On a problem of Katona on minimal completely separating systems with restrictions

Let S be a set of n elements, and k a fixed positive integer $\text{<}\text{1}\text{2}\text{n}$. Katona's problem is to determine the smallest integer m for which there exists a family $\text{A}$ = {A₁, …, A_m} of subsets of S with the following property: |_i| ? k (i = 1, …, m), and for any ordered pair x_i, x_i ∈ S (i ≠ j) there is A₁ ∈ $\text{A}$ such that x_i ∈ A₁, x_j ? A₁. It is given in this note that $\text{m = ?}\text{2n}\text{k}\text{?}\text{if}\text{1}\text{2}\text{k}{}^2\text{? 2}$.     

On the minimal basis of a completely separating matrix

We prove that there is essentially only onep×2 ^s incidence matrix satisfying a given separation condition, which is of minimal degree, and we find that matrix. This work is part of the author’s doctoral thesis, being written under the supervision of Dr. M. Maschler, at the Hebrew University     

On separating systems of graphs

Given a finite loopless graph G (resp. digraph D), let σ(G), ?(G) and ψ(D) denote the minimal cardinalities of a completely separating system of G, a separating system of G and a separating system of D, respectively. The main results of this paper are:

$\text{δ(G) =}\text{min}\text{m}{}^{\mathrm{m}}{}_{\mathrm{?m/2?}}\text{?γ(G)}\text{and}\text{?(G) = ?}\text{log}{}_2\text{γ(G)?}\text{where}\text{γ(G)}$

denotes the chromatic number of G. (ii) All the problems of determining σ(G), ?(G) and ψ(D) are NP-complete.     

On a problem on chamber systems

It is well known that a geometry belonging to a disconnected diagram is the direct sum of geometries corresponding to the connected components of the diagram. On the other hand, chamber systems with a disconnected diagram exist which do not split as direct products of components of smaller rank. Many finite examples of this kind are discussed in Groups of Lie Type and their Geometries (CUP, 1995, pp. 185–214), but none of them is simply connected. In this article, we construct a simply connected finite example.     

On a free boundary problem for minimal surfaces

ForC ⁴-embedded manifoldsS ³ which are differmorphic to the standard sphere in ³ the existence of non-constant minimal surfaces bounded byS and intersectingS orthogonally along their boundaries is deduced.This research was supported by the Sonderforschungsbereich 72 of the Deutsche-Forschungs-Gemeinschaft     

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41], [42] is optimal.     

On the thread problem for minimal surfaces

For a given one-dimensional fixed boundary $\Gamma$ in and a given constant we consider any one-dimensional free boundary $F$ in subject to the conditions that the length of is equal to , that and form a closed boundary, and that the minimal surface of dimension two being bounded by and minimizes the area among all comparison surfaces being bounded by and some with length equal to . This variational problem is known as the thread problem for minimal surfaces and stems from soap film experiments, in which the fixed boundary parts are pieces of wires and the free boundary parts are threads. The new result of this article will be that has no singular points in , provided the admissible surfaces and boundary parts are supposed to be rectifiable flat chains modulo two. Received February 16, 1995 / Accepted October 20, 1995     

Concentration on minimal submanifolds for a Yamabe-type problem

We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.     

On free actions, minimal flows, and a problem by Ellis

We exhibit natural classes of Polish topological groups such that every continuous action of on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup of the universal minimal -flow , being trivial, is not isomorphic with the greatest -ambit. Further refining our construction, we obtain a Polish topological group acting freely on the universal minimal flow yet such that and are not isomorphic. We also display Polish topological groups acting effectively but not freely on their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations of an infinite set, equipped with the pointwise topology, and the unitary group of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle forms the universal minimal flow for the topological group of orientation-preserving homeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.

     

On a separating solution of a recurrent equation

We consider the recurrent equation

On a problem of quasi-optimal impulse correction of stochastic systems

The problem of repeated correction of the solution of a stochastic system under a restriction on the total quantity of correction impulses is studied. A method for constructing nearly optimal corrections is proposed, and the error in the method is estimated.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 21–25, 1986.     

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation −ε²Δu+u=up in Ω⊆RN, where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N?3 and for k∈{1,…,N−2}. We impose Neumann boundary conditions, assuming 1<p<(N−k+2)/(N−k−2) and ε→⁰⁺. This result settles in full generality a phenomenon previously considered only in the particular case N=3 and k=1.