Hausdorff dimension of Cantor sets and polynomial hulls

We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

     

Removable sets for continuous solutions of quasilinear elliptic equations

We show that sets of Hausdorff measure zero are removable for -Hölder continuous solutions to quasilinear elliptic equations similar to the -Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.     

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension. We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables. We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then, we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for non-characteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step provides us with a result of independent interest: the trapping of the solution in self-similar variables near the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on a new analysis in the self-similar variable.     

Recession cones and asymptotically compact sets

The present note is concerned with the study of the relations between the notions of asymptotic cones introduced by Dedieu and that of recession cones introduced by Luc. Conditions under which these notions coincide are given, as well as the fact that the compactness condition used by Luc is related (more restrictively) to asymptotic compactness. As an application of these notions, a result on proper efficiency in the sense of Lampe, established by Luc in finite dimensions, is extended to the infinite-dimensional case.This work was partially done while the author was visiting the Department of Mathematics of the University of Pisa, Pisa, Italy.     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

On the blow-up rate of large solutions for a porous media logistic equation on radial domain

In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on the ball domain and can be further adapted in a more general domain.     

Hausdorff dimension of quadratic rational Julia sets

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.     

Non-simultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution (u₁,u₂,…,un) to heat equations coupled via nonlinear boundary (i=1,2,…,n). It is proved that there exist suitable initial data such that ui(i∈{1,2,…,n}) blows up alone if and only if qi+1<pi. All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k, i, n) of ui−k and ui leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that ui−k,ui−k+1,…,ui blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained.     

Boundaries of compact convex sets and fragmentability

If X is a compact convex set in a real locally convex space, B⊂X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set.     

Operations on approximatively compact sets

In the paper, the problem of preserving the property of approximative compactness under diverse operations is considered. In an arbitrary uniformly convex separable space, we construct an example of two approximatively compact sets whose intersection is not approximatively compact. An example of two linear approximatively compact sets for which the closure of their algebraic sum is not approximatively compact is constructed. In an arbitrary Banach space, we construct two nonlinear approximatively compact sets whose algebraic sum is closed but not approximatively compact. We also prove that any uniformly closed Banach space contains an approximatively compact cavity.     

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u_t = Δu + u^m(x, t)e^pv(0,t), v_t = Δv + u^q(0, t)e^nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u₀(v₀) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.     

The role of non-linear diffusion in non-simultaneous blow-up

We study a parabolic system of two non-linear reaction-diffusion equations completely coupled through source terms and with power-like diffusivity. Under adequate hypotheses on the initial data, we prove that non-simultaneous blow-up is sometimes possible; i.e., one of the components blows up while the other remains bounded. The conditions for non-simultaneous blow-up rely strongly on the diffusivity parameters and significant differences appear between the fast-diffusion and the porous medium case. Surprisingly, flat (homogeneous in space) solutions are not always a good guide to determine whether non-simultaneous blow-up is possible.     

Quantization dimension of probability measures supported on Cantor-like sets

Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied.     

Some criteria for the minimality of pairs of compact convex sets

Some criteria for minimality for a pair of compact convex subsets of a real locally convex vector space are proved. Moreover several examples are given.     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Extracting compact and information lossless sets of fuzzy association rules

Applying classical association rule extraction framework on fuzzy datasets leads to an unmanageably highly sized association rule sets. Moreover, the discretization operation leads to information loss and constitutes a hamper towards an efficient exploitation of the mined knowledge. To overcome such a drawback, this paper proposes the extraction and the exploitation of compact and informative generic basis of fuzzy association rules. The presented approach relies on the extension, within the fuzzy context, of the notion of closure and Galois connection, that we introduce in this paper. In order to select without loss of information a generic subset of all fuzzy association rules, we define three fuzzy generic basis from which remaining (redundant) FARs are generated. This generic basis constitutes a compact nucleus of fuzzy association rules, from which it is possible to informatively derive all the remaining rules. In order to ensure a sound and complete derivation process, we introduce an axiomatic system allowing the complete derivation of all the redundant rules. The results obtained from experiments carried out on benchmark datasets are very encouraging. They highlight a very important reduction of the number of the extracted fuzzy association rules without information loss.     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

Existence of blow-up solutions in the energy space for the critical generalized KdV equation

For the critical generalized Korteweg-de Vries equation, we establish blow-up in finite or infinite time in for initial data with negative energy, close to a soliton up to scaling and translation.