weak closedness of subspaces of fourier stieltjes algebras and weak continuity of the restriction map let g be a locally compact group and b g the fourier stieltjes algebra of g we study the problem of how weak8 closedness of some translation invariant subspaces of b g is related to the structure of g moreover we prove that for a closed subgroup h of g the restriction map from b g to b h is weak8 continuous only when h is open in g introduction let g be a locally compact group and let b g be the fourier stieltjes algebra of g as defined by eymard 8 recall that b g is the linear span of all continuous positive definite functions on g and can be identified with the banach space dual of c g the group c algebra of g the space b g with the xlorm as dual of c g is a commutative banach algebra with pointwise multiplication and complex conjugation the fourier algebra a g of g is the closed subalgebra of b g generated by the functions in b g with compact support in particular a g is contained in co g the algebra of complex valued continuous functions on g vanishing at infinity as is well known a g is weak dense in b g if and only if g is amenable in 3 translation invariant subalgebras a of b g were studied and it was shown that if such a is weak closed and point separating then it must contain a g however apart from this very little seems to be known about weak closed subspaces of b g the first purpose of this paper is to investigate the relation between weak closedness of certain interesting norm closed translation invariant subspaces of b g and the structure of g secondly we solve the problem of whexl for a closed subgroup h of g the restriction map from b g to b h is weak continuous a brief outline of the paper is as follows in section 2 we estsablish for almost connected locally compact groups g the relation between weclk closed ness of bo g b g n co g in e3 g and the structure of g theorem 2 10 the key result is that for a connected lie group g bo g is weak closed in b g if and only if g is a reductive lie group with compact centre and kazhdan s property t received by the editors december 1s 1995 199l mathematics subject classification primary 22d10 43a30 work supported by nato collaborative research grant crg 940184 g 1997 american mathematical society

non stationary parametric oscillations the response of a single degree of freedom system with cubic non linearity to a non stationary principal parametric excitation is investigated the method of multiple scales is used to derive two first order ordinary differential equations for the evolution of the amplitude and phase of the response the evolution equations are numerically integrated for various sweeping rates of the amplitude and frequency of the excitation the results show that the non stationary response penetrates the instability regions and the higher the sweeping rate is the deeper the penetration is

a generalized local linearization principle for non linear dynamical systems tangent spaces in non linear dynamical systems are state dependent hence it is generally not possible to exactly represent a non linear dynamical system by a linear one over finite segments of the evolving trajectories in the phase space it is known from the well known theorem of hartman and grobman that a non linear oscillator admits a linearization within a neighbourhood of any hyperbolic fixed point in the sense that the linearized flow bears a topological conjugacy to the non linear flow within the same neighbourhood this linearization is based on the construction of the tangent space to the non linear vector field at the hyperbolic fixed point even though useful in bifurcation analysis such a linearization procedure cannot be successfully applied to identically simulate the non linear flow over the phase space because the functional form of the topological conjugacy is not known with this in mind an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non linear dynamical system at that point in the state space where the solution vector is desired several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results it is shown that the locally transversal linearization ltl procedure finally reduces the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations with the desired solution vector as the unknown the variables in the state space appear as unknown quantities in this approach finally one arrives at non linear poincare maps for the non linear oscillator in this scheme the characteristic time interval for the map is arbitrarily fixed the methodology is found to be quite versatile for handling non linear dynamical systems in particular it is verified that these schemes are capable of accurately predicting a wide spectrum of typically non linear response characteristics such as limit cycles multi periodicity almost periodicity and chaos for a limited class of response patterns the principle proposed has another advantage in that the time step for integrating the non linear odes need not be small an improved and a more general higher order version of the ltl method is also considered a distinct advantage of this higher version is in its improved ability for a closer simulation of phase dependent time histories where a difference in the initial conditions leads to a phase difference in the realized solution history

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

application of the incremental harmonic balance method to cubic non linearity systems abstract in this paper the formulation of the incremental harmonic balance ihb method is derived for a general system of differential equations with cubic non linearity which governs a wide range of engineering problems such as large amplitude vibration of beams or plates an incremental arc length method combined with a cubic extrapolation technique is adopted to trace the response curve automatically the stability of its periodic solutions can also be evaluated from the ihb formulation by multi variable floquet theory hsu s method is adopted for computing the transition matrix at the end of one period two numerical examples are presented which demonstrate the effectiveness of the ihb method and hsu s method in treating the non linear vibration of a strongly non linear multiple degree of freedom system

the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance abstract the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes steady state responses and their stability are computed for selected values of the system parameters the effects of detuning the internal resonance detuning the parametric resonance the phase and magnitude of the second mode parametric excitation and the initial conditions are investigated the first order perturbation solution predicts qualitatively the trivial and non trivial stable steady state solutions and illustrates both the quenching and saturation phenomena in addition to the steady state solutions other periodic solutions are predicted by the perturbation amplitude and phase modulation equations these equations predict a transition from constant steady state non trivial responses to limit cycle responses hopf bifurcation some limit cycles are also shown to experience period doubling bifurcations the perturbation solutions are verified by numerically integrating the governing differential equations

on crossed product of algebras the concept of a crossed tensor product of algebras is studied from a few points of view some related constructions are considered crossed enveloping algebras and their representations are discussed applications to the noncommutative geometry and particle systems with generalized statistics are indicated

quantum group covariant anti symmetrizers tensors vielbein hodge map and laplacian glq n and soq n covariant deformations of the completely symmetric antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices the precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined adopting the glq n and soq n covariant differential calculi on the corresponding quantum group covariant noncommutative spaces we introduce a generalized notion of vielbein basis or frame based on differential operator valued 1 forms we then give a thorough definition of a soq n covariant bilinear hodge map acting on the bimodule of differential forms on introduce the exterior coderivative and show that the laplacian acts on differential forms exactly as in the undeformed case namely it acts on each component as it does on functions

the response of two degree of freedom systems with quadratic non linearities to a parametric excitation the method of multiple scales is used to analyze the response of two degree of freedom systems with quadratic non linearities to a parametric harmonic excitation having the frequency four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when 2 2 1 and either 2 1 or 2 2 where 1 and 2 are the linear undamped natural frequencies of the system two critical values 1 and 2 of the amplitude f of the excitation are identified in the analysis when f 2 the amplitude of the directly excited mode grows exponentially with time according to the linear analysis whereas the amplitudes of both modes achieve steady state constant values irrespective of the initial amplitudes according to the non linear analysis when f 1 the motion decays to zero according to both the linear and non linear analyses when 1 f 2 the motion decays to zero according to the linear analysis whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non linear analysis this is an example of subcritical instability when 2 2 the steady state value of the higher mode which is directly excited is a constant that is independent of the excitation of amplitude f whereas the amplitude of the lower mode which is indirectly excited through internal resonance grows with the excitation amplitude f this is another example of saturation

simultaneous combination principal parametric and internal resonances in a slender beam with a lumped mass three mode interactions abstract the non linear dynamics of a parametrically base excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined the mass and the position of the attached element are so adjusted that the system exhibits 1 3 5 internal resonance by virtue of which a combination resonance of type 1 3 occurs simultaneously in the system the method of multiple scales mms is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state periodic quasi periodic and chaotic responses of the system for different control parameters namely frequency and amplitude of base excitation damping internal detuning etc the results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion poincare section and lyapunov exponents are used to characterize the chaotic responses

the response of a single degree of freedom system with quadratic and cubic non linearities to a fundamental parametric resonance abstract the response of a one degree of freedom system with quadratic and cubic non linearities to a principal parametric resonance is investigated the method of multiple scales is used to determine the equations that describe to second order the modulation of the amplitude and phase with time about one of the foci these equations are used to determine the fixed points and their stability the perturbation results are verified by integrating the governing equation with use of a digital computer and an analogue computer for small excitation amplitudes the analytical results are in excellent agreement with the numerical solutions the large amplitude responses are investigated by using both a digital and an analogue computer the cases of single and double well potentials are investigated systems with double well potentials exhibit complicated dynamic behaviors including period multiplying and demultiplying bifurcations and chaos in some cases a bifurcated response coexists with another periodic attractor and a chaotic attractor coexists with a periodic attractor long time histories phase planes poincare maps fractal basin maps and spectra of the response are presented a bifurcation diagram of many solutions in the excitation amplitude excitation frequency plane is also presented

braided chains of q deformed heisenberg algebras given m copies of a q deformed weyl or clifford algebra in the defining representation of a quantum group we determine a prescription to embed them into a unique inclusive covariant algebra the different copies are coupled to each other and are naturally ordered into a chain in the case a modified prescription yields an inclusive algebra which is even explicitly covariant where is a symmetry relating the different copies by the introduction of these inclusive algebras we significantly enlarge the class of covariant deformed weyl clifford algebras available for physical applications

combination parametric resonance leading to periodic and chaotic response in two degree of freedom systems with quadratic non linearities the dynamic response of a parametrically excited non linear system with two degrees of freedom is studied the system of equations which describes the compliant motion of a robotic manipulator exhibits internal resonance and is excited at a frequency near a primary resonance the effect of coupling is studied in detail non zero periodic motions are observed to co exist with a stable equilibrium state over a range of detuning near the resonant frequency loss of stability of the periodic response is observed at points of vertical tangency corresponding to a jump in the response while for some of the cases hopf bifurcation in the response corresponding to amplitude modulated motions is observed in the latter case period doubling of the amplitude of response is observed with a monotonic change in detuning eventually leading to chaotic motion in the system near exact tuning to the parametric resonance a quenching of the chaotic motion is found leaving only the stable zero equilibrium state for the system over some detuning interval

geometrical tools for quantum euclidean spaces we apply one of the formalisms of noncommutative geometry to n q the quantum space covariant under the quantum group so q n over n q there are two so q n covariant differential calculi for each we find a frame a metric and two torsion free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature this generalizes results found in a previous article for the case of 3 q as in the case n 3 one has to slightly enlarge the algebra n q for n odd one needs only one new generator whereas for n even one needs two as in the particular case n 3 there is a conformal ambiguity in the natural metrics on the differential calculi over n q while in our previous article the frame was found by hand here we disclose the crucial role of the quantum group covariance and exploit it in the construction as an intermediate step we find a homomorphism from the cross product of n q with u q so n into n q an interesting result in itself

the response of two degree of freedom systems with quadratic and cubic non linearities to multifrequency parametric excitations abstract the response of two d o f systems with quadratic and cubic non linearities to multi frequency parametric excitations is determined by using the method of multiple scales four first order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously in all cases the steady state solutions and their stability are determined numerical results depicting the various resonances are presented

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

representation theory of lattice current algebras lattice current algebras were introduced as a regularization of the left and right moving degrees of freedom in the wznw model they provide examples of lattice theories with a local quantum symmetry their representation theory is studied in detail in particular we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra it is shown that for an arbitrary number of lattice sites the representation categories of the lattice current algebras agree with their continuum counterparts

bicovariant quantum algebras and quantum lie algebras a bicovariant calculus of differential operators on a quantum group is constructed in a natural way using invariant maps from fun mathfrak g q tou q g given by elements of the pure braid group these operators the reflection matrix y l sl being a special case generate algebras that linearly close under adjoint actions i e they form generalized lie algebras we establish the connection between the hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation fory inso q n

the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric resonance abstract an analysis is presented of the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type 3 2 1 where the n are the linear natural frequencies of the systems in the case of a fundamental resonance of the third mode i e 3 where is the frequency of the excitation one can identify two critical values 1 and 2 where 2 1 of the amplitude f of the excitation the value f 2 corresponds to the transition from stable to unstable solutions when f 2 the motion grows exponentially with time according to the linear theory but the non linearity limits the motion to a finite amplitude steady state the amplitude of the third mode which is directly excited is independent of f whereas the amplitudes of the first and second modes which are indirectly excited through the internal resonance are functions of f when 1 f 2 the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non linear theory whereas it decays to zero according to the linear theory this is an example of subcritical instability in the case of a fundamental resonance of either the first or second mode the trivial response is the only possible steady state when f 2 the motion decays to zero according to both linear and non linear theories when f 2 the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non linear theory experiments are being planned to check these theoretical results

the existence of inner invariant means on l g the purpose of this paper is to generalize the concepts of amenability for locally compact groups and inner amenability for discrete groups by considering the existence of inner invariant means based on this generalization locally compact groups can be classified as so called ia groups or non ia groups a number of equivalent conditions characterizing ia groups are given also the possibility of inner invariant extension of the dirac measure e is discussed

unbraiding the braided tensor product we show that the braided tensor product algebra a1 a2 of two module algebras a1 a2 of a quasitriangular hopf algebra h is isomorphic to the ordinary tensor product a1 a2 provided there exists a realization of h within a1 in other words under this assumption we construct a transformation of generators which decouples a1 a2 i e makes them commuting we apply the theorem to the braided tensor product algebras of two or more quantum group covariant quantum spaces deformed heisenberg algebras and q deformed fuzzy spheres

cross products by braided groups and bosonization let h be a braided cocommutative hopf algebra in a braided monoidal category c and b a hopf algebra in c on which h acts we construct a cross product hopf algebra b formula h in c as an application we show that every b in a certain class can be converted to an equivalent ordinary hopf algebra by a process of bosonization the class includes for example all super hopf algebras the constructions respect any quasitriangular structures on b as a corollary we show that if h r is an ordinary quasitriangular hopf algebra then the smash product by the adjoint action of h on itself had formula h can be given the structure of a hopf algebra we prove a similar result for any hopf algebra to which a quasitriangular hopf algebra maps

parametric excitation of two internally resonant oscillators abstract the response of two degree of freedom systems with quadratic non linearities to a principal parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential averaged equations governing the modulation of the amplitudes and the phases of the two modes these equations are used to determine steady state responses and their stability when the higher mode is excited by a principal parametric resonance the non trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude whereas the amplitude of the lower mode which is indirectly excited through the internal resonance increases with the amplitude of the excitation however in addition to poincare type bifurcations this response exhibits a hopf bifurcation leading to amplitude and phase modulated motions when the lower mode is excited by a principal parametric resonance the averaged equations exhibit both poincare and hopf bifurcations in some intervals of the parameters the periodic solutions of the averaged equations in the latter case experience period doubling bifurcations leading to chaos

parametrically excited non linear multidegree of freedom systems with repeated natural frequencies abstract a method for analyzing multidegree of freedom systems having a repeated natural frequency subjected to a parametric excitation is presented attention is given to the ordering of the various terms linear and non linear in the governing equations the analysis is based on the method of multiple scales as a numerical example involving a parametric resonance panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis some of the analytical results are verified by a numerical integration of the governing equations

realization of u q so n within the differential algebra on r q n we realize the hopf algebrauq 1 so n as an algebra of differential operators on the quantum euclidean spacer q n the generators are suitableq deformed analogs of the angular momentum components on ordinaryr n the algebra fun r q n of functions onr q n splits into a direct sum of irreducible vector representations ofuq 1 so n the latter are explicitly constructed as highest weight representations

response of two degree of freedom systems to multifrequency parametric excitations abstract the method of multiple scales is used to analyze the response of two degree of freedom systems to multifrequency parametric excitations the equations describing the modulation of the amplitudes with time are derived for the case of four simultaneous resonances main resonances of the two modes combination resonance of the difference type and combination resonance of the summed type when only two simultaneous resonances exist the modulation equations can be transformed into a system of four real first order differential equations with constant coefficients whose solution depends on the eigenvalues of a four by four constant coefficient matrix when three or more simultaneous resonances exist the modulation equations cannot be made autononous in general and hence one may not be able to solve them except numerically

a semi analytical locally transversal linearization method for non linear dynamical systems an numeric analytical implicit and local linearization methodology called the locally transversal linearization ltl is developed in the present paper for analyses and simulations of non linear oscillators the ltl principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non linear dynamical system at that particular point in the state space where the solution vector is sought for purposes of numerical implementation two different numerical schemes namely ltl 1 and ltl 2 schemes based on the ltl methodology are presented both ltl 1 and ltl 2 procedures finally reduce the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space while in the ltl 1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter in the ltl 2 scheme a set of unknown residues enters the linearized system as parameters a limited set of examples involving a few well known single degree of freedom sdof non linear oscillators indicate that the ltl methodology is capable of accurately predicting many complicated non linear response patterns including limit cycles quasi periodic orbits and even strange attractors copyright 2001 john wiley sons ltd

stability and primary simultaneous resonance of harmonically excited non linear spring pendulum system the considered system is a harmonically excited non linear spring pendulum which simulates the ship roll motion the method of multiple time scale perturbation is applied to solve the non linear differential equations describing the system up to and including the fourth order approximation all possible resonance cases were extracted at this approximation order this work is limited only to the primary resonance cases the stability of the system is investigated using both frequency response equations and phase plane method the effects of the different parameters on system behavior are studied numerically variation of some equations parameters leads to the bending of the frequency response curves and hence to the jump phenomenon it is quite clear that some of the simultaneous primary resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system such cases should be avoided as working conditions for the system some recommendations regarding the different parameters of the system are reported

on algebras associated to the conjugation representation of a locally compact group for a locally compact group g let yg denote the conjugation representation of g in l2 g in this paper we are concerned with nuclearity of c algebras associated to yg and the question of when these are of bounded representation type

on the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups we show that if there exists a realization of a hopf algebra h in a h module algebra then one can split their cross product into the tensor product algebra of itself with a subalgebra isomorphic to h and commuting with this result applies in particular to the algebra underlying inhomogeneous quantum groups like the euclidean groups which are obtained as cross products of the quantum euclidean spaces qn with the quantum groups of rotation uq so n of qn for which it has no classical analogue

compactness in dual spaces of locally compact groups and tensor products of irreducible representations abstract we deal with supports of tensor products of irreducible representations of locally compact groups the main result describes the structure of almost connected groups for which tensor products of representations in the reduced dual have compact supports an immediate consequence is that an infinite almost connected group cannot have a compact reduced dual furthermore the continuity problem for these supports is studied