Circulant preconditioners for solving differential equations with multidelays

AbstractWe consider the solution of differential equations with multidelays by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. If an Ak1,k2-stable BVM is used, we show that our preconditioner is invertible and the spectrum of the preconditioned matrix is clustered. It follows that when the GMRES method is applied to solving the preconditioned systems, the method would converge fast. Numerical results are given to show the effectiveness of our methods

Unraveling the Scotogenic Model at Muon Collider

The Scotogenic model extends the standard model with three singlet fermion $N_i$ and one inert doublet scalar $\eta$ to address the common origin of tiny neutrino mass and dark matter. For fermion dark matter $N_1$, a hierarchical Yukawa structure $|y_{1e}|\ll|y_{1\mu}|\sim|y_{1\tau}|\sim\mathcal{O}(1)$ is usually favored to satisfy constraints from lepton flavor violation and relic density. Such large $\mu$-related Yukawa coupling would greatly enhance the pair production of charged scalar $\eta^\pm$ at the muon collider. In this paper, we investigate the dilepton signature of the Scotogenic model at a 14 TeV muon collider. For the dimuon signature $\mu^+\mu^-+/ \hspace{-0.65em} E_T$, we find that most viable samples can be probed with $200~\text{fb}^{-1}$ data. The ditau signature $\tau^+\tau^-+/ \hspace{-0.65em}E_T$ is usually less promising, but is important to probe the small $|y_{1\mu}|$ region. Masses of charged scalar $\eta^\pm$ and dark matter $N_1$ can be further extracted by a binned likelihood fit of the dilepton energy.Comment: 29 pages, 11 figures, 4 table