Related papers: Decomposing functions of Baire class $2$ on Polish…

We define two recursive functions obtained by decomposition of a given interval into four close parts and prove two lemmas which determine features of these functions.

We comment on a recent paper that connects certain forms of machine learning to Set Theory. We point out that part of the set-theoretic machinery is related to a result of Kuratowski about decompositions of finite powers of sets and we show…

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…

We prove that the Hausdorff dimension of the set of points where a function in the Zygmund class in the euclidean space has bounded divided differences, is bigger or equal to 1. A similar result for functions in the Small Zygmund class is…

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…

In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called…

We prove that, for every n, the topological space {\omega}_n^{\omega} (where {\omega}_n has the discrete topology) can be partitioned into {\omega}_n copies of the Baire space. Using this fact, the authors then prove two new theorems about…

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to…

In this article, I present a conjecture on the number of independent sets on graph covers. I also show that the conjecture implies that the partition function of a binary pairwise attractive model is greater than that of the Bethe…

In this article, we prove that any pair of doubly commuting $2$-isometries on a Hilbert space has a Wold-type decomposition. Moreover, the analytic part of the pair is unitary equivalent to the pair of multiplication by coordinate function…

For a Tychonoff space X, we denote by B(X) the space of all Baire functions on X with the topology of pointwise convergence. In this paper we investigate selectors for sequences of countable dense and countable sequentially dense sets of…

We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space $W^{k,1}(\mathbb{R}^d)$ as the difference of two non-negative functions in the same space with control of their norms.

We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every…

The author has previously shown that solvable Lie A-algebras and complemented solvable Lie algebras decompose as a vector space direct sum of abelian subalgebras, and their ideals relate nicely to this decomposition. However, neither of…

If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results…

In this paper we consider some properties of a space B(X) of Borel functions on a set of reals X, with pointwise topology, that are stronger than separability.

We give, for each level of complexity L, a Hurewicz-like characterization of the Borel subsets with countable sections of a product of two Polish spaces that cannot become in L by changing the two Polish topologies.

We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $\kappa^{<\kappa}=\kappa$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to…

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to…

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a…