harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma ( a_1 ) = ( a_1 \varphi( \widetilde{a}_1 )^{-1}; \widetilde{a}_1 ) = ( a_1 \left( \varphi( 1,0 ) \varphi(0,1) \right)^{-1}; 1,1 )= ( b_1^{-1}; 1 , 1) = ( v^{-1}u^{-1}; 1 ,1), \gamma ( a_2 ) = ( a_2 \varphi( \widetilde{a}_2)^{-1}; \widetilde{a}_2 ) = ( a_2 \varphi (0,1) ; 0 , - 1)= ( b_2^{-1}; 0 , -1) = ( v ; 0 , -1),
(z-z_{0})^{-2}-\frac{z_{0}}{10}(z-z_{0})^{2}-\frac{1}{6}(z-z_{0})^{3}+h(z-z_{0})^{4}+\frac{z_{0}^{2}}{300}(z-z_{0})^{6}+\cdot\cdot\cdot
S_H = \{t_1-s_1,s_1-t_2,\dots,s_{n-1}-t_n,t_n-s_n,t_n+s_n\}, S_G = \{t_1-t_2,\dots,t_{n-1}-t_n,t_{n-1}+t_n\}\cup \{s_1-s_2,\dots,s_{n-1}-s_n,s_{n-1}+s_n\}.
\partial_t U - \nu \widetilde{\Delta}_t U + \widetilde{U}_0 \cdot \nabla U + U_{\neq}\cdot \nabla_t U = \begin{pmatrix}- U^2 0 0 \end{pmatrix}+ \nabla_t{\Delta}_t^{-1}2 \partial_X U^2 + \nabla_t \Delta_t^{-1}(\partial^t_i U^j \partial_j^t U^i).
( A_{+})^{\mu}\;_{\nu}\equiv \left( \begin{matrix}{{1}}&{{- l_{+}^{t}/ l_{+}^{x}}}&{{0}}\\{{0}}&{{- 1 / l_{+}^{x}}}&{{0}}\\{{0}}&{{l_{+}^{y}/ l_{+}^{x}}}&{{- 1}}\\ \end{matrix}\right) ,
|I_2| =\left|\left(h(n_1,E_1)-h(n_2,E_2)\right)\frac{(1-x)^\frac{1}{2}}{n_2-1}\right| &\le C\|b_1-b_2\|_{C[0,1]}+C\left(\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}}\right),\quad\exists\xi\in[x,1].
U(m\rtimes \Pi )\Big( \sum_{h \in \Pi}f_hu_h \Big) U^* \Big( \sum_{g \in \Pi}\xi_g \otimes \chi_g \Big) = \sum_{g,h \in \Pi}\varphi (f_hu_h)(\xi_g) \otimes \chi_{hg}, U (F_\Pi) U^* \Big( \sum_{g \in \Pi}\xi_g \otimes \chi_g \Big) =\sum_{g \in \Pi}WFW^*\xi_g \otimes \chi_g.
\begin{bmatrix}(r^{p_1}- 1) (r^{p_2}- 1) (r^{p_1}- 1) r^{p_1}(r^{p_2}- 1) r^{p_2}\end{bmatrix}\begin{bmatrix}C_{p_1}h^{p_1}C_{p_2}h^{p_2}\end{bmatrix}= \begin{bmatrix}e_{21}e_{32}\end{bmatrix},
\int_{A_i}g_i\,d\mu = \sum_{j=1}^{\infty}\frac{\mu(A_i\cap B_j^i)}{\mu(B_j^i)}\nu(\tau B_j^i) & \le c_o\nu^s(F_i)+\sum_{j=1}^{\infty}\frac{\mu(A_i\cap B_j^i)}{\mu(B_j^i)}\int_{\tau B_j^i}a\,d\mu.
\max\bigg\{\max_{1\leq j \leq n}\bigg(\bigg(1+\frac{M^{2}K^{2}b}{\delta^{j}}\bigg)\bigg(KK_{1}\gamma b+ KL_{\nu_{j}}\bigg)\bigg), &\qquad\qquad\bigg( \bigg(1+\frac{M^{2}K^{2}b}{\delta^{0}}\bigg)\bigg(K_{1}K\gamma b+KC_{\nu}\bigg)\bigg), \max_{1\leq j\leq n}L_{\nu_{j}}\bigg\}<1, \mbox{where}\ \gamma=\frac{b}{\beta}.
\Omega\left( G\right) =\bigcup\limits_{j=1}^{k}\Omega\left(H_{j}\right) ,\mathrm{core}(G)=\bigcup\limits_{j=1}^{k}\mathrm{core}\left( H_{j}\right) \mathrm{corona}(G) =\bigcup\limits_{j=1}^{k}\mathrm{corona}\left(H_{j}\right) \mathrm{\ker}(G)=\bigcup\limits_{j=1}^{k}\mathrm{\ker}\left( H_{j}\right) .
j_{1}=1\,\,{\rm and}\,\,I_{1}=\left\{j_{1}+1,j_{2},,j_{2}+1\dots,j_{l_{1}}+1\right\},\dots,I_{r}=\left\{j_{l_{r-1}},j_{l_{r-1}}+1,\dots,j_{\frac{m}{2}-1},j_{\frac{m}{2}-1}+1\right\}.
\tilde{Y}^{l+\alpha,l+\beta}=Z(I, J \setminus ( j_\alpha, j_\beta ), ( j_\alpha, j_\beta ) \sqcup K)\quad, \tilde{Y}^{l+1,l+2,l+3,l+4}=Z(I, J \setminus ( j_1, j_2, j_3, j_4 ), ( j_1, j_2, j_3, j_4 ) \sqcup K),
\tau_N= \sup_{\ell\geq 1}\;\inf \;\Big\{t\geq 0 \; : \; \exists \; i,j,k \; \hbox{pairwise different such that}&|X^{i,N}_{t}-X^{j,N}_{t}|+|X^{j,N}_{t}-X^{k,N}_{t}|+|X^{k,N}_{t}-X^{i,N}_{t}|\leq 1/\ell\Big\}.
\times \left\langle \exp i g \left[ \sum_{i = 1}^{r_{1}}\left\{\phi_{1}( x_{1 , i}) - \phi_{1}( y_{1 , i}) \right\}+ \sum_{i = 1}^{r_{2}}\left\{\phi_{2}( x_{2 , i}) - \phi_{2}( y_{2 , i}) \right\}+ \cdots \cdots \right] \right\rangle .
\|C^3_\infty\|_{L^1}- \|C^3_t\|_{L^1}\leq c\int_0^1 \frac{1}{t\big(m^2 +\frac{\alpha}{t}\big)^{4-d}}\, d\alpha &\leq \frac{c}{m^2(m^2t+1)}{\bf 1}_{d=2}+ c\log\Big(1+\frac{1}{m^2t}\Big){\bf 1}_{d=3}.
q^{2dk-d}\left| \sum_{{\bf m}\in S_0}\left(\prod_{j=1}^k \widehat{E_j}({\bf m})\right)\right|^2 -\nu^2_k(0) &\le 2 \left(q^{\frac{d}{2}-1}\left(\prod_{j=1}^k |E_j|\right)^{\frac{2k-1}{k}}+q^{d-1}\left(\prod_{j=1}^k |E_j|\right)^{\frac{2k-2}{k}}\right).
\kappa_0=\kappa_0(t,s):=\int_1^{+\infty}\frac{\kappa}{r^{2+s}}\,dr=\int_1^{+\infty}\frac{\left( 1+\frac{2\,\cos t}{r}+\frac{1}{r^2}\right)^{-\frac{2+s}{2}}-1}{r^{1+s}}\,dr.
\sum_{j=1}^n \prod_{i\neq j}a_i(x_0)\geq n\left( \prod_{j=1}^n \prod_{i\neq j}a_i(x_0) \right)^\frac{1}{n}= n\left(\prod_{i=1}^n a_i(x_0)\right)^\frac{n-1}{n}= nf(x_0)^\frac{n-1}{n}.
\begin{array}{lccclcc}{\mathrm Re}\Gamma^T_{+-}(p^2)\big|_{p^2 = M_W^2}= 0 \qquad{\mathrm Re}\Gamma^T_{ZZ}(p^2)\big|_{p^2 = M_Z^2}= 0 \phantom{\mathrm Re}\Gamma^T_{AA}(p^2)\big|_{p^2 = 0}= 0 \qquad{\mathrm Re}\Gamma_{HH}(p^2)\big|_{p^2 = m_H^2}= 0\end{array}
{f^{( 0 , 0 )}( \tau , \bar{\tau}) = \sum_{( m , n ) \ne ( 0 , 0 )}{\frac{\tau_{2}^{3 / 2}}{| m + n \tau |^{3}}}\equiv 2 \zeta ( 3 ) \, E_{\frac{3}{2}}( \tau ) ,}
Pr(\gamma_{i}<\gamma|i \in \mathcal{G}_{AF}) &\approx Pr\left(min\{\gamma_{1i}|_{i \in \mathcal{G}_{AF}},\gamma_{2i}\}< \gamma\right) &=1-(1-Pr(\gamma_{1i}<\gamma|i \in \mathcal{G}_{AF}))(1-Pr(\gamma_{2i}<\gamma) ),
\begin{cases}dx_i =( Ax_i +BR^{-1}B^Ty_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G) m_i +B\bar u +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [\mathbb{E}x_i- (\Gamma m_i+\eta)], dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma m_i+\eta)]\right\}dt +\zeta_i dW_i, \end{cases}
c_1(L) \bullet c_1(L') = [X \xleftarrow{i_{V(s)}}V(s) \xrightarrow{i_{V(s)}}X;{\bf 0}] \bullet [X \xleftarrow{i_{V(s')}}V(s') \xrightarrow{i_{V(s')}}X;{\bf 0}] = [X \xleftarrow{i_{V((s,s'))}}V((s,s')) \xrightarrow{i_{V((s,s'))}}X;{\bf 0}] = c_2(L \oplus L') = c_2(L' \oplus L) = c_1(L') \bullet c_1(L).
\sum_{k_{1}+\cdot\cdot\cdot+k_{p}=m}(\begin{matrix}n_{1}\\k_{1}\end{matrix})(\begin{matrix}n_{2}\\k_{2}\end{matrix})\cdot\cdot\cdot(\begin{matrix}n_{p}\\k_{p}\end{matrix})=(\begin{matrix}n_{1}+...+n_{p}\\m\end{matrix}).
\tau(u)^2=\tau(u^+,u^-)^2= ~\left(1-\frac{\Vert u^+\Vert^2-\Vert u^-\Vert^2}{4}\right)^2+\Vert u^+\Vert^2=\left(1-\frac{Q(u)}{2}\right)^2+\Vert u^+\Vert^2 = ~\left(1+\frac{\Vert u^+\Vert^2-\Vert u^-\Vert^2}{4}\right)^2+\Vert u^-\Vert^2=\left(1+\frac{Q(u)}{2}\right)^2+\Vert u^-\Vert^2.
- \mathfrak{A}_{s}=sN+sR+\partial^{\mu}N_{\mu}, N=i\delta A\cdot A\big(H^{2}+\phi^{2}\big),R=i\delta \frac{m_{H}^{2}}{m^{2}}\left(H^{2}\phi^{2}-\frac{1}{4}\phi^{4}\right).
{\sf t}_9 ={\sf t}_{9,1}+{\sf t}_{9,2}+{\sf t}_{9,3}\lesssim |x|^\frac{2(p-1)}{1-q}\Theta_{\sf a}{\bf 1}_{|\xi|>2{\sf l}_2}{\bf 1}_{|x|<{\sf r}_3}+{\sf U}_\infty^{q-2}\Theta_{\sf a}^2{\bf 1}_{|\xi|>2{\sf l}_2}{\bf 1}_{|x|<{\sf r}_3}.
\lim_{r\to0+}\sup_{s\leq r, \theta\in[-\infty,+\infty],w\in\{0,1\}}\frac{|X(x_0+y(s,\theta, w))-X(x_0)|}{s\sqrt{\log\log(1 + s^{-1})}}=C_{18}\qquad\mbox{a.s.},
f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+\cdot\cdot\cdot+\frac{f^{(k)}(a)}{k!}(x-a)^{k}+h_{k}(x)(x-a)^{k},
C_{2}[ \omega , \omega_{1}, \omega_{2}, B ] ={\frac{m^{8}}{4 B^{2}\omega \omega_{1}\omega_{2}}}\int_{0}^{\infty}{\frac{d s}{s}}e^{- m^{2}s}J_{2}( s , \omega , \omega_{1}, \omega_{2}, B ) ,
\Gamma(\pi) =\lim_{t\to\infty}\frac{1}{N}\sum_{\ell=1}^{N}\frac{\sum_{i=1}^{R_\ell^\pi(t)}\rho_\ell (\frac{(T_{\ell i}^\pi)^2}{2}+T_{\ell i}^\pi Z_{\ell i}^\pi)+c_\ell\tilde{R}_\ell^\pi(t)}{t}.
\|(\mathbf{g}^{k+1},\mathbf{g}^k)\|_G^2 = \frac52(\mathbf{g}^{k+1})^\top M_w \mathbf{g}^{k+1}- 2(\mathbf{g}^{k+1})^\top M_w\mathbf{g}^{k}+ \frac12(\mathbf{g}^{k})^\top M_w\mathbf{g}^{k},
\left\Vert \mathbb{H}_{\phi}\right\Vert_{\mathcal{B}^{p}\rightarrow\mathcal{B}^{p}}=\left\Vert \mathbb{H}_{\phi}\right\Vert_{\mathcal{CMO}^{p}\rightarrow\mathcal{CMO}^{p}}={\displaystyle\int\nolimits_{\left[ 0,1\right]^{n}}}\phi\left( t_{1},...,t_{n}\right) dt_{1}...dt_{n}
\chi(\widehat{\mathit{HL}}(L);t_1,\dots,t_\mu)\overset{\centerdot}{=}\begin{dcases*}\prod\limits_{k=1}^{\mu}(t_k^{1/2}-t_k^{-1/2})\, \Delta_L(t_1,\dots,t_\mu) if~$\mu>1$ \Delta_L(t_1) if~$\mu=1$\,.\end{dcases*}
\begin{array}{rcl}1 = \Vert d_{A(\tau_n)}W_n \Vert_{L^4(Z)}\leq \Vert d_{A_n'}W_n \Vert_{L^4(Z)}+ \Vert A(\tau_n) - A_n' \Vert_{L^\infty(Z)}\Vert W_n \Vert_{L^4(Z)}& & \leq \Vert d_{A_n'}W_n \Vert_{L^4(Z)}+ C_{PI}\Vert A(\tau_n) - A_n' \Vert_{L^\infty(Z)},\end{array}
\frac{d}{dt}\|\theta(t)\|_{H^{1+s}}^2+\|\partial_1\theta\|_{H^{1+s}}^2\leq C(1+\|\nabla u\|_{L^\infty}+\|\partial_1\theta\|_{L^\infty})\times(\|\omega\|_{H^s}^2+\|\theta\|_{H^{1+s}}^2).
\bar{\Phi}_{\rm{dn}}|^{T}_0 = \frac{1}{T}\int_0^T \sum_{\beta=1}^{\kappa}\Phi_{\rm{dn},\beta}(t) d t =\frac{1}{T}\left\{\int_0^T \sum_{\beta=1}^{\kappa}\Phi_{\rm{st},\beta}(t) d t + \frac{1}{E_{\rm{b}}}\sum_{i=0}^{N_\tau}\Omega_{\rm{m}}(\tau_i)\right\},
Pr(Y_{i}=y|X_{i})={p_{i}}^{y}(1-p_{i})^{1-y}=(\frac{e^{\beta\cdot X_{i}}}{1+e^{\beta\cdot X_{i}}})^{y}(1-\frac{e^{\beta\cdot X_{i}}}{1+e^{\beta\cdot X_{i}}})^{1-y}=\frac{e^{\beta\cdot X_{i}\cdot y}}{1+e^{\beta\cdot X_{i}}}
(II) \leq 16 \sum_{i=1}^m \frac1{\sqrt{\Delta_if_0(x_{i-1})}}\Big(\int_{x_{i-1}}^{x_i}\frac{(f(x)-f_0(x))^4}{f_0(x_{i-1})^3}dx\Big)^{1/2}&\leq 2^7 n\int_0^1 \frac{(f(x)-f_0(x))^4}{f_0(x)^3}dx + 16\sum_{i=1}^m \frac1{n\Delta_if_0(x_{i-1})}.
V = \sum_{\ell=1}^L \left( X_A^2 \Delta^2 |F_\ell|^2 + 2 X_A \Delta \Re[ e^{j \Phi_X}e^{j \Theta_\ell}F_\ell N_\ell^* ] + |N_\ell|^2 \right) &= X_A^2 \Delta G+ 2 X_A \Delta Z_1 + Z_0
\frac{S_{\boldsymbol\nu,\boldsymbol\mu}}{S_{N,M}}=O\left(\binom{N}{\boldsymbol\nu}\binom{M}{\boldsymbol\mu}\prod_{j=1}^3\left(\frac{\nu_j}{N}\right)^{\mu_{j,\boldsymbol\cdot}+\mu_{\boldsymbol\cdot,j}}\right).
Z ( \beta \, ; R ( S_{\nabla}^{{\bf C}}) , \langle \cdot \, , \cdot \rangle_{H}, \langle \cdot \, , \cdot \rangle ) = \pi^{\zeta}\, e^{- \frac{i \pi}{2}( ( \frac{2 \theta}{\pi}\mp 1 ) \zeta \, \pm \, \eta )}\, | \beta |^{- \zeta}\tau ( M , \chi , \langle \cdot \, , \cdot \rangle_{H})^{- 1}
{\cal A}_{\Lambda}^{a}\left( \phi , \phi^{*}, \overline{{\phi}}, \pi \right) = \left[ \overline{{\Delta}}^{a}W_{\Lambda}+ \frac i{2 \hbar}\left( W_{\Lambda}, W_{\Lambda}\right)^{a}\right] \left( \phi , \phi^{*}, \overline{{\phi}}, \pi \right) =
\Theta=\sum\limits_{\ell,n \in \left\{{{},{}}\right\}}{\rm{\mathbf{1}}}\left({\max \{{r_e},d\}}< \big(\frac{{P_t}{G_\ell}{G_n^e}\beta}{x \sigma_e^2}\big)^{\frac{1}{{\alpha_{\mathrm{LoS}}}}}\right){{{\Pr}_{\ell n}}},
{\cal L}= \bar{\psi}( i \! \not{\! \partial}+ J ) \psi + \frac{1}{\sqrt{N}}\bar{\psi}\not{\! \! A}\psi - \frac{1}{\sqrt{N}}\sigma \bar{\psi}\psi - \frac{1}{2 G_{S}}\sigma^{2}+ \frac{1}{2 G_{V}}A_{\mu}^{2}- \frac{1}{2 \xi}( \partial^{\mu}A_{\mu})^{2}\ ,
\begin{array}{l l}{{e^{2 \varphi}}}&{{= 1 - \frac{\mu^{2}}{r^{2}}\cosh^{2}\! \alpha}}\\{{e^{- 2 \lambda}}}&{{= 1 - \frac{\mu^{2}}{r^{2}}\cosh^{2}\! \beta ,}}\\ \end{array}
H_{\hat\mu \hat\nu \hat\rho}={1\over 6\sqrt{-G}}G_{\hat\mu \hat\mu'}G_{\hat\nu \hat\nu'}G_{\hat\rho \hat\rho'}\epsilon^{\hat\mu' \hat\nu'\hat\rho' \hat\lambda \hat\sigma \hat\eta}H_{\hat\lambda \hat\sigma \hat\eta},
P^8 v_2 = \langle \rho^4 v_3, \rho^3, P^6 v_2 \rangle P^8 v_2 = \langle \rho^{12}v_4, \rho^3, P^4 v_2 \rangle P^8 v_2 = \langle \rho^{28}v_5, \rho^3, v_2 \rangle.
\int_0^{2-i\Delta_n+s}|x-y|^{2\mathfrak H-1}dy\leq \int_0^2|x-y|^{2\mathfrak H-1}dy = \int_0^x(x-y)^{2\mathfrak H-1}dy+\int_{x}^{2}(y-x)^{2\mathfrak H-1}dy = \frac{x^{2\mathfrak H}+(2-x)^{2\mathfrak H}}{2\mathfrak H}\leq \frac{2^{2\mathfrak H}}{\mathfrak H},
C_{\mathbf{F}_N}^T C_{\mathbf{F}_N}=\begin{pmatrix}\sum_{k=0}^{N-1}\cos(2 \pi k/N)^2 \sum_{k=0}^{N-1}\cos(2 \pi k/N) \sin(2 \pi k/N) \sum_{k=0}^{N-1}\cos(2 \pi k/N) \sin(2 \pi k/N) \sum_{k=0}^{N-1}\sin(2 \pi k/N)^2\end{pmatrix}.
\left(\begin{array}{c}y_{1}(x,\lambda ) y_{2}(x,\lambda )\end{array}\right) =\left(\begin{array}{cc}b_{11}(\lambda ) b_{12}(\lambda ) b_{21}(\lambda ) b_{22}(\lambda )\end{array}\right) \left(\begin{array}{c}\phi_{1}(x,\lambda ) \phi_{2}(x,\lambda )\end{array}\right)
F(t;x,y,1,u,1,v) &=\frac{xvt^2(1-yr)}{(1-ytu+tuv(yr-1))(1-ytuv)(tux+y^{-1}-tu)}\quad+\frac{tx(1-ytu)(1-tuv(y-yr+1))}{(tux+y^{-1}-tu)(1-ytu+tuv(yr-1))(1-ytuv)}F(t;x,y-yr+1,1,u,1,v).
s_{\alpha_{n-1}}\dots s_{\alpha_{i+1}}s_{\alpha_i}w_2'(\gamma_1) = s_{\alpha_{n-1}}\dots s_{\alpha_{i+1}}s_{\alpha_i}w_2'(\gamma_2-(\alpha_i+\dots \alpha_{j-1})) = \beta +\alpha_{j-1}+\dots +\alpha_{n-2}+\alpha_{n-1}.
B_1^q = \{v\in A_q : \vec{v}_i=c(u_i)\textup{or}c(u_i)+1\textup{(mod}b)\}, B_r^q = \{v\in A_q : \vec{v}_i=c(u_i)+r\textup{(mod}b)\}\textup{, for}2\le r\le b-1.
{\left< T_{\sigma^{-}\sigma^{-}}^{\left( i n \right)}\left( \sigma^{-}\right) \right>}_{i n}= 0 -{\frac{1}{2 4}}D_{\sigma^{-}}^{S}\left( y^{-}\right) ={\frac{\lambda^{2}}{4 8}}\left( 1 -{\frac{1}{{\left( 1 + \lambda \Delta e^{\lambda \sigma^{-}}\right)}^{2}}}\right) \; ,
V_{i}\cdot V_{j}={\vec{U}}_{i}\cdot{\vec{U}}_{j}- \sum_{r}V_{i}^{r}V_{j}^{r}+ \sum_{\ell :{\mathrm{r i g h t}}}T_{i}^{\ell}T_{j}^{\ell}- \sum_{\ell :{\mathrm{l e f t}}}T_{i}^{\ell}T_{j}^{\ell}~ .
OE[a](t)=\prod_{0}^{t}e^{a(t^{\prime})dt^{\prime}}\equiv lim_{N\rightarrow\infty}(e^{a(t_{N})\Delta t}e^{a(t_{N-1})\Delta t}\cdot\cdot\cdot e^{a(t_{1})\Delta t}e^{a(t_{0})\Delta t})
P(x, y_s):={}\left|\begin{matrix}A_{2,1}- y_s A_{2,2}\cdots A_{2,r-1}A_{3,1}A_{3,2}- y_s \cdots A_{3,r-1}\vdots \vdots \ddots \vdots A_{r-1,1}A_{r-1,2}\cdots A_{r-1,r-1}- y_s \end{matrix}\right| ={}y_s^{r-2}+ B_1 y_s^{r-3}+\cdots+ B_{r-3}y_s + B_{r-4},
\widehat{K'}_\mathrm{a-cpt-d}^\mathbb{R}=K'+\sum_{i\in\mathcal{K}'}\frac{X_i/Y_i}{\sqrt{2N_1\varrho\vartheta_i}}+V=K'+\sum_{i\in\mathcal{K}'}\frac{Z_i}{\sqrt{2N_1\varrho\vartheta_i}}+V,
[y_1, y_1]=y_4, [y_1, y_2]=y_3, [y_2, y_1]=-y_3+(\alpha_5+\beta_2)y_5, [y_2, y_2]=\beta_4y_5, [y_3, y_1]=\alpha_3\gamma_2y_5, [y_2, y_3]=\alpha_3\gamma_3y_5=-[y_3, y_2], [y_1, y_4]=\alpha_1\gamma_6y_5.
I_\gamma(W,W) =\int_0^d[g_T(D_T^TW,D_T^TW)-g_T(W,W)\mathbf{K}(T,W)]dt &=\int_0^d[g_T(D_T^TW,D_T^TW)-g_T(W,W)\mathbf{H}(T_o)]dt &\leq \int_0^d[(s_\lambda'(t))^2-\lambda(s_\lambda(t))^2]dt=\frac{1}{2}\sqrt{\lambda}\sin(2\sqrt{\lambda}d)=0.
\frac{1}{2\pi i}\int_{a-i T}^{a+i T}F_{k, \beta}(s,z)\frac{x^s}{s}\ds =R_{k, \beta}(z)\frac{x^{1-z+\delta z}}{1-z+\delta z}+O(x^{(1-(1-\delta)b)}\exp(-f_k (\log x \log \log x)^{\frac{1}{2}}))
(u'\ast_{g_1g_2, g_2}w_1)\ast_{g_2, g_1g_2}a \intertext{by Lemma \ref{l3.4},}\in u'\ast_{g_1g_2, g_2}(w_1\ast_{g_2, g_1g_2}a)+O'_{g_1g_2, g_2}(M^1) \subseteq O''_{g_1g_2, g_2}(M^1)+O'_{g_1g_2, g_2}(M^1) = O_{g_1g_2, g_2}(M^1)
\phi_1(x)={p^3}({x_{m - 2}}), &\phi_2(x)=q({x_{m + 2}}){p^2}({x_{m-1}})+p^2({x_{m - 2}})q ({x_m}) \qquad\quad\quad+p({x_{m-2}})p ({x_{m - 1}})q({x_{m + 1}}), &\phi_3(x)=p({x_{m - 2}}){q^2}({x_{m + 1}})+q^2({x_{m + 2}})p ({x_m}) \qquad\quad\quad+ p({x_{m - 1}})q({x_{m + 1}})q({x_{m + 2}}), &\phi_4(x)={q^3}({x_{m + 2}}),
\hat{T}(w)\hat{T}(z) = \frac{\hat{c}/2}{(z-w)^4}+ \frac{2 \hat{T}(z)}{(z-w)^2}+ \frac{2 \partial_z \hat{T}(z)- F(z)}{w-z}+ O((z-w)^0) \hspace{.3in}.
A(\kappa, t) = \sum_{k \geq 1}\frac{e^{-kt}}{2^{2k}}L_{k-1}^{(1)}(2kt)\sum_{n \geq 0}\frac{1}{2^{2n}(k+n)}\binom{2n+2k}{n}[(1-\epsilon)w]^{n+k}& = \sum_{k \geq 1}\frac{e^{-kt}}{2^{2k}}[(1-\epsilon)w]^k L_{k-1}^{(1)}(2kt)\sum_{n \geq 0}\frac{1}{2^{2n+2k}(k+n)}\binom{2n+2k}{n}[(1-\epsilon)w]^n.
1=\int \! D B_{+}^1 \delta(B_{+}^1)D B_{+}^2 \delta(B_{+}^2) \exp\{i \int \!d^2 x\,[ \psi_-^{\dagger 1}B_{+}^1\psi_-^1 + \psi_-^{\dagger 2}B_{+}^2\psi_-^2] \}\quad ,
M_J =\begin{pmatrix}- d^{(J)}_1 1 0 \cdots 0 - d^{(J)}_2 0 1 \cdots 0 \vdots \vdots \vdots \cdots \vdots - d^{(J)}_{N-1}0 0 \cdots 1 - d^{(J)}_N 0 0 \cdots 0\end{pmatrix}~~~ : J \in \left\lbrace L,R \right\rbrace
c_{\tilde w_\mu,\tilde w_\lambda}^{\tilde w_\nu,k}= \sum_{\alpha: |\alpha|=\ell(\tilde w_\nu \tilde w_\lambda^{-1})}\mathcal K_{\tilde w_\nu \tilde w_\lambda^{-1},\alpha}\, \overline K_{\alpha,\mu}\,,
\int_\Omega\big(\mathrm{div}\,V\big)^2 \,dx=\int_\Omega\mathrm{tr}\big((\nabla\,V)^2\big)dx+\int_{\partial\Omega}\Big\{(\mathrm{div}_T V)\,V\cdot\nu- \nabla_T V\, V_T \cdot \nu \Big\}d\mathcal{H}^{n-1}.
\tilde{E}_I^{\nu}= \begin{cases}\{\prod_{i \in{I \cap\bar{m}}}(N_i -m_iN)=\tilde{N}_I \}, 0 \notin I \{\left( \begin{array}{cc}\{(\prod_{i \in{I \cap\bar{m}}}(N_i -m_iN)-N_{I \cap \bar{m}}\}N^{-1}-Var_{I \cap \bar{m}}can_{I \cap \bar{m}}N\end{array}\right) =\tilde{N}_I \}\end{cases}
\begin{array}{l c l}{{X_{n}^{( I I )}( \tau_{0})}}&{{=}}&{{{\displaystyle i \sum_{m = - \infty}^{\infty}\frac{1 - e^{i m \sigma_{0}}}{m \sigma_{0}- 2 \pi n}X_{m}^{( I )}( \tau_{0})}}}\\{{}}&{{}}&{{}}\\{{{\dot{X}}_{n}^{( I I )}( \tau_{0})}}&{{=}}&{{{\displaystyle i \sum_{m = - \infty}^{\infty}\frac{1 - e^{i m \sigma_{0}}}{m \sigma_{0}- 2 \pi n}{\dot{X}}_{m}^{( I )}( \tau_{0})}}}\\ \end{array}
(R_{12}S_{23}-R_{13}S_{12})^2S_3 = R_{12}{}^2S_{23}{}^2S_3-2R_{12}R_{13}S_{23}S_{12}S_3+R_{13}{}^2S_{12}{}^2S_3 = R_{1212}S_{23233}-2R_{1213}S_{12233}+R_{1313}S_{12123}.
\sum_{k=1}^K\sum_{j=1}^{J_k}\tilde x^{*,1}_{j,k}\bigl(\bigl(\nu^*\bigr)^-\bigr) \geq \min (\alpha, \overline\alpha) \quad\mbox{and}\sum_{k=1}^K\sum_{j=1}^{J_k}\tilde x^{*,1}_{j,k}\bigl(\bigl(\nu^*\bigr)^+\bigr) \leq\min(\alpha, \overline\alpha).
Equ> p^s+1-(\tau+1)p^{t}-(p-\tau)p^{s-t-1}(1+\frac{\delta-1}{\frac{p-\tau-1}{\tau+1}\delta+1}) \ge p^s + 1 - (\tau+1)p^{t}-(p-\tau)(\tau+2)p^{s-t-1}
I_L = -\int_{F[2,1,1;0]}\theta_{13}\theta_{24}I_M= \int_{F[1,2,1;0]}\theta_{12}\theta_{34}I_R= -\int_{F[1,1,2;0]}\theta_{13}\theta_{24}I_\tau= -\int_{F[1,1,1;1]}\theta_{14}\theta_{24}\theta_{34}
\frac{4}{\tilde{R}}\, h^{- 1 / 4}\, \pi_{i}\, h^{1 / 2}\, h^{i j}\, \pi_{j}\, h^{- 1 / 4}= - \frac{1}{\sin \theta}\, \frac{\partial}{\partial \theta}\, \sin \theta \, \frac{\partial}{\partial \theta}+ \frac{1}{\sin^{2}\theta}\left( i \, \frac{\partial}{\partial \varphi}+ \frac{\sigma^{3}}{2}\cos \theta \right)^{2}. \nonumber
r_n(\theta_0) v_n \to \overline{v}= \begin{pmatrix}\overline{v}_{11}\overline{v}_{12}\overline{v}_{21}\overline{v}_{22}\end{pmatrix}\; \; \det(\overline{v}) >0 \; \; r_n(\theta) = \begin{pmatrix}\frac{1}{\delta}\frac{\ln(n)}{\alpha^2}0 1\end{pmatrix}.
\begin{aligned}\check{b}(s) := \max_{\omega \in \Omega}\biggl\{a_{10}a_{2n}+ \sum_{t:\omega_t=1}a_{2(n-p^t_2)}\bigl(s_{1p^{t-1}_1}- s_{1p^{t}_1}\bigr) + & \sum_{t:\omega_t=2}a_{1p^t_1}\bigl(a_{2(n-p_2^t)}- a_{2(n-p_2^{t-1})}- s_{2p^{t}_2}+ s_{2p^{t-1}_2}\bigr)\biggr\},\end{aligned}
\delta{\cal E}= \delta{\cal E}_{0}- \frac{3 \hbar^{2}}{4 m}\int d^{3}\vec{r}\, \varphi^{\dagger}[ ( \vec{\sigma}\cdot \vec{\epsilon}) ( \vec{\sigma}\cdot \vec{\nabla}) + ( \vec{\sigma}\cdot \vec{\nabla}) ( \vec{\sigma}\cdot \vec{\epsilon}) ] \varphi \,{,}
\left[ B_a^{ij}\left( x\right) ,\pi_{kl}^b\left( y\right) \right]_{x^0=y^0}^{*}=\frac 12\delta_a^{\;\;b}\delta_{\;\;k}^{\left[ i\right.}\delta_{\;\;l}^{\left. j\right]}\delta^3\left({\bf x}-{\bf y}\right) .
a_{1}= x + v , \, \, \, \, \, \, \, \, \, \, a_{2}= 2 \, x - 4 \, v , \, \, \, \, \, \, \, \, \, \, a_{3}= - \frac{1}{3}\, x + v + \frac{y}{9}, \, \, \, \, \, \, \, \, \, \, a_{4}= z \zeta .
\varphi_{i}\left( x \right):=\frac{b_{i}x_{i}}{1+c_{ii}x_{i}+\sum\limits_{j\ne i}{c_{ij}x_{j}}}\ge \frac{b_{i}h_{i}}{1+c_{ii}h_{i}+\sum\limits_{j\ne i}{c_{ij}x_{j}}}\ge \frac{b_{i}h_{i}}{1+c_{ii}b_{i}+\sum\limits_{j\ne i}c_{ij}\frac{b_{j}}{c_{jj}}}\ge.
p_0^*(d^c_{I_1}(\frac{1}{2}f_{x_1}))(Y + aT) = d^c_{I_1}(\frac{1}{2}f_{x_1})(Y) = d^c_{I_1}(\hat{K}|_{\mu^{-1}(0)})(Y) = -d\hat{K}(I_1Y),
\tilde{g}_{l,l^{\prime}}^{j}\left( k\right) =4\pi\left( -i\right)^{-l+l^{\prime}}\left( 2j+1\right) \left[ \left( 2l+1\right) \left(2l^{\prime}+1\right) \right]^{1/2}a^{l+l^{\prime}+2}\left(\begin{array}[c]{ccc}l l^{\prime}j 0 0 0\end{array}\right) \frac{j_{l}\left( ka\right) j_{l^{\prime}}\left( ka\right)}{k^{2}}.
\begin{aligned}\Xi_{N}:=\{(a_1,a_2,\dots,a_N)\mid 1\leq a_1< a_2< \dots< a_N\leq K \}, \Xi_{N}^{>}:=\{(c_1,c_2,\dots,c_N)\mid K\geq c_1> c_2> \dots> c_N\geq 1 \}.\end{aligned}
X'(t) \stackrel{d}{=}f\bigl( X(t) \bigr) = \mathrm{V}' \begin{pmatrix}\begin{array}{l}1^T \mathrm{V}\end{array}\end{pmatrix}^{-1}\begin{pmatrix}\begin{array}{c}t X(t) \end{array}\end{pmatrix},
\Big{(}\mathcal{D}_t(u-u_h)^{\frac12},v_h\Big{)}-\gamma(\nabla (\sigma-\sigma_h)^{\frac12},\nabla v_h)+(\nabla (u-u_h)^{\frac12},\nabla v_h) &+(f^{\frac12}(u)-f^{\frac12}(u_{h}),v_h)=0,
\overline{B}_t = \int_{0}^{t}\langle v(s, \cdot), \phi'' \rangle \, \textrm{d}s, \overline{C}_t = \int_{0}^{t}\int_{0}^{\pi}f^2(v(s,x)) \phi^2(x) \, \textrm{d}s \, \textrm{d}x, \overline{A}_t = i \xi \overline{B}_t - \frac{1}{2}\xi^2 \overline{C}_t
\Xi(x) = \zeta(1)^{-1}\biggl(\sum_{i=1}^\infty\bigl(2i+v(\Delta'/\varpi)\bigr)it^i\biggr)\cdot (\log q)^2 = \frac{t \bigl(2 (1+ t) + v(\Delta'/\varpi)(1- t)\bigr)}{(1 - t)^2}\cdot (\log q)^2.
f_{\mu_{\zeta}^{\ell}}(x):=\begin{cases}\frac{1}{(2t)^{d}+2s^d}+\frac{\zeta \cdot j \cdot \ell}{s^d}x \in A_j j \in \{-1,0,1\}, 0 x \notin A_{-1}\cup A_0 \cup A_1.\end{cases}
\begin{array}{c}a_{11}u(0)-b_{11}u'(0)=H_{1}(\alpha_{1}[u]), \;\;a_{12}u(1)+b_{12}u'(1)=L_{1}(\beta_{1}[v]), a_{21}v(0)-b_{21}v'(0)=H_{2}(\alpha_{2}[v]), \;\;a_{22}v(1)+b_{22}v'(1)=L_{2}(\beta_{2}[u]),\end{array}
\alpha_{a,b}= M \Big ( f(b)-f(a)- \langle G(a), b-a \rangle\Big ) -\tfrac{1}{2}\|G(a)-G(b)\|^2 \,\,\, \textrm{and}& \alpha_{c,d}= M \Big ( f(d)-f(c)- \langle G(c), d-c \rangle \Big ) -\tfrac{1}{2}\|G(c)-G(d)\|^2
{\tilde{\Pi}}_{2 n}= \lim_{N \to \infty}\frac{1}{N^{1 + n}}\int D M t r ( M^{2 n}) \exp [ - \frac{1}{2}( 1 - 2 g{\tilde{\Pi}}_{2}) t r ( M^{2}) - \frac{g}{4 N}t r ( M^{4}) ] .
M=[\begin{matrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{6}\end{matrix}].
v_{a}^{+}= e^{i \alpha}\sqrt{\frac{M}{m}}\left( \begin{matrix}{e^{- w_{2}/ 2}u^{+}}\\{- e^{w_{2}/ 2}u^{+}}\\ \end{matrix}\right) v_{b}^{-}= e^{i \delta}\sqrt{\frac{M}{m}}\left( \begin{matrix}{- e^{w_{2}/ 2}u^{-}}\\{e^{- w_{2}/ 2}u^{-}}\\ \end{matrix}\right) ,
\Omega_{\rho i_{2}\ldots i_{2 m - 2}\sigma}^{( 2 m - 1 )}={\frac{1}{( 2 m - 3 ) !}}k_{\rho l_{1}\ldots l_{m - 1}}{C_{j_{2}j_{3}}}^{l_{1}}\ldots{C_{j_{2 m - 2}\sigma}}^{l_{m - 1}}\epsilon_{i_{2}\ldots i_{2 m - 2}}^{j_{2}\ldots j_{2 m - 2}}\, \, ,
\|T'\|_{S_1}=\left[\prod_{i=1}^N\left(1-\frac{1}{q_i}\right)^{-1}\right]\left\|\left[\prod_{i=1}^N\left(I-\frac{\tau_i}{q_i}\right)\right]T\right\|_{S_1}&\leq\left[\prod_{i=1}^N\left(1-\frac{1}{q_i}\right)^{-1}\right] \left[\prod_{i=1}^N\left(1+\frac{1}{q_i}\right)\right]\|T\|_{S_1}&=\left[\prod_{i=1}^N\frac{q_i+1}{q_i-1}\right]\|T\|_{S_1}.
\int_M{\left\langle{\nabla u,\nabla ({L_A}u)}\right\rangle dvo{l_g}}= - \int_M{({L_A}u)(\Delta u)dvo{l_g}}+ \int_{\partial M}{({L_A}u)\left\langle{\nabla u,\overrightarrow{n}}\right\rangle dvo{l_{\bar g}}}.
\mathfrak{X}= \frac{1}{\pi^{d/2-1/2}\prod_{j=1}^{d-1}j!}D(d/2 - 1/2, -1/2) & = \frac{1}{\pi^{d/2-1/2}\prod_{j=1}^{d-1}j!}I(d/2 - 1/2, 1/2).
(\partial_{1,0}^{\gamma}\eta)(u_{0},u_{1},\ldots,u_{p}):=\sum_{i=0}^{p}(-1)^{i}L_{\operatorname{hor}^{\gamma}(u_{i})}\eta(u_{0},u_{1},\ldots,\hat{u}_{i},\dots,u_{p}) & +\sum_{0\leq i<j\leq p}(-1)^{i+j}\eta([u_{i},u_{j}],u_{0},\ldots,\hat{u}_{i},\ldots,\hat{u}_{j},\ldots,u_{p}).
\beta_n (y)-c(x,y) \geq \gamma_n(x) \cdot (y-x) + \alpha_n(x)\,\, \forall x\in X_H, y \in Y_H \beta_n(y)-c(x,y) = \gamma_n(x) \cdot (y-x) + \alpha_n(x)\,\, \forall (x,y) \in H.
\begin{array}{rcl}u_{\lambda, l, m}(r,\theta,\phi) = N_{\lambda l}\v_{1,\lambda,l,m}^{(\circ)}= &= N_{\lambda l}\r^l \_2 F_1\left({l+1+i\Lambda\over 2},{l+1-i\Lambda\over 2}; l+{3\over 2}; -r^2\right) Y_{l,m}(\theta,\phi),\end{array}
\sum_{i=1}^{n}d(x_i/x_1) \frac{\partial \phi_1}{\partial x_i}=\sum_{i=1}^{n}\frac{dx_ix_1-x_idx_1}{x^2_1}\frac{\partial \phi_1}{\partial x_i}=\frac{d\phi_1}{x_1}-\frac{\phi_1 dx_1}{x_1^2}=0 \ \textrm{mod}\I(C),
H_{(Z_-,D_{Z,-})}(y)=e^{\frac{t_-}{2\pi i}}\sum_{d\in\mathbb K_{-}}y^d\left(\frac{\prod_{i=1}^2 \Gamma(1+\frac{v_i}{z}+v_i\cdot d)}{\prod_{i\in M_0}\Gamma (1+\frac{\bar D_i}{z}+D_i\cdot d)}\right)\textbf{1}_{[d]}[\textbf{1}]_{(D_i\cdot d)_{i\in I_-},v_1\cdot d}.
EV( X^n_{t\wedge \tau^n}, \alpha^n_{t\wedge \tau^n}) \leq e^{(\lambda_1 + \lambda_2)t}EV( X^n_0, \alpha^n_0) & \leq e^{(\lambda_1 + \lambda_2)T}EV( X^n_0, \alpha^n_0)=:\delta.
\left( \omega_{1},\omega_{2}\right) \longmapsto \chi_{x}^{(\Omega)}\left( \omega_{1},\omega_{2}\right) :=\left( \chi_{x}^{(\mathfrak{L})}\left( \omega_{1}\right) ,\chi_{x}^{(\mathfrak{b})}\left( \omega_{2}\right) \right) \ ,x\in \mathbb{Z}^{d}\ ,
\mathbb{E}[f(\Delta_{i,n}^{(1)}X^{toy,(1)},\Delta_{j,n}^{(2)}X^{toy,(2)})|\mathcal{S}]=\sum_{k=1}^N g_k(|\mathcal{I}_{i,n}^{(1)}|,|\mathcal{I}_{j,n}^{(2)}|,|\mathcal{I}_{j,n}^{(2)}\cap \mathcal{I}_{j,n}^{(2)}|)h_k(\sigma^{(1)},\sigma^{(2)},\rho)~~~~~~
\bigl(\mathcal{S}_N L_M\bigr)(\kappa\,q\,|\,\kappa\,a,\kappa\,b) = \bigl(\mathcal{S}_N L_M\bigr)(q\,|\,a, b) - \frac{\log\kappa}{M!}\bigl(\mathcal{S}_N B_{M,M}\bigr)(q\,|\,b)