A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

Solving Bilevel Linear Programs Using Multiple Objective Linear Programming

We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau. The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem. Results in multiple objective programming needed are presented. We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small.     

Numerical semigroups: Apéry sets and Hilbert series

Let a ₁,…,a _n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a ₁,…,a _n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. Dedicated to the memory of Ernst S. Selmer (1920–2006), whose calculations revealed the “Selmer group”.     

Proof of the Branner-Hubbard conjecture on Cantor Julia sets

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992. This work was supported by the National Natural Science Foundation of China     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

The existence of global attractors for 2D Navier-Stokes equations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.     

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.