Related papers: Orders of $\pi$-bases

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…

We collect and organise known results and add some new ones of the following nature: if A is a bounded operator in a Hilbert or Banach space, does there exist a nonconstant polynomial p(z) such that p(A) is "simpler", "nicer" than A. The…

We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional chaotic maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree n which are defined…

Explicit bases for the spaces of holomorphic cusp forms of all even positive weights and all orders are constructed. The dimensions of these spaces are computed.

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several…

The Lyubeznik numbers are invariants of a local ring containing a field that capture ring-theoretic properties, but also have numerous connections to geometry and topology. We discuss basic properties of these integer-valued invariants, as…

We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory T we put two structures on the type spaces of models of T in two languages, \mathcal{L}…

We compare the results of our two papers with the results of the paper Aratyn H., Gomes J.F., Zimerman A.H., Higher order Painlev\'e equations and their symmetries via reductions of a class of integrable models, J. Phys. A: Math. Theor., V.…

We affirmatively solve the main problems posed by Laczkovich and Paulin in \emph{Stability constants in linear spaces}, Constructive Approximation 34 (2011) 89--106 (do there exist cases in which the second Whitney constant is finite while…

We consider the cardinal invariant $bd$ defined by M. D\v{z}amonja and I. Juh\'asz concerning bidiscrete systems. Using the relation between bidiscrete systems and irredundance for a compact Hausdorff space $K$, we prove that ${w(K)\leq…

We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the…

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and…

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that…

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than…

We generalize a theorem of E. Michael and M. E. Rudin and a theorem of D. Preiss and P. Simon; we give, as well, some partial answers to a recent question of A. V. Arhangel'ski\v{\i}.

In this paper, we first introduce the concept of biholomorphic convex mapping of order alpha on the unit ball B in a complex Hilbert space X. Next we provide some sufficient conditions that a locally biholomorphic mapping f is a…

We develop a theory of \emph{reduced} Gromov-Witten and stable pair invariants of surfaces and their canonical bundles. We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP…

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof…

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a…

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to $\mathbb{P}_p(T)$ (polynomial space with total degree $p$) that are orthogonal to the lower-order subspace $\mathbb{P}_n(T)$, $n\leq p$,…