Cohomology of Coxeter groups with group ring coefficients: II

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H_*(U) and, in the case where the action is proper and cocompact, for H^*_c(U).Comment: This is the version published by Algebraic & Geometric Topology on 15 September 200

Echoes of the sea around us—Human hopes in the balance

Earth’s human life-support system shows signs of failing. Human capacity to alter landscapes and the atmosphere is reaching catastrophic levels. Only the oceans seemed to be beyond control, but still they are not beyond human influence. Limited experience in protecting nature’s integrity, health, and resilience in seascapes offers the potential to reverse sliding global environmental conditions by providing realistic expectations, offering moral fortitude, stimulating imagination, and proffering hope. The ocean’s capacity to evoke human awe and inspiration may be sufficient to focusmankind on the global existential threats we face. It is now vital to heed Rachael Carson’s 1937 prescient observation “Against this cosmic background the lifetime of a particular plant or animal appears not as a drama complete in itself but only as a brief interlude in a panorama of endless change.” The world will keep spinning, whether people are able to enjoy the ride or not

Coefficients in powers of the log series

We determine the p-exponent in many of the coefficients in the power series (log(1+x)/x)^t, where t is any integer. In our proof, we introduce a variant of multinomial coefficients. We also characterize the power series x/log(1+x) by certain zero coefficients in its powers.Comment: 8 page

Weighted L2L^2-cohomology of Coxeter groups

Given a Coxeter system $(W,S)$ and a positive real multiparameter \bq, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(\Sigma)$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group.Comment: minor change

The ℓ2\ell^2-homology of even Coxeter groups

Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\ell^2$-homology of Sigma vanishes in all but the middle dimension.Comment: 15 pages, 1 figur

On asymptotic dimension of groups

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asymptotic dimension: asdim A *_C B < infinity. B) Suppose that G' is an HNN extension of a group G with asdim G < infinity. Then asdim G'< infinity. C) Suppose that \Gamma is Davis' group constructed from a group \pi with asdim\pi < infinity. Then asdim\Gamma < infinity.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-4.abs.htm

Cosmic D-Strings and Vortons in Supergravity

Recent developments in string inspired models of inflation suggest that D-strings are formed at the end of inflation. Within the supergravity model of D-strings there are 2(n-1) chiral fermion zero modes for a D-string of winding n. Using the bounds on the relic vorton density, we show that D-strings with winding number n>1 are more strongly constrained than cosmic strings arising in cosmological phase transitions. The D-string tension of such vortons, if they survive until the present, has to satisfy 8\pi G_N \mu \lesssim p 10^{-26} where p is the intercommutation probability. Similarly, D-strings coupled with spectator fermions carry currents and also need to respect the above bound. D-strings with n=1 do not carry currents and evade the bound. We discuss the coupling of D-strings to supersymmetry breaking. When a single U(1) gauge group is present, we show that there is an incompatibility between spontaneous supersymmetry breaking and cosmic D-strings. We propose an alternative mechanism for supersymmetry breaking, which includes an additional U(1), and might alleviate the problem. We conjecture what effect this would have on the fermion zero modes.Comment: 11 page

Distributional Analyses of Revenue Options for Alaska

A new report commissioned by Rasmuson Foundation as part of its Plan4Alaska campaign finds that while strategies currently proposed to close Alaska's $4 billion budget gap would significantly improve the state's fiscal standing, a diversified revenue strategy is needed this year to close the gap and equitably distribute financial impact. Rasmuson Foundation commissioned the report in response to comments from lawmakers about the dearth of economic data available to gauge the impact of various revenue scenarios. "Distributional Analyses of Revenue Options for Alaska" was produced by the Institute on Taxation and Economic Policy (ITEP), a nonprofit, non-partisan research organization with a mission to ensure that elected officials, the media, and the general public have access to accurate, timely, and straightforward information that allows them to understand the effects of current and proposed tax policies. ITEP used Gov. Bill Walker's Sustainable Alaska Plan in its analysis, and evaluated its proposed reductions to the Permanent Fund dividend, and income, alcohol, tobacco, and motor fuel tax increases to determine effects on Alaskans at different income levels. ITEP found that a fiscal plan that relied heavily on Permanent Fund earnings without income tax and other forms of taxation would disproportionately impact middle-income working families and low-income Alaskans. The report also examines a variety of options to derive more revenue from the income tax and less from reductions to the dividend. Among the alternative income tax structures examined are a doubling of the governor's proposed tax, the implementation of a more progressive income tax proposed by Rep. Paul Seaton in 2015, and the enactment of a 6.4 percent flat tax on incomes over$100,000 (or over $200,000 for married couples)

What Inspires Leisure Time Invention?

This paper seeks to understand the intriguing but only sparsely explored phenomenon of “leisure time invention,” where the main underlying idea for the new product or process occurs when the inventor is away from the workplace. We add to previous research by focussing on the inventive creativity of the individual researcher, and reassessing the image of researchers inventing during unpaid time – who have often been dispatched as “hobbyists”. Based on the responses from a survey of over 3,000 German inventors, we tested hypotheses on the conditions under which leisure time invention is likely to arise. Results suggest that the incidence of leisure time invention is positively related to exposure to a variety of knowledge inputs – but, surprisingly, not to the quality of prior inventive output. Leisure time inventions are more frequently observed in conceptual-based technologies than in science-based technologies, in smaller R&D projects, and in externally financed R&D projects