+10)\frac{1}{2}(2g+3j-12i-20)\right\}_g \mbox{if $j\equiv 0\pmod{2}$.}\end{cases}
{\pi}^{-}= \frac{{\delta}L}{{\delta}{\partial}_{-}A_{-}}= 0 , \;{\pi}^{+}= \frac{{\delta}L}{{\delta}{\partial}_{-}A_{+}}= F_{- +}, \;{\pi}_{B}= \frac{{\delta}L}{{\delta}{\partial}_{+}B}= 0 , \;{\pi}_{\psi}= \frac{{\delta}L}{{\delta}{\partial}_{-}{\psi}}= i \bar{\psi}{\gamma}^{-}.
\tau^{*}(a,p;z):=\tau(a,p;q):=q^{C}\frac{\left[a,p;q^2\right]_{\infty}}{\left[a,p;q\right]_{\infty}}\textrm{,}C=\frac{a^2}{2p}-\frac{a}{2}+\frac{p}{12},
\alpha_n^i=\begin{cases}\varepsilon_n+\alpha_{n-1}^1 i=1, \varepsilon_n\alpha_{n-1}^{i-1}+\alpha_{n-1}^i n\geq 3 i=2,\ldots,n-1, \varepsilon_n\alpha_{n-1}^{n-1}i=n,\end{cases}
\sum_{k=0}^\infty(-1)^kq^{d\binom{k+1}{2}-rk}[2dk+r]\frac{(aq^r;q^d)_k(q^r/a;q^d)_k(q^r;q^d)_k}{(aq^d;q^d)_k(q^d/a;q^d)_k(q^d;q^d)_k}=[r]\,\frac{(q^{d-r};q^d)_\infty(q^{d+r};q^d)_\infty}{(aq^d;q^d)_\infty(q^d/a;q^d)_\infty}.
\delta{\cal L}_{1}={\frac{1}{1 - \lambda_{+ +}\lambda_{--}}}\left\{\left( 2 \lambda_{+ +}\lambda_{--}A_{+}\partial_{-}\alpha + 2 A_{-}\partial_{+}\alpha \right) - \delta \left( \lambda_{--}A_{+}^{2}+ \lambda_{+ +}A_{-}^{2}\right) \right\}
T_{\theta}=\left(\begin{array}{ccc}1 0 0 0 -\frac{i e^{-i \theta}}{\sqrt{2}}\frac{e^{i \theta}}{\sqrt{2}}0 -\frac{e^{-i \theta}}{\sqrt{2}}\frac{i e^{i \theta}}{\sqrt{2}}\end{array}\right)
\kappa^4 \int E R \overline{R}\left(\overline{\nabla}^2 R \right) \left(\nabla^2 \overline{R}\right) d^4\theta = -\frac{3}{2}\kappa^4\int \epsilon \left( \overline{\nabla}^2 +\frac{1}{3}\overline{R}\right)\left[R \overline{R}\left(\overline{\nabla}^2 R \right) \left(\nabla^2 \overline{R}\right) \right] d^2\theta + \mathrm{h.c.}
\mathbf{y}_{k,l}=\underset{N\times \tilde{M}}{\underbrace{\mathbf{G}_{k,l}\mathbf{\Lambda}_k^{\frac{1}{2}}\mathbf{U}_k \mathbf{P}_k}}\sum^{L}_{j=1}\mathbf{w}_{k,j}s_{k,j}+\mathbf{n}_{k,l},
K_{\nu}(x)=\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}2^{s-2}\Gamma\left(\frac{s-\nu}{2}\right)\Gamma\left(\frac{s+\nu}{2}\right)\,x^{-s}ds,\,\,\,\,\,x>0,\,\,\,\mu:=(s)>\max\{0,(\nu)\}.
\hat{a}{\cal P}= \frac{1}{q^{2}}{\cal P}\hat{a}- \frac{\mu}{q^{3}}\hat{a}, \hat{b}{\cal P}= \frac{1}{q^{2}}{\cal P}\hat{b}- \frac{\mu}{q^{3}}\hat{d}\hat{c}^{- 1}\hat{a},
(1,1)_{q,q^k}=W_0(q^k)\prod_{i=2}^n \genfrac[]{0pt}{0}{ik-1}{k-1}= \Big(\prod_{i=2}^n \frac{1-q^{ik}}{1-q^k}\Big) \prod_{i=2}^n \genfrac[]{0pt}{0}{ik-1}{k-1}=\prod_{i=2}^n \genfrac[]{0pt}{0}{ik}{k}.
{R^{\alpha}}_{\beta}={\frac{1}{4}}\left( 2{f^{\alpha}}_{\beta i}{f^{i}}_{\gamma \delta}+{f^{\alpha}}_{\beta \epsilon}{f^{\epsilon}}_{\gamma \delta}-{f^{\alpha}}_{\gamma \epsilon}{f^{\epsilon}}_{\beta \delta}\right) e^{\gamma}\wedge e^{\delta}.
Q_{F_{1}, V_{8}}^{M a x w e l l}= \int_{V_{8}}*_{s}j_{D 2}^{M a x w e l l}= \frac{T_{F 1}}{2}\left( 1 - \cos \theta_{0}{+ F_{2}\{\mathrm{f l u x \a b o v e \b r a n e}}\atop{- F_{2}\ \mathrm{f l u x \b e l o w \b r a n e}}}\right) ,
III=\frac{1}{n...(n-k+1)}\sum\limits_{i_{k},...,i_{l+1}}\|A_{i_{k}}^{*}...A_{i_{l+1}}^*A_{i_{l+1}}...A_{i_{k}}\|\leq e^{\frac{lk}{n-k-l}}\tilde{P}_{l}(A_{i_{1}},...,A_{i_{l}})
\tau^{2s+1}|T_{2j}(i\tau)| \lesssim \sqrt{A_p B_p}\Vert (u,v) \Vert^2_{H^s}\Vert (u,v) \Vert_{l^2_1(DU^2)}^{2j-2}\int_1^{\infty}\tau^{2s}|T_{2j}(i\tau)|d\tau \lesssim A_i \Vert (u,v) \Vert^2_{H^s}\Vert (u,v) \Vert_{l^2_1(DU^2)}^{2j-2}
\sum_{j=1}^{n+1}\| U^j \|_{C^{t_j,\alpha}_{\phi, \varphi_j}(\Omega)}\le C\left(\sum_{j=1}^{n+1}\| (L U)_j\|_{C^{-s_j, \alpha}_{\phi, \phi^{t_j+s_j}\varphi_j}(\Omega)}+ \sum_{j=1}^{n+1}\| U^j \|_{L^2_{\phi^{-n}\varphi^2_j}(\Omega)}\right)
\|\mathcal P(t) - \mathcal P(t_0)\| = \|C_U T_{(V^*b^0)_{[j]}}\mathcal P_{1}(t) C_V - C_U T_{(V^*b^0)_{[j]}}\mathcal P_{1}(t_0) C_V\| & \leq \|T_{(V^*b^0)_{[j]}}\| \|\mathcal P_{1}(t) - \mathcal P_{1}(t_0)\| \to 0, t \to t_0.
\begin{aligned}b_6 b_6 = b_0b_{12}-3^2\dfrac{b_1^2}{b_0^1}b_{10}+3\cdot 37\dfrac{b_1^4}{b_0^3}b_8-19\cdot 43 \dfrac{b_1^6}{b_0^5}b_6+59^2 \dfrac{b_1^8}{b_0^7}b_4-5\cdot 17\cdot 83 \dfrac{b_1^{10}}{b_0^9}b_2 +4289\dfrac{b_1^{12}}{b_0^{11}}b_0\end{aligned}
h([p,q,r][a,b,c]) = |r_1raa_1| - |pq_1| - |b_1c| &= |r_1ra| + |a_1|- |pq_1| - |b_1| - |c| &= |b| + |a_1| - |pq_1| - |c| &= |b| + |qq_1| - |r| -|a| - |p| - |q_1| - |c| &= |b| + |q| - |r| - |p| -|c| - |a| &=h[p,q,r]+ h[a,b,c].
\lambda^\star:= \inf_{\underset{w\in X^{1/2}_0(\mathcal{C}_\Omega)}{\Psi(w)>0}}\frac{\Phi(w)}{\Psi(w)}\geq \frac{\lambda_1^{1/2}}{c_f\left\|\beta\right\|_{L^{\infty}(\Omega)}}.
\rho\left( A_{a}\left( G\right) \right) \geq\rho\left( Q\right)-\left( \left( 1-\alpha\right) \rho\left( Q\right) +\left(2\alpha-1\right) \rho\left( A\right) \right) & =\alpha\rho\left( Q\right) +\left( 1-2\alpha\right) \rho\left(A\right) ,
D_i = \{(x_1,x_2)\mid 0\le x_1 \le x_2 \le \frac{1}{2},\{g_1(x_i)\}<\{g_2(x_i)\}\}\quad(i=1,2), D_i = \{(x_3,x_4)\mid \frac{1}{2}\le x_3 \le x_4 \le 1,\{g_1(x_i)\}<\{g_2(x_i)\}\}\quad(i=3,4),
\stackrel{\mathrm{H W}}{S^{\alpha \beta}}= \int \tau_{\mathrm{K G}}^{\alpha \beta 0}d^{3}x = \frac{1}{4 m i}\int \left[ \left( D^{0}\overline{{\Psi}}\right) \sigma^{\alpha \beta}\Psi - \overline{{\Psi}}\sigma^{\alpha \beta}\left( D^{0}\Psi \right) \right] d^{3}x .
0 = I_0 \leq I[\nu + t (\mu-\nu)] = I[\nu] + t \bigg( \iint f(\theta - \phi) d \nu(\theta) d(\mu-\nu) (\phi) + \iint f(\theta - \phi) d (\mu - \nu) (\theta) d\nu (\phi) \bigg) + t^2 I[\mu-\nu],
E_{jk}= \frac12 [ \tilde f_j^+, \tilde f_k^-] (j,k=1,\ldots,m),E_{m+j,m+k}= \frac12 \{\tilde b_j^+, \tilde b_k^-\}(j,k=1,\ldots,n); & E_{j,m+k}= \frac12 \{\tilde f_j^+, \tilde b_k^-\}, E_{m+k,j}= \frac12 \{\tilde b_k^+, \tilde f_j^-\}(j=1,\ldots,m; k=1,\ldots,n).