Related papers: Multivariate arithmetical functions and vector cal…
We outline an algorithm for construction of functional bases of absolute invariants under the rotation group for sets of rank 2 tensors and vectors in the Euclidean space of arbitrary dimension. We will use our earlier results for symmetric…
We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…
We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].
Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals,…
Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…
We introduce the notion of bilinear moment functional and study their general properties. The analogue of Favard's theorem for moment functionals is proven. The notion of semi-classical bilinear functionals is introduced as a generalization…
A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different…
Dynamic arrays, also referred to as vectors, are fundamental data structures used in many programs. Modeling their semantics efficiently is crucial when reasoning about such programs. The theory of arrays is widely supported but is not…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
In this research article, we formulate and prove multidimensional Widder--Arendt theorem and integrated form of multidimensional Widder--Arendt theorem for functions with values in sequentially complete locally convex spaces. Established…
Some formal analogies between the Differential Calculus in One Variable and the Differential Calculus in Several Variables are presented. It is studied and introduced the derivability of functions at several variables from the single…
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions…
This work considers the algebras of functions in the quantum matrix ball. An explicit formula for a positive invariant integral is presented.
In the paper a theorem of Piccard's type is proved and, consequently, the continuity of $\mathcal{D}$-measurable polynomial functions of $n$-th order as well as $\mathcal{D}$-measurable $n$-convex functions is shown. The paper refers to the…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…
This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and…
We consider certain scalar product of symmetric functions which is parameterized by a function $r$ and an integer $n$. One the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of…