Related papers: Multivariate Davenport series
We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
Pointwise-supported generalized wavelets are introduced, based on Dirac, doublet and further derivatives of delta. A generalized biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as…
We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of…
We introduce a {\it non-regular} generalisation of the N\"{o}rlund mean, and show its equivalence with a certain moving average. The Abelian and Tauberian theorems establish relations with convergent sequences and certain power series. A…
The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of…
We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new…
Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper…
We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by…
We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the Von Mangoldt function is presented, as well as a criteria for the Riemann Hypothesis.
The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $\delta$ and Dirichlet series…
We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Lo\`eve decomposition, consists in a series of…
In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which…
We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to…
We study perturbations of linear differential equations, deriving explicit series solutions, using Dyson-type expansions. We analyze the monodromy of deformed solutions in a number of examples, and relate this to cocycles in a cohomological…
This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises…
We study Dirichlet series arising as linear functionals on an inner product space of meromorphic functions and establish a relation between the discontinuities of the former on the boundary and the poles and zeros of the latter on the…
A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is…