Approximation and entropy numbers in Besov spaces of generalized smoothness

We determine the exact asymptotic behaviour of entropy and approximation numbers of the limiting restriction operator , defined by J(f)=f|_Ω. Here Ω is a non-empty bounded domain in , ψ is an increasing slowly varying function, , and is the Besov space of generalized smoothness given by the function t^sψ(t). Our results improve and extend those established by Leopold [Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, Marcel Dekker, New York, 2000, pp. 323–336].     

Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem

(1.1)

is considered, where , N≥2, KC²[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f^′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC^∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.     

Uniform asymptotic approximations for incomplete Riemann Zeta functions

An incomplete Riemann Zeta function Z₁(α,x) is examined, along with a complementary incomplete Riemann Zeta function Z₂(α,x). These functions are defined by and Z₂(α,x)=ζ(α)-Z₁(α,x), where ζ(α) is the classical Riemann Zeta function. Z₁(α,x) has the property that for and α≠1. The asymptotic behaviour of Z₁(α,x) and Z₂(α,x) is studied for the case fixed and , and using Liouville–Green (WKBJ) analysis, asymptotic approximations are obtained, complete with explicit error bounds, which are uniformly valid for 0x<∞.     

A generalized inductive limit strict topology on the space of bounded continuous functions

A generalized inductive limit strict topology β_∞ is defined on C_b(X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally -convex space E, where is a field with a nontrivial and nonarchimedean valuation, for which is a complete ultrametric space. Many properties of the topology β_∞ are proved and the dual of (C_b (X, E), β_∞) is studied.     

Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u^′(t)=A(t)u(t)+h(t) and on , assuming that A(t) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A(t) has exponential dichotomy, that R(λ₀,A()) is almost periodic, that B,C(t,s)_t≥s are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument and that h, f, F are Stepanov-like pseudo-almost periodic for p>1 and continuous. To illustrate our abstract result, a concrete example is given.     

Uniqueness of positive solutions for a boundary blow-up problem

In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem

where Ω is a C² bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/t^p is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x).     

On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field with a prime number p of elements, to . In this article we consider those functions f(X) for which there is a positive integer with the property that f(X)ⁱ, when considered as an element of , has degree at most p−2−n+i, for all i=1,…,n. We prove that every line is incident with at most t−1 points of the graph of f, or at least n+4−t points, where t is a positive integer satisfying n>(p−1)/t+t−3 if n is even and n>(p−3)/t+t−2 if n is odd. With the additional hypothesis that there are t−1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t−1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t−1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2(p−1)/3 directions.     

Interpolation and approximation in

Assume a standard Brownian motion W=(W_t)_t[0,1], a Borel function such that f(W₁)L₂, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a_X(f(X₁);τ) can be used to describe the L₂-error in discrete time simulations of the martingale generated by f(W₁) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.     

Region of variability for close-to-convex functions-II

For a complex number α with let be the class of analytic functions f in the unit disk with f(0)=0 satisfying in , for some convex univalent function in . For any fixed , and we shall determine the region of variability V(z₀,α,λ) for f(z₀) when f ranges over the class

In the final section we graphically illustrate the region of variability for several sets of parameters z₀ and α.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,