Variable Precision Rough Set Approximations in Concept Lattice

The notions of variable precision rough set and concept lattice are can be shared by a basic notion, which is the definability of a set of objects based on a set of properties. The two theories of rough set and concept lattice can be compared, combined and applied to each other based on definability. Based on introducing the definitions of variable precision rough set and concept lattice, this paper shows that any extension of a concept in concept lattice is an equivalence class of variable precision rough set. After that, we present a definition of lower and upper approximations in concept lattice and generate the lower and upper approximations concept of concept lattice. Afterwards, we discuss the properties of the new lower and upper approximations. Finally, an example is given to show the validity of the properties that the lower and upper approximations have

Convergence Rate and Locality of Improved Overlap Fermions

We construct new Ginsparg-Wilson fermions for QCD by inserting an approximately chiral Dirac operator - which involves ingredients of a perfect action - into the overlap formula. This accelerates the convergence of the overlap Dirac operator by a factor of 5 compared to the standard construction, which inserts the Wilson fermion as a point of departure. Taking into account the effort for treating the improved fermion, we are left with an total computational overhead of about a factor 3. This remaining factor is likely to be compensated by other virtues; here we show that the level of locality is clearly improved, so that the exponent of the correlation decay is doubled. We also show that approximate rotation invariance is drastically improved, but a careful scaling test has to be postponed.Comment: 33 pages, 16 figures, a Section on approximate rotation symmetry is adde

Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

In lattice QCD computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a precondtioner to solve the "clover improved" Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used seperately in lattice QCD before. We show in extensive numerical test conducted with a parallel production code implementation that considerable speed-up over conventional Krylov subspace methods, domain decomposition methods and other hierarchical approaches for realistic system sizes can be achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to mixed-precision odd-even preconditioned BiCGStab. Results of numerical experiments changed slightly due to more systematic use of odd-even preconditionin

The future prospects of muon colliders and neutrino factories

The potential of muon beams for high energy physics applications is described along with the challenges of producing high quality muon beams. Two proposed approaches for delivering high intensity muon beams, a proton driver source and a positron driver source, are described and compared. The proton driver concepts are based on the studies from the Muon Accelerator Program (MAP). The MAP effort focused on a path to deliver muon-based facilities, ranging from neutrino factories to muon colliders, that could span research needs at both the intensity and energy frontiers. The Low EMittance Muon Accelerator (LEMMA) concept, which uses a positron-driven source, provides an attractive path to very high energy lepton colliders with improved particle backgrounds. The recent study of a 14 TeV muon collider in the LHC tunnel, which could leverage the existing CERN injectors and infrastructure and provide physics reach comparable to the 100 TeV FCC-hh, at lower cost and with cleaner physics conditions, is also discussed. The present status of the design and R&D efforts towards each of these sources is described. A summary of important R&D required to establish a facility path for each concept is also presented.Comment: 29 pages, 17 figure