control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

on subordinacy and spectral multiplicity for a class of singular differential operators the spectral multiplicity of self adjoint operators h associated with singular differential expressions of the form is investigated based on earlier work of i s kac and recent results on subordinacy complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of i the boundary behaviour of titchmarsh weyl m functions and ii the asymptotic properties of solutions of lu u at the endpoints a and b in particular it is shown that h has multiplicity two if and only if l is in the limit point case at both a and b and the set of all for which no solution of lu u is subordinate at either a or b has positive lebesgue measure the results are completely general subject only to minimal restrictions on the coefficients p r q r and w r and the assumption of separated boundary conditions when l is in the limit circle case at both endpoints

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

deflated decomposition of solutions of nearly singular systems when solving the linear system ax b where a may be nearly singular and b is not consistent with a one is often interested in computing a deflated solution i e a unique solution to a nearby singular but consistent system a s x d tilde b kelley 14 15 has considered deflated solutions with a s corresponding to setting a small pivot of the lu factorization of a to zero stewart 22 proposed an iterative algorithm for computing a deflated solution with a s corresponding to setting the smallest singular value of a to zero kelley s approach explicitly uses submatrices of the lu factors whereas stewart s approach is implicit in that it only involves solving systems with a in this paper we generalize the concept of a deflated solution to that of a deflated decomposition which expresses the solution x in terms of x d and the null vectors of a s we treat such decompositions in a uniform framework that includes the approaches of kelley and stewart and introduce some new deflated

spectral concentration and rapidly decaying potentials abstract we consider the spectral function 0 for the sturm liouville equation y q y 0 on 0 with a boundary condition y 0 cos y 0 sin 0 and where q l 0 a point 0 is called a point of spectral concentration if has a local maximum there we give a new formula for in terms of the prufer transformation angle and we identify a rapid transitional property of in the neighbourhood of points of spectral concentration this property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by for example sledge our approach also leads to the possibility of a theoretical investigation

on the application of some of m g krein s results to the spectral analysis of sturm liouville operators abstract in one of his works m g krein discovered an analogy between polynomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems we used some ideas of this paper to obtain new results in spectral analysis of sturm liouville operators

the asymptotic form of the spectral function in sturm liouville problems with a large potential like xc c 2 a new formula is given for the spectral function in singular sturm liouville problems where the potential is like const x c 0

schrodinger operators with few bound states we show that whole line schrodinger operators with finitely many bound states have no embedded singular spectrum in contradistinction we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast we also prove the following result for one and two dimensional schrodinger operators h with bounded positive ground states given a potential v if both h v are bounded from below by the ground state energy of h then v 0

quadratic forms and solutions of the schrodinger equation the spectral theory for the schr dinger equation on the half line is treated through an analysis of the asymptotics of quadratic forms in a pair of solutions solutions of the second and third order differential equations for these forms are derived in the case of the second order de milne s equation it is shown that a single solution leads to the determination of the singular spectrum this generalizes previous results which applied only to isolated points of the discrete spectrum for the absolutely continuous spectrum it is shown that a single solution allows one not only to locate the spectrum but also to determine the spectral density function explicitly

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

an accelerated cyclic reduction based solvent method for solving quadratic eigenvalue problem of gyroscopic systems abstract the quadratic eigenvalue problem qep 2 m g k x 0 with m t m being positive definite k t k being negative definite and g t g is associated with gyroscopic systems in guo 2004 a cyclic reduction based solvent crs method was proposed to compute all eigenvalues of the above mentioned qep firstly the problem is converted to find a suitable solvent of the quadratic matrix equation qme m x 2 g x k 0 then using a cayley transformation and a proper substitution the qme is transformed into the nonlinear matrix equation nme z a t z 1 a q with a m k g and q 2 m k the problem finally can be solved by applying the cr method to obtain the maximal symmetric positive definite solution of the nme as long as the qep has no eigenvalues on the imaginary axis or for some cases where the qep has eigenvalues on the imaginary axis however when all eigenvalues of the qep are far away from or near the origin the cayley transformation seems not to be the best one and the convergence rate of the crs method proposed in guo 2004 might be further improved in this paper inspired by using a doubling algorithm to solve the qme we use a mobius transformation instead of the cayley transformation to present an accelerated crs acrs method for solving the qep of gyroscopic systems in addition we discuss the selection strategies of optimal parameter for the acrs method numerical results demonstrate the efficiency of our method

an iterative method for updating gyroscopic systems based on measured modal data abstract an efficient iterative method for updating the mass gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed by using the proposed iterative method the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

solutions spectrum and dynamics for schrodinger operators on infinite domains 1 introduction and main results in this paper we investigate the relations between the rate of decay of solutions of schrodinger equations continuity properties of spectral measures of the corresponding operators and dynamical properties of the corresponding quantum systems the first main result of this paper shows that in great generality certain upper bounds on the rate of growth of l 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures consider an operator h v defined by the differential expression h v v x

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

stokes phenomenon and the absolutely continuous spectrum of one dimensional schrodinger operators it is well known that the airy functions ai x and bi x form a fundamental set of solutions for the differential equation lu x u x xu x u x 0 x r and that the spectrum of the associated selfadjoint operator consists of the whole real axis and is purely absolutely continuous for any choice of boundary condition at x 0 also widely known is the fact that the semi axis is an anti stokes line for solutions of the differential equation lu z u z z c for each fixed value of the spectral parameter in this paper we show that this connection between the existence of anti stokes lines on the real axis and points of the absolutely continuous spectrum holds under much more general circumstances further correlations relating the stokes phenomenon to subordinacy properties of solutions of lu u at infinity and to the boundary behaviour of the titchmarsh weyl m function on the real axis are also deduced

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

what is variable bandwidth we propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator apf ddx p x ddx f where p 0 is a strictly positive function denote by c ap the orthogonal projection of ap corresponding to the spectrum of ap in the range of this projection is the space of functions of variable bandwidth with spectral set in we will develop the basic theory of these function spaces first we derive nonuniform sampling theorems second we prove necessary density conditions in the style of landau roughly for a spectrum 0 the main results say that in a neighborhood of x a function of variable bandwidth behaves like a band limited function with local bandwidth p x 1 2 although the formulation of the results is deceptively similar to the corresponding results for classical band limited functions the methods of proof are much more involved on the one hand we use the oscillation method from sampling theory and frame theoretic methods on the other hand we need the precise spectral theory of sturm liouville operators and the scattering theory of one dimensional schrodinger operators 2017 wiley periodicals inc

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

on the construction of asymptoticallly optimal controls for singularly perturbed systems we construct asymptotically optimal controls for singularly perturbed nonlinear systems where both the slow and the fast motions are controlled for this purpose we give necessary optimality conditions for an averaged system for the slow motion

absence of high energy spectral concentration for dirac systems with divergent potentials it is known that one dimensional dirac systems with potentials q which tend to or at infinity such that 1 q is of bounded variation have a purely absolutely continuous spectrum covering the whole real line we show that for the system on a half line there are no local maxima of the spectral density points of spectral concentration above some value of the spectral parameter if q satisfies certain additional regularity conditions these conditions admit thrice differentiable potentials of power or exponential growth the eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end point

absolute continuity and spectral concentration for slowly decaying potentials abstract we consider the spectral function 0 for the sturm liouville equation y qq y 0 on 0 with the boundary condition y 0 0 and where q has slow decay o x a a 0 as x we develop our previous methods of locating spectral concentration for q with rapid exponential decay this journal 81 1997 333 348 to deal with the new theoretical and computational complexities which arise for slow decay

uniqueness of solutions of the kdv hierarchy via dubrovin type flows abstract we consider the cauchy problem for the kdv hierarchy a family of integrable pdes with a lax pair representation involving one dimensional schrodinger operators under a local in time boundedness assumption on the solution for reflectionless initial data we prove that the solution stays reflectionless for almost periodic initial data with absolutely continuous spectrum we prove that under craig type conditions on the spectrum dirichlet data evolve according to a lipschitz dubrovin type flow so the solution is uniquely recovered by a trace formula this applies to algebro geometric finite gap solutions more notably we prove that it applies to small quasiperiodic initial data with analytic sampling functions and diophantine frequency this also gives a uniqueness result for the cauchy problem on the line for periodic initial data even in the absence of craig type conditions

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

a numerical method for quadratic eigenvalue problems of gyroscopic systems abstract we consider the quadratic eigenvalues problem qep of gyroscopic systems 2 m g k x 0 where m m g g and k k r n n with m being positive definite guo numerical solution of a quadratic eigenvalue problem linear algebra and its applications 385 2004 391 406 showed that all eigenvalues of the qep can be found by solving the maximal solution of a nonlinear matrix equation z a z 1 a q with quadratic convergence when the qep has no eigenvalues on the imaginary axis the convergence becomes linear or more slower guo 2004 when the qep allows purely imaginary eigenvalues having even partial multiplicities in this paper we consider the general case when the qep has eigenvalues on the imaginary axis we propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts while keeps other eigenpairs unchanged this transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically numerical examples illustrate the efficiency of our method

low energy properties of the random displacement model abstract we study low energy properties of the random displacement model a random schrodinger operator describing an electron in a randomly deformed lattice all periodic displacement configurations which minimize the bottom of the spectrum are characterized while this configuration is essentially unique for dimension greater than one there are infinitely many different minimizing configurations in the one dimensional case the latter leads to unusual low energy asymptotics for the integrated density of states of the one dimensional random displacement model for symmetric bernoulli distributed displacements it has a 1 log 2 singularity at the bottom of the spectrum in particular it is not holder continuous

spectral concentration and perturbed discrete spectra abstract we examine spectral concentration for a class of sturm liouville problems on 0 a typical example being y x x cx2 y x 0 the discrete spectrum for c 0 leads to spectral concentration in the continuous spectrum for c 0 and we use a new formula for the spectral function to make a detailed computational investigation of the way in which spectral concentration occurs in particular we find that as c decreases spectral concentration arises first from the lowest unperturbed eigenvalue and then in turn from these eigenvalues in increasing order of size

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

singular perturbation of optimal control problems on multidomains the goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi domains we consider an optimal control problem in which the controlled system contains a fast and a slow variables this problem is reformulated as an hamilton jacobi bellman hjb equation the main difficulty comes from the fact that the fast variable lives in a multi domain the geometric singularity of the multi domains leads to the discontinuity of the hamiltonian under a controllability assumption on the fast variables the limit equation as the velocity of the fast variable goes to infinity is obtained via a pde approache and by means of the tools of the control theory

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

on the absolutely continuous spectrum of the perturbed hill s equation on considere l equation de hill y q x y y avec x et complexe ou q x est reel continu par morceaux et periodique de periode 1 et l equation augmentee perturbee par une fonction l 1 y q x p x y y ou p x est reel et absolument integrable le long de l axe reel

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

numerical solution of a quadratic eigenvalue problem we consider the quadratic eigenvalue problem qep 2 m g k x 0 where m m t is positive denite k k t is negative denite and g g t the eigenvalues of the qep occur in quadruplets or in real or purely imaginary pairs we show that all eigenvalues of the qep can be found eciently and with the correct symmetry by nding a proper solvent x of the matrix equation mx 2 gx k 0 as long as the qep has no eigenvalues on the imaginary axis this solvent approach works well also for some cases where the qep has eigenvalues on the imaginary axis ams classication 15a18 15a24 65f30

the direct updating of damping and gyroscopic matrices in this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems this model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

some resolvent estimates for sturm liouville operators in this paper we derive some a priori estimates on the resolvent of one dimensional schrodinger operators from the solutions of the associated differential equation with real energy in particular this implies the existence of an absolutely continuous spectrum in some situations

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

half line schrodinger operators with no bound states acting in l2 0 c with the boundary condition r for convenience we require that the potential v be uniformly locally square integrable we write l l 2 for the banach space of such functions the free operators that is when v 0 can be diagonalized by the fourier transform this shows that they have spectra 2 2 and 0 co respectively and that in both cases the spectrum is purely absolutely continuous for a general discrete operator the mere fact that the spectrum is contained in 2 2 forces it to be purely absolutely continuous this is our first main result

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

an inverse problem for undamped gyroscopic systems linear undamped gyroscopic systems are defined by three real matrices m 0 k 0 and g g t g the mass stiffness and gyroscopic matrices respectively in this paper an inverse problem is considered given complete information about eigenvalues and eigenvectors l diag l 1 l 2 l 2 n 1 l 2 n c 2 n x 2 n and x x 1 x 2 x 2 n 1 x 2 n c n x 2 n where the diagonal elements of l are all purely imaginary x is of full row rank n and both l and x are closed under complex conjugation in the sense that l 2 email protected 2 j email protected c x 2 email protected 2 j email protected c n for j 1 n find m k and g such that email protected email protected kx 0 the solvability condition for the inverse problem and a solution to the problem are presented and the results of the inverse problem are applied to develop a method for model updating

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

on singular perturbations for differential inclusions on the infinite interval abstract we consider a differential inclusion subject to a singular perturbation i e part of the derivatives are multiplied by a small parameter 0 we show that under some stability and structural assumptions every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion obtained for 0 when tends to 0 the goal of the present paper is to provide a new result of tikhonov type on the time interval 0