stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

invariance of the strict hurwitz property for bivariate polynomials under coefficient perturbations a result is given that enables one to determine the interval within which the coefficients of a real bivariate polynomial might be allowed to vary centered around their respective nominal values so that the strict hurwitz property remains invariant these results are suitable for generalization to multivariate polynomials with complex coefficients applications can be found in branches of network and control theory concerned with robust stability analysis such analysis should be of interest because of the established role of multivariate realizability theory in systems research and also because of the increasing importance that is being attached to the design of multidimensional feedback control systems

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

robust controller synthesis using kharitonov s theorem the authors present a method based upon kharitonov s theorem for synthesizing robustly stabilizing feedback controllers the approach is based upon a multiple plant model formulation with a quadratic cost functional sufficient conditions are obtained for characterizing robustly stabilizing static output feedback proportional controllers for mimo multiple input multiple output plants with denominator polynomial uncertainty

stability of a hurwitz polynomial under coefficient perturbations necessary and sufficient conditions necessary and sufficient conditions for a perturbed polynomial to remain hurwitz are given the conditions allow considerable freedom in allocating different weights to various coefficients to reflect different levels of uncertainty in the coefficients the new conditions are an extension of a previous result of the author in which sufficient conditions for the same problem were obtained

conditions on the boundary of the zero set and application to stabilization of systems with uncertainty abstract we describe an analytic method for finding the location of the zero set of a vector valued function which depends on m real variables and n complex parameters we apply the method to robust stabilization of multivariable linear feedback systems we find exact measures of the extent of permissible perturbations in the plant and or the compensator that maintain feedback stability

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

strictly hurwitz property invariance of quartics under coefficient perturbation in this note it is shown how the maximum interval centered around each of the coefficients of a prescribed fourth degree strictly hurwitz polynomial may be determined so that the strictly hurwitz property remains invariant when each coefficient might be perturbed to any arbitrary value within its permissible interval

the fbi transform on compact manifolds we present a geometric theory of the fourier bros iagolnitzer transform on a compact c manifold m the fbi transform is a generalization of the classical notion of the wave packet transform we discuss the mapping properties of the fbi transform and its relationship to the calculus of pseudodifferential operators on m we also describe the microlocal properties of its range in terms of the scattering calculus of pseudodifferential operators on the noncompact manifold t m

strict hurwitz property of polynomials under coefficient perturbation a simple procedure for computing the allowable interval coefficient perturbations of a strictly stable polynomial is presented the amount of computations involved is very little while the procedure is able to handle the case where the relative uncertainty of the polynomial coefficients are not known it can also handle the case of linearly dependent parameter uncertainties the procedure can also be modified to treat the case of simple nonlinear dependent parameter uncertainties the details of the modified procedure will depend on the nature of the nonlinearity

characterization of allowable perturbations of a closed loop system with pole placement controller the problem of determining stability of a closed loop transfer function in the presence of coefficient perturbations is considered the results of interval polynomial theory are used to characterize the allowable interval of variation

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

sensitivities of stability constraints and their applications let the real polynomial a s a 0 a 1 s a n s n with the coefficients being known differentiable functions a k x be given and let the constraints g i x 0 determine the strictly hurwitz property of the polynomial a s a simple and efficient method to calculate the derivatives partial g i x partial x j is proposed then the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed also a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given

various quantum mechanical aspects of quadratic forms two of the oldest known classical integrable systems are i n decoupled harmonic oscillators constrained to the surface of the n sphere i e the classical c neumann system and ii geodesic flow on an n axial ellipsoid we integrate both these systems at the quantum level that is in both cases we construct n 1 pairwise commuting partial differential operators which in turn commute with the respective quantum hamiltonians moreover the joint eigenvalues of the commuting partial differential operators appear to be encoded linearly as parameters in an n parameter eigenvalue equation given by a second order complex ordinary differential equation with automorphic boundary conditions we then focus on the particular case of two degrees of freedom i e r3 via the microlocal wave averaging ansatz combined with explicit wkb constructions we give a spectral analysis both classical and semiclassical of the above systems and subsequently exhibit connections between the various spectral asymptotics in the course of this analysis we show that the above two systems are intimately related to yet another quantum integrable system the reduced quantum asymmetric rigid body in a vacuum

a note on kharitonov type results in the space of markov parameters c v hollot ibid vol 34 no 5 p 536 may 1989 used markov s theorem of determinants to obtain the largest stability box around a nominal hurtwitz polynomial under coefficient perturbation in the space of markov s parameters in the present work the authors use some previously established related research results to considerably simplify the proposed technique and make it computationally more efficient these results are also extended to the discrete time case and some illustrative examples are given

some applications of tristram levine signatures and relation to vassiliev invariants abstract tristram and levine introduced a continuous family of signature invariants for knots we show that any possible value of such an invariant is realized by a knot with given vassiliev invariants of bounded degree we also show that one can make a knot prime preserving alexander polynomial and vassiliev invariants of bounded degree finally the tristram levine signatures are applied to obtain a condition on signed unknotting number

semiclassical resonances generated by a closed trajectory of hyperbolic type we determine all the resonances in certain rectangular regions of the complex plane of the schrodinger operator h2 v whenh 0 under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding poincare map is hyperbolic

tunnelling between tori in phase space abstract this paper calculates the energy splitting e due to tunnelling between a pair of quantum states which correspond to classical motion on tori in phase space the calculation assumes that the states are separated in coordinate space however the result of the calculation e a 3 2 e 2 h has canonically invariant expressions for the tunnelling actions s and prefactor a and it is conjectured that the results apply even when the tori overlap in coordinate space this makes the results relevant to a very general problem connected with the einstein brillouin keller quantization scheme there is a surprising consequence of this calculation when the system is separable it is clear that e vanishes for most pairs of states this does not occur because the prefactor a vanishes but because the tunnelling action s is infinite as a parameter is varied to make the system separable s diverges logarithmically the results of this paper only apply to quasi integrable systems where s is finite p in the case of any exactly integrable system s is infinite and e vanishes in a more singular manner than e const h as 0

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

the semi classical trace formula and propagation of wave packets abstract we study spectral and propagation properties of operators of the form s h n j 0 h j p j where j p j is a differential operator of order j on a manifold m asymptotically as h 0 the estimates are in terms of the flow t of the classical hamiltonian h x p n j 0 p j x p on t m where p j is the principal symbol of p j we present two sets of results i the semiclassical trace formula on the asymptotic behavior of eigenvalues and eigenfunctions of s h in terms of periodic trajectors of h ii associated to certain isotropic submanifolds t m we define families of functions h and prove that t exp iths is a family of the same kind associated to t