structural properties of positive linear time invariant difference algebraic equations in this paper we analyze positive difference algebraic equations conditions regarding some of the matrices involved in the solution of this kind of systems are described geometrical conditions are used to characterize positive structural properties

stability analysis for positive singular systems with time varying delays this paper is concerned with the stability analysis for continuous time positive singular systems with time varying delays first an auxiliary system is introduced to establish the positivity condition for singular time delay systems based on the positivity condition the stability criterion is obtained for the positive singular systems with constant delays by analyzing the monotonic property of the system trajectory we extend the stability condition to the cases with time varying delays a numerical simulation is employed to illustrate the effectiveness of our results

the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

poisson structure of a modified hunter saxton equation both the well known korteweg de vries equation and the hunter saxton equation with a linear wux term utxx 2 ux 2uxuxx uuxxx 0 are bi hamiltonian in the sense that each equation is hamiltonian with respect to two compatible but distinct poisson brackets we present a nonlinear change of variables which maps between the two bi poisson structures giving a correspondence between the infinite hierarchies of conservation laws and commuting flows associated with the equations in particular under this change of variables this modified hunter saxton equation can be viewed as a member of the kdv hierarchy

construction of coupled harry dym hierarchy and its solutions from stackel systems in this paper we show how to construct the coupled multicomponent harry dym chd hierarchy from classical stackel separable systems both nonlocal and purely differential parts of hierarchies are obtained we also construct various classes of solutions of chd hierarchy from solutions of corresponding stackel systems

on the decay rate of discrete time linear delay systems with cone invariance this technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone invariant property in order to capture the decay rate of such systems we introduce a nondecreasing positive function whose reciprocal represents the decay rate under mild assumptions on the growth rate of this function an explicit condition is given to ensure that a cone preserving linear system with unbounded time varying delays is asymptotically stable with a given decay rate as typical cases necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear sublinear or logarithmic growth rate finally some numerical examples are given to illustrate the effectiveness of the theoretical results

looking for nonnegative solutions of a leontief dynamic model let a be an n n nonnegative real matrix we know that a is monotone if it has a nonnegative inverse moreover if a has an appropriate generalized inverse then we can obtain nonnegative solutions for the linear system ax b where b 0 in this paper using properties about nonnegatives matrices we present sufficient conditions in order that the leontief price model i f p k 1 1 r k fp k gp0 k 1 a0 where f i at 1bt and g i at 1 have nonnegative solutions

stability analysis for positive singular systems with distributed delays abstract this paper is concerned with the stability analysis for continuous time positive singular systems with distributed delays by introducing an auxiliary system a necessary and sufficient positivity condition is proposed for singular systems with distributed lags based on this condition we present a sufficient stability condition for positive singular systems with distributed delays and extend the result to systems with distributed delays over a bounded time varying interval in addition this stability condition is also necessary when strictly positive initial conditions are available numerical simulations are utilized to illustrate the effectiveness of our results

on the link between umbilic geodesics and soliton solutions of nonlinear pdes in this paper we describe a new class of soliton solutions called umbilic solitons for certain nonlinear integrable pdes these umbilic solitons have the property that as the space variable x tends to infinity the solution tends to a periodic wave and as x tends to minus infinity it tends to a phase shifted wave of the same shape the equations admitting solutions in this new class include the dym equation and equations in its hierarchy the methods used to find and analyse these solutions are those of algebraic and complex geometry we look for classes of solutions by constructing associated finite dimensional integrable hamiltonian systems on riemann surfaces in particular in this setting we use geodesics on n dimensional quadrics to find the spatial or x flow which together with the commuting t flow given by the equation itself defines new classes of solutions amongst these geodesics particularly interesting ones are the umbilic geodesics which then generate the class of umbilic soliton solutions this same setting also enables us to introduce another class of solutions of dym like equations which are related to elliptic and umbilic billiards

static output feedback stabilization with optimal l 1 gain for positive linear systems this paper is concerned with the static output feedback stabilization with optimal l1l1 gain for positive linear systems we aim to construct a static output feedback controller such that the closed loop system is positive asymptotically stable and the l1l1 gain from the exogenous input to the regulated output is minimized we show that when the control input or the measured output is a scalar this problem can be directly solved via linear programming by adding a one dimensional search moreover it is pointed out that the results can be extended to solve the l l optimal controller synthesis problem finally several illustrative examples are given to show the effectiveness of the theoretical results

time domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability transient fields arising from a general impulsive dipole above ground characterized by permittivity and permeability are studied a detailed analysis for arbitrary electric dipole excitation is given and the fields arising from the analog magnetic sources can be obtained by applying the duality transform the mathematical description of this problem is done using the hertz vector whose solution in the time domain is derived in a novel way using the exact image theory from this solution which involves a definite integral the actual fields are obtained through simple differentiations the efficiency of the method is demonstrated by solving 14 previously unknown closed form results for various field components in certain specific cases seven of these are obtained for electric dipole excitation and listed whereas the corresponding results in the case of magnetic excitation can be easily written by applying the duality transform

exact image theory for the calculation of fields transmitted through a planar interface of two media exact image theory recently developed for the calculation of fields reflected from a planar interface between two homogenous media is extended for transmitted field calculation the image of a point source is found to be a line source located in complex space for proper convergence unlike for the reflection image the transmission image line is seen to be curved in complex space for efficient computation expressions are derived for the transmission image of the most general three dimensional electric and magnetic sources and for media with both epsilon and mu discontinuous across the interface the new theory is verified with several asymptotic tests and properties of the transmission image function are discussed as an application radiation from a resonant horizontal loop antenna into the ground is calculated and results are seen to agree with those given in the literature

new variational and multisymplectic formulations of the euler poincare equation on the virasoro bott group using the inverse map we derive a new variational principle leading to a new momentum map and a new multisymplectic formulation for a family of euler poincare equations defined on the virasoro bott group by using the

time domain analysis of the sommerfeld vmd problem based on the exact image theory a novel time domain solution of the sommerfeld half space problem is constructed through the exact image theory the radiation from a vertical magnetic dipole vmd above the planar interface of two lossless and dispersionless dielectric media is analyzed the reflection field is given as a laplace transform when the refraction index of the lower medium is greater the converse case of refraction indexes is also studied and the reflection field is written as originating from an image source located in homogeneous space the present theory is seen to lead to a numerically efficient result which is demonstrated with examples

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

conservative solutions to a system of asymptotic variational wave equations abstract this paper is concerned with an asymptotic variational wave system which models weakly nonlinear waves for a system of variational wave equations arising in the theory of nematic liquid crystals and a few other physical contexts by constructing a global semigroup we establish the well posedness of the initial boundary value problem within the class of energy conservative solutions for initial data of finite energy

exact image theory for the sommerfeld half space problem part iii general formulation the general sommerfeld problem with both epsilon and mu discontinuous and a source consisting of arbitrarily oriented electric and or magnetic dipoles at the same location is considered in terms of image theory the problem is formulated with electric and magnetic fields instead of potential quantities resulting in a vector transmission line interpretation of the fourier transformed problem the image sources are seen to be located in complex space expressable in terms of a certain basic image current function which was encountered in part ii of this paper on the vertical electric dipole problem the horizontal electric magnetic dipole image is solved and found to consist of both vertical and horizontal current components the image concept is generalized to the most general three dimensional sources as a check the well known reflection coefficient method is obtained as the far field approximation of the present theory

on negative flows of the akns hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type a deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parametrize different extensions of the akns hierarchy to include negative flows this construction establishes a purely algebraic link between on the one hand two realizations of the first negative flow of the akns model and on the other two component generalizations of camassa holm and dym type equations the two component generalizations of camassa holm and dym type equations can be obtained from the negative order hamiltonians constructed from the lenard relations recursively applied on the casimir of the first poisson bracket of hydrodynamic type the positive order hamiltonians which follow from the lenard scheme applied on the casimir of the second poisson bracket of hydrodynamic type are shown to coincide with the hamiltonians of the akns model the akns hamiltonians give rise to charges conserved with respect to equations of motion of two component camassa holm and two component dym type equations

scattering from dielectric structures above impedance surfaces and resistive sheets interest in understanding of electromagnetic interaction with rough surfaces has prompted the study of scattering from typical dielectric humps over impedance surfaces it is shown that the green s function of the problem for a resistive sheet resembles that of the impedance surface hence both problems are considered here a numerical solution for the scattered field of a two dimensional dielectric object possibly inhomogeneous with arbitrary cross section above the impedance surface or resistive sheet is sought first the green s function of the problem is derived based on the exact image theory this form of the green s function is amenable to numerical computation then the induced polarization currents are calculated by casting the integral equations into a matrix equation via the method of moments numerical problems in calculation of the green s function when both source and observation points are close to the surface are discussed comparison of numerical results with a perturbation solution shows excellent agreement between the two methods

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

convex searches for discrete time zames falb multipliers in this article we develop and analyze convex searches for zames falb multipliers we present two different approaches infinite impulse response iir and finite impulse response fir multipliers the set of fir multipliers is complete in that any iir multipliers can be phase substituted by an arbitrarily large order fir multiplier we show that searches in discrete time for fir multipliers are effective even for large orders as expected the numerical results provide the best ell 2 stability results in the literature for slope restricted nonlinearities in particular we establish the equivalence between the state of the art lyapunov results for slope restricted nonlinearities and a subset of the fir multipliers finally we demonstrate that the discrete time search can provide an effective method to find suitable continuous time multipliers

exact image method for impedance computation of antennas above the ground the principle of exact images recently developed by two of the present authors for the sommerfeld problem is here applied for the classical impedance problem of antennas above the ground in the present method the effect of the ground is taken into account with an image source which is located in complex space leading to a modified green function for the antenna current integral equation the image of the antenna can be effectively determined through new expressions evaluated for the image current function the stationary functional for the antenna impedance is extended to take into account the effect of the ground as examples of the theory impedance expressions for small dipoles and half wave dipoles are formulated with numerical results calculated and compared with previous results a simple method applicable for small microcomputers is suggested for calculating the impedance of wire antennas with finite thickness from that of infinitesimally thin antenna if the height of the antenna from the ground is not too small also a practical method is suggested for the determination of ground parameters from horizontal lambda 2 dipole minimum resistance and height measurements

on an integrable hierarchy derived from the isentropic gas dynamics in this paper we study a new hierarchy of equations derived from the system of isentropic gas dynamics equations where the pressure is a nonlocal function of the density we show that the hierarchy of equations is integrable we construct the two compatible hamiltonian structures and show that the first structure has three distinct casimirs while the second has one the existence of casimirs allows us to extend the flows to local ones we construct an infinite series of commuting local hamiltonians as well as three infinite series related to the three casimirs of nonlocal charges we discuss the zero curvature formulation of the system where we obtain a simple expression for the nonlocal conserved charges which also clarifies the existence of the three series from a lie algebraic point of view we point out that the nonlocal hierarchy of hunter zheng equations can be obtained from our nonlocal flows when the dynamical variables are properly constrained

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

on the n 2 supersymmetric camassa holm and hunter saxton equations we consider n 2 supersymmetric extensions of the camassa holm and hunter saxton equations we show that they admit geometric interpretations as euler equations on the superconformal algebra of contact vector fields on the 1 2 dimensional supercircle we use the bi hamiltonian formulation to derive lax pairs moreover we present some simple examples of explicit solutions as a by product of our analysis we obtain a description of the bounded traveling wave solutions for the two component hunter saxton equation

coupled camassa holm equations n peakons and infinitely many conservation laws abstract three coupled camassa holm equations are proposed which are associated with a 4 4 matrix spectral problem with two potentials with the aid of the zero curvature equation we derive a hierarchy of new nonlinear evolution equations and establish their bi hamiltonian structures the three coupled camassa holm equations are exactly negative flows in the hierarchy and exact solutions with n peakons and infinite sequences of conserved quantities are discussed

an integrable equation governing short waves in a long wave model the dynamics of a nonlinear and dispersive long surface capillary gravity wave model equation is studied analytically in its short wave limit we exhibit its lax pair and some non trivial conserved quantities through a change of functions an unexpected connection between this classical surface water wave model and the sine gordon or sinh gordon equation is established numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time these features can be traced to the fact that the surface tension term in the energy involves second order derivatives it would be interesting to see in an experiment whether such singularities actually appear for which surface tension would be specifically responsible

characterization and stability of autonomous positive descriptor systems this note addresses two problems the characterization of positivity of autonomous descriptor systems and the problem of finding conditions for their stability for these fundamental issues we provide necessary and sufficient conditions complemented with a numerically reliable computational approach the given conditions can be checked by means of linear programming lp also we derive a necessary and sufficient condition for stability that is connected to the underlying eigenstructure of the system

stability analysis of positive descriptor systems abstract in this paper we discuss stability properties of positive descriptor systems in the continuous time as well as in the discrete time case we present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems we show that if the spectral projector onto the finite deflating subspace of the matrix pair e a is nonnegative then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case furthermore we provide sufficient conditions that guarantee entry wise nonnegativity along with positive semidefiniteness of solutions of generalised projected lyapunov equations as an application of the framework established throughout this paper we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

nonnegativity stability and regularization of discrete time descriptor systems in this paper for a given descriptor system we construct a feedback in order to obtain a nonnegative stable and regular closed loop system the required conditions impose structure on the state matrices of the descriptor system and thereby allow expression of the solution of the new system in terms of smaller matrices for that we exploit the structure of the state matrices of the descriptor system we also design an algorithm to construct the aforementioned feedback and the solution of the closed loop system in addition some examples illustrating the results are presented

a note on multi dimensional camassa holm type systems on the torus we present a 2n component nonlinear evolutionary pde which includes the n dimensional versions of the camassa holm and the hunter saxton systems as well as their partially averaged variations our goal is to apply arnold s geometric formalism arnold 1966 ann inst fourier grenoble 16 319 61 ebin and marsden 1970 ann math 92 102 63 to this general equation in order to obtain results on well posedness conservation laws or stability of its solutions following the line of arguments of kohlmann 2011 j phys a math theor 44 465205 we present geometric aspects of a two dimensional periodic b equation on the diffeomorphism group of the torus in this context

positivity and stability analysis for linear implicit difference delay equations abstract this paper deals with positivity and stability of linear implicit difference delay equations being different from the lyapunov function approach commonly used in stability analysis the method employed in this paper gives a way to solve the exponential stability of linear implicit difference equations with time varying delay by decomposition state space and mathematical induction method new necessary and sufficient conditions for positivity and stability of such systems are derived numerical examples are given to illustrate the proposed results

integrable vs nonintegrable geodesic soliton behavior abstract we study confined solutions of certain evolutionary partial differential equations pde in 1 1 space time the pde we study are lie poisson hamiltonian systems for quadratic hamiltonians defined on the dual of the lie algebra of vector fields on the real line these systems are also euler poincare equations for geodesic motion on the diffeomorphism group in the sense of the arnold program for ideal fluids but where the kinetic energy metric is different from the l2 norm of the velocity these pde possess a finite dimensional invariant manifold of particle like measure valued solutions we call pulsons we solve the particle dynamics of the two pulson interaction analytically as a canonical hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum the result of this two pulson interaction for rear end collisions is elastic scattering with a phase shift as occurs with solitons the results for head on antisymmetric collisions of pulsons tend to be singularity formation numerical simulations of these pde show that their evolution by geodesic dynamics for confined or compact initial conditions in various nonintegrable cases possesses the same type of multi soliton behavior elastic collisions asymptotic sorting by pulse height as the corresponding integrable cases do we conjecture this behavior occurs because the integrable two pulson interactions dominate the dynamics on the invariant pulson manifold and this dynamics dominates the pde initial value problem for most choices of confined pulses and initial conditions of finite extent

model order reduction with preservation of passivity non expansivity and markov moments a new model order reduction technique is presented which preserves passivity and non expansivity it is a projection based method which exploits the solution of linear matrix inequalities to generate a descriptor state space format which preserves positive realness and bounded realness in the case of both non singular and singular systems solving the linear matrix inequality can be replaced by equivalently solving an algebraic riccati equation which is known to be a more efficient approach a new algebraic riccati equation and a frequency inversion technique are also presented to specifically deal with the important singular case the preservation of markov moments is also guaranteed by the judicious choice of a projection matrix three pertinent examples comparing the present approach with positive real balanced truncation show the strength and accuracy of the present approach

an algorithm to check the nonnegativity of singular systems in the literature an important class of generalized inverse matrices corresponds to the group inverse that is matrices of index 1 recently the nonnegativity of a singular system has been applied to different fields in this paper an algorithm to check the nonnegativity of a singular linear control system of index 1 is presented to this purpose the nonnegativity of this kind of systems is characterized using a block partition of the original matrices in this way we can work with matrices having smaller sizes and keeping the original information finally numerical examples illustrating the obtained results are shown

efficient calculation of the fields of a dipole radiating above an impedance surface in this paper the classic problem of field computation for an infinitesimal dipole radiating above an impedance half space is revisited the expressions for the traditional solution consist of integrals of the sommerfeld type that cannot be evaluated in closed form and due to their highly oscillatory nature are difficult to evaluate numerically a method known as exact image theory which has previously been applied to vertical electric and magnetic dipoles is used to derive explicit expressions for dipoles of arbitrary orientation above impedance surfaces starting from the spectral representation of the field the reflection coefficients are cast in the form of exact laplace transforms and then by changing the order of integrations field expressions in terms of rapidly converging integrals are obtained these expressions are exact and valid for any arbitrary source alignment or observation position it is shown that the formulation for a horizontal dipole contains an image in the conjugate complex plane resulting in a diverging exponential term not previously addressed in the literature it is shown through further mathematical manipulations that the diverging term is a contribution of the mirror image which can be extracted comparison of numerical results from exact image theory and the original sommerfeld type expressions shows good agreement as well as a speedup in computation time of many orders of magnitude which depends on the distance between the transmitter and the receiver this formulation can effectively replace the approximate asymptotic expressions used for predicting wave propagation over a smooth planar ground having different regions of validity the exact image formulation is also of practical use in evaluation of the green s function for various applications in scattering problems where approximate solutions are not sufficient

maximal dissipation in hunter saxton equation for bounded energy initial data abstract in 13 it was conjectured by zhang and zheng that dissipative solutions of the hunter saxton equation which are known to be unique in the class of weak solutions dissipate the energy at the highest possible rate the conjecture of zhang and zheng was proven in 4 by dafermos for monotone increasing initial data with bounded energy in this note we prove the conjecture in 13 in full generality to this end we examine the evolution of the energy of any weak solution of the hunter saxton equation our proof shows in fact that for every time t 0 the energy of the dissipative solution is not greater than the energy of any weak solution with the same initial data

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

robust analysis and control of parameter dependent uncertain descriptor systems in this paper we address the robust admissibility analysis and control of descriptor systems with parameter dependent possibly time varying uncertainties first we propose a new necessary and sufficient condition for the admissibility of linear time invariant lti descriptor systems this condition uses a strict linear matrix inequality lmi formulation and involves two additional matrix variables usually called slack variables then we present an extension of this condition to robust admissibility of time varying parameter dependent descriptor systems using parameter dependent lyapunov functions and employing two slack variables this extension states robust admissibility lmi conditions that are less restrictive as supported by our numerical examples than the existing ones our robust admissibility analysis result which can be directly used for state estimation cannot be so for state feedback control design this is due to the absence of duality between state estimation and state feedback control in the case of parameter dependent systems with uncertain parameters as recently reported in the literature in order to overcome this problem we also present in this paper a new robust admissibility analysis result involving only one slack variable which we apply to state feedback control design

lie symmetries and infinite dimensional lie algebras of certain 1 1 dimensional nonlinear evolution equations in this paper we discuss the lie symmetries symmetry algebra and similarity reductions of two different equations introduced in the recent literature namely i a new coupled integrable dispersionless equation and ii a new coupled hyperbolic variational equation we point out that both the systems admit in contradistinction to conventional lie algebras in 1 1 dimensional systems infinite dimensional lie algebras furthermore we find physically interesting solutions for special choices of the symmetry parameters

nonlinear asymptotic short wave models in fluid dynamics nonlinear monochromatic short surface waves in ideal fluids are studied and by the general consideration of wave dynamics and perturbative methods a simple and effective multiscale approach is devised for nonlinear asymptotic short wave dynamics in dispersive systems in particular the evolution of a monochromatic surface wave in an ideal fluid is shown to lead to a modified green naghdi system of equations and a green naghdi system with surface tension short surface waves exist in these systems and the nonlinear asymptotic analysis produces the nonlinear model equations that govern their dynamics particular solutions are shown moreover the method allows for a general classification of classical model equations as boussinesq benjamin bona mahony peregrine and camassa holm equations their related nonlinear short wave model limits are then derived a relation between the short wave limit of the integrable camassa holm equation and the harry dym hierarchy is finally unveiled

stability analysis of linear delay systems with cone invariance this paper is concerned with the stability and input output gain analysis of linear delay systems with cone invariance based on the partial ordering over a cone the monotonicity of the trajectory of the cone preserving systems with constant delays is first studied then by comparing the trajectory of the constant delay systems and that of time varying delay systems we prove that a cone preserving system with interval time varying delays is asymptotically stable if and only if the corresponding delay free system is asymptotically stable this implies that the stability of a cone preserving system is insensitive to the magnitude of the delays moreover based on the cone induced norms an explicit characterization on the cone induced gain of an input output cone preserving system is given in terms of system matrices finally numerical examples are provided to illustrate the theoretical results

existence and singularities of solutions to an integrable equation governing short waves in a long wave model this paper is concerned with the cauchy problem of a new periodic integrable equation governing short waves in a long wave model we first establish the local well posedness for its derivative equation which correspondingly implies the local existence and multivaluedness of solutions to the model then we obtain a blow up scenario for these solutions finally we show that the equation has strong solutions that develop singularities in finite time

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

global existence of dissipative solutions to the hunter saxton equation via vanishing viscosity abstract using the vanishing viscosity method we prove the global existence of dissipative weak solutions to the hunter saxton equation that describes the propagation of waves in a massive director field of a nematic liquid crystal our main tool is the l p young measure theory we also derive the upper bound on the convergence rate for the vanishing viscosity approximations

periodic solitons of an equation for short capillary gravity waves we consider the periodic solitons of an equation for short capillary gravity waves although their profiles cannot be given a description in terms of a mathematical expression in closed form we show that they are transformed by a natural change of variables to the well known cnoidal periodic solitons of the kdv equation

well posedness and peakons for a higher order μ camassa holm equation abstract in this paper we study the cauchy problem of a higher order camassa holm equation by employing the green s function of x 2 2 we obtain the explicit formula of the inverse function x 2 2 w and local well posedness for the equation in sobolev spaces h s s s 7 2 then we prove the existence of global strong solutions and weak solutions moreover we show that the data to solution map is holder continuous in h s s s 4 equipped with the h r s topology for 0 r s finally the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives

stability and stabilizability criteria for discrete time positive switched systems in this paper we consider the class of discrete time switched systems switching between p autonomous positive subsystems first sufficient conditions for testing stability based on the existence of special classes of common lyapunov functions are investigated and these conditions are mutually related thus proving that if a linear copositive common lyapunov function can be found then a quadratic positive definite common function can be found too and this latter in turn ensures the existence of a quadratic copositive common function secondly stabilizability is introduced and characterized it is shown that if these systems are stabilizable they can be stabilized by means of a periodic switching sequence which asymptotically drives to zero every positive initial state conditions for the existence of state dependent stabilizing switching laws based on the values of a copositive linear quadratic lyapunov function are investigated and mutually related too finally some properties of the patterns of the stabilizing switching sequences are investigated and the relationship between a sufficient condition for stabilizability the existence of a schur convex combination of the subsystem matrices and an equivalent condition for stabilizability the existence of a schur matrix product of the subsystem matrices is explored

input output gain analysis for linear systems on cones abstract this paper is concerned with the input output gain analysis for dynamic systems that are defined on cones for such monotone systems we characterize their input output gain induced by the so called cone linear absolute norm in terms of generalized inequalities it is further shown that for a linear time invariant system represented by a convolution operator that maps one proper cone to another in the l 1 space its operator norm is governed by the static gain of the system as a byproduct the spectral radius of a cone preserving convolution operator turns out to be the same as that of the static gain matrix moreover dual results based on the cone max norm are presented finally the theoretical results are illustrated via linear systems over polyhedral cones or second order cones as well as a class of stochastic systems

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

analyticity of the cauchy problem for two component hunter saxton systems abstract this paper is mainly concerned with the periodic cauchy problem for a generalized two component hunter saxton system with analytic initial data the analyticity of its solutions is proved in both variables globally in space and locally in time the obtained result can be also applied to its special cases the classical integrable two component hunter saxton system the generalized hunter saxton equation and the classical hunter saxton equation

evaluation of sommerfeld integrals using chebyshev decomposition antenna analysis sommerfeld integrals ensue when one uses a spectral domain transformation to calculate fields of a dipole source in a homogeneous space or in a layered medium with planar boundaries a new and efficient method of evaluating these integrals is presented this method still converge in the case of unbounded homogeneous media or when source and observer are both on the boundary itself as in microstrip lines the integration is based on decomposition of the integrand into chebyshev polynomials numerical results are given and compared with published literature

exact closed form expression of a sommerfeld integral for the impedance plane problem a sommerfeld integral for an impedance half plane is one of the classical problems in electromagnetic theory in this paper the integral is evaluated into two series representations which are expressed in terms of the exponential integral and the lommel function respectively it is shown that the series expansions absolutely converge for any parameters such as distance or surface impedance then based on the lommel function expansion an exact closed form expression of the integral is formulated this expression is written in terms of incomplete weber integrals which are directly related to incomplete cylindrical functions such as incomplete lipschitz hankel integrals additionally a complete asymptotic series of the integral is obtained based on the exponential integral expansion with conventional methods such as steepest descent method it is cumbersome to derive the divergent series the validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

positivity and stability of coupled differential difference equations with time varying delays this paper studies the asymptotic stability of a special class of coupled delay differential difference equations with internal positive property an explicit characterization on the positivity of coupled differential difference equations is firstly given then based on the positivity of coupled differential difference equations with constant delays we investigate the entrywise monotonicity and asymptotic property of their state trajectories starting from appropriately chosen initial conditions furthermore the time varying delay system is analyzed through comparing with the corresponding constant delay system it turns out that an internally positive coupled differential difference equation with bounded time varying delays is asymptotically stable as long as the corresponding delay free system is asymptotically stable

disturbance impulse controllability in descriptor system the descriptor system ex ax gu hv with external disturbance v is considered to characterize the possibility to cancel the impulses produced by external disturbance v through using control u the concept of disturbance impulse controllability is introduced

supersymmetric extensions of the harry dym hierarchy we study the supersymmetric extensions of the harry dym hierarchy of equations we obtain the susy b extension the doubly susy b extension as well as the n 1 and the n 2 supersymmetric extensions for this system the n 2 supersymmetric extension is particularly interesting since it leads to new classical integrable systems in the bosonic limit we prove the integrability of these systems through the bi hamiltonian formulation of integrable models and through the lax description we also discuss the supersymmetric extension of the hunter zheng equation which belongs to the harry dym hierarchy of equations

weak geodesic flow on a semidirect product and global solutions to the periodic hunter saxton system abstract we give explicit solutions for the two component hunter saxton system on the unit circle moreover we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re expression of the geodesic flow on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle these spatially and temporally periodic solutions turn out to be conservative

exact image theory for the sommerfeld half space problem part ii vertical electrical dipole applying the laplace transform the exact distributed image current function is obtained for the classical sommerfeld half space problem with vertical magnetic current source the resulting field integral is well behaved when the image current is situated in complex space unlike previous approximate images the present theory is valid for any distance height of the source frequency and half space parameters it is demonstrated that the present image theory reduces to the well known dipole image at complex depth for large dielectric parameters of the half space also the reflection coefficient method is obtained as a farfield approximation calculation of fields through exact image integration is seen to be simple and accurate and require modest computer capacity and time in an appendix some properties of the multivalued green s function arising from a dipole source in complex space are also studied

convergent difference schemes for the hunter saxton equation we propose and analyze several finite difference schemes for the hunter saxton equation hs u t uu x 1 2 x 0 u x 2 d x x 0 t 0 this equation has been suggested as a simple model for nematic liquid crystals we prove that the numerical approximations converge to the unique dissipative solution of hs as identified by zhang and zheng a main aspect of the analysis in addition to the derivation of several a priori estimates that yield some basic convergence results is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u which is achieved by analyzing various renormalizations in the sense of diperna and lions of the numerical schemes finally we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results

imaging quasi layered conductive structures by simple processing of transient electromagnetic data an imaged conductivity section of a layered earth can be obtained by simple transformation of step response electromagnetic data measured in the quasi static zone this method of data transformation is presented as an alternative to conventional apparent conductivity transformations at each delay time the variation of the step response as a function of geometry transmitter and receiver location is transformed to an equivalent reference depth h which can be related to the depth of electromagnetic field diffusion the behavior of h as a function of delay time is nearly independent of the source receiver geometry the slowness dt dh divided by the magnetic permeability is almost exactly proportional to the cumulative conductance measured from the surface down to a depth h thus we can estimate an apparent conductivity which we call the imaged conductivity at depth to be d2t 0dh2 the cost of this transformation is a fraction of the cost of conventional data inversion and it does not require an a

continued fractions and integrable systems stieltjes solution of the classical moment problem is the forerunner of inverse spectral theory in the modern theory of completely integrable systems the method of inverse scattering is used to obtain explicit solutions of a class of hamiltonian systems which arise as isospectral deformations of a linear operator in this lecture we explain how stieltjes s formulas are used to obtain explicit solutions to a number of completely integrable systems including discrete reductions of some nonlinear partial differential equations and the isospectral deformations of jacobi matrices a k a toda flows

a characterization of the drazin inverse abstract if a is a nonsingular matrix of order n the inverse of a is the unique matrix x such that rank a i i x rank a in this paper we generalize this fact to any matrix a of order n over the complex field to obtain an analogous result for the drazin inverse of a we then give an algorithm for computing the drazin inverse of a

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

positivity preserving balanced truncation for descriptor systems we propose a model reduction method for positive systems that ensures the positivity of the reduced order model in the standard as well as in the descriptor case for continuous time and discrete time systems our approach is based on constructing diagonal solutions of lyapunov inequalities these are linear matrix inequalities lmis which are shown to be feasible positivity and stability are preserved and an error bound in the mathcal h infty norm is provided

positivity of discrete singular systems and their stability in this paper we present an efficient approach to the analysis of discrete positive singular systems one of our main objectives is to investigate the problem of characterizing positivity of such systems previously this issue was not completely addressed we provide easily checkable necessary and sufficient conditions for such problem to be solved on the other hand we study the stability of discrete positive singular systems note that this is not a trivial problem since the set of admissible initial conditions is not the whole space but it is represented by a special cone all the conditions we provide are necessary and sufficient and are based on a reliable computational approach via linear programming

bosonic reduction of susy generalized harry dym equation in this paper we construct the two component supersymmetric generalized harry dym equation which is integrable and study various properties of this model in the bosonic limit in particular in the bosonic limit we obtain a new integrable system which under a hodograph transformation reduces to a coupled three component system we show how the hamiltonian structure transforms under a hodograph transformation and study the properties of the model under a further reduction to a two component system we find a third hamiltonian structure for this system which has been shown earlier to be a bi hamiltonian system making this a genuinely tri hamiltonian system the connection of this system to the modified dispersive water wave equation is clarified we also study various properties in the dispersionless limit of our model

locating all the modes of green s function for a three layered medium based on the path tracking algorithm through a mathematical analysis on the relationship between the characteristic equations of the microstrip geometry and a configuration with a three layered medium a method based on the path tracking algorithm is proposed for precisely locating all the modes for a three layered medium with moderate thickness including surface wave modes leaky wave modes and improper real modes with this method the difficulty caused by branch points and cuts in the contour integral method could be avoided this method can be applied not only in the discrete complex image method dcim but also in the steepest descent path sdp method the proposed method can be extended to more complicated configurations

global weak solutions and smooth solutions for a two component hunter saxton system in this paper we mainly study the initial value problem of a two component hunter saxton system by the method of characteristics we first show that the system has global smooth solutions and blowing up smooth solutions by using the obtained a priori estimates on smooth solutions we then prove the existence of global weak solutions to the system

complete closed form expression of dyadic green s function and its far and near field approximations for an impedance half plane a closed form expression of the dyadic green s function is formulated for an impedance half plane which is written in terms of the incomplete cylindrical function of poisson form due to the branch cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function the closed form representation consists of two formulations since the closed form expression contains a singularity at 0 the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at 0 the previously reported complete asymptotic expansion for the sommerfeld integral for an impedance half plane is not accurate for practically important cases such as near earth propagation and or when the surface is highly conductive hence in this paper a new complete asymptotic series of the sommerfeld integral are derived for the case that the existing asymptotic series is not accurate the two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios based on the two asymptotic series the complete asymptotic series of the dyadic green s function is derived all derived formulations are numerically verified and their accuracies are investigated

conservative solutions to a nonlinear variational sine gordon equation abstract this paper is concerned with a nonlinear variational sine gordon equation u t t c u c u u x x 2 2 s i n 2 u 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena we establish the global existence of an energy conservative weak solution to its cauchy problem for initial data of finite energy to deal with the possible blowup of solutions we introduce a new set of variables depending on the energy whereby all singularities are resolved

deformed harry dym and hunter zheng equations we study the deformed harry dym and hunter zheng equations with two arbitrary deformation parameters these reduce to various other known models in appropriate limits we show that both these systems are bi hamiltonian with the same hamiltonian structures they are integrable and belong to the same hierarchy corresponding to positive and negative flows we present the lax pair description for both the systems and construct the conserved charges of negative order from the lax operator for the deformed harry dym equation we construct the nonstandard lax representation for two special classes of values of the deformation parameters in general we argue that a nonstandard description will involve a pseudodifferential operator of infinite order

stability analysis and l1 gain characterization for switched positive systems under dwell time constraint abstract this paper deals with the problems of stability analysis and l 1 gain characterization for continuous time switched systems consisting of positive subsystems with the aid of a discretized copositive lyapunov function a sufficient condition ensuring the asymptotic stability of continuous time switched positive systems under dwell time constraint is obtained which can be checked via linear programming furthermore the conservatism of the proposed approach is studied and the result with least conservatism in the framework of discretized copositive lyapunov function is obtained the result is then extended to l 1 gain characterization for switched positive systems with a prescribed dwell time an unweighted l 1 gain can be computed via solving a linear programming problem a numerical example and a practical traffic example are given to illustrate our results