Related papers: Virtual Calculus - Part II
A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some…
We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the…
The Whittaker function and its diverse extensions have been actively investigated. Here we introduce an extension of the Whittaker function by using the known extended confluent hypergeometric function $\Phi_{p,v}$ and investigate some of…
Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total,…
We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…
In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that a certain class of non-integrable real functions can be represented…
We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of…
We show that the second central extension of the Galilei group in (2+1) dimensions corresponds to spin, which can be any real number.
Some aspects of analysis involving fields with absolute value functions are discussed, which includes the real or complex numbers with their standard absolute values, as well as ultrametric situations like the p-adic numbers.
Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
In this paper, we have studied first the idea of rough continuity of real valued functions of real variables and then we have discussed some important properties of rough continuity. Then we study the idea of rough $I$-continuity of real…
We discuss the history and uses of the parallel census technique---an elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel will discuss advances (including Cai, Naik, and…
We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real…
We classify finite dimensional division real associative $\mathcal{Z}_2$-algebras, introduce composition $\mathcal{Z}_2$-algebras, and extend the Campbell-Baker-Hausdorff series and Lie correspondence in the context of linear Hu-Liu Leibniz…
This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain…
The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number $n$ the first one is the proper subclass of the second one. To reach this aim new tools…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
In this article the notion of virtual double category (also known as fc-multicategory) is extended as follows. While cells in a virtual double category classically have a horizontal multi-source and single horizontal target, the notion of…