twisted dual group algebras equivariant deformations of c0 g abstract we consider a family of twisted fourier algebras a g of a locally compact group g which in the case of a abelian group g are the fourier transforms of the usual twisted group algebras of over cap g o the corresponding c algebras c g o are deformations of c 0 g which are equivariant in the sense that g still acts by left translation the main examples come from cocycles on the dual of an abelian subgroup h of g we prove that for such cocycles the twisted dual group algebras c g o are induced from the twisted group algebras c h o and we give detailed formulas for the multiplication on a g which extend to larger dense subalgebras of c 0 g and c b g we anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces c 0 g

further results on global stabilization of discrete nonlinear systems in this paper we show how recent advances 11 12 in global stabilization of continuous time nonaffine nonlinear systems with stable free dynamics can be nontrivially generalized to their discrete time counterparts by means of passivity and bounded state feedback as a consequence global stabilization results of discrete time nonlinear systems with triangular structure are established

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

reflection of an acoustic line source by an impedance surface with uniform flow abstract an exact analytic solution is derived for the 2d acoustic pressure field generated by a time harmonic line mass source located above an impedance surface with uniform grazing flow closed form asymptotic solutions in the far field are also provided the analysis is valid for both locally reacting and nonlocally reacting impedances as is demonstrated by analyzing a nonlocally reacting effective impedance representing the presence of a thin boundary layer over the surface the analytic solution may be written in a form suggesting a generalization of the method of images to account for the impedance surface the line source is found to excite surface waves on the impedance surface some of which may be leaky waves which contradict the assumption of decay away from the surface predicted in previous analyses of surface waves with flow the surface waves may be treated either correctly as unstable waves or artificially as stable waves enabling comparison with previous numerical or mathematical studies which make either of these assumptions the computer code for evaluating the analytic solution and far field asymptotics is provided in the supplementary material it is hoped this work will provide a useful benchmark solution for validating 2d numerical acoustic codes

on the exponential stability of discrete time systems with applications in observer design exponential stability of discrete time systems stability domains and state bounds are obtained with the use of vector norm lyapunov functions for linear systems necessary and sufficient conditions are given for asymptotic stability and exponential bounds are derived for nonlinear systems sufficient conditions for exponential stability also yield a stability domain and exponential bounds applications are demonstrated in observer design for uncertain discrete time nonlinear systems

notes on the stability of dynamic economic systems the lyapunov function method is the focus of this paper in proving marginal stability asymptotical or globally asymptotical stability of discrete dynamic systems we show that the slightly relaxed versions of the well known sufficient conditions are also necessary

globally asymptotical stability of discrete time analog neural networks some globally asymptotical stability criteria for the equilibrium states of a general class of discrete time dynamic neural networks with continuous states are presented using a diagonal lyapunov function approach the neural networks are assumed to have the asymmetrical weight matrices throughout the paper the resulting criteria are described by the diagonal stability of some matrices associated with the network parameters some novel stability conditions represented by either the existence of the positive diagonal solutions of the lyapunov equations or some inequalities are given using the equivalence between the diagonal stability and the schur stability for a nonnegative matrix some simplified global stability conditions are also presented finally some examples are provided for demonstrating the effectiveness of the global stability conditions presented

sound radiation by a moving line source above an impedance plane with frequency dependent properties abstract an analytic solution for the problem of sound radiation by a harmonic line source moving at a uniform subsonic speed parallel to an impedance plane is proposed the main originality of this work is that the variation of the impedance with the frequency is taken into account compared to the case of a constant impedance the reflection coefficient and the location of its poles in the complex plane are modified a uniform asymptotic expression is then developed for moderate mach numbers and a closed form expression corresponding to a weyl van der pol formula is proposed for a grazing incidence for hard grounds and for low mach numbers unlike previous analytical solutions derived in the literature for a point source the impedance is evaluated at the doppler frequency instead of at the source frequency the analytical solution and asymptotic expressions are then compared satisfactorily to a numerical solution obtained from a time domain solver of the linearized euler equations finally a parametric study is carried out showing that the assumption of a constant impedance is valid if the source mach number remains small typically less than 0 2 and if the source is not too close to the ground

grothendieck rings of universal quantum groups abstract we determine the grothendieck ring of finite dimensional comodules for the free hopf algebra on a matrix coalgebra and similarly for the free hopf algebra with bijective antipode and other related universal quantum groups the results turn out to parallel those for wang and van daele s universal compact quantum groups and bichon s generalization of those results to universal cosovereign hopf algebras in all cases the rings are isomorphic to those of non commutative polynomials over certain sets these sets varying from case to case in most cases we are able to give more precise information about the multiplication table of the grothendieck ring

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

on the long range propagation of sound over irregular terrain abstract the theory of sound propagation over randomly irregular nominally plane terrain of finite impedance is discussed the analysis is an extension of the theory of coherent scatter originally proposed by biot 1 for an irregular rigid surface it combines biot s approach wherein the surface irregularities are modeled by a homogeneous distribution of hemispherical bosses with more conventional analyses in which the ground is modeled as a smooth plane of finite impedance at sufficiently low frequencies it is shown that the interaction of the surface irregularities with the near field of a ground based source leads to the production of surface waves which are effective in penetrating the ground shadow zone predicted for a smooth surface of the same impedance

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

global exponential asymptotic stability in nonlinear discrete dynamical systems abstract for the nonlinear discrete dynamical system x k 1 tx k on bounded closed and convex set d r n we present several sufficient and necessary conditions under which the unique equilibrium point is globally exponentially asymptotically stable the infimum of exponential bounds of convergent trajectories is also derived

a new method for stability analysis of nonlinear discrete time systems we address the problem of global lyapunov stability of discrete time systems with known coefficients we develop a method for reduction of dissipativity domain effectively testing if the system has a convex lyapunov function our implementation is immediately applicable to differentiable systems with bounded nonlinearities but the method proposed is more general and applicable to nondifferentiable systems with bounded right hand sides our main application emphasis is on stability analysis of recurrent neural networks we illustrate how to use our approach with examples

note on a result of morse and bolt abstract a result given without derivation by morse and bolt 1 pertaining to the reflection of a spherical sound wave from an absorbent surface is investigated it is shown that the result as given is not quite accurate

fundamental problems with the model of uniform flow over acoustic linings abstract the walls of ducts containing flow are often acoustically lined in order to reduce sound many simple models of acoustic linings assume the lining to be linear and locally reacting examples considered here include the three parameter mass spring damper helmholtz resonator and enhanced helmholtz resonator models all of these models have been found to have stability issues with uniform mean grazing flow and there has been some confusion over the existence of hydrodynamic instability surface waves over such linings mathematically the standard proven briggs bers stability analysis is not applicable computationally the hydrodynamic modes are routinely ignored in the frequency domain and instabilities filtered out in the time domain this confusion also causes significant problems for mode matching green functions and scattering analyses in this paper it is shown that any situation not capable of being analysed using the briggs bers criterion is illposed and that this is the root cause of the confusion over hydrodynamic instabilities a large class of lining models including all those mentioned above are shown to be illposed with uniform mean flow an explanation is given of the effects of this illposedness in practice and it is argued that illposed models should not be used an alternative stability criterion to briggs bers has been occasionally used in the literature this alternative criterion involving analytic continuation to purely imaginary frequencies was recently christened the crighton leppington stability criterion however this stability criterion is incorrect and several counter examples are given here

a controllable canonical form implementation of time domain impedance boundary conditions for broadband aeroacoustic computation a new method which can be effectively and efficiently applied in the simulations of broadband noise problems is proposed for time domain impedance boundary condition implementations by using the so called controllable canonical form that is well known in linear system usually the impedance boundary condition can be defined in frequency domain as a rational polynomial function with poles in the negative half of the complex plane to guarantee stability otherwise causality might be violated in the corresponding time domain implementation to address this issue various methodologies have been proposed previously that usually lead to complicated polynomials whose numerical implementations are often indirect and intricate the proposed method with a controllable canonical form on the other hand directly transforms the frequency domain transfer function a quotient of rational polynomials to an equivalent state space model which consists of a series of first order ordinary differential equations that can be numerically implemented in a straightforward way the proposed method is demonstrated by using two benchmark problems a two dimensional gaussian pulse propagating in a uniform flow with a lined wall and the test cases from the nasa langley grazing incidence tube experiments good agreements demonstrate the potential of the proposed computational method

quantitative characterizations of exponential convergence property of nonlinear discrete dynamical systems in this technical note we consider the exponential convergence property of the trajectories of nonlinear discrete dynamical systems with exponential stability based on an invariant quantity derived from lyapunov functions of the systems the essential and quantitative relationship between exponential convergence property of trajectories and the lyapunov functions is revealed

an invariance principle for nonlinear discrete autonomous dynamical systems this note proposes an extension of lasalle s invariance principle for nonlinear discrete autonomous dynamical systems the invariance principle is extended to allow the first difference of the auxiliar scalar function usually a lyapunov function to be positive in some bounded regions moreover a uniform version is proposed to deal with nonlinear discrete dynamical systems that vary with parameters both extensions have the original invariance principle as a particular case as a consequence a larger class of systems can be treated with this new theory the extensions are very useful to obtain attractor estimates as well as their corresponding stability regions the uniform version in particular is useful to obtain estimates that are uniform regarding parameters

time domain implementation of an impedance boundary condition with boundary layer correction a time domain boundary condition is derived that accounts for the acoustic impedance of a thin boundary layer over an impedance boundary based on the asymptotic frequency domain boundary condition of brambley 2011 25 a finite difference reference implementation of this condition is presented and carefully validated against both an analytic solution and a discrete dispersion analysis for a simple test case the discrete dispersion analysis enables the distinction between real physical instabilities and artificial numerical instabilities the cause of the latter is suggested to be a combination of the real physical instabilities present and the aliasing and artificial zero group velocity of finite difference schemes it is suggested that these are general properties of any numerical discretization of an unstable system existing numerical filters are found to be inadequate to remove these artificial instabilities as they have a too wide pass band the properties of numerical filters required to address this issue are discussed and a number of selective filters are presented that may prove useful in general these filters are capable of removing only the artificial numerical instabilities allowing the reference implementation to correctly reproduce the stability properties of the analytic solution

a note on reflection of spherical waves in 1909 sommerfeld gave an exact solution for the reflection of a spherical wave from a plane surface in terms of an oscillatory integral and also presented an asymptotic solution for the case where both source and receiver are at the boundary weyl 1919 presented an alternative solution and also an asymptotic solution for the case where the source is at the boundary it is known that the general case is solved if a general solution for the case where the source is at the boundary is known here it is demonstrated that it is sufficient to have the general solution for the case where both source and receiver are at the boundary this is mainly of theoretical interest but may have practical applications as an example it is demonstrated that sommerfeld s approximate solution gives ingard s 1951 approximate solution which is valid for arbitrary source and receiver heights

stability analysis of hybrid composite dynamical systems descriptions involving operators and differential equations we address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in fig 1 consisting of a block usually the plant which is described by an operator l and of a finite dimensional block described by a system of ordinary differential equations usually the controller we establish results for the well posedness attractivity asymptotic stability uniform boundedness asymptotic stability in the large and exponential stability in the large for such systems the hypotheses of these results are phrased in terms of the i o properties of l and in terms of the lyapunov stability properties of the subsystem described by the indicated ordinary differential equations the applicability of our results is demonstrated by means of general specific examples involving c 0 semigroups partial differential equations or integral equations which determine l

diffraction of a spherical wave by a thin infinite plate a thin elastic plate extending to infinity is immersed in a perfect gas the response of this system to a spherical harmonic sound source is studied and an exact representation of the solution is developed from this result a series representation useful to calculate the farfield is established expressions for the various mean powers are given

rieffel deformation of homogeneous spaces let g1 g be a closed subgroup of a locally compact group g and let x g g1 be the quotient space of left cosets let x c0 x x be the corresponding g c algebra where g c0 g suppose that is a closed abelian subgroup of g1 and let be a 2 cocycle on the dual group let g be the rieffel deformation of g using the results of the previous paper of the author we may construct g c algebra x the rieffel deformation of x on the other hand we may perform the rieffel deformation of the subgroup g1 obtaining the closed quantum subgroup g1 g which in turn by the results of s vaes leads to the g c algebra g g1 in this paper we show that g g1 x we also consider the case where g is not a subgroup of g1 for which we cannot construct the subgroup g1 then generically x cannot be identified with a quantum quotient what may be shown is that it is a g simple object in the category of g c algebras

hopf galois systems abstract we introduce the concept of hopf galois system a reformulation of the notion of galois extension of the base field for a hopf algebra the main feature of our definition is a generalization of the antipode of an ordinary hopf algebra we present several examples which indicate that although our axiomatic is slightly more complicated than the classical one it is also more natural and easier to handle with the main application of hopf galois systems is the construction of monoidal equivalences between comodule categories

brief paper growth rate of switched homogeneous systems we consider discrete time homogeneous systems under arbitrary switching and study their growth rate the analogue of joint spectral radius for switched linear systems we show that a system is asymptotically stable if and only if its growth rate is less than unity we also provide an approximation algorithm to compute growth rate with arbitrary accuracy

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

extended sources radiation and laplace type integral representation application to wave propagation above and within layered media abstract in has been shown that the sound field reflected by the plane boundary of a layered ground can always be described by a specularly reflected wave and layer potentials despite its generality this representation is not quite suitable for numerical computation dealing with a very simple case weyl and later on ingard and thomasson proposed a representation of the solution in which the layer potential terms are replaced by the sum of surface wave and a laplace type integral such an integral is very convenient for numerical purposes in this paper it is shown that this kind of representation can be obtained for a very wide class of sound propagation problems above or within layered media a half space bounded by a locally reacting surface a finite layer of porous medium a porous medium with depth varying porosity a thin elastic plate wave propagation in shallow water with an impedance bottom many other applications could be developed

accurate fft based hankel transforms for predictions of outdoor sound propagation without modification the fft based hankel transform that is used in the fast field program developed for applications in underwater acoustics is not sufficiently accurate for predictions of outdoor sound propagation near to finite impedance boundaries numerical problems arise from singularities in the integrand and from representation of an infinite integral by a finite sum modifications to the basic algorithm are proposed to improve its accuracy these modifications include integration below the real axis subtraction of an analytic function and addition of terms to represent the difference between the finite sum and the sum to infinity predictions of the modified algorithm of propagation in a calm isothermal atmosphere from a point source near to a boundary are compared both with those of the unmodified algorithm and with exact solutions or analytic approximations

improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow we present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment the method relies on a variational formulation of the problem which allows to derive a multimodal formulation within a rigorous mathematical framework notably to properly account for the boundary conditions on the walls being locally the myers condition and the neumann condition otherwise also the method uses an enriched basis with respect to the usual cosine basis able to absorb the less converging part of the modal series and thus to improve the convergence of the method using the cosine basis the modal method has a low convergence 1 n with n the order of truncation using the enriched basis the improvement in the convergence is shown to depend on the mach number from 1 n5 to roughly 1 n1 5 for m 0 to m close to unity the case of a continuously varying wall impedance is considered and we discuss the limiting case of piecewise constant impedance which defines pressure edge conditions at the impedance discontinuities

losslessness feedback equivalence and the global stabilization of discrete time nonlinear systems in this paper a necessary and sufficient condition for a nonlinear system of the form spl sigma given by x k 1 f x k g x k u k y k h x k j x k u k to be lossless is given and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero state observable then we investigate conditions under which spl sigma can be rendered lossless via smooth state feedback in particular we show that this is possible if and only if the system in question has relative degree spl lcub 0 0 spl rcub and has lossless zero dynamics under suitable controllability like rank conditions we prove that nonlinear systems having relative degree spl lcub 0 0 spl rcub and lossless zero dynamics can be globally stabilized by smooth state feedback as a consequence we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable the global stabilization problem of the nonlinear system spl sigma without output is also investigated in this paper by means of feedback equivalence some of the results are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

c algebres de kac et algebres de kac s baaj and g skandalis have proved that to every kac algebra as studied by j m schwartz and the first author corresponds a canonical c kac algebra as studied by the second author this article proves the converse result so we have then a complete proof that the c version and the von neumann version of this theory are exactly equivalent as are in the commutative case thanks to a weil s theorem locally compact groups and measured groups with a left invariant measure

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

growth rate of switched homogeneous systems we consider discrete time homogeneous systems under arbitrary switching and study their growth rate the analogue of joint spectral radius for switched linear systems we show that a system is asymptotically stable if and only if its growth rate is less than unity we also provide an approximation algorithm to compute growth rate with arbitrary accuracy

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

long distance sound propagation over an impedance discontinuity abstract a calculation method is presented for sound propagation over an impedance discontinuity which is an improvement over previous methods being particularly efficient for long distances a homogeneous still atmosphere and flat ground are assumed the starting point is to make an approximation in a two dimensional boundary integral equation formulation of the problem this leads to an expression of the solution as the solution for homogeneous ground plus an integral over part of the boundary this integral is recognized as a generalized fourier integral to which standard techniques of asymptotic analysis can be applied an asymptotic approximation is found which can replace the part of the integral which is most expensive to evaluate by numerical quadrature the remaining integral is evaluated by numerical quadrature monofrequency excess attenuation results for propagation from a point source above rigid ground to a receiver above absorbing ground are discussed when the source and receiver are low enough very orderly trends are observed the results are applied with air attenuation and a weighting to a notional jet engine noise source simple trends are noted

well posed boundary condition for acoustic liners in straight ducts with flow doi 10 2514 1 j050723 the myers boundary condition for acoustics within flow over an acoustic lining has been shown to be ill posed leading to numerical stability issues in the time domain and mathematical problems with stability analyses this paper gives a modification for flat or cylindrical straight ducts to make the myers boundary condition well posed and indeed more accurate by accounting for a thin inviscid boundary layer over the lining and correctly deriving the boundary condition to first order in the boundary layer thickness the modification involves two integraltermsovertheboundarylayer the firstmaybewrittenintermsofthemass momentum andkinetic energy thicknessesoftheboundarylayer whichareshownto physicallycorrespondtoamodifiedboundarymass modified grazingvelocity andatensionalongtheboundary thesecondintegral termisrelatedtothecriticallayerwithinthe boundary layer a time domain version of the new boundary condition is proposed although not implemented the modified boundary condition is validated against high fidelity numerical solutions of the pridmore brown equation for sheared inviscid flow in a cylinder absolute instability boundaries are given for certain examples though convective instabilities appear to always be present at certain frequencies for any boundary layer thickness

an elementary result in the stability theory of time invariant nonlinear discrete dynamical systems the stability of the equilibria of time invariant nonlinear dynamical systems with discrete time scale is investigated we present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function all eigenvalues of the jacobian computed at the equilibrium must be inside or on the unit circle we also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle then the equilibrium maybe unstable or marginally stable or even asymptotically stable which show that the necessary condition cannot be further restricted in general in addition the necessary condition is given in terms of spectral radius and matrix norms

absolute stability analysis of discrete time systems with composite quadratic lyapunov functions a generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and or uncertainty this paper will further enhance absolute stability analysis by using the composite quadratic lyapunov function whose level set is the convex hull of a family of ellipsoids the absolute stability analysis will be approached by characterizing absolutely contractively invariant aci level sets of the composite quadratic lyapunov functions this objective will be achieved through three steps the first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems the second step establishes stability conditions for linear difference inclusions and then for saturated linear systems the third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability based on the conditions for absolute stability optimization problems are formulated for the estimation of the stability region numerical examples demonstrate that stability analysis results based on composite quadratic lyapunov functions improve significantly on what can be achieved with quadratic lyapunov functions

on characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions in this paper we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space through introducing two new concepts the lip constant of a nonlinear operator and strongly equivalent metrics of a norm we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics or if and only if the lip constant of the nonlinear operator is less than one in the latter case we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the lip constant based on the obtained results we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the lip constant and strongly equivalent metrics the obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems

bounds on nonlinear operators in finite dimensional banach spaces we consider lipschitz continuous nonlinear maps in finite dimensional banach and hilbert spaces boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals these functionals are used to assess qualitative properties such as invertibility and also enable a generalization of some well known matrix results directly to nonlinear operators closely related to the numerical range of a matrix the gerschgorin domain is introduced for nonlinear operators this point set in the complex plane is always convex and contains the spectrum of the operator s jacobian matrices finally we focus on nonlinear operators in hilbert space and hint at some generalizations of the von neumann spectral theory

ground effect analysis surface wave and layer potential representations abstract this paper presents two kinds of analytical exact expressions of the sound field reflected by a plane boundary as obtained by using either surface wave or layer potentials representations both solutions are first expressed as a sum of integrals which have a form suitable for numerical computation then these integrals are expanded into convergent series which provide analytical approximations of the solution numerical techniques are proposed for computing a the surface wave representation and b the approximation deduced from the layer potentials representation this approximation and the classical one sum of the surface wave and the first two terms of the asymptotic series are compared with the exact solution several examples show that the approximate formulas established here are valid on a range much wider than the validity domain of the classical ones

a uniformly valid far field asymptotic expansion of the green function for two dimensional propagation above a homogeneous impedance plane abstract a generalized asymptotic expansion in the far field for the problem of cylindrical wave reflection at a homogeneous impedance plane is derived the expansion is shown to be uniformly valid over all angles of incidence and values of surface impedance including the limiting cases of zero and infinite impedance the technique used is a rigorous application of the modified steepest descent method of ott 1

the field of a spherical wave reflected from a plane absorbent surface expressed in terms of an infinite series of legendre polynomials abstract the acoustic field of a spherical sound wave reflected from a locally reacting plane surface is expressed as an asymptotic expansion whose terms consist of the product of legendre polynomials and powers of the reciprocal of the distance from the image source

the complex equivalent source method for sound propagation over an impedance plane the sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane originally such an approach for representing the half space green s function was described by sommerfeld at the beginning of the last century in order to treat half space problems of heat conduction however the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances the singular part of the line integral can be transformed into a hankel function which shows that surface waves are contained in the whole solution unfortunately this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence the main idea of the present method is to use also a superposition of equivalent point sources but to allow that these sources can be located at complex source points the corresponding form of the half space green s function is suitable for both masslike and springlike surface impedances and can be used as a cornerstone for a boundary element method

surface modes in sheared boundary layers over impedance linings abstract surface modes being duct modes localized close to the duct wall are analysed within a lined cylindrical duct with uniform flow apart from a thin boundary layer as well as full numerical solutions of the pridmore brown equation simplified mathematical models are given where the duct lining and boundary layer are lumped together and modelled using a single boundary condition a modification of the myers boundary condition previously proposed by the author from which a surface mode dispersion relation is derived for a given frequency up to six surface modes are shown to exist rather than the maximum of four for uniform slipping flow not only is the different number and behaviour of surface modes important for frequency domain mode matching techniques which depend on having found all relevant modes during matching but the thin boundary layer is also shown to lead to different convective and absolute stability than for uniform slipping flow numerical examples are given comparing the predictions of the surface mode dispersion relation to full solutions of the pridmore brown equation and the accuracy with which surface modes are predicted is shown to be significantly increased compared with the uniform slipping flow assumption the importance of not only the boundary layer thickness but also its profile tanh or linear is demonstrated a briggs bers stability analysis is also performed under the assumption of a mass spring damper or helmholtz resonator impedance model

numerical prediction of aft radiation of turbofan tones through exhaust jets this paper describes a numerical methodology for calculating tonal noise propagation and radiation through turbomachinery exhaust ducts including non uniform background jet flows the numerical method is based on solution of the linearized euler equations directly in the frequency domain employing a direct sparse matrix solver in parallel acoustic sources are introduced into the computational domain via the perfectly matched layer equations various test cases including propagation through infinite ducts and propagation and radiation through semi infinite ducts with and without liners are solved and results are compared with analytical solutions it is demonstrated that solutions even with thin shear layers separating the exterior and jet streams at reasonably high mach numbers can be obtained quite successfully the method is also validated by simulating the radiated sound waves from two actual engine exhaust cowl geometries and the results are compared with experimental data

on the robustness of mathcalkl stability for difference inclusions smooth discrete time lyapunov functions we consider stability with respect to two measures of a difference inclusion i e of a discrete time dynamical system with the push forward map being set valued we demonstrate that robust stability is equivalent to the existence of a smooth lyapunov function and that in fact a continuous lyapunov function implies robust stability we also present a sufficient condition for robust stability that is independent of a lyapunov function toward this end we develop several new results on the behavior of solutions of difference inclusions in addition we provide a novel result for generating a smooth function from one that is merely upper semicontinuous

existence and stability of a unique equilibrium in continuous valued discrete time asynchronous hopfield neural networks it is shown that the assumption of d stability of the interconnection matrix together with the standard assumptions on the activation functions guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes for the synchronous mode these assumptions are also shown to imply local asymptotic stability of the equilibrium for the asynchronous mode of operation two results are derived first it is shown that symmetry and stability of the interconnection matrix guarantee local asymptotic stability of the equilibrium under a class of asynchronous modes this is referred to as local absolute asymptotic stability second it is shown that under the standard assumptions if the nonnegative matrix whose elements are the absolute values of the corresponding elements of the interconnection matrix is stable then the equilibrium is globally absolutely asymptotically stable under a class of asynchronous modes the results obtained are discussed from the points of view of their applications robustness and their relationship to earlier results

on a bilinear operator free third order method on riemannian manifolds we present a semilocal convergence analysis of a bilinear operator free third order method on riemannian manifolds using a combination of generalized lipschitz and center lipschitz conditions we provide a convergence analysis which expands the applicability of the method even in the setting of nonlinear equations considered in earlier studies such as cf 4 9 38

quantitative studies on asymptotic growth behaviors of trajectories of nonlinear discrete dynamical systems this technical note studies quantitatively asymptotic growth behaviors of trajectories agbt of nonlinear autonomous discrete dynamical system that has unbounded domain non lipschitz continuous nonlinear operator and stable or unstable equilibrium point we explain how trajectory motion speed is quantitatively determined in the system and study exact computation and sharp estimation of the smallest exponential bound of trajectories we characterize exponential stability and asymptotic stability of the system from a new point of view and provide a simple condition to distinguish them from each other these results extend existing results that were obtained in some special cases of the system and are helpful for quantitative analysis and understanding of agbt of the system