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In this paper, we show that the set of continuous functions defined on R n that approach zero at infinity and attain their maximum at precisely one (and only one) point is n-lineable but not (n + 2) -lineable. This result complements some recent published works on an open question originally posed by Vladimir I. Gurariy (1935–2005) in 2003. [ABSTRACT FROM AUTHOR]

Important probabilistic problems require to find the limit of a sequence of random variables. However, this limit can be understood in different ways and various kinds of convergence can be defined. Among the many types of convergence of sequences of random variables, we can highlight, for example, that convergence in L p -sense implies convergence in probability, which, in turn, implies convergence in distribution, besides that all these implications are strict. In this paper, the relationship between several types of convergence of sequences of random variables will be analyzed from the perspective of lineability theory. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set of infinite differentiable and integrable functions, for any order p ≥ 1 , on R N which are unbounded in a pre-fixed set. [ABSTRACT FROM AUTHOR]

Since the concept of lineability was coined by Gurariy in the early 2000s, only a few results provide a general criterion that guarantees the existence of large algebraic structures. Furthermore, most results in lineability theory are generally positive results, and there are a small amount of negative results. In this paper, we provide a non-constructive general (α , β) -lineability criterion in the context of complements of vector spaces, and we present two criteria of non- (α , β) -spaceability. Several applications are presented. [ABSTRACT FROM AUTHOR]

We generalize the classical Olivier’s theorem which says that for any convergent series ∑ n a n with positive nonincreasing real terms the sequence (n a n) tends to zero. Our results encompass many known generalizations of Olivier’s theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of (n a n) is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier’s theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families. [ABSTRACT FROM AUTHOR]

The class D of generalized continuous functions on R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property (CIVP) or the Strong Cantor Intermediate Value Property (SCIVP). The goal of this paper is to show that all Darboux-like subclasses of (PC \ D) ∪ (AC \ Ext) in the algebra generated by D are 2 c -lineable, that is, have maximal lineability. [ABSTRACT FROM AUTHOR]

A function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC , if each open set G ⊂ R 2 containing the graph of f contains also the graph of a continuous function g : R → R ; Sierpiński–Zygmund, f ∈ SZ (or, more generally, f ∈ SZ (Bor) ), provided its restriction f ↾ M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC ) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov (M) = c , that is, that R cannot be covered by less than c -many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ ∩ AC is also consistent with ZFC plus the negation of cov (M) = c . More precisely, we show that it is consistent with ZFC+ cov (M) < c (follows from the assumption that non (N) < cov (N) = c ) that there is an f ∈ SZ (Bor) ∩ AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov (M) = c or non (N) < cov (N) = c , the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated. [ABSTRACT FROM AUTHOR]

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem. [ABSTRACT FROM AUTHOR]

We provide an elegant argument showing, in ZFC, that the class PES \ Conn of all functions from R to R that are perfectly everywhere surjective (so Darboux) but not connectivity is 2 c -lineable, that is, that there exists a linear subspace of R R of dimension 2 c that is contained in (PES \ Conn) ∪ { 0 } . This solves a problem from a 2020 paper of Albkwre, Ciesielski, and Wojciechowski. The construction utilizes a transcendental basis of R . [ABSTRACT FROM AUTHOR]

Recently Tomasz Natkaniec in (Natkaniec in Rev R Acad Cenc Exactas Fís Nat Ser A Mat RACSAM, 115(1):10, 2021) studied the lineability problem for several classes of non-measurable functions in two variables. In this note we improve his results in the direction of algebrability. In particular, we show that most of the classes considered by Natkaniec contain free algebras with 2 c many generators. [ABSTRACT FROM AUTHOR]

We investigate the existence of algebraic structures in the set of continuous, unbounded and integrable functions in 0 , ∞ , continuing the work initiated by Calderón-Moreno et al. (J. Math. Anal. Appl. 470:348-359, 2019). We explore the stricter notions of α , β -lineability/spaceability and pointwise lineability/spaceability, which were recently introduced. For example, we prove that the aforementioned set is pointwise spaceable, in particular, spaceable. On the other hand, negative results are also obtained. For instance, we prove the non ℵ 0 , c -spaceability of the family of unbounded, continuous and integrable functions. [ABSTRACT FROM AUTHOR]

The space whose subsets we analyse with respect to lineability is L 0 (Ω , P) N consisting of random variables sequences on probability space (Ω , P) with atomless probability measure P. We study lineability and algebrability of L 0 (Ω , P) N -subsets of independent random variables with additional properties connected with various types of convergence, laws of large numbers, and Markov and Kolgomorov conditions. [ABSTRACT FROM AUTHOR]

This work is a contribution to the ongoing search for algebraic structures within a nonlinear setting. Here, we shall focus on the study of lineability of subsets of continuous functions on the one hand and within the setting of Sobolev spaces on the other (which represents a novelty in the area of research). [ABSTRACT FROM AUTHOR]

Consider the set R R of all functions from R to R as a vector space over R . A family G ⊂ R R is κ -lineable provided there exists a κ -dimensional linear subspace of R R contained in G ∪ { 0 } . The goal of this work is to show 2 c -lineability of all non-empty classes of functions in the algebra A (D) of Darboux-like maps that are contained in the family D \ Conn of Darboux functions that have disconnected graphs. The arguments for these classes stand apart from the proofs of 2 c -lineability of other classes in the algebra A (D) : the proofs we present below are considerably more delicate and heavily rely on the existence of algebraically independent subsets of R having different structures. The presented results generalize recently published proof of 2 c -lineability of the class D \ Conn . [ABSTRACT FROM AUTHOR]