6,376 research outputs found
On a discrete version of Tanaka's theorem for maximal functions
In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for
the Hardy-Littlewood maximal operator in dimension , both in the
non-centered and centered cases. For the discrete non-centered maximal operator
we prove that, given a function
of bounded variation,
where represents the total variation of . For the discrete
centered maximal operator we prove that, given a function such that , This provides a positive solution to a question
of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.Comment: V4 - Proof of Lemma 3 update
Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
Montgomery's pair correlation conjecture predicts the asymptotic behavior of
the function defined to be the number of pairs and
of ordinates of nontrivial zeros of the Riemann zeta-function
satisfying and as . In this paper, assuming the Riemann hypothesis,
we prove upper and lower bounds for , for all , using
Montgomery's formula and some extremal functions of exponential type. These
functions are optimal in the sense that they majorize and minorize the
characteristic function of the interval in a way to minimize
the -error. We give a complete solution for this extremal problem
using the framework of reproducing kernel Hilbert spaces of entire functions.
This extends previous work by P. X. Gallagher in 1985, where the case was considered using non-extremal majorants and
minorants.Comment: to appear in J. Reine Angew. Mat
On Useful Conformal Tranformations In General Relativity
Local conformal transformations are known as a useful tool in various
applications of the gravitational theory, especially in cosmology. We describe
some new aspects of these transformations, in particular using them for
derivation of Einstein equations for the cosmological and Schwarzschild
metrics. Furthermore, the conformal transformation is applied for the
dimensional reduction of the Gauss-Bonnet topological invariant in to the
spaces of lower dimensions.Comment: 17 pages, LaTeX. The paper is intended mainly for pedagogical
purposes and represents a collection of exercises concerning local conformal
transformations and dimensional reduction. To be published in "Gravitation
and Cosmology
- …