More Lowenheim-Skolem results for admissible sets

It is shown that ifA is a countable, admissible set and ø ∈{ie53-1}, then under certain conditions, ø has a model inA. In general, however, ifT is a consistent theory of {ie53-2}, Σ-definable onA, then there is an admissible setB?A, with the same ordinals asA, containing a model ofT.     

Countably decomposable admissible sets

The known results about Σ₁-completeness, Σ₁-compactness, ordinal omitting etc. are given a unified treatment, which yields many new examples. It is shown that the unifying theorem is best possible in several ways, assuming V = L.     

Forcing in admissible sets

Translated from Algebra i Logika, Vol. 29, No. 6, pp. 648–658, November–December, 1990.     

Some dimensional results for homogeneous Moran sets

Let ?(¦n _k ¦k?1,¦c _k ¦k?1) be the collection of homogeneous Moran sets determined by ¦n _k ¦k?1 and ¦c _k ¦k?1, where ¦n _k ¦k?1 is a sequence of positive integers and ¦c _k ¦k?1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in ? are determined. The result is proved that for any values between the maximal and minimal values, there exists an element in ?(¦n _k ¦k?1,¦c _k¦k?1) such that its Hausdorff dimension is equal tos. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.     

Principal numerations of functionals on admissible sets

Translated from Algebra i Logika, Vol. 29, No. 4, pp. 398–420, July–August, 1990.     

Hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is applied to establish the existence of solutions to a large class of nonconvex constrained problems in a reflexive Banach space. The admissible sets are supposed to be star-shaped with respect to a ball. Due to a discontinuity property of the Clarke directional differential related to the corresponding distance functions, the proposed method permits one to attain the solution without passing to zero with the penalization parameter. Some applications to nonconvex constrained variational problems illustrate the theory.     

Some results on projection of planar sets

In this paper we define certain types of projections of planar sets and study some properties of such projections.     

Some non-existence results on divisible difference sets

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:

1. character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
2. involution arguments and
3. multipliers in conjunction with results on ordinary difference sets.

Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.     

Some results about integration on regular stratified sets

Summary In this paper we prove the existence of the integral of differential forms with cornpact support on stratified subsets of Rⁿ with locally finite volume and toe give some sufficient conditions to have Stokes' formula on regular stratified sets.     

Some results on topical functions and upward sets

The purpose of this paper is to introduce and discuss the concept of topical functions on upward sets.We give characterizations of topical functions in terms of upward sets.     

Some results on Ramsey numbers using sumfree sets

Maximal sumfree sets chosen from the non-zero elements of galois group of order 2⁶ (i.e. GF(2⁶)?{0}) are found to be of size 13, 17, 18, 20 and 32 only. This gives a negative result to an investigation which would otherwise have delivered an exact value of the Ramsey number R(3,3,3,3;2).     

Optimal problems for parabolic-type systems with aspheric sets of admissible controls

We consider an optimization problem for one class of parabolic systems with aspheric sets of admissible controls and establish an existence theorem for its optimal solution. We apply the obtained results to the study of one mathematical heat-exchange model.     

Some dimensional results for a class of special homogeneous Moran sets

We construct a class of special homogeneous Moran sets, called {m_k}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {m_k}_k?1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works.     

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439-3444].     

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is used in order to establish the existence of solutions to a large class of noncoercive constrained problems in a reflexive Banach space, in which the set of all admissible elements is not convex but fulfills some star-shaped property.Chargé de Recherches du Fonds National Belge de la Recherche Scientifique.