Related papers: Bounded ranges of cardinal functions
The sequence space of all real-valued sequences, denoted $Seq(\mathbb{R})$, is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a…
This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.
A family of real functions {g_\alpha} defining a spectral regularization method with optimal qualification is considered. Sufficient condition on the family and on the optimal qualification guaranteeing the existence of saturation are…
Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give…
Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means…
We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large…
In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The…
Considering the $B$-branes over the complex projective space ${\mathbb P}^n$ as the objects of the bounded derived category $D^b({\mathbb P}^n)$, we prove that the cardinal of the set of holomorphic gauge fields on a given $B$-brane…
We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot…
Let $\lambda_{1},\ldots,\lambda_{n}$ be real numbers in $(0,1)$ and $p_{1},\ldots,p_{n}$ be points in $\mathbb{R}^{d}$. Consider the collection of maps $f_{j}:\mathbb{R}^{d}\to\mathbb{R}^{d} $ given by $$f_{j}(x)=\lambda_{j} x…
For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…
Rational functions of total degree $l$ in n variables have a representation in the Bernstein form defined over $n$ dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein…
A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of…
In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite…
Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the…
Let $f\in \mathbb{Q}(x)$ be a non-constant rational function. We consider "Waring's Problem for $f(x)$," i.e., whether every element of $\bbq$ can be written as a bounded sum of elements of $\{f(a)\mid a\in \mathbb{Q}\}$. For rational…
In this paper we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind, and is hence not overly technical. We consider answers to questions like: how…
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…