Linear Arboricity and Linear k-Arboricity of Regular Graphs

 We find upper bounds on the linear k-arboricity of d-regular graphs using a probabilistic argument. For small k these bounds are new. For large k they blend into the known upper bounds on the linear arboricity of regular graphs. Received: December 21, 1998 Final version received: July 26, 1999     

Large induced forests in sparse graphs

For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree Δ. Our results include: (a) if Δ ≤ 3, and G ≠ K₄, then a(G) ≥ n ? e/4 ? 1/4 and this is sharp for all permissible e ≡ 3 (mod 4); and (b) if Δ ≥ 3, then a(G) ≥ α(G) + (n ? α(G))/(Δ ? 1)². Several problems remain open. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 113–123, 2001     

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

Additive Patterns in Multiplicative Subgroups

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q ^3/4 in the finite field F _q must contain an additive relation x + y = z. Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p ^1/3) in prime fields F _p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q ^3/4+o(1) of F _q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.     

Counting sum-free sets in abelian groups

In this paper we study sum-free sets of order m in finite abelian groups. We prove a general theorem about independent sets in 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order m in abelian groups G whose order n is divisible by a prime q with q ≡ 2 (mod 3), for every m ? \(C(q)\sqrt {n\log n} \) , thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sumfree subsets of size m are contained in a maximum-size sum-free subset of G. We also give a completely self-contained proof of this statement for abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of a fixed size in an (n, d, λ)-graph.     

On the concentration of eigenvalues of random symmetric matrices

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e ⁻ t ²32 ^s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.     

A theorem on electromagnetic properties of leptons

The following theorem is proven: Every lepton with the mass m, electric charge q and spin J belonging to any representation of a non-abelian gauge group must have the magnetic moment μ = qJm^?1, electric mean squared radius r² = qJ(J + 1)m^?2 and electric quadrupole moment Q = qJ(2J ? 1)m^?2 in the first order of the electromagnetic effects in an arbitrary renormalizable theory with the non-abelian gauge group symmetry which permits the validity of the Gerasimow-Drell-Hearn and Cabibbo-Radicati sum rules. The formula for the magnetic moment applies also for an abelian symmetry and remains valid even if the gauge symmetry is spontaneously broken.     

Laser annealing and theory of photostimulated structures

It is shown that a circularly polarized laser light passing through a disordered system represented by a thin film of an amorphous semiconductor gives rise to a self-organization of new ordered states. Laser annealing of implanted semiconductors, emergence of charge density waves, light-induced transmittance oscillations and optical stopping effect are explained on a unified ground within the framework of quantum field theory.Dedicated to Professor Ivan Úlehla on the occasion of his sixtieth birthday.The authors thank Dr. E. Majerníková for fruitful conversations. One of us (M. Noga) wishes to express his sincere indebtness and gratidude to Prof. Stig Stenholm and Christopher Cronström for valuable discussions at Research Institute for Theoretical Physics, University of Helsinki, where he was given the oportunity to start the research of these problems.