Related papers: The Three Gap Theorem and Periodic Functions
In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition…
As a cornerstone of functional analysis, Hahn Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics,…
This paper consider the functional mechanics as one of modern approaches to a problem of the correspondence between classical mechanics and the statistical physics. Deviations from classical trajectories are calculated and evolution of the…
For a transcendental entire function, a partial affirmative answer to Baker's question on the boundedness of its Fatou components is given. In addition, we have addressed Wang's question on Fej\'er gaps. Certain results about functions with…
Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist…
We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function…
In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth,…
In the present paper the two and three point functions which occur at the study of the various physical processes are considered. The investigation has dan in the framework of the perturbative theory at the one loop level. The general and…
The scalar three-point function appearing in one-loop Feynman diagrams is compactly expressed in terms of a generalized hypergeometric function of two variables. Use is made of the connection between such Appell function and dilogarithms…
In measure theory, Steinhaus theorem is a result that deals with a property of the difference between two sets of positive measure. We give a simple elementary proof of the result.
In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a…
We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of…
In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We…
We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…
Based on the closed time path formalism, a new Feynman rule for directly calculating the retarded and advanced Green functions is deduced. This Feynman rule is used to calculate the two-point self-energy and three-point vertex correction in…
Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.
The theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich of results, is sometimes surprisingly different from the theory of…
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…
In this paper, a simple proof of the divergence theorem is given by using the Dirac operator and noncommutative residues. Then we extend the divergence theorem to compact manifolds with boundary by the noncommutative residue of the…
In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in…