harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}t -\mu^2 (2 t \mathbf M^{-1}) =t^o z_1 + \mu^2 (2 z_1 \mathbf M^{-1}) = z_1^o z_2 + \mu^2 (2 z_2 \mathbf M^{-1}) =z_2^o.\end{cases}
\begin{aligned}p_{abr}\approx p\{\phi_{abr}\geq \max \{\phi_a, \phi_{b}\}\}p_{ar}\approx p\{\phi_{a}> \phi_{abr}\cap \phi_a \geq \phi_b \}p_{br}\approx p\{\phi_{b}> \phi_{abr}\cap \phi_b \geq \phi_a \}\end{aligned}
\int_{\mathbb{R}^n}e^{-\frac12 b \| x \|_K^2}e^{-\frac12a|x|^2}\, \frac{dx}{(2\pi)^\frac{n}{2}}=\frac1{|{\rm B}_2^n|}\int_K \frac{b}{( b+ a|x|^2 )^{\frac{n+2}{2}}}\, dx
((\upsilon \ltimes \pi) T(a))\xi = \sum_{g\in G}\pi \left(T(a)(g)\right)\upsilon_{g}\xi &=\sum_{g\in G}\pi\left(\frac{1}{k}\sum_{j = 1}^{k}\alpha_{h_j}(a_{h_{j}^{-1}g h_{j}})\right)\upsilon_g \xi\;.
\begin{aligned}{\mathbb E}\left\{{\left|{\beta_{n,{\rm{R}},{l_n}}^*{\beta_{m,{\rm{R}},{l_m}}}}\right|}\right\}= \frac{1}{2}{\mathbb E}\left\{\sqrt{{\left|\sqrt{2}{\beta_{n,{\rm{R}},{l_n}}}\right|^2}}\right\}{\mathbb E}\left\{\sqrt{{\left|\sqrt{2}{{\beta_{m,{\rm{R}},{l_m}}}}\right|^2}}\right\}.\end{aligned}
\begin{array}{c c}{\lambda^{a \alpha}= i \textstyle{\frac{1}{6}}\bar{D}^{a \beta}\mathrm{h}^{\alpha}{}_{\beta},}&{\bar{\lambda}_{a \alpha}= - i \textstyle{\frac{1}{6}}D_{a \beta}\mathrm{h}^{\beta}{}_{\alpha},}\\ \end{array}
Z ( u ) = \prod_{p = 1}^{\infty}\frac{\sin \left[ \widetilde{\gamma}( ( 2 p - 1 ) i \pi - u ) \right] \sin \left[ \widetilde{\gamma}( 2 p i \pi + u ) \right]}{\sin \left[ \widetilde{\gamma}( ( 2 p - 1 ) i \pi + u ) \right] \sin \left[ \widetilde{\gamma}( 2 p i \pi - u ) \right]}.
\left|\begin{array}{ccc}A B C D f g E h \boxed{i}\end{array}\right|=\left|\begin{array}{cc}A C E \boxed{i}\end{array}\right|-\left|\begin{array}{cc}A B E \boxed{h}\end{array}\right|\left|\begin{array}{cc}A B D \boxed{f}\end{array}\right|^{-1}\left|\begin{array}{cc}A C D \boxed{g}\end{array}\right|,
L^*_5:={}-t^5 \frac{\partial^{5}}{\partial t^{5}}-15t^{4}\frac{\partial^{4}}{\partial t^{4}}+5 t^3 (4 t^2-13)\frac{\partial^{3}}{\partial t^{3}}+90 t^2 (2 t^2-1)\frac{\partial^{2}}{\partial t^{2}}{}-t (64 t^4-392 t^2+31)\frac{\partial}{\partial t}-(192 t^4-184 t^2+1).
(I-G_0) US_{a,b}U^* G_0 + G_0 U^* S_{a,b}^* U (I-G_0) = U \left(\begin{array}{cc}0 0 0 I_{N-K}\end{array}\right)S_{a,b}\left(\begin{array}{cc}I_K 0 0 0 \end{array}\right) U^* & - U \left(\begin{array}{cc}I_K 0 0 0 \end{array}\right)S_{a,b}\left(\begin{array}{cc}0 0 0 I_{N-K}\end{array}\right) U^*
\begin{aligned}[x_1^{\mu_1}]f_{k_1}(x_1)^{\nu}\,[x^{\mu_{1,2}+\mu_2}]f_{k_1}(x)^{N-\nu}&\le \frac{\gamma_{k_1}^2}{\sqrt{\mu_1(\mu_{1,2}+\mu_2)}}\frac{f_{k_1}(z_i)^N}{z_i^{M+1}}\!\left(\frac{M}{N}\right)^{M+1}\!\!\left(\frac{\nu}{\mu_1}\right)^{\mu_1}\!\!\left(\frac{N-\nu}{\mu_{1,2}+\mu_2}\right)^{\mu_{1,2}+\mu_2}\!.\end{aligned}
\begin{matrix}\begin{aligned}C_{0}^{\vee}=\mathrm{Cone}\left\{\begin{matrix}(0,1,0,0),(1,0,0,0),(0,0,1,0),(0,0,0,1) (1,1,0,-1),(-1,0,1,1)\end{matrix}\right\}, C_{NE}^{\vee}=\mathrm{Cone}\left\{(1,1,0,0),(0,1,1,0),(0,0,1,1),(1,1,1,0),(0,1,1,1)\right\}.\end{aligned}\end{matrix}
\omega= \sum_{i=1}^n \frac{du_i}{u_i}\wedge d\beta_i-\sum_{i<j}\frac{d\beta_i\wedge d\beta_j}{\beta_i-\beta_j}+\sum_{i=1}^m \frac{dv_i}{v_i}\wedge d\kappa_i-\sum_{i<j}\frac{d\kappa_i\wedge d\kappa_j}{\kappa_i-\kappa_j}.
\xi_{1/2}f(z) = \frac{1}{2}c_0^- u_{3/2,0}^{[0],+}(y) - \sum_{0<n \equiv 0,3 (4)}c_{-n}^- u_{3/2,n}^{[0],-}(y)e^{2\pi inx}&= \frac{1}{2}c_0^- - \sum_{0<n \equiv 0,3 (4)}c_{-n}^- (4\pi n)^{3/4}q^n,
s_{i,j}^{(-1)}= -\sum_{k=1}^i \sum_{k_2=1}^{k-1}\sum_{k_3=1}^{k_2-1}s_{i,k-1}\cdot s_{k-1,k_2-1}\cdot s_{k_2-1,k_3-1}\cdot s_{k_3-1,j}^{(-1)}\phantom{=\sum\sum}+ \sum_{k=j+2}^i s_{i,k-1}\cdot s_{k-1,j}- s_{i,j}+ \delta_{i,j}.
\hat{K}_{\varepsilon}(\rho ,t)=\frac{1}{2\pi \mathrm{i}}\int_{s_{0}-\mathrm{i}\infty}^{s_{0}+\mathrm{i}\infty}\widehat{\tilde{K}}_{\varepsilon}(\rho,s)\mathrm{e}^{st}\mathrm{d}s,\;\;a\geq 0,
d\big(\tilde J_\rho\rho\big) = \, \, q_1e^{1234}+q_2e^{1235}+q_3e^{1245}+q_4e^{1246}+q_5(e^{1256}-e^{2346})+q_6e^{1345}begin{align*}2pt] + q_7(e^{1346}-e^{2356})+q_8e^{1356}+q_9e^{2345},
X \triangleright (\mathsf{M}^n_m)^i_j = (X_{(1)}\triangleright u^{m*}_i) (X_{(2)}\triangleright u^n_j) = \sum_{k, \ell}\pi(S(X_{(1)}))^i_k \pi(X_{(2)})^\ell_j u^{m*}_k u^n_\ell.
\nu\left(Q_{i'}^{s-q}\left(\partial_{j_0}a_{s,i'}\right)\prod\limits_{t=1}^q\left(\partial_{j_t}Q_{i'}\right)\right)>\nu_i\left(a_{j,'i}Q_{i'}^j\right)-p^e\left(\beta_{i'}-\nu\left(\partial_{b_\infty}Q_{i'}\right)\right).
\log\Bigg(\int e^{\sum_{m=4}^NJ_m^l(\psi+\psi^1)}d\mu_{C_l}(\psi^1)\Bigg) &=\sum_{n=1}^{\infty}\frac{1}{n!}\left(\frac{d}{dz}\right)^n\log\Bigg(\int e^{z\sum_{m=4}^NJ_m^l(\psi+\psi^1)}d\mu_{C_l}(\psi^1)\Bigg)\Bigg|_{z=0}&=J^{l-1}(\psi),\ (\forall l\in \{0,-1,\cdots,N_{\beta}\}),
J_N(z, u) = \mathbf{E}_{z}^u\!\left[\sum_{j=0}^{\infty}e^{-\beta t_j}\left( r(z^{u,N}(t_j), \nu_j) + \int_{t_j}^{t_{j+1}}\phi(z^{u,N}(s), \nu_j)\, ds \right)\right]\,,
I^{(\alpha)}(B_R) = \sum_{n \in \mathbb{Z}}\frac{\left|\left\langle w^\infty_{\alpha\alpha}, \varphi^{(n)}_{B_R, \alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_R, \alpha}\right|}< + \infty
\begin{array}{l}{{C l ( \eta_{1}) > 0 \; \; \Longrightarrow C l ( \eta_{1}\oplus \eta_{2}) = C l ( \eta_{1}) \otimes C l ( \eta_{2})}}\\{{C l ( \eta_{1}) < 0 \; \; \Longrightarrow C l ( \eta_{1}\oplus \eta_{2}) = C l ( \eta_{1}) \otimes C l ( - \eta_{2}) \, .}}\\ \end{array}
G_{\ell}^{(\underline{a},N)}(\tau):=\sideset{}{'}\sum_{\substack{m_1,m_2\in\mathbb{Z}m_1\equiv a_1\!\!\!\!\!\!\pmod{N}m_2\equiv a_2\!\!\!\!\!\!\pmod{N}}}\dfrac{1}{(m_1\tau+m_2)^{\ell}}
Q({\bf X}_N,{\bf Y}_N,{\bf Y}_N^{-1})=P_1({\bf X}_N,Y_N){\bf Y}_N^{-1}P_2({\bf X}_N,{\bf Y}_N){\bf Y}_N^{-1}\ldots P_n({\bf X}_N,{\bf Y}_N){\bf Y}_N^{-1}
ds^2= \alpha'\left [\frac{u^2}{\tilde{R}^2}(\cos^{2}\varphi (- \tilde{f}dx_0^2 + dx_1^2) + \cos^{-2}\theta (dx_2^2+dx_3^2)) + \frac{\tilde{R}^2}{u^2}(\tilde{f}^{-1}du^2 +u^2 d\Omega^2)\right],
\frac{\mathbb{E}\{\mathrm{I}_{ij}^\nu\}}{\mathbb{E}\{\mathrm{I}_{ij}\}^\nu}= \frac{\nu!}{\alpha_{ij}^\nu(1+\rho_{ij})^\nu}\sum_{k=0}^{\nu}\binom{\nu}{k}\frac{\Gamma(\alpha_{ij}+\nu-k)}{\Gamma(\alpha_{ij})}\frac{(\alpha_{ij}\rho_{ij})^\nu}{\nu!}
\frac{4}{\tilde R}\,h^{-1/4}\,\pi_i\,h^{1/2}\, h^{ij}\,\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 .
\delta := \min_{i=1,\ldots,m}\mu(F_i)+\varepsilon -\mu''(F_i) ,\delta' := \min_{i=1,\ldots,m'}\mu'(F'_i)+\varepsilon' -\mu''(F'_i), \gamma := \min_{j=1,\ldots,n}\varepsilon - |\mu''(\overline{B}_{r_j}) - \mu(\overline{B}_{r_j})| ,\gamma' := \min_{j=1,\ldots,n'}\varepsilon' - |\mu''(\overline{B}_{r'_j}) - \mu'(\overline{B}_{r'_j})|;
\vert\min{\bf I_l^s}(A_1)-\min{\bf I_l^s}(A_2)\vert >0\hbox{and, if}\min{\bf \Sigma^s}({\bf I_l^s}(A_1))=\min{\bf \Sigma^s}({\bf I_l^s}(A_2)), \hbox{their respective multiplicities in the vectors}\quad{\bf \Sigma^s}({\bf I_l^s}(A_1))\,\,\hbox{and}\,\,\,{\bf \Sigma^s}({\bf I_l^s}(A_2)) \hbox{are different,}
\xi (-J) m'(B^*) + \delta_{n, 3}\tilde \xi (-J) \rho_{\frac \pi 2}m'(B) = \mu_* [B,-J] &= \theta^t \mu_* [\theta B, J] &= \theta^t \left( \xi (J) m'((\theta B)^*) + \delta_{n, 3}\tilde \xi (J) \rho_{\frac \pi 2}m'(\theta B) \right) &= \xi (J) m'(B^*) - \delta_{n, 3}\tilde \xi (J) \rho_{\frac \pi 2}m'(B) ,
\sqrt{- g}\left( -{\frac{2}{{\kappa^{\prime}}^{2}}}R ( \eta_{\mu \nu}+ \kappa^{\prime}\sqrt{Z_{g}}\phi_{\mu \nu}) +{\frac{1}{2}}Z_{\varphi}g^{\mu \nu}\partial_{\mu}\varphi \partial_{\nu}\varphi \right) .
\mathcal{W}_\sigma(z)\equiv -\,W\left[w_\sigma,\,w_{\rm reg}\right](0)=\delta_{\sigma,0}\,\left(\sum_{j=0}^\infty j\,b_j(z)\right) + \delta_{\sigma,1}\,\left(\sum_{j=0}^\infty b_j(z)\right)\,,
\lim_{N \to \infty}\frac{1}{N \theta}\log \left( \int_{\mathbb{R}^{d \times N}}\exp \left( - \beta \mathcal{H}^{*}_{N}\right) d \pi^{\otimes N}\right) = \inf_{\mu \in \mathcal{P}(\mathbb{R}^{d})}\{\mathcal{E}^{*}(\mu) + \frac{1}{\theta}{\rm ent}[\mu | \pi] \}.
P(B|A)=\frac{\sum_{F\subset A\cap B}|\int D\varphi O_{in}[\varphi]e^{iS[\varphi]}F[\varphi]|^{2}}{\sum_{F\subset A}|\int D\varphi O_{in}[\varphi]e^{iS[\varphi]}F[\varphi]|^{2}},
\alpha: =\max(\delta^{1+\frac{p-2}{2}}2^{\frac{p}{2}-1}\eta^{p}C^{p}_f+2^{\frac{p}{2}-1}4^{p-1}\gamma^p_1, \delta^{1+\frac{p-2}{2}}\eta^{p}C^{p}_f+2^{\frac{p}{2}-1}4^{p-1}\gamma^p_2, \qquad\delta^{1+\frac{p-2}{2}}\eta^{p}C^{p}_f+2^{\frac{p}{2}-1}4^{p-1}\kappa^p_1, \delta^{1+\frac{p-2}{2}}\eta^{p}C^{p}_f+2^{\frac{p}{2}-1}4^{p-1}\kappa^p_2).
\begin{aligned}R(M_s,M_p)\Big|_{\textrm{Scheme2}}= \frac{\displaystyle\sum_{n=1}^{\Lambda-t_s}\binom{\Lambda-n}{t_s}\left[\binom{\mathcal{L}_1}{t_p+1}-\binom{\mathcal{L}_1-\mathcal{L}_n}{t_p+1}\right]}{\binom{\Lambda}{t_s}\binom{\mathcal{L}_1}{t_p}}\end{aligned}
\# \left(K_n\cap F_n\right) \leq \frac{1}{\mathsf{c}\log(n)}\sum_{j\in F_n(w)}l(n,j)\mathbf{1}_{K_n(w)}\leq \frac{1}{\mathsf{c}\log(n)}\# \left\{0\leq i\leq n-1:\ \mathbf{z}(a)_{s_i(w)}\neq \mathbf{z}(a')_{s_i(w)}\right\}= \frac{n}{\mathsf{c}\log(n)}\bar{d}_n \left( a,a' \right).
\max_{v\in \mathcal{V}}\mathcal{H}(t,{X}_{t},v,\hat{p}, \hat{q}, \hat{w})=\mathcal{H}(t, \hat{X}_{t}, \hat{u}_{t},\hat{p}, \hat{q}, \hat{w}) , \mbox{}\mbox{}dt \times P \mbox{}a.s.
\lim_{\varepsilon \to +0}\sum_{\beta>0, n\in \mathbb{Z}}L_{n, \beta}^{t_i+\varepsilon}q^n t^{\beta}&=\exp\left( \sum_{n/\omega \cdot \beta=t_i}(-1)^{n-1}n N_{n, \beta}q^n t^{\beta}\right) \cdot \left(\lim_{\varepsilon \to +0}\sum_{\beta>0, n\in \mathbb{Z}}L_{n, \beta}^{t_i-\varepsilon}q^n t^{\beta}\right).
W_1(\upsilon_x , \upsilon_y) = \sup_{\|g\|_{Lip_0}\leq 1}\left\{S\circ\delta_x(g) - S\circ\delta_y(g)\right\}&= \sup_{\|g\|_{Lip_0}\leq 1}\left\{E(g)(x)-E(g)(y)\right\}\hbox{(since $\|E(g)\|_{Lip}\leq K$)}\ \leq K W_1(\delta_x, \delta_y) &= K d(x, y).
I_n^{(2)}\leq c'_1 \left\{\int_{|\eta'|>R/2}d^3 \eta' \bigl| V_{13}(\alpha' \eta')\bigr| \right\}\int d^3 p_\zeta \;{\tilde t}^{\; 2}_n (p_\zeta) |p_\zeta|^{-4}= \hbox{o}(R),
\sum_{x_i, x_j}\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 + 2\big\langle x_i - x_j, \xi_2 (x_i - x_j)\big\rangle = 0, \sum_{x_i, x_j}\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 + 2\big\langle x_i - x_j, \xi_2 (x_i - x_j)\big\rangle = 0.
B(x_{1},x_{2},x_{3}) = \begin{cases}|\theta x_{1}^{1/p}-(1-\theta)x_{2}^{1/p}|^{p}, x_{1}^{1/p}+x_{2}^{1/p}=x_{3}^{1/p}, (\theta x_{3}^{1/p}+x_{2}^{1/p})^{p}, x_{2}^{1/p}+x_{3}^{1/p}=x_{1}^{1/p}, (x_{1}^{1/p}+(1-\theta)x_{3}^{1/p})^{p}, x_{3}^{1/p}+x_{1}^{1/p}=x_{2}^{1/p},\end{cases}
X_{i}^{-}V_{\vec{\lambda}, \vec{m}}( z ) = \frac{1}{1 - q_{i}^{- 1}}\int_{G}d t J_{i}( t ) V_{\vec{\lambda}, \vec{m}}( z ) ~ \sim ~ Q_{i}( \infty , z ) V_{\vec{\lambda}, \vec{m}}( z ) ,
\frac{\mu^{s + 1}}{2}\left( \frac{2 \pi}{L}\right)^{- s}\sum_{n = - \infty}^{\infty}\frac{1}{\Gamma \left( \frac{s}{2}\right)}\int_{0}^{\infty}d t \, t^{( \frac{s}{2}- 1 )}e^{- ( n^{2}+ ( \frac{m L}{2 \pi})^{2}) t}\rfloor_{s = - 1}\, =
\|\nabla u\|_{L^\infty(\Omega_\delta)}=\left( \frac{\mathcal{R}_o}{\mathcal{C}_{12}}\right)^{\frac{1}{p-1}}\delta^{-\frac{d-1}{2(p-1)}}\left[1+o(1)\right], p>2, d=2,3, \mbox{for}~\delta\ll 1,
(1-x)^{m}H_{m}(x) =\left(\sum_{j=0}^{m}{m\choose j}(-1)^{j}x^{j}\right)\left(\sum_{n=0}^{\infty}b_{m}(n)x^{n}\right) =\sum_{n=0}^{\infty}\left(\sum_{j=0}^{m}{m\choose j}(-1)^{j}b_{m}(n-j)\right)x^{n}=\sum_{n=0}^{\infty}b_{m}(n)x^{2n}.
Z [ J ] = \mathcal{N}\int \big [ \mathcal{D}\Phi^{\prime}\big ] \; \mathrm{e x p}\Big ( \frac{\mathrm{i}}{\hbar}\Gamma_{\mathit{c l}}[ \Phi^{\prime}] + \frac{\mathrm{i}}{\hbar}J^{i}\Phi_{i}^{\prime}\Big ) \; .
{\cal L}= \frac{m}{2}\left((\dot x_1)^2 + (\dot x_2)^2\right) +\frac{Be}{2}\epsilon_{ij}\left(\dot x_1^i x_1^j - \dot x_2^i x_2^j\right) -\frac{K}{2}(x_1 - x_2)^2
c_n^q(r) =\sum_{k=0}^n\:\sum_{i=k}^n \binom{n}{i}\:(-1)^{n-i}\:q^{i-k}\:[r|q]_{n-i}\:s(i,k)\:\frac{1}{k+1}=\sum_{i=0}^n\:\sum_{k=0}^i \binom{n}{i}\:(-1)^{n-i}\:q^{i-k}\:[r|q]_{n-i}\:s(i,k)\:\frac{1}{k+1}.
G^{[d]}(\xi) =\lim_{\rho\to0}\left.\rho^d\,\frac{\partial^d}{\partial\Theta^d}p_\zeta(\Upsilon_\Theta S^{-1}(\rho\xi))\right|_{\Theta=0}&=\frac{\partial^d}{\partial\xi_2^d}\lim_{\rho\to0}p_\zeta(S^{-1}(\rho\xi))=\frac{\partial^d}{\partial\xi_2^d}G^{[0]}(\xi).
\sum_{n=1}^\infty P(Y_n\geq u^{-1}(n)) =\sum_{n=1}^\infty P(Y_1\geq u^{-1}(n))\leq \int_0^\infty P(Y_1\geq u^{-1}(x))dx &= \int_0^\infty P(u(Y_1)\geq x)dx=E[u(Y_1)]<\infty\,.
f(n\tau) \approx S_0(\tau) (S_0(\tau) P)^{n-1}f_0 + S_0(\tau) (P-1) (S_0(\tau) P)^{n-1}f_0 = \cdots\cdots\cdots = S_0(n\tau)f_0 + \sum_{k=1}^n S_0(k\tau) (P-1) (S_0(\tau) P)^{n-k}f_0 \approx S_0(n\tau)f_0 + \sum_{k=1}^n S_0(k\tau) (P-1) f[ (n-k) \tau]\,.
\langle \hat{T}_{\mu}^{\nu}( x ) \rangle_{R e g .}= u_{1}( \alpha , \gamma ) d i a g ( 1 , 1 , - 3 , 1 ) + \left( \xi - 1 / 6 \right) u_{2}( \alpha , \gamma ) d i a g ( 1 , - 1 / 2 , 3 / 2 , 1 ) \ ,
\widehat{\lambda}_{s,\mu}(E)=\left( D_{s,1}\sup_{\delta\neq 0}\sup_{u\in \mathbb{Z}^d}\sum_{v\in \mathbb{Z}^d}|G_0(u,v;E+i\delta)|^se^{\mu|u-v|}\right)^\frac{-1}{s}.
{\cal H}= \frac{1}{2}\left[ \vec{E}_{\bot}^{2}+ \vec{B}^{2}\right] + \frac{1}{2}{\cal E}_{3}^{2}+ \frac{1}{2 L^{2}}e_{3}^{2}\left( \vec{x}_{\perp}\right) + \frac{1}{2}\vec{\eta}^{2}\left( \vec{x}_{\perp}\right) \ .
S_{3}(\lambda) = \frac{2}{1+2\lambda}\sum\limits_{I_{13}=-\lambda}^\lambda \sum\limits_{I_{23}=-\lambda}^\lambda (-1)^{I_{13}+I_{23}}\binom{2\lambda}{\lambda+I_{13}}\binom{2\lambda}{\lambda+I_{23}}\frac{1}{2+4\lambda+I_{13}+I_{23}}\frac{1}{1+2\lambda-I_{13}-I_{23}}\times & \times\frac{1}{\binom{1+3\lambda+I_{13}}{1+2\lambda}}.
\partial_s \binom{q_1}{q_2}= \left( \begin{matrix}\mathcal{L}+ V 0 0 \mathcal{L}+ V\end{matrix}\right) \binom{q_1}{q_2}+ \left( \begin{matrix}V_{1,1}V_{1,2}V_{2,1}V_{2,2}\end{matrix}\right) \binom{q_1}{q_2}+ \binom{B_1(q_1, q_2)}{B_2(q_1,q_2)}+ \binom{R_1}{R_{2}},
g^{\mu \nu}( t , \vec{x}) \equiv{\frac{1}{c^{2}}}\left[ \begin{matrix}{{- 1}}&{{\vdots}}&{{- v \; \hat{r}^{j}}}\\{{\cdots \cdots}}&{{\cdot}}&{{\cdots \cdots \cdots \cdots}}\\{{- v \; \hat{r}^{i}}}&{{\vdots}}&{{( c^{2}\; \delta^{i j}- v^{2}\; \hat{r}^{i}\; \hat{r}^{j})}}\\ \end{matrix}\right] .
\gamma_{\textrm{c},k}=\frac{{\lvert \mathbf{h}_{k}^{H}\mathbf{p}_{\textrm{c}}\rvert}^{2}}{\sum_{i=1}^{K}{\lvert \mathbf{h}_{k}^{H}\mathbf{p}_{\textrm{p},i}\rvert}^{2}+\sigma_{z}^{2}}.
\displaystyle \frac{{\cal J}_{-}}{{\cal J}_{+}}= 2 \displaystyle \frac{J_{l - \xi}( k a ) J_{l + \Phi}( k a ) - J_{l}( k a ) J_{l + \Phi - \xi}( k a )}{J_{l - \xi}( k a ) H_{l + \Phi}^{+}( k a ) - J_{l}( k a ) H_{l + \Phi - \xi}^{+}( k a )}- 1 .
T(A_1(n))=\{(\mu,(c^d))\in B(n) \ | \c+d+1\in \mu\}; T(A_2(n))=\left\{(\mu,(c^d))\in B(n) \ \Big | \ \begin{array}{ll}c+d+1\not \in\mu, \, \mu_1\neq 3(c+d+1), \mu_1-\mu_2 >2(c+d+1)\end{array}\right\}; T(A_3(n))=\{(\mu,(c^d))\in B(n) \ | \ \mu=(c,c-2)\}.
\mathcal{S}_{q}(z)-\mathcal{S}_{q}(qz)=z\mathcal{C}_{q}(q^{1/2}z) \mbox{and}\mathcal{C}_{q}(z)-\mathcal{C}_{q}(qz)=-zq^{1/2}\mathcal{S}_{q}(q^{1/2}z).
2 \left( D_{x}^{\dagger}D_{x}\right)^{- 1}( z , z^{\prime}) = ( \sigma_{0}+ i \sigma_{3}) e^{- \phi ( z )}K_{+}( z , z^{\prime}; x ) e^{- \phi ( z^{\prime})}+ ( \sigma_{0}- i \sigma_{3}) e^{\phi ( z )}K_{-}( z , z^{\prime}; x ) e^{\phi ( z^{\prime})},
P(s) := \sum_{i=1}^h \frac{k_{i}\omega_{n,i}^2}{s^2 +2\zeta_i\omega_{n,i}s + \omega^2_{n,i}}\Delta(s) :=\!\!\sum_{i=(h+1)}^H \frac{k_{i}\omega_{n,i}^2}{s^2 +2\zeta_i\omega_{n,i}s + \omega^2_{n,i}}.
\sum_{n=1}^{\infty}|\partial_x[(h_{\alpha}\circ F_{\alpha, n}^{-1}) (\hat \Psi_{\alpha}\circ F_{\alpha, n}^{-1}) G_{\alpha, n}]| \leq C\sum_{n=1}^{\infty}n^{-\left(1+\frac{1}{\alpha}\right)}(n+1) < \infty
(\min_{j}\beta_{j}^{2})g_{E}\leq g^{k}_{\beta}\leq g_{E},\g_{E}=\Sigma_{j=1}^{k}(dr_{j}^2+r_{j}^2d\theta_{j}^2)+\Sigma_{j=k+1}^{n}dz_{j}\otimes d\bar{z}_{j}.
\int_{\Omega}\phi_1(q+\varepsilon h) (-\Delta \phi_1(q))\, dx + \int_{\Omega}(q+\varepsilon h)\phi_1(q+\epsilon h)\phi_1(q) \,dx = &\lambda_1(q+\epsilon h) \int_{\Omega}\phi_1(q+\epsilon h)\phi_1(q) \,dx.
I^{0+}_\textup{CC}(\bar x,\bar y,\bar z) =I^{-0}(\bar x)\cup I^{-+}(\bar x)\cup I^{0+}(\bar x), I^{+0}_\textup{CC}(\bar x,\bar y,\bar z) =I^{0-}(\bar x)\cup I^{+-}(\bar x)\cup I^{+0}(\bar x), I^{00}_\textup{CC}(\bar x,\bar y,\bar z) =I^{--}(\bar x)\cup I^{00}(\bar x).
C\left\|\,g,\, b_{\,2},\, \cdots,\, b_{\,n}\,\right\|_{2}^{\,2}\, \leq\,\sum\limits_{\,j \,=\, 1}^{\,\infty}\,\left<\,g,\, g_{\,j}\,|\, b_{\,2},\, \cdots,\, b_{\,n}\,\right>_{2}\,\left<\,C_{2}\,g_{\,j},\, g \,|\, b_{\,2},\, \cdots,\, b_{\,n}\,\right>_{2}&\leq\, D\,\left\|\,g,\, b_{\,2},\, \cdots,\, b_{\,n}\,\right\|_{2}^{\,2}\; \;\forall\; g \,\in\, K_{G}.
\kappa^{2}\delta \rho = - 6 \left( \dot{\alpha_{0}}\dot{\xi}_{0}^{y}- \dot{\alpha}_{0}^{2}\xi_{0}^{y}- \dot{\alpha}_{0}^{2}\alpha_{1}l^{2}\phi_{0}+ \dot{\alpha_{0}}\ddot{\alpha}_{0}e^{\alpha_{0}}l b_{0}- \frac{\dot{\alpha}_{0}\ddot{\alpha}_{0}}{\alpha_{1}}e^{2 \alpha_{0}}\dot{\hat{E}}_{0}- \hat{\Psi}_{1}\right) ,
0 < \phi(n^*n) = \alpha_n(\phi)(d_1)\phi(n^*n) = \phi(n^* d_1 n) = \phi\begin{pmatrix}n_1^*d_1n_1 n_1^*d_1\cdot\xi \xi^* \cdot d_1 n_1 \langle\xi, d_1\cdot \xi\rangle_{A_2}\end{pmatrix}= \phi(n_1^*d_1n_1) = \alpha_{n_1}(\phi)(d_1) \phi(n_1^*n_1).
\int_t^{\varepsilon_n^\alpha}a'(s)\frac{(U)^2(s)}{2}ds=a(\varepsilon_n^\alpha)\frac{(U)^2(\varepsilon_n^\alpha)}{2}-a(t)\frac{(U)^2(t)}{2}-\int_t^{\varepsilon_n^\alpha}a(s)U(s)U'(s)ds
{\frac{\xi_{0}}{\lambda}}= 2 ~ i ~ \xi_{0}\left({\frac{1}{2 L}}\sum_{n = \pm \frac{1}{2}, \pm \frac{3}{2}, \cdots ~}\right) \int_{- \infty}^{\infty}{\frac{d p^{-}}{2 \pi}}{\frac{1}{2 p_{n}^{+}p^{-}- \xi_{0}^{2}+ i \epsilon}}~ ,
\langle Q_{\lambda}, P_{\mu}P_{\nu}\rangle:=\langle Q_{\lambda/\mu}, P_{\nu}\rangle,Q_{\lambda/\mu}(x_1,\dots,x_n;t)=\frac{b_{\lambda}(t)}{b_{\mu}(t)}P_{\lambda/\mu}(x_1,\dots,x_n;t).
\lim\limits_{h \rightarrow 0}\frac{\int_B G(x + h,z)P_2(y,z)dz - \int_B G(x,z)P_2(y,z)dz}{h}= \lim\limits_{h \rightarrow 0}\int_B\frac{G(x + h,z)- G(x,z)}{h}P_2(y,z)dz = \int_B \partial_x G(x,z)P_2(y,z)dz.
p(h_{l}^{(2m)}=1|h^{(1m)},h^{(3)})=g(\sum_{j=1}^{F_{1}^{m}}W_{jl}^{(2m)}h_{j}^{(1m)}+\sum_{p=1}^{F_{3}}W_{lp}^{(3m)}h_{p}^{(3)}+b_{l}^{(2m)})
\gamma_a(2s) = v^{-2s^2-s}\prod_{k=1}^s \frac{1-v^{-2a-4k}}{1-v^{-4k}}, \gamma_a(2s+1) = v^{-a-2s^2-3s-1}\prod_{k=1}^s \frac{1-v^{-2a-4k}}{1-v^{-4k}}.
I_-^\mu \big\{(1+x)^{-\mu}x\hat h_j(x)\big\}\big|_{x=1}= 2(1-\mu)\Gamma(2-\mu) \hat \varrho_{1j}+ 2 \sum_{l=1}^{N-2}\frac{\Gamma(l+2-\mu)}{(l+1)!}\hat \varrho_{l+1,j}&=I_-^\mu \big\{(1+x)^{-\mu}\hat h_j(x)\big\}\big|_{x=1}- 2\mu\,\Gamma(2-\mu) \hat \varrho_{1j}.
\min_{\tilde{\psi}_{t,n}, \tilde{\psi}_{r,n}}\mathrm{Re}( \tilde{\vartheta}_{t,n}^* \tilde{\psi}_{t,n}) + \mathrm{Re}( \tilde{\vartheta}_{r,n}^* \tilde{\psi}_{r,n}) begin{align*}-0.5em] \mathrm{s.t.}\cos(\tilde{\phi}_{t,n}- \tilde{\phi}_{r,n}) = 0, |\tilde{\psi}_{t,n}| = 1, |\tilde{\psi}_{r,n}| = 1.
S_{i,j}(y_a,u_a,d_b) = \begin{dcases}\smashoperator[r]{\sum_{u_b}}W_2(d_{bj}|u_b;y_a, 1), \begin{aligned}u_a =0, begin{align*}-0.1cm] d_b =\pm d_{bi}\end{aligned}\smashoperator[r]{\sum_{u_b}}W_2(d_{bi}|u_b;y_a, 0), \begin{aligned}u_a =1, begin{align*}-0.1cm] d_b =\pm d_{bj}\end{aligned}0, \end{dcases}
F_{L}^{( \alpha )}( \bar{z}) = - \overline{{\Theta}}_{\Lambda_{2 4}}\partial_{\bar{z}}\overline{{\rho}}- \overline{{\Theta}}_{\Lambda_{2 4}}\overline{{\hat{G}}}_{2}+ \frac{\pi^{2}}{6 \zeta ( 1 4 )}\overline{{G}}_{1 4}.
a_{t,\tau}(x,s,\xi) = \psi(\tau |s|)\,a_t(x,s,\xi)+ [1-\psi(\tau |s|)]\,|\xi|^{p-2}\xi\,, &b_{t,\tau}(x,s,\xi) = \Psi_{\tau}(b_t(x,s,\xi))\,,
|\Delta\bar v_h(x)|^2=\frac{E^2}{(1-\nu^2)^2}\frac{s^2}{16\pi^2}\log^2\frac{\big(x_1-\frac{h}{2}\big)^2+x_2^2}{\big(x_1+\frac{h}{2}\big)^2+x_2^2}\,.
Nr^{2}(JO_{n}^{(3)}) =(J_{n}^{(3)})^{2}+(J_{n+1}^{(3)})^{2}+\cdots +(J_{n+7}^{(3)})^{2}&=\frac{1}{49}\left((2^{n+1}-(a\omega_{1}^{n}+b\omega_{2}^{n}))^{2}+\cdots +((2^{n+8}-(a\omega_{1}^{n+7}+b\omega_{2}^{n+7}))^{2}\right),
\big(\overset{\rightarrow}{L}_{-1}\phi_+\big)(z)=\frac{d}{d\epsilon}\big\vert_{\epsilon=0}\left(\phi_+\left(\frac{\lambda(\epsilon)z+\omega(\epsilon)}{1+\omega(\epsilon)^*\lambda(\epsilon)z}\right)+\epsilon\right) \phantom{\big(\overset{\rightarrow}{L}_{-1}\phi_+\big)(z)}=\phi_+'(z)\big(\dot{\lambda}(0)z+\dot{\omega}(0)-z\dot{\omega}(0)^* z\big)+1.
\frac{\ddot{a}}{\operatorname{c o s}^{2}{\frac{\tau}{r_{0}}}}+ \frac{2 a}{r_{0}^{2}\operatorname{c o s}^{4}{\frac{\tau}{r_{0}}}}- 2 \lambda v^{2}a + \frac{8}{9 v^{2}}a^{3}= 0 .
\gamma = \begin{bmatrix}a\beta x c\beta y\end{bmatrix}\begin{bmatrix}(\alpha\beta)^{-1}0 0 D\beta\end{bmatrix}\begin{bmatrix}1 \alpha (by-xd) 0 1\end{bmatrix}\begin{bmatrix}\alpha 0 0 1\end{bmatrix}.
s_p(h) = \begin{cases}(c,d)_p cd \neq 0, \mathrm{ord}_p(c) , 1 \end{cases}x(h) = \begin{cases}c c \neq 0, d c = 0, \end{cases}x(h\alpha_n) = \begin{cases}cp^n c \neq 0, dp^{-n}c = 0. \end{cases}
\frac{\int_E u^{\alpha(x)}t^{\alpha(x)}\rho(dx)}{\int_E t^{\alpha(x)}\rho(dx)}\leq \frac{\int_E u^{\alpha_0}t^{\alpha(x)}\rho(dx)}{\int_E t^{\alpha(x)}\rho(dx)}= u^{\alpha_0}, t\in (0,\infty).
(\mathrm{d}X)=\prod_{j<p}|\lambda(D^{(k)}_{pp})-\lambda(D^{(k)}_{jj})|\,(\mathrm{d}T)(Q^T\mathrm{d}Q)\prod_{j=1}^k\mathrm{d}\lambda_j\prod_{s=1}^{(N-k)/2}\mathrm{d}G_s,
Z(A_{m,n}) = Z(C(a_d,a_{d-1},\ldots,a_3,1,a_3,\ldots,a_{d-1},a_d))+2a_2(m-n)^2 = (m-n)(Z(C(1,a_3,\ldots,a_d))+ Z(C(a_4,\ldots,a_{d-1},a_d))+2a_2(m-n)).
T_{X}Y = \mathcal{H}\nabla^M_{\mathcal{V}X}\mathcal{V}Y + \mathcal{V}\nabla^M_{\mathcal{V}X}\mathcal{H}Y A_{X}Y = \mathcal{H}\nabla^M_{\mathcal{H}X}\mathcal{V}Y + \mathcal{V}\nabla^M_{\mathcal{H}X}\mathcal{H}Y,
{\frac{\partial t_{1}}{\partial \phi^{1}}}= -{\frac{1}{\sqrt 3}}\left({\frac{4}{3}}\right)^{2 / 3}e^{{\frac{\phi}{3}}}\ , \qquad{\frac{\partial t_{0}}{\partial \phi^{1}}}= - \sqrt{\frac{2}{3}}\left({\frac{4}{3}}\right)^{2 / 3}e^{-{\frac{2 \phi}{3}}}\ .
V_{1}^{\prime}( z ) \, S^{\prime}( x ) = ( - 1 )^{\alpha_{1}^{\prime}}\, S^{\prime}( x ) \, V_{1}^{\prime}( z ) \, , \qquad V_{1}( z ) \, S ( x ) = ( - 1 )^{\alpha_{1}}\, S ( x ) \, V_{1}( z ) \, ,
\sum_{i_1,\ldots i_{d}=0}^{\infty}W_{i_1,\ldots,i_{d}}^2 C_{n_1,\ldots,n_{d}}(i_1,\ldots,i_{d})C_{m_1,\ldots,m_{d}}(i_1,\ldots,i_{d})=\delta_{n_1m_1}\delta_{n_2m_2}\cdots \delta_{n_{d}m_{d}}.
\| \Pi_r\theta-\widehat \Pi_r\theta\|^2 \leq \sum_{i=r+1}^d \left( \sum_{k=1}^r \langle \theta, q_k \rangle^2\right) \left( \sum_{k=1}^r \langle q_k,p_i\rangle^2\right) = \sum_{k=1}^r \sum_{i=r+1}^d \langle q_k,p_i\rangle^2.
B''' (\tau ')=\sum_{j,k}(D_j-V_j)g^{jk}(D_k-V_k)+W-\tau ' \intertext{with the new potential}W=2MV-(V^0)^2\intertext{and with the modified spectral parameter}\tau'=\tau \bigl(1-\frac{3\tau}{2M}\bigr)
\omega^{(0)}_{(*)}=-y(z_*)\,dx(z_*),\omega^{(0)}_{(\bar *)}=-x(z_*)\,dy(z_*), \omega^{(0)}_{(1,2)}=-\omega^{(0)}_{(1,\bar 2)}=\omega^{(0)}_{(\bar 1,\bar 2)}=\tfrac{dz_1dz_2}{(z_1-z_2)^2}.
\sigma(|D|) =\sqrt{\xi^2+2A_{\mu}\xi_{\lambda}\otimes\gamma^{\mu}\gamma^{\lambda}+\partial_{\mu}(A_{\lambda})\otimes\gamma^{\mu}\gamma^{\lambda}+A_{\mu}A_{\lambda}\otimes\gamma^{\mu}\gamma^{\lambda}}\sim \sigma_1(\xi)+\sigma_0(\xi)+\sigma_{-1}(\xi)+\cdots,
T_2(\omega) =2\mathrm{ord}(\omega)n \sum_{k'=0}^{\frac{n}{\mathrm{ord}(\omega)}}\sum_{j=0}^{\frac{n}{\mathrm{ord}(\omega)}-1}\binom{\frac{n}{\mathrm{ord}(\omega)}}{k'}^2 k' \binom{\frac{n}{\mathrm{ord}(\omega)}-1}{j}\binom{k'-1}{j}\binom{k'+j}{\frac{n}{\mathrm{ord}(\omega)}}\sum_{i=1}^{\frac{n}{\mathrm{ord}(\omega)}-1}\frac{\omega^i}{(\omega^i-1)^2}.
\psi_{\varphi \varphi \pi \varphi \ldots \pi_{N}}( \stackrel{\rightharpoonup}{x_{1}}, \ldots , \stackrel{\rightharpoonup}{x_{N}}) \rightarrow ( - 1 )^{\# ( \pi - \mathrm{i n d i c e s})}\psi_{\varphi \varphi \pi \varphi \ldots \pi_{N}}( - \stackrel{\rightharpoonup}{x_{1}}, \ldots , \stackrel{\rightharpoonup}{x_{N}}) \ .
X_{1}= \begin{pmatrix}x_{1,11}\hdots x_{1,1d}\vdots \ddots \vdots x_{1,d1}\hdots x_{1,dd}\end{pmatrix}\dots X_{n}= \begin{pmatrix}x_{n,11}\hdots x_{n,1d}\vdots \ddots \vdots x_{n,d1}\hdots x_{n,dd}\end{pmatrix}
\begin{aligned}J_2(\beta, N,T) &\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}
x \cdot y = \left( \overline{x}+ \overline{y}, \widehat{x}_{1}+ \widehat{y}_{1}+ \frac{1}{2}\langle U^{(1)}\overline{x}, \overline{y}\rangle, \ldots, \widehat{x}_{n}+ \widehat{y}_{n}+ \frac{1}{2}\langle U^{(n)}\overline{x}, \overline{y}\rangle \right),