Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L _∞(0, T; L ₂(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h ^k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h ^k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.     

On a conjecture about the Randi? index

A conjecture of Delorme, Favaron and Rautenbach [On the Randi? index, Discrete Math. 257 (2002) 29-38] about the Randi? index of a graph, in relation to its order and minimum degree, is refuted by the AutoGraphiX 2 system. Moreover, a modified conjecture is derived from presumably extremal graphs obtained with that system.     

On Asymptotic Stability and Sweeping of Collisionless Kinetic Equations

We study a collisionless kinetic equation describing density distribution function of the position and velocity of particles moving in a slab with finite thickness and with a partly diffusive boundary reflection. In particular, we deal with existence of an invariant density and with the convergence to the equilibrium. We also study the long time behavior of densities when the equilibrium does not exists.     

Sweeping process with right uniformly lower semicontinuous mappings

Positivity - In this paper, we establish, in the setting of infinite dimensional Hilbert space, a new existence result for nonconvex sweeping process with right uniformly lower semicontinuous sets....     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Linear recurrence relations in the algebra of matrices and applications

We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for Aⁿ (nr) and e^tA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices A^j, where 0jr−1.     

Stokes flow in a junction of two-dimensional orthogonal channels

The purpose of this study is to calculate Stokes flow structures in relation to flow rate distribution in a junction of four orthogonal channels. Particular attention was paid to the existence of recirculation flow inside the channels for low Reynolds numbers. The presence of this recirculation flow may be observed, but only when the flow rate is relatively low in at least one of the branches.     

Isotopic production cross sections of residues in reactions induced by relativistic heavy ions with protons and deuterons

The isotopic production cross sections of heavy residues in relativistic heavy-ion collisions have been investigated in inverse kinematics. The primary reaction products were fully identified in mass and atomic number prior to beta decay using the fragment separator FRS. The huge collection of data obtained helps in the understanding of the two main reaction mechanisms involved: fragmentation and fission. These data provide basic information for future radioactive ion beam facilities and for technical applications like intense neutron sources by means of spallation targets.     

Cross sections of spallation residues produced in 1A GeV 208Pb on proton reactions

Spallation residues produced in 1 GeV per nucleon 208Pb on proton reactions have been studied using the Fragment Separator facility at GSI. Isotopic production cross sections of elements from 61Pm to 82Pb have been measured down to 0.1 mb with a high accuracy. The recoil kinetic energies of the produced fragments were also determined. The obtained cross sections agree with most of the few existing gamma-spectroscopic data. The data are compared with different intranuclear-cascade and evaporation-fission models. Drastic deviations were found for a standard code used in technical applications.