An upper bound for theα-height of (0, 1)-matrices

We obtain an upper bound for theα-height of an arbitrary matrix of zeros and ones. We apply the result to a number of known combinatorial problems.     

A constant bound on the number of columns (1,k − 2, 1) in the intersection array of a distance-regular graph

We show that the number of columns (1,k – 2, 1) in the intersection array of distance-regular graphs is at most 20.     

Packing chromatic number, $$mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings,and characterizing the Petersen graph

The packing chromatic number $$chi _{rho }(G)$$     

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial.”     

Random perturbations of 1:1-resonant systems with SO(2) symmetry

^** Email: narayananramakrishnan{at}maxtor.com^*** Email: navam{at}uiuc.edu The near-resonant motion of trajectories of an integrable systemsubject to small dissipation and random perturbations is analysed.Under an appropriate change of time, we identify a reduced model.Our principal technique of dimensional reduction will be themethod of stochastic averaging for non-linear systems with smallnoise. A scheme to obtain the averaged motion of trajectoriesnear resonances is developed. The method of reduction is presentedusing the example of a two-degree-of-freedom Hamiltonian systemwith SO(2) symmetry subject to random perturbations. The averagedmotion of trajectories close to the 1:1-resonance surface isobtained and the effects of random perturbations on the near-resonantdynamics are analysed.     

A lower bound for r(5, 5)

A K₅-free two-coloring of the edges of K₄₂ is described, establishing 43 as a lower bound for R(5, 5).     

Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case

We investigate the problem of bound states for bosons and fermions in the framework of the relativistic configurational representation with the kinetic part of the Hamiltonian containing purely imaginary finite shift operators e^±ihd/dx instead of differential operators. For local (quasi)potentials of the type of a rectangular potential well in the (1+1)-dimensional case, we elaborate effective methods for solving the problem analytically that allow finding the spectrum and investigating the properties of wave functions in a wide parameter range. We show that the properties of these relativistic bound states differ essentially from those of the corresponding solutions of the Schrödinger and Dirac equations in a static external potential of the same form in a number of fundamental aspects both at the level of wave functions and of the energy spectrum structure. In particular, competition between ? and the potential parameters arises, as a result of which these distinctions are retained at low-lying levels in a sufficiently deep potential well for ? ? 1 and the boson and fermion energy spectra become identical.     

典型群PSU(3,q~2)与2-(v,k,1)设计

讨论自同构群是酉群 PSU(3 ,q2 ) (q=2 l)的区 -本原的 2 -(v,k,1 )设计 .首先证明了它必是点 -本原的 ,然后确定了这种类型的设计 ,即它只能为 2 -(q3+1 ,q+1 ,1 )设计     

Ramsey Number of K_(2,s 1) vs.K_(1,n)

It is shown that the Ramsey number r(K_(2,s 1),K_(1,n))(?)n sn~(1/2) (s 3)/2 o(1) for large n,and r(K_(2,s 1),K_(1,n))∈{((q-1)~2/s) 1,((q-1)~2/s) 2},where n=((q-1)~2/s)-q 2 and q is a prime power such that s|(q-1).     

Ree群2G2(q)与2-(v,k,1)区组设计(Ⅱ)

本文证明了自同构群的基柱为Ree群2G2(q)的区-本原2-(v,k,1)设计必为Ree unital,即2-(q3+1,q+1,1)设计,从而部分地回答了Praeger问题.