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

Noncommutative Uncertainty Principles

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's $\lambda$-lattices, modular tensor categories etc.Comment: 41 pages, 71 figure

On decompositions and Connes\u27s embedding problem of finite von Neumann algebras

A longstanding open question of Connes asks whether every finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy Connes\u27s embedding property belong to just a few special classes (e.g. amenable algebras and free group factors). In this dissertation we establish Connes\u27s embedding property for von Neumann algebras satisfying Popa\u27s co-amenability condition. Some decomposition properties of finite von Neumann algebras are also investigated. Chapter 1 reviews von Neumann algebras, completely bounded mappings, conditional expectations, tensor products, crossed products, direct integrals, and Jones basic construction. Chapter 2 introduces new decompositions of finite von Neumann algebras which we call F-thin, strongly F-thin, and weakly F-thin, etc. We also consider the singly-generated problem, and compute the cohomology in such decompositions of finite von Neumann algebras. In Chapter 3 we show by estimation of free entropy that free group factors lack the type of decompositions discussed in Chapter 2

Dimensional change of wool fabrics in the process of a tumble-drying cycle

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Currently domestic tumble dryers are popularly used for drying garments; however, excessive drying and the inappropriate way of tumble agitation could waste energy and cause damage to or the dimensional change of garments. Shrinkage of wool fabrics during tumble drying causes a serious problem for wool garments. The current study investigated the shrinkage of untreated and Chlorine-Hercosett–finished wool fabrics at different drying times. Temperature of air in the tumble dryer, temperature of fabric, moisture content of fabric, and dimensional change at different drying times were measured. For the duration of the tumble drying, the rise of fabric temperature and the reduction of moisture content on the wool fabric were investigated to explore their relationship to the shrinkage of wool fabrics in the tumble-drying cycle. It was found that the tumble-drying process can be divided into different stages according to the temperature change trend of wool fabrics. The shrinkage mechanisms of the untreated and the treated fabrics were different. The dimensional change of untreated wool fabric was caused mainly by felting shrinkage during tumble drying. Chlorine-Hercosett–finished wool fabric can withstand the tumble-drying process without noticeable felting shrinkage due to the surface modification and resin coating of surface scales of wool fibers. The finding from the current research provides further understanding of the shrinkage behavior of wool fabrics during the tumble-drying process, leading to optimizing operational parameters at specific stages of a tumble-drying cycle