vector invariants of sylp gl n fq and their hilbert ideals abstract we describe the hilbert ideal of the vector invariants of a p sylow subgroup of the general linear group

rings of invariants for modular representations of the klein four group we study the rings of invariants for the indecomposable modular representations of the klein four group for each such representation we compute the noether number and give minimal generating sets for the hilbert ideal and the field of fractions we observe that with the exception of the regular representation the hilbert ideal for each of these representations is a complete intersection

practical criterion for delay estimation using random perturbations many systems contain an internal time delay which significantly influences their dynamical properties methods to estimate this delay from times series in the presence of dynamical noise are not systematically studied addressing this problem we demonstrate that it is sufficient to analyze the system s response to short correlated external disturbances or internal noise following this idea it is shown for linear and nonlinear systems as well as for periodic dynamics that the delay can be estimated by analyzing the correlation function this method covers the case of strong noise and multiple delays

vector invariant fields of finite classical groups abstract let w be an n dimensional vector space over a finite field f q of any characteristic and mw denote the direct sum of m copies of w let f q m w gl w and f q m w gl w denote the vector invariant ring and vector invariant field respectively where gl w acts on w in the standard way and acts on mw diagonally we prove that there exists a set of homogeneous invariant polynomials f 1 f 2 f m n f q m w gl w such that f q m w gl w f q f 1 f 2 f m n we also prove the same assertions for the special linear groups in any characteristic and the orthogonal unitary and symplectic groups in odd characteristic

extraction of delay information from chaotic time series based on information entropy using information entropy and embedding methods for reconstructing chaotic attractors a technique for estimating time delay from experimental time data is proposed for chaotic systems governed by a delay differential equation the effectiveness of the proposed approach is demonstrated with numerical experiments with and without measurement noise the quantization effect on information entropy is also discussed the number of quantization levels is found not to effect the accuracy of the proposed delay extraction technique significantly

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

symmetric powers of modular representations for groups with a sylow subgroup of prime order abstract let v be a representation of a finite group g over a field of characteristic p if p does not divide the group order then molien s formula gives the hilbert series of the invariant ring in this paper we find a replacement of molien s formula which works in the case that g is divisible by p but not by p 2 we also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of v into indecomposables our methods can be applied to determine the depth of the invariant ring without computing any invariants this leads to a proof of a conjecture of the second author on certain invariants of gl 2 p

invariants of the diagonal cp action on v3 abstract let c p denote the cyclic group of order p where p 3 is prime we denote by v 3 the indecomposable three dimensional representation of c p over a field f of characteristic p we compute a set of generators in fact a sagbi basis for the ring of invariants f v 3 v 3 c p our main result confirms the conjecture of shank r j shank classical covariants and modular invariants in h e a campbell d l wehlau eds invariant theory in all characteristics crm proc lecture notes vol 35 amer math soc 2004 pp 241 249 for this example that all modular rings of invariants of c p are generated by rational invariants norms and transfers

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

vector invariants for the two dimensional modular representation of a cyclic group of prime order in this paper we study the vector invariants of the 2 dimensional indecomposable representation v2 of the cyclic group cp of order p over a field f of characteristic p f mv2 cp this ring of invariants was first studied by david richman 1990 20 who showed that the ring required a generator of degree m p 1 thus demonstrating that the result of noether in characteristic 0 that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group does not extend to the modular case he also conjectured that a certain set of invariants was a generating set with a proof in the case p 2 this conjecture was proved by campbell and hughes 1997 in 3 later shank and wehlau 2002 in 24 determined which elements in richman s generating set were redundant thereby producing a minimal generating set we give a new proof of the result of campbell and hughes shank and wehlau giving a minimal algebra generating set for the ring of invariants f mv2 cp in fact our proof does much more we show that our minimal generating set is also a sagbi basis for f mv2 cp further our results provide a procedure for finding an explicit decomposition of f mv2 into a direct sum of indecomposable cp modules finally noting that our representation of cp on v2 is as the p sylow subgroup of sl2 fp we describe a generating set for the ring of invariants f mv2 sl2 fp and show that p m 2 p 1 is an upper bound for the noether number for m 2

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

quandle cocycles from invariant theory abstract let g be a group any g module m has an algebraic structure called a g family of alexander quandles given a 2 cocycle of a cohomology associated with this g family topological invariants of handlebody knots in the 3 sphere are defined we develop a simple algorithm to algebraically construct n cocycles of this g family from g invariant group n cocycles of the abelian group m we present many examples of 2 cocycles of these g families using facts from modular invariant theory

the module structure of a group action on a polynomial ring abstract for any representation of a p group g on a vector space of dimension 3 over a finite field k of characteristic p we show how the symmetric algebra regarded as a kg module can be expressed as a direct sum of kg modules each one of which is isomorphic to a summand in low degree it follows that for any group g only a finite number of isomorphism classes of summands can occur

reconstruction of time delayed feedback systems from time series for various classes of time delay systems we propose the methods of their model delay differential equation reconstruction from time series the methods are based on the characteristic location of extrema in the time series of time delay systems and the projection of infinite dimensional phase space of these systems to suitably chosen low dimensional subspaces we verify our methods by using them for the recovery of time delay differential equations from their chaotic solutions and for modelling experimental systems with delay induced dynamics from their chaotic time series

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes we propose a set of methods for the estimation of the parameters of time delay systems with a linear filter and nonlinear delay feedback performing periodic oscillations the methods are based on an analysis of the system response to regular external perturbations and are valid only for systems whose dynamics can be perturbed the efficiency of the methods is illustrated using both numerical and experimental data

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

negative temperatures and the definition of entropy the concept of negative temperature has recently received renewed interest in the context of debates about the correct definition of the thermodynamic entropy in statistical mechanics several researchers have identified the thermodynamic entropy exclusively with the volume entropy suggested by gibbs and have further concluded that by this definition negative temperatures violate the principles of thermodynamics we disagree with these conclusions we demonstrate that volume entropy is inconsistent with the postulates of thermodynamics for systems with non monotonic energy densities while a definition of entropy based on the probability distributions of macroscopic variables does satisfy the postulates of thermodynamics our results confirm that negative temperature is a valid extension of thermodynamics

structure identification based on steady state control experimental results and applications we report experimental results on structure identification of nonlinear systems by a steady state control method the idea underlying the method is to drive the nonlinear system to steady state by applying a suitable feedback control input it turns out experimentally that this control based structure identification method can be used for some applications such as estimation of initial conditions and state variables of nonlinear systems and structure identification of some special elements two attractors of the chua oscillator are presented to illustrate the reliability of the suggested techniques under the hypotheses of measurable state variables and physical access to the system for implementing the proportional feedback

structure theorems over polynomial rings given a polynomial ring r over a field k and a finite group g we consider a finitely generated graded rg module s we regard s as a kg module and show that various conditions on s are equivalent such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of d karagueuzian p symonds the module structure of a group action on a polynomial ring a finiteness theorem preprint http www ma umist ac uk pas preprints

comparison of canonical and microcanonical definitions of entropy abstract for more than 100 years one of the central concepts in statistical mechanics has been the microcanonical ensemble which provides a way of calculating the thermodynamic entropy for a specified energy a controversy has recently emerged between two distinct definitions of the entropy based on the microcanonical ensemble 1 the boltzmann entropy defined by the density of states at a specified energy and 2 the gibbs entropy defined by the sum or integral of the density of states below a specified energy a critical difference between the consequences of these definitions pertains to the concept of negative temperatures which by the gibbs definition cannot exist in this paper we call into question the fundamental assumption that the microcanonical ensemble should be used to define the entropy we base our analysis on a recently proposed canonical definition of the entropy as a function of energy we investigate the predictions of the boltzmann gibbs and canonical definitions for a variety of classical and quantum models our results support the validity of the concept of negative temperature but not for all models with a decreasing density of states we find that only the canonical entropy consistently predicts the correct thermodynamic properties while microcanonical definitions of entropy including those of boltzmann and gibbs are correct only for a limited set of models for models which exhibit a first order phase transition we show that the use of the thermodynamic limit as usually interpreted can conceal the essential physics

the fundamental theorems of vector invariants abstract new proofs are given of the first fundamental theorems of vector invariants for the special linear and special orthogonal groups the novelty of the proofs is that they rely on reducing the theorems to the case of 2 2 matrices there is a close connection between the vector invariants of the special linear group or special orthogonal group of n n matrices for all n 2 and those of the analogous group of 2 2 matrices this connection is described using the notion of leading monomials properties of leading monomials are also used to give new proofs of the second fundamental theorems of vector invariants for the general linear and orthogonal groups

group actions on rings and the cech complex abstract we have previously shown that when a finite group g acts on a polynomial ring s in n variables over a finite field k only finitely many isomorphism classes of indecomposable k g modules occur as summands of s we have also shown that the regularity of the invariant subring s g is at most zero which has various consequences for example s g is generated in degrees at most n g 1 provided n g 2 both of these theorems depend on the structure theorem of karagueuzian and the author which is proved by means of a long and complicated calculation the aim of this paper is to prove these results using a more conceptual method

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

the second main theorem vector for the modular regular representation of c2 we study the ring of invariants for a finite dimensional representation v of the group c2 of order 2 in characteristic 2 let denote a generator of c2 and x1 y1 xm ym a basis of v then xi xi and yi yi xi to our knowledge this ring for any prime p was first studied by david richman 12 in 1990 he gave a first main theorem for v2 c2 that is he proved that the ring of invariants when p 2 is generated by xi ni yi2 xiyi tr a 2 a m where a 0 1 m ya y1a1y2a2 ymam and tr a ya y1 x1 a1 y2 x2 a2 ym xm am in this paper we prove the second main theorem for v2 c2 that is we show that all relations between these generators are generated by relations of type i i axitr a i 0 and of type ii tr a tr b l ixi lnltr i l j k ni l jxj ltr l k for all m we also derive relations of type iii which are simpler and can be used in place of the relations of type ii

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

in defense of negative temperature this pedagogical comment highlights three misconceptions concerning the usefulness of the concept of negative temperature being derived from the usual often termed boltzmann definition of entropy first both the boltzmann and gibbs entropies must obey the same thermodynamic consistency relation second the boltzmann entropy does obey the second law of thermodynamics third there exists an integrating factor of the heat differential with both definitions of entropy

reconstruction of time delay systems from chaotic time series we propose a method that allows one to estimate the parameters of model scalar time delay differential equations from time series the method is based on a statistical analysis of time intervals between extrema in the time series we verify our method by using it for the reconstruction of time delay differential equations from their chaotic solutions and for modeling experimental systems with delay induced dynamics from their chaotic time series

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

recovery of the time evolution equation of time delay systems from time series we present a method for time series analysis of both scalar and nonscalar time delay systems if the dynamics of the system investigated is governed by a time delay induced instability the method allows one to determine the delay time in a second step the time delay differential equation can be recovered from the time series the method is a generalization of our recently proposed method suitable for time series analysis of scalar time delay systems the dynamics is not required to be settled on its attractor and this also makes transient motion accessible to the analysis if the motion actually takes place on a chaotic attractor the applicability of the method does not depend on the dimensionality of the chaotic attractor one main advantage over all time series analysis methods known until now for a demonstration we analyze time series which are obtained with the help of the numerical integration of a two dimensional time delay differential equation after having determined the delay time we recover the nonscalar time delay differential equation from the time series in agreement with the original time delay equation finally possible applications of our analysis method in such different fields as medicine hydrodynamics laser physics and chemistry are discussed

galois structure of homogeneous coordinate rings suppose g is a finite group acting on a projective scheme x over a commutative noetherian ring r we study the rg modules h 0 x f l n when n 0 and f and are coherent g sheaves on x such that l is an ample line bundle we show that the classes of these modules in the grothendieck group g 0 rg of all finitely generated rg modules lie in a finitely generated subgroup under various hypotheses we show that there is a finite set of indecomposable rg modules such that each h 0 x f l n is a direct sum of these indecomposables with multiplicities given by generalized hilbert polynomials for n 0

estimation of delay in coupling from time series we demonstrate that a time delay in weak coupling between two self sustained oscillators can be estimated from the observed time series data we present two methods which are based on the analysis of interrelations between the phases of the signals we show analytically and numerically that irregularity of the phase dynamics due to the intrinsic noise or chaos is essential for determination of the delay we compare and contrast both methods to the standard cross correlation analysis

the ring of upper triangular invariants as a module over the dickson invariants in this paper we study the ring of invariants hn of un fp the upper triangular group with l s on the main diagonal as a module over the ring of invariants d of gin fp the full general linear group over the finite field fp for p a prime both groups acting as algebra automorphisms of the polynomial algebra fp xl xn both of these rings of invariants are known to be polynomial algebras hn fp hl hn and dn fp dl d see sect 3 the latter ring of invariants is known as the ring of dickson invariants now d serves as a homogeneous system of parameters for h in other words h is a finitely generated module over d and since am is a polynomial algebra and hence cohen macaulay h is in fact a free dn module see sect 2 in this situation we have a lot of information concerning this module structure in particular there is a poincar polynomial which gives the number and dimension of the elements in a free basis this polynomial admits a factorization sect 5 which suggests a free basis of a certain form and we are able to show that the suggested basis is indeed a basis see theorem 6 3 once we have a free basis we may define a natural epimorphism of d modules hn dn as follows any free basis contains 1 since dn c h and so any polynomial h c h admits a unique expression h d 9 1 f where d e d and f is a dn linear combination of the non trivial basis elements we define h d given our particular choice of basis we think of as rewriting since the value of on h is calculated from the unique expression for h in terms of our basis on the other hand it is well known that u fp is a p sylow subgroup of gl fp since this is the case we may define another natural epimorphism of d modules p hn d by averaging h hn over a set of coset

vector invariants ofu2 fp a proof of a conjecture of richman abstract in this paper we prove a conjecture of d r richman concerning the vector invariants of the group u 2 f p let v be a vector space of dimension 2 with basis x y over the field f p and let f p x y be the symmetric algebra of v over f p if denotes a generator of u 2 f p then we may assume x x and y x y let a n be the symmetric algebra of v n we obtain an automorphism of a n of order p by using the diagonal action of extended to the whole of a n the subalgebra of polynomials left invariant by this action is called the ring of vector invariants of u 2 f p richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p 2 we prove his conjecture for all primes