Linear and Non-Linear Waves
http://hdl.handle.net/20.500.11824/4
Wed, 25 May 2022 09:48:14 GMT2022-05-25T09:48:14ZOn the Hausdorff dimension of Riemann's non-differentiable function
http://hdl.handle.net/20.500.11824/1454
On the Hausdorff dimension of Riemann's non-differentiable function
Eceizabarrena, D.
Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14542021-01-01T00:00:00ZA pseudospectral method for the one-dimensional fractional Laplacian on R
http://hdl.handle.net/20.500.11824/1453
A pseudospectral method for the one-dimensional fractional Laplacian on R
Cayama, J.; Cuesta, C.M.; De la Hoz, F.
In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric series. Therefore, the central point of this paper is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we also do the simulation of Fisher's equation with fractional Laplacian in the monostable case.
Fri, 15 Jan 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14532021-01-15T00:00:00ZDiscrepancy of Minimal Riesz Energy Points
http://hdl.handle.net/20.500.11824/1451
Discrepancy of Minimal Riesz Energy Points
Marzo, J.; Mas, A.
We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz s-energy on the sphere Sd. Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in S2 were obtained. Our result improves previously known bounds for 0 ≤ s< 2 and s≠ 1 in S2, where s= 0 is Wolff’s result, and for d- t< s< d with t≈ 2.5 when d≥ 3 and s≠ d- 1.
Wed, 01 Dec 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14512021-12-01T00:00:00ZSelf-adjointness of two-dimensional Dirac operators on corner domains
http://hdl.handle.net/20.500.11824/1450
Self-adjointness of two-dimensional Dirac operators on corner domains
Pizzichillo, F.; Van Den Bosch, H.
We investigate the self-adjointness of the two-dimensional Dirac operator D, with quantum-dot and Lorentz-scalar i-shell boundary conditions, on piecewise C2 domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space H1=2, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of the adjoint operator D in terms of the domain of D and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14502021-01-01T00:00:00Z