Related papers: Virtual Calculus - Part II
We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified…
Numerical approaches to higher-order calculations often employ subtraction terms, both for the real emission and the virtual corrections. These subtraction terms have to be added back. In this paper we show that at NLO the real subtraction…
The derivatives with respect to order {\nu} for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series…
This paper presents a graded hierarchy or chain of binary operations on the reals and the complex numbers. The operations are related distributively in the sense that any one of them distributes over the next lower operation in the chain.…
This note replaces two earlier preprints (1101.3737 by Koll\'ar) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…
A quantum theoretic representation of real and complex numbers is described here as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are 4 types of qubits each with associated single qubit…
In an earlier paper we introduced the notion of 'bifurcating continued fractions' in a heuristic manner. In this paper a formal theory is developed for the 'bifurcating continued fractions'.
We propose definitions of regular and exact (virtual) double categories, proving a number of results which parallel many basic results in the theory of regular and exact categories. We show that any regular virtual double category admits a…
The variant of calculation of functions of set and their application is offered. In particular: the new measure of system of sets generalizing classical concept of a measure is entered; the variation of set that has allowed to construct a…
In the paper new representations are obtained for duals and dual hulls of the classes of analytic functions. The Ruscheweyh duality principle is shown to hold under somewhat weaker assumptions. For a compact class of functions its subclass…
The motivation of this paper is to construct the theory of vector calculus of multivariate arithmetical functions. We prove analogues of integral theorems and Poincare's lemma.
We introduce a natural class of functions, the {\em pseudomultipliers}, associated with a general Hilbert function space, prove an extension theorem which justifies the definition, give numerous examples and establish the nature of the…
Virtual double categories provide an effective framework for formal category theory. Recent work has investigated the question of higher morphisms between virtual double categories, following on from work on higher morphisms between double…
We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model…
We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…
We suggest a construction of virtual fundamental classes of certain types of moduli spaces.
We discuss an extension of classical combinatorics theory to the case of spatially distributed objects.
We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras, and describe their associated reproducing kernel spaces. The case of entire functions is of special interest,…
Operator $k$-tone functions on an open interval of the real line, which are higher order extensions of operator monotone and convex functions, are characterized via certain inequalities for the real and imaginary parts of analytic…