Let P be a Markov operator on a general state space (S, Σ) with an invariant probability measure m, assumed to be ergodic. We study conditions which yield that for every centered non-zero f ∈ L²(m) a non-degenerate annealed CLT and an L²-normalized CLT hold.

In this paper, the almost everywhere convergence of Cesàro means of Walsh–Kaczmarz–Fourier series in a varying parameter setting is investigated. In particular, we define subsequence ℕ_{α_n,q} of natural numbers and prove that the maximal operator
sup_{n∈ℕ_{α_n,q}}|σ_n^α_nf|
is of strong type (H¹,L¹), where H¹ is a Hardy space.

Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation
f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y),   x,y ∈ S,
in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations
-div[A(x,u,∇u)ω₁ + B(x,u,∇u)ν₁] + ????(x,u,∇u)ν₂ + |u|^p-2uω₂ - Σ_i,j=1ⁿD_j(a_ij(x)D_iu(x)) = f₀(x) - Σ_j=1ⁿD_jf_j(x)   in Ω,
u(x) = 0   on ∂Ω,
in the setting of the weighted Sobolev spaces.

The sequence spaces _ruℓ_∞(O, ∇_q), _ruℓ_p(O, ∇_q), ruc(O, ∇q), _ruc₀(O, ∇_q), _rum_φ(O, ∇_q, p), _run_φ(O, ∇_q, p), _rum_φ(O, ∇_q), _run_φ(O, ∇_q) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity on the space _rum_φ(O, ∇_q, p) are also examined in this article.

We generalize a classical result about derivation pairs on function algebras. Specifically, we describe the forms of derivation pairs on rings and rngs (non-unital rings) which are not assumed to be commutative. The proofs are based on knowledge of the solutions of the sine addition formula on a semigroup. Examples are given to illustrate the results.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

Let R be an associative ring not necessarily with unity. We say that R is a semi-direct sum of rings S and I, if R = S+I, where S is a subring of a ring R, I is an ideal of R and S ∩ I = {0}.
The aim of this paper is to investigate certain algebraic properties of semidirect sums of associative rings with applications to amalgamated rings. We generalize several results from the literature to associative rings without unity. In particular we show that the class of semi-direct sums of rings is equal to the class of amalgamated rings, we provide a description of the Jacobson radical of semi-direct sums and we offer a characterization of semi-direct sums that are left Steinitz rings.

In this paper we introduce a new kind of generalized Jacobsthal numbers in a distance sense. We give the identities and matrix representations for them and their connections with the Fibonacci and the Pell numbers. We also describe the interpretations of these numbers in terms of some kind of (k₁A₁, k₂A₂, k₃A₃)-edge colouring and quasi colouring.

In this paper, we study the local existence of weak solutions for parabolic problem involving the fractional p-Laplacian. Our technique is based on the Galerkin method combined with the theory of Young measures. In addition, an example is given to illustrate the main results.

We will be concerned with deformations of a free elastic top rim of a parachute of a gas balloon. The top rim is connected with the circular deflation port of the balloon envelope by heavy duty flexible load tapes. The inside part of the balloon is filled with compressed gas. Equilibrium forms of the parachute may be found as solutions of a certain nonlinear functional-differential equation with two physical parameters: an elasticity coefficient of tapes and a physical parameter describing compressed gas. This equation possesses radially symmetric solutions corresponding to circular shapes of the top rim. Our goal is to study the existence of symmetry breaking bifurcation of the top rim of parachute.

Report of Meeting. The Twenty-fourth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 6 – 9, 2025.