Coinvariants and the regular representation of a cyclic P-group

Cataloged from PDF version of article.We consider an indecomposable representation of a cyclic p-group Zpr over a field of characteristic p. We show that the top degree of the corresponding ring of coinvariants is less than. This bound also applies to the degrees of the generators for the invariant ring of the regular representation. © 2012 Springer-Verlag

Explicit separating invariants for cyclic p-groups

Cataloged from PDF version of article.We consider a finite-dimensional indecomposable modular representation of a cyclic p-group and we give a recursive description of an associated separating set: We show that a separating set for a representation can be obtained by adding, to a separating set for any subrepresentation, some explicitly defined invariant polynomials. Meanwhile, an explicit generating set for the invariant ring is known only in a handful of cases for these representations. © 2010 Elsevier Inc. All rights reserved

Locally finite derivations and modular coinvariants

We consider a finite dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic this yields that the algebra k[V]_G of coinvari-ants is a free module over its subalgebra generated by kG-module generators of V^∗ . This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank

Constructing modular separating invariants

Cataloged from PDF version of article.We consider a finite dimensional modular representation V of a cyclic group of prime order p. We show that two points in V that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation. Simultaneously, we describe an explicit construction for a separating set consisting of polynomials with these properties. (C) 2009 Elsevier Inc. All rights reserved

Degree bounds for modular covariants

Let V,W be representations of a cyclic group G of prime order p over a field k of characteristic p. The module of covariants k[V,W]^G is the set of G-equivariant polynomial maps from V to W, and is a module over the algebra of invariants k[V]^G. We give a formula for the Noether bound of k[V,W]^G over k[V]^G, i.e. the minimal degree d such that k[V,W]^G is generated over k[V]^G by elements of degree at most d

Separating Invariants for the Klein Four Group and Cyclic Groups

Cataloged from PDF version of article.We consider indecomposable representations of the Klein four group over a field of characteristic 2 and of a cyclic group of order pm with p, m coprime over a field of characteristic p. For each representation, we explicitly describe a separating set in the corresponding ring of invariants. Our construction is recursive and the separating sets we obtain consist of almost entirely orbit sums and products. © 2013 World Scientific Publishing Compan

On the top degree of coinvariants

Cataloged from PDF version of article.For a finite group G acting faithfully on a finite-dimensional F-vector space V, we show that in the modular case, the top degree of the vector coinvariants grows unboundedly: lim(m ->infinity) topdeg F[V-m](G) = infinity. In contrast, in the nonmodular case we identify a situation where the top degree of the vector coinvariants remains constant. Furthermore, we present a more elementary proof of Steinberg's theorem which says that the group order is a lower bound for the dimension of the coinvariants which is sharp if and only if the invariant ring is polynomial

Associative memory design using overlapping decompositions

Cataloged from PDF version of article.This paper discusses the use of decomposition techniques in the design of associative memories via arti"cial neural networks. In particular, a disjoint decomposition which allows an independent design of lower-dimensional subnetworks and an overlapping decomposition which allows subnetworks to share common parts, are analyzed. It is shown by a simple example that overlapping decompositions may help in certain cases where design by disjoint decompositions fails. With this motivation, an algorithm is provided to synthesize neural networks using the concept of overlapping decompositions. Applications of the proposed design procedure to a benchmark example from the literature and to a pattern recognition problem indicate that it may improve the e!ectiveness of the existing methods. ( 2001 Published by Elsevier Science Ltd

Separating invariants for arbitrary linear actions of the additive group

We consider an arbitrary representation of the additive group G_a over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants

Robust adaptive sampled-data control of a class of systems under structured nonlinear perturbations

Cataloged from PDF version of article.A robust adaptive sampled-data feedback stabilization scheme is presented for a class of systems with nonlinear additive perturbations. The proposed controller generates a control input by using high-gain static or dynamic feedback from nonuniform sampled values of the output. A simple adaptation rule adjusts the gain and the sampling period of the controller