Approximation by Shifts of Compositions of Dirichlet L-Functions with the Gram Function

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions L ( s + i a 1 t τ , χ 1 ) , … , L ( s + i a r t τ , χ r ) , where a 1 , … , a r are non-zero real algebraic numbers linearly independent over the field Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.

On joint approximation of analytic functions by nonlinear shifts of zeta-functions of certain cusp forms

In the paper, joint discrete universality theorems on the simultaneous approximation of a collection of analytic functions by a collection of discrete shifts of zeta-functions attached to normalized Hecke-eigen cusp forms are obtained. These shifts are defined by means of nonlinear differentiable functions that satisfy certain growth conditions, and their combination on positive integers is uniformly distributed modulo 1.

Joint approximation of analytic functions by shifts of the Riemann and periodic Hurwitz zeta-functions

We present some new results on the simultaneous approximation with given accuracy, uniformly on compact subsets of the critical strip, of a collection of analytic functions by discrete shifts of the Riemann and periodic Hurwitz zeta-functions. We prove that the set of such shifts has a positive lower density. For this, we apply the linear independence over the field of rational numbers of certain sets related to the zeta-functions.

A method for creating interactive, user-resembling avatars

Virtual reality (VR) applications have disseminated throughout several fields, with a special quest for immersion. The avatar is one of the key constituents of immersive applications, and avatar resemblance can provoke diverse emotional responses from the user. Yet a lot a virtual reality systems struggle to implement real life-like avatars. In this work, we propose a novel method for creating interactive, user-resembling avatars using available commercial hardware and software. Avatar visualization is possible with a point-cloud or a contiguous polygon surface, and avatar interactions with the virtual scenario happens through a body joint-approximation for contact. In addition, the implementation could be easily extended to other systems and its modular architecture admits improvement both on visualization and physical interactions. The code is under Apache License 2.0 and is freely available as Supplemental Information 1 in this article.

The joint distribution of periodic zeta-functions

In this paper, the joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions with periodic coefficients is obtained. This is applied to prove the functional independence for these zeta-functions.