harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mbox{det}M - (\tilde B^{A_1 A_2\ldots A_{N_f-N_c}}M_{A_1}{}^{B_1}M_{A_2}{}^{B_2}\ldots M_{A_{N_f-N_c}}{}^{B_{N_f-N_c}}B_{B_1 B_2\ldots B_{N_f-N_c}}),
f\left( S_{\alpha}\left( t\right) \phi \left( s\right) ,S_{\alpha}\left(t\right) \phi \left( s-r\right) ,\alpha \right) =\sum\limits_{j+k\geq 2}\frac{1}{j!k!}F_{jk}\left( s,\alpha \right) z^{j}\overset{\_}{z}^{k}.
d s_{v o l c a n o}^{2}= \eta_{\mu \nu}d Z^{\mu}d Z^{\nu}= - d t^{2}+ \frac{1 + A^{2}/ r^{2 p - 2}}{1 - ( r_{0}/ r )^{2 p - 2}}d r^{2}+ r^{2}( d \theta^{2}+ s i n^{2}\theta d \Omega_{p - 2}^{2})
(v_t+\mu v_x+\tfrac{\sigma^2}{2}v_{xx}-rv)(t,x) &=\int_0^x (U_t-rU)(t,y)dy+\mu U(t,x)+\tfrac{\sigma^2}{2}U_x(t,x) &=-\int_0^x (\tfrac{\sigma^2}{2}U_{xx}+\mu U_x)(t,y)dy+\mu U(t,x)+\tfrac{\sigma^2}{2}U_x(t,x) &=\tfrac{\sigma^2}{2}U_x(t,0)+\mu U(t,0)=0
\dim_{H}\left(\left\{x\in X\mid \overline{\lim}|A_{N}f(x)|>0 \right\}\right) = \dim_{H}\left(\cup_{i\in\mathbb{N}}\left\{x\in X_{i}\mid \overline{\lim}|A_{N}f(x)|>0 \right\}\right) \leq~3-~\frac{\alpha'}{d},
\Sigma^0(p_f) ={e^2 \over \beta}\int{d^2k \over (2 \pi)^2}{{\cal M}({\bf k},p_{0f}) \over{\bf k}^2+p^2_{0f}+{\cal M}^2({\bf k},p_{0f})}[D^0_L({\bf q})+D^0_T({\bf q})]
a\times b=|\begin{matrix}e_{1}&e_{2}&e_{3}\\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\ \end{matrix}|=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\epsilon_{ijk}e_{i}a^{j}b^{k}
I_3 = \int_{B_{2|x|}\backslash B_{|x|/2}}|z|^{\frac{1-d}2}\langle x-z\rangle^{-\nu}\, dz &= |x|^{\frac{d+1}2-\nu}\int_{B_{2}\backslash B_{1/2}}\frac{|\bar z|^{\frac{1-d}2}\, d\bar z}{(\frac1{|x|^2}+|e_1-\bar z|^2)^{\frac\nu2}}\,.
\hat V_i^L=\bigoplus_{l_0+\ldots+l_{i-1}\le L}{\cal V}^{l_0}\otimes{\cal V}^{l_1}_\#\otimes\ldots\otimes{\cal V}^{l_{i-1}}_\#\otimes V^{L-(l_0+\ldots+l_{i-1})}_\#,
\left|\int_0^h \langle \varphi(t), \eta_\tau(t) - \overline \eta_\tau(t) \rangle \,dt \right| \le \lim_{\tau \to 0}\sqrt{\frac{C_0 \tau}{c}}\int_0^h \|\varphi(t)\|_{(W^{k_0, 2})^*}\,dt = 0
\left\|u_n(t) -\sum_{k=1}^K \frac{\iota_k}{(\lambda_k^\infty)^{3/2}}W_{\boldsymbol{\ell}_k}\left(\frac{.-{\boldsymbol{\ell}_k}t-\mathbf{y}_k^\infty}{\lambda_k^\infty}\right)\right\|_{\dot H^1}\lesssim \frac 1 t,
\left( \sqrt[N]{\alpha_k}\sqcup_{E_k / F}\Lambda_k \right)(\sigma) = \prod_{\tau \in \mathrm{Gal}(E_k / F)}\sqrt[N]{\alpha_k}(\sigma, \tau) \otimes \sigma \tau ( \Lambda ) = \prod_{\tau \in \mathrm{Gal}(E_k / F)}\prod_{w \in S'_{E_k}}\sqrt[N]{\alpha_k}(\sigma, \tau)(w) \otimes \sigma \tau (\Lambda)(w).
T(e_1\otimes e_1)= e_1\otimes e_1, T(e_1\otimes e_2)=\frac{\lambda_1}{1-\lambda_2}e_1\otimes e_1, & T(e_2\otimes e_1)= e_2\otimes e_1, T(e_2\otimes e_2)=\frac{\lambda_1}{1-\lambda_2}e_2\otimes e_1.
G_\Pi f(x) =\frac{1}{x}\int_0^\infty[f(xe^{-y})-f(x)+yxf'(x)]\Pi(dy) &=\frac{1}{x}\int_0^\infty[f(xe^{-y})-f(x)-(e^{-y}-1)xf'(x)]\Pi(dy) &+f'(x)\int_0^\infty (e^{-y}-1+y)\Pi(dy) &=~ A_1(x)+A_2(x).
M_{I_{1}}=\left[\begin{array}{cccc}* 4 0 2 4 * 3 1 0 3 * 5 2 1 5 * \end{array}\right] M_{I_{2}}=\left[\begin{array}{cccc}* 4 2 0 4 * 1 3 2 1 * 5 0 3 5 * \end{array}\right],
\dim\big(Z_{p}(I,I')\setminus Z_{p}^{\alpha}(I,I')\big) - \dim S_{p}\leq \binom{k_{0}+\delta_{p}}{k_{0}}+ 1 - 2\binom{k_{0}+\delta_{p}}{k_{0}}- (k_{1}-k_{0}) + k_{0}+k_{1}-1 \leq 2k_{0}-\binom{k_{0}+\delta_{p}}{k_{0}}.
\begin{aligned}w^\delta_{A_i}(t)=\frac{\sin(a\pi)}{\sqrt{\delta}\pi}\int_{0}^{t}(t-z)^{a-1}S_i\bigg(\frac{t-z}{\delta}\bigg) M^{\delta}_{a}(0,z,z;i)dz,\; a\in(0,1/2),\end{aligned}
S^{(\mathrm{soft})}(\kappa, \pi;u,v)= \int_{\mathcal{C}_{>}}\frac{dz}{2\pi i}\int_{\mathcal{C}_{<}}\frac{dw}{2\pi i}e^{\frac{1}{3}(w-\kappa)^3-u(w-\kappa)-\frac{1}{3}(z-\kappa)^3+ v(z-\kappa)}& \times \frac{1}{\sqrt{4zw}}\frac{z+w}{z-w}\prod_{k = 1}^{m}\frac{z-\pi_{k}}{z+ \pi_{k}}\frac{w+\pi_{k}}{w-\pi_{k}},
\beta/\sigma =\tfrac{d(1+2c\cos{\psi}+c^2)}{d(1+c\cos{\phi})(1+c\cos{\psi})}=\tfrac{(1+2c\cos{\psi}+c^2\pm c^2\cos^2{\psi})}{(1+2c\cos{\phi}+c^2\cos^2{\psi})}= &=1+\tfrac{c^2(1-\cos^2{\psi})}{(1+c\cos{\psi})^2}=1+\tfrac{c^2\sin^2{\psi}}{(1+c\cos{\psi})^2}=1+\tfrac{\beta^2 c^2\sin^2{\psi}}{d^2}.
g=[\begin{matrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{matrix}]org=[\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}].
\sup_{\eta\in (0, 1]}|J_1^\eta (s_1, s_2)|^p\le \sup_{\eta\in (0, 1]}\int_0^{1}\int_0^{1}|s_2-s_1+\eta (r_1'-r_2')|^{(2H_0 -2)p}dr_1'dr_2'<\infty\,.
\mathcal{D}= \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}\delta(n, p) H_{m, n}&= \frac{L(1, \tilde f) \bigl(A(p, 1) p - 1\bigr)}{p^{\frac{3}{2}}\pi}\int_0^\infty k(t) \tanh(\pi t) t \, d t + O(M T^{\frac{1}{7}+ \varepsilon}p^{\varepsilon}).
S^{( 2 )}= - 3 \int d^{4}x{\frac{- \lambda}{4 !}}\left[ G ( x , x ) \right]^{2}- 3 ! \int d^{4}x d^{4}y ({\frac{- \lambda}{3 !}})^{2}\phi_{c l}( x ) \phi_{c l}( y ) \left[ G ( x , y ) \right]^{3}\, \, .
|F(x,t, |y'|)y'- F(x,t,|y|)y| \le(1+\alpha_N)(N+1) \max_{i=0,\ldots, N}a_i |y'-y|\Big(1+ \int_0^1 |\gamma(\tau)|^{\alpha_N}d\tau\Big) &\le (1+\alpha_N)(N+1) \max_{i=0,\ldots, N}a_i |y'-y|\Big(1+ \int_0^1 (|y'|+ |y|)^{\alpha_N}d\tau\Big) &\le 2^{\alpha_N}(1+\alpha_N)(N+1) \max_{i=0,\ldots, N}a_i \Big(1+ |y'|^{\alpha_N}+ |y|^{\alpha_N}\Big)|y'-y|,
q_{n}^{m}\left( \gamma\right) =\left({\frac{\gamma\alpha^{2}}{2e}}\right)^{\gamma\alpha^{2}/2}\left({\frac{\pi^{2}}{2\gamma}}\right)^{1/2}\exp\left\{{-2\gamma\int_{\sigma}^{1}{\left({\frac{t^{2}-\sigma^{2}}{1-t^{2}}}\right)^{1/2}dt}}\right\}.
e^{|\rho|_t}\delta_\theta Y_t = e^{|\rho|_T}\delta_\theta Y_T +\int_t^T e^{|\rho|_s}F^\theta_sdA_s +\int_t^T e^{|\rho|_s}\big(g_s d\langle N \rangle_s -\theta g^\prime_s d\langle N^\prime \rangle_s\big) &-\int_t^T e^{|\rho|_s}\big(\delta_\theta Z_s dM_s+ d\delta_\theta N_s\big),
P_{A_{r}, \, s}^{{\bf V}_{i}}( x ) = \prod_{1 \leq j_{1}< \cdots < j_{i}\leq r + 1}\Bigl ( x - \sin ( \bar{q}_{j_{1}}+ \cdots \bar{q}_{j_{i}}) \Bigr ) = P_{A_{r}, \, s}^{{\bf V}_{r + 1 - i}}( x ) .
|D_{x}\tilde{v}_t^{(k,i)}(\mu)| \leq t^{i/d}V_i(B^d(x,R(x,\mu))) \mu(B^d(x,2R(x,\mu)))^{d+1-k}& = t^{i/d}V_i(B^d(0,1)) R(x,\mu)^{i}\mu(B^d(x,2R(x,\mu)))^{d+1-k}.
\begin{array}{l}{{[ J_{1}^{\prime}, J_{2}^{\prime}]_{G P B}= - \displaystyle \displaystyle \frac{\sinh 2 \gamma J_{3}^{\prime}}{2 \sinh \gamma}~ ,}}\\{{[ J_{2}^{\prime}, J_{3}^{\prime}]_{G P B}= - J_{1}^{\prime}~ ,{}[ J_{3}^{\prime}, J_{1}^{\prime}]_{G P B}= - J_{2}^{\prime}~ ;}}\\ \end{array}
{\cal P}_{0,n}(\kappa) \triangleq \Big\{p_t(dx_t|x^{t-1}, y^{t-1}), t=0,1, \ldots, n: \frac{1}{n+1}{\bf E}^{p}\Big( c_{0,n}(X^n, Y^{n-1}) \Big)\leq \kappa\Big\}\subset{\cal P}_{0,n},~ \kappa \in [0,\infty)
y_{\alpha}(\chi^{-1})=\begin{dcases}c_{\alpha}(\chi^{-1})c_{\alpha}(\chi) \frac{(1+q^{-1/2}\epsilon_{\varrho,k}\chi^{-1/2}(a_{\alpha}))(1-q^{-1/2}\epsilon_{\varrho,k}\chi^{1/2}(a_{\alpha}))}{1-\chi(a_{\alpha})}\end{dcases}
f_{1}( t ) f_{2}( t ) = \frac{\sqrt{\pi}}{\sqrt{3 2}N^{3 / 2}\Sigma^{4}t t_{c}}I_{0}\left( \Sigma u_{p}\sqrt{1 / t_{c}- 1 / t}\right) \frac{e^{- N \left( \frac{t}{t_{c}}- 1 - \log \frac{t}{t_{c}}\right)}}{t / t_{c}- 1}
(1+t)\sum_{0\leq p < \tau}n_p(\theta(q_{p+1})-\theta(q_p))+\sum_{\tau \leq p \leq k+1}n_p(\theta(q_{p+1})-\theta(q_p) ) &=\sum_{0\leq p\leq k}m_p(\theta(q_{p+1})-\theta(q_p)) &=\int_0^1 \xi''(q)x(q)dq.
0 \leq F_i(Y(0)) -F_i(\xi) = (2p_i-(n_i+1)q_i^2(Y_i(0)+\xi_i)) \cdot (Y_i(0)-\xi_i) - \sum_{j \neq i}n_j q_j^2 (Y_j(0)-\xi_j)(Y_j(0) + \xi_j).
H = \big\{[\varphi(p)]_0 - [\varphi(q)]_0 : \ \p,q \in \mathcal{P}_{\infty}(C(X)) \big\}&= \big\{[p \circ h]_0 - [q \circ h]_0 : \ \p,q \in \mathcal{P}_{\infty}(C(X)) \big\}.
\mathbf{D}_{i,j}=\left(\begin{array}{cccccc}2d_{1}^{'}c_{1}c_{1}2d_{2}^{'}c_{2}& c_{2}2d_{3}^{'}c_{3}& \ddots \ddots \ddots & c_{m-2}2d_{m-1}^{'}c_{m-1}& c_{m-1}2d_{m}^{'}\end{array}\right).
\left(\begin{array}{ccc}\gamma^{00}\gamma^{01}\gamma^{02}\gamma^{10}\gamma^{11}\gamma^{12}\gamma^{20}\gamma^{21}\gamma^{22}\end{array}\right)= \left(\begin{array}{ccc}\eta \eta^\prime \eta^\prime \eta^\prime \eta \eta^\prime \eta^\prime \eta^\prime \eta \end{array}\right)
\zeta_{T}( \nu | \beta ) ={\frac{\varrho^{- 2 \nu}}{\pi i}}\int_{\mu}^{\infty}d \omega \int_{C_{+}}d z{\frac{d}{d z}}\ln \left( 1 - e^{i \beta z}\right) \breve{\Phi}( \omega ; z ) ( z^{2}+ \omega^{2})^{- \nu},
P_{\texttt{b};\mu}(\boldsymbol{\xi};q,q_0) := \sum_{\substack{\sigma\in S_n \epsilon\in \{1,-1\}^n}}C_{\texttt{b}}(\epsilon_1 \xi_{\sigma_1},\ldots , \epsilon_n \xi_{\sigma_n};q,q_0)\exp (i\epsilon_1 \xi_{\sigma_1}\mu_1+\cdots +i \epsilon_n \xi_{\sigma_n}\mu_n) ,
\frac{3}{2}s -\frac{(s+\frac{1}{2})^2}{s+t+1}= \frac{2s^2 + 6st +2s-1}{4(s+t+1)}\geq\frac{2s + 4st +2t +1}{4(s+t+1)}= \frac{2s+1}{2}- \frac{(2s+1)^2}{2(2s+2t+2)}.
G_{I_{k+1;1}}:=\sum_{i\subset I_{k+1;1}}H_i+t\sum_{I_{1;i}\subset I_{k+1;1}}V^{(k, N-k)}_{I_{1;i}}+\dots+t\sum_{I_{k;i}\subset I_{k+1;1}}V^{(k, N-k)}_{I_{k;i}}
A_0 := \i\sum_{i=1}^d \left(2^{-1}(S_i^*+S_i) - (S_i^*-S_i)N_i\right) = \i \sum_{i=1}^d 2^{-1}\big( (S_i-S_i^*)N_i + N_i(S_i-S_i^*) \big)
\frac{\partial}{\partial t}\phi_{\alpha \bar{\beta}}=\Delta_{b}\phi_{\alpha \bar{\beta}}+2R_{\alpha \bar{\gamma}\mu \bar{\beta}}\phi_{\gamma \bar{\mu}}-(R_{\gamma \bar{\beta}}\phi_{\alpha \bar{\gamma}}+R_{\alpha \bar{\gamma}}\phi_{\gamma \bar{\beta}}).
\left| \sum_{k=0}^{n-1}\frac{\Delta W_{k-1}L_Bs_k}{2\sqrt{x_k}e^{I_k}}-\int_{0}^{1}\frac{L_B s(x)}{2\sqrt{x}e^{I(x)}}d\overleftarrow{W}(x) \right| \leq C t n^{-0.49}
E^0_{n_x,n_y}=m\omega^{2}\hbar^{2}\left(\frac{1}{m^2 \omega^2\hbar^2}+\frac{\theta^2}{4\hbar^4}\right)^{\frac{1}{2}}\left(n_x+n_y+1\right)-\frac{m\omega^2\theta}{2}\left(n_x-n_y\right).
H_{}- d! (L^{(2)})^d = \sum_{\alpha!> 1}{\binom{d}{\alpha}}^2|r|^{2\alpha}(1-\alpha!) &= -\frac{(d!)^2}4 \sum_{|\alpha|= d,\atop \alpha!=2}|r|^{2\alpha}+ \sum_{\alpha!> 2}{\binom{d}{\alpha}}^2 |r|^{2\alpha}(1-\alpha!).
\exp \left[ - \frac{i}{2}\int d^{d + 1}X \hat{F}_{I}A_{I J}\hat{F}_{J}\right] = \int \left[ \prod_{X > 0}\left[ \det A_{I J}( X ) \right]^{- 1}\prod_{I}{\cal D}\hat{\tilde{F}}_{I}( X ) \right]
\mathcal{R}_2^- = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{c \leq \frac{C}{m}}\frac{S(-n, p; c)}{c}H_{m, n}^- \Bigl ( \frac{4 \pi \sqrt{n p}}{c}\Bigr )
\phi_{\kappa}( \zeta ) = F \left( \begin{array}{c c c c}{d_{1 , \kappa},}&{\dots ,}&{d_{n - 1 , \kappa},}&{d_{n , \kappa}}\\{e_{1 , \kappa},}&{\dots ,}&{e_{n - 1 , \kappa},}&{\zeta}\\ \end{array}\right)
{\cal L}_{e f f}= \frac{1}{4 \pi}\mathrm{I m}\left[ \int{d^{4}\theta \frac{\partial{\cal F}( A )}{\partial A}\bar{A}}+ \int{d^{2}\theta \frac{1}{2}\frac{\partial^{2}{\cal F}( A )}{\partial A^{2}}W^{\alpha}W_{\alpha}}\right] ,
r_{i,j,k}=\left[E_{i,j},E_{i,k}\right], r_{i,j,k}'=\left[E_{i,j},E_{j,k}\right]E_{i,k}^{-1}\intertext{are the Steinberg relations ($i,j$ and $k$ are distinct), and}r = \left(E_{1,2}E_{2,1}^{-1}E_{1,2}\right)^4.
\int_0^1 x \psi_n(x) \zeta_{n+k}(x) dx = \frac{\kappa_n \theta^{-n}}{2\pi i}\oint_{\Sigma_n'}\frac{\prod_{j=0}^{n-1}(\theta w + j +a )}{\prod_{j=0}^{n}(w-j)}\Upsilon_{n+k}(w)dw
a = a_{3}^{*}+ a_{4}^{*}n_{43}^{2}\leq \frac{4p}{9}+ \frac{p^{3/4}}{3 \sqrt{3}}\leq 0.54p |b| = 2a_{4}^{*}|n_{43}| \leq \frac{4 p^{3/4}}{3 \sqrt{3}}c = a_{4}^{*}\leq \frac{4 p^{3/4}}{3 \sqrt{3}}|b^{2}- 4ac| = |-4a_{3}^{*}a_{4}^{*}| \leq \frac{64 p^{7/4}}{27 \sqrt{3}}
(Se)e =\underset{a\in S}{\cup}(ae)e=\underset{a\in S}{\cup}(ae)(ee)=\underset{a\in S}{\cup}(ee)(ea) =\underset{a\in S}{\cup}e(ea)=\underset{a\in S}{\cup}a(ee)=\underset{a\in S}{\cup}ae=Se.
e(G) \le\binom{2(k-1)}{2}+(2(k-1)-c)(2k-1)+c(n-2(k-1))+\frac{k-1-c}{2}(n-2(k-1)) &=(k-1)(6k-5)-c(2k-1)+\frac{k-1+c}{2}(n-2(k-1)) &\le (k-1)(6k-5)+(k-1)(n-2(k-1)) &= n(k-1)+(k-1)(4k-3).
\Gamma\left( a+1,z\right) =z^{a}\exp\left( -z\right) \left\{1+\frac{a}{z}+\frac{a\left( a-1\right)}{z^{2}}+\cdots+\frac{a\left(a-1\right) \cdots2}{z^{a-1}}\right\}& +a\left( a-1\right) \cdots2\cdot1\cdot\Gamma\left( 1,z\right) ,
\frac{\lambda}{6}g(Y,W)-\frac{r}{6}g(Y,W)- \frac{1}{2}Ric(V,W)g(Y,\nabla f)-\frac{1}{f}Hessf(Y,W) + \frac{1}{f}g(Y,\nabla f)Hessf(V,W)+\frac{\Delta f}{3f}g(Y,W)+Ric(Y,W)=0, \ \ \forall \Y \ \in \TM.
\zeta_{T}( \nu | \beta ) ={\frac{\varrho^{- 2 \nu}}{\pi i}}\int_{\mu}^{\infty}d \omega \int_{C_{+}}d z \ln \left( 1 - e^{i \beta z}\right) \left[{\frac{2 \nu z \breve{\Phi}( \omega ; z )}{( z^{2}+ \omega^{2})^{\nu + 1}}}-{\frac{\partial_{z}\breve{\Phi}( \omega ; z )}{( z^{2}+ \omega^{2})^{\nu}}}\right] ~ ~ .
\Delta(u_{ij}) = \sum_k u_{ik}\otimes u_{kj}, \Delta(v_{ij}) = \sum_k v_{ik}\otimes v_{kj}, \varepsilon (u_{ij}) = \varepsilon (v_{ij}) = \delta_{ij}, S(u) ={v^t}, S(v) = F{u^t}F^{-1}.
\frac{w_{p q}^{C P}( n )}{w_{p q}^{C P}( 0 )}= w_{p R ( q )}( n ) , ~ \frac{\bar{w}_{p q}^{C P}( n )}{\bar{w}_{p q}^{C P}( 0 )}= \frac{1}{w_{p q}( - n )};
H(v, s) = \exp\Big( \int_{\max\{v - R_{\tau}, R_{\tau}\}}^{v + R_{\tau}}\frac{2 y\lambda_J'(y)}{1 + \frac{y^{\alpha}}{s P_J}}dy \Big), f(v) = 2 \pi \lambda_U^s(r, v) v \exp(-2\pi \int_{0}^{v}\lambda_U^s(r, y) y dy), \lambda_J'(y) = \lambda_J\arccos(\frac{y^2 + v^2 - R_{\tau}^2}{2 y v}).
\int_{\partial\Omega}e^{-i\beta(\lambda z+\frac{\overline{z}}{\lambda})}[\frac{\partial u}{\partial w}+\beta(\lambda\frac{dz}{ds}-\frac{1}{\lambda}\frac{d\overline{z}}{ds})u]ds=0,\lambda\in\mathbb{C}\backslash\{0\},
\rho^{+}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{\! \gamma^{0}}}&{{0}}\\ \end{array}\right) \quad , \quad \rho^{-}= \left( \begin{array}{c c}{{0}}&{{\! \gamma^{0}}}\\{{0}}&{{0}}\\ \end{array}\right)
x_1^\prime = j x_1 \, , x_2^\prime = x_2 \, , y_1^\prime = \phantom{j}y_1 \, , y_2^\prime = y_2 \, ; x_1^\prime = \phantom{j}x_1 \, , x_2^\prime = x_2 \, , y_1^\prime = j y_1 \, , y_2^\prime = y_2 \, .
\Gamma^{A,B,C}(uv)(X,Y) = u_1(A)X(u_2v_1)(B)Yv_2(C) & =u_1(A)Xu_2(B)v_1(B)Yv_2(C) =\Gamma^{A,B}(u)(X)\Gamma^{B,C}(v)(Y).
A_1(\mathbf{p},q,\alpha_1,\alpha_2)= n\alpha_1(1/p_1-1/p_2), A_2(\mathbf{p},q,\alpha_1,\alpha_2)= n\alpha_{2}(1-1/p_{2})-n\alpha_{1}(1-1/p_{1})-n(\alpha_{2}-\alpha_{1})/q, A_3(\mathbf{p},q,\alpha_1,\alpha_2)= n(\alpha_2-\alpha_1)(1/p_2-1/q)+n\alpha_1(1/p_1-1/p_2).
\mu_{n}^{2}\phi_{n}( z ) = \left( m^{2}- \frac{g^{2}N}{2 \pi}\right) \left( \frac{1}{z}+ \frac{1}{1 - z}\right) \phi_{n}( z ) - \frac{g^{2}N}{2 \pi}\int_{0}^{1}d z^{\prime}\frac{\phi_{n}( z^{\prime})}{( z - z^{\prime})^{2}},
\|f\|_{C^{0,\mu}H_\xi^s}:=\|f\|_{L^\infty H_\xi^s}+\sup_{\xi\neq \xi'\in Q_\ell^*}\frac{\|f(\xi,\cdot,\cdot)-f(\xi',\cdot,\cdot)\|_{H^s(Q_\ell\times Q_\ell)}}{|\xi-\xi'|^\mu}.
\tau\partial_{j}^{2}+\bar{\tau}\bar{\partial}_{j}^{2}=t(\partial_{j}^{2}+\bar{\partial}_{j}^{2})+iu(\partial_{j}^{2}-\bar{\partial}_{j}^{2})=\frac{t}{2}(\tilde{X}_{j}^{2}-\tilde{Y}_{j}^{2})+u\tilde{X}_{j}\tilde{Y}_{j}.
\begin{array}{ll}\|\alpha_1\|_{\mathcal{C}_0^{k-2}}=\|\frac{u^s_{yt}}{u^s_y}\|_{\mathcal{C}_0^{k-2}}\leq C_{\ell}\|\frac{u^s_{yt}}{u^s_y}\|_{\mathcal{C}_{\ell}^{k-2}}\leq C\|\alpha\|_{\mathcal{C}_{\ell}^{k-1}}\leq C(T).\end{array}
\det \left( 1 + K_{\alpha}(E) \right) = ( 1 + e^{- \beta_r E \alpha - (1 - \alpha ) \beta_l E}) ( 1 + e^{- \beta_l E \alpha - (1 - \alpha ) \beta_r E}) &= \frac{4 \cosh ( ( \beta_l ( 1 - \alpha ) + \beta_r \alpha ) E /2 ) \cosh ( ( \beta_r ( 1 - \alpha ) + \beta_l \alpha )E /2 )}{e^{( \beta_l + \beta_r ) E /2}},
\sum_{u,v,n\geq 0}\sharp \mathcal{D}(u,v,n) a^u b^v q^n = \sum_{u,v,n\geq 0}\sharp \mathcal{C}(u,v,n) a^u b^v q^n =(-aq;q)_{\infty}(-bq;q)_{\infty}\,\cdot
\begin{aligned}J_2(\beta, N,T) &:= CN^2 \int_0^T\int_0^T \int_{\mathbf{B}_2(0)}f^{(k,\ell,\alpha)}_{t,s}(z) dz ds dt &\leq CN^2 \int_0^T\int_0^T \left( \left( \frac{1}{(t+s)^{1/2}}\int_{|z_1|<2}e^{-\frac{(z_1-\beta^{1/3}N^{1/3}( t - s))^2}{2(t+s)}}dz_1 \right) \wedge 1 \right)ds dt.\end{aligned}
\Psi ( x_{1}, \cdots , x_{M}| z_{1}, \cdots , z_{N}) = \vartheta ( x_{1}, \cdots , x_{M}| z_{1}, \cdots , z_{N}) \prod_{\mu = 1}^{M}\prod_{j = 1}^{N}\psi \Bigl ( \frac{x_{\mu}}{z_{j}}\Bigr ) ,
1 =\sum_{v\in W}\rho(S_{e_v})^*\rho(S_{e_v}) + \sum_{v\in V\setminus W}\sum_{r(e)=v}\rho(S_e) \rho(S_e)^* &=\sum_{v\in W}\begin{bmatrix}\pi(S_{e_v})^* \pi(S_{e_v}) +Y_{e_v}^* Y_{e_v}* * *\end{bmatrix}&\quad\quad+\sum_{v\in V\setminus W}\sum_{r(e)=v}\begin{bmatrix}\pi(S_e) \pi(S_e)^* +X_e X_e^* * * *\end{bmatrix},
-i\left((S_2 \widetilde{F})^a \bar{G}^b - (S_2 \widetilde{F})^a (\bar{F}S_1)^b- \widetilde{G}^a(\bar{F}S_2)^b + (S_1 \widetilde{F})^a (\bar{F}S_2)^b\right).
E_\lambda(x):= \begin{pmatrix}e^{-i\lambda x}0 0 e^{i\lambda x}\end{pmatrix},R:= \begin{pmatrix}i 0 0 -i\end{pmatrix},\Phi:= \begin{pmatrix}0 \varphi_1 \varphi_2 0\end{pmatrix}\,.
\begin{array}{llll}\mbox{for}\phi=+1 ~:~ (i^k,1,\pm1,\pm1,\pm1,1) ,(k=0,...,3) \mbox{for}\phi=-1 ~:~ (i^k e^{i\pi/4},1,\pm1,\pm1,\pm1,1) ,(k=0,...,3)\; ,\end{array}
R_{l j k}(i_n):=\{\omega\in\mathcal{G}_n : \lvert \mathrm{i}\omega\cdot l+\mu_j^{\infty}(i_n)-\mu_k^{\infty}(i_n) \rvert< 2\,\gamma_n\, \lvert j^3-k^3 \rvert \langle l \rangle^{-\tau}\}.
\left( A+P-\varkappa_{\mathsf{k}}\right)^{\mathsf{n}^{\ast}}\mathsf{q}_{\mathsf{k},\gamma}=\gamma \left( 1-\varkappa_{\mathsf{k}}\right)^{\mathsf{n}^{\ast}}\mathsf{\hat{p}}_{\beta ,\mathsf{Gibbs}}
\sum_{n_1 + n_2 = 2g+1-D}{2g+1-D \choose n_1}\frac{(2g-3+ n+n_1)!}{(2g-3+n)!}\cdot n_2! \cdot{n_2 \choose 2c_i + 2 - e_i - e_{n+1}}=\left(2g+2 - \sum b_j - 1\right)! \cdot{4g -1 + n - \sum b_j \choose 2g-1 - \sum b_j - 2c_i+ e_i + e_{n+1}},
\big\lVert\omega_{1,\infty}\big\rVert^2_{Z_{1,0}}+\big\lVert\omega_{2,\infty}\big\rVert^2_{Z_{2,0}}= 1, D^F_{Z_{j,\infty}}\omega_{j,\infty}= \lambda_\infty\omega_{j,\infty}, \omega_{j,\infty}^\mathrm{zm}= 0, j=1,2.
w(\pi)=\begin{cases}(1+\delta(s_1+s_2))((1+t_1)(1+t_2))^{c_1'/2}(1+t_1+t_2)^{c_2'/2}, q\equiv 1 \pmod 4, (1+v_1)^{c_1'}(1+v_2)^{c_1'}(1+v_1+v_2)^{c_2'}, q \equiv 3 \pmod 4.\end{cases}
q^{(n-m)(n-j)}\left[ \begin{array}{l}j-m n-m\end{array}\right]^{1/2}_{q^{2}}\left[ \begin{array}{l}j+n n-m\end{array}\right]^{1/2}_{q^{2}}P^{n-m,m+n}_{j-n}(-c^{*}c;q^2) (-c^{*})^{n-m}(a^{*})^{m+n},
\omega^{2n-1}(\Phi)\omega^{2n-1}(\Psi) +\omega^{2n}(\Phi)\omega^{2n}(\Psi)= &=\sin\phi^1\sin\phi^2\dots\sin \phi^{2n-2}\sin\psi^1\sin\psi^2\dots\sin \psi^{2n-2}\cos(\phi^{2n-1}-\psi^{2n-1}) & \ge \frac{1}{\sqrt 2}\sin\phi^1\sin\phi^2\dots\sin \phi^{2n-2}\sin\psi^1\sin\psi^2\dots\sin \psi^{2n-2},
m^{2}\geq 4\pi v^{2}(n_{e},n_{m}) \, \frac{1}{{\rm Im}\, \tau}\left(\begin{array}{cc}1 -{\rm Re}\, \tau -{\rm Re}\, \tau |\tau|^{2}\end{array}\right) \left(\begin{array}{c}n_{e}n_{m}\end{array}\right)
\tilde{f}(H(s))=f( \tilde{H}(s)) \ \ \ \tilde{H}(s) \ \frac{\|H(s)-\tilde{H}(s) \|_{H_{2}\ \H_{\infty}}}{\|H(s)\|_{H_{2}\ \H_{\infty}}}= \ \mathcal{O}( ||Z||),
\dot{x}(t,s) = \lambda x(t,s) + \partial_s^2 x(t,s), s\in [0,1], &\dot{\hat x}(t,s) = \lambda \hat x(t,s) + \partial_s^2 \hat x(t,s) + l(s) \left(\partial_s x(t,1) - \partial_s \hat x(t,1)\right), &\partial_s x(t,0) = 0, x(t,1) = 0, &\partial_s \hat x(t,0) = 0, \hat x(t,1) = 0.
\begin{array}{c}\mathbf{E}_{n}^{{(k_1,...,k_r)}}(x;a,b)= \sum\limits_{k=0}^{n}\sum\limits_{j=0}^{k}r^{n-k}\binom{n}{k}\binom{k}{j}(\ln a)^{k-j}(\ln a+\ln b)^j\mathbf{E}_{j}^{{(k_1,...,k_r)}}x^{n-k}\end{array}.
|P_b(y)-P(y)| =\bigg{|}b\int_0^1\frac{\partial^2v}{\partial b^2}(tb,p,y)\chi(y)dt-P(y)\big{(}1-\chi(y)\big{)}\bigg{|}&\lesssim b_c|y|\mathbf{1}_{[-2b_c^{-1},0]}(y)+b_c\mathbf{1}_{[-2b_c^{-1},2b_c^{-1}]}(y)+\mathbf{1}_{\{|y|>1/b_c\}}(y),
{\cal B}^{r}( t ) \sim \frac{1}{( \Omega ! )^{2}}\left[ \Phi^{*}( t ) \right]^{r}\sum_{s = 0}^{r}{\binom{r}{s}}( \Omega - s ) ! \left[ \Omega - ( r - s ) \right] ! x^{2 ( r - s )}\Phi^{s}( t ) \Phi^{r - s}( t - 1 ) \; .
T = \sum_{a = 1}^{3}T^{a}\otimes T^{a}= \left( \begin{array}{c c c c}{1}&{0}&{0}&{0}\\{0}&{- 1}&{2}&{0}\\{0}&{2}&{- 1}&{0}\\{0}&{0}&{0}&{1}\\ \end{array}\right)
\sum_{\rho + \Lambda_{i}\in{\cal F}_{l}}S_{\beta \rho + \Lambda_{i}}{\cal U}_{\rho + \Lambda_{i}\rho}^{( p_{i}, q_{i})}= \exp \biggl ({\frac{\pi i k d}{4 l}}\mathrm{s i g n}\Bigl ({\frac{q_{i}}{p_{i}}}\Bigr ) \biggr ){\cal U}_{\beta \rho}^{( - q_{i}, p_{i})},
\begin{cases}\begin{aligned}\lim_{m\rightarrow\infty}H^2(p_m)=\sup H^2=\bar H^2,\lim_{m\rightarrow\infty}|\nabla H^2(p_m)|=0, &0\geq\lim_{m\rightarrow\infty}|\nabla^{\perp}\vec H|^2(p_m)-\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2+\dfrac12(\bar H^2-S)(\bar H^2-3S+2),\end{aligned}\end{cases}
\cot \delta = \cot \pi \nu -{\frac{\Gamma ( 1 + \nu )^{2}}{\pi \nu}}\left({\frac{k \ell}{2}}\right)^{- 2 \nu}-{\frac{\Gamma ( 1 + \nu )^{2}}{\pi^{2}}}\left({\frac{k}{2}}\right)^{- 2 \nu}\left[{C}+{\frac{k^{2}}{\xi^{2}}}\left({\frac{1 - \eta^{2 - 2 \nu}}{2 \pi ( 1 - \nu^{2})}}-{\frac{\eta^{2 - 2 \nu}- \eta^{2}}{2 \pi \nu ( 1 + \nu )}}\right) \right] .
tr_{L^2(N)}(T+z^2)^{-d}= \frac{1}{(d-1)!}\left(-\frac{1}{2z}\frac{\partial}{\partial z}\right)^{d-1}tr_{L^2(N)}(T+z^2)^{-1}= \frac{r^{2d-1}}{(d-1)!}\left(-\frac{1}{2rz}\frac{\partial}{\partial(rz)}\right)^{d-1}\sum_{\nu}I_\nu(rz)K_\nu(rz).
\begin{cases}w_1<\cdots <w_{k_2}, w_{k_2+1}<\cdots < w_{k_3}, \cdots, w_{k_p+1}< \cdots < w_n \mbox{if}\; k_1=0, 0<w_1<\cdots <w_{k_1}, w_{k_1+1}<\cdots < w_{k_2}, \cdots, w_{k_p+1}< \cdots < w_n \mbox{if}\; k_1\not=0.\end{cases}
< \overline{{{T}}}( k_{1}) T ( k_{2}) \phi_{a}( k_{3}) \phi_{b}( k_{4}) >_{t r u n c}= - i ( 2 \pi )^{D}\delta \left( \sum k_{i}\right) \left( 2 \nu_{a b}+ \frac{\nu_{a}\nu_{b}}{u - m^{2}}+ \frac{\nu_{a}\nu_{b}}{t - m^{2}}\right)
[\begin{matrix}Y^{\prime}\\C_{B}\\C_{R}\end{matrix}]=\frac{1}{256}[\begin{matrix}66&129&25\\ -38&-74&112\\ 112&-94&-18\end{matrix}][\begin{matrix}R^{\prime}\\G^{\prime}\\B^{\prime}\end{matrix}]+[\begin{matrix}16\\ 128\\ 128\end{matrix}]
z\frac{d}{dz}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z)=G_{p,q}^{m,n}(\begin{matrix}a_{1}-1,a_{2},...,a_{p}\\b_{q}\end{matrix}|z)+(a_{1}-1)G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z),n\ge1.
G''_{i,j}(\xi)=\frac{\varphi''}{\varphi}-\left( \frac{\varphi'}{\varphi}\right)^2=\pm \frac{r(r-1) \xi^{r-2}}{\varphi(i,j,\xi)}-\frac{r^2 \xi^{2r-2}}{\varphi(i,j,\xi)^2}\prec \frac{\xi^{r-2}}{\varphi(i,j,\xi)},
(\alpha(\beta))^{(s)}=\sum_{t=1}^s\sum_{A_1\dot\cup\ldots\dot\cup A_t}\beta^{(|A_1|)}\ldots\beta^{(|A_t|)}\alpha^{(t)}(\beta) &=\sum_{(1^{a_1},\ldots,s^{a_s})\in P(s)}\frac{s!}{(1!)^{a_1}\ldots (s!)^{a_s}a_1!\ldots a_s!}(\beta')^{a_1}\ldots(\beta^{(s)})^{a_s}(\alpha(\beta))^{(a_1+\ldots+a_s)},
\sum_{k=1}^\infty \left(C_1\mathbb{E}\left[\sum_{i=1}^N\alpha_{k,i}^2\mid \mathcal{F}_{k}\right] + 2C_2\mathbb{E}\left[\sum_{i=1}^N\left|\alpha_{k,i}- \frac{1}{kp_i}\right|\mid \mathcal{F}_{k}\right]\right) < \infty.
|-\frac{c_{b}}{2}\int_{0}^{\infty}d\xi \sin(t\xi) \xi J_{2}(r\xi) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}| \leq \frac{C r^{2}}{t^{4}\log^{b}(t)}, r \leq \frac{t}{2}
(u \delta_e) (k \delta_{g^{-1}}) (u \delta_e) (a_g \delta_g) (u \delta_e) = u \delta_e \Longleftrightarrow(uk \delta_{g^{-1}}) (u a_g \delta_g) (u \delta_e) = u \delta_e &\Longleftrightarrow\alpha_{g^{-1}}( \alpha_g(uk) u a_g ) \delta_e (u \delta_e) = u \delta_e
\Psi^k = \frac{1}{\eta_x}\|x^k - x^*\|^2+\frac{1}{\eta_y}\|y^k - y^*\|^2+\frac{2}{\sigma_x}B_f(x_f^k, x^*)+\frac{2}{\sigma_y}B_g(y_f^k, y^*) &+\frac{1}{4\eta_y}\|y^k - y^{k-1}\|^2-2\langle y^k-y^{k-1}, A(x^k-x^*) \rangle.
\vert e \vert = \vert \pi_{P_x}^\perp (e) \vert \leq \sphericalangle\left(P_x,P_{F(z)}\right) \vert e \vert + \vert \pi_{P_{F(z)}}^\perp (e)\vert \leq C_2(m) \tilde \delta_{x,k}\vert e \vert + \vert \pi_{P_{F(z)}}^\perp (e)\vert.
- k^{2}\varepsilon^{\nu \lambda}- k^{\nu}k_{\mu}\varepsilon^{\lambda \mu}- k^{\lambda}k_{\mu}\varepsilon^{\mu \nu}+ m^{2}( \varepsilon^{\nu \lambda}+ i k^{\nu}\varepsilon^{\lambda}- i k^{\lambda}\varepsilon^{\nu}) = 0 .
Z_\mu^{(v),\lambda}(\xi,\eta) = \frac{K_\mu(\xi\sqrt{2\lambda+|v|^2})}{K_\mu(\eta\sqrt{2\lambda+|v|^2})}\intertext{and}Z_\mu^{(v),\lambda}(\xi,\eta) = \frac{I_\mu(\xi\sqrt{2\lambda+|v|^2})}{I_\mu(\eta\sqrt{2\lambda+|v|^2})}.
\left( \int_{1}^{\infty}e^{-\delta z^{2}}dz\right)^{2}=\int_{1}^{\infty}\int_{1}^{\infty}e^{-\delta(z^{2}+w^{2})}dzdw & \geq\int_{\pi/6}^{\pi/3}\int_{2}^{\infty}e^{-\delta r^{2}}rdrd\varphi & =\frac{\pi}{6}\frac{e^{-4\delta}}{2\delta}= \frac{\pi e^{-4\delta}}{12\delta},
\eqref{eq:cont1}= \, -\frac{q-1}{2q}\sum_{x\in C_{1}^{(2,q^2)}}\psi(x\omega^{\frac{q+1}{2}})+\frac{(q-1)(q^2-1)}{4q}= \,-\frac{q-1}{2q}\left(\frac{-1+G_{q^2}(\eta)}{2}\right)+\frac{(q-1)(q^2-1)}{4q}=\frac{q^2-1}{4}.
H(q)=(q^{2};q^{5})_{\infty}^{-1}(q^{3};q^{5})_{\infty}^{-1}=(q^{5};q^{5})_{\infty}^{1/2}(q;q)_{\infty}^{-1/2}[\frac{H(q)}{G(q)}]^{1/2}