The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ,ψ:I→R and Borel probability measures μ,ν on the interval [0,1] such that

     

Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set

In this paper, we establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let X be a Hausdorff topological space, I⊆R an interval and . Assume that the function Ψ(x,⋅) is lower semicontinuous and quasi-concave in I for all x∈X, while the function Ψ(⋅,q) has compact sublevel sets and one local minimum at most for each q in a dense subset of I. Then, one has

     

Coincidence of strict s-numbers of weighted Hardy operators

We establish the equality of all the so-called strict s-numbers of the weighted Hardy operator T:Lp(I)→Lp(I), where 1<p<∞, I=(a,b)⊂R and

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

Invariance in a class of Bajraktarevi? means

Let I⊂R be a non-trivial interval, s:I→(0,∞) be a function, and let φ,ψ be real continuous strictly monotonic functions defined on I. We consider the equation

     

Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v) such that the geometric mean operator G₁, defined for positive functions f on (0,∞) by

     

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q?3 and k?1, define the exponential sum

     

Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices

Let S={Si}_i∈I be an arbitrary family of complex n-by-n matrices, where 1?n<∞. Let denote the joint spectral radius of S, defined as

     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

Computer aided solution of the invariance equation for two-variable Stolarsky means

We solve the so-called invariance equation in the class of two-variable Stolarsky means {S_p,q:p,q∈R}, i.e., we find necessary and sufficient conditions on the six parameters a, b, c, d, p, q such that the identity

     

Exponential elliptic boundary value problems on a solid torus in the critical of supercritical case

In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus of R³, when data are invariant under the group G=O(2)×I⊂O(3). The model problems of interest are stated below:

     

On the size of maximal chains and the number of pairwise disjoint maximal antichains

For each integer k≥3, we find all maximal intervals I_k of natural numbers with the following property: whenever the number of elements in every maximal chain in a finite partially ordered set P lies in I_k, then P contains k pairwise disjoint maximal antichains. All such I_k are of the form

     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),