The sum of degrees in cliques

We investigate lower bounds on the average degree in r-cliques in graphs of order n and size greater than t(r,n), where t(r,n) is the size of the Turan graph on n vertices and r color classes. Continuing earlier research of Edwards and Faudree, we completely prove a conjecture of Bollobas and Erdoes from 1975.Comment: 10 page

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

Ewald summation on a helix : a route to self-consistent charge density-functional based tight-binding objective molecular dynamics

We explore the generalization to the helical case of the classical Ewald method, the harbinger of all modern self-consistent treatments of waves in crystals, including ab initio electronic structure methods. Ewald-like formulas that do not rely on a unit cell with translational symmetry prove to be numerically tractable and able to provide the crucial component needed for coupling objective molecular dynamics with the self-consistent charge density-functional based tight-binding treatment of the inter-atomic interactions. The robustness of the method in addressing complex hetero-nuclear nano- and bio-systems is demonstrated with illustrative simulations on a helical boron nitride nanotube, a screw dislocated zinc oxide nanowire, and an ideal DNA molecule

Subthreshold photoproduction of charm

Charm photoproduction rates off nuclei below the nucleon threshold are estimated using the phenomenologically known structure functions both for x>1 and x<1. The rates rapidly fall below the threshold from values of the order 10 pb for Pb close to the threshold (at 7.5 GeV) down to values of the order 1 pb at 6 GeV.Comment: 11 p[ages, 7 figure

Motion of vortices in ferromagnetic spin-1 BEC

The paper investigates dynamics of nonsingular vortices in a ferromagnetic spin-1 BEC, where spin and mass superfluidity coexist in the presence of uniaxial anisotropy (linear and quadratic Zeeman effect). The analysis is based on hydrodynamics following from the Gross-Pitaevskii theory. Cores of nonsingular vortices are skyrmions with charge, which is tuned by uniaxial anisotropy and can have any fractal value between 0 and 1. There are circulations of mass and spin currents around these vortices. The results are compared with the equation of vortex motion derived earlier in the Landau-Lifshitz-Gilbert theory for magnetic vortices in easy-plane ferromagnetic insulators. In the both cases the transverse gyrotropic force (analog of the Magnus force in superfluid and classical hydrodynamics) is proportional to the charge of skyrmions in vortex cores.Comment: 19 pages, 2 figures, to be published in the special issue of Fizika Nizkikh Temperatur dedicated to A.M.Kosevich. arXiv admin note: substantial text overlap with arXiv:1801.0109

Dephasing due to electron-electron interaction in a diffusive ring

We study the effect of the electron-electron interaction on the weak localization correction of a ring pierced by a magnetic flux. We compute exactly the path integral giving the magnetoconductivity for an isolated ring. The results are interpreted in a time representation. This allows to characterize the nature of the phase coherence relaxation in the ring. The nature of the relaxation depends on the time regime (diffusive or ergodic) but also on the harmonics $n$ of the magnetoconductivity. Whereas phase coherence relaxation is non exponential for the harmonic $n=0$, it is always exponential for harmonics $n\neq0$. Then we consider the case of a ring connected to reservoirs and discuss the effect of connecting wires. We recover the behaviour of the harmonics predicted recently by Ludwig & Mirlin for a large perimeter (compared to the Nyquist length). We also predict a new behaviour when the Nyquist length exceeds the perimeter.Comment: 21 pages, RevTeX4, 8 eps figures; version of 10/2006 : eqs.(100-102) of section V.C correcte

A search on Dirac equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.Comment: 9 page

Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.Comment: 25 pages, LaTe