harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
t_1 =2s_1, t_2 =\frac{-s_1-7s_1^3+2s_1s_2+2s_1^3s_2+(1+5s_1^2+2s_1^4-s_2-3s_1^2s_2)\sqrt{-3}}{(s_1^2-1)(1-s_2+s_1\sqrt{-3})}, t_3 =\frac{4(s_1^2+1)s_2s_3}{(s_1^2-1)(1-s_2+s_1\sqrt{-3})}.
a_{j-1}w =(a_{n-1}a_1)^{i_1}\dotsm(a_{s+2}a_{s-2})^{i_{s-2}}a_{j-1}a_s^{i_s}v\intertext{and}a_jw =(a_{n-1}a_1)^{i_1}\dotsm(a_{s+2}a_{s-2})^{i_{s-2}}a_ja_s^{i_s}v.
\Omega_{2 1}= Q \bar{K}_{1}Q \bar{K}_{1}Q = \left( \begin{array}{c c}{{- d_{1}d_{2}^{2}}}&{{u_{2}d_{1}d_{2}}}\\{{- d_{2}^{3}}}&{{u_{2}d_{2}^{2}}}\\ \end{array}\right)
S = \int d^{3}x \left[ \frac{1}{4}F_{\mu \nu}^{2}+ \bar{\psi}_{1}( \gamma \cdot ( \nabla + i e A ) + m ) \psi_{1}+ \bar{\psi}_{2}( \gamma \cdot ( \nabla + i e A ) - m ) \psi_{2}\right]
\partial_{x_i}G_t(x) =\dfrac{x_i}{8\pi d^2t^2}\mathrm{e}^{-|x|^2/4dt},\, \partial_{x_i}\partial_{x_j}G_t(x) = \dfrac{1}{8\pi d^2t^2}\mathrm{e}^{-|x|^2/4dt}\left[\delta_{ij}-\dfrac{x_ix_j}{2dt}\right],
m(t)=\int_{A(t)}\rho(x,t)x \textrm{(total mass)}, &M(t)=\int_{A(t)}\rho(x,t)|x|^{2}x \textrm{(second moment)}, &F(t)=\int_{A(t)}\rho(x,t)u(x,t)\cdot x x \textrm{(radial component of momentum)}, &\varepsilon(t)=\int_{A(t)}\Big(\frac{1}{2}\rho |u|^2+\frac{P}{\gamma-1}\Big)(x,t) x \textrm{(total energy)}.
A_{j i}^{- 1}\frac{| \alpha_{i}|^{2}}{2}= \left\langle \lambda_{j}, \lambda_{i}\right\rangle = \sum_{x = 1}^{h}\frac{x}{h}\, \left\langle \lambda_{j}, \sigma^{x + \frac{c_{i}- 1}{2}}\gamma_{i}\right\rangle \; ,
\mathrm{d}\equiv \mathrm{d}z^{{\cal A}}\frac{\partial}{\partial z^{{\cal A}}}= e^{{\cal A}}D_{{\cal A}}= e^{A}\partial_{A}+ \mathrm{d}\theta^{i \alpha}D_{i \alpha}+ \mathrm{d}\tilde{\theta}_{\alpha}^{\tilde{\imath}}{\tilde{D}}_{\tilde{\imath}}^{\alpha}
\Phi_{0,quantum}^X(y_0, \ldots, y_m, q) = \sum_{k_0, \ldots, k_m}\sum_{0 \neq \beta \in H_2(X)}\langle\gamma_0^{k_0}, \ldots, \gamma_m^{k_m}\rangle_{0,\beta}^X \frac{y_0^{k_0}}{k_0!}\cdots \frac{y_m^{k_m}}{k_m!}q^\beta
j_{\mu}= \overline{{{\Psi}}}_{2}\gamma_{\mu}\Psi_{1}\, , \, \, \, \partial^{\mu}j_{\mu}= 0 , \, \, \, \, \, \left( \overline{{{\Psi}}}_{2}\Psi_{1}\right) : = \int j_{0}d^{3}x = 2 m ( \psi_{2}, \psi_{1}) \,
\begin{array}{rrrr}((Q_0 \cdot Q_{A_{\psi_0}})(\iota(\pi)) | L_{\psi_0}) = 0, begin{align*}2mm]\bigwedge^{n+1}(Q_{A_{\psi_0}}(\iota(\pi) | L_{\psi_0})) = 0, begin{align*}2mm](Q_{A_{\psi}}(\iota(\pi)) | L_{\psi}) = 0, \;\psi \neq \psi_0.\end{array}
x=[\frac{e^{y_{1}}}{1+\sum_{i=1}^{D-1}e^{y_{i}}},...,\frac{e^{y_{D-1}}}{1+\sum_{i=1}^{D-1}e^{y_{i}}},\frac{1}{1+\sum_{i=1}^{D-1}e^{y_{i}}}]^{\top}
\begin{cases}-\kappa \zeta_\beta''(r) = \mu \zeta_\beta(r) r\in (-d,0) \kappa \zeta_\beta'(0) + \beta_+ \zeta_\beta(0) =0 -\kappa \zeta_\beta'(-d) + \beta_- \zeta_\beta(-d) =0 \end{cases}
p = \begin{cases}\frac{1}{\lambda_1-\omega_3}(\omega_1 \omega_2+\omega_1 \omega_3+\omega_2 \omega_3 - \lambda_1 \lambda_2-\lambda_1 \lambda_3-\lambda_2 \lambda_3), \lambda_1 \ne \omega_3; 0, \lambda_1 = \omega_3, \end{cases}
\phi_g^{*}(e_1,e_2,e_3) =\begin{cases}2(a-b)^2(e_{\{5,1,2\}}-e_{\{3,4,5\}}) , 0 \end{cases}&=\begin{cases}2(a-b)^2(e_1+e_2-e_3-e_4) , 0 .\end{cases}
1-A_{2}^*A_{1}^*A_{1}A_{2}=1-A_{2}^*A_{2}+A_{2}^*A_{2}-A_{2}^*A_{1}^*A_{1}A_{2}&=(1-A_{2}^*A_{2})+A_{2}^*(1-A_{1}^*A_{1})A_{2}
\sum_{j=1}^{p-1}\omega_q^{\left(a^t_{\sigma^j, r}-a^t_{\sigma^j, s}\right)-\left(a^t_{\sigma, r}-a^t_{\sigma, s}\right)}=\sum_{j=1}^{p-1}\omega_p^{j\left({r}_{\pi_\beta(1)}-{s}_{\pi_\beta(1)}\right)}=-1.
\mathcal{C}=-\frac{1}{\,\Delta_p(\{\hat{v}^{-1}\})}\sum_{\mu=1}^{m}(-1)^{\mu}\det\begin{bmatrix}\left[\mathcal{G}_{\mu,j}(\hat{v})\right]_{j=1,...,p}\left[\hat{v}_j^{-i+1}\right]_{\substack{i,j=1,...,p (i\neq \mu)}}\end{bmatrix},
{\cal E}^{\mu}(f,f)=\min_{h\in\ell_2(\Gamma)}{\cal E}^{\mu}(f+h,f+h) \asymp \min_{h\in\ell_2(\Gamma)}{\cal E}^{\mu'}(f+h,f+h) ={\cal E}^{\mu'}({\it HD}(\mu\rightarrow\mu')f,{\it HD}(\mu\rightarrow\mu')f).
\hat{R}F(v_{1}\otimes v_{1}) =[2]^{1/2}(q^{-2}v_{1}\otimes v_{0}+\hat{R}(v_{1}\otimes v_{0})), F\hat{R}(v_{1}\otimes v_{1}) =[2]^{1/2}(v_{0}\otimes v_{1}+q^{2}v_{1}\otimes v_{0}).
\def\arraystretch{2.2}\left.\begin{array}{rl}\limsup\limits_{t\to 1-}\int_{|x|\leq 1-t}\frac{|u|^2(x,t)}{|x|^2}\,dx \leq C\,\limsup\limits_{t\to 1-}\left(\int_{|x|\leq 1-t}|\nabla u|^2(x,t)\,dx+\frac{1}{1-t}\int_{|x|=1-t}|u|^2(x,t)\,d\sigma(x)\right) \leq CM^2. \end{array}\right.
f^{\lambda}= \left( f_{1}\delta_{\sigma}^{\lambda}g^{\mu \nu}+ f_{2}g^{\mu \lambda}\delta_{\sigma}^{\nu}+ f_{3}g^{\nu \lambda}\delta_{\sigma}^{\mu}\right) \bar{\Gamma}_{\mu \nu}^{\sigma}+ G_{\sigma}^{\lambda \rho \lambda \alpha \beta}\nabla_{\rho}\nabla_{\lambda}\bar{\Gamma}_{\alpha \beta}^{\sigma}
V_{2r-2}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv \sum_{l=1}^{n-1}\frac{V_{2r-2}n^2(2n-3l)}{l^2}\binom{n-1}{l-1}^2\binom{2l-1}{l-1}\sum_{j=l}^{n-1}(2j+1)^{2r-2}\pmod{2n^2}.
V_{\rm np}\rightarrow\frac{A^2N^2_{\rm cr}\Sigma_0^3}{32}\left[-{10\over3}-2\log\frac{\Lambda_{\rm np}e^{\delta_0}}{\Sigma_0}+{3\over2}\log^2\frac{\Lambda_{\rm np}e^{\delta_0}}{\Sigma_0}\right]\rightarrow\infty
\frac{1}{N}\sum_{\ell=0}^{L-1}X_{\ell}(a_N T) \leq C_2 e^{-s(a_N T - \gamma_L)}\leq C_2 e^{-s a_N/k_N}= C_2 \bigg( \frac{s}{\mu}\bigg)^{-1/k_N},
S_1 :=\{(t,x,y)\in\overline{P}:\,-\ell< x < -\ell+\kappa_0, \; y=-\ell, \; t\in (0,\tau]\}, S_2 :=\{(t,x,y)\in\overline{P}:\,x=\ell, \; \ell-\kappa_0 < y<\ell, \; t\in (0,\tau]\}, S_3 :=\{(t,x,y)\in\overline{P}:\,x=y, \; t\in (0,\tau]\}, S_4 :=\{(t,x,y)\in\overline{P}:\,x-y=\kappa_0, \; t\in (0,\tau]\},
g_{ij}=\begin{pmatrix}2u^3-2\lambda(u^3)^2 -1-2\lambda u^3 -u^1+\lambda u^2+2\lambda u^1 u^3 -1-2\lambda u^3 -2\lambda \lambda u^1 -u^1+\lambda u^2+2\lambda u^1 u^3 \lambda u^1 -2\lambda(u^1)^2\end{pmatrix}, \omega_{12}=0,\omega_{23}=\frac{\sqrt{3}\lambda^2u^1}{\sqrt{-\det g}},\omega_{31}=\frac{\sqrt{3}\lambda^2u^2}{\sqrt{-\det g}},
\frac{\partial (ee^*)}{\partial \textbf{w}^*}= \frac{\partial (d-\textbf{w}^{H}\textbf{X})(d^*-\textbf{w}^{T}\textbf{X}^*)}{\partial \textbf{w}^*}= (-d^*\textbf{X}+ \textbf{XX}^{H}\textbf{w})
e_q'=e_q-\frac{S_t(e_p,e_q)}{S_t(e_p,e_p)}e_p=\begin{pmatrix}1 \ \frac{4\sqrt{\nu}\;S^2(t)}{(1-e^{-t})+2S^2(t)+2S(t)C(t)}\end{pmatrix}\;,
S^p = \int \sqrt{g}\delta_{[\alpha_1 \cdots \alpha_{2p}]}^{[\beta_1\cdots \beta_{2p}]}R^{\alpha_1 \alpha_2}_{\;\;\;\beta_1\beta_2}\cdots R^{\alpha_{2p-1}\alpha_{2p}}_{\;\;\;\beta_{2p-1}\beta_{2p}}d^{d-2}x \ ,
\sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n^{(k)}(\theta) \right\}& = \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n^{(k)}(\theta) x_i < x_i^* i = 1,\ldots, k+1\right\}.
(z-T)^{-1}e_n =\sum\limits_{k=0}^{\infty}\frac{1}{z^{k+1}}T^k e_n=\sum\limits_{k=0}^{\infty}\frac{w_nw_{n+1}\cdots w_{n+k-1}}{z^{k+1}}e_{n+k}&=\sum\limits_{k=0}^{\infty}\frac{\beta_{n+k}}{\beta_n z^{k+1}}e_{n+k}=\beta_n^{-1}z^{n-1}\sum\limits_{k=n}^{\infty}\frac{\beta_k}{z^k}e_k.
\frac{1}{\sqrt{3!}}\varepsilon_{b_1 b_2 b_3}\overline{\psi}_{1 d_1}(y_1)\overline{\psi}_{2 d_2}(y_2) \overline{\psi}_{3 d_3}(y_3)U^{d_1 b_1}(y_1,y_M)U^{d_2 b_2}(y_2,y_M) U^{d_3 b_3}(y_3,y_M)| 0 \rangle
\vert \partial_{p}^{\omega}G_{l , s}^{\alpha}( \vec{\alpha}, \vec{\lambda}, \vec{p}) \vert \ \leq \e^{- m^{2}\sum_{j = 1}^{s}\alpha_{j}}P ( \vert \vec{p}\vert ) Q ( \sqrt \alpha_{1}, \ldots , \sqrt \alpha_{s}) \quad ,
\delta \chi_{a}\; = \; - i \, \sqrt{\, \frac{1}{2}}\, \gamma^{\mu}\, ( \partial_{\mu}\, \Phi ) \, \varepsilon_{a}\; + \; A ( \Gamma_{4 5})_{a b}\, \varepsilon^{b}\; + \; \frac{1}{2}\, \sqrt{\, \frac{1}{6}}\, \gamma^{\, \mu \nu}\, \left( \, H_{\, \mu \nu a b}\; - \; \sqrt{\, 2}\, h_{\, \mu \nu a b}\, \right) \, \varepsilon^{b}
\left(\begin{array}{ccc}-1 0 0 0 0 -1 0 -1 0\end{array}\right),\quad\left(\begin{array}{ccc}0 -1 0 -1 0 0 0 0 -1\end{array}\right),\quad\left(\begin{array}{ccc}0 0 -1 0 -1 0 -1 0 0\end{array}\right)
\alpha=\left[\begin{array}[c]{cc}L_{1}0 0 L_{2}\end{array}\right] ,R=\left[\begin{array}[c]{cc}0 0 0 \ell\end{array}\right] ,\eta=\left[\begin{array}[c]{cc}\frac{1}{C_{1}}+\frac{1}{C_{12}}-\frac{1}{C_{12}}-\frac{1}{C_{12}}\frac{1}{C_{2}}+\frac{1}{C_{12}}\end{array}\right] .
W \left[ f_{\ell}( k , R ) , \phi_{\ell}( k^{\prime}, R ) \right] = \left( - k \right)^{\frac{1}{2}- \nu}F_{\ell}( k ) + \left( k^{2}- k^{\prime 2}\right) \int_{0}^{R}d r \, f_{\ell}( k , r ) \phi_{\ell}( k^{\prime}, r ) \, .
[D,[D,1\otimes M_f]] = -[\gamma_1\otimes\partial_1+\gamma_2\otimes \partial_2,\gamma_1\otimes M_{\partial_1 f}+\gamma_2\otimes M_{\partial_2 f}] = -1\otimes M_{\partial_1^2f+\partial_2^2f}+2\gamma_1\gamma_2\otimes (M_{\partial_2f}\partial_1+M_{\partial_1f}\partial_2)
\Delta t_{+}=\int_{0}^{2\pi}\frac{r_{0}d\phi}{1-\omega r_{0}}=\frac{2\pi r_{0}}{1-\omega r_{0}},\Delta t_{-}=\int_{0}^{-2\pi}-\frac{r_{0}d\phi}{1+\omega r_{0}}=\frac{2\pi r_{0}}{1+\omega r_{0}}
\sum_{1\leq i,j\leq n}a^{ij}u_{ijk}= -2u_1u_{1k}\Delta u+2u_1\sum_{1\leq i\leq n}u_{1i}u_{ik}+v^3D_{k}f+3fvu_1u_{1k}, = \frac{2u_1}{v^2}u_{11}u_{1k}+2u_1\sum_{2\leq i\leq n}u_{1i}u_{ik}+v^3D_{k}f+fvu_1u_{1k}= \frac{2u_1}{v^2}u_{11}u_{1k}+2u_1\sum_{2\leq i\leq n}u_{1i}u_{ik}+f_uv^3u_k+f_{x_k}v^3+fvu_1u_{1k}.
C_2 \; := \;\Big\langle \prod_{b=1}^N\prod_{i=1}^{n_b}\overline{\psi}^{(b)}(x^{(b)}_i )P_+ \psi^{(b)}(x^{(b)}_i )\prod_{i=1}^{m_b}\overline{\psi}^{(b)}(y^{(b)}_i )P_- \psi^{(b)}(y^{(b)}_i ) \Big\rangle_0
G_k(\alpha c_k, c_{k+1})=\frac{\mu^2\alpha c_{k+1}}{c_k}G_k(\alpha c_k, \alpha c_{k+1}), G_k(\alpha c_k, c_{k+1})=\frac{\mu^2 c_{k+1}}{c_k}G_k( c_k, c_{k+1})
\mathcal{I}_{\tilde{\delta}}(2\varepsilon)~=~\Big\lbrace \delta \in \Delta_{h,N_{1}}~\Big|~\rho_{\bf{L}^1}(g_\delta, g_{\tilde{\delta}}) \leq 2\varepsilon \Big\rbrace,\eta(\delta, \tilde{\delta})~=~\left(\left\lbrace i \in \overline{0,N_1-1}~\big|~ \delta_i \neq \tilde{\delta}_i \right\rbrace\right).
[\mathbf{m}_e]_{e\in In(t)}\mathbf{D}_t = \left([\mathbf{m_d}]_{d\in Out(s)}[\mathbf{F}_e]_{e\in In(t)}\right) \mathbf{D}_t &=[\mathbf{m_d}]_{d\in Out(s)}\left([\mathbf{F}_e]_{e\in In(t)}\mathbf{D}_t\right) &= [\mathbf{m_d}]_{d\in Out(s)}\mathbf{I}= [\mathbf{m_d}]_{d\in Out(s)}.
\dot\gamma_W(t) = \frac{d}{dt}(\Psi^{-1}\circ \delta_t)(x,z) &= \Psi^* \left( \sum x^\alpha \frac{\partial}{\partial x^\alpha}+ \sum 2t z^i \frac{\partial}{\partial z^i}\right)_{\Psi(\gamma_W(t))}&= t^{-1}(\Psi^*\tilde{P})_{\Psi(\gamma_W(t))}&= t^{-1}P_{\gamma_W(t)}.
\|\varphi\|_{\mathrm{b}\sim}= \sup_{\mathbb{K}_6}\left\{\frac{1}{|\mathbb{K}_6|}\int_{\mathbb{K}_6}\left|\varphi(x) - N_{(\mathrm{a})}^{\mathbb{K}_6}\varphi\right|dx\right\}< \infty, \enspace 1 < p < \infty.
\begin{aligned}\mathbb{E}[XY] =(t-s)^{-H}(r-u)^{-H}\left(R(t,r)-R(t,u)-R(s,r)+R(s,u)\right) =\frac{1}{2(t-s)^{H}(u-r)^{H}}\left[(t-u)^{2H}-(t-r)^{2H}-\left((s-u)^{2H}-(s-r)^{2H}\right)\right],\end{aligned}
v_B^{-1}\triangleright \chi^+_1 = \big(\varphi_{v^{-1}}\big(\chi^+_1\big)^v\big)^{v^{-1}}= \mu^l(v)^{-1}\mu^l\big(K^{p-1}v ?\big)^{v^{-1}}= \mu^l(v)^{-1}\mu^r\big(K^{p+1}?\big).
{\cal F}= \frac{1}{2}\sum_{n_{1}\geq n_{2}\geq \cdots \geq n_{N}\geq 0}^{\infty}\det\left[d_{n_{j}+N-j,i}\right] \det\left[t_{i}^{n_{j}+N-j}+t_{i}^{-(n_{j}+N-j)}-\delta_{jN}\delta_{n_{N}0}\right].
\bigg\|\;\sum_{k=0}^\infty\sum_{\ell=1}^{\ell(k)}T_{k,\ell}\,u\;-\,\sum_{k=0}^\infty\sum_{\ell=1}^{\ell(k)}\sum_{j=1}^{n_k}T_{k,\ell}\,u_j\;\bigg\|_1\leq\;\frac{\alpha}{\delta}\,\left(\frac{1+q_1}{1-q_1}\right)^2\,\varepsilon,
M_{n-1}^{(x^2+b,bx^2)}= \sum_{k=0}^{n-1}\binom{n-1}{k}M_k^{(ax,bx^2)}(x^2-ax+b)^{n-k}= \sum_{k=0}^{n-1}\binom{n-1}{k}M_k^{(a,b)}x^k(x^2-ax+b)^{n-k}.
d{\tilde{\rho}}_K\left( X \right)=\sum\limits_{k=1}^{2K}{\overbrace{{\tilde{\rho}}_{1/ 2}\left( 1 \right)\otimes \cdots \otimes{\tilde{\rho}}_{1/ 2}\left( 1 \right)}^{k-1}\otimes d{\tilde{\rho}}_{1/ 2}\left( X \right)\otimes \overbrace{{\tilde{\rho}}_{1/ 2}\left( 1 \right)\otimes \cdots \otimes{\tilde{\rho}}_{1/ 2}\left( 1 \right)}^{2K-k}}\ ,
d^{\nabla^{\omega}_{\alpha}}= \Upsilon_{\alpha}\circ D^{\omega}\circ \Upsilon^{-1}_{\alpha}\,,d^{\widehat{\nabla}^{\omega}_{\alpha}}= \widetilde{\Upsilon}_{\alpha}\circ \ast\circ D^{\omega}\circ \ast \circ \widetilde{\Upsilon}^{-1}_{\alpha}
\int_{A_{r_2,r_3}}\left(\mathfrak u^2-\lambda \phi_{\mathfrak u}-\frac{1}{2C_2}\right)\mathfrak u^2\, dx> \delta,\int_{A_{r_i,r_{i+1}}}\left(\mathfrak u^2+\lambda \phi_{\mathfrak u}+\frac{1}{2C_2}\right)\mathfrak u^2\, dx< \frac{\delta}{4},
J_{1}( - 1 , - 1 , - 1 , - 1 ; m ) = \frac{\pi^{2}}{6 m^{4}}F_{3}\left( 1 , 1 , 1 , 1 ; \frac{5}{2}\left| \frac{s}{4 m^{2}}, \frac{t}{4 m^{2}}\right) \right.
\varphi_{i j}^{a b}( \theta ) = \int_{- \infty}^{\infty}d t \left[ \delta_{a b}\delta_{i j}- \left( 2 \cosh \frac{\pi t}{h}- \tilde{I}\right)_{a b}\left( 2 \cosh \frac{\pi t}{h}- I^{s u ( k )}\right)_{i j}^{- 1}\, \right] \, \, e^{- i t ( \theta + \sigma_{a b})}\; ,
-c^1_+(c_2-c_1)+(-c^1+2-\partial_{\beta}W_{v^1}-\sigma^1)(c_2-c_1)_+-c^1_+(\sigma_2-\sigma_1)-\partial_{\beta}\partial_{\beta}W_{v^1}c^1_+(v_2-v_1)=-\tilde{R}^2.
T_1^{(n)}(x) = \frac{s \cdot 2^s}{(x-1)^2}\sqrt{\frac{x - 1}{x + 1}}\int_{-\infty}^{\infty}e^{\tau}\left( 1 + \sqrt{\frac{x + 1}{x - 1}}e^{\tau}\right)^{-s-1}\times \times \Big( x + (x^2 - 1)^{\frac{1}{2}}\cosh \tau \Big)^{- n - 1}d\tau
\int_T^{2T}|\log |\zeta(\sigma+it)-a||^{2k}\, dt \leq 4^k \int_T^{2T}\bigg( \sum_{|\rho_a-s_0| \leq r-\delta}|\log |\sigma+it-\rho_a||\bigg)^{2k}\, dt &+O\bigg( 4^k \int_T^{2T}(\log M_{r}(s_0))^{2k}\, dt \bigg).
\langle x^\ell P_n^{(\ell)}, x^n\rangle=\left|\begin{array}{ccc}m_\ell \cdots m_{\ell+n}\vdots \vdots m_{\ell+n}\cdots \boxed{m_{\ell+2n}}\end{array}\right|=\left|\begin{array}{cc}\Lambda^{(\ell)}_{n-1}(\theta_{n+\ell}^n)^\top \theta_{n+\ell}^n \boxed{m_{\ell+2n}}\end{array}\right|:=H_{n}^{(\ell)}.
\int{\partial X \bar{\partial}X^{\ast}+ \psi \bar{\partial}\psi^{\ast}+ \bar{\psi}\partial \bar{\psi}^{\ast}+ F F^{\ast}+ W^{\prime}( X ) F + \psi \bar{\psi}W^{\prime \prime}( X ) + W^{\prime}( X^{\ast}) F^{\ast}+ \psi^{\ast}\bar{\psi}^{\ast}W^{\prime \prime}( X^{\ast})}
P_{s, f}r_{s, f}(v) = \int_{\Gamma_{\mu}}\frac{\langle f, g v \rangle^s}{\langle f, v \rangle^s}\frac{1}{\langle f, g \cdot v \rangle^s}r_s(g \cdot v) \mu(dg) = \int_{\Gamma_{\mu}}\frac{1}{\langle f, v \rangle^s}|gv|^s r_s(g \cdot v) \mu(dg) = \kappa(s) \frac{1}{\langle f, v \rangle^s}r_s(v) & = \kappa(s) r_{s, f}(v).
e^{M ( t - t_{i})}= \left( \begin{matrix}{c o s ( \omega{}( t - t_{i}) )}&{s e n ( \omega{}( t - t_{i}) )}&{0}\\{- s e n ( \omega{}( t - t_{i}) )}&{c o s ( \omega{}( t - t_{i}) )}&{0}\\{0}&{0}&{1}\\ \end{matrix}\right) .
C_{1+s}\left(X;Y\right):=\min_{P_{XYW}:\,P_{XY}=\pi_{XY},\,X-W-Y}\qquad\sum_{w}P_{W}\left(w\right)D_{1+s}\left(P_{X|W}(\cdot|w)P_{Y|W}(\cdot|w)\|P_{XY}\right)
\alpha ' := \alpha - \left\lfloor \frac{b_1}{2}\right\rfloor(\sigma_{2,1}- 2E^{[3]}) = \left(a_{2,1}- \left\lfloor \frac{b_1}{2}\right\rfloor\right)\sigma_{2,1}+ a_3\sigma_3 - \sum_{i=2}^{r}b_i E_i^{[3]}
R_2 := R_1[Y_{22};\alpha_2,\delta_2], \alpha_2: Y_{11}\mapsto Y_{11},\Y_{12}\mapsto t^{-1}Y_{12}, \Y_{21}\mapsto t^{-1}Y_{22}; \delta_2: Y_{11}\mapsto (t^{-1}-t)Y_{12}Y_{21}, \Y_{12}\mapsto 0, \Y_{21}\mapsto 0.
C_6^{\frac{pq}{p-q}}\lesssim \sum_{k=-\infty}^{M-1}2^{-k}\sup_{i \leq k}2^{i}\bigg(\int_{x_i}^{x_{i+1}}u(t)\bigg(\int_t^{\infty}u \bigg)^{\frac{q}{p-q}}dt\bigg) \Phi(0,x_i) & + \sum_{k=-\infty}^{M-1}2^{-k}\sup_{i \leq k}2^{i}\sup_{y \in (x_i, x_{i+1})}\bigg(\int_y^{x_{i+1}}u(t)\bigg(\int_t^{\infty}u \bigg)^{\frac{q}{p-q}}dt\bigg) \Phi(x_i,y).
A_{ij}= \begin{cases}0, \mbox{if}i=j, \mbox{or}\left\lbrace v_i, v_j \right\rbrace \notin E(G) w(\left\lbrace v_i, v_j \right\rbrace),\mbox{if}i \neq j, \left\lbrace v_i, v_j \right\rbrace \in E(G). \end{cases}
\widehat{\chi}_{[0,1)}( \xi) = 2^{-1/2}D_2 \widehat{\chi}_{[0,1)}(\xi) + 2^{-1/2}E_{-2^{-1}}D_2 \widehat{\chi}_{[0,1)}(\xi) = 2^{-1/2}\pi^+ (2^{-1},0)\widehat{\chi}_{[0,1)}(\xi) +2^{-1/2}\pi^+(2^{-1}, -2^{-1}) \widehat{\chi}_{[0,1)}(\xi).
\epsilon_{a,0}(\beta) = T \sum_{b=1}^{N-1}I_{ab}^{(N)}\int_{-\infty}^\infty\frac{d\beta'}{2\pi}\frac{N/2}{\cosh[N(\beta-\beta')/2]}\ln\left(1+e^{-\epsilon_{b,0}(\beta')/T}\right)
{\cal Z}_{\delta}(x,z;p,\epsilon)\equiv\langle\alpha|\delta\rangle\,\theta_{\delta}(0)^{-1}\langle\psi(x_1)\!\cdot\!\psi(x_2)\prod_{i=1}^{4}p_i\!\cdot\!\psi(z_i)\,\epsilon_i\!\cdot\!\psi(z_i)\rangle_{\delta}\, .
\tilde{\Psi}^{+}= - i \frac 1 r ( 2 k + 1 )^{- 1 / 2}\left( \begin{array}{c}{{( k + m + 1 / 2 )^{1 / 2}Y_{k}^{m - 1 / 2}( \vartheta , \varphi ) B ( r )}}\\{{( k - m + 1 / 2 ) Y_{k}^{m + 1 / 2}( \vartheta , \varphi ) B ( r )}}\\ \end{array}\right)
{\cal H}_{j , j + 1}= \sigma_{j}^{+}\sigma_{j + 1}^{-}+ \sigma_{j}^{-}\sigma_{j + 1}^{+}+ \Gamma \left( \sigma_{j}^{+}\sigma_{j}^{+}+ \sigma_{j}^{-}\sigma_{j + 1}^{-}\right) - \frac{h}{2}\left( \sigma_{j}^{z}+ \sigma_{j + 1}^{z}\right) \; .
{2i(j_{1,c}+j_{2,c}+1)(j_{1,c}-j_{2,c})\over ic}=2c \left ({j_{1,c}^2+j_{2,c}^2+j_{1,c}+j_{2,c}\over c^2}-2{j_{2,c}^2+j_{2,c}\over c^2}\right )
E_1(h_{12}^3) =\frac{h_{13}^3}{\sqrt{3}}-2 h_{12}^3 h_{13}^3 \left(\cot (2 \Lambda )+\frac{2 \sin (2 \Lambda )}{\cos (2 \Lambda )-\cos (3 c)}\right), E_1(h_{13}^3) =\frac{1}{3}\Big(\cot \left(\Lambda +\frac{3 c}{2}\right)\big( 1-\cos (2 \Lambda +3 c)+6 (h_{12}^3)^2 + 6 (h_{13}^3)^2 )+6 (h_{12}^3)^2 \cot (2 \Lambda )-\sqrt{3}h_{12}^3\Big).
\langle\prod_{i=1}^{4}V_{i}(z_{i})\rangle=\sum_{s}C_{12s}C_{s34}F_{\Delta_{s}}^{(s)}(\{\Delta_{i}\},\{z_{i}\})F_{\overline{\Delta}_{s}}^{(s)}(\{\overline{\Delta}_{i}\},\{\overline{z}_{i}\}),
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(-1)^{n-1}z^nq^{\frac{n(n+1)}{2}}(q)_{n}}{(1-cq^{n})(zq)_n}=\frac{z}{c}\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(zq/c)_{n-1}(q)_{n}(cq)_{N-n}(cq)^{n}}{(zq)_{n}(cq)_{N}}.
S_{f i}^{( 2 )}= - \int \int d t_{1}d t_{2}\, \theta ( t_{2}- t_{1}) \int d{\bf r_{2}}\, \Psi_{{\bf p^{\prime}}}^{( + ) \, \dagger}({\bf r_{2}}) \, \tau_{3}\, H_{i n t}({\bf r_{2}}, t_{2}) \, H_{i n t}({\bf r_{2}}, t_{1}) \, \Psi_{{\bf p}}^{( + )}({\bf r_{2}}) \; ,
\frac{\partial \iota}{\partial \theta^{i}}\Big|_{k(\theta)}=\Bigl\{\frac{\partial}{\partial \theta^{i}}k(\theta)_t\Bigr\}_{0 \le t \le 1}\mbox{and}\frac{\partial^2 \iota}{\partial \theta^{i}\partial \theta^{j}}\Big|_{k(\theta)}=\Bigl\{\frac{\partial^2}{\partial \theta^{i}\partial \theta^{j}}k(\theta)_t\Bigr\}_{0 \le t \le 1}.
\frac{\partial}{\partial t}\left(\nabla_i\nabla_{\bar{j}}L\right)= \Delta\left(\nabla_i\nabla_{\bar{j}}L\right)+R_{i\bar{j}l\bar{k}}\nabla_k\nabla_{\bar{l}}L &-\frac{1}{2}\left(R_{l\bar{j}}\nabla_i\nabla_{\bar{l}}L+R_{i\bar{l}}\nabla_l\nabla_{\bar{j}}L\right)+\nabla_i\nabla_{\bar{j}}|\nabla L|^2+\epsilon \nabla_i\nabla_{\bar{j}}R.
\Psi_E (r,\theta) = \sum_{n=-\infty}^\infty e^{i n\theta}\left(\begin{array}{c}A_n\,J_{n-\kappa}\left(|E|r\right)+ B_n\,J_{\kappa-n}\left(|E|r\right) -i\frac{|E|}{E}\left(A_n\,J_{n+1-\kappa}\left(|E|r\right)- B_n\,J_{\kappa-n-1}\left(|E|r\right)\right) e^{i \theta}\end{array}\right)
\int_{s}^{b}\int_{|x|\geq\rho}|K(t,s,x)|dxdt \leq\int_{s}^{b}\int_{\mathbf{R}^{d}}\frac{|x|^{\mu}}{\rho^{\mu}}|K(t,s,x)|dxdt \leq N\int_{s}^{b}\rho^{-\mu}(t-s)^{\frac{\mu}{\gamma}-1}dt =N\rho^{-\mu}(b-s)^{\frac{\mu}{\gamma}}.
(t\mu_1+(1-t)\mu_2)( \{x: \mathrm{stab}_{ta+(1-t)b}(x) \in A\}) = (t\mu_1 + (1-t)\mu_2)( \{x \in X_1: \mathrm{stab}_a(x) \in A \}& \sqcup \{x \in X_2: \mathrm{stab}_b(x) \in A\}) = t \mu(\{x: \mathrm{stab}_a(x) \in H \}) + (1-t) \mu(\{x: \mathrm{stab}_b(x) \in A\}).
D_{\varphi}(fg)=D(\varphi\circ fg)= D((\varphi\circ f)(\varphi\circ g)) = \varphi\circ f(x) D(\varphi\circ g)+ \varphi\circ g(x)D(\varphi\circ f) = \varphi(f(x)) D_{\varphi}g+ \varphi(g(x)) D_{\varphi}f.
\Delta(\lambda) = (-1)^{\alpha(1-\beta)}\frac{\varphi_{|\beta-\alpha|}(\pi \rho)}{\rho^{1-\alpha-\beta}}+ \frac{1}{\rho^{2-\alpha-\beta}}\int_0^\pi \varphi_{1-|\beta-\alpha|}(t \rho) \, W(t)\,dt, W \in L_2(0, \pi),
\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{FDF=0}\right\rbrace\to\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}\left(2^{R}-1\right)(1+k)}{\lambda_{\textrm{BF}}\left(k-\left(2^{\textrm{R}}-1\right)\right)}\frac{1}{P_{\textrm{B}}}.
S_{I , \theta}= - \frac{\sigma}{2}\int \frac{d \nu}{2 \pi}\int_{\bar{R}_{1}}^{\bar{R}_{2}}d r \tilde{\theta}_{I}^{*}\left[ - \frac{\partial}{\partial r}\gamma^{- 1}r^{2}\frac{\partial}{\partial r}- ( \nu^{2}- \omega^{2}) \gamma r^{2}\right] \tilde{\theta}_{I},
(\delta_t \hat{v}^1_h,\bar{v}_h) +({A}_h (\hat{v}^1_h - \hat{v}^0_h),\bar{v}_h) +({A}_h \hat{v}^0_h,\bar{v}_h) =((\hat{u}^1_h + 2m_0) \hat{u}^1_h,\bar{v}_h), \ \ \forall\bar{v}_h\in V_h.
I_2 = \int (1+|\xi-\eta|^2)^{-N}(1+|\eta|^2)^{-N_1}d \eta \leq C (1+|\xi|^2)^{-N}\int (1+|\eta|^2)^{N - N_1}d \eta = B_N (1 + |\xi|^2)^{- N},
\partial_{c_k^V}\left( \varepsilon_{q,m}^{V,s}\right) = \left\langle u_{q,m}^{V,s}, \partial_{c_k^V}A^V_q, u_{q,m}^{V,s}\right\rangle = \langle u_{q,m}^{V,s}, \cos(k\cdot) u_{q,m}^{V,s}\rangle_{L^2_{\rm per}}.
\sum_{q\in [N]^k}\sum_{\substack{s\in [N]^k \{s_c, s_d\}= \{q_a, q_b\}}}\gamma_{q_aq_b}D(q, \{a,b\})D(s, \{c,d\}) &=\sum_{q\in [N]^k}\sum_{\substack{s\in [N]^k s_c=q_a,\, s_d=q_b}}D(q, \{a,b\})D(s, \{c,d\}) &+ \sum_{q\in [N]^k}\sum_{\substack{s\in [N]^k s_c=q_b,\, s_d=q_a}}D(q, \{a,b\})D(s, \{c,d\})\, .
P_v(\varepsilon):= 2\|v\|^2 \varepsilon-2^*(s_1)\mu|v|_{2^*(s_1),s_1}^{2^*(s_1)}\big[t(\varepsilon)\big]^{2^*(s_1)-2}|\varepsilon|^{2^*(s_1)-2}\varepsilon &-\kappa 2^*(s_2)\beta c(u,v)\big[t(\varepsilon)\big]^{2^*(s_2)-2}|\varepsilon|^{\beta-2}\varepsilon.
m_A =-\frac{5t}{2}-\frac{5|J|}{4}-\frac{|\alpha_J|}{2}+\frac{3}{4}-\frac{5(g_0-t)}{2}-\frac{5|I_0\setminus J|}{4}-\frac{|\alpha_{I_0\setminus J}|}{2}+\frac{3}{4}=e_1
\sigma_\varepsilon'(y) + \frac{p}{p-1}\sigma_\varepsilon(y)^{(p-1)/p}= \frac{p}{p-1}(1-y) \left[ \frac{1}{2^{(p-1)/p}\varepsilon^{2-p}}- \frac{(1-y)^{(2-p)/(p-1)}}{2 \varepsilon^{p(2-p)/(p-1)}}\right] & \ge \frac{p}{p-1}(1-y) \frac{2^{1/p}- 1}{2 \varepsilon^{2-p}}& \ge \frac{p a^{2-p}}{p-1}(1-y) = \psi'(y) + \frac{p}{p-1}\psi(y)^{(p-1)/p}\ ,
-ig\frac{\tau^{\gamma}}{2}A_{{\sf GI}\,i}^{\gamma}({\bf{r}})=-igV_C({\bf{r}})\frac{\tau^{\gamma}}{2}A_{i}^{\gamma}({\bf{r}})V_C^{-1}({\bf{r}})+V_C({\bf{r}}){\partial}_iV_C^{-1}({\bf{r}}),
T_{\mu}= \frac{( 2 \imath )^{\frac{D - 2}{2}}}{( D - 2 ) !}\epsilon_{\nu \mu \rho_{2}\ldots \rho_{D - 1}}\pi^{\mu}\psi^{\rho_{2}}\ldots \psi^{\rho_{D - 1}}+ \frac{\alpha}{2^{\frac{D - 2}{2}}}\pi_{\nu}= 0 \; .
\lim_{m\rightarrow+\infty}\frac{x_{2}^{\left(t_{0}+m\right)}}{x_{1}^{\left(t_{0}+m\right)}}=\begin{cases}0 \mbox{if}\gamma_{2}=0,\delta_{1}\neq0, \frac{x_{2}^{\left(t_{0}\right)}}{x_{1}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, +\infty \mbox{if}\gamma_{2}\neq0,\delta_{1}=0.\end{cases}
\stackrel{{\rm HW}}{S^{\alpha\beta}}=\int \Psi^\dagger S^{\alpha\beta}_{\rm op}\Psi d^3x, \quad S^{\alpha\beta}_{\rm op}= \frac{1}{2}\,\sigma^{\alpha\beta}+ \frac{i}{2m}\left(p^\alpha \gamma^\beta-p^\beta \gamma^\alpha \right) \, .
W_{|m\rangle}(x,p)=\frac{(-1)^{m}}{\pi^{3/2}2^{m}m!}e^{-x^{2}-p^{2}}\int_{-\infty}^{\infty}d\zeta e^{-\zeta^{2}}H_{m}(\zeta-ip+x)H_{m}(\zeta-ip-x).
\langle H_y \rangle_{u_{\alpha}\otimes \Omega}= e_{\alpha}+ \alpha\|\lambda_y \|^2 + \langle \alpha V\rangle_{u_{\alpha}}&= e_{\alpha}+ \alpha\|\lambda_y \|^2 -\frac{\alpha^2}{L^3}+ O\Big( \frac{\alpha^2}{L^5}\Big) + O\big( \alpha^2 L e^{-L/2}\big).
\nu_0^{(m,-)}\big[ \sum_{j=1}^k Z_j (0) \big] \le\nu_0^{(m,+)}\big[ \sum_{j=1}^k Z_j (0) \big] = \sum_{j=1}^{k}(z_j + \alpha_j^{-1}) \le \sum_{j=1}^{k}(z_j + 1) \,,
S_{W Z}={\frac{1}{4 \pi}}\int_{0}^{1}d r \int d^{2}x \epsilon^{\mu \nu}\mathrm{t r}\left( \tilde{g}^{- 1}\partial_{r}\tilde{g}\tilde{g}^{- 1}\partial_{\mu}\tilde{g}\tilde{g}^{- 1}\partial_{\nu}\tilde{g}\right) . \vspace{2 m m}
\|R_j F\|_{L^1(\mathcal G)}= \int_{B(g, c_* \eta r)}|R_j F(\tilde g)| \,d\tilde g+\int_{B(g_*, c_* \eta r)}|R_j F(\tilde g)| \,d\tilde g &\quad+ \int_{\mathcal G\backslash\big( B(g, c_* \eta r) \cup B(g_*, c_* \eta r)\big)}|R_j F(\tilde g)| \,d\tilde g &= I+II+III.
f \star g = \exp \Bigg [ \hbar \Bigg ( \frac{\partial ~}{\partial q}\frac{\partial ~}{\partial \tilde{p}}- \frac{\partial ~}{\partial p}\frac{\partial ~}{\partial \tilde{q}}\Bigg ) \Bigg ] f ({\bf x}) g ({\bf \tilde{x}}) \vert_{{\bf x}={\bf \tilde{x}}},
H_{0}= \begin{pmatrix}1 0 i\frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}r^{\frac{\sigma_{3}}{4}}G_{0}(0;r\vec{x},\vec{s})\widehat{c}^{\sigma_{3}}= R(0)\begin{pmatrix}1 0 id_{1}1\end{pmatrix}x_{1}^{-\frac{\sigma_{3}}{4}}N.
N = \frac{1}{4 \beta \left( \frac{2}{\sqrt{d_{\perp}}}\right)^{d_{\perp}}}~ \int_{y_{N}}^{y_{e n d}}\frac{( 1 - y )^{d_{\perp}- 1}d y}{y^{3}\left( 1 + ( \frac{d_{\perp}}{4}- \frac{3}{2}) y \right)}