On the geometric form of free boundaries satisfying a bernoulli condition

In the x–y plane, let Ω denote an annular domain bounded by the simple closed curves ? and ?^*, such that the capacity potential U in Ω satisfies ∣ ▽ U∣ = 1 on ?. Assuming ?^* to be known, we obtain qualitative properties of ?. For example, the number of local maxima or minima of the y-co-ordinate relative to ? cannot exceed the corresponding number for ?^*.     

On the curvature of free boundaries with a Bernoulli-type condition

In this paper we study the classical external Bernoulli problem set in an annular domain Ω$\Omega$ of the plane.     

On the geometric form of solutions of a free boundary problem involving galvanization

In [6], T. I. Vogel studied a free boundary problem originating in the galvanization process. He showed that if the given boundary Γ* is starlike or convex, then so is the free boundary solution Γ. Our purpose is to generalize Vogel's second result by showing (under certain assumptions) that Γ cannot have more (local) maxima or minima (relative to a given direction) than Γ*; also that Γ cannot have more inflection points or greater total curvature than Γ*. The author has already proven analogous results for the Bernoulli free boundary problem in [1], [2] and [3].     

On graphs satisfying a local ore-type condition

For an integer i, a graph is called an L_i-graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w (element of) N(u) (intersection) N(v), d(u) + d(v) ≥ | N(u) (union) N(v) (union) N(w)| —i. Asratian and Khachatrian proved that connected L_o-graphs of order at least 3 are hamiltonian, thus improving Ore's Theorem. All K_1,3-free graphs are L₁-graphs, whence recognizing hamiltonian L₁-graphs is an NP-complete problem. The following results about L₁-graphs, unifying known results of Ore-type and known results on K_1,3-free graphs, are obtained. Set K = lcub;G|K_p,p+1 (contained within) G (contained within) K_p V K_p+1 for some ρ ≥ } (v denotes join). If G is a 2-connected L₁-graph, then G is 1-tough unless G (element of) K. Furthermore, if G is as connected L₁-graph of order at least 3 such that |N(u) (intersection) N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ϵ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L₁-graph of even order, then G has a perfect matching. © 1996 John Wiley & Sons, Inc.     

On pro-unipotent groups satisfying the Golod-Shafarevich condition

We prove that a pro-unipotent group satisfying the Golod-Shafarevich condition contains a free non-abelian pro-unipotent group. Together with the result of A. Magid this implies that such a group is not linear.

     

On associative algebras satisfying the Engel condition

It is shown that every finitely generated associative algebra over a field of characteristicp>0 satisfying the Engel condition is Lie-nilpotent. It follows that the Engel condition is inherited from an algebraA to its group of units,U(A).     

On Artinian rings satisfying the Engel condition

Let R be an Artinian ring, not necessarily with a unit, and let R ^o be the group of all invertible elements of R with respect to the operation a o b = a + b + ab. We prove that the group R ^o is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring R by its Jacobson radical is commutative. In particular, R ^o is nilpotent if it is a weakly nilpotent group or an n-Engel group for some positive integer n. We also establish that the ring R is strictly Lie-nilpotent if and only if it is an Engel ring and the quotient ring of the ring R by its Jacobson radical is commutative. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1264–1270, September, 2006.     

On power-bounded operators and operators satisfying a resolvent condition

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.     

On positive solutions of the heat equation satisfying a nonlinear boundary condition

We consider a boundary value problem for the heat equation in the exterior of a bounded domain of space variables. On the boundary of the domain, we pose a nonlinear boundary condition. We find sharp nonlinearity exponents for which there exists no global solution.     

On groups satisfying the minimal condition on certain subgroups

Periodica Mathematica Hungarica -     

On the geometric form of Bernoulli configurations

In References 4 and 5, the author studied the geometric form of the free boundary Γ of an annular domain Ω, characterized by the Bernoulli condition that |?U|=1 on Γ, where U is the capacity potential in Ω. It was shown, for example, that Γ has at most as many ν-extrema (for a given direction ν) or inflection points as the other boundary component Γ* of Ω, which is assumed given. Our purpose here is to extend the results in References 4 and 5 (wherever possible) to multiply connected regions for which some boundary components are given and others are free boundaries characterized by the Bernoulli condition. We present both positive results and counterexamples.     

Classification of sets satisfying the geometric characterization

The geometric characterization identifies the sets of nodes such that the Lagrange polynomials are products of factors of first degree. We offer a detailed classification of all known sets satisfying the geometric characterization in the plane. The defect, which takes into account the number of lines containing more nodes than the degree, plays a fundamental role in this classification. Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.     

On additive maps of prime rings satisfying the engel condition

Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n₁, n₂,…,n_k be natural numbers. Suppose that […[[(x), xn¹], xn²],…, xn^k]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n₁+n₂+ …+n_k Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.     

Clones satisfying the term condition

The term condition considered here is the property of an operation ? that holds iff ? and all of its variants obtained by permuting the variables satisfy (for all x,y,u₁,…v₁,…)$\text{?(x,u}{}_1\text{,…) = ?(x,v}{}_1\text{…)??(y,u}{}_1\text{,…) = ?(y,v)}{}_1\text{,…)}$. Clones consisting entirely of operations satisfying this term condition are called TC clones; algebras whose clone of term operations is a TC clone are called TC algebras; varieties such that every algebra in the variety is a TC algebra are called TC varieties. The paper is a systematic study of these notions, giving primary attention to operations and algebras on finite base sets, and to varieties generated by finite algebras. It is proved, among other results, that the number of n-ary TC operations on a k-element set is logarithmically asymptotic to k^(k?1)n when n increases without bound and k is held fixed; that there exist only countably many TC clones on any finite set; that the maximal TC clones on a finite set are finite in number (for each set). Some necessary conditions for an algebra to generate a TC variety are given, also some sufficient conditions.     

Contact metric manifolds satisfying a nullity condition

This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric. Dedicated to Professor Chorng-Shi Houh on his 65th birthday     

On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Continuing our previous investigations, we give simple sufficient conditions for the global stability of the zero solution of the difference equation x _n+1 = qx _n + ƒ_n(x _n, …, x _n−k), n ∈ ℤ, where the nonlinear functions ƒ_n satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2k + 2)/3] disjoint subintervals, and, for q from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 73–80, January, 2008.