acoustic propagation over an impedance plane a new solution is obtained for the acoustic pressure field generated by a point source and scattered by a locally reacting impedance covered plane the new solution is obtained in the form of an asymptotic series which has been shown to agree well with other studies one of the most important features of this solution is that higher order terms can be calculated from preceding terms in the series by the use of recursion formulas comparing data predicted from this solution with that from a numerical integration of the exact expression shows the asymptotic series to be extremely accurate even for relatively low values of the parameter kr not surprisingly the plane wave solution often shows major deviations from the exact integral solution

sound propagation above an impedance boundary an asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method numerical calculation indicates that the relatively simple solution is more accurate than the thomasson asymptotic solution and the chien soroka solution furthermore it is shown that another solution simplified still more is also in good agreement with the exact solution

on an integral type operator from ω bloch spaces to µ zygmund spaces let h b denote the space of all holomorphic functions on the unit ball b c n we study the boundedness and compactness of the following integral type operatorp g f z 0 1f tz g tz dtt f h b z b where g h b and g 0 0 from w bloch spaces to m zygmund spaces

compactification and desingularization of spaces of polynomial lienard equations abstract the paper deals with polynomial lienard equations of type m n i e planar vector fields associated to a scalar second order differential equation x f x x g x 0 with f and g polynomials of respective degree m and n it is shown that besides compactifying the phase plane or the lienard plane one can also compactify and desingularize the space of lienard equations of type m n for each m n separately by adding both singular perturbation problems and hamiltonian perturbation problems

regular and slow fast codimension 4 saddle node bifurcations abstract using geometric singular perturbation theory including the family blow up as one of the main techniques we prove that the cyclicity i e maximum number of limit cycles in both regular and slow fast unfoldings of nilpotent saddle node singularity of codimension 4 is 2 the blow up technique enables us to use the well known results for slow fast codimension 1 and 2 hopf bifurcations slow fast bogdanov takens bifurcations and slow fast codimension 3 saddle and elliptic bifurcations

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

limit cycles near hyperbolas in quadratic systems abstract in this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane we study a strip of hyperbolas occurring in some quadratic systems we deal with the cyclicity of the degenerate graphics d i 2 a from the programme set up in f dumortier r roussarie c rousseau hilbert s 16th problem for quadratic vector fields j differential equations 110 1994 86 133 to solve the finiteness part of hilbert s 16th problem for quadratic systems techniques from geometric singular perturbation theory are combined with the use of the bautin ideal we also rely on the theory of darboux integrability

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

extended cesaro operators between bloch type spaces in the unit ball of cn abstract in this paper we characterize those holomorphic symbols in the unit ball of c n for which the induced extended cesaro operator t b b respectively b 0 b 0 is bounded or compact where and are normal functions on 0 1 in addition we obtain some properties of those spaces b and b 0

products of volterra type operator and composition operator from h and bloch spaces to zygmund spaces the boundedness and compactness of the products of volterra type operators and composition operators from the space of bounded analytic functions and the bloch space to the zygmund space are discussed in this paper

sound propagation above an inhomogeneous plane boundary integral equation methods abstract a boundary integral equation method is used to compute the sound pressure emitted by a harmonic source above an inhomogeneous plane first the theoretical aspects of the problem behaviour of the pressure around the discontinuities are studied then a comparison between theoretical levels and experimental levels obtained in an anechoic room is presented it shows that the boundary integral equation bie method is quite convenient for solving this kind of problem two interesting results are pointed out i if only a prediction of maximum sound levels is needed the attenuation is the same for a cylindrical source a spherical source and n spherical sources and so it is possible to transform some three dimensional problems into two dimensional ones ii a numerical method of computation of the sound field above an inhomogeneous plane does not provide a correct prediction if each part of the plane is not accurately described by the boundary condition chosen

generalized composition operators on zygmund spaces and bloch type spaces the boundedness and compactness of the generalized composition operator on zygmund spaces and bloch type spaces are investigated in this paper

a boundary element method for propagation over absorbing boundaries abstract two dimensional sound propagation over an arbitrarily shaped barrier situated on a locally reacting infinite plane in a homogeneous medium is treated using the boundary element method bem the boundary integral equation bie is formulated in terms of a half space green s function which is constructed to satisfy the impedance boundary condition on the ground plane the integral in the resulting green s function is transformed using the steepest descent method to a suitable form for effective numerical evaluation even for a relatively small distance between the image source and the receiving point comparison of the bem results with an available analytic solution and experimental results by kearns and by habault shows good agreement to the sound field using the bem

composition operators on zygmund spaces of the unit ball abstract in this paper we give some new characterizations of the zygmund space and the little zygmund space of the unit ball in c n we further characterize the boundedness and compactness of composition operators on these spaces

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

sound radiation from ducts theory and experiment abstract the paper deals with a method of prediction of noise radiated from finite length ducts with arbitrary shape the computation of the internal and external acoustic field is based upon a new variational formulation by integral equations numerical results of far and near field radiation patterns are compared with experimental data for various higher order mode source conditions

canard phenomenon for an sis epidemic model with nonlinear incidence abstract by using the singular perturbation theory especially on canard cycles the canard phenomenon for an sis epidemic model with nonlinear incidence is investigated the phenomenon suggests the existence of disease outbreak in the infection transmission it is proved that the cyclicity of any possible slow fast cycle is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the slow fast cycle by small perturbations we also indicate the regions in parameter space where the corresponding slow fast cycle has cyclicity at most one or at most two

singular perturbations and vanishing passage through a turning point abstract the paper deals with planar slow fast cycles containing a unique generic turning point we address the question on how to study canard cycles when the slow dynamics can be singular at the turning point we more precisely accept a generic saddle node bifurcation to pass through the turning point it reveals that in this case the slow divergence integral is no longer the good tool to use but its derivative with respect to the layer variable still is we provide general results as well as a number of applications we show how to treat the open problems presented in artes et al 2009 1 and dumortier and rousseau 2009 13 dealing respectively with the graphics di 2 a and df 1 a from dumortier et al 1994 14

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

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

closed form solutions for the acoustical impulse response over a masslike or an absorbing plane the transient sound field caused by a dirac delta impulse function above an infinite locally reacting plane can be calculated by applying the inverse fourier transform of the corresponding half space green s function in frequency domain as a starting point the representation given by ochmann j acoust soc am 116 6 3304 3311 2004 is used which consists of discrete and continuous superposition of point sources for a locally reacting plane with masslike character and also with pure absorbing behavior it is possible to express the resulting impulse response in closed form such a result is surprising because corresponding formulations in the frequency domain are not available yet hence the first main result is the closed form solution eq 22 for an impulse response over an infinite plane with a pure imaginary impedance the second main result is the closed form solution eq 53 for an impulse response over an infinite plane with a pure real impedance as a particular application of both ma

composition operators on the normal weight zygmund spaces in high dimensions abstract let be a normal function on 0 1 suppose that is a holomorphic self map on the unit ball b of c n in this paper the authors give a new kind of equivalent norm of the normal weight zygmund space z b on the basis of recent work the authors further study boundedness criteria of composition operators c on z b and the essential norm of c for all normal weights when 0 1 0 r d t t d r or 0 1 d r r the authors further give a brief compactness criteria of c on z b or b

variational formulation using integral equations to solve sound scattering above an absorbing plane the boundary element method is used to model two dimensional acoustic radiation and scattering from a body of arbitrary shape above an infinite plane of flat surface and homogeneous impedance the particularity of the study is the use of an indirect integral representation of the solution given in terms of the jumps of pressure and its normal derivative through the boundaries a variational formulation is associated with the boundary indirect integral equations modeling our problem the major difficulty in the formulation is the infinite feature of the plane which is avoided by introducing an appropriate green s function numerical results of the attenuation of sound by noise barriers are presented they show good agreement with other results in the literature

slow divergence integral on a mobius band abstract the slow divergence integral has proved to be an important tool in the study of slow fast cycles defined on an orientable two dimensional manifold e g r 2 the goal of our paper is to study 1 canard cycle and 2 canard cycle bifurcations on a non orientable two dimensional manifold e g the mobius band by using similar techniques our focus is on smooth slow fast models with a hopf breaking mechanism the same results can be proved for a jump breaking mechanism and non generic turning points the slow fast bifurcation problems on the mobius band require the study of the 2 return map attached to such 1 and 2 canard cycles we give a simple sufficient condition expressed in terms of the slow divergence integral for the existence of a period doubling bifurcation near the 1 canard cycle we also prove the finite cyclicity property of singular 1 and 2 homoclinic loops regular 1 homoclinic loops of finite codimension have been studied by guimond

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

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

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

the equivalent norms and the gleason s problem on μ zygmund spaces in cn abstract let be a normal function on 0 1 in this paper the authors give four equivalent norms of the functions in the zygmund space z b on the unit ball b in c n at the same time the authors prove that the gleason s problem is solvable on z b