AbstractThe Kerr–Schild–Kundt (KSK) metrics are known to be one of the universal metrics in general relativity, which means that they solve the vacuum field equations of any gravity theory constructed from the curvature tensor and its higher-order covariant derivatives. There is yet no complete proof that these metrics are universal in the presence of matter fields such as electromagnetic and/or scalar fields. In order to get some insight into what happens when we extend the “universality theorem” to the case in which the electromagnetic field is present, as a first step, we study the KSK class of metrics in the context of modified Horndeski theories with Maxwell’s field. We obtain exact solutions of these theories representing the pp-waves and AdS-plane waves in arbitrary D dimensions.
Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),⋯,ζ(s+ihrτ,F)) is proved. Here, h1,⋯,hr are algebraic numbers linearly independent over the field of rational numbers.
We prove explicit formulae for [Formula: see text]-points of [Formula: see text]-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function [Formula: see text] to the case of [Formula: see text]-points of the aforementioned [Formula: see text]-functions. This result implies the uniform distribution of subsequences of [Formula: see text]-points and from this a discrete universality theorem in the spirit of Voronin is derived.
In 2007, H. Mishou obtained a joint universality theorem for the Riemann and Hurwitz zeta-functions ζ(s) and ζ(s,α) with transcendental parameter α on the approximation of a pair of analytic functions by shifts (ζ(s+iτ),ζ(s+iτ,α)), τ R. In the paper, the Mishou theorem is generalized for the set of above shifts having a weighted positive lower density. Also, the case of a positive density is considered.
We represent the universal Menger curve as the topological realization [Formula: see text] of the projective Fraïssé limit [Formula: see text] of the class of all finite connected graphs. We show that [Formula: see text] satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only [Formula: see text]-dimensional topology and constructions on finite graphs. Using the topological realization [Formula: see text], we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson’s finite homogeneity theorem, and prove a variant of Anderson–Wilson’s theorem. The finite homogeneity theorem is the first instance of an “injective” homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.
It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iτ), τ ϵ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if τ take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when τ takes values from the arithmetic progression {kh : k ϵN} with arbitrary fixed h > 0. For this, two types of h are considered.
Kerr–Schild–Kundt metrics in generic gravity theories with modified Horndeski couplings
