Given a bounded polyhedral domain Ω ⊂ R3, we construct a sequence of tetrahedralizations (i.e., meshes) T ′ k that provides quasi-optimal rates of convergence with respect to the dimension of the aproximation space for the Poisson problem with data f ∈ Hm−1(Ω), m ≥ 2. More precisely, let Sk be the Finite Element space of continuous, piecewise polynomials of degree m ≥ 2 on T ′ k and let uk ∈ Sk...

and Applied Analysis 3 In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point u = 0. Definition 1. The equilibrium point u = 0 of problems (1)(2) is said to be globally exponentially stable if there exist constants κ > 0 andM ≥ 1 such that 󵄩󵄩󵄩󵄩u (t, x; t0, φ) 󵄩󵄩󵄩󵄩Ω ≤ M 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩Ω e −κ(t−t 0 ) , t ≥ t 0 , (10) where ‖φ‖ 2 Ω = sup t 0 −τ≤s≤t 0 ∑ n i=1 ∫ Ω φ...

We study a variant of the Whitney extension problem [21, 22] for the space C k,ω (R n). We identify C k,ω (R n) with a space of Lipschitz mappings from R n into the space P k × R n of polynomial fields on R n equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C k,ω (R n) as a Lip-schitz selection problem for set-valued mappings into a certain fa...

For every k ∈ N0, we consider graphs in which for any induced subgraph, ∆ ≤ ω−1+k holds, where ∆ is the maximum degree and ω is the maximum clique number of the subgraph. We give a finite forbidden induced subgraph characterization for every k. As an application, we find some results on the chromatic number χ of a graph. B. Reed stated the conjecture that for every graph, χ ≤ ⌈ ∆+ω+1 2 ⌉ holds....

We consider a two-dimensional Fermi liquid in the vicinity of a spin-densitywave transition to a phase with commensurate antiferromagnetic long-range order. We assume that near the transition, the Fermi surface is large and crosses the magnetic Brillouin zone boundary. We show that under these conditions, the self-energy corrections to the dynamical spin susceptibility, χ(q, ω), and to the quas...

The dynamical structure factor S(q, ω) of the SU(K) (K = 2, 3, 4) Haldane-Shastry model is derived exactly at zero temperature for arbitrary size of the system. The result is interpreted in terms of free quasi-particles which are generalization of spinons in the SU(2) case; the excited states relevant to S(q, ω) consist of K quasi-particles each of which is characterized by a set of K − 1 quant...

We study the existence of positive solutions to the following nonlocal boundary value problem −K ‖u‖2 Δu λus−1 f x, u inΩ, u 0 on ∂Ω, where s ∈ 1, 2 , f : Ω×R → R is a Carathéodory function,K : R → R is a positive continuous function, and λ is a real parameter. Direct variational methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spher...

A classical result of Alexsandrov [1] asserts that convex functions in R are twice differentiable a.e., (see also [3], [8] for more modern treatments). It is well known that Sobolev functions u ∈ W , for p > n/2 are twice differentiable a.e.. The following weaker notion of convexity known as k-convexity was introduced by Trudinger and Wang [12, 13]. Let Ω ⊂ R be an open set and C(Ω) be the clas...

Let Ω be a Cartan domain of rank r and genus p and Bν , ν > p−1, the Berezin transform on Ω; the number Bνf(z) can be interpreted as a certain invariant-mean-value of a function f around z. We show that a Lebesgue integrable function satisfying f = Bνf = Bν+1f = · · · = Bν+rf , ν ≥ p, must be M-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordi...

For Ω an infinite set, k ≥ 2 and W the set of k-sets from Ω, there is a natural closed permutation group Γk which is a non-split extension 0 → Z 2 → Γk → Sym(Ω) → 1. We classify the closed subgroups of Γk which project onto Sym(Ω). The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms.