Homology of Distributive Lattices

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology of a finite distributive lattice. We also indicate the answer for unital spindles and conjecture the general formula for semi-lattices and some skew lattices. Then we propose a generalization of a lattice as a set with a number of idempotent operations satisfying the absorption law.Comment: 30 pages, 3 tables, 3 figure

A model to estimate the lifetime health outcomes of patients with Type 2 diabetes: the United Kingdom Prospective Diabetes Study (UKPDS) Outcomes Model (UKPDS no. 68)

<i>Aims/hypothesis</i> The aim of this study was to develop a simulation model for Type 2 diabetes that can be used to estimate the likely occurrence of major diabetes-related complications over a lifetime, in order to calculate health economic outcomes such as quality-adjusted life expectancy. <i>Methods</i> Equations for forecasting the occurrence of seven diabetes-related complications and death were estimated using data on 3642 patients from the United Kingdom Prospective Diabetes Study (UKPDS). After examining the internal validity, the UKPDS Outcomes Model was used to simulate the mean difference in expected quality-adjusted life years between the UKPDS regimens of intensive and conventional blood glucose control. <i>Results</i> The models forecasts fell within the 95% confidence interval for the occurrence of observed events during the UKPDS follow-up period. When the model was used to simulate event history over patients lifetimes, those treated with a regimen of conventional glucose control could expect 16.35 undiscounted quality-adjusted life years, and those receiving treatment with intensive glucose control could expect 16.62 quality-adjusted life years, a difference of 0.27 (95% CI: –0.48 to 1.03). <i>Conclusions/interpretations</i> The UKPDS Outcomes Model is able to simulate event histories that closely match observed outcomes in the UKPDS and that can be extrapolated over patients lifetimes. Its validity in estimating outcomes in other groups of patients, however, remains to be evaluated. The model allows simulation of a range of long-term outcomes, which should assist in informing future economic evaluations of interventions in Type 2 diabetes

Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

An assessment of western North Pacific ozone photochemistry based on springtime observations from NASA's PEM-West B (1994) and TRACE-P (2001) field studies

The current study provides a comparison of the photochemical environments for two NASA field studies focused on the western North Pacific (PEM-West-B (PWB) and TRACE-P (TP)). These two studies were separated in calendar time by approximately 7 years. Both studies were carried out under springtime conditions, with PWB being launched in 1994 and TP being deployed in 2001 (i.e., 23 February - 15 March 1994 and 10 March-15 April 2001, respectively). Because of the 7-year time separation, these two studies presented a unique scientific opportunity to assess whether evidence could be found to support the Department of Energy\u27s projections in 1997 that increases in anthropogenic emissions from East Asia could reach 5%/yr. Such projections would lead one to the conclusion that a significant shift in the atmospheric photochemical properties of the western North Pacific would occur. To the contrary, the findings from this study support the most recent emission inventory data [Streets et al., 2003] in that they show no significant systematic trend involving increases in any O3 precursor species and no evidence for a significant shift in the level of photochemical activity over the western North Pacific. This conclusion was reached in spite of there being real differences in the concentration levels of some species as well as differences in photochemical activity between PWB and TP. However, nearly all of these differences were shown to be a result of a near 3-week shift in TP\u27s sampling window relative to PWB, thus placing it later in the spring season. The photochemical enhancements seen during TP were most noticeable for latitudes in the range of 25-45°N. Most important among these were increases in J(O1D), OH, and HO2 and values for photochemical ozone formation and destruction, all of which were typically two times larger than those calculated for PWB. A comparison of these airborne results with ozonesonde data from four Japanese stations provided further evidence showing that the 3-week shift in the respective sampling windows of PWB and TP was a likely cause for the differences seen in O3 levels and in photochemical activity between the two airborne studies. Copyright 2003 by the American Geophysical Union

Taxonomies of networks from community structure

The study of networks has become a substantial interdisciplinary endeavor that encompasses myriad disciplines in the natural, social, and information sciences. Here we introduce a framework for constructing taxonomies of networks based on their structural similarities. These networks can arise from any of numerous sources: They can be empirical or synthetic, they can arise from multiple realizations of a single process (either empirical or synthetic), they can represent entirely different systems in different disciplines, etc. Because mesoscopic properties of networks are hypothesized to be important for network function, we base our comparisons on summaries of network community structures. Although we use a specific method for uncovering network communities, much of the introduced framework is independent of that choice. After introducing the framework, we apply it to construct a taxonomy for 746 networks and demonstrate that our approach usefully identifies similar networks. We also construct taxonomies within individual categories of networks, and we thereby expose nontrivial structure. For example, we create taxonomies for similarity networks constructed from both political voting data and financial data. We also construct network taxonomies to compare the social structures of 100 Facebook networks and the growth structures produced by different types of fungi

Quasi-Quantum Groups, Knots, Three-Manifolds, and Topological Field Theory

We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group $G$, and a 3-cocycle \om, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data G, \,\om.Comment: 30 page