Fat subsets of P kappa (lambda)

For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal

Relative Invariants, Ideal Classes and Quasi-Canonical Modules of Modular Rings of Invariants

We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings. We use an explicit classification of all reflexive rank one $R$-modules, which is given in terms of the class group of $R$, or in terms of $R$-semi-invariants. This result is implicitly contained in a paper of Nakajima (\cite{Nakajima:rel_inv}).Comment: 16 page

Geometric information in eight dimensions vs. quantum information

Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basis elements of geometric Clifford algebra for 3D Euclidean space as the states of a geometric byte in a given frame of reference. Two layers of information, available in real numbers, are distinguished. The first layer is a continuous one. It is used to identify spatial orientations of similar geometric objects in the same computational basis. The second layer is a binary one. It is used to manipulate with 8D structure elements inside the computational basis itself. An oriented unit cube representation, rather than a matrix one, is used to visualize an inner structure of basis multivectors. Both layers of information are used to describe unitary operations -- reflections and rotations -- in Euclidian and Hilbert spaces. The results are compared with ones for quantum gates. Some consequences for quantum and classical information technologies are discussed.Comment: 14 pages, presented at International Symposium "Quantum Informatics 2007", October 3rd - 5th, 2007, Moscow Zvenigorod, Russi

Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflectio

How can I use video to improve teacher engagement with my school’s abundant ICT equipment?

This paper examines how video can be used to enhance teacher engagement with ICT as a teaching tool. The Connect School Project, South Dublin County Council’s ubiquitous computing initiative, equipped St. Aidan’s with a vast amount of information and communication technologies (ICT). A review of appropriate literature indicated that the use of such technologies in class could lead to effective teaching and learning environments, increase student self-esteem and could bridge the ‘digital-divide’ that might exist between students in disadvantaged areas and those in more affluent communities. Through two cycles of enquiry, I planned and produced a video which modelled how the laptops and associated hard and software could be used effectively in class. The video aimed to represent all stakeholders in the school: management, teachers and students. In this way a top-down endorsement of the use of ICT as a teaching and learning tool was communicated. I feel that if teachers are to embrace the use of technology in class, pedagogical as well as technical support must be provided. Using video to model effective practice could be an ideal way to provide such support