Hausdorff dimension of the set of almost convergent sequences

The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

An Infinite System of Fractional Order with p-Laplacian Operator in a Tempered Sequence Space via Measure of Noncompactness Technique

In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional derivatives. In a new sequence space associated with a tempered sequence and the sequence space c0 (the space of convergent sequences to zero), a suitable new Hausdorff measure of noncompactness form is provided. This formula is applied to discuss the existence of a solution to our infinite system through applying Darbo’s theorem which extends both the classical Banach contraction principle and the Schauder fixed point theorem.

On a Characterization of Convergence in Banach Spaces with a Schauder Basis

We extend the well-known characterizations of convergence in the spaces l p ( 1 ≤ p < ∞ ) of p -summable sequence and c 0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences.“The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications.” Jacob T. Schwartz

Hyperspace of finite unions of convergent sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

On projections of products of spaces

We consider dense sets of products of topological spaces. We prove that in the product $Z^c=\prod\limits_{\alpha\in 2^\omega} Z_{\alpha},$ where $Z_\alpha=Z$ $(\alpha\in 2^\omega),$ there are dense sets such that their countable subsets have projections with additional properties. These properties entail that these dense sets contain no convergent sequences. By these properties we prove that the character of closed sets of the product is uncountable.