In this paper a billiard problem in nonlinear and nonequilibrium systems is investigated. This is an interesting problem where a traveling pulse solution behaves as if it is a billiard ball at a glance in some kind of reaction-diffusion system in a rectangular domain. We would like to elucidate the characteristic properties of the solution of this system. For the purpose, as the first step, we try to make a reduced model of discrete dynamical system having the important properties which the original system must have. In this paper we present a discrete toy model, which is reduced intuitively as one of the candidates by use of numerical experiments and careful observation of the solutions. Moreover, we discuss about the similar and important points between the solution in the original ordinary differential equation (which describes the pulse behavior) and the one in the toy model by computing numerically the characteristic quantities in view of the dynamical system, for example, global and local Lyapunov exponents and Lyapunov dimensions. As a result, we elucidate that the system possesses an intermittent-type chaotic attractor.

The definition of p-injectivity motivates us to generalize the notion of injectivity and projectivity, noted respectively C-injectivity and C-projectivity. Noetherian, semi-simple Artinian, quasi-Frobenius, regular hereditary and self-injective regular rings are considered in terms of C-injectivity and C-projectivity. Partial answers are given to Matlis’ Problem and Boyle’s Conjecture.

We study the uniqueness of meromorphic functions when two nonlinear differential polynomials generated by two meromorphic functions share the same simple and double $1$-points and improve an earlier result given by Fang-Fang and a recent result of Lahiri-Mandal.

We construct a polynomial invariant of a virtual magnetic graph diagram by defining an index of an enhanced state. For a virtual link diagram, it equals the Miyazawa polynomial and then the maximal degree on $t$ of the polynomials not only gives a lower bound of the real crossing number but also that of the virtual crossing number. Moreover, by definition we can calculate the polynomial for a link in a thickened surface or a Gauss chord diagram directly without transforming it into a virtual link diagram.

In this paper, we study the Dempster trace criterion. When the number of variables and the dimension of the null hypothesis are large relative to sample size, we derive the asymptotic distribution and Cornish-Fisher expansion of the Dempster trace criterion in the cases such that the covariance matrix is known and that the covariance matrix is unknown. Finally, we study the accuracy of the asymptotic expansion by the numerical simulation

We consider a singular perturbation problem of Modica-Mortola functional as the thickness of diffused interface approaches to zero. We assume that sequence of functions have uniform energy and square-integral curvature bounds in two dimension. We show that the limit measure concentrates on one rectifiable set and has square integrable curvature.

The aim of this paper is to establish an equivalent criterion for certain expansive diffeomorphisms of the 2-torus to admit an invariant Borel probability measure that is absolutely continuous with respect to the Riemannian volume. Our result is closely related to the well known Livšic-Sinai theorem for Anosov diffeomorphisms.