Large Algebraic Structures Research Articles

Modes of convergence in nets, counterexamples, and lineability

In this work we continue with the ongoing search for what are often large algebraic structures of mathematical objects (functions, sequences, etc.) which enjoy certain special properties. This type of study belongs to the recent area of research known as lineability. On this occasion, and among several other results, we shall show that there are large algebraic structures within (i) the set of nets which are weakly convergent, but are not bounded, (ii) nets that are weakly convergent, but are not convergent in norm, or (iii) the set of nets of measurable functions which converge pointwise to a function that is not measurable and that are bounded in $[0,1]$.

Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence (a(n)) the sequence (na(n)) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset {mathcal {AOS}} of ell _{1} consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in {mathcal {AOS}} and its subsets; this kind of study is known as lineability. Finally we show that {mathcal {AOS}} is a residual mathcal {G}_{delta sigma } but not an {mathcal {F}}_{sigma delta } text {-set} .

Discontinuous, although “highly” differentiable, real functions and algebraic genericity

We exhibit a class of functions f:R→R which are bounded, continuous on R∖Q, left discontinuous on Q, right differentiable on Q, and upper left Dini differentiable on R∖Q. Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed using probabilistic methods. We also show that the families of functions satisfying similar properties contain large algebraic structures (obtaining lineability, algebrability and coneability).

Lineability, differentiable functions and special derivatives

The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (3) the family of functions from (0, 1) to $$\mathbb {R}$$ that are not derivatives, or (4) the family of mappings that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results.

Anti M-Weierstrass function sequences

Large algebraic structures are found inside the space of sequences of continuous functions on a compact interval having the property that, the series defined by each sequence converges absolutely and uniformly on the interval but the series of the upper bounds diverges. So showing that there exist many examples satisfying the conclusion but not the hypothesis of the Weierstrass M-test.

Families of feebly continuous functions and their properties

Let f:R2→R. The notions of feeble continuity and very feeble continuity of f at a point 〈x,y〉∈R2 were considered by I. Leader in 2009. We study properties of the sets FC(f) (respectively, VFC(f)⊃FC(f)) of points at which f is feebly continuous (very feebly continuous). We prove that VFC(f) is densely nonmeager, and, if f has the Baire property (is measurable), then FC(f) is residual (has full outer Lebesgue measure). We describe several examples of functions f for which FC(f)≠VFC(f). Then we consider the notion of two-feeble continuity which is strictly weaker than very feeble continuity. We prove that the set of points where (an arbitrary) f is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.

Lineability and modes of convergence

In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.

Holomorphic functions with distinguished properties on infinite dimensional spaces

Abstract In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

Lineability in the sets of Baire and continuous real functions

Discovering large algebraic structures within various sets of exotic real-valued (or complex-valued) functions or sequences is a very active research program now. For a cardinal number η, a subset A of a vector space X is said to be η-lineable if there is a vector subspace Z⊂X of dimension η with Z⊂A∪{0}. In this article, we investigate lineability properties in certain vector spaces of Baire-two functions and continuous real-valued functions.

Non-Lipschitz differentiable functions on slit domains

It is proved the existence of large algebraic structures—including large vector subspaces or infinitely generated free algebras—inside the family of non-Lipschitz differentiable real functions with bounded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the mean value theorem.

Lineability in sequence and function spaces

It is proved the existence of large algebraic structures \break --including large vector subspaces or infinitely generated free algebras-- inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

Anti-L’Hôpital Differentiable Functions

Large algebraic structures are found inside the family of those real differentiable functions f on the real line having the property that, for a prescribed subset Z, the continuity of \(f'\) fails at the points of Z, so showing that, even in the easiest case, the L’Hopital rule should be used carefully.

Large Algebras of Singular Functions Vanishing on Prescribed Sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.

Noetherianity of some degree two twisted commutative algebras

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that certain degree two twisted commutative algebras are noetherian. This example appears to have some fundamental differences from previous examples, and is therefore especially interesting. Reflective of this, our proof introduces new methods for establishing noetherianity that are likely to be applicable in other situations. The algebras considered in this paper are closely related to the stable representation theory of classical groups, which is one source of motivation for their study.

Linear subsets of nonlinear sets in topological vector spaces

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

Large algebraic structures inside the set of surjective functions

In this note we study large linear structures inside the set of Jones functions, which is a {\em highly pathological class} of surjective functions. We show that there exists an infinite dimensional linear space inside this set of functions. Moreover, we show that this linear space is isomorphic to \(\mathbb{R}^\mathbb{R}\), that is, it has the {\em biggest} possible dimension. The result presented in this note is an improvement of several recent results in the topic of {\em lineability}.

Sierpiński-Zygmund functions and other problems on lineability

We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpiński-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results.

Moduleability, algebraic structures, and nonlinear properties

We show that some pathological phenomena occur more often than one could expect, existing large algebraic structures (infinite dimensional vector spaces, algebras, positive cones or infinitely generated modules) enjoying certain special properties. In particular we construct infinite dimensional vector spaces of non-integrable, measurable functions, completing some recent results shown in García-Pacheco et al. (2009) [13], García-Pacheco and Seoane-Sepúlveda (2006) [15], Muñoz-Fernández et al. (2008) [20]. We prove, as well, the existence of dense and not barrelled spaces of sequences every non-zero element of which has a finite number of zero coordinates (giving partial answers to a problem originally posed by R.M. Aron and V.I. Gurariy in 2003).

Composite Signal Representation for Fast and Storage-Efficient Processing of Encrypted Signals

Signal processing tools working directly on encrypted data could provide an efficient solution to application scenarios where sensitive signals must be protected from an untrusted processing device. In this paper, we consider the data expansion required to pass from the plaintext to the encrypted representation of signals, due to the use of cryptosystems operating on very large algebraic structures. A general composite signal representation allowing us to pack together a number of signal samples and process them as a unique sample is proposed. The proposed representation permits us to speed up linear operations on encrypted signals via parallel processing and to reduce the size of the encrypted signal. A case study-1-D linear filtering-shows the merits of the proposed representation and provides some insights regarding the signal processing algorithms more suited to work on the composite representation.

Lineability and algebrability of pathological phenomena in analysis

We show that, in analysis, many pathological phenomena occur more often than one could expect, that is, in a linear or algebraic way. We show this by means of the construction of large algebraic structures (infinite dimensional vector spaces or infinitely generated algebras) enjoying some special or pathological properties.