A problem of Zarankiewicz

Zarankiewicz, in problem P 101, Colloq. Math., 2 (1951), p. 301, and others have posed the following problem: Determine the least positive integer k_α,β(m, n) so that if a 0,1-matrix of size m by n contains k_α,β(m, n) ones then it must have an α by β submatrix consisting entirely of ones. This paper improves upon previously known upper bounds for k_α,β(m, n) by proving that $\text{k}{}_{\mathrm{\alpha \beta }}\text{(m,n)?}\text{1+((β?1)}\text{(}\text{p}\text{α?1}\text{))(}\text{m}\text{α}\text{)}\text{+}\text{((p+1)(α?1)}\text{α}\text{)n}$ for each integer p greater than or equal to α ? 1. Each of these inequalities is better than the others for a specific range of values of n. Equality is shown to hold infinitely often for each value of p. Finally some applications of this result are made to arrangements of lines in the projective plane.     

A contribution to the Zarankiewicz problem

Given positive integers let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m×n that does not contain an all ones submatrix of size s×t. We show that if s?2 and t?2, then for every k=0,…,s-2,

z(m,n,s,t)?(s-k-1)^1/tnm^1-1/t+kn+(t-1)m^1+k/t.     

A new result on the Pompeiu problem

A nonempty bounded open set () is said to have the Pompeiu property if and only if the only continuous function on for which the integral of over is zero for all rigid motions of is . We consider a nonempty bounded open set with Lipschitz boundary and we assume that the complement of is connected. We show that the failure of the Pompeiu property for implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in , , has the Pompeiu property. So far the result was proved only for solid tori in . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in bounded by hypersurfaces of revolution.

     

The Zarankiewicz problem in 3‐partite graphs

Let be a graph, be an integer, and write for the maximum number of edges in an ‐vertex graph that is ‐partite and has no subgraph isomorphic to . The function has been studied by many researchers. Finding is a special case of the Zarankiewicz problem. We prove an analog of the Kövári‐Sós‐Turán theorem for 3‐partite graphs by showing for . Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case that and is odd, that is, for . In the cases of and , we use a result of Allen, Keevash, Sudakov, and Verstraëte, to show that a similar upper bound holds for all and gives a better constant when . Finally, we point out an interesting connection between difference families from design theory and .     

A new result on Davenport constant

Let G be a finite abelian group of order n and Davenport constant D(G). Let S=⁰h(S)_∏g∈Ggvg(S)∈F(G) be a sequence with a maximal multiplicity h(S) attained by 0 and t=|S|?n+D(G)−1. Then 0∈k_∑(S) for every 1?k?t+1−D(G). This is a refinement of the fundamental result of Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103].     

A multiplicity result for the jumping nonlinearity problem

We consider a class of nonlinear problems of the form Au+g(x,u)=f, where A is an unbounded self-adjoint operator on a Hilbert space H of L2(Ω)-functions, an arbitrary domain, and is a “jumping nonlinearity” in the sense that the limits , exist and “jump” over the principal eigenvalue of the operator −A. Under rather general conditions on the operator L and for suitable a<b, we prove some multiplicity results. Applications are given to the wave equation, and elliptic equations in the whole space .     

A stability result for the magnetic shaping problem

We consider the bidimensional magnetic shaping problem without surface tension and study its stability when the boundary of the domain has cusp points. We show in particular that one has stability when the curvature of the smooth parts of is negative.     

A new result about panconnectivity on graphs

LetG be a simple graph such that the sum of the degrees of any two independent vertices ofG is at leastn–1. We shall prove thatG is [6,n]-panconnected except for four kinds of graphs.     

A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem

In this study, we consider the exponential utility maximization problem in the context of a jump–diffusion model. To solve this problem, we rely on the dynamic programming principle to express the value process of this problem in terms of the solution of a quadratic BSDE with jumps. Since the quadratic BSDE¹ under study is driven by both a Wiener process and a Poisson random measure having a Lévy measure with infinite mass, our main task is therefore to establish a new existence result for the specific BSDE introduced.     

A result about a selection problem of Michael

It is shown that a continuum that is an space in the sense of Michael must be hereditarily decomposable. This improves known results, thereby providing more evidence that such continua must be dendrites.

     

Steiner's problem and fagnano's result on the sphere

This note describes the nature of optimal solutions for the spherical Steiner-Weber location problem for the case of unit weights and either 3 or 4 demand points (requireing 4 demand points to lie in an open hemisphere). Geometrically appealing results which are necessary conditions for optimum solutions and spherical analogs of known planar results are obtained.     

A perturbation result for the Webster scalar curvature problem on the CR sphere

We consider the problem of prescribing the Webster scalar curvature on the unit sphere of $\text{C}{}^{\mathrm{n+1}}$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant and satisfying an assumption of Bahri–Coron type.     

Existence result for a new variational problem in one-dimensional segmentation theory

In order to study some segmentation problems in dimension one, we propose a new functional, whose leading term includes the second order derivative of the unknown function. We prove an existence result for the associated minimization problem, by relying on the compactness and the lower semicontinuity of the functional with respect to theL ¹-convergence.     

A stability result concerning the obstacle problem for a plate

It is proposed here to study the free boundary of the obstacle problem in the case of an elastic plate. Under a nondegeneracy assumption, we prove a stability theorem which relates the variations of the contact zone to the variations the external forces. The statement of this result obtained and the steps in the proof are very close to those given by D.G. Schaeffer in 1975, except for the very important fact that the present study deals with the biharmonic operator.     

A blow-up result in a Cauchy viscoelastic problem

In this work we consider a Cauchy problem for a nonlinear viscoelastic equation. Under suitable conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.     

A stability result concerning the obstacle problem for a plate

It is proposed here to study the free boundary of the obstacle problem in the case of an elastic plate. Under a nondegeneracy assumption, we prove a stability theorem which relates the variations of the contact zone to the variations the external forces. The statement of this result obtained and the steps in the proof are very close to those given by D.G. Schaeffer in 1975, except for the very important fact that the present study deals with the biharmonic operator.     

A new approximation result for BV-functions

This Note is devoted to obtaining an approximation result for BV-functions by means of a quasi-polyhedral sequence of BV-functions. This approximation could have interesting applications in some problems of the Calculus of Variations. To cite this article: M. Amar, V. De Cicco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).