passivity preserving model reduction by analytic interpolation abstract antoulas and sorensen have recently proposed a passivity preserving model reduction method of linear systems based on krylov projections the idea is to approximate a positive real rational transfer function with one of lower degree the method is based on an observation by antoulas in the single input single output case that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros then the approximant is also positive real however this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by byrnes georgiou and lindquist namely the maximum entropy central solution by tuning the interpolation points and the spectral zeros as prescribed by this theory one is able to obtain considerably better reduced order models we also show that in the multi input multi output case sorensen s algorithm actually amounts to tangential nevanlinna pick interpolation

stochastic balancing and approximation stability and minimality a new method of stochastic model reduction has recently been introduced by desai et al 1 2 the stability of the reduced order model has not previously been considered in this paper we show that the stability of the reduced order model follows directly from the results of pernebo and silverman 3 it is also shown that the reduced order model is minimal in the controllability observability sense the relevance of this notion of minimality to stochastic minimality is made clear

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

the adi method for bounded real and positive real lur e equations we propose an algorithm for the numerical solution of the lur e equations in the bounded real and positive real lemma for stable systems the algorithm provides approximate solutions in low rank factored form we prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness if the shift parameters are chosen appropriately the sequence is proven to be convergent to the minimal solution of the lur e equations the algorithm is based on the ideas of the recently developed adi iteration for algebraic riccati equations massoudi et al siam j matrix anal appl 2016 in particular the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

a behavioral characterization of the positive real and bounded real characteristic values in balancing in passivity preserving and bounded realness preserving model reduction by balanced truncation an important role is played by the so called positive real pr and bounded real br characteristic values both for the positive real as well as the bounded real case these values are defined in terms of the extremal solutions of the algebraic riccati associated with the system more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the riccati equation in this paper we will establish a representation free characterization of these values in terms of the behavior of the system we will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior we will then show that both for the pr and the br case the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input output behavior its unique maximal supply extracting future continuation we will explain that the term singular values should be interpreted here in a generalized sense since in our setup the future behavior is only an indefinite inner product space c 2010 elsevier b v all rights reserved

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

eigenvalue based characterization and test for positive realness of scalar transfer functions an eigenvalue based characterization of positive realness of transfer functions of a single input single output time invariant linear system is derived based on this characterization we propose an efficient computational procedure to determine if a given transfer function is positive real the input for this eigenvalue based test is any given not necessarily minimal state space representation of the linear system the test only involves standard matrix computations such as computing eigenvalues of a matrix or a matrix pencil results of numerical experiments are reported

the tremblay turbiner winternitz system as extended hamiltonian we generalize the idea of extension of hamiltonian systems developed in a series of previous articles which allows the explicit construction of hamiltonian systems with additional non trivial polynomial first integrals of arbitrarily high degree as well as the determination of new superintegrable systems from old ones the present generalization that we call modified extension of hamiltonian systems produces the third independent first integral for the complete tremblay turbiner winternitz system as well as for the caged anisotropic oscillator in dimension two

extensions of hamiltonian systems dependent on a rational parameter the technique of extension allows to build d 2 dimensional hamiltonian systems with a non trivial polynomial in the momenta first integral of any given degree starting from a suitable d dimensional hamiltonian until now the application of the technique was restricted to integer values of a certain fundamental parameter determining the degree of the additional first integral in this article we show how the technique of extension can be generalized to any rational value of the same parameter several examples are given among them the two uncoupled oscillators and a special case of the tremblay turbiner winternitz system

superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

system characterization of positive real conditions necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs by a particular transform of these conditions a direct algorithm for testing positive realness is determined that requires only checking a set of simple algebraic conditions this provides an alternative procedure to the positive real lemma and to the s domain inequalities based on this algorithm a synthesis of a positive real system via output feedback is presented

passivity preserving model reduction with finite frequency h approximation performance this paper is concerned with model reduction for passive systems for a given linear time invariant system that is stable and positive real pr our goal is to find a pr reduced order model to approximate it and our attention is focused on reducing the error with respect to a finite frequency h performance which is the most remarkable difference between the proposed approach and the existing ones first by applying multiplier expansion new conditions in terms of linear matrix inequalities are derived for characterizing the positive realness of the reduced order model and the finite frequency h performance of the error system a necessary and sufficient condition is then established for parameterizing a pr reduced order model with finite frequency h approximation performance based on which an iterative algorithm is constructed for numerically exploring such a reduced order model particularly a partial multiplier expansion treatment is introduced which greatly reduces the decision variables but does not cause conservatism to the derived conditions the proposed method is also extended to robust passivity reserving model reduction with polytopic uncertainty finally we provide two numerical examples about rlc circuits to show the effectiveness and advantages of the proposed model reduction method

zeta invariants for sequences of spectral type special functions and the lerch formula we discuss the spectral properties of a class of sequences of what we call spectral type we introduce an effective method to calculate the zeta invariants for this type of sequence such invariants are given in terms of some new and old special functions and we consider a number of examples in which we study the properties of these special functions

infinite families of superintegrable systems separable in subgroup coordinates a method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly solvable it is shown that in two dimensional euclidean or pseudo euclidean spaces the method also preserves superintegrability two infinite families of classical and quantum superintegrable systems are obtained in two dimensional pseudo euclidean space whose classical trajectories and quantum eigenfunctions are investigated in particular the wave functions are expressed in terms of laguerre and generalized bessel polynomials

higher order superintegrability of a holt related potential in a recent paper post and winternitz 2011 j phys a math theor 44 162001 studied the properties of two dimensional euclidean potentials that are linear in one of the two cartesian variables in particular they proved the existence of a potential endowed with an integral of third order and an integral of fourth order in this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related to the holt potential it is proved that the existence of a potential with higher order superintegrability is a direct consequence of the integrability of the family of holt type potentials

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

families of quasi bi hamiltonian systems and separability it is shown how to construct an infinite number of families of quasi bi hamiltonian qbh systems by means of the constrained flows of soliton equations three explicit qbh structures are presented for the first three families of the constrained flows the nijenhuis coordinates defined by the nijenhuis tensor for the corresponding families of qbh systems are proved to be exactly the same as the separated variables introduced by mean of the lax matrices for the constrained flows

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

necessary conditions for classical super integrability of a certain family of potentials in constant curvature spaces we formulate the necessary conditions for the maximal super integrability of a certain family of classical potentials defined in the constant curvature two dimensional spaces we give examples of homogeneous potentials of degree 2 on as well as their equivalents on and for which these necessary conditions are also sufficient we show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

stability preserving rational approximation subject to interpolation constraints a quite comprehensive theory of analytic interpolation with degree constraint dealing with rational analytic interpolants with an a priori bound has been developed in recent years in this paper we consider the limit case when this bound is removed and only stable interpolants with a prescribed maximum degree are sought this leads to weighted h 2 minimization where the interpolants are parameterized by the weights the inverse problem of determining the weight given a desired interpolant profile is considered and a rational approximation procedure based on the theory is proposed this provides a tool for tuning the solution to specifications the basic idea could also be applied to the case with bounded analytic interpolants

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

spectral factorization with imaginary axis zeros abstract we examine the problem of the existence and calculation of hermitian solutions p of a linear matrix inequality corresponding to the spectral factorization of a proper rational spectral density in particular zeros are permitted on the finite imaginary axis or at infinity as part of our construction of solutions we show explicitly that each such imaginary axis eigenvalue either finite or infinite defines a fixed part of all solutions p

generating functions bi hamiltonian systems and the quadratic hamiltonian theorem the concept of bi hamiltonian systems and its connection with the theory of canonoid transformations is shown from a geometrical viewpoint the relations between symplectic and canonoid diffeomorphisms are studied using an approach based on the theory of generating functions these results are used for obtaining a new theorem which will represent an intrinsic and coordinate free generalization of the quadratic hamiltonian theorem

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

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

canonoid transformations from a geometric perspective the concept of canonoid transformation for a locally hamiltonian vector field is introduced and its relation with the existence of non noether constants of the motion is shown from a geometrical viewpoint the equations determining generating functions for such canonoid transformations are obtained and applications to some particular problems given

steady state matching and model reduction for systems of differential algebraic equations the problem of model reduction for nonlinear differential algebraic systems is addressed using the notions of moment and of steady state response these notions are formally introduced for this class of systems and families of nonlinear differential algebraic reduced order models achieving moment matching with additional properties are presented stronger results for the special class of linear singular systems are provided two simple examples illustrate the proposed technique

spectral characterization and enforcement of negative imaginariness for descriptor systems abstract systems with counterclockwise input output dynamics or negative imaginary transfer functions arise in various applications such as the modeling of flexible mechanical structures or electrical circuits when certain kinds of measurements are taken in this paper we introduce descriptor systems with such an additional structure we state various of their properties and prove algebraic characterizations of negative imaginariness in terms of spectral conditions of certain structured matrix pencils for this purpose we also analyze particular boundary cases which are characterized by properties of a structured kronecker canonical form finally we describe a method which can be used to restore the negative imaginary property in case that it is lost this happens e g when a system with theoretically negative imaginary transfer function is obtained by e g model order reduction methods linearization or other approximations the method is illustrated by numerical examples

canonical projectors for linear differential algebraic equations abstract this paper aims at improving the decoupling projector chain approach for daes to provide complete decouplings these special canonical projectors are shown to form exactly the spectral projection related to the matrix coefficient pair given by the dae further they prove their value when describing the constraint manifolds of the dae

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

two degrees of freedom quasi bi hamiltonian systems starting from the classical example of the henon heiles integrable hamiltonian system we show that it admits a slightly different formulation from the classical bi hamiltonian system we introduce the general notion of a quasi bi hamiltonian system qbhs and study some of its basic properties only in the case of two degrees of freedom in particular we give the general form of this structure in the nijenhuis coordinates which are constructed using the eigenvalues of the nijenhuis operator of the system and show how they serve to separate the hamilton jacobi equation in the last section we look at the problem of the semi local existence of such structures

index aware model order reduction for linear index 2 daes with constant coefficients a model order reduction method for index 2 differential algebraic equations daes is introduced which is based on the intrinsic differential equations and on the remaining algebraic constraints this extends the method introduced in a previous paper for index 1 daes this procedure is implemented numerically and the results show numerical evidence of its robustness over the traditional methods

quasi bi hamiltonian systems and separability two quasi bi hamiltonian systems with three and four degrees of freedom are presented these systems are shown to be separable in terms of nijenhuis coordinates moreover the most general pfaffian quasi bi hamiltonian system with an arbitrary number of degrees of freedom is constructed in terms of nijenhuis coordinates and its separability is proved

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

n 2 supersymmetric extension of the tremblay turbiner winternitz hamiltonians on a plane the family of tremblay turbiner winternitz hamiltonians hk on a plane corresponding to any positive real value of k is shown to admit an supersymmetric extension of the same kind as that introduced by freedman and mende for the calogero problem and based on an superalgebra the irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion xk of hk bases for them are explicitly constructed the ground state of each supersymmetrized hamiltonian is shown to belong to an atypical lowest weight state irreducible representation

superintegrable deformations of superintegrable systems quadratic superintegrability and higher order superintegrability the superintegrability of four hamiltonians hr hr r a b c d where hr are known hamiltonians and is a certain function defined on the configuration space and depended on a parameter is studied the new hamiltonians and the associated constants of motion jri i 1 2 3 are continous functions of the parameter the first part is concerned with separability and quadratic superintegrability the integrals of motion are quadratic in the momenta and the second part is devoted to the existence of higher order superintegrability the results obtained in the second part are related with the tremblay turbiner winternitz and the post winternitz systems

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

matrix nearness based guaranteed passive system approximation abstract in this paper we present a new approach towards global passive approximation in order to find a passive real rational transfer function g s that is an arbitrarily close approximation of the passive transfer function nearest to a non passive square transfer function h s it is based on existing solutions to pertinent matrix nearness problems it is shown that the key point in constructing the passive real rational transfer function g s is to find a good rational approximation of the well known ramp function over an interval defined by the minimum and maximum dissipation of h s the proposed algorithms rely on the stable anti stable decomposition of a given transfer function pertinent examples are given to show the scope and accuracy of the proposed algorithms

the eta function of the localised dirac operator for the universal cover of sl2 r abstract recently brodzki niblo plymen and wright determined a closed explicit description of the spectrum of the dirac operator d for the universal cover of sl 2 r localised at a representations in the principal series and in the discrete series 4 using this description we introduce an eta function loc s d for the localised dirac operator and we study its analytic properties we show that the localised eta function has an analytic extension to the complex plane with simple poles at negative integers and at s 1 and it is regular at s 0 and we give a formula for loc 0 d we discuss a possible eta invariant for the dirac operator

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

third order superintegrable systems separable in parabolic coordinates in this paper we investigate superintegrable systems which separate in parabolic coordinates and admit a third order integral of motion we give the corresponding determining equations and show that all such systems are multi separable and so admit two second order integrals the third order integral is their lie or poisson commutator we discuss how this situation is different from the cartesian and polar cases where new potentials were discovered which are not multi separable and which are expressed in terms of painleve transcendents or elliptic functions

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

error bounds in the gap metric for dissipative balanced approximations abstract we derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation we prove these results by working in the context of dissipative driving variable systems as in behavioral and state signal systems theory in such a framework no prior distinction is made between inputs and outputs dissipativity preserving balanced truncation of dissipative driving variable systems is addressed and a gap metric error bound is obtained bounded real and positive real input state output systems are manifestations of a dissipative driving variable system through particular decompositions of the signal space under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

bi quasi hamiltonian systems a general notion of bi quasi hamiltonian systems is introduced and is related to previous work on various special cases of such systems

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

third order superintegrable systems separating in polar coordinates a complete classification of quantum and classical superintegrable systems in e2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum new quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth painleve transcendent or in terms of the weierstrass elliptic function

model reduction for large scale dynamical systems via equality constrained least squares in this paper we present a new method of model reduction for large scale dynamical systems which belongs to the svd krylov based method category it is a two sided projection where one side reflects the krylov part and the other side reflects the svd observability gramian part the reduced model matches the first r i markov parameters of the full order model and the remaining ones approximate in a least squares sense without being explicitly computed where r is the order of the reduced system and i is a nonnegative integer such that email protected i

bi hamiltonian structure of the bi dimensional superintegrable nonlinear isotonic oscillator the higher order superintegrability of the two dimensional isotonic oscillator noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator with rational ratio of frequencies is directly related with the existence of some complex functions with interesting poisson bracket properties first the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi hamiltonian complex structure in the second part several real symplectic structures are obtained and the properties of the recursion operators are studied

quasi bi hamiltonian structures complex functions and superintegrability the tremblay turbiner winternitz ttw and the post winternitz pw systems the existence of quasi bi hamiltonian structures for the tremblay turbiner winternitz ttw and the post winternitz pw systems is studied we first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting poisson bracket properties and then we prove the existence of several quasi bi hamiltonian structures making use of these complex functions this is done in two steps first with complex 2 forms wedge product of the differentials of the complex functions and then with several real 2 forms the properties of these geometric structures and the associated recursion operators are also analyzed the paper can be considered as divided in two parts in the first part a we study the ttw system related to the harmonic oscillator and in the second part we study the pw system related to the kepler problem

bi hamiltonian representation of stackel systems it is shown that linear separation relations are fundamental objects for integration by quadratures of stackel separable liouville integrable systems the so called stackel systems these relations are further employed for the classification of stackel systems moreover we prove that any stackel separable liouville integrable system can be lifted to a bi hamiltonian system of gel fand zakharevich type in conjunction with other known result this implies that the existence of bi hamiltonian representation of liouville integrable systems is a necessary condition for stackel separability

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

periodic orbits for an infinite family of classical superintegrable systems we show that all bounded trajectories in the two dimensional classical system with the potential are closed for all integer and rational values of k the period is and does not depend on k this agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable

non symplectic symmetries and bi hamiltonian structures of the rational harmonic oscillator the existence of bi hamiltonian structures for the rational harmonic oscillator non central harmonic oscillator with rational ratio of frequencies is analysed by making use of the geometric theory of symmetries we prove that these additional structures are a consequence of the existence of dynamical symmetries of non symplectic non canonical type the associated recursion operators are also obtained

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems

families of classical subgroup separable superintegrable systems we describe a method for determining a complete set of integrals for a classical hamiltonian that separates in orthogonal subgroup coordinates as examples we use it to determine complete sets of integrals polynomial in the momenta for some families of generalized oscillator and kepler coulomb systems hence demonstrating their superintegrability the latter generalizes recent results of verrier and evans and rodriguez tempesta and winternitz another example is given of a superintegrable system on a non conformally flat space

uniform asymptotic expansions for the barnes double gamma function we develop uniform asymptotic approximations for the barnes double gamma function for various parameter values this involves the use of the method of matched asymptotic expansions to solve the defining functional difference equation in various asymptotic regions the resulting solutions provide extremely accurate approximations to the double gamma function we also give applications of our results to functional difference equations that arise in the theory of moving contact lines and acoustic wave diffraction