759,918 research outputs found
Fusion Bialgebras and Fourier Analysis
We introduce fusion bialgebras and their duals and systematically study their
Fourier analysis. As an application, we discover new efficient analytic
obstructions on the unitary categorification of fusion rings. We prove the
Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and
their duals. We show that the Schur product property, Young's inequality and
the sum-set estimate hold for fusion bialgebras, but not always on their duals.
If the fusion ring is the Grothendieck ring of a unitary fusion category, then
these inequalities hold on the duals. Therefore, these inequalities are
analytic obstructions of categorification. We classify simple integral fusion
rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less
than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be
eliminated by applying the Schur product property on the dual. In general,
these inequalities are obstructions to subfactorize fusion bialgebras.Comment: 39 pages; 8 figures; addition of a classification in Subsection 9.2;
the long lists in Subsection 9.3 are now more pleasant to read; addition of
Section 7 providing a categorical proof of Schur Product Theore
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and
representation theory, we give a unified and very accessible approach to solve
three important functional equations -- the d'Alembert equation, the Wilson
equation, and the d'Alembert long equation, on compact groups.Comment: 8 pages, to appear in CM
Discrete Fourier analysis of multigrid algorithms
The main topic of this report is a detailed discussion of the discrete Fourier multilevel analysis of multigrid algorithms. First, a brief overview of multigrid methods is given for discretizations of both linear and nonlinear partial differential equations. Special attention is given to the hp-Multigrid as Smoother algorithm, which is a new algorithm suitable for higher order accurate discontinuous Galerkin discretizations of advection dominated flows. In order to analyze the performance of the multigrid algorithms the error transformation operator for several linear multigrid algorithms are derived. The operator norm and spectral radius of the multigrid error transformation are then computed using discrete Fourier analysis. First, the main operations in the discrete Fourier analysis are defined, including the aliasing of modes. Next, the Fourier symbol of the multigrid operators is computed and used to obtain the Fourier symbol of the multigrid error transformation operator. In the multilevel analysis, two and three level h-multigrid, both for uniformly and semi-coarsened meshes, are considered, and also the analysis of the hp-Multigrid as Smoother algorithm for three polynomial levels and three uniformly and semi-coarsened meshes. The report concludes with a discussion of the multigrid operator norm and spectral radius. In the appendix some useful auxiliary results are summarized
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