harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathsf Z(a(bm_1+cm_2))=\Big\{\left(b-\left\lfloor\tfrac{b}{m_2}\right\rfloor m_2,c+\left\lfloor\tfrac{b}{m_2}\right\rfloor m_1,0\right),\ldots,(b-m_2,c+m_1,0),(b,c,0), (b+m_2,c-m_1,0)\ldots, \left(b+\left\lfloor\tfrac{c}{m_1}\right\rfloor m_2,c-\left\lfloor\tfrac{c}{m_1}\right\rfloor m_1,0\right),(0,0,a)\Big\}
\mathbb{E}\left[\Vert f_1-\hat{f}_1^*\Vert^2\right]\leq \mathbb{E}\left[1_{\{\hat{J}_0= J_0\}}\Vert f_1-\hat{f}_1^*\Vert^2\right]+(\left\|f_1\right\|+k_n)^2\mathbb{P}\left( J_0\neq \hat{J}_0 \right)
\| F_{v_0}(\eta) - F_{v_0}(\xi) \|_{1,\alpha}= \big\| \int_0^1 \frac{d}{dr}F_{v_0}\big( r\eta + (1-r)\xi \big) dr \big\|_{1,\alpha}\leq \int_0^1 \| \delta_{\eta - \xi}F_{v_0}\big( r\eta + (1-r)\xi \big) \|_{1,\alpha}dr \lesssim \| v_0\|_{1,\alpha}\| \eta - \xi \|_{1,\alpha},
\begin{array}{l l}{\displaystyle{\sum_{s = 1}^{D / 2}\, [ Z_{s}, \bar{Z}_{s}] = \sum_{s = 1}^{D / 2}\, \frac{1}{\, \theta_{s}}}\, , ~ ~ ~}{~ ~ ~ ~ [ Z_{s}, Z_{r}] = 0 \, ,}\end{array}
u(t) - w_n (t) \geq \left( w(x_n) + 3\theta_n g (b_n) - 2((x_n + b_n) - t)\theta_n g' (\beta_n)\right) & \phantom{=}{}- \left( w(x_n) + 2 \theta_n g'(\beta_n) (t- (x_n + b_n)) + 2 \theta_n g (b_n)\right) & = \theta_n g(b_n).
P_{\lambda,d}(\eta_{t}^{\delta_1}(O)=1) =P_{\lambda,d}(\zeta_t(O)>0) &\geq\frac{[E_{\lambda,d}\zeta_t(O)]^2}{E_{\lambda,d}\zeta_t^2(O)}\geq\frac{C^2(\lambda,d,\zeta)t^{-d}}{F_t(O)}.
\mathcal{L}_j[w_j]= \left[ \mathcal X_j -2|D^2 u_j|^2+ 4 \alpha_j a_{0,j}\frac{\langle \nabla u_j,\nabla \phi\rangle}{\phi}+ 2\left(\mathcal Q_j-\frac{h_j'(u_j)}{h_j(u_j)}z_j\right) a_{1,j}\right] \phi^{2 \alpha_j},
\check{R}(z_1 -z_2 ) \phi_{ad}(z_1 )\otimes \phi_{dc}(z_2 )=\sum_{b}W_{z_1 -z_2}\left( \begin{array}{cc}a b d c \end{array}\right)\phi_{ab}(z_2 )\otimes \phi_{bc}(z_1 ).
\sum_{j,l}\langle h_jf(h_l),\tau|_{g_1X_l\cap X_j}\pi \rangle =\sum_l\langle \pi^{**}(f(h_l)),(g_1^{-1}\tau)_l \rangle, \sum_{j,l}\langle f(h_jh_l),\tau|_{g_1X_l\cap X_j}\pi \rangle =\sum_s\langle \pi^{**}(f(h_s)), \tau_s \rangle, \sum_{j, l}\langle f(h_j),\tau|_{g_1X_l\cap X_j}\pi \rangle =\sum_j\langle \pi^{**}(f(h_j)),\tau_j \rangle.
K = *{(\kappa^2)\over 2}={i \over 2}\left[ f_{\mu_1\bar\nu_1}\, f_{\mu_2\bar\nu_2}\right] \epsilon^{\mu_1\mu_2}_{\hspace{0.25in}\alpha}\,\epsilon^{\bar\nu_1\bar\nu_2}_{\hspace{0.25in}\bar\beta}\,\, dz^\alpha \wedge d\bar{z}^\beta
\Gamma_{1}^{( I )}= \frac{1}{2}\mathrm{T r}\ln \left[ \delta ( x_{1}- x_{2}) - 2 g^{2}\int_{y}\Delta_{\varphi}( x_{1}, y ){\bar{\chi}}( y ){\mathcal D}^{- 1}( y , x_{2}) \chi ( x_{2}) \right] .
T \approx ( m t^{3 ( 1 - \delta ) / 2})^{- 1 / 2}\left( D_{1}( k ) \exp \left[ - i \frac{2 m t^{3 ( 1 - \delta ) / 2}}{3 ( 1 - \delta )}\right] + D_{2}( k ) \exp \left[ i \frac{2 m t^{3 ( 1 - \delta ) / 2}}{3 ( 1 - \delta )}\right] \right) ,
\frac{1 - F_{\rho}(tk)}{t^{-\gamma}(1 - F_{\rho}(k))}= \left(\frac{1 - F_{\rho}(tk)}{1 - F(tk)}\right) \left(\frac{1 - F(k)}{1 - F_{\rho}(k)}\right) \left(\frac{1 - F(tk)}{t^{-\gamma}(1 - F(k))}\right) &= \left(\frac{1 - F_{\rho}(tk)}{1 - F(tk)}\right) \left(\frac{1 - F(k)}{1 - F_{\rho}(k)}\right) \frac{\mathcal{L}(tk)}{\mathcal{L}(k)}.
\lambda_{L}^{( 3 )}= \bar{\lambda}_{L}\biggr [ 1 + \bar{\lambda}_{L}\frac{2}{\pi}\sum_{i = 0}^{L}\frac{k^{2 ( L - i )}\Lambda^{2 i + 1}}{2 i + 1}+ \bar{\lambda}_{L}^{2}\biggr ( \frac{2}{\pi}\sum_{i = 0}^{L}\frac{k^{2 ( L - i )}\Lambda^{2 i + 1}}{2 i + 1}\biggr )^{2}\biggr ]
\mathbb{H}_{\rm Bog}= \sum_{p\neq 0}\left(p^2+8\pi\mathfrak{a}\frac{n}{\ell}\right) a_p^*a_p + \frac{1}{2}\sum_{p \neq 0}n\widehat{\epsilon}_{\ell,\lambda}(p)(a_p^*a_p^*+a_pa_p) + \frac{1}{2}\sum_{p \neq 0}\frac{|n\widehat{\epsilon}_{\ell,\lambda}(p)|^2}{2p^2}.
K_Q = - |\nabla F_Q|^2 -\Delta \Big\{\sum_{j\not\in Q}\tfrac{Z}{2}\sqrt{|x_j|^2+1}-\sum_{j,k\not\in Q,j<k}\tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}-\sum_{j,k\in Q,j<k}\tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}\,\Big\}\,.
\mathcal{P}_{1}= - (s_{0}^{2}M+s_{0}D+K)^{-1}(2s_{0}M+ D), \mathcal{P}_{2}= -(s_{0}^{2}M+s_{0}D+K)^{-1}M, \mathsf{Q}= \ (s_{0}^{2}M+s_{0}D+K)^{-1}F,
e_\leq(A) = \frac{\mathcal E(\Omega) + \mathcal E(G_n)}{|\Omega|+|G_n|}+ o(1)\geq \frac{\mathcal E(\Omega)}{|\Omega|+|G_n|}+ \frac{|G_n|}{|\Omega|+|G_n|}\,e_\leq(A) + o(1) \,.
\lim_{N\rightarrow\infty}\Psi_{m,N}(a,b,q) = \lim_{N\rightarrow\infty}G_{m,N}(a,b,q)=(-aq^{3-(-1)^{m+1}m},-bq^{3-(-1)^m m};q^6)_\infty(-q^3;q^3)_\infty.
\partial_{\chi_{( A )}}\Gamma_{Y_{1}}( 0 ) \Gamma_{\varphi_{1}\varphi_{1}}( 0 ) + \partial_{\chi_{( A )}}\Gamma_{Y_{1}\varphi_{1}}( 0 ) \Gamma_{\varphi_{1}}( 0 ) = - \partial_{\xi_{( A )}}\Gamma_{\varphi_{1}}( 0 )
\begin{aligned}v_{s+k+1}^* = v_{s+k+1}- \omega_{k+1}\omega_{k+1}\cdot\left( v_{s+k+1}- v_{i_{k+1}}^\prime (\tau;0) \right) & v_{i_{k+1}}^{\prime *}= v_{i_{k+1}}^\prime (\tau;0) +\omega_{k+1}\omega_{k+1}\cdot \left(v_{s+k+1}- v_{i_{k+1}}^\prime (\tau;0) \right)\end{aligned}
\mathbf{E}\Big(\sup_{0 \le t' \le t}\big|\xi(x(t')) - z(t')\big|^2\Big) \le \Big(\frac{3\kappa_1^2\,t^2}{\beta \rho}+ \frac{96\kappa_2^2 t}{\beta^2 \rho}\Big) +\Big(3L_b^2\,t+ \frac{48L_\sigma^2}{\beta}\Big) \, \int_0^t \Big(\mathbf{E}\sup_{0 \le t' \le s}\big|\xi(x(t')) - z(t')\big|^2\Big)\,ds\,,
\left(\sum^{\infty}_{n,m=-\infty}q^{2n^2+2nm+2m^2}\right)^2=-\frac{1}{2}T_{3,r}=\left(\frac{\theta_3(q)^4+3\theta_3(q^3)^4}{4\theta_3(q)\theta_3(q^3)}\right)^2.
\log\left(\frac{\Gamma(t+1)}{\Gamma(r+1)}\right) = -\gamma(t-r) + (t-r)\sum_{l \geq 2}\frac{(-1)^l}{l}\zeta(l)\left(\sum_{0 \leq k \leq l-1}t^kr^{l-1-k}\right) &= (t-r)\left(-\gamma +\sum_{l \geq 2}\frac{(-1)^l}{l}\left(\sum_{0 \leq k \leq l-1}t^kr^{l-1-k}\right) \zeta(l)\right).
u_0=R^{-1}\log\bigl(\tfrac{1-R}{1+R}\bigr),\ \u_1=R^{-2}\log\bigl(\tfrac{1-R}{1+R}\bigr)+2 R^{-1},\ \u_2=\tfrac{3-R^2}{2 R^3}\log\bigl(\tfrac{1-R}{1+R}\bigr)+3 R^{-2};
u_T(x)\prod_{k=1}^{{T\over 2}}(f(x)-v_{2k-2}^2(x)) \sim u_0(x)\prod_{k=1}^{{T\over 2}}(f(x)-v_{2k-1}^2(x)) u_T(x)u_0(x)\prod_{k=1}^{\lfloor{T\over 2}\rfloor}(f(x)-v_{2k-1}^2(x)) \sim \prod_{k=0}^{\lfloor{T\over 2}\rfloor}(f(x)-v_{2k}^2(x)) .
f_{-}^{n}= \frac{D_{0}^{n - 1}}{D_{0}^{n}}, f_{0}^{n}= \frac{\dot{D}_{0}^{n}}{D_{0}^{n}}+ \tau , f_{+}^{0}= \frac{D_{0}^{n + 1}}{D_{0}^{n}}.
P_{_0}(y^2) = \frac{1}{\Gamma(1/2)}\gamma\left(\frac{1}{2}, \frac{y^2}{2 \sigma_w^2}\right), P_{_1}(y^2) = \frac{1}{\Gamma(1/2)}\gamma\left(\frac{1}{2}, \frac{y^2}{2(P_x+\sigma_w^2)}\right),
\psi_{\Lambda_{1}}(b_{\Lambda})=\sum_{i,j\in I}\prod_{x\in \Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, j}^* b_{x}\right)\prod_{y\in \Lambda_{1}\setminus\Lambda}\hbox{Tr}_{\mathcal{H}_{y}}\left(h_{y, i}h_{y, j}^* \right).
m\rq_{1,n}(x,y)+\phi_n(m\rq_{1,0}(x,y))=m_{1,n}(x,y)+m_{1,0}(\phi_n(x),y)+m_{1,0}(x,\phi_n(y)), m\rq_{1,n}(x,y)-m_{1,n}(x,y)=m_{1}(\phi_n(x),y)+m_1(x,\phi_n(y))-\phi_n(m\rq_1(x,y))=\delta_1 \phi_n(x,y),
1_{L_1 (t)}^* = 1_{( \phi_1^* )^{-1}( \mathbb{R}_{> t})}= 1_{( - J_1 (t) , J_1 (t) )}, 1_{L_2 (t)}^* = 1_{( \phi_2^* )^{-1}( \mathbb{R}_{> t})}= 1_{( - J_2 (t) , J_2 (t) )}
I (s, t) =\frac{3^9}{2^{12}}\int_0^\infty dT \; e^{T}\;\frac{\det (1+S \tilde{X}(T))}{\det (1-S \tilde{N}(T))^{13}}\exp \left(-\frac{1}{2}p_iQ_{ij}p_j\right)
\frac{\kappa}{2}\frac{d}{dt}\int_{\Omega}|\nabla\theta|^2 x+\int_{\Omega}\rho |\theta_t|^2 x = -\int_{\Omega}\rho u\cdot \nabla \theta \theta_tx-\int_{\Omega}Pu \theta_tx+\int_{\Omega}Q(u) \theta_tx= \sum_{i=13}^{15}L'_i.
\begin{array}{l}{\nabla^{2}\phi_{e}= - \rho_{e}( \mathbf{x}) , \nabla^{2}\phi_{m}= \rho_{m}( \mathbf{x}) ,}\\{\nabla^{2}\mathbf{A_{m}}= - \mathbf{J_{e}}( \mathbf{x}) , \nabla^{2}\mathbf{A_{e}}= \mathbf{J_{m}}( \mathbf{x})}\\ \end{array}
a \left( \eta \right) = a_{\rm min}\, \sqrt{1+ \gamma^2 \, \eta^2}\;\;,\;\; a_{\rm min}= \sqrt{\frac{3 \, Q^2 \, \omega_4}{16 \, M}}\;,\; \gamma=\frac{4\,M}{\sqrt{3}\,Q}\,\,.
\left(\frac{c(x) + c(y)}{2}\right)^N e\left(x\left(i-\frac{a}{2}\right) + y\left(j-\frac{b}{2}\right)\right) = \exp\left(-\frac{x_2^2 + y_2^2}{2}-\frac{\left(i - \frac{a}{2}\right)^2 + \left(j - \frac{b}{2}\right)^2}{N}\right) \times \left(1 + O\left(\frac{x_2^4 + y_2^4}{N}+ \frac{i^4 + j^4+1}{N^3}\right) \right).
{M}_{\phi}(h_{1}+h_{2}\cdot n)(x) = L_{\phi}(h_{1})(x) + L_{\phi}(h_{2})(x) \cdot n &= \phi(x)\cdot h_{1}(x)+ \phi(x) \cdot h_{2}(x) \cdot n &= \phi(x)(h_{1}(x) + h_{2}(x) \cdot n) &= \phi \cdot (h_{1}+ h_{2})(x).
\mathbf{T}_i\mathbf{G}_j =\mathbf{G}_j\mathbf{T}_i|i-j|>1 \mathbf{T}_i \mathbf{G}_i =\mathbf{G}_i \mathbf{T}_{i+1}\mathbf{T}_{i+1}\mathbf{G}_i =\mathbf{G}_i \mathbf{T}_i.
2E(\phi) =\|(-\Delta)^{\frac{s_2}{2}}\phi\|_2^2+\frac{Np-2N-4s_1}{N(p-2)}\|(-\Delta)^{\frac{s_1}{2}}\phi\|_2^2 &\leq \frac{2(Np-2N-2s_1)}{N(p-2)}\|(-\Delta)^{\frac{s_1}{2}}\phi\|_2^2=\frac{2(Np-2N-2s_1)}{2s_1p-Np+2N}\|\phi\|_2^2.
\begin{array}{lcl}\tilde{Z}_{jnlm}\left(v\right)\equiv Z_{jnlm}\left(-v\right) = \int_{0}^{+\infty}d\omega'\,\frac{e^{-i\omega'v}}{\sqrt{\omega'}}\hat Z_{jnlm}\left(\omega'\right)\left(-e^{-{{\pi\omega_j}\over\kappa}}\right) \ + \int_{0}^{+\infty}d\omega'\,\frac{e^{i\omega'v}}{\sqrt{\omega'}}\hat Z_{jnlm}\left(\omega'\right)\;.\end{array}
p''(\eta_1)+p''(\eta_3)=~ -3\eta_1(1+\eta_1^2)^{-5/2}+3\eta_3(1+\eta_3^2)^{-5/2}=~ \frac{-3\eta_1(1+\eta_3^2)^{5/2}+3\eta_3(1+\eta_1^2)^{5/2}}{(1+\eta_1^2)^{5/2}(1+\eta_3^2)^{5/2}},
d_2^o \in [ \mid r_1 + r_2 -r_3 \mid, r_1 + r_2 + r_3] \cap [ r_3 - r_2 +r_1, - r_1 + r_2 + r_3] = [ r_3 - r_2 +r_1, - r_1 + r_2 + r_3] \neq \emptyset,
\tilde{\xi}^t = \left( \begin{array}{cccc|ccccc}\frac{t}{d'}x_t 0 0 0 \cdots \cdots \cdots 0 0 \frac{t}{d'}0 0 0 y_t \cdots y_t 0 0 0 \frac{t}{d'}0 0 z_t \cdots z_t \frac{t}{d'}0 0 0 \frac{t}{d'}\frac{t}{d'}z_t \cdots z_t 0 \end{array}\right)
\mathbf{Y}=\frac{1}{2}\mathbf{H}^{-1}\mathbf{W}=\frac{1}{2(H_{33}H_{44}-H_{34}^2)}\begin{pmatrix}\delta_3H_{33}H_{44}-\gamma H_{34}^2 H_{44}H_{34}(\gamma - \delta_4) H_{33}H_{34}(\gamma - \delta_3) \delta_4H_{33}H_{44}-\gamma H_{34}^2\end{pmatrix},
P_1^{(a,b)}P_n^{(a,b)}= \frac{1}{a(b-1)}P_{n-1}^{(a,b)}+ \frac{b-2}{a(b-1)}P_n^{(a,b)}+\frac{a-1}{a}P_{n+1}^{(a,b)}\quad\quad(n\ge1) .
\mathcal{C}^{++-}= \mathcal{C}^{++-}(v,v,\bar{v}) := 2M_{++}( v, Q_{+-}(v,\bar{v}))+ M_{+-}( Q_{++}(v,v), \bar{v}) &+ M_{+-}(v, \bar{Q_{+-}(v,\bar{v})})+ 2M_{--}( \bar{Q_{--}(\bar{v},\bar{v})}, \bar{v}) + M_{++-}(v,v, \bar{v}) ,
\frac1{(n+k)n^{r_2}}=\sum_{m=2}^{r_2}\frac{(-1)^{r_2-m}}{n^m k^{r_2+1-m}}+ \frac{(-1)^{r_2+1}}{k^{r_2}}\Big(\frac{1}{n}-\frac{1}{n+k}\Big).
[a_l,a_m,a_k,a_{\infty}]= \exp(4 \pi i (\eta_{m}-\eta_{l})_1(\eta_k)_2)\left( \frac{\theta[\eta_{U_\eta\circ(V \cup \{k,l\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{k,l\})}](\Omega)}{\theta[\eta_{U_\eta \circ (V \cup \{k,m\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{k,m\})}](\Omega)}\right)^2.
\tau^{(1)}= \frac{\nu_{\min}+ 1}{2}, \rho^{(1)}= \frac{1}{r + 1}, & \tau^{(2)}= \frac{\nu_{\min}+ 1}{2}, \rho^{(2)}= \frac{1}{r - q + 1}, & \tau^{(3)}= \frac{1}{2}, \rho^{(3)}= \frac{\theta}{\theta + 1},
\beta(B_1,B_2) \stackrel{def}{=}\sup_{{\bf P}(A_1 \in B_1) \{\bf P}(A_2 \in B_2) > 0}\left| \ \frac{{\bf P}(A_1 \cap A_2) -{\bf P}(A_1){\bf P}(A_2)}{{\bf P}(A_1){\bf P}(A_2)}\ \right|.
I^{\ast}_{W}=\frac{L}{2}\int_{\bf R}dt\int_{\Sigma}\eta_{ab}\{-^{(2)}A^{(+)a}\wedge^{\mbox{}(2)}\dot{A}^{(+)b}+^{(2)}A^{(-)a}\wedge^{\mbox{}(2)}\dot{A}^{(-)b}\}.
c^k_{\mathrm{b2s},1}\left( N_{\mathrm{sc}}\right) = \mathsf{E}\left\lbrace B N_{\mathrm{sc}}\log_2\left[ \frac{P_{\mathrm{bs}}}{BN_{\mathrm{bs}}\sigma^2_{\mathrm{sc}}}\left(\max \{\|\mathcal{H}^{\dagger}_{\mathrm{b2s},i}\| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}+1 \right] \right\rbrace,
\sum_{f\in M_{n}}\alpha_{2}(f)=\frac{q^{n/2}}{\zeta_{q}(3)}\sum_{\substack{j=n\mod 2 0\leq j \leq \lfloor\dfrac{n}{3}\rfloor-2}}q^{\frac{-j}{2}}+q^{\lfloor{\frac{n-\lfloor n/3 \rfloor}{2}\rfloor}}
E_{\lambda}(t)-E_{\infty}(t)=\int_{0}^{t}iU_{\varepsilon}(t-s)\Big(\big[\{{I_{\varepsilon}}|E_{\lambda}|^{2}\}E_{\lambda}-\{{I_{\varepsilon}}|E_{\infty}|^{2}\}E_{\infty}\big] -(Q_{\lambda}E_{\lambda})\Big)ds,
L = i \left( w_{\alpha}\nabla_{-}\varphi^{\alpha}- \tilde{w}_{\overline{{{\alpha}}}}\nabla_{-}\tilde{\varphi}^{\overline{{{\alpha}}}}\right) ( \varphi , \tilde{\varphi}) - q A_{-}^{a}{W}_{a}( \varphi , \tilde{\varphi})
\nu_{\alpha \beta}^{g_{1}g_{2}}\: = \: \nu_{\alpha \beta}^{g_{2}}\, g_{2}^{*}\nu_{\alpha \beta}^{g_{1}}\, \left( h_{\alpha}^{g_{1}, g_{2}}\right) \, \left( h_{\beta}^{g_{1}, g_{2}}\right)^{- 1}
\left\{Q_{\alpha a},Q_{\beta b}\right\}=\gamma_{\alpha \beta}^\mu \gamma_{ab}^{ij}P_{\mu ij}+\gamma_{\alpha \beta}^{\mu_1\mu_2\mu_3}\delta_{ab}T_{\mu_1\mu_2\mu_3}^{+}+\gamma_{\alpha \beta}^{\mu_1\mu_2\mu_3}\gamma_{ab}^{ijkl}Z_{\mu_1\mu_2\mu_3ijkl}^{+}
( Q_{2 5}^{0}A )_{i j}= ( \phi_{2 5})_{i k}\star A_{k j}- ( - 1 )^{A}A_{i k}\star ( \phi_{2 5})_{k j}= ( Q_{2 5}^{R}A )_{i j}+ ( Q_{2 5}^{L}A )_{i j}= ( Q_{2 5}^{0}A )_{i j}
={\cal E}+ \left( \begin{array}{c c c c}{{( ( p^{2}- q^{2}) a + 2 p q b )}}&{{( p a + q b )}}&{{( p b - q a )}}&{{( ( p^{2}- q^{2}) a + 2 p q b )}}\\{{( p a + q b )}}&{{0}}&{{0}}&{{( p a + q b )}}\\{{( p b - q a )}}&{{0}}&{{0}}&{{( p b - q a )}}\\{{( ( p^{2}- q^{2}) a + 2 p q b )}}&{{( p a + q b )}}&{{( p b - q a )}}&{{( ( p^{2}- q^{2}) a + 2 p q b )}}\\ \end{array}\right) .
0= \int Ku^{\frac{4}{n-2}}\phi_{k,i}v=\int K(u_{\alpha, \beta}+\alpha^{j}\varphi_{j}+v)^{\frac{4}{n-2}}\phi_{k,i}v = \int_{[u_{\alpha, \beta}+\alpha^{j}\varphi_{j}\geq v]}K(u_{\alpha, \beta}+\alpha^{j}\varphi_{j})^{\frac{4}{n-2}}\phi_{k,i}v +O(\Vert v \Vert^{2})
\lim_{T \rightarrow \infty}\frac{\sum_{\gamma \in P_*(T,\Delta)}e^{\ell_\psi(\gamma) + it\ell_\phi(\gamma)/\sqrt{T}}}{\sum_{\gamma \in P_*(T,\Delta)}e^{\ell_\psi(\gamma)}}= e^{-\sigma_{\phi,\psi}^2t^2/2}.
\left\lfloor \frac{t-m\alpha}{v}\right\rfloor = \lfloor(t-m\alpha)(\beta-1/u) \rfloor & = \left\lfloor (t\beta-m\alpha\beta)-\frac{(t-m\alpha)}{u}\right\rfloor & = (t\beta-m\alpha\beta)+\left\lfloor -\frac{(t - m\alpha)}{u}\right\rfloor.
\nu(n,r) = \frac{1}{\sqrt{2\pi nr}}\exp\left\{-\frac{n\log n}{r}+\frac{n}{r}+\sum_{N=1}^rd_{rN}n^{(r-N)/r}\right\}\left(1+O\left(n^{-1/r}\right)\right) &\ll \exp\left\{-\frac{n\log n}{r}+\frac{n}{r}+\sum_{N=1}^rd_{rN}n^{(r-N)/r}-\log r \right\},
\left[\begin{array}[c]{ccc}l_{1}l_{2}l_{3}m_{1}m_{2}m_{3}\end{array}\right] \!\!{[j]}\ & =\sqrt{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}\left(\begin{array}[c]{ccc}l_{1}l_{2}l_{3}-m_{1}-m_{2}-m_{3}\end{array}\right) \left\{\begin{array}[c]{ccc}l_{1}l_{2}l_{3}j j j\end{array}\right\}
K_{\lambda \nu}^{\hspace{.3 true cm}\alpha}=-\frac{1}{2}(S_{\lambda \hspace{.2 true cm}\nu}^{\hspace{.1 truecm}\alpha}+ S_{\nu \hspace{.2 true cm}\lambda}^{\hspace{.1 truecm}\alpha}- S_{\lambda \nu}^{\hspace{.3 true cm}\alpha}) \; .
\lim_{v \to x_{i}}\left[ ( \partial_{x}+ \partial_{u})^{2}f ( x , x_{i}, u , v ) \right] = - ( 2 \pi )^{2}\frac{( x - u )^{2}}{( x - x_{i})^{2}( u - x_{i})^{2}}\: g ( x , x_{i}, u ) .
(\widetilde{V}_1 \widetilde{V}_2 \widetilde{V}_c)^2 \widetilde{C^\prime}_5 \widetilde{C}_5 = \partial, \widetilde{C}_1^2 \widetilde{W}_1 \widetilde{W}_2 \widetilde{W}_c \widetilde{W_1^\prime}\widetilde{W_2^\prime}\widetilde{W_c^\prime}= \partial.
A_{\perp}^{i c}( x ) = \sum_{\lambda}\int_{k^{+}> 0}{\frac{d k^{+}d^{2}k_{\perp}}{1 6 \pi^{3}k^{+}}}\Bigl [ a^{c}( k , \lambda ) \epsilon_{\perp}^{i}( \lambda ) e^{- i k \cdot x}+ a^{c \dagger}( k , \lambda ) \epsilon_{\perp}^{i *}( \lambda ) e^{i k \cdot x}\Bigr ] .
G_\pm (\alpha_\pm) = e^{\alpha_\pm \bar{T}_\pm}= 1 + \alpha_\pm \bar{T}_\pm = \left(\begin{array}{ccc}1 \frac{\alpha_\pm}{\sqrt{2}}\mp i \frac{\alpha_\pm}{\sqrt{2}}0 1 0 0 0 1\end{array}\right)
I_1+I_2 +I_3 \leq \gamma hk{\rm log}\delta (b-a) +\frac{\gamma kN^4}{2\tau h}D_{i,j_{k+1}}^{-1}+C_1 +C_0C^*_\phi h & =\frac{1}{2N}{\rm log}\delta (b-a) +\frac{N^3}{4\tau h^2}D_{i,j_{k+1}}^{-1}+C_1+C_0C^*_\phi h <0
=\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\chi_{E_{f}(r)}(x)\chi_{E_{g}(s)}(x)dxdrds=\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\chi_{E_{f}(r)\cap E_{g}(s)}(x)dxdrds.
\frac{J_{\alpha}(x)}{(\frac{x}{2})^{\alpha}}=\frac{e^{-t}}{\Gamma(\alpha+1)}\sum_{k=0}^{\infty}\frac{L_{k}^{(\alpha)}(\frac{x^{2}}{4t})}{(\begin{matrix}k+\alpha\\k\end{matrix})}\frac{t^{k}}{k!}.
{\hat{\pi}}^{i}( \varepsilon ) \simeq m{\frac{({\hat{x}}^{i}( \varepsilon ) -{\hat{x}}^{i}( 0 ) )}{\varepsilon}}+{\frac{e}{2}}({\hat{x}}^{k}( \varepsilon ) -{\hat{x}}^{k}( 0 ) ){\hat{F}}^{i k}\, .
2^{j(s-\sigma)}W^{\frac{1}{p}}\left[\varphi_j*\left(\dot I_\sigma\vec f\right)\right] =2^{j(s-\sigma)}W^{\frac{1}{p}}\left[\varphi_j*\left(|\cdot|^\sigma\widehat{\vec f}\right)^\vee\right] &=2^{j(s-\sigma)}W^{\frac{1}{p}}\left(\widehat{\varphi_j}|\cdot|^\sigma\widehat{\vec f}\right)^\vee.
\tilde{R}_{11}^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1/2}\frac{l_p^3}{R_8 R_9}\ \ , \ \ \tilde{l}_p = \left( \frac{\tilde{R}}{R}\right)^{1/3}\frac{l_p^2}{(R R_8 R_9 )^{1/3}}\ \ , \ \ \tilde{R}_i^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1/2}R_i .
\left( \begin{array}{c c}{{\hat{K}}}&{{g}}\\{{- g}}&{{\hat{F}}}\\ \end{array}\right) \longrightarrow \left( \begin{array}{c c}{{\Lambda \hat{K}\Lambda^{t}}}&{{\Lambda g \Lambda^{t}}}\\{{- \Lambda g \Lambda^{t}}}&{{\Lambda \hat{F}\Lambda^{t}}}\\ \end{array}\right) .
\int_{[\sigma^{-1}_e(t),t]}|\Phi^e_{t}(0,s)|dm_{x=x_i}^j(s) \leq V_{max}\int_{[\sigma^{-1}_e(t),t]}|t-s|dm_{x=V}^j(s) &\leq V_{max}\Delta t^N m_{x=V}^j( I_{n-1}^N),
\mathrm{corank}_{\Lambda(G)}(J_v^{\pm}(E_{p^{\infty}}/L_\infty)) =\mathrm{corank}_{\Omega(G)}((J_v^{\pm}(E_{p^{\infty}}/L_\infty))_p) &=\mathrm{corank}_{\Omega(G)}({^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}).
u_2 = 0, u_1 ={\rm i}\nu \langle K , h \rangle , f_1(t) = -{\rm i}\nu \langle K, h'(-t+\cdot) \rangle, \psi_1(t) = u_1 + (1*f_1)(t) ={\rm i}\nu \langle K(t+\cdot), h \rangle , f_2 (t) ={\rm i}(\frac{\nu}2 - \lambda) \langle K(t+\cdot), h \rangle, \psi_2 = K*F(\psi_1, \psi_2),
\sum_{s_1=0}^{\infty}\cdots \sum_{s_k=0}^{\infty}\Phi_{{\bf S}_1}^{s_1}\circ\cdots \circ \Phi_{{\bf S}_k}^{s_k}(\Lambda_t)=\widetilde\psi_t({\bf R}_{i,j}) \widetilde\psi_t({\bf R}_{i,j})^*.
\frac{\partial V}{\partial \Lambda}=\frac{1}{\Lambda^{2}}\frac{\partial V}{\partial\psi^{\phantom{a}}_{\alpha a}}C^{\prime}(0)\frac{\partial V}{\partial\bar{\psi}_{a}^{\alpha}}-\tau\Lambda^{D-2}\frac{\partial^{2}V}{\partial\psi^{\phantom{a}}_{\alpha a}\partial\bar{\psi}_{a}^{\alpha}},
\left. \frac{\partial}{\partial a_{\ell}}\langle \lambda_{\mathrm{s}}\rangle \right|_{g_{j}= 0}= \left. \frac{\partial}{\partial a_{\ell}}\left\langle \frac 1 N \mathrm{t r}\frac{1}{z - \phi}\right\rangle \right|_{g_{j}= 0}= 0
\begin{aligned}\psi(\xi) \equiv \varpi \chi_{R}(|\xi|) +\frac{1}{4}(\xi,\sigma\xi)-\frac{1}{2}\nabla_\xi \cdot \left[\sigma\xi\right] &= \varpi \chi_{R}(|\xi|)+ \left\{\frac{3}{4}\lambda_1(\xi) \left|\xi\right|^2 -\lambda_2(\xi) -\frac{1}{2}\lambda_1(\xi)\right\}.\end{aligned}
h^{n-1}=(\rho^{n-1})^{-1}\Bigl( \Delta b^{n-1}\nabla a^{n-1}\,+\, u^{n-1}\cdot \nabla^2 a^{n-1}\, + \,\nabla b^{n-1}\cdot\nabla^2 a^{n-1}\Bigr)\,,a^{n-1}=a(\rho^{n-1}).
\begin{array}{ccc}(Au)_1 f = u_{+}\theta_{1}\frac{\partial f}{\partial\theta_{1}}begin{align*}.5em](Au)_2 f = u_{+}\theta_{2}\frac{\partial f}{\partial\theta_{2}}\vdots (Au)_{d-1}f = u_{+}\theta_{d-1}\frac{\partial f}{\partial\theta_{d-1}}\end{array}
\Big|\sum_{\substack{L(\rho,\pi)=0 |s-\rho|\leq 200\eta}}\frac{1}{(s-\rho)^{k+1}}+\sum_{\substack{L(\rho',\pi\otimes\chi)=0 |s-\rho'|\leq 200\eta}}\frac{1}{(s+1-\beta_{\chi}-\rho)^{k+1}}\Big|\geq\Big(\frac{1}{50|s-\rho_0|}\Big)^{k+1}\geq\frac{1}{(100\eta)^{k+1}}.
[D,a_1]=[F|D|,a_1] =F\delta(a_1)+[F,a_1]|D|=[F,\delta(a_1)]+\delta(a_1)F+[F,a_1]|D| &=([F,\delta(a_1)]|D|^{-1}+\delta(a_1)D^{-1}+[F,a_1])|D|.
\Big( \sum_{e\in E(\mathbb Z^d)}q_e p_e \Big)^d =\mu^d \le C (8/3^d)^k\|q\|^{d-1}\sum_{e\in E(\mathbb Z^d)}q_e \le C\|q\|^{d-1}\sum_{e\in E(\mathbb Z^d)}q_e,
\left\{\sup_{0\leq s<t\leq T, t-s<\delta}\left|M^{[f]}_t-M^{[f]}_s\right|\geq \rho\right\}=\left\{\sup_{0\leq s<t\leq T, t-s<\delta}\left| B_{\langle M^{[f]}\rangle_t}-B_{\langle M^{[f]}\rangle_s}\right|\geq \rho\right\},
R_{\hat{\frac 1 2}\otimes j}( u ) = \left( \begin{array}{c c c c}{\sin ( u + ( \frac{1}{2}+ \hat{H}) \eta )}&{\sin ( \eta ) \hat{F}}\\{\sin ( \eta ) \hat{E}}&{\sin ( u + ( \frac{1}{2}- \hat{H}) \eta )}\\ \end{array}\right) ,
D^\alpha(\xi,\zeta^{(k)})= ( 2\pi i R)^{|\alpha|}\left( \left(\frac{\xi_1}{\sqrt{E}}\right)^{\alpha_1}\left(\frac{\xi_2}{\sqrt{E}}\right)^{\alpha_2}e\left(\left\langle R\frac\xi{\sqrt{E}},y\right\rangle\right) - \left(\zeta_1^{(k)}\right)^{\alpha_1}\left(\zeta_2^{(k)}\right)^{\alpha_2}e\left(\left\langle R\zeta^{(k)},y\right\rangle\right) \right)
-J_4 = x^{ij}y_{jk}x^{kl}y_{li}-{x^{ij}y_{ij}x^{kl}y_{kl}\over 4}+{\epsilon_{ijklmnop}\over 96}(x^{ij}x^{kl}x^{mn}x^{op}+ y^{ij}y^{kl}y^{mn}y^{op})
\begin{cases}y_0=\frac{x_0}{x_1}=\frac{1}{z_1}y_2=\frac{x_2}{x_1}=\frac{z_2}{z_1}y_3=\frac{x_3}{x_1}=\frac{z_3}{z_1}\end{cases}\begin{cases}z_1=\frac{1}{y_0}z_2=\frac{y_2}{y_0}z_3=\frac{y_3}{y_0}\end{cases}\begin{cases}dy_2=\frac{z_1dz_2-z_2dz_1}{z_1^2}dy_3=\frac{z_1dz_3-z_3dz_1}{z_1^2},\end{cases}
T(z)=1+\frac{z^{2}}{2!}-\frac{z^{4}}{4!}+3\frac{z^{6}}{6!}+17\frac{z^{8}}{8!}+9\frac{z^{10}}{10!}+111\frac{z^{12}}{12!}+\cdot\cdot\cdot,z\in\mathbb{C}.
\Delta_{\tilde \phi}\tilde L (\tilde \psi) - \Delta_\phi \tilde L(\tilde \psi) = - \biggl (\int_0^1 g_s^{i \bar \ell}g_s^{k \bar \jmath}ds \biggr ) \cdot (\tilde \phi - \phi)_{k \bar \ell}\cdot (\tilde L(\tilde \psi))_{i \bar \jmath}.
[f]_{Q}^{(m,\frac{m}{2})}=\sum_{2s+|k|=[m]}[\partial_t^{s}\partial_x^{k}f]^{(\mu,\frac{\mu}{2})}_{Q}, &|f|_{Q}^{(m,\frac{m}{2})}=\sum_{2s+|k|\leq [m]}||\partial_t^{s}\partial_x^{k}f||_{L^{\infty}(Q)}+[f]_{Q}^{(m,\frac{m}{2})}.
\sum_{k=0}^{\infty}\sum_{r_1=1}^{n}\frac{S_{0r_1r_1}}{(k+1)(r_1+k+1)^2}=\sum_{r_1=1}^n\frac{S_{0r_1r_1}H_{r_1}}{r_1^2}-\sum_{r_1=1}^n\frac{S_{0r_1r_1}}{r_1}\left(\zeta(2)-H^{(2)}_{r_1}\right);
{\cal L}={\frac{1}{2}}\partial_\mu \phi_x \partial^\mu \phi_x-{\frac{1}{2}}m_0^2\phi_x^2-{\frac{m^2}{\beta^2}}(1-cos(\beta\phi_x)) \equiv{\frac{1}{2}}\partial_\mu \phi_x \partial^\mu \phi_x-U(\phi_x),
\Psi_0^{(n,m,i)}(y)=y^{\frac{m}{2}}\begin{pmatrix}y^{\frac12}1 y^{\frac12}\frac{(m+1)-y(m+n-i+1)}{i-n}\end{pmatrix}, T=\begin{pmatrix}1 0 0 \frac{n-i}{i}\end{pmatrix},
\dot{\Xi}= - \frac{\partial p}{\partial x}= -\Bigl(g^{tt}_{,x}\tau^2 + 2g^{t\phi}_{,x}\tau \Phi + 2g^{t\psi}_{,x}\tau\Psi +g^{\phi\phi}_{,x}\Phi^2 + g^{\psi\psi}_{,x}\Psi^2 + 2g^{\phi\psi}_{,x}\Phi\Psi +g^{xx}_{,x}\Xi^2 + g^{\theta\theta}_{,x}\Theta^2\Bigr).
\int_{\mathbb{R}^{n+2}}\Big|\prod_{l=1}^{n+1}K_{k_l}(y_l,\rho)\Big|\,d\rho d\underline{y}\leq C^{n+1}2^{\widetilde{k}_1+\ldots+\widetilde{k}_n}(1+|\widetilde{k}_{n+1}-\widetilde{k}_n|),
\Omega_1 = \{(x,y) \,|\, x>0,\, y>0 \}, \Omega_2 = \{(x,y) \,|\, x>0,\, y<0 \}, \Omega_3 = \{(x,y) \,|\, x<0,\, y<0 \}, \Omega_4 = \{(x,y) \,|\, x<0,\, y>0 \}.
P_{{\mathop{\textrm{int}}}}^i = \frac{{\zeta ({m_i},{k_i},{\alpha_{ie}})}}{{{m_i}\Gamma ({m_i})}}{\left({\frac{{{m_i}}}{{{\alpha_{is}}}}}\right)^{{m_i}}}\cdot{\left({\frac{1}{{{\lambda_{me}}}}}\right)^{{m_i}}},
\begin{aligned}|f(x)-f(x')| \le C \Big(\frac{\|x-x'\|}{r}\Big)^\beta \Big\{\frac{1}{V(x,x_0,r+d(x,x_0))}\Big(\frac{r}{r+{\|x-x_0\|}}\Big)^\gamma \qquad+ \frac{1}{V(x',x_0,r+d(x',x_0))}\Big(\frac{r}{r+{\|x'-x_0\|}}\Big)^\gamma\Big\}; \end{aligned}
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n(q)_{n-1}(a)_{N-n}a^n}{(a)_n(1-q^n)(a)_{N}}=\displaystyle\sum_{n=1}^{N}\frac{aq^{n-1}}{(1-aq^{n-1})^2}.
T_\epsilon^\chi ( z ) = - \frac{i}{\epsilon}\left( \chi ( Q_{\epsilon , \theta}- z )^{-1}\chi - \chi ( Q_{0, \theta}- z )^{-1}\chi \right) & = \chi ( Q_{\epsilon , \theta}- z )^{-1}x_\theta^2 ( Q_{0, \theta}-z )^{-1}\chi ,
\frac{p_i(y_n)}{p_i(\tilde{y}_n)}= \frac{q_i(x_n')}{p_i(\tilde{y}_n)}\left(\frac{q_i(x_n')}{p_i(y_n)}\right)^{-1}\to \frac{\mu_i}{\lambda_i}, \i \in I, \n \to \infty.
\Xi'' + 3 W' \Xi' + 2 \sqrt{2}\frac{b}{\kappa}\left( \frac{\Delta}{8}+ \delta \right) \lambda_{0}^{2}e^{2W - 2 \sqrt{2}b \kappa \Xi}+ 2 \frac{b}{\kappa}\lambda_{0}e^{W - \sqrt{2}b \kappa \Xi}\delta(z - z_{0}) = -4 \sqrt{2}\frac{b}{\kappa}\frac{\delta}{\Delta}\lambda_0^2,
\int_{\mathbb{R}^{2n}}\widetilde{U}(t,x) \, \mathrm{d}\gamma(x) = (2\tau)^{\frac{n}{2}(\alpha_1 + \alpha_2-1)}\int_{\mathbb{R}^{2n}}u_1(\tau, L_1y)^{\alpha_1}u_2(\tau,L_2y)^{\alpha_2}\times \frac{\exp(\tfrac{1}{4\tau}(\alpha_1|L_1y|^2 + \alpha_2|L_2y|^2 - |y|^2))}{(4\pi \tau)^{\frac{n}{2}}}\, \mathrm{d}y.