Constructing modular separating invariants

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.     

Separating invariants and local cohomology

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.     

The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree

We study the q-state Potts model on a Cayley tree of order k ≥ 2. In the group representation of the Cayley tree for the ferromagnetic Potts model, we single out a set of index-2 subgroups under which each weakly periodic Gibbs measure is translation invariant. For the anti-ferromagnetic Potts model with k ≥ 2 and q ≥ 2, we show that a weakly periodic Gibbs measure that is not translation invariant is not unique.     

Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.     

Gibbs <Emphasis Type="Italic">u</Emphasis>-states for the foliated geodesic flow and transverse invariant measures

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow. p]On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. p]On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ ^u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions.     

Nielsen classes of orientation reversing homeomorphisms and complementary domains of plane separating invariant continua

Let h be an orientation reversing planar homeomorphism and X be an invariant plane separating continuum. We show that there is a natural linear order on the invariant components of R²?X that resemble the one found in connected unions of circles invariant under the reflection r(x,y)=(−x,y). The main result relates to the Nielsen fixed point theory and work of Krystyna Kuperberg on fixed points of planar homeomorphisms in invariant continua.     

On a Class of Graded Ideals of Semigroup <Emphasis Type="Italic">C</Emphasis>*-Algebras

We present general results about graded C*-algebras and continue the previously initiated research of the C*-algebra generated by the left regular representation of an abelian semigroup. We study the invariant ideals of this C*-algebra invariant with respect to the representation of a compact group G in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded C*-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.     

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.     

Irregular Partially Invariant Solutions of Rank 2 and Defect 1 to Equations of Gas Dynamics

We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels.     

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

Galois ring extensions and localized modular rings of invariants of p-groups

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].     

Wythoff partizan subtraction

We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation.     

On the structure of invariant attracting sets of systems of finite-difference equations generating Hénon-Lozi mappings

For Hénon-Lozi mappings F, we find sufficient conditions under which on the plane there exists a domain U such that its closure is mapped by F strictly inside U. This ensures the existence of a compact invariant set in U. We prove the existence of an open set of parameter values for which this invariant set contains a zero-dimensional locally maximal topologically transitive Markov set such that the restriction of the mapping to this set is topologically conjugate to the shift automorphism in the space of sequences of two symbols. We show that if this Markov set is hyperbolic, then the above-mentioned compact invariant set coincides with the closure of the unstable manifold of F at a fixed point lying in that set and is a topologically indecomposable one-dimensional continuum. We present the parameter values for which these results hold for the Hénon mapping. We thereby prove the existence of a parameter range in which the invariant set of the Hénon mapping is a one-dimensional topologically indecomposable Brauer-Janiszewski continuum that contains a zero-dimensional locally maximal set and lies in the attraction domain of itself.     

A Finite Separating Set for Daigle and Freudenburg's Counterexample to Hilbert's Fourteenth Problem

This article gives the first explicit example of a finite separating set in an invariant ring which is not finitely generated, namely, for Daigle and Freudenburg's 5-dimensional counterexample to Hilbert's Fourteenth Problem.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

Characterization and computation of control invariant sets for linear impulsive control systems

Impulsive control systems are suitable to describe and control a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they evolve freely in-between impulsive actions, which makes it difficult to guarantee its permanence in a given state-space region. In this work, we develop a method for characterizing and computing approximations to the maximal control invariant sets for linear impulsive control systems, which can be explicitly used to formulate a set-based model predictive controller. We approach this task using a tractable and non-conservative characterization of the admissible state sets, namely the states whose free response remains within given constraints, emerging from a spectrahedron representation of such sets for systems with rational eigenvalues. The so-obtained impulsive control invariant set is then explicitly used as a terminal set of a predictive controller, which guarantees the feasibly asymptotic convergence to a target set containing the invariant set. Necessary conditions under which an arbitrary target set contains an impulsive control invariant set (and moreover, an impulsive control equilibrium set) are also provided, while the controller performance are tested by means of two simulation examples.