linear stability of symplectic maps a general method is presented for analytically calculating linear stability limits for symplectic maps of arbitrary dimension in terms of the coefficients of the characteristic polynomial and the krein signatures explicit results are given for dimensions 4 6 and 8 the codimension and unfolding are calculated for all cases having a double eigenvalue on the unit circle the results are applicable to many physical problems including the restricted three body problem and orbital stability in particle accelerators

an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

products of three triangular matrices over commutative rings abstract we characterize commutative rings with 1 over which any n n matrix a can be written as lum where l m are lower triangular u is upper triangular and l and u have all their diagonal entries 1

phase pulling and breather instability in an ac driven damped one dimensional sine gordon system abstract an ac driven damped sine gordon system in the nonlinear schrodinger regime may converge into various nontrivial space time attractors one of them is breather riding on top of flat profile both locked to the driver frequency using collective coordinate ansatz for this attractor and a multiple scale perturbation technique we study the location and nature of its instability among the various characteristics this study includes understanding the role of the breather background coupling the structure of the associated unstable modes the scaling structure of the instability boundary etc most strikingly we find the instability mechanism and the above features to be strongly sensitive to the asymptotic phase difference between the spatially localized breather and the spatially extended uniform background we call this phenomenon phase pulling

stability of water waves we apply some general results for hamiltonian systems depending on the notion of signature of eigenvalues to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability up to non degeneracy assumptions a new superharmonic instability is predicted and verified

involutary expressions for elements in gln z and sln z abstract we shall give upper bounds for the sets of minimal numbers of involutary factors of elements in the general linear group gl n z and the special linear group sl n z over the ring of rational integers z 0 1 2 for n 2 respectively

expressing infinite matrices as products of involutions abstract we take into consideration ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 we show that for any field k every element of this group can be expressed as a product of at most five involutions moreover we prove that if k is a field of characteristic different from 2 then four involutions suffice

travelling solitons in the externally driven nonlinear schrodinger equation we consider the undamped nonlinear schr dinger equation driven by a periodic external force in the absence of damping solitons do not have undulations on their tails yet we show that they can bind into stationary multisoliton complexes using two previously known stationary solitons and two newly found stationary complexes as starting points we obtain classes of localized travelling waves by the numerical continuation in the parameter space two families of stable solitons are identified one family is stable for sufficiently low velocities while solitons from the second family stabilize when travelling faster than a certain critical speed the stable solitons of the former family can also form stably travelling bound states

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

transition probability preserving maps on a grassmann space in a semifinite factor abstract in this paper the surjective transition probability preserving transformations on the grassmann space of infinite projections in an infinite semifinite factor are characterized as in the classical wigner s theorem all these maps are induced by isomorphisms or anti isomorphisms of the semifinite factor

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

bifurcation and chaotic motion of a soliton in a long josephson junction oscillator the fluxon dynamics of a long josephson junction ljj is investigated numerically the effect of the surface resistance of the superconductor on the bifurcation phenomenon is examined in detail we show the occurrence of a period doubling route to chaos and the reappearance of low order periodic motions in the ljj by varying the magnitude of the external force the chaotic attractors of the ljj are characterized using maximal lyapunov exponents and a fractal correlation dimension we show the existence of noise induced oscillations in a force free ljj system finally the influence of noise on the periodic and chaotic dynamics is also presented

period doubling solitons yes or no we investigate the possibility of period doubling transition to temporal chaos of a coherent structure in a finite dimensional hamiltonian system with moderately small friction a solitary wave in a periodically driven chain of particles provides a typical example to this end the equations of motion are transformed to those of one driven dominant oscillator with one degree of freedom weakly coupled to a set of oscillators representing the other degrees of freedom the equations for the dominant oscillator alone confirm the possibility of such a transition to chaos for well chosen driving however taking into account coupling to the other degrees of freedom one must conclude that the supposed infinite period doubling sequence for small or no dampling survives only in a heavily damaged state in practice a reproducable completed sequence cannot be expected unless the damping is strong enough

the random phase property and the lyapunov spectrum for disordered multi channel systems a random phase property establishing in the weak coupling limit a link between quasi one dimensional random schrodinger operators and full random matrix theory is advocated briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a markov process with a unique invariant measure which is of geometric nature on the elliptic part of the transfer matrices this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study while the random phase property can up to now only be proved in special models or in a restricted sense we provide strong numerical evidence that it holds in the anderson model of localization a main outcome of the random phase property is a perturbative calculation of the lyapunov exponents which shows that the lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary orthogonal and symplectic ensemble differ by a factor 2 each in an anderson ando model on a tubular geometry with magnetic field and spin orbit coupling the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the lyapunov spectrum

transformations on the sets of states and density operators in this paper we consider two physically relevant numerical quantities namely the concurrence and fidelity and an operation called product defined on the set of all quantum states or density operators our main aim is to present the complete descriptions of all bijective transformations which preserve any of these quantities or operation

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

conjugacy classes in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n n over fq the finite field with q pt elements higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we develop the algorithm given in j algebra 177 1995 899 introducing new ideas and theoretical properties which lead us to get the conjugacy vector of gn for n 13 these vectors for n 5 resp for n 6 7 8 were given in j algebra 152 1 1992 1 resp j algebra 177 1995 899 in particular we conclude that for these values the number of conjugacy classes of gn is a polynomial in q with integral coefficients and degree n n 6 12 thus higman s conjecture holds for n 13 on the other hand for each positive integer n we find explicitly a set of q n n 6 12 different conjugacy classes

hierarchy of chaotic bands results of a detailed numerical study on the structure of chaotic bands in a forced limit cycle oscillator the brusselator are presented embedded in the chaotic bands of primary bifurcation sequence there are many secondary sequences with both direct and inverse segments within secondary chaotic bands tertiary sequences of similar structure exist this has been shown by using subharmonic stroboscopic sampling combined with power spectra analysis

on exact solitary wave solutions of the nonlinear schrodinger equation with a source we use a fractional transformation to connect the travelling wave solutions of the nonlinear schrodinger equation nlse phase locked with a source to the elliptic equations satisfying f af f3 0 the solutions are necessarily of rational form containing both trigonometric and hyperbolic types as special cases bright and dark solitons as well as singular solitons are obtained in a suitable range of parameter values

on products of matrices of a fixed order abstract we study products of matrices of fixed orders we show that if g is an upper triangular matrix finite or infinite over a field of q elements then g can be expressed as a product of at most four triangular matrices whose orders are divisors of q 1 this result can be applied to the general linear and to the vershik kerov group we also present some facts about conjugacy of elements of orders dividing q 1

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

the center manifold and bifurcations of damped and driven sine gordon breathers abstract the generic bifurcations of breathers in the damped and driven sine gordon equation are investigated both numerically and analytically the linear stability analysis and information from periodic spectral theory suggest that three modes are relevant for the system they correspond to frequency and temporal phase changes and to the flat pendulum using these modes nonautonomous amplitude equations are derived and compared with numerical simulations of the perturbed sine gordon equation

transition probabilities of normal states determine the jordan structure of a quantum system let x1d516 m1 x1d516 m2 be a bijection not assumed affine nor continuous between the sets of normal states of two quantum systems modelled on the self adjoint parts of von neumann algebras m1 and m2 respectively this paper concerns with the situation when preserves or partially preserves one of the following three notions of transition probability on the normal state spaces the transition probability pu introduced by uhlmann rep math phys 9 273 279 1976 the transition probability pr introduced by raggio lett math phys 6 233 236 1982 and an asymmetric transition probability p0 as introduced in this article it is shown that the two systems are isomorphic i e m1 and m2 are jordan isomorphic if preserves all pairs with zero uhlmann respectively raggio or asymmetric transition probability in the sense that for any normal states and we have p 0 if and only if p 0 where p stands for pu respectively pr or p0 furthermore as an extension of wi

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

conjugacy in groups of upper triangular matrices abstract we show that in the group u n f 2 a sylow 2 subgroup of gl n f 2 there are elements not conjugate to their inverses if n 13 it follows that there are irreducible characters of this group that are not real valued and that a conjecture of kirillov is not correct as stated we give a partial explanation of why these group elements do not exist if n 12 and an example to show that analogous phenomena can occur for odd primes

breather dynamics in the nonlinear schrodinger regime of perturbed sine gordon systems abstract a possible route to temporal chaos with coherent stable spatial structures is proposed for the driven damped sine gordon equation for near conservative perturbations the dynamics of a breather is investigated numerically and semi analytically in the presence of an ac driver and a simple damping term for moderate driver strengths a flat space independent attractor exists whereas above a threshold a phase locked breather co exists the latter can undergo a period doubling route to temporal chaos as is shown here for a certain parameter regime relations to other works which operate in a different parameter regime are discussed the near conservative perturbations and the low driver strengths allow to interpret the results within a simple model originating from the so called nonlinear schrodinger limit in fact within this limit small amplitude breather the chaotic or not transitions are dominated by interactions between breather like solutions and radiation mostly k 0 mode therefore three collective coordinates i e the amplitude of the phase locked breather its phase as well as the complex amplitude of the linear k 0 mode are sufficient to construct a system of four ordinary differential equations of first order which reveal the basic features of the partial differential equations in a satisfactory manner

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

wigner type theorem on transition probability preserving maps in semifinite factors abstract the wigner s theorem which is one of the cornerstones of the mathematical formulation of quantum mechanics asserts that every symmetry of quantum system is unitary or anti unitary this classical result was first given by wigner in 1931 thereafter it has been proved and generalized in various ways by many authors recently g p geher extended wigner s and molnar s theorems and characterized the transformations on the grassmann space of all rank n projections which preserve the transition probability the aim of this paper is to provide a new approach to describe the general form of the transition probability preserving not necessarily bijective maps between grassmann spaces as a byproduct we are able to generalize the results of molnar and g p geher

symmetry transformations in indefinite metric spaces a generalization of wigner s theorem we formulate and prove a generalization to indefinite metric spaces of uhlhorn s version of wigner s theorem

on the attracting set for duffing s equation ii a geometrical model for moderate force and damping abstract after a brief review of some earlier work on duffing s equation in the small force and damping regions we use the results of numerical integrations to construct a geometrically defined poincare map which captures the qualitative features of the attracting set of larger force and damping levels this map has a small constant jacobian determinant and can be regarded as a perturbation of a non invertible one dimensional map we give a partial analysis of the map and pose some important open questions regarding perturbations of one dimensional maps and the creation of strange attractors during bifurcation to horseshoes

low dimensional chaos in a driven damped nonlinear schro dinger equation abstract we study bifurcations of attractors in a driven damped nonlinear schrodinger equation which models nonlinear responses of a plasma driven by a rf field and a sequence of period doubling bifurcations is shown to lead to a chaotic attractor in a low dimensional subspace spanned by a soliton and long wavelength radiation

upper triangular similarity of upper triangular matrices abstract we consider the following equivalence relation in the set of all complex upper triangular n n matrices a and b are called u similar if there exists an invertible upper triangular matrix s such that a s 1 bs if a b are u similar then they must have the same diagonal and the same jordan form it is known that for n 6 there are infinitely many mutually non u similar nilpotent upper triangular matrices with the same jordan form we introduce an appropriate generalization of the jordan block called an irreducible matrix and we prove that each upper triangular matrix is u similar to a generalized direct sum of irreducible blocks where the location and the order of the blocks is fixed and each block is determined uniquely up to u similarity

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

a generalization of wigner s unitary antiunitary theorem to hilbert modules let h be a hilbert c module over a matrix algebra a it is proved that any function t h h which preserves the absolute value of the generalized inner product is of the form tf f uf f h where is a phase function and u is an a linear isometry the result gives a natural extension of wigner s classical unitary antiunitary theorem for hilbert modules

on the stability of periodic orbits of high dimensional autonomous hamiltonian systems we study the stability of periodic orbits of autonomous hamiltonian systems with n 1 degrees of freedom or equivalently of 2n dimensional symplectic maps with n 1 we classify the different stability types introducing a new terminology which is perfectly suited for systems with many degrees of freedom since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix on the complex plane the different stability types correspond to different regions of the n dimensional parameter space s defined by the coefficients of the characteristic polynomial of the monodromy matrix all the possible direct transitions between different stability types are classified and the corresponding transition hypersurface in s is determined the dimension of the transition hypersurface is an indicator of how probable to happen is the corresponding transition as an application of the general results we consider the well known cases of hamiltonian systems with two and three degrees of freedom we also describe in detail the different stability regions in the three dimensional parameter space s of a hamiltonian system with four degrees of freedom or equivalently of a six dimensional symplectic map 2001 elsevier science b v all rights reserved

wigner s theorem on grassmann spaces wigner s celebrated theorem which is particularly important in the mathematical foundations of quantum mechanics states that every bijective transformation on the set of all rank one projections of a complex hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator this vital theorem has been generalised in various ways by several scientists in 2001 molnar provided a natural generalisation namely he provided a characterisation of not necessarily bijective maps which act on the grassmann space of all rank n projections and leave the system of jordan principal angles invariant see 17 and 20 in this paper we give a very natural joint generalisation of wigner s and molnar s theorems namely we prove a characterisation of all not necessarily bijective transformations on the grassmann space which fix the quantity trpq i e the sum of the squares of cosines of principal angles for every pair of rank n projections p and q

products of three triangular matrices abstract we prove that any square matrix over a field is the product of at most three triangular matrices we also give explicit lul factorizations for all 2 2 and 3 3 matrices over fields

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

maximum likelihood estimators for generalized cauchy processes maximum likelihood estimator mle for a generalized cauchy process gcp is studied with the aid of the method of information geometry in statistics our gcp is described by the langevin equation with multiplicative and additive noises the exact expressions of mles are given for the two cases that the two types of noises are uncorrelated and mutually correlated it is shown that the mles for these two gcps are free from divergence even in the parameter region wherein the ordinary moments diverge the mle relations can be regarded as a generalized fluctuation dissipation theorem for the present langevin equation availability of them and of some other higher order statistics is demonstrated theoretically and numerically

generalization of wigner s unitary antiunitary theorem for indefinite inner product spaces we present a generalization of wigner s unitary antiunitary theorem for pairs of ray transformations as a particular case we get a new wigner type theorem for non hermitian indefinite inner product spaces

quasiperiodic bifurcation in nonlinearly coupled oscillators near a point of strong resonance a pair of uncoupled linear oscillators have natural frequencies delta and gamma and damping coefficient proportional to lambda when they are coupled through a class of cubic nonlinearities the null solution bifurcates at lambda lambda c the bifurcating solutions are analyzed near the strong resonance delta gamma by relating these delta gamma frac 1 2 k varepsilon 2 to the amplitude varepsilon and using multiple scale theory the resulting approximation is uniformly valid even for the far from resonance case k o varepsilon 2 an entrainment parameter rho is defined rho 2 delta 2 lambda lambda c for which a critical value rho c exists for rho leqq rho c there are mixed mode frequency locked periodic oscillations rho rho c the oscillations depend on two frequencies and so are periodic or quasiperiodic depending on the ratio delta gamma