harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{dH(x(t),\xi_0(t),\xi(t))}{dt}=\frac{\partial H}{\partial x}\frac{\partial H}{\partial \xi}+\frac{\partial H}{\partial \xi}\Big(-\frac{\partial H}{\partial x}\Big)+\frac{\partial H}{\partial\xi_0}\frac{d\xi_0}{dt}=0.
\int_{S^2}|\nabla u|^2 dv_{g_0}= 2\pi \int_{0}^\infty t |v'(t)|^2 dt = (1+{\alpha_1})\int_{0}^{+\infty}s |v'_{\alpha_1}(s)|^2 ds = (1+{\alpha_1})\int_{S^2}|\nabla u_{\alpha_1}|^2 dv_{g_0},
\Bigl[\{\nabla g_i(\bar x)\,|\,i\in I^g(\bar x)\}\cup\{\nabla G_l(\bar x)\,|\,l\in I^{0+}(\bar x)\cup I^{00}(\bar x)\}&\cup\{\nabla H_l(\bar x)\,|\,l\in I^{+0}(\bar x)\cup I^{00}(\bar x)\}\Bigr]\cup\{\nabla h_j(\bar x)\,|\,j\in\mathcal P\}
\left( 0 , a , 1 \right) = \left( \begin{array}{c c}{{\mathrm{s g n}\left( a \right) \cdot \left( \begin{array}{c c}{{\cosh \phi}}&{{\sinh \phi}}\\{{\sinh \phi}}&{{\cosh \phi}}\\ \end{array}\right)}}&{{0}}\\{{0}}&{{{\bf 1}_{n - 2}}}\\ \end{array}\right) \quad ,
- \frac{\alpha^{\prime}}{2}u \sum_{n = - \infty , n \neq 0}^{\infty}X_{- n}^{1}X_{n}^{1}= - \frac{1}{2 \pi}\int_{0}^{2 \pi}d \tau u X^{1}( \tau ) X^{1}( \tau ) \ ,
(u_1,u_2,\cdots,u_k)\preceq (v_1,v_2,\cdots,u_k)\ \{\rm if}\ \u_i\leq v_i,\ \ (i=1,2,\cdots,k), &u:=(u_1,u_2,\cdots,u_k)\prec v:=(v_1,v_2,\cdots,v_k)\ \{\rm if}\ \u\preceq v \ \{\rm and}\ \u\not= v, &(u_1,u_2,\cdots,u_k)\ll (v_1,v_2,\cdots,v_k)\ \{\rm if}\ \u_i< v_i,\ \ (i=1,2,\cdots,k),
~~~~~\|x\textbf{*}x\|_\mathbb D=\|x\|^2_\mathbb D &\Leftrightarrow \|e_1x_1\textbf * x_1 +e_2x_2\textbf * x_2 \|_\mathbb D=\|e_1x_1+e_2x_2\|^2_\mathbb D &\Leftrightarrow e_1\|x_1\textbf * x_1\| +e_2\|x_2\textbf * x_2\|=e_1\|x_1\|^2+e_2\|x_2\|^2 &\Leftrightarrow \|x_1\textbf * x_1\|=\|x_1\|^2\;\;\|x_2\textbf * x_2\|=\|x_2\|^2.
\langle w_1(\det(D_\bullet)), v \rangle_{C^{\infty}_{X, L}}= \langle w_2(TL), v_*[S^1 \times \partial D] \rangle_L + (I_{\mu_L}(A) - 1)\langle w_1(TL), v_*[S^1 \times \{1\}]\rangle_L.
c_0(\tau_0,\omega_0) = -\frac{1}{24\omega_0^2}E_2(\tau_0)+ \frac{1}{8\pi\omega_0^2{\rm Im}(\tau_0)}= -\frac{1}{24\omega_0^2}\left( E_2(\tau_0)-\frac{3}{\pi{\rm Im}(\tau_0)}\right).
\phi(A) =\phi(A_{11}+A_{12}+A_{21}+A_{22}) &=\phi(P_1)(A_{11}+A_{12}+A_{21}+A_{22})+\phi(P_2)(A_{11}+A_{12}+A_{21}+A_{22}) &=\phi(P_1+P_2)(A_{11}+A_{12}+A_{21}+A_{22}) &=\phi(I)A.
\begin{aligned}\kappa_1(t)=\frac{1}{2}R_n(t)-\frac{R_n'(t)}{2R_n(t)}-\frac{\sigma_n(t)}{t},\kappa_2(t)=\frac{R_n'(t)}{2 R_n(t)}+\frac{R_n'(t)}{2(1-R_n(t))},\end{aligned}
(\Lambda^2)_R =\nu^2\exp\Bigg\{-2\int\limits^{g(\nu)}\frac{dg'}{\left[\beta(g')\right]_R}\Bigg\}=\nu^2\exp\left\{-\frac{(4\pi)^2}{\beta_0 g(\nu)^2}\left[1+{\cal O}(g^2)\right]_R\right\},
\Delta s = \left( \frac{g_{\star \mathrm{R H}}}{g_{\star \mathrm{d o m}}}\right)^{1 / 4}\left( \frac{H_{\mathrm{d o m}}}{H_{\mathrm{R H}}}\right)^{[ 4 - 3 ( 1 + w ) ] / [ 2 ( 1 + w ) ]}\Omega_{r}^{- 3 / 4}.
\prod_{\substack{l,m,n \geq 1 l,m \leq n ; \, \gcd(l,m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{l m^2}{n^4}}= \sqrt[3]{\left( \frac{1}{1-z}\right)}\; \exp\left\{\frac{1}{12}\left( Li_2(z) + \frac{z(7-5z)}{(1-z)^2}\right) \right\},
\Omega(r\xi, r\eta)=\sum_{j,k=1}^n [(1-r^2)\eta_j-\xi_j(1-r^2\bar \xi\cdot \eta)][(1-r^2)\xi_k-\eta_k(1- r^2\xi\cdot \bar \eta)] \gamma_j(r\xi)\gamma_k(r\eta).
A_{\mathrm{p l a n a r}}= -{\frac{2 e^{2}p^{2}}{( 4 \pi )^{2}}}\, \left[ \left({\frac{1 3}{6}}-{\frac{\xi}{2}}\right) \left({\frac{1}{\epsilon}}- \log{\frac{- p^{2}}{4 \pi \mu^{2}}}- \gamma \right) +{\frac{3 1}{9}}- ( 1 - \xi ) +{\frac{( 1 - \xi )^{2}}{4}}\right]
\bar{\mathbf{x}}^{[\sf bs]}_{i}[m]=\gamma^{[\sf d]}_{i}\sum_{j=1}^N\sum_{a=1}^{\frac{\lambda_2}{\lambda_1}T^{KN(KM+KN)}}\mathbf{\bar{v}}_{ij}^{[{\sf bs}](a)}[m]c^{[{\sf d}](a)}_{ij}[m],
{\bf C}_{[ \alpha_{1}] , [ \alpha_{2}] , ( \alpha_{3})}^{\epsilon}= G_{R}( \alpha_{1}) \epsilon{\bf C}_{[ Q - \alpha_{1}] , [ \alpha_{2}] , ( \alpha_{3})}^{\epsilon}= G_{N S}( \alpha_{3}){\bf C}_{[ \alpha_{1}] , [ \alpha_{2}] , ( Q - \alpha_{3})}^{\epsilon},
M^{\pm}_2(x ; z)-e_{2}=\mathcal{P}^{\pm}\left(M_{-}(x ; \cdot) R(x ; \cdot)\right)_{21}, z \in \mathbb{R}, &2ik(M^{\pm}_2(x ; z)-e_{2})=\mathcal{P}^{\pm}\left(2ikM_-(x ; \cdot) R(x ; \cdot)\right)_{21}, z \in \mathbb{R},
\zeta_k^{(i)}(t) = p_k - \int_0^t r_k(\boldsymbol{\zeta}^{(i)}(s))\,ds, k \in \mathbb{N}, \psi^{(i)}(t) = \sum_{k=0}^\infty (k-2) \int_0^t r_k(\boldsymbol{\zeta}^{(i)}(s))\,ds.
\begin{array}{lcll}\mbox{floor}\mbox{SU(2/1)}\mbox{field}\mbox{helicity}\mid \mbox{gnd}> (0~-\frac{7}{2}) e_\mu^a +2 \mid \mbox{1st}> (1~-\frac{7}{2}) 2 \Psi_\mu +\frac{3}{2}\mid \mbox{2nd}> (0~-\frac{5}{2}) A_\mu +1 \end{array}
(\d a)^\circ= - \d a \d b \d c,\quad (\d b)^\circ= - \d a \d b\d d,\quad(\circ \d c)=\d a \d c\d d q^{-2},\quad (\d d)^\circ=\d b \d c\d d - \d a \d b\d c(1-q^{-2})
P^{(t)}_{ij}+ P^{(t)}_{ji}={P_{ij}^{loss}}^{(t)}Q^{(t)}_{ij}+ Q^{(t)}_{ji}= -\frac{b_{ij}}{g_{ij}}{P_{ij}^{loss}}^{(t)}\big( ={Q_{ij}^{loss}}^{(t)}\big)
\big\|\otimes_{i=1}^n K(Q_{N_i}*Q_{(Y_1,\varepsilon_i)})-\otimes_{i=1}^n Q_{N_i}\big\|_{TV}\leq \sqrt{2\sum_{i=1}^n\bigg(\frac{6}{\sigma_i}\varphi\Big(\frac{1}{6\sigma_i}\Big)+4\phi\Big(\frac{-1}{6\sigma_i}\Big)\bigg)}
\sup \limits_{j = 1, \ldots, n}\left|G\left(\tfrac{j}{n}\right) + \tfrac{x_{n, j}}{n}\right| \xrightarrow{n \to \infty}0~, \sup \limits_{j = 1, \ldots, n}\left|G\left(\tfrac{j}{n}\right) + \tfrac{y_{n, j}}{n}\right| \xrightarrow{n \to \infty}0 ~.
H_{0}= \frac{1}{2}{\bf b_{0}}^{2}+ \frac{e^{2}}{2 \pi}{\bf a_{0}}^{2}+ \frac{\pi}{4 L e^{2}}( Q^{2}+ Q_{5}^{2}) + \frac{1}{\sqrt{2 L}}{\bf a_{0}}( Q - Q_{5}) \ .
\operatorname{tr}(L) =\prod_{i=1}^{2k+1}(1+(b_{\rho_{i},\rho_{i}+k}-\lambda)\frac{\partial^2}{\partial x_{\rho_i}\partial x_{\rho_{i+1}}})\prod_{j=1}^{2k+1}x_{\rho_i}&=\prod_{i=1}^{2k+1}(1+(b_{i,i+k}-\lambda)\frac{\partial^2}{\partial x_i\partial x_{i+k}})\prod_{j=1}^{2k+1}x_i.
\varepsilon(X, \mathcal K(n)) =\prod_{j}\det(-{\rm Frob}_k; H^j(X_{\bar k},\mathcal K(n)))^{-1-j}=\prod_{j}\left(q^{-n\cdot (-1)^{-1-j}\cdot \dim_\Lambda H^j(X_{\bar k},\mathcal K)}\cdot\det(-{\rm Frob}_k; H^j(X_{\bar k},\mathcal K))^{-1-j}\right) =q^{n\cdot\chi(X_{\bar k},\mathcal K)}\cdot \varepsilon(X,\mathcal K).
p(y_{ij}) H\left(\frac{p(y_i)P_{S|Y=y_j}+p(y_j)P_{S|Y=y_j}}{p(y_{ij})}\right) & - \left( p(y_i) H(P_{S|Y=y_i})+ p(y_j) H(P_{S|Y=y_j}) \right).
A_{i,k}:= \{(3i) a \}, B_{i,k}:= [(3i) a + 1, (3i) a + 3i], C_{i,k}:= [(3i) a + 3k+1, (3i) a + 3k+2 + 3 (i-1)], D_{i,k}:= A_{i,k}+ \{a \}, E_{i,k}:= B_{i,k}+ \{a \}, F_{i,k}:= [(3i) a + b, (3i) a + c + 3 i], G_{i,k}:= D_{i,k}+ \{a \}, I_{i,k}:= E_{i,k}+ \{a \}, J_{i,k}:= F_{i,k}+ \{a \},
|\begin{matrix}x&x^{\prime}&x^{\prime\prime}&x^{\prime\prime\prime}\\x_{1}&x_{1}^{\prime}&x_{1}^{\prime\prime}&x_{1}^{\prime\prime\prime}\\x_{2}&x_{2}^{\prime}&x_{2}^{\prime\prime}&x_{2}^{\prime\prime\prime}\\x_{3}&x_{3}^{\prime}&x_{3}^{\prime\prime}&x_{3}^{\prime\prime\prime}\\ \end{matrix}|=0.
0 = \biggl | \begin{matrix}{{A_{j m}( x )}}&{{A_{j m}( y )}}\\{{A_{m k}( x )}}&{{A_{m k}( y )}}\\ \end{matrix}\biggr | - \biggl | \begin{matrix}{{\psi_{j m}( x )}}&{{\psi_{j m}( y )}}\\{{\psi_{m k}( x )}}&{{\psi_{m k}( y )}}\\ \end{matrix}\biggr | A_{j m k}( x + y ) .
\widetilde{Q}_{GL_{n-1}}^{GL_{n}}(a_1,\cdots,a_n;t_1,\cdots, t_{n-1}|\chi_n)=\psi\left(\sum_{k=1}^{n-1}\frac{a_k}{t_{k}}-\sum_{k=1}^{n-1}\frac{t_k}{a_{k+1}}\right).
\sum_{n=1}^{\infty}\frac{1}{n^k}P_k\left(\frac{x}{n^2}\right) = \sum_{n=1}^{\infty}\frac{1}{n^k}\sum_{m=0}^\infty \frac{(-1)^m x^m}{m! n^{2m}\zeta(k+2m)}=\sum_{m=0}^\infty \frac{(-1)^m x^m}{m!}= e^{-x}.
p_{0 \alpha}= p_{\alpha}- \frac{g}{2}( \tau^{+}A^{+}{}_{\alpha}+ \tau^{-}A^{-}{}_{\alpha}) + \frac{g g^{\prime}}{\sqrt{g^{2}+ g^{2}}}Q A_{\alpha}+ \frac{g^{2}}{\sqrt{g^{2}+ g^{2}}}Q^{\prime}Z_{\alpha}.
\phi_{\alpha, x}(f) = \partial^\alpha f(x), |\alpha|\le k-1,\x\in B'\,, \phi_{\alpha, \gamma, x}(f) = \frac{\partial^\alpha f(x) - \partial^\alpha f(y)}{|x-y|^\gamma}\,, |\alpha|= k-1,\x,y\in B'\,, x\ne y\,.
R(\delta,\sigma):={\rm{max}}\{\Vert (e_\sigma,z^l(e_\delta-\delta))\Vert_{L^\infty},\Vert z^l (e'_\delta,e'_\sigma)\Vert_{L^\infty},\Vert z^l (e'_\delta,e'_\sigma,e''_\delta,e''_\sigma)\Vert_{L^2}\}.
\Gamma^{(p)}_\pm = \frac{- i}{2 \omega_\pm}[(1 + N_\pm) \Sigma_{1 2 (2 1)}- N_\pm \Sigma_{2 1 (1 2)}] \, \rule[-3mm]{.14mm}{8.5mm}\raisebox{-2.85mm}{\scriptsize{$\; p_0 = \pm \omega_\pm$}},
c_1(\overline{M}_{j,0,v})^{n-1}=\lim_{\epsilon\to 0}c_1(\overline{M}_{j,\epsilon,v})^{n-1}=\lim_{\epsilon\to 0}\frac{\deg_{M_{j,\epsilon}}(H) c_1(\overline{M}_{k,\epsilon,v})^{n-1}}{\deg_{M_{k,\epsilon}}(H)}=\frac{\deg_{M_{j,0}}(H)}{\deg_{M_{k,0}}(H)}c_1(\overline{M}_{k,0,v})^{n-1}
\begin{array}{lll}s_{uvu'}:= r_1u'-r_5v+r_6u = u'r_1-vr_5+ur_6 , s_{uvv'}:= r_1v'+ r_3v+ r_4u = -v'r_1- vr_3 +ur_4 + \lambda ur_5 , s_{uu'v'}:= r_2u - r_3u' + r_5v' = ur_2 - u'r_3 + v'r_5 , s_{vu'v'}:= r_2v + r_4u' - r_6v' +\lambda r_5u' = -vr_2 + u'r_4 + v'r_6 .\end{array}
\Phi_{c}[\rho ]=\int \frac{d^{2s}\mathbf{z}}{\pi^{s}c^{2s}}\;\mbox{Tr}[\rho D(c^{-1}\mathbf{z})\rho_{0}D^{\ast}(c^{-1}\mathbf{z})]\;\rho_{\mathbf{z}}.
M:=\begin{bmatrix}0 0 0 \cdots 0 a_0 1 0 0 \cdots 0 a_1 0 1 0 \cdots 0 a_2 \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots 1 a_{n-1}\end{bmatrix}.
S_{k}f(x) = V(x') \int{1\over{\big( 1+{{d(x,y)}\over{\rho(x')}}\big)^{k}}}{1\over{d(x,y)^{n}}}f(y) dy, \mbox{and}S_{k,a}f(x) = V(x') \int{1\over{\big( 1+{{d(x,y)}\over{\rho(x')}}\big)^{k}}}{1\over{d(x,y)^{n}}}|a(y)-a(x)| f(y) dy,
P_{\min}^{{\rm{N}}}\left({\overline S,\overline \varepsilon ,\overline R}\right) = \bigcap{\left\{{P \in{\mathbb{R}^+}:\left({\overline S,\overline R}\right) \in{{\mathcal{R}}}_{{\rm{SvB}}}^{{\rm{N}}}\left({W,P,K,\overline \varepsilon}\right)}\right\}}.
[ \beta_{q^{\prime}}^{( \sigma )}, \hat{\beta}_{q^{\prime}}^{( \sigma )}] = \sqrt{q^{\prime}}e^{- \frac{\epsilon}{2}( \beta^{( \sigma ) 2}- \beta^{( \sigma ) \dagger 2})}\equiv \sqrt{q^{\prime}}{\cal S}_{2}( \epsilon ) , \quad q^{\prime}\equiv e^{- \epsilon}= 1 / q{,}
r_n^{(k+1)}=\frac{\iint_{\, \mathbb{C}_n^{(k)}}(x\cos{\phi_n}+y\sin{\phi_n}) \, \mathrm{e}^{-\frac{x^2+y^2}{\sigma^2}}\, \mathrm{d}x \, \mathrm{d}y}{\iint_{\, \mathbb{C}_n^{(k)}}\mathrm{e}^{-\frac{x^2+y^2}{\sigma^2}}\, \mathrm{d}x \, \mathrm{d}y}, \phi_n =2\pi \varphi n.
{}^\omega\Delta(E_j)=F_j \otimes 1 + K_j'\otimes E_j,{}^\omega\Delta(F_j)=1\otimes F_j + E_j\otimes K_j',{}^\omega\Delta(K_j)=K_j' \otimes K_j,{}^\omega\Delta(K_j')=K_j \otimes K_j'.
\mathcal{G}_0 (U) := \bigg\{f \in \mathcal{O}_{X\times \mathbb{B}^1}(U \times \mathbb{B}^1) = \mathcal{O}_X (U) \langle T \rangle \, \bigg| \, f = 1+ \sum_{i=1}^\infty f_iT^i \vert f_i(u) \vert < 1 \ \forall i \geq 1 \ \forall u \in U \bigg\}
E_{l}^{*}E_{l}u = -r^{l}\frac{d}{dr}r^{-2l}\frac{d}{dr}(r^{l}u) = T_{l}E_{l}E_{l}^{*}u = -r^{-l}\frac{d}{dr}r^{2l}\frac{d}{dr}(r^{-l}u) = T_{-l}u = T_{l-1}u .
T^{\delta_{l}(k_{1}+1)}\partial_{T}^{\delta_l}= (T^{k_{1}+1}\partial_{T})^{\delta_l}+\sum_{1 \leq p \leq \delta_{l}-1}A_{\delta_{l},p}T^{k_{1}(\delta_{l}-p)}(T^{k_{1}+1}\partial_{T})^{p}
c_{q_i,p_j}(tq,tp) = c_{q_i,p_j}(0,0) + c_{q_iq_k,p_j}(\tau q,\tau p)q_k + c_{q_i,p_jp_k}(\tau q, \tau p)p_k = c_{q_i,p_j}(0,0) + c_{q_iq_k,p_j}(\tau q,0)q_k +a^{(3)}_{ij,kl}(q,p)q_kp_l \quad\quad+ c_{q_i,p_jp_k}(0, \tau p)p_k + a^{(4)}_{ij,kl}(q,p)q_kp_l.
{\cal U}_{u}( \delta ( \tau ) , 0 ) = \exp [ ( i \hbar )^{- 1}\int_{\bar{M}_{\kappa}^{4}}d^{4}x{\cal H}( \phi ( x ) , \partial \phi ( x ) ) \mp i \pi n ] , n = 0 , 1 , 2 , \ldots .
(x^-_\alpha\otimes t^{b_\alpha}(t-z_1)^{s_{\alpha}^1}\cdots(t-z_m)^{s_{\alpha}^m}(t-z_{m+1})^{r_\alpha})(v_1\otimes\cdots \otimes v_m\otimes v_{m+1}) =\sum_{j=1}^{m+1}\left(v_1\otimes\cdots\otimes(x^-_\alpha\otimes t^{s^j_{\alpha}}g_j(t) v_j)\otimes\cdots\otimes v_{m+1}\right)=0,
\mathfrak{B}_{3}\left( r,\theta \right) =\left( \begin{array}{c}\frac{r^{2}}{2}\cos \left( 2\theta \right) +\frac{r^{4}}{4}\cos \left(4\theta \right) -\frac{r^{2}}{2}\sin \left( 2\theta \right) +\frac{r^{4}}{4}\sin \left(4\theta \right) \frac{2}{3}r^{3}\cos \left( 3\theta \right)\end{array}\right)
Z_{\mu}= \frac{(x-z)_{\mu}}{(x-z)^2}- \frac{(y-z)_{\mu}}{(y-z)^2}\; \mbox{,}\hspace{0.8cm}Z^2 = \frac{(x-y)^2}{(x-z)^2 (y-z)^2}\; \mbox{.}
\Sigma(Y,\theta)=~ \frac{1}{\#D_1\#D_2}\sum_{\eta_1\bmod{M}}\bar{\eta_1(b_0)}\sum_{\chi \bmod{x^J}}\Sigma_0(\eta_1,\chi;Y) &\times\sum_{a\in D_2}\bar{\chi(a)}I_{t^Kad_M}(\theta;\c).
f\left( M+m-\overline{a}\right)\le \sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{\overline{a}-{{a}_{i}}}\int_{M+m-\overline{a}}^{M+m-{{a}_{i}}}{f\left( t \right)dt}}\le f\left( M \right)+f\left( m \right)-\sum\limits_{i=1}^{n}{{{w}_{i}}f\left({{a}_{i}}\right)}
\sum_{k=0}^{(n+1)/2}\beta_{\zeta}(k)=\sum_{j=0}^{(n/t-3)/2}\beta_{\zeta}(jt)\sum_{k=0}^{t-1}\beta_{\zeta}(k)+\sum_{k=0}^{(t+1)/2}\beta_{\zeta}((n-t)/2+k)=0.
u = q \prod_{\substack{n \geq 1 n \equiv \pm 1,\pm 5 (16)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 3,\pm 7 (16)}}(1-q^n) v = q \prod_{\substack{n \geq 1 n \equiv \pm 14 (16)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 10(16)}}(1-q^n).
V_{3}^{a}+ \sum_{i = 0}^{2}\alpha_{i}V_{i}^{a}= V_{4}^{a}+ \sum_{i = 0}^{2}\beta_{i}V_{i}^{a}={\frac{1}{2}}~ , ~ ~{\mathrm{f o r ~ s o m e}}~ a = 1 , 2 , \dots , 2 0
R_0(\lambda\pm i0)f(x)=\pm\frac{i\pi}{2\sqrt{\lambda}}\widehat{d\sigma_{\sqrt{\lambda}}}\ast f(x)+p.v.\int_{\mathbb{R}^n}e^{ix\cdot\xi}\frac{\widehat{f}(\xi)}{|\xi|^2-\lambda}d\xi.
\begin{bmatrix}-1 \alpha 0 \overline\alpha -\overline\alpha 0 -\alpha J(T^{-2r},T^{-3r}) 0 0 -\overline\alpha J(T^{-r},T^{-r}) 0 -\alpha J(T^{-3r},T^{-3r}) -\alpha 0 -\overline\alpha J(T^{-2r},T^{-r}) 0\end{bmatrix}
N \Big(\frac{N_1}{N_{+}}\Big)^{(5-2\alpha)+\varepsilon}\sum_{L\geqslant N_+^{(5-2\alpha)+\varepsilon}}L^{-\frac{1}{2}}\|\mathbf{1}_{D_{N,L}}(f_{N_1,L_1}\ast g_{N_2,L_2})\|_{L^2}\lesssim N_1^{-c_1(\varepsilon)}[\min(N, 1)]^{c_2(\varepsilon)}L_1^{\frac{1}{2}}\|f_{N_1,L_1}\|_{L^2}~L_2^{\frac{1}{2}}\|g_{N_2,L_2}\|_{L^2}.
\dot Q_s(t) = \frac{s (1-t)^{n-1}}{((1-t)^2+s^2t^2)^{1/4}(1+s^2)^{1/2}}- \frac{st^2(1-t)^{n-1}(1+s^2)^{1/2}}{2((1-t)^2+s^2t^2)^{5/4}}
\frac{x^{n}-1}{x^{n}(x^{2}-x)}=\frac{(1+x+x^{2}+\cdot\cdot\cdot+x^{n-1})(x-1)}{x^{n+1}(x-1)}=\frac{1}{x^{n+1}}+\frac{1}{x^{n}}+\frac{1}{x^{n-1}}+\cdot\cdot\cdot+\frac{1}{x^{2}}.
\begin{array}{r l l l l l l}{\omega_{2}}{=}{\omega_{3}}{=}{\omega_{5}}{=}{1 \; ,}{\mathrm{a n d}\hspace{1 0 m m}}{}{\omega_{1}^{2}}{=}{\omega_{4}}{=}{- ( q - q^{- 1})^{2}\; ,}\end{array}
Z[j]=e^{\frac{i}{\hbar}\int dx^{4}{\cal{L}}_{I}(\frac{\hbar}{i}\frac{\delta}{\delta j})}Z_{0}[j]\textrm{with}Z_{0}[j]=\int_{\Lambda}{\cal{D}}\eta \; e^{\frac{i}{\hbar}\int dx^{4}\left( \frac{1}{2}[(\partial \eta )^{2}-m^{2}\eta^{2}]+j\eta \right)},
G_\Lambda(k) = G_\Lambda(k)\chi_{|k|\leq 1}- \frac{1}{|k|^2}[p\cdot k, G_\Lambda (k)]\chi_{|k|> 1}= G_\Lambda(k)\chi_{|k|\leq 1}- \frac{1}{|k|^2}(p\cdot k G_\Lambda(k) - G_\Lambda(k) k\cdot p )\chi_{|k|>1}
K_i^{\mathrm{top}}(\mathrm{Perf}^{\mathrm{gr}}(\mathcal{X}\times \mathbb{C}^{\ast})) =K_i^{\mathrm{top}}(\mathrm{Perf}(\mathcal{X})) =K_i^{\mathrm{top}}(BG)=\begin{cases}K(BG)_{\mathbb{Q}}, i=0, 0, i=1, \end{cases}
\sum_{k=0}^{n}S_B[n,k](t)_{k,q}^D = \sum_{k=0}^{n}S_D[n,k](t)_{k,q}^D+\sum_{k=0}^{n-1}n\cdot[2]_q^{n-k-1}q^{n-k-1}S[n-1,k]_{q^2}(t)_{k,q}^D.
\left|\prod_{j=k_1+1}^{k_2}a_{j-1,j}\right|= \left|\prod_{j=k_1+1}^{k_2}(1-b_{j-1,j})\right|\le \prod_{j=k_1+1}^{k_2}\left(1+\left|b_{j-1,j}\right|\right) \le \prod_{j=k_1+1}^{k_2}\exp\left(\left|b_{j-1,j}\right|\right)\le \exp\left(\sum_{j=k_1+1}^{k_2}\left|b_{j-1,j}\right|\right).
|\psi(t)| \leqslant C_{\delta_0}\|u_t(t)\|_{L^2(\Omega)}^2+\delta_0 \| \theta(t)\|_{L^2(\Omega)}^2 &\leqslant \max\{C_{\delta_0},\delta_0\}\left(\|\Delta u(t)\|_{L^2(\Omega)}^2+\|u_t(t)\|_{L^2(\Omega)}^2+\|\theta(t)\|^2_{L^2(\Omega)}\right) &\leqslant \tilde{C}_3\mathcal{E}(t)+\tilde{C}_4,
\prod_{1\leq i < j\leq m}{(-1)^{\tau_{i, j}}T_{q, x_i}^{\tau_{i, j}}T_{q, x_j}^{1 - \tau_{i, j}}}= (-1)^{\sum\limits_{1\leq i < j\leq m}{\tau_{i, j}}}\prod_{1\leq i < j\leq m}{T_{q, x_i}^{\tau_{i, j}}T_{q, x_j}^{1 - \tau_{i, j}}}.
\bar p=\underset{p \in \mathcal{C}(u)}{\mbox{argmax}}\ \frac{1}{r}\Vert d(p) \Vert^r -\Vert d(p) \Vert^{r-2}\langle \frac{-p}{\Vert d(p) \Vert^{r-2}}, \mbox{proj}_{T_K(u)}\left( \frac{-p}{\Vert d(p) \Vert^{r-2}}\right) \rangle.
\begin{array}{c}\omega \epsilon E_B^1-\partial_2H_A^3+\partial_3H_A^2+\partial_1E_A^0=0, \omega \epsilon E_B^2-\partial_3H_A^1+\partial_1H_A^3+\partial_2E_A^0=0, \omega \mu H_A^1-\partial_2E_B^3+\partial_3E_B^2-\partial_1H_B^0=0, \omega \mu H_A^2-\partial_3E_B^1+\partial_1E_B^3-\partial_2H_B^0=0.\end{array}
\hat{H}= \frac{1}{2}\hat{p}_{z}^{2}+ \hbar \omega \left[ a_{+}^{\dagger}a_{+}\, + \, a_{-}^{\dagger}a_{-}\right] \ \ \ , \ \ \ \hat{\phi}= \hat{p}_{z}+ \hbar g \left[ a_{+}^{\dagger}a_{+}\, - \, a_{-}^{\dagger}a_{-}\right] \ \ \ .
{( \mathcal{M}(\tilde u_0,a,b)u)(t,\cdot)}= e^{-A(t-s)}\tilde u_0 -\chi \int_s^t e^{-A(t-\tau)}\nabla \cdot \left(\frac{u(\tau,\cdot)}{v(\tau,\cdot)}\nabla v(\tau,\cdot) \right) d\tau + \int_s^t e^{-A(t-\tau)}u(\tau,\cdot)\big[{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u(\tau,\cdot)\big] d\tau,
4(s-1)P=\frac{4F_{s-1}V_s(\rho)}{(1-|x|^2)|x-y|^2}\Big[(s-1)(1-2x\cdot y+|x|^2|y|^2)(1-|x|^2) &\qquad\qquad+(1-|y|^2)\Big(-\frac{N}{2}(|y|^2-2x\cdot y+1)(1-|x|^2)+ (s-1)|x-y|^2 &\qquad\qquad-(2s-2-N)(1-x\cdot y)(1-|x|^2)+ (s-1-\frac{N}{2})(1-|y|^2)(1-|x|^2)\Big)\Big].
x\left( y+z\right) =\left( p+\oslash \right) \left( q+\oslash+r+\oslash \right) =\left( p+\oslash \right) \left( q+r+\oslash \right) & =p\left( q+r\right) +p\oslash+\left( q+r\right) \oslash+\oslash\oslash & =p\left( q+r\right) +\oslash.
\mu_2(x) = \frac{-1}{2}(([a,b]+[a^{\dagger},b^{\dagger}+ij-j^{\dagger}i^{\dagger}+fg-g^{\dagger}f^{\dagger}]), & ([a',b']+[a^{\dagger},b^{\dagger}]-gf+f^{\dagger}g^{\dagger}))
(f^2\circ \phi)' = (2f f')(\phi(x)) \phi'(x), (f^2\circ \phi)'' = (2ff')(\phi(x)) \phi''(x) + (2ff''+ 2(f')^2)(\phi(x)) \phi'(x), \|(f^2\circ \phi)'\|^2 \leq C |f|_{C^1}^2(\|f''\|^2 + \|f'\|^2),
s_0 =\sqrt{d_0}, t_0=r_0, s_{n+1}= \frac{1-\sqrt[4]{1-s_n^4}}{1+\sqrt[4]{1-s_n^4}}, t_{n+1}=(1+s_{n+1})^4 t_n - c_0 2^{2n+1}s_{n+1}(1+s_{n+1}+s_{n+1}^2),
\eta(t)=\sum_{k=1}^v\,\sum_{s:\, i_{k_A-1}<r_s\leq i_k}\eta_s(c_{i_{k-1}+1,r_s}t)=\sum_{A\in\mathcal A}\,\sum_{s: q_{r_s}\in A}\eta_s(c_{i_A,r_s}t)
\sum_{j=1}^N \left(\max_{i \in \mathcal{A}_j}\ell_i +\max_{i \in ([1:N]\backslash \{j\})\backslash \mathcal{A}_j}r_i \right) \geq \sum_{j=1}^{N-1}\left(\max_{i \in \mathcal{A}_{{\rm{F}}j}}\ell_i + \max_{i \in [1:N] \backslash \mathcal{A}_{{\rm{F}}j}}r_i \right).
\Phi(t,y)-y = \int_0^t U(\tau, y) d\tau + \int_0^t \Bigl( U(\tau,\Phi(\tau,y)) -U(\tau, y) \Bigr) d\tau & = U(0,y) t & + \int_0^t (t-\tau) (\partial_{\tau}U)(\tau, y) d\tau & + \int_0^t \Bigl( U(\tau,\Phi(\tau,y)) -U(\tau, y) \Bigr) d\tau.
\frac{d}{t}I(t)= \frac{d}{dt}M(t)-2(t+1)\frac{d}{dt}F(t)+2(t+1)^{2}\frac{d}{dt}\varepsilon(t)-2F(t)+4(t+1)\varepsilon(t) \displaystyle= \frac{2}{\gamma-1}(2-3(\gamma-1))(t+1)\int_{A(t)}P x+\overline{J}_1+\overline{J}_2,
\mathrm{e}^{\mathrm{i}\vartheta}(z_1,z_2,y) :=(\mathrm{e}^{\mathrm{i}\vartheta}z_1,\mathrm{e}^{\mathrm{i}\vartheta}z_2,y), \qquad\qquad\tau(z_{1},z_{2},y):=(z_{2},z_{1},y), \alpha(z_1,z_2,y) :=(z_1,z_2,\alpha y)\quad\alpha\in O(N-4).
\left( \lambda -A\right) \left( x\right) =\lambda \left( \begin{array}{c}u\left( x\right) R^{\prime}\left( u\right)\end{array}\right) -\left( \begin{array}{c}\tau \left( u\right) -R\left( u\right)\end{array}\right) =\left( \begin{array}{c}f\left( x\right) f_{1}\end{array}\right)
\begin{array}{ll}\displaystyle \max_{\{A_{k}\}_{k =1}^{K}}\displaystyle \sum_{k = 1}^{K}\frac{1}{A_{k}^{-1}+ \rho(\bar{\tau_{k}}, \alpha)}\mbox{subject to:}\displaystyle \sum_{k = 1}^{K}A_{k}= 1 & A_{k}\geq 0 \; \; k = 1,\cdots K,\end{array}
K = i \int_{0}^{\infty}d T \; e^{- i T ( \gamma^{\mu}P_{\mu}^{\prime}- m )}\delta^{4}( P^{\prime \prime}- P^{\prime}) ={\frac{\delta^{4}( P^{\prime \prime}- P^{\prime})}{\gamma^{\mu}P_{\mu}^{\prime}- m - i 0_{+}}}\, .
\nu_{\pi_k}= \nu^{S_k}_{\pi_k}\leq \nu^{S_k}_{\pi_l}\leq \nu^{S_{k+1}}_{\pi_l}\leq \cdots \leq \nu^{S_{l-1}}_{\pi_{l}}\leq \nu^{S_l}_{\pi_l}= \nu_{\pi_l}.
R^{(n)}_{|n_{1},n_{2},\dots ,n_{k}\rangle}(\eta ,\vartheta,u)=R^{(n)}(\eta ,\vartheta ,u)\prod_{i\textrm{}\mathrm{odd}}b^{n}_{n_{i}}(\eta ,u)\prod_{i\textrm{}\mathrm{even}}b^{n}_{n_{i}}(\bar{\eta},u)\, \, \,
\|(uf_t)_T\|= \|\big((uf_t)_{T-t}\big)_{t}\|\lesssim \|(uf_t)_{T-t}\|\leq \|(uf_t)_{T-t}-u f_{T}\|+\|uf_{T}\| &\lesssim [u]_\alpha [f]_\beta (T^{1/3})^{\alpha+\beta}+\|u\|[f]_\beta T^{\beta/3}\lesssim ([u]_\alpha+\|u\|) [f]_\beta T^{\beta/3}.
\int_0^t e^{A(t-s)}Bdu(s)=\int_0^{t_{j+1}}\! e^{A(t-s)}Bdu(s) +\int_{t_{j+1}}^t \! A \int_0^s e^{A(s-r)}Bdu(r) \, ds + \int_{t_{j+1}}^t B du(s)
{\frac{d}{d \bar{t}}}\left( \bar{H}_{\mathrm{e l}}+ \bar{H}_{\mathrm{m a g}}\right) = -{\frac{d}{d \bar{t}}}\left( \bar{H}_{\mathrm{k i n}}+ \bar{H}_{\mathrm{p o t}}\right) = \sum_{x , i}\bar{E}_{x , i}^{a}{\bar{j}}_{x , i}^{a}
\varepsilon ( 0 ) = p_{1}( 0 ) = p_{2}( 0 ) = \frac{1}{8 \pi^{2}a^{4}}\int_{0}^{\infty}{d z z^{3}\left[ \frac{K_{0}( z )}{I_{0}( z )}+ \frac{K_{0}^{\prime}( z )}{I_{0}^{\prime}( z )}\right]}= - 0 . 0 1 6 8 a^{- 4}
\begin{array}{l l}{\nabla \cdot \mathbf{E}^{\prime}= \rho_{e}^{\prime},}{\nabla \times \mathbf{B}^{\prime}= \mathbf{j}_{e}^{\prime}+ \frac{\partial \mathbf{E^{\prime}}}{\partial t},}{\nabla \cdot \mathbf{B}^{\prime}= \rho_{m}^{\prime}= 0 ,}{\nabla \times \mathbf{E}^{\prime}= - \frac{\partial \mathbf{B^{\prime}}}{\partial t}.}\end{array}
\! \! D_{2}\eta ({\xi}_{1},..,{\xi}_{q-1}) ({\phi}_{1},..,{\phi}_{p+1}) ={\displaystyle \sum_{i}}\eta\left ({\phi}_{i},{\xi}_{1},..,{\xi}_{q-1}\right) \left({\phi}_{1},..,\widehat{{\phi}_{i}},..,{\phi}_{p+1}\right ),
P^nA_u V (x) =~ \varphi_{u,n}(x) + \sum_{k=1}^{n}B_kP^{n- k}A_u V (x) &= \varphi_{u,n}(x) + \sum_{k=1}^{n}\sum_{y\in E}b_k(x,y) P^{n- k}A_u V (y).
F_4(a,b,c_1,c_2;x_1,x_2)=\sum_{n_1,n_2=0}^\infty\frac{(a,n_1+n_2)(b,n_1+n_2)}{(c_1,n_1)(c_2,n_2)n_1!n_2!}x_1^{n_1}x_2^{n_2},
\left(U_{\Delta}f\!\left(\alpha q^{-\Delta};\alpha^{-1}q^{m}\right)\right)_{j}=f_{j-\Delta}\!\left(\alpha q^{-\Delta};\alpha^{-1}q^{m}\right) \hskip20pt=(-1)^{j+\Delta}\alpha^{\Delta-j}q^{-\frac{1}{2}\Delta(\Delta+1)+\frac{1}{2}j(j+1)}\,_{1}\tilde{\phi}_{1}\left(0;q^{-m+j+1};q,\alpha^{-2}q^{m+j+1}\right)
\begin{cases}D_{A}\psi =0 F_{A}^{+}= \frac{r}{2}(q(\psi,\psi)- i\Omega) + i\mu *d*b -2^{-\frac{1}{2}}r^{\frac{1}{2}}(\eta^* \psi_I-\psi_I^* \eta)=0. \end{cases}
\lim_{n\to \infty}\rho_r((x_1^{(n)},y_1^{(n)}),(x_2^{(n)},y_2^{(n)}),...,(x_r^{(n)},y_r^{(n)})=\det[\mathcal{B}_\Omega(m_{ij},l_{ij})]_{i,j=1}^r
\delta \psi_{\mu a}\;=\; D_{\mu}\, \varepsilon_a\, +\,i\, \gamma_{\mu}\,T \left(\Gamma_{45}\right)_{ab}\varepsilon^b\;+\,i\;\frac{1}{6}\;\sqrt{\, \frac{1}{2}}\, \left(\,\gamma_{\mu}^{\;\nu \rho}\;-\;4\,\delta_{\mu}^{\nu}\,\gamma^{\rho}\,\right) \left(\,H_{\,\nu \rho a b}\;+\; \sqrt{\,\frac{1}{2}}\,h_{\,\nu \rho a b}\, \right)\varepsilon^b,
\gamma_{g , 9}\gamma_{S_{1}, 9}\gamma_{g , 9}^{- 1}= \gamma_{S_{2}, 9}~ , \gamma_{g , 9}\gamma_{S_{2}, 9}\gamma_{g , 9}^{- 1}= \gamma_{S_{1}, 9}~ , \gamma_{g , 9}\gamma_{S_{3}, 9}\gamma_{g , 9}^{- 1}= - \gamma_{S_{3}, 9}~ .
v^{(i,m)}_1\bigl(\varepsilon(j+b_{i,m}),x_2,\xi_1-j,t\bigr) = - \bigl( \xi_1 - j- b_{i,m}\bigr)\, \partial_{x_1}v^{(i,m)}_{0}\bigl(\varepsilon(j+b_{i,m}),x_2,t\bigr)\,.
\begin{array}{c}p_{(1)}\cdot p_{(2)}= \tilde{p}_{(1)}\cdot \tilde{p}_{(2)}= 2t/\alpha', \qquad p_{(1)}\cdot D\cdot \tilde{p}_{(2)}= \tilde{p}_{(1)}\cdot D \cdot p_{(2)}begin{align*}3mm]\mbox{and}\qquad p_{(1)}\cdot D\cdot\tilde{p}_{(1)}=p_{(2)}\cdot D \cdot\tilde{p}_{(2)}=2s/\alpha',\end{array}
\bigcap_{m\geq 0}(\mathbb{Q}(m) \Rightarrow \ominus_D \bigcup_{\ell\geq 0}(\mathbb{Q}(\ell) \wedge \ominus_D \|m=\ell\|)) \leq \ominus_D \bigcup_{\ell\geq 0}(\mathbb{Q}(n) \wedge \ominus_D \|n=\ell\|)
\max_{0 \leq l \leq x}\lbrace l + v_{b^*}(x-l) - v_{b^*}(x) \rbrace = \begin{cases}0, x \in [0,b^*], x-b^* + v_{b^*}(b^*) - v_{b^*}(x) , x \in (b^*,\infty).\end{cases}
\Vert \tilde h \Vert^{2}=\Vert \Pi_{W_{2}}\tilde h\Vert^{2}+\Vert \Pi_{W_{2}^{\perp}}\tilde h\Vert^{2},\;\Vert\Pi_{W_{2}^{\perp}}\tilde h\Vert=o_{\varepsilon}(\Vert \tilde h \Vert),
| p_{0 N}\rangle = \lim_{\tau \rightarrow - \infty}V_{T}( \sigma , \tau ; p_{0}) | \mathbf{0}\rangle = e^{i p_{0 \mu}\hat{x}_{0}^{\mu}}| \mathbf{0}\rangle , \quad \langle - p_{0 N}| = \langle \mathbf{0}| e^{i p_{0 \mu}\hat{x}_{0}^{\mu}}.