An application of group theory to matrices and to ordinary differential equations

AbstractA result concerned with groups is proved, from which several applications can be derived. We estimate e.g. the number of distinct eigenvalues of the Kronecker product and sum of two given matrices A, B, when A as well as B has distinct eigenvalues. We also discuss the order of the linear ODE whose solutions are the products of solutions of two given linear ODEs, when such ODEs are in certain classes

Canonical forms and discrete Liouville–Green asymptotics for second-order linear difference equations

Abstract Liouville–Green (WKB) asymptotic approximations are constructed for some classes of linear second-order difference equations. This is done starting from certain "canonical forms" for the three-term linear recurrence. Rigorous explicit bounds are established for the error terms in the asymptotic approximations of recessive as well as dominant solutions. The asymptotics with respect to parameters affecting the equation is also discussed. Several illustrative examples are given

Geometric effects in the design of catalytic converters in car exhaust pipes

Abstract Introduction We solve the gas dynamics (Euler) equations, augmented by adding a fourth equation governing the fraction of unburnt gas, in a number of cylindrically symmetric configurations of the pipe system. Case description The purpose is to test several duct profiles to see which one favors a higher reduction of the residual noxious gases, at the end of a car's exhaust pipe. Discussion and evaluation It is found that this purely geometric factor does play a role in the environment's purification accomplished by the catalytic converter. This is possibly due to the longer time spent by the noxious gases resident inside the device when this has certain profiles, though at the price of a little higher temperature attained. Conclusions It seems that geometric factors play a role in reducing cars' noxious gases by means of catalytic converters. A more precise analysis should be formulated as a mathematical inverse problem

How sharp is the Jensen inequality?

We study how good the Jensen inequality is, that is, the discrepancy between $\int_{0}^{1} \varphi(f(x)) \,dx$ , and $\varphi ( \int_{0}^{1} f(x) \,dx )$ , φ being convex and $f(x)$ a nonnegative $L^{1}$ function. Such an estimate can be useful to provide error bounds for certain approximations in $L^{p}$ , or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of $C^{2}$ functions, as well as for merely Lipschitz continuous convex functions φ, are established. Some examples are given to illustrate how sharp our results are, and a comparison is made with some other estimates existing in the literature. Finally, some applications involving the Gamma function are obtained

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration

The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included

Peer Reviewing and Electronic Publishing

Several issues about organizing worldwide open archives, refereeing system, and quality control and certification of research publications are reviewed. Advantages and drawbacks of using the web are discussed, and peculiarities of different research areas pointed out

Peer review e pubblicazione elettronica

The Internet and information technology permit an easier production and dissemination of scientific research results. In this situation, some problems remain still open: the validation and certification of these results and the problems linked to copyright rules, that in fact allow publishers to limit the circulation and the use of scientific results