Non-linear Group Actions with Polynomial Invariant Rings and a Structure Theorem for Modular Galois Extensions

Let $G$ be a finite $p$-group and $k$ a field of characteristic $p>0$. We show that $G$ has a \emph{non-linear} faithful action on a polynomial ring $U$ of dimension $n=\mathrm{log}_p(|G|)$ such that the invariant ring $U^G$ is also polynomial. This contrasts with the case of \emph{linear and graded} group actions with polynomial rings of invariants, where the classical theorem of Chevalley-Shephard-Todd and Serre requires $G$ to be generated by pseudo-reflections. Our result is part of a general theory of "trace surjective $G$-algebras", which, in the case of $p$-groups, coincide with the Galois ring-extensions in the sense of \cite{chr}. We consider the \emph{dehomogenized symmetric algebra} $D_k$, a polynomial ring with non-linear $G$-action, containing $U$ as a retract and we show that $D_k^G$ is a polynomial ring. Thus $U$ turns out to be \emph{universal} in the sense that every trace surjective $G$-algebra can be constructed from $U$ by "forming quotients and extending invariants". As a consequence we obtain a general structure theorem for Galois-extensions with given $p$-group as Galois group and any prescribed commutative $k$-algebra $R$ as invariant ring. This is a generalization of the Artin-Schreier-Witt theory of modular Galois field extensions of degree $p^s$.Comment: 20 page

Growth of Black Holes in the interior of Rotating Neutron Stars

Mini-black holes made of dark matter that can potentially form in the interior of neutron stars have been always thought to grow by accreting the matter of the core of the star via a spherical Bondi accretion. However, neutron stars have sometimes significant angulal velocities that can in principle stall the spherical accretion and potentially change the conclusions derived about the time it takes for black holes to destroy a star. We study the effect of the star rotation on the growth of such black holes and the evolution of the black hole spin. Assuming no mechanisms of angular momentum evacuation, we find that even moderate rotation rates can in fact destroy spherical accretion at the early stages of the black hole growth. However, we demonstrate that the viscosity of nuclear matter can alleviate the effect of rotation, making it possible for the black hole to maintain spherical accretion while impeding the black hole from becoming maximally rotating.Comment: 9 page

10 Reasons to Oppose Virginia Sales Tax Increases

Northern and southeastern Virginians will vote in referenda this November to approve or reject increases in the retail sales tax to fund transportation projects. Northern Virginians will decide whether to increase the sales tax from 4.5 percent to 5.0 percent, an 11 percent increase. Virginians in the Hampton Roads area will decide whether to increase the sales tax from 4.5 percent to 5.5 percent, a 22 percent increase. Proponents of tax increases point to unmet transportation needs to support their cause. Yet state spending increased 13 percent in 1999, 7 percent in 2000, and 9 percent in 2001. If key transportation needs have not been met, the problem is not a lack of funds but legislators who have not properly prioritized the budget. If the sales tax referenda are passed, the state government will have a strong incentive to reduce what it would otherwise spend on transportation in northern Virginia and Hampton Roads. By some measures, northern Virginia already gets the short end of the stick with regard to the state budget. Tax increases are not just bad budget policy; they are also bad economic policy. Since higher taxes reduce economic growth, an added cost of higher sales taxes would be lower incomes for Virginians. During the 1990s Virginia taxes grew faster than incomes, and local property taxes have soared recently. Even modest restraint in nontransportation spending could save enough money to fund priority highway projects without tax increases. Further, the state could adopt a spending growth cap that channels excess future tax revenues to transportation needs and tax cuts

The implementation of a multimedia learning environment for graduate civil engineers

This paper examines the underpinning theory, design and implementation of a computer‐based Multimedia Learning Environment (MLE) for graduate civil engineers. The MLE brings together multimedia technology and Intelligent Tutoring Systems techniques for the purpose of developing in the graduate engineer the skills and understanding needed to produce initial design proposals for real‐world dam spillway design problems

Earth's middle age

This research was funded through Natural Environment Research Council (grant NE/J021822/1) and the APC was paid through the RCUK OA block grant.Earth's middle age, extending from 1.7 to 0.75 Ga, was characterized by environmental, evolutionary, and lithospheric stability that contrasts with the dramatic changes in preceding and succeeding eras. The period is marked by a paucity of preserved passive margins, an absence of a significant Sr anomaly in the paleoseawater record and in the εHf(t) in detrital zircon, a lack of orogenic gold and volcanic-hosted massive sulfide deposits, and an absence of glacial deposits and iron formations. In contrast, anorthosites and kindred bodies are well developed and major pulses of Mo and Cu mineralization, including the world's largest examples of these deposits, are features of this period. These trends are attributed to a relatively stable continental assemblage that was initiated during assembly of the Nuna supercontinent by ca. 1.7 Ga and continued until breakup of its closely related successor, Rodinia, ca. 0.75 Ga. The overall low abundance of passive margins is consistent with a stable continental configuration, which also provided a framework for environmental and evolutionary stability. A series of convergent margin accretionary orogens developed along the edge of the supercontinent. Abundant anorthosites and related rocks developed inboard of the plate margin. Their temporal distribution appears to link with the secular cooling of the mantle, at which time the overlying continental lithosphere was strong enough to be thickened and to support the emplacement of large plutons into the crust, yet the underlying mantle was still warm enough to result in widespread melting of the lower thickened crust.Publisher PDFPeer reviewe

Poor-Inclusive Urban Sanitation: An Overview

This paper provides an overview of urban sanitation while highlighting the need to address this challenge with emphasis on including slum dwellers and poor communities that have typically been neglected

Networked learning in higher education: Practitioners’ perspectives

There is a growing use of a variety of communications media to provide networked learning in higher education. The practitioners in the field vary from experienced educators who have many years’ experience to early adopters who have begun to use networked technology for teaching and learning recently. Using interviews informed by a phenomenographic approach, this paper investigates the varieties of experience of practitioners of networked learning. It reports initial findings that represent an early stage of analysis. The findings point towards a common philosophy held by current practitioners of networked learning but a lack of ‘rules of thumb’. Practitioners expressed ideas close to a new paradigm in education but were cautious about specific design outcomes meeting expectations. This finding raises questions about design and whether networked learning is yet stable enough a field to provide guidance on best practice. The paper also reflects on criticisms of the phenomenographic method, in particular its reliance on interview data, and offers some possible ways of dealing with the criticisms