Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results corrected, in particular added 5.2.3-5.2.

The Correlation Between Parental Involvement and Student Academic Achievement

This study investigates whether or not there is a correlation between parental involvement and student academic achievement. A sample of 103 students at Madison Preparatory Academy and CSAL Middle School (CSAL, Inc) were tested for correlations between the degree with which their parents are engaged in their academic lives and the success that they achieve as a result. Small correlations were found to exist between some of the variables tested. The test group consisted of a group of high school students from Madison Preparatory Academy (MPA) and their siblings who attended CSAL Middle which is the feeder school for MPA. The control group consisted of the remaining middle school students, whose parents volunteered for the study. It was assumed that because the demographic make-up of the two schools was similar that, untreated, their results would be similar as well. The high school parents received a handbook, which gave them tips for improving academic achievement, whereas the parents of the middle school did not receive the handbook. The analysis of the data showed a correlation in several categories for both groups. A correlation existed between student’s views of the assistance that they received from their parents and their level of confidence in their parents’ abilities to assist them. There was also a correlation between the parents’ self-efficacy and the amount of assistance they offered. I conclude that, parents who are more involved in the academic lives of their children have students who have a higher probability of being successful academically, as a result of that additional assistance

A preliminary checklist of the benthic marine macroalgae of Rotuma

To the author' s knowledge, no phycological investigation of Rotuma took place prior to the present research, although the nearby Fijian algal flora has been fairly well studied (vide South and Kasahara, 1992). Hence this present checklist represents the first record of the Rotuman algal flora, which is dealt with in more detail by N'Yeurt (1993), South and N'Yeurt (1992) and South et al. (1993)

Working 6m from Rotuma the ‘Island Brew’ Way

Amateur Radio Operator Antoine De Ramon N’Yeurt (3D2AG) from Fiji relates a tale of make do to allow him to operate on the VHF 6m band while on the remote Pacific island of Rotuma, using locally-available materials. A fully-functional 5-element yagi antenna was constructed on the island using wood, domestic metal parts and accessory cables and connectors. This allowed some 1,000 radio contacts to be made on the 50 MHz Amateur band using Sporadic-E (Es), Trans-Equatorial Propagation (TEP) and F2 layer ionospheric propagation to countries as varied as Australia, New Zealand, South America, Japan and North America

An investigation into the biology and spawning maturity of Anadara Spp. (Bivalvia : Arcidae) in Fiji

The anatomy and histology of the Fijian Anadare mollusc are described. Its structure is similar to other lamellibranchs of the family Arcidae. The gender determination of live specimens was deemed impossible, due to the inaccessibility of the gonads. Peculiar white patches were found on the mantle of all specimens, suggesting a reaction to disease or pollution which warrants further investigation

DP-SGD with weight clipping

Recently, due to the popularity of deep neural networks and other methods whose training typically relies on the optimization of an objective function, and due to concerns for data privacy, there is a lot of interest in differentially private gradient descent methods. To achieve differential privacy guarantees with a minimum amount of noise, it is important to be able to bound precisely the sensitivity of the information which the participants will observe. In this study, we present a novel approach that mitigates the bias arising from traditional gradient clipping. By leveraging public information concerning the current global model and its location within the search domain, we can achieve improved gradient bounds, leading to enhanced sensitivity determinations and refined noise level adjustments. We extend the state of the art algorithms, present improved differential privacy guarantees requiring less noise and present an empirical evaluation