Spectrum is periodic for n-intervals

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set Ω is periodic, with the period an integral multiple of the measure of Ω. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.     

Regular saturated graphs and sum-free sets

In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular F-saturated graphs. One of the essential questions is given F, for which n does a regular n-vertex F-saturated graph exist. They proved that for all sufficiently large n, there is a regular $K_3$-saturated graph with n vertices. We extend this result to both $K_4$ and $K_5$ and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all $k\ge 2$, there is a regular $C_{2k+1}$-saturated with n vertices for infinitely many n. Studying the sum-free sets that give rise to $C_{2k+1}$-saturated graphs is an interesting problem on its own and we state an open problem in this direction.     

Structural, electronic and optical properties of SrCl2 under hydrostatic stress

The results of first-principles theoretical study of the structural, electronic and optical properties of SrCl₂ in its cubic structure, have been performed using the full-potential linear augmented plane-wave method plus local orbitals (FP-APW+lo) as implemented in the WIEN2k code. In this approach both the local density approximation (LDA) and the generalized gradient approximation (GGA) are used for the exchange-correlation (XC) potential. Also we have used the Engel-Vosko GGA formalism, which optimizes the corresponding potential for band structure calculations. We performed these calculations with and without spin-orbit interactions. Including spin-orbit coupling cause to lifts the triple degeneracy at Γ point and a double degeneracy at X point. Results are given for structural properties. The pressure dependence of elastic constants and band gaps are investigated. The dielectric function, reflectivity spectra and refractive index are calculated up to 30 eV. Also we calculated the pressure and volume dependence of the static optical dielectric constant.     

基于粗糙集与小波变换的图像融合算法

粗糙集理论是处理不确定性问题的数学方法，本文提出了基于粗糙集与小波变换相结合的图像融合算法。该方法首先将粗糙集理论应用于图像滤波中，对含有椒盐噪声的图像进行粗糙中值滤波，然后对滤波后的图像进行小波融合。实验结果表明，粗糙中值滤波有较强的去噪能力，且较好地保持了图像的细节信息，在此基础上进行小波融合，使得融合结果图像具有良好的效果。     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

On Clique-Transversals and Clique-Independent Sets

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by _C (G) and _C (G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when _C (H)= _C (H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters _C (G) and _C (G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference _C (G)– _C (G) is arbitrarily large.     

Metric entropy of convex hulls in type spaces--The critical case

Given a precompact subset of a type Banach space , where , we prove that for every and all

holds, where is the absolutely convex hull of and denotes the dyadic entropy number. With this inequality we show in particular that for given and with for all the inequality holds true for all . We also prove that this estimate is asymptotically optimal whenever has no better type than . For this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.

     

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.

     

On translations of subsets of the real line

In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpinski. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

     

Soft almost disjoint families

An almost disjoint family is said to be soft if there is an infinite set that meets each in a nonempty but finite set. We consider the associated cardinal invariant defined to be the minimal cardinality of an almost disjoint family that is not soft. We show that this cardinal coincides with J. Brendle's cardinal .