the effect of noise on β cell burst period bursting electrical behavior is commonly observed in a variety of nerve and endocrine cells including that in electrically coupled cells located in intact pancreatic islets however individual cells usually display either spiking or very fast bursting behavior and the difference between isolated and coupled cells has been suggested to be due to stochastic fluctuations of the plasma membrane ion channels which are supposed to have a stronger effect on single cells than on cells situated in clusters the channel sharing hypothesis this effect of noise has previously been studied using numerical simulations we show here how the application of two recent methods allows an analytic treatment of the stochastic effects on the location of the saddle node and homoclinic bifurcations which determine the burst period thus the stochastic system can be analyzed similarly to the deterministic system but with a quantitative description of the effect of noise this approach supports previous investigations

ensemble averages of power flow in randomly excited coupled beams the transmission of transverse vibrational energy between two coupled beams subjected to random excitation is considered extensive numerical results are given for the frequency decomposition of power flow between the beams the power flow is a function of the three non dimensional parameters describing frequency 1 where 1 is the fundamental resonance frequency of beam one length ratio l 2 l 1 and damping or modal overlap ratio single frequency and octave band averages are shown for various values of l 2 l 1 and means and variances are calculated for the ensemble of systems corresponding to a uniform distribution of 1 l 2 l 1 2 various approximate results based on modal analysis and on sea are compared with the exact results

the quantum poincare birkhoff witt theorem we define and study a wide class of associative algebras in which the poincare birkhoff witt theorem is valid this class includes numerous quantum algebras which recently appeared

quantum lorentz and braided poincare groups quantum lorentz groups h admitting quantum minkowski space v are selected the natural structure of a quantum space is introduced defining a quantum group structure on g only for triangular h q 1 we show that it defines a braided quantum group structure on g for q 1

random point excitation of coupled beams numerical results are presented for the power flow between two coupled beams one of which is excited at a point by a random force frequency spatial over the excitation point and ensemble by varying the relative lengths of the beams averages are compared with each other and with the results from simple statistical models such as sea the mean values agree well and the standard deviations are surprisingly small

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

confinement of vibration by one dimensional disorder ii a numerical experiment on different ensemble averages abstract when producing theoretical estimates of the effect of irregularity on the vibrational behaviour of a structure as has been described previously 1 3 one aims to predict the behaviour of a typical irregular structure within the set considered it might be thought that this would correspond to the ensemble averaged behaviour over the whole set but this need not be the case in fact there are several different possible ways of taking the ensemble average which can produce remarkably different results some of these do indeed correspond to the typical member of the ensemble but some of them certainly do not since rare configurations giving very large contributions to the average distort the answer this problem is investigated here via a numerical experiment on a very simple system the object is to complement the discussion in a companion paper 4 which gives theoretical analysis the aims here are threefold to know which average is the most appropriate in a given context to understand why the other averages are different and to identify the anomalous configurations making them different so that these can be avoided in designing real structures a by product is a better understanding of the averaging procedures used in the past it has not always been recognized that these various averages can give radically different answers and one can find instances where what was used was perhaps the easiest approach rather than the most appropriate one in addition the numerical experiment described here gives a good illustration of the basic phenomenon of anderson localization in an acoustic context

confinement of vibration by one dimensional disorder i theory of ensemble averaging abstract the use of ensemble averaging procedures in studying the dynamics of structurally irregular mechanical systems is discussed it is found that taking a linear average over the ensemble does not always yield the behaviour of a typical member of that ensemble under certain circumstances there are occasional members with properties so exceptional as to weight the average appreciably and the linear ensemble average can then be seriously misleading the effect is illustrated by the response and transmission properties of two simple mechanical models a system of coupled oscillators with varying natural frequencies and a vibrating string or bending beam with irregularly spaced mass and spring constraints the distinction between the ensemble average and the typical value is important in understanding the phenomenon of anderson localization described previously 1 3 by using these two models this paper provides some analytic results while an accompanying paper 4 provides complementary results from a numerical study of the coupled oscillator model

phase spaces related to standard classical r matrices fundamental representations of real simple poisson lie groups are poisson actions with a suitable choice of the poisson structure on the underlying real vector space we study these mostly quadratic poisson structures and corresponding phase spaces symplectic groupoids

fast subsystem bifurcations in a slowly varying lie nard system exhibiting bursting a perturbed lienard differential system is examined using local stability and hopf bifurcation analyses asymptotic techniques and melnikov s method the results of these analyses are applied to a simple cubic model that exhibits a variety of different oscillatory behaviors for different parameter values for a bounded region in fast parameter space the model exhibits square wave bursting patterns analogous to the bursting electrical activity observed in pancreatic beta cells under certain hypotheses solutions of the cubic model are known to have square wave patterns by using the theory developed for the more general lienard system each hypothesis is shown to correspond to a curve in parameter space together the curves bound a region in which the model exhibits square wave bursting patterns since the model is simple the curves that bound this region can all be determined analytically

derivation of statistical energy analysis from radiative exchanges abstract this paper is concerned with the derivation of sea equations from structural ray equations rays are assumed to be uncorrelated leading to the additivity of energy inside all subsystems the energy density is the sum of a direct field from driving forces a reflected field from the boundary and a transmitted field from adjacent subsystems assuming a rain on the roof excitation and a compact shape for subsystems actual and fictitious sources on the boundary are found to be constant furthermore if the attenuation of rays during a mean free path normalized attenuation factor is light the field becomes diffuse i e homogeneous and isotropic the net exchanged power between two adjacent subsystems is then proportional to the difference of energy densities and therefore to the difference of modal energies the derived proportionality coefficient is consistent with the well known formula for coupling loss factor in terms of transmission factors these results are illustrated by a numerical simulation for a multi plate system finally the validity domains of sea and the ray theory are discussed and particularly the diffuse field assumption

on the structure of inhomogeneous quantum groups we investigate inhomogeneous quantum groups g built from a quantum group h and translations the corresponding commutation relations contain inhomogeneous terms under certain conditions which are satisfied in our study of quantum poincare groups 12 we prove that our construction has correct size find the r matrices and the analogues of minkowski space for g

power flow between two continuous one dimensional subsystems a wave solution abstract the power flow between two point coupled wave bearing subsystems is determined by using a wave approach the subsystems are one dimensional in that each supports just one wave mode and are described by a reflection coefficient from which in certain circumstances the modal overlap and modal density can be defined as continuous functions of frequency the coupling element is described by a reflection coefficient and attention is focused on the case of conservative coupling time harmonic point forces or rain on the roof excitations may act on the subsystems uncertainty exists in the properties of the subsystems which are assumed to be drawn from ensembles of similar subsystems the exact power flow and the ensemble statistics are seen to depend on the magnitudes of the subsystem and coupling reflection coefficients for some particular cases the maximum and minimum power flows are examined together with the ensemble mean and geometric mean the cumulative probability distribution and the probability density function for a given power incident upon the coupling the ensemble mean power flow through the coupling is always less than that expected from a normal sea wave approach due to power re radiation the differences are largest when the damping is light and the coupling strong for a given level of excitation power re radiation effects are also apparent for the ensemble mean power flow from an infinite subsystem to a finite subsystem in the opposite case the ensemble mean power flow from a finite to an infinite subsystem is greater than that between two infinite subsystems due to power re injection the differences are largest for light damping and for weak coupling finally the case where there is uncertainty in one subsystem only is considered

an approach to the theoretical background of statistical energy analysis applied to structural vibration rayleigh s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis sea the classical method has advantages of simplicity and rigor over previous approaches to the background of sea in certain cases and provides extensions and simplifications in several areas of the theory it also suggests modifications to sea modeling strategy depending on the type of coupling involved even when that coupling is weak so that earlier analyses might be thought to apply

vibration isolation from irregularity in a nearly periodic structure theory and measurements this article describes the theory and a simple experiment carried out to demonstrate the phenomenon of anderson localization in an acoustical context this is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities giving rise on the average to an exponential decay of vibration level away from the driving point even in the absence of any dissipation the structure used in the experiment was a stretched string with masses attached to it this string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity in both cases the transmission of energy from end to end of the string was measured as a function of frequency and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon in which the individual modes making up each passband change from being extended throughout the structure in the regular case to being localized in specific areas of the structure in the presence of irregularity all measurements yielded satisfactory agreement with the theoretical predictions

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

in plane free vibration analysis of combined ring beam structural systems by wave propagation this paper presents a systematic wave propagation approach for the free vibration analysis of networks consisting of slender straight and curved beam elements and complete rings the analysis is based on a ray tracing method and a procedure to predict the natural frequencies and mode shapes of complex ring beam networks is demonstrated in the wave approach the elements are coupled using reflection and transmission coefficients and these are derived for discontinuities encountered in a mems rate sensor consisting of a ring supported on an array of folded beams these are combined taking into account wave propagation and decay to provide a concise and efficient method for studying the free vibration problem a simplification of the analysis that exploits cyclic symmetry in the structure is also presented the effects of decaying near field wave components are included in the formulation ensuring that the solutions are exact to illustrate the effectiveness of the approach several numerical examples are presented the predictions made using the proposed approach are shown to be in excellent agreement with a conventional fe analysis

comparison of statistical and interval analysis for the energy flow uncertainties in structural vibrating systems in this paper the energy flow confidence between two structural multimodal systems coupled by a joint with uncertain parameters is computed by two different methods the first one assumes that the joint parameters perturbed randomly the statistical moments of the energy flow are calculated by an analytical procedure the second one uses interval analysis the joint parameters are considered interval variables and the interval of the energy flow is determined the properties of the statistical and interval solution are investigated and compared

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

the statistics of power flow between two continuous one dimensional subsystems abstract the power flow between two continuous subsystems is examined using a wave solution 1 one of the subsystems is excited by time harmonic forces and both subsystems can support only one power transmitting wave mode uncertainty exists in the subsystems which are assumed to be drawn at random from ensembles those ensembles members that have a natural frequency at the excitation frequency are determined the particular members that are resonant are seen to depend on the strength of coupling the statistics of power flow are considered in particular the maximum minimum ensemble arithmetic and geometric means the variance and the cumulative probability distribution of the power flow the maximum power flow undergoes a qualitative change at a particular coupling strength for weak coupling the peak power flows occur at the uncoupled natural frequencies while for stronger coupling they occur at the coupled frequencies the ensemble average power flow may be significantly less than that expected from a normal sea approach due to power re radiation especially for a lightly damped receiving subsystem and strong coupling conversely it may be much greater due to power re injection which occurs particularly for a lightly damped excited subsystem and weak coupling the power flow variance is largest for small damping and weak coupling under which circumstances the cumulative probability distribution has an infrequent but dominant tail of very large power flows

the statistics of waves propagating in a one dimensional random medium we consider the one dimensional scattering of waves in a time independent random medium the waves considered are time harmonic it is assumed that the wavelength of the waves and the correlation length of the scatterers are small compared with the distance required for significant scattering stochastic process theory is used to investigate the statistics of the wavefield the problem of a wave incident on a length of random medium is investigated in two cases the first is where the medium is backed by a perfectly reflecting boundary here the intensity is shown to be a product of two factors an exponential term that decays into the medium and the exponential of a brownian motion that describes the fluctuations of intensity with different realizations of the random medium because a brownian motion has a normal distribution the intensity has a log normal distribution at any fixed point in the interior of the medium for a typical realization of the random medium the exponential decay leads to most of the wave energy being near the front of a long medium however it is shown that the average intensity is independent of position in the medium this is because of the long tail of the log normal distribution and comes about because the average is heavily weighted by exceptional realizations of the medium thus the average value of the intensity unlike the average of the logarithm of the intensity is not representative of the intensity in a typical realization the exponential decay of intensity is a result of the phenomenon of anderson localization which has received much attention in solid state physics the second case considered is where there is no barrier at the back of the medium for large lengths of the random medium it is shown that the transmitted intensity has an approximately log normal distribution the typical transmitted intensity is exponentially small as a result of localization the decay rate with length being the same as the decay rate for the previous case the average transmitted intensity is also exponentially small but with a different decay rate because of weighting by exceptional realizations the third problem discussed is that of the response of a random medium to time harmonic forcing in the interior the boundaries are taken to be perfectly reflecting and the response is found to be localized near the source for a typical realization this result is related to the existence of localized normal modes in a long medium

wave derivation of the energetics of driven coupled one dimensional dynamic systems abstract the equation of statistical energy analysis was originally derived on the basis of modal concepts in this paper a derivation of the equation on the basis of wave concepts is outlined although the new formalism is limited at present to the analysis of coupled one dimensional dynamic systems it carries details that were a priori forfeited by the previous formalism the wave approach to sea as developed in the paper reveals a formalism that can be naturally used for the analysis of net power and intensity in coupled dynamic systems this aspect of the formalism is briefly discussed

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

propagation of decaying waves in periodic and piecewise periodic structures of finite length abstract a new method is described for the analysis of long and complicated structures which are composed of spatially periodic units or sections of spatially periodic units the response of such a structure to external excitations is treated as a superposition of wave motions with account taken of the effects of wave reflection due to change in the construction pattern along the structure and boundary conditions since damping usually exists in a real structure a wave motion always decays as it propagates along the structure however this can cause numerical difficulties if the structure is rather long because a decaying wave appears as a growing wave in the opposite direction and since propagations in both directions must be accounted for in the analysis computational errors associated with the apparent growing waves also grow and the numerical results become meaningless in the proposed new scheme use is made of wave reflection and transmission matrices making it possible for all computations to proceed in the direction of wave propagation so that the results are always stable numerically furthermore the required computing time is greatly reduced in comparison with that of an all encompassing finite element analysis for the entire structure application of the new method is illustrated by examples

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

q deformed poincare algebra theq differential calculus for theq minkowski space is developed the algebra of theq derivatives with theq lorentz generators is found giving theq deformation of the poincare algebra the reality structure of theq poincare algebra is given the reality structure of theq differentials is also found the real laplacian is constructed finally the comultiplication counit and antipode for theq poincare algebra are obtained making it a hopf algebra

response of multiple coupled dynamic systems abstract analytical procedures are developed to investigate the nature of the response of multiple coupled dynamic systems the dynamic systems are one dimensional and are coupled at junctions the spatial extents of the dynamic systems are determined by these junctions a dynamic system terminates at these junctions the junctions are characterized by assigning reflection and transmission coefficients at the terminal positions of the dynamic systems in addition a dynamic system is characterized by a single propagation wavenumber the wavenumber may be complex accounting for possible spatial attenuation and for distributed damping in the dynamic system further damping edge damping may be accounted for by assigning non conservative coefficients at the junction quadratic forms describing the dynamics of multiple coupled dynamic systems are also presented and discussed relationship of such forms to the statistical energy analysis sea of complex dynamic systems is touched upon this paper is concerned chiefly with the development of analytical procedures examples of application of these procedures will be dealt with in subsequent publications

comparison of two non probabilistic approaches for the energy flow uncertainty in structural vibrating system two non probabilistic set theoretical methods for determining the energy flow between two structural multimodal systems coupled by a joint with uncertain parameters are presented they are based on the theories of interval mathematics and convex models the uncertain parameters of the joint are assumed to be a convex set hyper rectangle or ellipsoid for both non probabilistic methods less prior information about the uncertain nature is required than that which is required concerning the probabilistic model the properties of the standard interval arithmetic the interval taylor expansion the convex models and the monte carlo method solutions are investigated and compared

return map characterizations for a model of bursting with two slow variables various physiological systems display bursting electrical activity bea there exist numerous three variable models to describe this behavior however higher dimensional models with two slow processes have recently been used to explain qualitative features of the bea of some experimentally observed systems t chay and d cook math biosci 90 1988 pp 139 153 p smolen and j keizer j memb biol 127 1992 pp 9 19 r bertram et al biophys j 79 2000 pp 2880 2892 r bertram et al biophys j 68 1995 pp 2323 2332 j keizer and p smolen proc nat acad sci usa 88 1991 pp 3897 3901 in this paper we present a model with two slow and two fast variables for some parameter values the system has stable equilibria while for other values there exist bursting solutions singular perturbation methods are used to define a one dimensional return map wherein fixed points correspond to singular bursting solutions we analytically demonstrate that bursting solutions may exist

analytical solution for the power exchange between strongly coupled plates under random excitation a test of statistical energy analysis concepts the response of the structural components of a complex structure subjected to random loads is often estimated by approximate methods based on the concepts of statistical energy analysis sea to examine the fundamental sea postulates and assumptions the problem of two coupped plates is rigorously formulated and solved in terms of usual sea variables for randomly distributed loads analytical results are presented for the power flow and plate vibrational energies the formulation allows the extraction of apparent quantifies that correspond to the coupling loss factor and modal density ratio of statistical energy analysis it is demonstrated that this model can yield an understanding of the implications of basic sea assumptions results for simply supported plates clearly show the effects of internal damping and frequency on the coupling parameters