A survey of stratified domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we survey recent results on the F-domination number for various 2-stratified graphs F.     

On (s,t)-supereulerian graphs in locally highly connected graphs

Given two nonnegative integers s and t, a graph G is (s,t)-supereulerian if for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y|≤t, there is a spanning eulerian subgraph H of G that contains X and avoids Y. We prove that if G is connected and locally k-edge-connected, then G is (s,t)-supereulerian, for any pair of nonnegative integers s and t with s+t≤k−1. We further show that if s+t≤k and G is a connected, locally k-edge-connected graph, then for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y≤t, there is a spanning eulerian subgraph H that contains X and avoids Y, if and only if G−Y is not contractible to K₂ or to K_2,l with l odd.     

Decomposition of balanced complete bipartite multigraphs into multistars

In this paper, we obtain a criterion for the decomposition of the λ-fold balanced complete bipartite multigraph λK_n,n into (not necessarily isomorphic) multistars with the same number of edges. We also give a necessary and sufficient condition of decomposing 2K_n,n into isomorphic multistars.     

Improved bounds on the supremum of autoconvolutions

We give an improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions.     

Spin polarization and charge transfer of Co nanoclusters coated with CO molecules

We have investigated magnetic properties of Co clusters coated with CO molecules by first-principles density functional calculations. Total magnetic moment of the system decreases with the increase of CO molecule concentration regardless of adsorption sites for CO molecules. Spin polarization slightly increases as CO molecules are adsorbed on the most stable sites of Co cluster, which is caused by the increased local spin polarization of p-electrons due to charge transfer from Co clusters to CO molecules.     

Raman intensities of methanes from electrooptical parameters

This paper deals with the calculation of the intensities of Raman active fundamental vibrations of methane and its isotopical derivatives by the electrooptical theory of valence and with the “least squares” refinement of a set of electrooptical parameters from the experimental data. First some general considerations on the electrooptical theory of valence and on the computational method are presented. Then we give the expressions of the Raman intensities and depolarization ratios as functions of the electrooptical parameters and discuss some features of the “least squares” refinement technique. The results of our calculations on methane and deuteroderivatives are then reported. Particular attention has been given to the study of the type and combinations of parameters affecting each intensity.     

Spin-dependent nuclear weak processes and nucleosynthesis in stars

Spin dependent nuclear weak processes and nucleosynthesis in stars are investigated based on recent advances in shell model studies of stable and unstable exotic nuclei. Three topics on (1) neutrino-nucleus reactions in supernova explosions and nucleosynthesis of light elements as well as Mn, (2) electron capture reaction rates on Ni and Co isotopes at high densities and temperatures in the core-collapse process, and (3) new β-decay half-lives of N=126 isotones obtained by including both the Gamow-Teller and the first-forbidden transitions, and the effects on the element abundance in the r-process at the third peak region (A∼195), are studied with the use of new shell model Hamiltonians.     

Set systems without a 3-simplex

A 3-simplex is a collection of four sets A₁,…,A₄ with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on n elements without a 3-simplex is for all n≥1, with equality only achieved by the family of sets containing a given element or of size at most 2. This extends a result of Keevash and Mubayi, who showed the conclusion for n sufficiently large.     

Convolutions of singular measures and applications to the Zakharov system

Uniform L²-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.