harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\langle \varphi_{n},\varphi_{k}\right\rangle_{L^{2}_{G_{\alpha}\otimes G_{\alpha}}(\mathbb{R}^{2d})}=\int_{\mathbb{R}^{2d}}\varphi_{n}(\boldsymbol{\lambda}_{1},\boldsymbol{\lambda}_{2})\varphi_{k}(\boldsymbol{\lambda}_{1},\boldsymbol{\lambda}_{2})G_{\alpha}(d\boldsymbol{\lambda}_{1})G_{\alpha}(d\boldsymbol{\lambda}_{2})=\delta_{n,k},
\int_{-\textup{i}\infty}^{\textup{i}\infty}dz \prod_{i=1}^3\mathcal{B}(a_i - z,b_i + z) = \mathcal{B}(a_2+b_1,a_3+b_2)\mathcal{B}(a_1+b_2,a_3+b_1).
2\ell A^2+\sum\limits_{m=1}^{d}AR^{m}AR_{m*}^{*}+\left(\sum_{r,s=1}^d(AR^i)_{rs}(AR^j)_{sr})\right)+\left(\sum_{r,s=1}^{d}(AR^s)_{ir}(AR^j)_{sr})\right)=A,
\begin{aligned}\sum_{\nu_2\cdots\nu_r\leq \tau^{-1/(2m)}}\tau^{-1/(2m)}/(\nu_2\cdots\nu_r) = \tau^{-1/(2m)}\left(\sum_{\nu\leq \tau^{-1/(2m)}}1/\nu\right)^{r-1}& \asymp \tau^{-1/(2m)}(\log 1/\tau)^{r-1}.\end{aligned}
1 = \sin^2\!\varphi_4 + \cos^2\!\varphi_4 = \frac{1}{d^2}\, [ a^2+ b^2 + c^2 + 2ab \cos(\varphi_1 - \varphi_2) &+ 2bc \cos(\varphi_2 - \varphi_3) + 2ca \cos(\varphi_3 -\varphi_1)] = A(a,b,c) \;.
\Omega_2 f(\rho)= \sum_{x\in \mathbb{Z}^d}\big[f(\rho^x)-f(\rho)\big]+\delta\sum_{x\in \mathbb{Z}^d}\big[f(\rho^{x,+})-f(\rho)\big] &+\gamma\sum_{x\in \mathbb{Z}^d}\big[f(\rho^{x,-})-f(\rho)\big]+\lambda\sum_{x\in \mathbb{Z}^d}\sum_{y\sim x}\big[f(\rho^{x,y})-f(\rho)\big]
\left( \begin{array}{c}{{\hat{\xi}}^{t}}\\{{\hat{\xi}}^{r}}\\{{\hat{\xi}}^{\varphi}}\\ \end{array}\right) ={\cal G}\left( \begin{array}{c}{{\hat{\Sigma}}^{t}}\\{{\hat{\Sigma}}^{r}}\\{{\hat{\Sigma}}^{\varphi}}\\ \end{array}\right) ;{\cal G}= \mathrm{E x p}\left( - \int^{\tau}{\cal D}( \tau^{\prime}) d \tau^{\prime}\right) ,
i \frac{\partial}{\partial t}\langle \vec q\,',t\mid \vec q,0 \rangle = -\frac{1}{2}\sum_{i=1}^3 \frac{\partial^2}{\partial q'^2_i}\langle \vec q\,',t\mid \vec q,0 \rangle + V(\vec q\,') \langle \vec q\,',t\mid \vec q,0 \rangle
{\cal L}_{\mathrm{m a s s}}^{W}= -{\frac{1}{2}}\left( \begin{array}{c c c}{{\psi_{X}^{\dot{\alpha}}}}&{{\psi_{Y}^{\dot{\alpha}}}}&{{\psi_{S}^{\dot{\alpha}}}}\\ \end{array}\right) M_{\psi}^{W}\left( \begin{array}{c}{{\psi_{X \dot{\alpha}}}}\\{{\psi_{Y \dot{\alpha}}}}\\{{\psi_{S \dot{\alpha}}}}\\ \end{array}\right) + \mathrm{h . c .},
L_{0}=-\rho\{\zeta[\frac{\partial\Phi}{\partial t}]_{z=0}+\int_{-h}^{0}\frac{1}{2}[(\frac{\partial\Phi}{\partial x})^{2}+(\frac{\partial\Phi}{\partial y})^{2}+(\frac{\partial\Phi}{\partial z})^{2}]dz+\frac{1}{2}g\zeta^{2}\}.
\delta_{n}^{(3)}= \delta_{n}^{(3)}(v,A,b,\sigma,q,B,\eta,\gamma) & = \frac{b K_n}{\gamma^{1 + 1/q}n^{1/2-1/q}}+ \frac{(b\sigma^2 K_n^{2})^{1/3}}{\gamma^{1/3}n^{1/6}}+ \frac{(b \sigma K_n^{3/2})^{1/2}}{\gamma^{1 + 1/q}n^{1/4}}.
\|(\prod_{i=1}^{10}\sum_{\atop{l(\Delta)=N^{-1/2}}}|E_{\Delta}g_i|^2)^{1/20}\|_{L^{p}(w_{B_N})}\lesssim_{\nu,\epsilon}N^{-\frac{15}{2p}+\epsilon}(\prod_{i=1}^{10}\sum_{\atop{l(\Delta)=N^{-1/2}}}\|E_{\Delta}g_i\|_{L^{\frac{2p}5}(w_{B_N})}^2)^{\frac1{20}}.
W\supset\frac{1}{\tilde{N}_c}\textup{Tr}(q\tilde{q})(\sum_{k=1}^{N_c-\tilde{N}_c}\psi_k)-\sum_{k=1}^{N_c-\tilde{N}_c}\psi_ke_k\tilde{e}_k+\mu\Lambda\sum_{k=1}^{N_c-\tilde{N}_c}x_k\psi_k.
\sum_{i=1}^k{b_i}^u\leq\sum_{i=1}^k{b_i}^v{b_i}^l\leq b^v\sum_{i=1}^k{b_i}^l\leq\left(\sum_{i=1}^kb_i\right)^v\left(\sum_{i=1}^kb_i\right)^l\leq\left(\sum_{i=1}^kb_i\right)^u
\rho_t(t,x)= -\xi'\bar h' K_4\epsilon(t)V^*+\xi K_4\epsilon'(t)V^* \succeq -\xi_*K_4\epsilon(t)V^*- K_4\beta(t+\theta)^{-1}\epsilon(t)V^* \succeq -(\xi_* +\beta\theta^{-1})K_4\epsilon(t)V^*,
\frac{\rm d}{{\rm d}t}\left(\frac{1}{2}\|u\mathbf 1_{A_{\alpha^*}}\|_{L^2(d\nu)}^2\right)= (u\mathbf 1_{A_{\alpha^*}},\,Lu)_{L^2(d\nu)}+(u\mathbf 1_{A_{\alpha^*}},\,\tilde{R}(u))_{L^2(d\nu)}+\int_{\mathbb{R}}u\mathbf 1_{A_{\alpha^*}}\alpha\, b_1 d\nu +R(t),
\int_{\mathbb{R}}\frac{\mathrm{d}\mu_{\omega}(x)}{z-x}=-\frac{\psi_{a}(qz)\omega(z)+\varphi_{a}(qz)}{a\psi_{a}(z)\omega(z) +\varphi_{a}(z)}\,,\ \z\in\mathbb{C}\setminus\mathbb{R}.
\sum_{b=0}^a \xi_3^{b}(b+1) = \sum_{b=0}^{\frac{a-2}{3}}\xi_3^{3b}(3b+1) + \xi_3^{3b+1}(3b+2) + \xi_3^{3b+2}(3b+3) &= \sum_{b=0}^{\frac{a-2}{3}}\xi_3^{3b+1}+ 2\xi_3^{3b+2}= \frac{a+1}{3}(\xi_3 + 2\xi_3^{2}).
[r^is,r^j]=(r^is)^{-1}(r^{j})^{-1}r^isr^j=s^{-1}r^{-i-j+i}sr^{j}=s^{-1}sr^{j}r^{j}=r^{2j}, [r^is,r^js]=s^{-1}r^{-i}s^{-1}r^{-j}r^isr^js=s^{-1}s^{-1}r^{i}r^{-j}r^ir^{-j}ss=r^{-N+i-j+i-j+N}=r^{2(i-j)},
V=\frac{-e^{2}}{16\pi\epsilon_{0}Z^{3}}(\frac{x^{2}+y^{2}}{2}+z^{2})+\frac{3e^{2}}{32\pi\epsilon_{0}Z^{4}}(\frac{x^{2}+y^{2}}{2}z+z^{3})+O(\frac{1}{Z^{5}}).
\sum_{\nu\in{\cal R}}(L^-)_{\mu\nu}\,e^W= \left(p\cdot\mu-i\omega q\cdot\mu +i\sum_{\rho\in\Delta_+}{\rho\cdot\mu\over{\rho\cdot q}}\right)\,e^W= \mu\cdot\!\left(p+i{\partial W\over{\partial q}}\right)e^W=0,
f(z)-f(0) = \int_{0}^1 \left( \frac{\partial f}{\partial x}(tz) x - \frac{\psi_0}{\psi_1}y \frac{\partial f}{\partial x}(tz) \right) dt = \int_{0}^1 \left( x - \frac{\psi_0}{\psi_1}y \right) \frac{\partial f}{\partial x}(tz) dt = \left( x - \frac{\psi_0}{\psi_1}y \right) h_1(z),
\tilde P(\tilde\sigma_{n+1}=\tilde\sigma ',\,t\leq t_{n+1}<t+dt\,|\,\tilde\sigma_{n}=\tilde\sigma ,\,t_{n}=s) =c_M\sum_{x\in\mathcal{S}_{\gamma}}\mathbf 1_{\tilde\sigma'=\tilde\sigma^{x}}e^{-c_M|\mathcal{S}_{\gamma}|(t-s)}\mathbf{1}_{\{t>s\}}dt.
X_\alpha(t.u_i) =\sum\limits_{\chi\in A_Q}t^{d(i,\chi)}X_{(\alpha,\chi)}(u_{(i,\chi)})=\sum\limits_{\chi\in A_Q}t^{d(i,\chi)}u_{(j,\chi+e_\alpha)}=t^{-c_\alpha}\sum\limits_{\chi\in A_Q}t.u_{(j,\chi+e_\alpha)}=t^{-c_\alpha}t.u_j.
=C_{m_{1},m_{2},m}^{l_{1},l_{2},l}C_{0,0,0}^{l_{1},l_{2},l}+\frac{1}{n}C_{m_{1},m_{2},m}^{l_{1},l_{2},l}P(l_{1},l_{2},l)+\frac{1}{n^2}D_{m_{1},m_{2}}^{l_{1},l_{2},l}(n) \ ,
x_rx_t=x_tx_r,x_rc_t=c_tx_r,c_rc_t=c_tc_r,c_r^2=0; \tau_k^2=1,\tau_k\tau_{k+1}\tau_k=\tau_{k+1}\tau_k\tau_{k+1},\tau_k\tau_l=\tau_l\tau_k \mbox{if}|l-k|>1; \tau_kc_k=c_{k+1}\tau_k,\qquad\tau_kc_r=c_r\tau_k \mbox{if}r\ne k,k+1; \tau_kx_k-x_{k+1}\tau_k=-c_k-c_{k+1}=x_{k}\tau_k-\tau_kx_{k+1},\tau_kx_r=x_r\tau_k \mbox{if}r\ne k,k+1.
\lim_{x\rightarrow 0+}x^{(\ell-1)/\ell}\frac{d\mu}{dx}(x) =\lim_{x\rightarrow 0+}\frac{x^{(\ell-1)/\ell}}{2\pi i x}(w_{-}(x)-w_{+}(x)) &= \frac{b_1}{2\pi i}\left(e^{2\pi i/\ell}-1\right)>0,
K= A (y^0\frac{\partial}{\partial y^4}-y^4\frac{\partial}{\partial y^0})+ B_i( y^0\frac{\partial}{\partial y^i}+y^i\frac{\partial}{\partial y^0})+ C_j (y^4\frac{\partial}{\partial y^j}+y^j\frac{\partial}{\partial y^4})+ D_p \epsilon_{pqr}y^q\frac{\partial}{\partial y^r}.
\Re\Big\{\sum_{n=1}^{\infty}z_n^m \Big\}\leq \frac{1}{\alpha^m}- \frac{1}{(\alpha+1-\beta_1)^{2m}}+ \Re\Big\{\frac{\delta(\chi)}{(\alpha+i\gamma')^{2m}}- \frac{\delta(\chi)}{(\alpha +1+i\gamma' -\beta_1)^{2m}}\Big\}
Q = \frac{6 \cdot 1 6}{4 ! ( 2 \pi )^{4}}\left( \frac{\zeta \pi}{2}\right)^{2}\left( \frac{\zeta^{\prime}\pi}{2}\right)^{2}\sum \biggl \{\frac{- 2 \zeta^{2}}{\delta ( \delta + \zeta / 2 )^{2}( \delta + \zeta )}\biggr \}\biggl \{\frac{- 2 \zeta^{\prime 2}}{\delta^{\prime}( \delta^{\prime}+ \zeta^{\prime}/ 2 )^{2}( \delta^{\prime}+ \zeta^{\prime})}\biggr \}= 1 \, ,
\mathcal{C}'(z_1, z_2) = (q_2/\delta)^{-2}|z_1z_2|^{-1/2}\sum_{h_1}\sum_{h_2}F\left(\frac{h_1}{|\Delta z_2|^{1/2}q_2/\delta}, \frac{h_2}{|\Delta z_1|^{1/2}q_2/\delta}\right)G'(h_1, h_2);
\begin{array}{rcl}\displaystyle\int_{\Gamma_1}dt_1\int_{\Gamma_2}dt_2\cdots\int_{\Gamma_n}dt_n \int_{\Gamma'_1}dt'_1 \int_{\Gamma'_2}dt'_2\cdots \int_{\Gamma'_m}dt'_m\times & \times\langle V_{\alpha_1}(z_1)V_{\alpha_2}(z_2)V_{\alpha_3}(z_3)V_{\alpha_4}(z_4)J_+(t_1) J_+(t_2)\cdots J_+(t_n)J_-(t'_1) J_-(t'_2)\cdots J_-(t'_m) \rangle~.\end{array}
(j+1) (\ell+2) d_{j+1}+ (2n-2j+1) p (\ell+1) d_j + (\ell+3) c_{j-1}\qquad{}+ \binom{n}{j}(-2)^j (p(\ell+1))^{j+1}(\ell+2)^{n-j}= 0,j = 1, 2, \dots, n
\pi \left( a , b , c \right) + J \pi \left( a , b , c \right) J^{\dagger}= d i a g \left( \begin{array}{c}{{2 b}}\\{{a + b 1_{2}}}\\{{c + \left( u b - b \right) 1_{3}}}\\{{c + \left( u b + b \right) 1_{3}}}\\{{a \otimes 1_{3}+ 1_{2}\otimes c + u b \otimes 1_{6}}}\\ \end{array}\right) \otimes 1_{N}
\gamma (\tilde\gamma^{-1}(\tilde\gamma^{-1})^T)=\tilde\gamma^{-1}-D, D\equiv \left( \begin{array}{ccc}0 -B_0/(\gamma^2+B_0^2) 0 B_0/(\gamma^2+B_0^2) 0 0 0 0 0 \end{array}\right).
\begin{cases}m_{1}= -k^2\pi^2 - 2c_k k\pi - 2id_k k\pi + O(1), \forall k\geq k_0 , m_{2}=-\frac{1}{2}+d_k - i(k\pi +c_k) + O(k^{-1}), \forall k\geq k_0 , m_{3}= - \frac{1}{2}-d_k + i(k\pi +c_k)+O(k^{-1}) , \forall k\geq k_0 . \end{cases}
\frac{d+\varepsilon\sqrt{ad}}{d+\varepsilon\sqrt{bd}-1}<\frac{d+\sqrt{bd}}{d-\sqrt{bd}}=1+\frac{2\sqrt{bd}}{d-\sqrt{bd}}=1+\frac{2}{\frac{d}{\sqrt{bd}}-1}<1+\frac{2}{\sqrt{B_0}-1}=\frac{\sqrt{B_0}+1}{\sqrt{B_0}-1},
\int K_{1 1}G_{0}K_{1 1}\frac{d^{3}k^{\prime}}{\varepsilon_{k^{\prime}}}\sim \int \frac{1}{k^{\prime}}\frac{1}{k^{2}}\frac{1}{k^{\prime}}\frac{k^{2}d k^{\prime}}{k^{\prime}}= \int^{L}\frac{d k^{\prime}}{k^{3}}\propto c o n s t .
\langle\Delta\widetilde{X}_{\overline{\gamma}c},\Delta\widetilde{\Gamma}_{\overline{\gamma}c}\rangle =\sum_{l=1}^{r_0}\frac{\theta}{\overline{\nu}_l}\|\Delta\widetilde{\Gamma}_{a_lc}\|^2 +\sum_{l=r_0+1}^{r_1}\frac{\theta\overline{u}_{l'}}{\overline{\nu}}\|\Delta\widetilde{\Gamma}_{a_{l'}c}\|^2.
M^2_2=\dfrac{2(1+{\beta^2_r}\tan^2{\phi_r})(1-\tilde{b}{\beta_i}\beta_r)}{(\gamma+1){\beta_r}-(\gamma-1+2\tilde{b}{\beta_i}\beta_r)},~~~\tan{\delta_r}=\dfrac{({\beta_r}-1)\tan{\phi_r}}{1+{\beta_r}\tan^2{\phi}_r},
x :=[[a,b^{-1}],b]=(b^{-1}b^{-a}, b^2)\equiv (b^{-2}[b,a]^{-1}, b^2) x^{b^{2n}}= x^{(a^n, a^n)}\equiv (b^{-2}[b,a]^{-2n-1}, b^2[b,a]^{2n}) x^{b^{2n+1}}= (x^b)^{(a^n, a^n)}\equiv (b^{2}[b,a]^{2n+2}, b^{-2}[b,a]^{-2n-1}).
\begin{aligned}\left\{\varsigma_{Q_j,m}=1, \varsigma_{Q_j,i}=1 \right\}\bigcap_{r\in [i-1]}\left\{\varsigma_{Q_j,r}=0\right\}&=\left\{\varsigma_{Q_j,i}=1,\varsigma_{Q_j,i-1}=0\right\}&=\left\{\varsigma_{Q_j,i}=1\right\}/\left\{\varsigma_{Q_j,i-1}=1\right\}.\end{aligned}
\left( I_{2}\right)_{\mu}= - \frac{1}{2}( k_{1}+ k_{2})_{\alpha}\left[ \Delta_{\alpha \mu}\right] - \frac{1}{2}( k_{1}+ k_{2})_{\mu}\left\{I_{\log}( m^{2}) - \left( \frac{i}{( 4 \pi )^{2}}\right) Z_{0}( m^{2}, m^{2}, ( k_{1}- k_{2})^{2}; m^{2}) \right\}.
\frac{p-1}{p}\Delta_{p,f}|\nabla u|^{p}= |\nabla u|^{2p-4}\big[|{\rm Hess}\,u|_A^2+{\rm Ric}_f(\nabla u,\nabla u)\big] &+|\nabla u|^{p-2}\langle\nabla u,\nabla\Delta_{p,f}u\rangle,
S = - \frac{1}{4 \pi \alpha^{\prime}}\int \!{{d^{2}}\xi}[{R^{2}}{\partial_{\alpha}}{y}{\partial^{\alpha}}{y}+ 2 k R{A_{\mu}}{\partial_{\alpha}}{y}{\partial^{\alpha}}{x^{\mu}}+ ( g_{\mu \nu}+{k^{2}}{A_{\mu}}{A_{\nu}}){\partial_{\alpha}}{x^{\mu}}{\partial^{\alpha}}{x^{\nu}}] .
E\int_0^{T+K}\left(e^{\beta t}|Y(t)|^{2}+ \int_0^{T+K}e^{\beta s}\|Z(t,s)\|^{2}ds \right) dt \leq 2E\int_0^{T+K}\left(e^{\beta t}|Y(t)|^{2}+ \int_t^{T+K}e^{\beta s}\|Z(t,s)\|^{2}ds \right) dt,
\tilde\lambda=\frac{9\lambda (\mu_1^3-\mu_1^2\mu_2+\mu_1\mu_2^2-\mu_2^3 )^2\, (\mathbb E [(\xi_i-\eta_i)^2 ] )^3}{2 (\mu_1^2-\mu_1\mu_2+\mu_2^2 )^3\, (\mathbb E [(\xi_i-\eta_i)^3 ] )^2}.
0 =n D(n-i+\ell+1, n-i+j+2)=n + k_{n-i+\ell+1}-k_{n-i+j+2}-k_{n+\ell-j-1}=n-n+k_{n-i+\ell}+r - k_{n-i+j}-2r-k_{n+\ell-j-1}=k_{n+\ell-j}-r-k_{n+\ell-j-1}=-D(n+\ell-j-1).
\bigl < q | \hat{\eta}| \eta ; t \bigr > = \Bigl [ 2 i \hbar f ( t ) \tilde{\varphi}( t ) \frac{\partial}{\partial q}+ \Bigl ( \frac{1}{f ( t )}- 2 g ( t ) \tilde{\varphi}( t ) \Bigr ) q \Bigr ] \bigl < q | \eta ; t \bigr > = \eta \bigl < q | \eta ; t \bigr > .
a^{11}_{11}=\overline{a^{11}_{11}},a^{11}_{12}=\overline{a^{11}_{21}},a^{11}_{22}=\overline{a^{11}_{22}}, & a^{12}_{11}=\overline{a^{12}_{11}},a^{12}_{12}=\overline{a^{21}_{21}},a^{22}_{11}=\overline{a^{22}_{11}}, & a^{22}_{11}=\overline{a^{22}_{11}},a^{22}_{12}=\overline{a^{22}_{21}},a^{22}_{22}=\overline{a^{22}_{22}}.
\delta L_{LOS}^{s}(z)=\frac{10^{-6}}{cos\theta}\{\frac{k_{1}R_{d}}{g_{m}}(P(z)-P(z_{ref}))+\int_{z}^{z_{re}}((k_{2}-\frac{R_{d}}{R_{v}}k_{1})\frac{e}{T}+k_{3}\frac{e}{T^{2}})dz\}
S^2\backslash\Omega=S^2\backslash\bigcup_{r\in(1,2)}\gamma_r=\bigcap_{r\in(1,2)}(S^2\backslash\gamma_r)=\bigcap_{r\in(1,2)}(A_r\cup B_r)=\Bigl(\bigcap_{r\in(1,2)}A_r\Bigr)\cup\Bigl(\bigcap_{r\in(1,2)}B_r\Bigr).
\Gamma_{\mathrm{m i n}}={\frac{\sigma_{\mathrm{a b s}}^{\mathrm{m i n}}}{e^{\frac{\omega}{T_{H}}}- 1}}={\frac{\pi l^{2}\omega}{( e^{\frac{\omega}{2 T_{R}}}- 1 ) ( e^{\frac{\omega}{2 T_{L}}}- 1 )}}.
\frac{d (a)_n}{d a}= (a)_n \left(\psi^{(0)}(a+n)-\psi^{(0)}(a) \right) = (a)_{n}\sum_{k=0}^{n-1}\frac{1}{a+k}=(a)_{n}\frac{1}{a}\sum_{k=0}^{n-1}\frac{(a)_{k}}{(a+1)_{k}}
K_{n}=[\begin{matrix}0&0&0&\cdot\cdot\cdot&0&0\\ 1&1&0&\cdot\cdot\cdot&0&0\\ 1&0&1&\cdot\cdot\cdot&0&0\\ &&&\vdots&&\\ 1&0&0&\cdot\cdot\cdot&1&0\\ 1&0&0&\cdot\cdot\cdot&0&1\end{matrix}].
\left(\partial_{\tau}-\triangle_{\Gamma_{\tau}}\right)\left|A_{\Gamma_{\tau}}\right|^{2}\,=\,-2\left|\nabla_{\Gamma_{\tau}}A_{\Gamma_{\tau}}\right|^{2}\,+\,2\left|A_{\Gamma_{\tau}}\right|^{4}\,-\,\frac{1+2\sigma}{2\sigma\tau}\left|A_{\Gamma_{\tau}}\right|^{2}
\left[\sum_{j=1}^3 |x_{j1}|^2 a_j - \left(b_1|\alpha|^2+b_2|\beta|^2\right)\right]i, \left[\sum_{j=1}^3 |x_{j2}|^2 a_j - \left(b_1|\beta|^2+b_2|\alpha|^2\right)\right]i, &\left[\sum_{j=1}^3 |x_{j3}|^2 a_j - b_3\right]i
[a][b]-[b][a]=\sum_{\varepsilon_i\in\{0,\infty\}}(q^{\varepsilon+\frac{1}{2}N}-q^{-\varepsilon-\frac{1}{2}N})[(\cdots((D(a,b)_{\varepsilon_1}^{p_1})_{\varepsilon_2}^{p_2})\cdots)_{\varepsilon_n}^{p_n}]
{\bf 7 0}\stackrel{U s p ( 4 ) \times S U ( 4 ) \times U ( 1 )}{\longrightarrow}\left({\bf 1}+{\bar{\bf 1}},{\bf 1},{\bf 1}\right) \oplus \left({\bf 1},{\bf 5},{\bf 6}\right) \oplus \left({\bf 1},{\bf 1},{\bf 6}\right)
\bigg| \sum_{k=0}^N |\xi|\hat\phi_k(\xi)\bigg|^{r'}\leq \sum_{j}\bigg| \sum_{k=0}^N \chi_{[2^{j},2^{j+1}]}(\xi) \, |\xi|\hat\phi_k(\xi) \bigg|^{r'}\leq |\Phi_{1,0}^N(\xi)|^{r'}+|\Phi_{2,0}^N(\xi)|^{r'}+ \cdots + |\Phi_{N,0}^N(\xi)|^{r'}+ \sum_{j>N}|\Phi_{j,0}^N(\xi)|^{r'}+ \sum_{j<1}|\Phi_{j,0}^N(\xi)|^{r'}= I_1(\xi) + I_2(\xi) + I_3(\xi)
+ \frac{a^{2 l + 2}}{b^{2 l + 2}- a^{2 l + 2}}\Big ( \frac{r^{l}}{r^{l + 2}}+ \frac{r^{l}}{r^{l + 2}}+ \frac{l}{l + 2}\frac{b^{2 l + 2}}{r^{l + 2}r^{l + 2}}\Big ) \Bigg ) \frac{\sin ( ( l + 1 ) \theta )}{\sin \theta}.
\langle0|TA_{\mu}(x)A_{\nu}(x^{\prime})|0\rangle=-i\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p_{\alpha}p^{\alpha}+i\epsilon}[\eta_{\mu\nu}-(1-\xi)\frac{p_{\mu}p_{\nu}}{p_{\sigma}p^{\sigma}-\xi m^{2}}]e^{ip^{\mu}(x_{\mu}-x_{\mu}^{\prime})}
\ \ \ \frac{d}{dt}\left( \int_{\mathbb{T}}(|\omega_{s2}^{\nu}|^{2}-|\partial_{y}\psi_{s2}^{\nu}|^{2})dxdy+\left\Vert \omega_{n2}^{\nu}\right\Vert_{X}^{2}\right) & =-2\nu\left( \int_{\mathbb{T}}(|\partial_{y}\omega_{s2}^{\nu}|^{2}-|\omega_{s2}^{\nu}|^{2})dxdy+\left\Vert \omega_{n2}^{\nu}\right\Vert_{X^{1}}^{2}\right) .
\frac{1}{N^{h-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^h}\Big(\prod\limits_{j=1}^d f_j(\xi_j(\mathbf{n}) + \widetilde{\mathbf{r}}_j)\Big) \frac{1}{C_{\Xi,\chi}\eta^h}\int\limits_{\mathbf{y}\in \mathbb{R}^h}(F(\mathbf{n}) + O_C(\eta/\sigma_F N))G(L\mathbf{y}) \boldsymbol{\chi}(\Xi(\mathbf{y}- \mathbf{n})) \, d\mathbf{y}.
\int\limits_{{\mathbb R}^{2n}}h(z) d \nu_y^A(z) = \int\limits_{{\rm SO}(2n)}h(f(O))d\mu_{2n}(O) = \int\limits_{S^{2n-1}}\Bigl ( \int\limits_{G_y}h(f(O_vU)) d \mu_{2n-1}(U) \Bigr ) d \sigma_{2n-1}(v).
C_{F}( d - \tilde{\eta}_{o}) = C_{F}^{- 1}( \tilde{\eta}_{o}) = \frac{\Gamma ( \tilde{\eta}_{o}) \Gamma ( \tilde{\eta}_{o}- \mu ) \Gamma^{4}( \mu - \frac{1}{2}\tilde{\eta}_{o}+ \frac{1}{2})}{\Gamma ( d - \tilde{\eta}_{o}) \Gamma ( \mu - \tilde{\eta}_{o}) \Gamma^{4}( \frac{1}{2}\tilde{\eta}_{o}+ \frac{1}{2})}\, .
{\sum_{i , j , k , l = 1}^{h}\omega_{l}^{2}\partial_{z}\left({\omega_{i}/ \omega_{l}}\right) A_{i j}A_{l k}\overline{\omega}_{k}^{2}\partial_{\bar{z}}\left({\overline{\omega}_{j}/ \overline{\omega}_{k}}\right)}= \left( \sum_{i , j = 1}^{h}\omega_{i}A_{i j}\overline{\omega}_{j}\right)^{3}.
[\theta_{tt}(t), \, \chi]_{\mathcal{C}_h}- [\Delta_h\, \theta(t) , \, \chi]_{\mathcal{C}_h}= -[ \rho_{tt}(t), \, \chi]_{\mathcal{C}_h}- [ f'(u_h(t))- f'(u(t))^I, \, \chi]_{\mathcal{C}_h}
\begin{aligned}\frac{|v|}{\lambda(t)}\int\rho'\left(\frac{x-vt}{\lambda(t)}\right) \left| \partial_x \Lambda\partial_t \Lambda+4\partial_x \phi \partial_t \phi\sinh^2(\Lambda) \right| = ~{}\frac{|v|}{\lambda(t)}\int\rho'\left(\frac{x-vt}{\lambda(t)}\right) |p(t,x)| \leq ~{}\frac{|v|}{\lambda(t)}\int\rho'\left(\frac{x-vt}{\lambda(t)}\right) e(t,x).\end{aligned}
\boldsymbol{f}^{\mathrm{dvm}}= (f^{\mathrm{dvm}}_0, \cdots, f^{\mathrm{dvm}}_M)^T, \boldsymbol{S}^{\mathrm{dvm}}= (S^{\mathrm{dvm}}_0, \cdots, S^{\mathrm{dvm}}_M)^T,{\bf \Xi}= \mathrm{diag}\{\xi_0, \cdots, \xi_M \}.
\frac{X_t}{t}= \frac{X_{\varrho_0+t-\varrho_0}}{t}= \frac{X_{\varrho_0^{-\infty}+t-\varrho_0}^{-\infty}}{t}= \frac{X_{t-\Delta}^{-\infty}}{t-\Delta}\frac{t-\Delta}{t}.
\gamma_{1}=\frac{\lambda^{-1/2\nu}}{\sqrt{\nu}}(1-\frac{5(\nu^{2}-1)}{48\nu^{2}}\lambda^{-2/\nu}-\frac{7(\nu^{2}-1)}{24\nu^{3}}\lambda^{-3/\nu}+O(\lambda^{-4/\nu})),
\phi_{\triangle \omega}( t , \rho , z ) ={\frac{1}{2 \pi}}\int_{0}^{\infty}d \omega \int d^{2}k ~ u_{\omega , k}( t , \rho ) e^{- i k_{j}z^{j}}\tilde{\phi}_{\triangle \omega}( \omega ) \tilde{\varphi}( k ) .
K = \sigma_{\mu}x^{\mu}= \left[ \begin{array}{c c}{q ( x^{0}- x^{3})}&{- \left[ 2 \right]^{1 / 2}q^{- 1 / 2}x^{+}}\\{- \left[ 2 \right]^{1 / 2}q^{1 / 2}x^{-}}&{q x^{0}+ q^{- 1}x^{3}}\\ \end{array}\right] \equiv \left[ \begin{array}{c c}{q D}&{B}\\{A}&{C / q}\\ \end{array}\right] ,
\sum\limits_{i=3}^{n+1}M_{Q_i}= \sum\limits_{i=3}^{n+1}\sum\limits_{Q\in \P[D]_1^*(Q_i)}(m_Q-m_{Q_{n+1}})\cdot 1_Q +(m_{Q_2}-m_{Q_{n+1}})\cdot 1_{Q_2}
\tau_c(h_i)=\tau_0(h_i)=-h_i, 1\leq i\leq l; &\tau_c(e_{{\varphi}})=\tau_0(e_{{\varphi}})=-e_{{-\varphi}}, \varphi\in \Delta_{\mathfrak{k}}; &\tau_0(e_{{\varphi}})=-\tau_c(e_{{\varphi}})=e_{{-\varphi}}, \quad\varphi\in \Delta_{\mathfrak{p}},
\Omega_4= G_m \sum_{z\in \mathbb{F}_p^*}\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-\frac{m_p}{4z}y^2-y}\sum_{\delta \in \mathbb{F}_p^*}\zeta_p^{- \frac{A}{4z}\delta^2 -\big( \frac{By}{2z}+\rho \big)\delta}.
\hat{\alpha}_k(x_k^i) = \sum_{j \in I_{k+1}}\hat y_{k+1}(x_{k+1}^j) \, p_k^{ij}\textrm{and}\hat{\beta}_k(x_k^i) = \frac{1}{\sqrt{\Delta_n}}\sum_{j\in I_{k+1}}\hat y_{k+1}(x_{k+1}^j) \, \Lambda_k^{ij},
\frac{1}{2}\Delta\big(|du|^{2(p-1)}\big) =|du|^{p-1}\Delta(|du|^{p-1})+\Big|\nabla |du|^{p-1}\Big|^2 &\leq |du|^{p-1}\Delta(|du|^{p-1})+\Big|\nabla\big(|du|^{p-2}du \big) \Big|^2.
\sum_{\ell=0}^{p-1}\frac{1}{(p-\ell)}\frac{e^{\ell z}}{(e^{z}-1)^{\ell+1}}= \sum_{\ell=0}^{p-1}\frac{1}{(p-\ell)}\sum_{j=-\ell - 1}^{\infty}B_{j+\ell+1}^{(\ell+1)}(\ell) \frac{z^{j}}{(j+\ell+1)!}.
\big\|\exp\Bigg(\int_{t_0}^t\widetilde{A}_\nu (\tau )d\tau \Bigg)\widetilde{\mathbf{x}}(t_0)\big\|\leq e^{-\frac L2(t-t_0)}\|\widetilde{\mathbf{x}}(t_0)\|,-t_0\,,t\geq T_\omega, \nu>0.
\begin{array}{l c r}{x_{0}= X_{0}[ R_{1}+ | Y_{0}|^{2}+ | Y_{1}|^{2}]^{-{\frac{1}{2}}}}\\{x_{1}= X_{1}[ R_{1}+ | Y_{0}|^{2}+ | Y_{1}|^{2}]^{-{\frac{1}{2}}},}\\ \end{array}
\big[s(a')-s(a)\big]\wedge\big[s(c)-s(b)\big]=\kern-10pt\sum_{1\le j<k\le n}\kern-5pt\Big\{\big\{[s_{j,k}(a')-s_{j,k}(a)]\wedge[s_{j,k}(c)-s_{j,k}(b)]\big\}\cdot\mathbb{I}_2\Big\}h_j\wedge h_{k}\ .
\P \bigl[ A_{k,\ell}| \mathcal T_{u}^N \bigr] \leq (1 - e^{-\rho (v-u) kS_k}) \prod_{m=\ell+1}^{k-1}e^{-mS_m}(1-e^{-\ell S_\ell}) \leq \rho (v-u) kS_k \ell S_\ell \prod_{m=\ell+1}^{k-1}e^{-mS_m}.
\begin{cases}D_{x_i r_j}g= 2 \sin \frac{2\pi}{n}+2n \sin^2 \frac{\pi}{n}\tan \frac{\pi}{n},\ \mbox{for}\j=i,i+1, D_{x_i r_j}g= 2 \sin \frac{2\pi}{n}, \ \mbox{otherwise}.\end{cases}
S(v)=B\left[dc_1(v)+\frac{d}{\lceil\Delta^{\frac{1}{6}}\ln^{-\frac{1}{6}}\Delta\rceil}{\lceil\Delta^{\frac{1}{6}}\ln^{-\frac{1}{6}}\Delta\rceil+1 \choose 2}+\frac{d}{\lceil\Delta^{\frac{1}{3}}\ln^{-\frac{1}{3}}\Delta\rceil}{\lceil\Delta^{\frac{1}{3}}\ln^{-\frac{1}{3}}\Delta\rceil+1 \choose 2}\right]
\displaystyle{\lim_{j\to +\infty}\lim_{\varepsilon \to 0^+}\frac{1}{\varepsilon}{\cal L}^N\left(\left\{x \in A^\ast:{\rm dist}(x, \partial^\ast E\cap \overline{A^\ast})< \varepsilon,{\rm dist}(x, \partial A^\ast)<\tfrac{1}{j}\right\}\right)=0.}
\left( x_{k}\right)_{j}\left( x_{k}\right)_{m}=\frac{\left( -1\right)^{j+m+1}}{c_{A}^{\prime}\left( \lambda_{k}\right)}\det\left(A_{\backslash\operatorname{row}j\backslash\operatorname{col}m}-\lambda_{k}I\right)
\sum_{\substack{1\leq j\leq g d_i\geq 0}}(-1)^j<\lambda_j \tau_0\prod_{i=1}^n\tau_{d_i}>u^{2j}\prod_{i=1}^n\widetilde{\phi_{d_i}}(u,q) = \sum_{\substack{1\leq j\leq g d_i\geq 0}}\sum_{k=1}^n(-1)^j<\lambda_j \tau_{d_k-1}\prod_{i\neq j}\tau_{d_i}>u^{2j}\prod_{i=1}^n\widetilde{\phi_{d_i}}(u,q).
DS^{(\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{w-z}{z+w}\prod_{k = 1}^{m}\frac{z+\pi_{k}}{z - \pi_{k}}\frac{w+\pi_{k}}{w - \pi_{k}},
\varphi_t(\bar{x}, \bar{y}) = \begin{pmatrix}x_i \otimes_t (0 \oplus_t x_{i-1}\otimes_t y_{i-1}) - 0 \oplus_t x_{i+1}\otimes_ty_{i+1}& y_{i+1}\otimes_t (0 \oplus_t x_{i+2}\otimes_t y_{i+2}) - 0 \oplus_t x_{i}\otimes_t y_{i}\end{pmatrix}
\sup \limits_{t>0}\int_0^{\infty}\bigl( \lambda_j^{-\frac 2{m+2}}+|t-\tau|\bigr)^{-(m+2) \left(\frac{1}{p}-\frac{1}{2}\right)\frac{n-1}{2}- \frac{m}{2\mu}}\, d\tau < \infty,
P(\tau\in[t'_L,t_L]) \le e\int_{t'_L}^{t_L+1}P(Z^x_t=0)\le Ce\int_{t'_L}^{L^2\wedge t'_L}t^{-d/2}dt + Ce\int_{t'_L\wedge L^2}^{t'_L+1}L^{-d}dt.
\bullet \forall \, p \in [-N,0] \left| \psi'(p) \right| = \left| \frac{x}{t}- f'(p) \right| \geqslant \min \big\{f'(-N) - a , b - f'(0) \big\}:= m_{-N}> 0 \; , & \bullet \forall \, p \in [0,N] \left| \psi'(p) \right| = \left| \frac{x}{t}- f'(p) \right| \geqslant \min \big\{f'(0) - a , b - f'(N) \big\}:= m_{+N}> 0 \; ,
(1+t)E[u](t) + \int_{\mathbb{R}^n}a(x) |u(t,x)|^2 \,dx \le C \begin{dcases}(1+t)^{-\frac{4}{2-\alpha}\left( \frac{1}{p-1}- \frac{n-\alpha}{4}\right)}(p > p_{subc}(n,\alpha)), (1+t)^{-\frac{2}{p-1}}\log(2+t) (p = p_{subc}(n,\alpha)), (1+t)^{-\frac{2}{p-1}}(p<p_{subc}(n,\alpha)), \end{dcases}
\big [ M_{\mu \nu}, Q_{\alpha}\big ] = -{\frac{1}{2}}( \sigma_{\mu \nu})_{\alpha}{}^{\beta}Q_{\beta}, \qquad \big [ M_{\mu \nu}, \bar{Q}_{\dot{\alpha}}\big ] ={\frac{1}{2}}( \tilde{\sigma}_{\mu \nu})^{\dot{\beta}}{}_{\dot{\alpha}}\bar{Q}_{\dot{\beta}}\; ,
D_{\! x}^{\sigma}\sum_{k = 0}^{\infty}a_{k}x^{k}= \! \! \sum_{k = \left\lceil \sigma \right\rceil}^{\infty}\! a_{k}\frac{\Gamma ( 1 \! + \! k )}{\Gamma ( 1 \! + \! k \! - \! \sigma )}x^{k - \sigma}\quad ( \sigma > 0 )
P = \sum_{n = 0}^{\infty}\sum_{\mu_{1}, \cdots , \mu_{n}= 1}^{N}\int{\frac{{d \theta_{1}}\cdots{d \theta_{n}}}{{n !}( 2 \pi )^{n}}| V_{\mu_{1}}( \theta_{1}) \cdots V_{\mu_{n}}( \theta_{n}) \rangle \langle V_{\mu_{1}}( \theta_{1}) \cdots V_{\mu_{n}}( \theta_{n}) |}
I=( x_1x_2x_3, x_1x_3x_5, x_1x_4x_5, x_2x_3x_4, x_2x_4x_5, x_1x_2x_6, x_1x_4x_6, x_2x_5x_6, x_3x_4x_6, x_3x_5x_6 ).
\mbox{\textbf{P3-3:}}\;\; \max_{\{x_{k_j},\beta_{k_j}\}}\sum_{k=1}^K\sum_{j=1}^J x_{k_j}{\log(c_{k_j}\beta_{k_j})}\mbox{s.t.}\;\; \mbox{same constraints as problem{\bf P3-1}}.
\frac{\big(q^{\theta N - z}; q\big)_{\infty}}{(q^{- z}; q)_{\infty}}= \frac{\big(1 - q^{\theta N - z}\big)\cdots \big(1 - q^{\theta N - z + M}\big)}{(1 - q^{- z})\cdots \big(1 - q^{- z + M}\big)}\cdot\frac{\big(q^{\theta N - z + M + 1}; q\big)_{\infty}}{\big(q^{- z + M + 1}; q\big)_{\infty}},
f_0=f_Y|_{\mathbb{P}\mathrm{Hom}(F,C_0,L_0)_e}: \mathbb{P}\mathrm{Hom}(F,C_0,L_0)_e \to\overline{\mathcal{C}(d_1,d_2,c)}: ([F],C_0,[L_0],\mathbf{k}\varphi) \mapsto [\ker(\varphi:F\twoheadrightarrow L_0(2))].
C(x_k \mid x_V, p_{[k+1..\ell]}) \stackrel{(a)}{\leq}C(x_{V}, x_k, p_{[k+1..\ell]}) - C(x_{V}, p_{[k+1..\ell]}) + c \log n \stackrel{(b)}{\leq}A(x_{V}, x_k, p_{[k+1..\ell]}) - A(x_{V}, p_{[k+1..\ell]})+ 8 \gamma_{k+1}+ c \log n \stackrel{(c)}{\leq}n_k + 16 \gamma_{k+1}+ 2c \log n \stackrel{(d)}{\leq}n_k + \eta_k.
m - \frac{( - \det h )}{2}\frac{v}{\mu}- \frac{\Lambda_{1}^{3}}{4}\left( \frac{\mu}{v}\right)^{2}= 0 , \qquad - ( - \det h ) + \Lambda_{1}^{3}\left( \frac{\mu}{v}\right)^{3}= 0 .
+\sum\limits_{n=1}^{N-1}\frac{\Lambda^+_n}{\sqrt{\pi}2^{n}}t^{-n/2}\left[ \int\limits_{1}^{+\infty}z^{-n}e^{-(z-\eta)^2}dz + \int\limits_{0}^{1}\Psi_n(z,\eta)dz \right] +
\widetilde{Z}_F\stackrel{\mathit{law}}{=}\sum_{k=1}^K\frac{\partial F}{\partial x_k}(J) \bigl(\sqrt{U_k}a(T_k,X_{T_k-})N_k^{-}+ \sqrt{1-U_k}a(T_k,X_{T_k})N_k^{+}\bigr)+ \widetilde{R}_F,