Related papers: A note on rational numbers and certain operators
In this paper we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. In its own turn, the achieved generalization allows us to spread the obtained…
A new class of Semantic Numeration Systems, namely, positive rational Semantic Numeration Systems is introduced. For cardinal semantic operators, differences in the formation of carry (common carry) and remainders are defined. The…
This note aims to present novel positive linear operators involving the Wright function. Furthermore, the present research established the moments of these newly defined operators and estimated the convergence rate using the classical…
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on…
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.
In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation…
A new understanding of the notion of regularizer is proposed. It is argued that this new notion is more realistic than the old one and better fits the practical computational needs. An example of the regularizer in the new sense is given. A…
One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to…
In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators…
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and a combinatorial formula. Our main tool is…
The classical $q$-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give $q$-analogues of rational numbers. Some combinatorial interpretations are already known, namely as the rank generating functions for…
This paper is mostly a survey, with a few new results. The first part deals with functional equations for q-exponentials, q-binomials and q-logarithms in q-commuting variables and more generally under q-Heisenberg relations. The second part…
Using the recent theory of Krein--von Neumann extensions for positive functionals we present several simple criteria to decide whether a given positive functional on the full operator algebra is normal. We also characterize those…
In these notes we generalize the theory of graphical functions from scalar theories to theories with spin.
In this paper, we give some necessary and sufficient conditions for weighted conditional expectation type operators on L2 to be centered. Also, we investigate the relation between normal and centered weighted con- ditional type operators.…
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various…
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L…