On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

Hyperfine interactions in titanates: Study of orbital ordering and local magnetic properties

Hyperfine magnetic fields induced on the nuclei of nonmagnetic ions 139La and 89Y in LaTiO3 and YTiO3, respectively, have been microscopically calculated. The dependence of the hyperfine fields on the orbital and magnetic structures of the compounds under study has been analyzed. The comparative analysis of the calculated and known experimental data confirms the existence of the static orbital structure in lanthanum and yttrium titanates. © 2013 Pleiades Publishing, Ltd

Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

Crystal structure of mixed fluorites Ca(1-x)Sr(x)F(2) and Sr(1-x)Ba(x)F(2) and luminescence of Eu(2+) in the crystals

Within the framework of the virtual crystal method implemented in the shell model and pair potential approximation the crystal structure of mixed fluorites Ca(1-x)Sr(x)F(2) and Sr(1-x)Ba(x)F(2) has been calculated. The impurity center Eu(2+) and the distance Eu(2+)-F in this crystals have been also calculated. The low level position of excited 4f65d configuration of the Eu(2+) ion has been expressed using phenomenological dependence on distance E(2+)-F. The dependences of Stokes shift and Huang-Rhys factor on concentration x have been received for yellow luminescence in Sr(1-x)Ba(x)F(2):Eu(2+). The value x, for which the eg -level of Eu(2+) ion will be in conduction band in Sr(1-x)Ba(x)F(2):Eu(2+) has been calculated.Comment: 8 pages, 3 figures. The manuscript is sent to journal 'Physics of the solid state'. The results will be submitted on inernational conference SCINTMAT'2002 in oral session (june,20-22,2002,Ekaterinburg,Russia). Corresponding author e-mail: [email protected]

Galactic Rotation Parameters from Data on Open Star Clusters

Currently available data on the field of velocities Vr, Vl, Vb for open star clusters are used to perform a kinematic analysis of various samples that differ by heliocentric distance, age, and membership in individual structures (the Orion, Carina--Sagittarius, and Perseus arms). Based on 375 clusters located within 5 kpc of the Sun with ages up to 1 Gyr, we have determined the Galactic rotation parameters Wo =-26.0+-0.3 km/s/kpc, W'o = 4.18+-0.17 km/s/kpc^2, W''o=-0.45+-0.06 km/s/kpc^3, the system contraction parameter K = -2.4+-0.1 km/s/kpc, and the parameters of the kinematic center Ro =7.4+-0.3 kpc and lo = 0+-1 degrees. The Galactocentric distance Ro in the model used has been found to depend significantly on the sample age. Thus, for example, it is 9.5+-0.7 kpc and 5.6+-0.3 kpc for the samples of young (50 Myr) clusters, respectively. Our study of the kinematics of young open star clusters in various spiral arms has shown that the kinematic parameters are similar to the parameters obtained from the entire sample for the Carina-Sagittarius and Perseus arms and differ significantly from them for the Orion arm. The contraction effect is shown to be typical of star clusters with various ages. It is most pronounced for clusters with a mean age of 100 Myr, with the contraction velocity being Kr = -4.3+-1.0 km/s.Comment: 14 pages, 4 figures, 2 table

Generalized modularity matrices

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral-type algorithms for community detection