Related papers: Distributive properties of the rationals
We investigate the additive theory of the set $S = \{1^c, 2^c, \dots, N^c\}$ when $c$ is a real number. In the language of additive combinatorics, we determine the asymptotic behaviour of the additive energy of $S$. When $c$ is rational,…
The $(q,r)$-Whitney numbers were recently defined in terms of the $q$-Boson operators, and several combinatorial properties which appear to be $q$-analogues of similar properties were studied. In this paper, we obtain elementary and…
Let $m$ and $n$ be positive integers. For the quantum integer $[n]_q = 1 + q + ... + q^{n-1}$ there is a natural polynomial addition such that $[m]_q \oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that $[m]_q…
One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric…
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm…
In the setting of distributions taking values in a $C^\ast$-algebra $\mathcal{B}$, we define generalized Jacobi parameters and study distributions they generate. These include numerous known examples and one new family, of…
Given the family $P$ of all nonempty subsets of a set $U$ of alternatives, a choice over $U$ is a function $c \colon \Omega \to P$ such that $\Omega \subseteq P$ and $c(B) \subseteq B$ for all menus $B \in \Omega$. A choice is total if…
We give a classification of {\texttt{e.a.b.}} semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
Normalisation in probability theory turns a subdistribution into a proper distribution. It is a partial operation, since it is undefined for the zero subdistribution. This partiality makes it hard to reason equationally about normalisation.…
Selective rationalization has become a common mechanism to ensure that predictive models reveal how they use any available features. The selection may be soft or hard, and identifies a subset of input features relevant for prediction. The…
We reduce the set of classic relational algebra operators to two binary operations: natural join and generalized union. We further demonstrate that this set of operators is relationally complete and honors lattice axioms.
The subject of this thesis are various properties of quantum states that make them "non-classical" and their behaviour under unitary operations. In chapter 2 some basic concepts of quantum mechanics and quantum information are reviewed. In…
A certain analysis of all possible associative binary operations on N is presented. This is equivalent with an analysis of all possible monoid structures on N. Several results and a conjecture in this regard are given.
When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to…
A new algebra, hitherto not encountered in the usual Lie algebraic varieties or supervarieties, is introduced. The paper explores the rich and novel structure of the algebra, and it compares it on the one hand with the Jordan-Lie…
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
An important global property of a bit string is the number of ones in it. It has been found that the parity (odd or even) of this number can be found by a sequence of deterministic, translational invariant cellular automata with parallel…