commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

the poset of the nilpotent commutator of a nilpotent matrix abstract let b be an n n nilpotent matrix with entries in an infinite field k the nilpotent commutator n b of b is known to be an irreducible algebraic variety assume that b is in jordan canonical form with the associated jordan block partition p and let q p be the jordan partition of a generic element of n b p oblak associated to a given partition p another partition resulting from a recursive process she conjectured that this partition is the same as q p r basili a iarrobino and the author later generalized the process introduced by oblak the main result of this paper states that all such processes give rise to the same partition a partition that is defined in terms of the lengths of unions of special symmetric chains in a poset associated to n b this poset was defined by p oblak as the digraph determined by the non zero entries of a generic matrix in a maximal nilpotent subalgebra of n b

a resolution of paz s conjecture in the presence of a nonderogatory matrix abstract let m n f be the algebra of n n matrices over the field f and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space it is a long standing conjecture of paz that the length of any generating set of m n f cannot exceed 2 n 2 we prove this conjecture under the assumption that the generating set s contains a nonderogatory matrix in addition we find linear bounds for the length of generating sets that include a matrix with some conditions on its jordan canonical form finally we investigate cases when the length 2 n 2 is achieved

a class of hamiltonian systems with increasing periods on considere un systeme hamiltonien de la forme x g y y f x ou h f et h g satisfont certaines hypotheses on montre qu il y a un intervalle maximal 0 a a 0 tel que chaque solution x t y t avec x 0 a 0 a y 0 0 est periodique avec son orbite enfermant 0 0 soit p a la plus petite periode de cette solution alors p 0 a r est differentiable avec p a 0

parameters spline methods for the solution of hyperbolic equations abstract in this paper by using a parameters cubic spline in space and compact finite difference in time direction we get a class of finite difference schemes for solving second order hyperbolic equations with mixed boundary conditions stability analysis of the methods have been carried out it has been shown that by suitable choosing the cubic spline parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our methods we also obtain new high accuracy schemes of o h 2 2 h 2 and o h 4 2 h 4 numerical comparison with rashidinia s method j rashidinia et al spline methods for the solutions of hyperbolic equations appl math comput 190 2007 882 886 shows the superiority of our presented schemes

gorenstein subrings of invariants under hopf algebra actions abstract this paper concerns conditions on the action of a finite dimensional semisimple hopf algebra on an artin schelter regular algebra that force the subring of invariants to satisfy the artin schelter gorenstein condition classical results such as watanabe s theorem and stanley s theorem are extended from the case of a group action to the context of a hopf algebra action a hopf algebra version of the homological determinant is introduced and it becomes an important tool in the generalization from group actions to hopf algebra actions

doubly stochastic matrices whose powers eventually stop abstract in this note we characterize doubly stochastic matrices a whose powers a a 2 a 3 eventually stop i e a p a p 1 for some positive integer p the characterization enables us to determine the set of all such matrices

a new unconditionally stable compact difference scheme of o τ2 h4 for the 1d linear hyperbolic equation abstract in this paper we propose a three level compact difference scheme of o 2 h 4 for the difference solution of one dimensional second order non homogeneous linear hyperbolic equation u tt 2 u t 2 u u xx f x t 0 x 1 t 0 where 0 stability analysis of the method has been carried out finally numerical examples are used to illustrate the efficiency of the new difference scheme

a class of difference scheme for solving telegraph equation by new non polynomial spline methods abstract in this paper by using a new non polynomial parameters cubic spline in space direction and compact finite difference in time direction we get a class of new high accuracy scheme of o 4 h 2 and o 4 h 4 for solving telegraph equation if we suitably choose the cubic spline parameters meanwhile stability condition of the difference scheme has been carried out finally numerical examples are used to illustrate the efficiency of the new difference scheme

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

periodic solutions of a forced lotka volterra equation abstract for a 2 species predator prey model the effect of a periodic forcing dependent on two parameters is studied a description is given for the local bifurcation curves of periodic solutions near a fixed family of solutions of the free equation

reflexivity defect of kernels of the elementary operators of length 2 abstract let x be a finite dimensional complex vector space we give an explicit formula for the reflexivity defect of the kernel of an arbitrary elementary operator of length 2 i e an elementary operator of the form t a 1 tb 1 a 2 tb 2 t l x where a 1 a 2 and b 1 b 2 are linearly independent

lengths of quasi commutative pairs of matrices abstract in this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras we consider mainly the sets of two matrices which are quasi commuting it is shown that in this case the length function is linearly bounded we also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product nilpotent pairs all possible numbers are realizable however there are non realizable values between lower and upper bounds for the other quasi commuting pairs in conclusion we also present several related open problems

hopf actions and nakayama automorphisms abstract let h be a hopf algebra with antipode s and let a be an n koszul artin schelter regular algebra we study connections between the nakayama automorphism of a and s 2 of h when h coacts on a inner faithfully several applications pertaining to hopf actions on artin schelter regular algebras are given

on the irreducibility of commuting varieties of nilpotent matrices abstract given an n n nilpotent matrix over an algebraically closed field k we prove some properties of the set of all the n n nilpotent matrices over k which commute with it then we give a proof of the irreducibility of the variety of all the pairs a b of n n nilpotent matrices over k such that a b 0 if either char k 0 or char k n 2 we get as a consequence a proof of the irreducibility of the local hilbert scheme of n points of a smooth algebraic surface over k if either char k 0 or char k n 2

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

the minimal dimension of maximal commutative subalgebras of full matrix algebras abstract let f be a field and a a maximal commutative subalgebra of the full matrix algebra mn f it is shown that dim a 2n 2 3 1 it is also shown that if the radical of a has cube zero then dim a 3n 2 3 4 and that this result is best possible for infinitely many natural numbers n

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

reflexivity defect of spaces of linear operators abstract for a finite dimensional linear subspace s l v w and a positive integer k the k reflexivity defect of s is defined by rd k s dim ref k s s where ref k s is the k reflexive closure of s we study this quantity for two dimensional spaces of operators and for single generated algebras and their commutants

the commutator algebra of a nilpotent matrix and an application to the theory of commutative artinian algebras we show a number of properties of the commutator algebra of a nilpotent matrix over a field in particular we determine the simple modules of the commutator algebra then the results are applied to prove that certain artinian complete intersections have the strong lefschetz property

an operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions a new three level implicit unconditionally stable operator splitting method of o k 2 h 2 is proposed for the numerical solution of two space dimensional linear hyperbolic equation u t email protected x y t u email protected 2 x y t u a x y t u x x b x y t u y y f x y t 00 subject to appropriate initial and dirichlet boundary conditions where a x y t b x y t 0 a x y t 0 b x y t 0 the resulting system of algebraic equations is solved by two step split method the proposed method is applicable to the problems having singularity at x 0 numerical results are provided to demonstrate the utility of the new method

nilpotent commuting varieties of reductive lie algebras let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 and lie g in positive characteristic suppose in addition that p is good for g and the derived subgroup of g is simply connected let denote the nilpotent variety of and nil x y x y 0 the nilpotent commuting variety of our main goal in this paper is to show that the variety nil is equidimensional in characteristic 0 this confirms a conjecture of vladimir baranovsky see 2 when applied to gl n our result in conjunction with an observation in 2 shows that the punctual local hilbert scheme n hilb n 2 is irreducible over any algebraically closed field

spline methods for the solutions of hyperbolic equations abstract second order hyperbolic equations with mixed boundary conditions are solved by using a non polynomial cubic spline in space and finite difference in time direction we develop new classes of three level methods stability analysis of the methods have been carried out it has been shown that by suitably choosing the cubic spline parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our methods we also obtain new high accuracy schemes of o k 2 h 2 and o k 2 h 4 numerical example is given to illustrate the applicability and efficiency of the new methods

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

fourth order approximations at first time level linear stability analysis and the numerical solution of multidimensional second order nonlinear hyperbolic equations in polar coordinates in this article three level implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively are proposed for the numerical solution of one two and three space dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and dirichlet boundary conditions we also discuss fourth order approximation at first time level for more general case we also obtain the stability range of the difference scheme when applied to a test equation utt urr aruu ar2u g r t a 1 and 2 numerical examples are provided to demonstrate the required order of convergence of the methods

on nilpotent commuting varieties and cohomology of frobenius kernels abstract the paper studies the dimensions of irreducible components of commuting varieties of restricted nilpotent r tuples in a classical lie algebra g defined over an algebraically closed field k as applications we obtain some new results on the structure of the even cohomology ring of frobenius kernels g r for each r 1 where g is the simply connected simple algebraic group such that lie g g explicit calculations for rank two groups are also presented

on varieties of commuting triples abstract the problem of approximating triples of commuting n n matrices by triples of generic matrices is equivalent to that whether the variety c 3 n of triples of commuting matrices is irreducible it is known that the variety is irreducible for n 7 and reducible for n 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that c 3 8 is irreducible

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

perturbation from an asymmetric cubic hamiltonian this paper concerns with the number of limit cycles from an asymmetric hamiltonian of degree three under cubic perturbation eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis two of which are new

chaos in periodically perturbed planar hamiltonian systems using linked twist maps abstract we present a simple topological approach for the search of fixed points and the detection of chaotic dynamics for two dimensional maps satisfying a twist condition on linked annuli applications to planar hamiltonian systems are given

an unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients we propose a three level implicit unconditionally stable difference scheme of o k 2 h 2 for the difference solution of second order linear hyperbolic equation u t email protected x t u email protected 2 x t u a x t u x x f x t 00 subject to appropriate initial and dirichlet boundary conditions where a x t 0 a x t b x t 0 the proposed formula is applicable to the problems having singularity at x 0 the resulting tri diagonal linear system of equations is solved by using gauss elimination method numerical examples are provided to illustrate the unconditionally stable character of the proposed method

nilpotent matrices having a given jordan type as maximum commuting nilpotent orbit abstract the jordan type of a nilpotent matrix is the partition giving the sizes of its jordan blocks we study pairs of partitions p q where q q p is the jordan type of a generic nilpotent matrix a commuting with a nilpotent matrix b of jordan type p t kosir and p oblak have shown that q has parts that differ pairwise by at least two such partitions which are also known as super distinct or rogers ramanujan are exactly those that are stable or self large in the sense that q q q in 2012 p oblak formulated a conjecture concerning the cardinality of q 1 q when q has two parts and proved some special cases r zhao refined this to posit that the partitions in q 1 q for q u u r with u r 1 could be arranged in an r 1 u r table t q where the entry in the k th row and l th column has k l parts we prove this table theorem and then generalize the statement to propose a box conjecture for the set of partitions q 1 q for an arbitrary partition q whose parts differ pairwise by at least two

upper bounds for the length of non associative algebras abstract we introduce the notion of length for non associative finite dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class we also put forward a new method of characteristic sequences based on linear algebra technique which provides an efficient tool for computing the length function in non associative case then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra in the last case the length is bounded in terms of the fibonacci sequence we present concrete examples demonstrating the sharpness of our bounds

locally linearly dependent operators and reflexivity of operator spaces we obtain two results on the existence of large subspaces of operators of small rank in locally linearly dependent spaces of operators as a consequence we obtain an upper bound for the rank of operators belonging to a minimal locally linearly dependent space of operators it has been known that the only obstruction to the reflexivity of a finite dimensional operator space comes from the operators with small ranks our results improve known bounds on the minimal rank that guarantees the reflexivity

the effect of assuming the identity as a generator on the length of the matrix algebra abstract let m n f be the algebra of n n matrices and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space traditionally the identity matrix is assumed to be automatically included in all generating sets s and counted as a word of length 0 in this paper we discuss how the problem changes if this assumption is removed

principal nilpotent pairs in a semisimple lie algebra 1 this is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple lie algebra that enjoy quite remarkable properties and which are expected to play a major role in representation theory the properties of these pairs and their role is similar to those of the principal nilpotents each principal nilpotent pair gives rise to a harmonic polynomial on the cartesian square of the cartan subalgebra that transforms under an irreducible representation of the weyl group in the special case of n the conjugacy classes of principal nilpotent pairs and the irreducible representations of the symmetric group s n are both parametrised in a compatible way by young diagrams in general our theory provides a natural generalization to arbitrary weyl groups of the classical construction of simple s n modules in terms of young s symmetrisers first results towards a complete classification of all principal nilpotent pairs in a simple lie algebra are presented at the end of this paper in an appendix written by a elashvili and d panyushev

a characterization of centres via first integrals on considere l existence d une integrale premiere lisse et l existence d une mesure invariante equivalente a la mesure de lebesgue pour des systemes differentiels dans le plan

skew calabi yau algebras and homological identities abstract a skew calabi yau algebra is a generalization of a calabi yau algebra which allows for a non trivial nakayama automorphism we prove three homological identities about the nakayama automorphism and give several applications the identities we prove show i how the nakayama automorphism of a smash product algebra a h is related to the nakayama automorphisms of a graded skew calabi yau algebra a and a finite dimensional hopf algebra h that acts on it ii how the nakayama automorphism of a graded twist of a is related to the nakayama automorphism of a and iii that the nakayama automorphism of a skew calabi yau algebra a has trivial homological determinant in case a is noetherian connected graded and koszul

reflexivite d une extension d un operateur normal par un operateur nilpotent let t be an extension of a one to one normal operator a by a nilpotent operator n in this paper we calculate the defect of reflexivity of t we give a necessary and sufficient condition to insure the reflexivity of such extensions in particular it is shown that t is reflexive when n is reflexive

global existence of a restoring force realizing a prescribed half period abstract we consider an inverse problem to determine a nonlinearity of an autonomous differential equation so that each solution of the equation has a half period that is a prescribed function of a half amplitude of the solution a global existence of the nonlinearity is proved under the assumption that the prescribed function is lipschitz continuous a reconstructive way of the nonlinearity is also given

length realizability for pairs of quasi commuting matrices abstract for the pairs of quasi commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators

generic and maximal jordan types for a finite group scheme g over a field k of characteristic p 0 we associate new invariants to a finite dimensional kg module m namely for each generic point of the projectivized cohomological variety text proj mathop h bullet g k we exhibit a generic jordan type of m in the very special case in which g e is an elementary abelian p group our construction specializes to the non trivial observation that the jordan type obtained by restricting m via a generic cyclic shifted subgroup does not depend upon a choice of generators for e furthermore we construct the non maximal support variety g m a closed subset of text proj mathop h bullet g k which is proper even when the dimension of m is not divisible by p

coherent algebras and noncommutative projective lines abstract a well known conjecture says that every one relator group is coherent we state and partly prove a similar statement for graded associative algebras in particular we show that every gorenstein algebra a of global dimension 2 is graded coherent this allows us to define a noncommutative analogue of the projective line p 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgra of such an algebra a that is the category of coherent a modules modulo the torsion ones this category is always abelian ext finite hereditary with serre duality like the category of coherent sheaves on p 1 in this way we obtain a sequence p n 1 n 2 of pairwise non isomorphic noncommutative schemes which generalize the scheme p 1 p 2 1

an operator difference scheme for abstract cauchy problems an abstract cauchy problem for second order hyperbolic differential equations containing the unbounded self adjoint positive linear operator a t with domain in an arbitrary hilbert space is considered a new second order difference scheme generated by integer powers of a t is developed the stability estimates for the solution of this difference scheme and for the first and second order difference derivatives are established in hilbert norms with respect to space variable to support the theoretical statements for the solution of this difference scheme the numerical results for the solution of one dimensional wave equation with variable coefficients are presented

an upper bound for the length of a finite dimensional algebra abstract letfbe a field and letabe a finite dimensionalf algebra writed dimf a and letebe the largest degree of the minimal polynomial for anya a define the function f d e e 2d e 1 1 4 e 2 2 we prove that ifsis any finite generating set foraas anf algebra the words insof length less thanf d e spanaas anf vector space in the special case ofn by nmatrices this bound becomes f n 2 n n 2n 2 n 1 1 4 n 2 2 o n 3 2 this is a substantial improvement over previous bounds which have all beeno n2 we also prove that for particular setssof matrices the bound can be sharpened to one that is linear inn as an application of these results we reprove a theorem of small stafford and warfield about semiprime affinef algebras

numerical methods for nonlinear second order hyperbolic partial differential equations i time linearized finite difference methods for 1 d problems abstract a three parameter family of finite difference methods for one dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms is presented and shown to result in two time level semi discrete equations second order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three time level linear finite difference equations which are second order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1 2 1 to the past current and future time levels second order accurate in time time linearized three point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth order accurate in space are also presented and their linear stability analyzed an exhaustive comparison between analytical and numerical results indicates that the accuracy of time linearized methods increases as the time step and grid spacing are decreased and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure and the best accuracy is achieved when the three implicitness parameters are equal to 1 2 it is also shown that the accuracy of time linearized methods is a strong function of the wave speed as the wave speed is increased and the wave equation tends to a parabolic one the methods are still three time level techniques which may not preserve the positivity of the solution whereas a crank nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions it is also shown that for one dimensional wave equations with monotonic kinetics and time dependent forcing there are very few differences between second and first order time accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small and the discretization of the reaction terms plays a less important rule than that of the diffusion terms

some applications of doubly stochastic matrices abstract we discuss primarily two applications of doubly stochastic matrices and related matrices the first concerns a topic in communication theory called satellite switched time division multiple access systems and we attempt to illuminate some results which have occured in the engineering literature the second concerns a recent notion of a doubly stochastic automorphism of a graph we prove several new theorems about doubly stochastic automorphisms of certain classes of graphs and in particular obtain a theorem of which birkhoff s theorem for doubly stochastic matrices is a special case

quantum rigidity of negatively curved manifolds we show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected riemannian manifold m g with strictly negative curvature is automatically classical in the sense that it factors through the action of the isometry group of m g this partially answers a question by d goswami

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

commutative matrix subalgebras and length function it is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion the examples are given showing that the converse does not hold to establish this result we prove several fundamental properties of the length function

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

subharmonic solutions of periodically forced predator prey systems in this paper we consider a class of planar ordinary differential equations consisting of a basic predator prey system with an autonomous perturbation that may represent a damping or frictional force and a time periodic perturbation the problem we study is that of the existence of subharmonic solutions that is periodic solutions whose period is an integral multiple of the forcing period it is not our intention to describe any specific ecosystem but rather to give some rather general mathematical results and discuss the numerical computation of these subharmonics we mention however that predator prey cycles have been observed in nature for a long time and their mechanism remains a major problem in mathematical biology some classic early studies were those of volterra l lotka 2 and kolmogoroff 3 and the examples in section 4 of this paper are perturbations of volterra lotka systems considerable work has been done with various autonomous predator prey models see 4 chapter 121 for a very readable introduction but much less with time dependent systems from the point of view of applications however it seems reasonable to introduce periodic perturbations and indeed it is difficult to imagine any high level ecosystem in which the reproduction rates of at least some species are not affected by some cyclic factor e g yearly daily lunar it is hoped that the results given here may contribute to the understanding of such systems for discussions of predator prey cycles from the biological point of view see 5 71 the main tool here is a variant of the liapunov schmidt method as given by chow and hale 8 the liapunov schmidt method first seems to have been used to prove the existence of subharmonics for some classes of ordinary differential equations by hale and taboas 9 since then a fairly extensive theory has been developed for single second order equations see lo and references therein subharmonics for a certain perturbed volterra lotka system were examined from a different point of view in 111 we will now make a few comments on our notation whenever a function is introduced on the right hand side of a differential equation it is assumed to be smooth smooth here means c it will be evident that a hypothesis of c in what follows can usually be replaced by a hypothesis of c for some finite n but we wanted to avoid the distraction of counting derivatives ordinary derivatives are denoted in the usual way by primes dots or d partial derivatives by subscripted variable names or a and derivative matrices by ds the 2 x 2 identity matrix is denoted by i a wedge is a section of a disk lying between two smooth curves through the origin whose tangents at the origin are different

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

dualising complexes and twisted hochschild co homology for noetherian hopf algebras we show that many noetherian hopf algebras a have a rigid dualising complex r with r a1 d here d is the injective dimension of the algebra and is a certain k algebra automorphism of a unique up to an inner automorphism in honour of the finite dimensional theory which is hereby generalised we call the nakayama automorphism of a we prove that s2 where s is the antipode of a and is the left winding automorphism of a determined by the left integral of a the hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non zero in the top dimension d when a is an artin schelter regular noetherian hopf algebra of global dimension d twisted poincare duality holds in this setting as is deduced from a theorem of van den bergh calculating for a using also the opposite coalgebra structure we determine a formula for s4 generalising a 1976 formula of radford for a finite dimensional applications of the results to the cases where a is pi an enveloping algebra a quantum group a quantised function algebra and a group algebra are outlined

the period in the lotka volterra system is monotonic abstract periodic solutions in a class of hamiltonian systems with one degree of freedom containing the lotka volterra system are considered their period is represented as an integral appropriate for numerical evaluation as well as for theoretical investigations the monotonicity of the period in the lotka volterra system follows as a simple consequence

periodic solutions for planar autonomous systems with nonsmooth periodic perturbations abstract in this paper we consider a class of planar autonomous systems having an isolated limit cycle x 0 of smallest period t 0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1 we consider two functions defined by means of the eigenfunctions of the adjoint of the linearized system and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of t periodic solutions of the autonomous system when it is perturbed by nonsmooth t periodic nonlinear terms of small amplitude we also show the convergence of these periodic solutions to x 0 as the perturbation disappears and we provide an estimation of the rate of convergence the employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach commonly employed in the existing literature on this subject based on various versions of the implicit function theorem

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

semisimple hopf actions on commutative domains abstract let h be a semisimple so finite dimensional hopf algebra over an algebraically closed field k of characteristic zero and let a be a commutative domain over k we show that if a arises as an h module algebra via an inner faithful h action then h must be a group algebra this answers a question of e kirkman and j kuzmanovich and partially answers a question of m cohen the main results of this article extend to working over k of positive characteristic on the other hand we obtain results on hopf actions on weyl algebras as a consequence of the main theorem

on quantum groups associated to non noetherian regular algebras of dimension 2 we investigate homological and ring theoretic properties of universal quantum linear groups that coact on artin schelter regular algebras a n of global dimension 2 especially with central homological codeterminant or central quantum determinant as classified by zhang the algebras a n are connected mathbb n graded algebras that are finitely generated by n indeterminants of degree 1 subject to one quadratic relation in the case when the homological codeterminant of the coaction is trivial we show that the quantum group of interest defined independently by manin and by dubois violette and launer is artin schelter regular of global dimension 3 and also skew calabi yau homologically smooth of dimension 3 for central homological codeterminant we verify that the quantum groups are noetherian and have finite gelfand kirillov dimension precisely when the corresponding comodule algebra a n satisfies these properties that is if and only if n 2 we have similar results for arbitrary homological codeterminant if we require that the quantum groups are involutory we also establish conditions when hopf quotients of these quantum groups that also coact on a n are cocommutative

a stable noniterative prediction correction domain decomposition method for hyperbolic problems in this paper a noniterative domain decomposition algorithm for solving hyperbolic partial differential equations is presented the algorithm includes a prediction to estimate the values at the interface and these values are corrected by a scheme that improves the accuracy it has been shown that the algorithm is unconditionally stable efficiency is analyzed in terms of speedup and operation ratio numerical experiments illustrate that the method is stable and efficient