kolmogorov sinai entropy of many body hamiltonian systems the kolmogorov sinai ks entropy is a central measure of complexity and chaos its calculation for many body systems is an interesting and important challenge in this paper the evaluation is formulated by considering n dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem this approach makes explicit a duality relation that is exactly analogous to one found in a generalized anderson tight binding model and leads to a formally exact expression for the finite time ks entropy within this formalism there is a hierarchy of approximations the final one being a diagonal approximation that only makes use of instantaneous hessians of the potential to find the ks entropy by way of a nontrivial illustration the ks entropy of n identically coupled kicked rotors standard maps is investigated the validity of the various approximations with kicking strength particle number and time are elucidated an analytic formula for the ks entropy within the diagonal approximation is derived and its range of validity is also explored

nonlinear dynamics in coupled fuzzy control systems i coherence and chaos frustration in triangle configuration abstract nonlinear dynamics and chaos are studied in a system for which a complete set of equations of motion such as equations of newton navier stokes and van der pol is not available as a very general system as such we consider coupled classical spins pendulums each of which is under control by a fuzzy system that is designed to align the spin to an unstable fixed point the fuzzy system provides a deterministic procedure to control an object without use of a differential equation the positions and velocities of the spins are monitored periodically and each fuzzy control gives a momentum to its associated spin in the reverse directions if the monitoring is made with an interval short enough the spin spin interactions are overwhelmed by the fuzzy control and the system converges to a state as designed however a long interval monitoring induces dynamics of too late response and thereby results in chaos a great variety of dynamics are generated under very delicate balance between the fuzzy control and the spin spin interaction in which two independent mechanisms of creating negative and positive liapunov exponents interact with each other

chaos in the hamiltonian mean field model rst show that the largest lyapunov exponent remains strictly positive in the innite size limit converging to its asymptotic value with 1 lnn corrections we then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered clustered phase and the disordered phase present at high energy densities we also show that the full spectrum of lyapunov exponents consists of a bulk component converging to the zero value taken by a test oscillator forced by the mean eld plus subextensive bands of o lnn exponents taking nite values we nally investigate the robustness of these results by studying a 2d extension of the hmf model where each particle is endowed with 4 degrees of freedom thus allowing the emergence of chaos at the level of single particle altogether these results illustrate the subtle eects of global or long range coupling and the importance of the order in which the innite time and innite size limits are taken for an innite size hmf system represented by the vlasov equation no chaos is present while chaos exists and subsists for any nite system size

discriminants of brauer algebras in this paper we compute gram determinants associated to all cell modules of brauer algebras fancyscript b n partial theoretically we know when a cell module of fancyscript b n partial is equal to its simple head this gives a solution of this long standing problem

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

characterizing weak chaos in nonintegrable hamiltonian systems the fundamental role of stickiness and initial conditions abstract weak chaos in high dimensional conservative systems can be characterized through sticky effect induced by invariant structures on chaotic trajectories suitable quantities for this characterization are the higher cummulants of the finite time lyapunov exponents ftles distribution they gather the whole phase space relevant dynamics in one quantity and give information about ordered and random states this is analyzed here for discrete hamiltonian systems with local and global couplings it is also shown that ftles plotted versus initial condition ic and the nonlinear parameter are essential to understand the fundamental role of ics in the dynamics of weakly chaotic hamiltonian systems

clustered motion in symplectic coupled map systems clustered motion of particles is found in hamiltonian dynamics of symplectic coupled map systems particles assemble and move with strong correlation the motion is chaotic but is distinguishable from random chaotic motion lyapunov analysis distinguishes global instability from local fluctuations clustered motions have finite lifetime they have fractal geometric structure in the phase space as the orbits are trapped to ruins of kam tori and islands

strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset nonlinear dynamics in coupled fuzzy control systems ii abstract a new type of onset of chaos in a multidimensional dissipative system is reported that has been observed under a circumstance in which a delicate balance of order formation and chaos is materialized two limit cycles are connected directly to a sequence of strange attractors the widths in the bifurcation feigenbaum diagram of which converge to zero in a range close to the onset the measures of the extent of chaos are also zero there and the trajectories behave regularly in a macroscopic scale the chaos zones appear alternately with the windows of resonance frequency locking in the bifurcation diagram whose number of periodicity decreases as a system parameter is increased from an infinity in the manner of arithmetical progression 28 n 2 with n being integers down to unity concomitantly the extent of chaos develops and thereby grows measurable also the ranges of both the chaos and resonance zones in the bifurcation diagram become wider two lorenz plots of quantum numbers n and n 1 are always necessary to characterize the n th chaos zone for a chaotic state they appear in such a manner as to sandwich the identity map y x this chaos is readily interrupted by a limit cycle since one of the lorenz maps is crossed with y x by a small change of the control parameter moreover the tangential parts of the individual lorenz plots to y x have as many folds as their quantum numbers they form clusters each of which consists of many stable points being located densely at the limit of onset n both the intervals between these stable points and the folds lying between two neighboring stable points converge to zero this is the origin of the unmeasurable degree of chaos

the local stationary presentation of the alternating groups and the normal form abstract we give the canonical normal form for the elements of the finite or infinite alternating groups using local stationary presentation of these groups

on the structure of quantum super groups glq m n abstract we show that a quantum super matrix in standard format is invertible if and only if its block matrices of even entries are invertible we prove the q analog of the well known formula for the berezinian

liapunov spectra for infinite chains of nonlinear oscillators we argue that the spectrum of liapunov exponents for long chains of nonlinear oscillators at large energy per mode may be well approximated by the liapunov exponents of products of independent random matrices if in addition statistical mechanics applies to the system the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone under a certain isotropy hypothesis which is not always satisfied we argue that the liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix this construction uses an integral equation first derived by newman we then derive and discuss a method to compute the spectrum of a typical random matrix putting the pieces together we see that the liapunov spectrum can be computed from the potential between the oscillators

the largest liapunov exponent for random matrices and directed polymers in a random environment we study the largest liapunov exponent for products of random matrices the two classes of matrices considered are discrete d dimensional laplacians with random entries and symplectic matrices that arise in the study ofd dimensional lattices of coupled nonlinear oscillators we derive bounds on this exponent for all dimensions d and we show that ifd 3 and the randomness is not too strong one can obtain an explicit formula for the largest exponent in the thermodynamic limit our method is based on an equivalence between this problem and the problem of directed polymers in a random environment

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

globally coupled maps phase transitions and synchronization abstract bifurcations in a system of coupled maps are investigated using symbolic dynamics it is shown that for coupled shift maps the well known space time mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state for coupled non hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour the anomalous dependence of fluctuations on the system size is attributed to these bifurcations

irreducible representations of quantum linear groups of type a1 0 a hecke symmetry is an invertible operator defined on the tensor square of a finite dimensional vector space which satisfied the quantized yang baxter equation and the hecke equation x 1 x q 0 see 4 6 for more details given such a matrix one can construct a hopf algebra which is the function algebra over an algebraic matrix quantum group one is interested in the representations of this quantum group that is the comodules over the hopf algebra it turns out that the dimension of the vector space on which the hecke symmetry is defined does not play a significant role but rather a certain inner characteristics of the hecke symmetry the birank a pair of non negative integers a hecke symmetry is called even if its birank is of the form r 0 and odd if its birank is of the form 0 r in these cases the representation category of the corresponding quantum group is like the representation category of the matrix group gl r it is semisimple and generally does not depend on the hecke symmetry but only on the birank if the hecke symmetry is neither even nor odd the representation category of the corresponding quantum group seems to be similar to the representation category of the super group gl r s this problem has not been settled yet the aim of this paper is to classify irreducible representation of the matrix quantum group corresponding to a hecke symmetry of birank 2 1 such a quantum group is usually called matrix quantum group of type 1 0 in this work we show that irreducible representations of a quantum group of type 1 0 can be indexed by tuples m n p of integers with m n we exhibit basic decomposition rules of the tensor product of these representations and compute their dimensions recall that representations of a quantum group are by definition comodules over the corresponding hopf algebra of functions using the koszul komplex we construct for each tuple m n p m n a comodule and prove that they are simple the difficulty here is that the comodule category is not semisimple our main technique is based on the theory of hopf algebras with integral co frobenius hopf algebras for such hopf algebras there is a special class of simple comodules that split in any comodule we show that our hopf algebra is co frobenius and use a criterion for a comodule to be splitting in any comodule which is equivalent to being injective and projective in kac s theory of representation of lie superalgebras irreducible representations that split in any representations are called typical 9

modular representations of hecke algebras of type g p p n hecke algebras for the complex reflection groups g r p n where p 1 and p r were introduced in the work of ariki j algebra 177 1995 164 185 broue and malle asterisque 212 1993 119 189 in this paper we consider modular representation theory for these algebras in the case where r p we assume that the field k contains a primitive pth root of unity e our method is to study the restrictions of specht modules s for hecke algebras hp n of type g p 1 n with parameters q 1 e ep 1 suppose that f q e 0 in k see 4 7 for definition of f q e for any multipartition 1 p of n we prove that s hp p n s k hp p n for any 1 k p 1 where k k 1 k 2 p 1 2 k and if k is the smallest positive integer such that k hence k p we explicitly decompose s hp p n into a direct sum of p k smaller hp p n submodules with the same dimensions as a result when f q e 0 in k we show that hp p n is split over k and get a complete classification of all the absolutely irreducible hp p n modules this generalizes earlier work of c pallikaros j algebra 169 1994 20 48 and j hu manuscripta math 108 2002 409 430 on hecke algebras of type dn which are included as a special case of our main results

the schur algebra we study a class of endomomorphism algebras of certain q permutation modules over the hecke algebra of type b whose summands involve both parabolic and quasi parabolic subgroups and prove that these algebras are integrally free and quasi hereditary and are stable under base change some consequences for decomposition numbers are discussed the notion of a q schur algebra was introduced by dipper and james dj2 who used these algebras to parametrize the irreducible representations of the finite general linear groups in non describing characteristics with hindsight these algebras had already appeared earlier in an entirely different quantum group context ji inspired by physics in ji jimbo considered the endomorphism algebras of tensor spaces as hecke algebra modules in his context a q schur algebra can be viewed as a quotient of the quantized enveloping algebra associated to gln in pw these algebras were shown to be quasi hereditary the quasi heredity property is an embodiment in classical algebra of the geometric derived category stratification exhibited by perverse sheaves ps it means more applications can be deduced from a ring theoretic point of view see e g dps3 and the possibility is raised of even deeper results in the future as suggested by cps2 certainly these algebras play a central role in the representation theories of the finite and quantum general linear groups naturally one asks are there such algebras for types other than a our paper ds showed that there were similar quasi hereditary quotients of quantized enveloping algebras for all types of root systems however no connection with hecke algebras and finite groups of lie type was found there this paper aims at the same question and constructs possible algebras directly from hecke algebras hence from finite groups of lie type we restrict attention to the type b case imitating the definition of a q schur algebra we introduce the notion of a q schur algebra these algebras are the endomorphism algebras of certain modules over the hecke algebra of type b called tensor spaces whose summands involve not only parabolic subgroups but also quasi parabolic subgroups of the weyl group of type b a main result of our paper shows that qschur algebras are quasi hereditary we speculate that similar constructions exist for other classical types only type d remains actually since type c is equivalent received by the editors march 3 1997 and in revised form october 28 1998 2000 mathematics subject classification primary 20c08 20g05 20c33 the authors would like to thank arc for support under the large grant a69530243 as well as nsf and the universities of virginia and new south wales for their cooperation the first author also thanks the newton institute at cambridge for its hospitality c 2000 american mathematical society

the number of simple modules for the hecke algebras of type g r p n with an appendix by xiaoyi cui abstract we derive a parameterization of simple modules for the cyclotomic hecke algebras of type g r p n with p 1 and n 3 over fields of any characteristic coprime to p we give explicit formulas for the number of simple modules over these cyclotomic hecke algebras

on the decomposition numbers of the hecke algebra of type dn when n is even abstract let n 4 be an even integer let k be a field with char k 2 and q an invertible element in k such that i 1 n 1 1 q i 0 in this paper we study the decomposition numbers over k of the iwahori hecke algebra h q d n of type d n we obtain some equalities which relate its decomposition numbers with certain schur elements and the decomposition numbers of various iwahori hecke algebras of type a with the same parameter q when char k 0 this completely determine all of its decomposition numbers the main tools we used are the morita equivalence theorem established in j hu a morita equivalence theorem for hecke algebra h q d n when n is even manuscripta math 108 2002 409 430 and certain twining character formulae of weyl modules over a tensor product of two q schur algebras

infinitely many periodic attractors for holomorphic maps of 2 variables an important development in the study of discrete dynamical systems was newhouse s use of persistent homoclinic tangencies to show that a large set of c2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors or sinks 6 where large refers to a residual subset of an open set of diffeomorphisms in the present paper we obtain this result for various spaces of holomorphic maps of two variables newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency 7 more recently palis and viana extended this latter result to higher dimensions when the stable manifold has codimension one 10 and romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks 12 in each case however the construction reduces to the study of intersecting cantor sets in the line under an appropriate projection the stable and unstable manifolds of a basic set are mapped to cantor sets in the line and a tangency between these manifolds corresponds to a point of intersection of the cantor sets the generic unfolding of tangencies then gives rise to periodic attractors or saddles this reduction to linear cantor sets depends heavily on the fact that there is only one expanding eigenvalue even romero s result for higher codimension stable manifolds involves a reduction to this case in the holomorphic setting eigenvalues come in conjugate pairs from a real point of view so this reduction is not possible instead stable and unstable manifolds are riemann surfaces and after extending the stable and unstable manifolds of a basic set to foliations these foliations will be tangent in a real 2 dimensional disk and the stable and unstable manifolds correspond to cantor sets in this disk hence persistent tangencies between basic sets correspond to the stable intersection of two cantor sets in the plane

localized behavior in the lyapunov vectors for quasi one dimensional many hard disk systems we introduce a definition of a localization width whose logarithm is given by the entropy of the distribution of particle component amplitudes in the lyapunov vector different types of localization widths are observed for example a minimum localization width where the components of only two particles are dominant we can distinguish a delocalization associated with a random distribution of particle contributions a delocalization associated with a uniform distribution and a delocalization associated with a wavelike structure in the lyapunov vector using the localization width we show that in quasi one dimensional systems of many hard disks there are two kinds of dependence of the localization width on the lyapunov exponent index for the larger exponents one is exponential and the other is linear differences due to these kinds of localizations also appear in the shapes of the localized peaks of the lyapunov vectors the lyapunov spectra and the angle between the spatial and momentum parts of the lyapunov vectors we show that the krylov relation for the largest lyapunov exponent ensuremath lambda ensuremath sim ensuremath ensuremath rho mathrm ln ensuremath rho as a function of the density ensuremath rho is satisfied apart from a factor in the same density region as the linear dependence of the localization widths is observed it is also shown that there are asymmetries in the spatial and momentum parts of the lyapunov vectors as well as in their x and y components

remarks on the mean field dynamics of networks of chaotic elements abstract fluctuations of the mean field of a globally coupled dynamical systems are discussed the origin of hidden coherence is related with the instability of the fixed point solution of the self consistent perron frobenius equation collective dynamics in globally coupled tent maps are re examined both with the help of direct simulation and the perron frobenius equation collective chaos in a single band state and bifurcation against initial conditions in a two band state are clarified with the return maps of the mean field lyapunov spectra and also the newly introduced lyapunov exponent for the perron frobenius equation future problems on the collective dynamics are discussed

the q analogue of the alternating group and its representations we define the new algebra this algebra has a parameter q the defining relations of this algebra at q 1 coincide with the basic relations of the alternating group we also give the new subalgebra of the hecke algebra of type a which is isomorphic to this algebra this algebra is free of rank half that of the hecke algebra hence this algebra is regarded as a q analogue of the alternating group all the isomorphism classes of the irreducible representations of this algebra and the q analogue of the branching rule between the symmetric group and the alternating group are obtained

henon like mappings in c we study the dynamics of a class of nonalgebraic holomorphic diffeomorphisms topo logical analogues in the unit bidisk of complex h6non mappings in c2 in particular a dynamical degree is defined which is related to topological entropy and we construct stable unstable invariant currents and prove there is a unique mixing measure of maximal entropy with product structure 1 introduction one of the most famous 2 dimensional dynamical systems is smale s horseshoe it is a basic example of a simple map with complicated dynamics and has served as a guideline for the whole modern development of discrete time dynamical systems a very natural problem in this context is to understand the dynamical phenomena occuring when uniform hyperbolicity of the horseshoe degenerates this was a motivation for the study of the so called henon map

su n meander determinants we propose a generalization of meanders i e configurations of non self intersecting loops crossing a line through a given number of points to su n this uses the reformulation of meanders as pairs of reduced elements of the temperley lieb algebra a su 2 related quotient of the hecke algebra with a natural generalization to su n we also derive explicit formulas for su n meander determinants defined as the gram determinants of the corresponding bases of the hecke algebra

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

stability and bifurcations for dissipative polynomial automorphisms of mathbb c 2 we study stability and bifurcations in holomorphic families of polynomial automorphisms of mathbb c 2 we say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there we first show that this defines a meaningful notion of stability which parallels in many ways the classical notion of j stability in one dimensional dynamics define the bifurcation locus to be the complement of the weak stability locus in the second part of the paper we prove that under an assumption of moderate dissipativity the parameters displaying homoclinic tangencies are dense in the bifurcation locus this confirms one of palis conjectures in the complex setting the proof relies on the formalism of semi parabolic bifurcation and the construction of critical points in semi parabolic basins which makes use of the classical denjoy carleman ahlfors and wiman theorems

hamiltonian mean field model effect of network structure on synchronization dynamics the hamiltonian mean field model of coupled inertial hamiltonian rotors is a prototype for conservative dynamics in systems with long range interactions we consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix by studying the linear stability of the incoherent state we find that the transition to synchrony begins when the coupling constant k is inversely proportional to the largest eigenvalue of the adjacency matrix we derive a closed system of equations for a set of local order parameters to study the effect of network heterogeneity on the synchronization of the rotors when k is just beyond the transition to synchronization we find that the degree of synchronization is highly dependent on the network s heterogeneity but that for large k the degree of synchronization is robust to changes in the degree distribution our results are illustrated with numerical simulations on erd os renyi networks and networks with power law degree distributions

composition factors of specht modules for hecke algebras of type an abstract we examine the composition factors of specht modules for hecke algebras of type a n at roots of zero and their positions in the jantzen schaper filtration each specht module is decomposed into a direct sum of orthogonal subspaces corresponding to residue classes of standard tableaux similarly for the gram matrix we show that for a given subset of these classes the corresponding invariant factors of the gram matrix over a local ring completely determine the decomposition matrix and jantzen schaper filtration from this we deduce elementary proofs for a number of well known results notably the james first column theorem together with its generalisation by donkin and the determination of the decomposition matrices for two part partitions for the symmetric group algebra and for general linear groups we extend these to analogous results for the jantzen schaper filtration in particular we derive a closed formula for the jantzen indices for two part partititions two column diagrams in our formulation in the case of the hecke algebra

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

permutation resolutions for specht modules of hecke algebras abstract in boltje and hartmann 2011 bh a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the dual of the integral specht module for the symmetric group in terms of permutation modules in this paper we extend the definition of the chain complex to the integral iwahori hecke algebra and prove the same partial exactness results that were proven in the symmetric group case

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

the representation dimension of hecke algebras and symmetric groups abstract we establish a lower bound for the representation dimension of all the classical hecke algebras of types a b and d for all the type a algebras and most of the algebras of types b and d we also establish upper bounds moreover we establish bounds for the representation dimension of group algebras of some symmetric groups

irreducible specht modules for hecke algebras of type a abstract let f be a field n a non negative integer a partition of n and s the corresponding specht module for the iwahori hecke algebra h f q s n james and mathas conjecture a necessary and sufficient condition on for s to be irreducible we prove the sufficiency of this condition in the case where f has infinite characteristic and also in the case where q 1

koszul property and poincare series of matrix bialgebras of typean abstract bialgebras defined by means of yang baxter operators which verify the hecke equation are considered it is shown that they are koszul algebras their poincare series are calculated via the poincare series of the corresponding quantum planes

semisimplification of the category of tilting modules for gln abstract we describe the semisimplification of the monoidal category of tilting modules for the algebraic group g l n in characteristic p 0 in particular we compute the dimensions of the indecomposable tilting modules modulo p

on the relaxation dynamics of the kuramoto oscillators with small inertia for the kuramoto oscillators with small inertia we present several quantitative estimates on the relaxation dynamics and formational structure of a phase locked state pls for some classes of initial configurations in a super critical regime where the coupling strength is strictly larger than the diameter of natural frequencies we present quantitative relaxation dynamics on the collision numbers and the structure of pls in a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies we provide a sufficient condition for an asymptotically pls solution in particular we show the existence of slow relaxation to a pls when there are exactly two natural frequencies this generalizes the earlier results of choi et al asymptotic formation and orbital stability of phase locked states for the kuramoto model physica d 241 735 754 2012 10 1016 j physd 2011 11 011 choi et al complete synchronization of kuramoto oscillators with finite inertia physica d 240

a frobenius formula for the characters of the hecke algebras this paper uses the theory of quantum groups and the quantum yang baxter equation as a guide in order to produce a method of computing the irreducible characters of the hecke algebra this approach is motivated by an observation of m jimbo giving a representation of the hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group u q mathfrak s l n by rewriting the solutions of the quantum yang baxter equation for u q mathfrak s l n in a different form one can avoid the quantum group completely and produce a frobenius formula for the characters of the hecke algebra by elementary methods using this formula we derive a combinatorial rule for computing the irreducible characters of the hecke algebra this combinatorial rule is aq extension of the murnaghan nakayama for computing the irreducible characters of the symmetric group along the way one finds connections apparently unexplored between the irreducible characters of the hecke algebra and hall littlewood symmetric functions and kronecker products of symmetric groups

on tensor spaces over hecke algebras of type bn abstract let w b n be the weyl group of type b n and h b n be the associated iwahori hecke algebra in this paper we study the n tensor space v n where dim v 2 m with natural actions introduced in r m green hyperoctahedral schur algebras j algebra 192 1997 418 438 of w b n and of h b n for each composition 1 m of n let e be the corresponding initial basis element of v n see 3 8 for definition we show that if d is a distinguished right coset representative of s in w b n then the action of the natural basis element t d on e coincides with the permutation action of d up to a scalar as an application we prove that the n tensor space decomposes at the integral level into a direct sum of some permutation modules over hecke algebra h b n with respect to certain standard parabolic subalgebras

chaos and relaxation to equilibrium in systems with long range interactions in the thermodynamic limit systems with long range interactions do not relax to equilibrium but become trapped in nonequilibrium stationary states for a finite number of particles a nonequilibrium state has a finite lifetime so that eventually a system will relax to thermodynamic equilibrium the time that a system remains trapped in a quasistationary state qss scales with the number of particles as n with 0 and diverges in the thermodynamic limit in this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a qss we discover that chaos measured by the lyapunov exponents favors faster relaxation to equilibrium surprisingly weak chaos favors faster relaxation than strong chaos

newhouse phenomenon for automorphisms of low degree in c3 abstract we show that there exists a polynomial automorphism f of c 3 of degree 2 such that for every automorphism g sufficiently close to f g admits a tangency between the stable and unstable laminations of some hyperbolic set as a consequence for each d 2 there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense to prove these results we give a complex analogous to the notion of blender introduced by bonatti and diaz

quantum weyl reciprocity for cohomology we study the relationship between the cohomology of q schur algebras and hecke algebras in type a a key tool which has independent interest involves a specific hecke algebra resolution introduced in a special case by deodhar this resolution is somewhat akin to the koszul resolution for lie algebra cohomology in that it has an explicit description yet applies for all modules an application of the properties of this resolution establishes a quantum weyl reciprocity connecting in a range of degrees the cohomology of the q schur algebra with that of the hecke algebra this work also provides independently of the resolution cohomological inequalities which apply in all degrees and which sometimes can be related to kazhdan lusztig polynomials finally we extend to cohomology a decomposition number method of erdmann and illustrate its usefulness in computations this method enables us to exhibit some properties of symmetric group cohomology which if true would imply the sl n version of lusztig s characteristic p character conjecture

macroscopic chaos in globally coupled maps abstract we study the coherent dynamics of globally coupled maps showing macroscopic chaos with this term we indicate the hydrodynamical like irregular behavior of some global observables with typical times much longer than the times related to the evolution of the single or microscopic elements of the system the usual lyapunov exponent is not able to capture the essential features of this macroscopic phenomenon using the recently introduced notion of finite size lyapunov exponent we characterize in a consistent way these macroscopic behaviors basically at small values of the perturbation we recover the usual microscopic lyapunov exponent while at larger values a sort of macroscopic lyapunov exponent emerges which can be much smaller than the former a quantitative characterization of the chaotic motion at hydrodynamical level is then possible even in the absence of the explicit equations for the time evolution of the macroscopic observables

representations and traces of the hecke algebras hn q of type an 1 the notion of the connectivity class of minimal words in the algebra hn q is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented guided by these results the connection between the ocneanu trace on hn q and schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of hn q they appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions their evaluation involves the use of the littlewood richardson rule which is proved to apply to hn q just as it does to sn both representation matrices and characters are tabulated

peeling the onion of order and chaos in a high dimensional hamiltonian system abstract coexistence of various ordered chaotic states in a hamiltonian system is studied with the use of a symplectic coupled map lattice besides the clustered states for the attractive interaction a novel chaotic ordered state is found for a system with repulsive interaction characterized by a dispersed state of particles the dispersed and clustered states form an onion like structure in phase space the degree of order increases towards the center of the onion while chaos is enhanced at the edge between ordered and random chaotic states for a longer time scale orbits itinerate over ordered and random states the existence of these ordered states leads to anomalous long time correlation for many quantifiers such as the global diffusion

routes to chaos in high dimensional dynamical systems a qualitative numerical study this paper examines the most probable route to chaos a high dimensional dynamical systems function space time delay neural networks endowed with a probability measure in a computational setting the most probable route to chaos relative to the measure we impose on the function space as the dimension is increased is observed to be a sequence of neimark sacker bifurcations into chaos the analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn a scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high dimensional dissipative dynamical systems is also presented c 2006 elsevier b v all rights reserved

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

quiver hecke algebras for alternating groups the main result of this paper shows that over large enough fields of quantum characteristic different from 2 the alternating hecke algebras are mathbb z graded algebras that are isomorphic to fixed point subalgebras of the quiver hecke algebra of the symmetric group mathfrak s n as a special case this shows that the group algebra of the alternating group over large enough fields of characteristic different from 2 is a mathbb z graded algebra we give a homogeneous presentation for these algebras compute their graded dimension and show that the blocks of the quiver hecke algebras of the alternating group are graded symmetric algebras

z2 graded clifford system in the hecke algebra of type bn abstract we extend the representation theory of the q analogue of the alternating group to the hecke algebra of type bn we show that the hecke algebra of type bn has a z 2 graded clifford system using this result we introduce a new subalgebra in the hecke algebra of type bn this subalgebra is an extension of the q analogue of the alternating group the branching rule for the hecke algebra of type bn is given by making use of the generalized clifford theory

canonical basic sets in type bn more than 10 years ago dipper james and murphy developed the theory of specht modules for hecke algebras of type bn more recently using lusztig s a function geck and rouquier showed how to obtain parametrizations of the irreducible representations of hecke algebras of any finite type in terms of so called canonical basic sets for certain values of the parameters in type bn combinatorial descriptions of these basic sets were found by jacon based on work of ariki and foda leclerc okado thibon welsh here we consider the canonical basic sets for all the remaining choices of the parameters

specht modules labelled by hook bipartitions i abstract brundan kleshchev and wang equip the specht modules s over the cyclotomic khovanov lauda rouquier algebra h n with a homogeneous z graded basis in this paper we begin the study of graded specht modules labelled by hook bipartitions n m 1 m in level 2 of h n which are precisely the hecke algebras of type b with quantum characteristic at least three we give an explicit description of the action of the khovanov lauda rouquier algebra generators 1 n 1 on the basis elements of s n m 1 m introducing certain specht module homomorphisms we construct irreducible submodules of these specht modules and thereby completely determine the composition series of specht modules labelled by hook bipartitions

generic family with robustly infinitely many sinks we show for every r d ge 0 or r d ge 2 the existence of a baire generic set of c d families of c r maps f a a in mathbb r k of a manifold m of dimension ge 2 so that for every a small the map f a has infinitely many sinks when the dimension of the manifold is ge 3 the generic set is formed by families of diffeomorphisms when m is the annulus this generic set is formed by local diffeomorphisms this is a counter example to a conjecture of pugh and shub

lyapunov analysis and information flow in coupled map lattices abstract lyapunov analysis for the coupled map lattices is presented the co moving lyapunov exponent is calculated which is related with the propagation of the disturbance in space the propagation speed agrees with the zero crossing point of the co moving lyapunov exponent the co moving mutual information flow is introduced which shows the selective transmission of the information at some speed lyapunov spectra and vectors are calculated spatial structures of the vectors are investigated possible analogy with the anderson localization problem is discussed

an algorithm for semistandardising homomorphisms abstract suppose is a partition of n and a composition of n and let s m denote the specht module and permutation module defined by dipper and james for the iwahori hecke algebra h n of the symmetric group w n we give an explicit fast algorithm for expressing a tableau homomorphism a s m as a linear combination of semistandard homomorphisms along the way we provide a utility result related to removing rows from tableaux

bounds on the dimension of ext for finite groups of lie type abstract let g be a finite group of lie type defined in characteristic p and let k be an algebraically closed field of characteristic r 0 we will assume that r p so we are in the non defining characteristic case let v be a finite dimensional irreducible left kg module in 2011 guralnick and tiep found bounds on the dimension of h 1 g v in non defining characteristic which are independent of v the aim of this paper is to generalize the work of gurlanick and tiep we assume that g is split and use methods of modular harish chandra theory to find bounds on the dimension of ext k g 1 y v where y and v are irreducible kg modules we then use dipper and du s algorithms to illustrate our bounds in a series of examples

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

w graph ideals abstract we introduce a concept of a w graph ideal in a coxeter group the main goal of this paper is to describe how to construct a w graph from a given w graph ideal the principal application of this idea is in type a where it provides an algorithm for the construction of w graphs for specht modules

the affineq schur algebra abstract we introduce an analogue of the q schur algebra associated to coxeter systems of type a n 1 we give two constructions of this algebra the first construction realizes the algebra as a certain endomorphism algebra arising from an affine hecke algebra of type a r 1 where n r this generalizes the original q schur algebra as defined by dipper and james and the new algebra contains the ordinary q schur algebra and the affine hecke algebra as subalgebras using this approach we can prove a double centralizer property the second construction realizes the affine q schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module analogous to the construction of the ordinary q schur algebra as a quotient of u g l n