Bounding the Compression Loss of the FGK Algorithm

An important issue related to coding schemes is their compression loss. A simple measure ε of the compression loss due to a coding scheme different than Huffman coding is defined by ε = A_C − A_H where A_H is the average code length of a static Huffman encoding and A_C is the average code length of an encoding based on the compression scheme C. When the scheme C is the FGK algorithm, Vitter conjectured that ε ≤ K for some real constant K. Here, we use an amortized analysis to prove this conjecture. We show that ε < 2. Furthermore, we show through an example that our bound is asymptotically tight. This result explains the good performance of FGK that many authors have observed through practical experiments.     

The Bunce-Deddens algebras as crossed products by partial automorphisms

We describe both the Bunce-DeddensC ^*-algebras and their Toeplitz versions, as crossed products of commutativeC ^*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers , fitted together in such a way that is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on . From this we deduce, by taking quotients, that the Bunce-DeddensC ^*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.     

Continuous Fell bundles associated to measurable twisted actions

Given a twisted action of a second-countable, locally compact group on a separable -algebra , we prove the existence of a topology on making it a Fell bundle, whose cross sectional -algebra is isomorphic to the Busby-Smith-Packer-Raeburn crossed product.

     

Partial actions of groups and actions of inverse semigroups

Given a group , we construct, in a canonical way, an inverse semigroup associated to . The actions of are shown to be in one-to-one correspondence with the partial actions of , both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, and have the same representation theory. We show that governs the subsemigroup of all closed linear subspaces of a -graded -algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A ``partial' version of the group -algebra of a discrete group is introduced. While the usual group -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group -algebra of the two commutative groups of order four, namely and , are not isomorphic.

     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

Fluoropolymer piezoelectrets with tubular channels: resonance behavior controlled by channel geometry

Ferro- or piezoelectrets are dielectric materials with two elastically very different macroscopic phases and electrically charged interfaces between them. One of the newer piezoelectret variants is a system of two fluoroethylenepropylene (FEP) films that are first laminated around a polytetrafluoroethylene (PTFE) template. Then, by removing the PTFE template, a two-layer FEP structure with open tubular channels is obtained. After electrical charging, the channels form easily deformable macroscopic electric dipoles whose changes under mechanical or electrical stress lead to significant direct or inverse piezoelectricity, respectively. Here, different PTFE templates are employed to generate channel geometries that vary in height or width. It is shown that the control of the channel geometry allows a direct adjustment of the resonance frequencies in the tubular-channel piezoelectrets. By combining several different channel widths in a single ferroelectret, it is possible to obtain multiple resonance peaks that may lead to a rather flat frequency-response region of the transducer material. A phenomenological relation between the resonance frequency and the geometrical parameters of a tubular channel is also presented. This relation may help to design piezoelectrets with a specific frequency response.     

The K-Theory of Cuntz–Krieger Algebras for Infinite Matrices

We compute the K-theory groups of the Cuntz–Krieger C ^*-algebra O _A associated to an infinite matrix A of zeros and ones.