Related papers: Integrals for functions with values in a partially…
In courses on integration theory, Chasles property is usually considered as elementary and so "natural" that this is sometimes left to the reader. When the functions take their values in finite dimensional spaces, the property is always…
$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring…
We consider multi-polylogarithm functions which are slightly different from the ordinary ones. These functions have two integral representations and an order structure similar to those of multiple zeta star values. We also give a necessary…
Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…
Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $\phi$ on a lattice $L$ which are modular, i.e.,…
We prove that continuous reducibility is a well-quasi-order on the class of continuous functions between separable metrizable spaces with analytic zero-dimensional domain. To achieve this, we define scattered functions, which generalize…
In this paper, we introduce a method of converting implicit equations to the usual forms of functions locally without differentiability. For a system of implicit equations which are equipped with continuous functions, if there are unique…
We develop integration theory for integrating functions taking values into a Dedekind complete unital $f$-algebra $\mathbb{L}$ with respect to $\mathbb{L}$-valued measures. We then discuss and prove completeness results of…
In this article, we study spectral Barron spaces whose elements are made up of some vector-valued functions on a compact group whose Fourier transforms admit a certain summability property. We investigate their functional properties and…
The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em…
In this paper, new classes of functions are defined. These spaces generalize Morrey spaces and give a refinement of Lebesgue spaces. Some embeddings between these new classes are also proved. Finally, the authors apply these classes of…
In this work we introduce the spaces $L^p_{w}(\nu; X; Y)$ for the case where $\nu$ is a vector measure and the functions are vector-valued. We establish fundamental properties for such spaces.
I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…
In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case.…
In the present paper we extend the multiplicative integral to complex-valued functions of complex variable. The main difficulty in this way, that is the multi-valued nature of the complex logarithm, is avoided by division of the interval of…
We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…
We consider the problem of numerically integrating functions with hyperplane discontinuities over the entire Euclidean space in many dimensions. We describe a simple process through which the Euclidean space is partitioned into simplices on…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…