A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation $$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {\mathrm{??}}^2(?).$$ “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”     

A note on circular dimension

In [1] Feinberg conjectures that the maximum circular dimension of all graphs having n vertices is attained by a complete partite graph. In this note we show that this is not so.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

A note on fractal strings and spacetime

A short discussion is presented on the role of Mersenne primes and even perfect numbers in fractal strings, Cantorian-fractal spacetime, spin structures on Riemann surfaces and the classification of consistent string theories. Since in principle the number of Mersenne primes should be infinite this entails that the family of consistent quantum string theories in higher dimensions than 26 could be infinite as well and, consequently, this could be the physical reason why the number of quantum-spacetime dimensions is infinite. The Tsallis entropy of fractal strings is constructed followed by the exact expression for the fractal string mass spectrum.     

A note on complete manifolds with finite volume

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

A note on the Conley attractors of lower semicontinuous multifunctions

We discuss Conley-type approach to attractive sets for lower semicontinuous multifunctions. Since every iterated function system induces a Barnsley–Hutchinson multifunction which is l.s.c. in such a case it is much more natural to consider a multifunctions of that type then closed relations on compact spaces earlier considered by some authors. We use topological (Kuratowski’s) limit instead of commonly used Hausdorff metric.     

The liapunov dimension of strange attractors

Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.     

Resistance dimension,random walk dimension and fractal dimension

In this paper, we generalize a theorem due to Telcs concerning random walks on infinite graphs, which describes the relation of random walk dimension, fractal dimension and resistance dimension. Moreover, we obtain a reasonable upper bound and lower bound on the hitting time in terms of resistance for some nice graphs. In fact, the conditions given in this paper are weaker than those obtained by A. Telcs.Partly supported by National Natural Science Foundation and State Educational Committee of China.     

A note on finite element wavelets

1. IotroductionThe study of mu1ti-wavelets was initiated by Goodman, Lee and Tang[2]. Later, acharacterization of sca1ing functions and wavelets was established by Goodman and Lee[3].In [4] GoodInan constructed multi-wavelets that satisfy certain interpolating conditions.Geronimo, Hardin and Massopust[5] used fractal interpolation to construct orthogonal scal-ing functions gh1 and gh2 (with multiplicity r = 2) that are both symmetric and colltinuousbut nondifferentiable. Strang and Strela…     

A note on finite abelian covers

We prove that any abelian cover over a smooth variety is defined by some cyclic equations. From the defining equations, we compute explicitly the normalization, branch locus, ramification indices, global invariants, and the resolution of singularities. As an application, we construct a new algebraic surface which is the quotient of ball.     

A note on finite local rings

We prove that a finite local ring with one minimal left ideal is a QF-ring with one minimal right ideal, thus answering a question of Claasen and Goldbach in the negative.