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We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property, and the asymptotic average shadowing property, in the conservative case. [ABSTRACT FROM AUTHOR]

We give some sufficient conditions that a point can be approximated by an entropy point in terms of the sensitivity and the shadowing property. More precisely, we prove that for a continuous self-map f of a compact metric space X and a closed f -invariant subset S ⊂ X , if eventually sensitive points of f | S are dense in S , then any point of S can be approximated by an entropy point with an accuracy corresponding to that of the shadowing. Moreover, its homeomorphisms version and two corollaries are proved. [ABSTRACT FROM AUTHOR]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

We propose an efficient rendering method for dynamic scenes under all-frequency environmental lighting. To render the surfaces of objects illuminated by distant environmental lighting, the triple product of the lighting, the visibility function and the BRDF is integrated at each shading point on the surfaces. Our method represents the environmental lighting and the BRDF with a linear combination of spherical Gaussians, replacing the integral of the triple product with the sum of the integrals of spherical Gaussians over the visible region of the hemisphere. We propose a new form of spherical Gaussian, the integral spherical Gaussian, that enables the fast and accurate integration of spherical Gaussians with various sharpness over the visible region on the hemisphere. The integral spherical Gaussian simplifies the integration to a sum of four pre-integrated values, which are easily evaluated on-the-fly. With a combination of a set of spheres to approximate object geometries and the integral spherical Gaussian, our method can render object surfaces very efficiently. Our GPU implementation demonstrates realtime rendering of dynamic scenes with dynamic viewpoints, lighting, and BRDFs. [ABSTRACT FROM AUTHOR]

In ad hoc networks, the analysis of connectivity performance is crucial. The waiting time to deliver message M from source S to destination D is a measure of connectivity that reflects the effects of mobility, and some approximate methods have been proposed to theoretically analyze the mean waiting time in one-dimensional ad hoc networks that consist of mobile nodes moving along a street. In this paper, we extend these approximate methods to analyze the mean waiting time in two-dimensional networks with a lattice structure with various flows of mobile nodes. We discuss how the mean waiting times behave in such complicated street networks and how to approximate two kinds of mean waiting times. We show that our approximate methods can successfully compute the mean waiting times for even traffic patterns and roughly estimate them for uneven traffic patterns in two-dimensional lattice networks. In these analyses, we consider two shadowing models to investigate how shadowing affects the waiting time. We also discuss the effect of different positions of S on the mean waiting time. [ABSTRACT FROM AUTHOR]

Abstract: Shadow systems are often used to approximate reaction–diffusion systems when one of the diffusion rates is large. In this paper, we study the global existence and blow-up phenomena for shadow systems. Our results show that even for these fundamental aspects, there are serious discrepancies between the dynamics of the reaction–diffusion systems and that of their corresponding shadow systems. [Copyright &y& Elsevier]

Molecular dynamics refers to the computer simulation of a material at the atomic level. An open problem in numerical analysis is to explain the apparent reliability of molecular dynamics simulations. The difficulty is that individual trajectories computed in molecular dynamics are accurate for only short time intervals, whereas apparently reliable information can be extracted from very long-time simulations. It has been conjectured that long molecular dynamics trajectories have lowdimensional statistical features that accurately approximate those of the original system. Another conjecture is that numerical trajectories satisfy the shadowing property: they are close over long time intervals to exact trajectories but with different initial conditions. We prove that these two views are actually equivalent to each other, after we suitably modify the concept of shadowing. A key ingredient of our result is a general theorem that allows us to take random elements of a metric space that are close in distribution and embed them in the same probability space so that they are close in a strong sense. This result is similar to the Strassen-Dudley theorem except that a mapping is provided between the two random elements. Our results on shadowing are motivated by molecular dynamics but apply to the approximation of any dynamical system when initial conditions are selected according to a probability measure. [ABSTRACT FROM AUTHOR]