fine hierarchy of regular ω languages by applying descriptive set theory we get several facts on the fine structure of regular languages considered by k wagner we present quite different shorter proofs for main his results and get new results our description of the fine structure is new very clear and automata free we prove also a closure property of the fine structure under boolean operations our results demonstrate deep interconnections between descriptive set theory and theory of languages

delay independent stability test for systems with multiple time delays delay independent stability dis of a general class of linear time invariant lti multiple time delay system mtds is investigated in the entire delay parameter space stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays we build an analytical approach that requires the inspection of the roots of finite number of single variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays excluding infinite delays the approach enables to test the necessary and sufficient conditions of dis of lti mtds technically known as weak dis as well as the robust stability of single delay systems against all variations in delay ratios the proposed approach which does not require any parameter sweeping and graphical display becomes possible by establishing a link between the infinite spectrum and algebraic geometry case studies are provided to show the effectiveness of the approach

borel classes and closed games wadge type and hurewicz type results for each countable ordinal and pair ao al of disjoint analytic subsets of 2 we define a closed game j aos a1 and a complete iit subset ht of 2 such that i a winning strategy for player i constructs a 24 set separating ao from a1 and ii a winning strategy for player ii constructs a continuous map p 2 a u a1 with p l ao h applications of this construction include a proof in second order arithmetics of the statement every iit non ss set is ii complete an extension to all levels of a theorem of hurewicz about 22 sets a new proof of results of kunugui novikov bourgain and the authors on borel sets with sections of given class extensions of results of stern and kechris our results are valid in arbitrary polish spaces and for the classes in lavrentieff s and wadge s hierarchies introduction let e 2 be the usual cantor space and following the standard modern terminology of addison let e and fit denote respectively the borel subsets of e of additive and multiplicative borel class starting with e open and ti closed the aim of this paper is to characterize given a set a c 2 and one of the borel classes say r whether a is in r or not by a characterization we have in mind the finding of some mathematical object associated with a and which positively witnesses that a e r or that a 0 r let r be the class e t if the set a is in e a natural candidate for such a witness is simply one particular construction of a as a e set a list of sequences of open sets together with instructions on the operations to be performed to get a in t steps the problem of how to find such constructions which is deeply connected to the structural properties of sets in the plane with e sections has been extensively investigated in papers by novikov n and kunugui kun for 1 see also dellacherie de saint raymond sr 1 for t 2 bourgain b 1 b 2 and louveau lo 1 for t 3 louveau lo 2 for t 3 in case a is known to be borel in case a ranges over larger classes the problem is studied in papers by stern st kechris ke and louveau lo 3 received by the editors december 26 1985 and in revised form november 19 1986 these results have been presented at the annual meeting of the american mathematical society new orleans january 1986 in a special session on determinacy and large cardinals 1980 mclthemcltics subject clclssificcltion 1985 revision primary o2k30 04a15 28a05 secondary 54h05 26a21

amplitude control of a limit cycle in a coupled van der pol system abstract in order to control the amplitude of a limit cycle of a coupled van der pol system feedback controllers are designed the control equations of weakly nonlinear systems are obtained by using an approximate method and the relationship between the amplitude of the limit cycle and the control parameter is acquired hence the amplitude of the limit cycle can be controlled effectively the method may also be applied to other coupled van der pol systems for some systems it is difficult to find the function in the amplitude of the limit cycle and the control parameter directly by means of numerical analysis we can design effective controllers therefore we can preferentially control the amplitude of the limit cycle of coupled van der pol systems

on the number of permutatively inequivalent basic sequences in a banach space let x be a banach space with a schauder basis en n n the relation e0 is borel reducible to permutative equivalence between normalized block sequences of en n n or x is c0 or lp saturated for some 1 p if en n n is shrinking unconditional then either it is equivalent to the canonical basis of c0 or lp 1 p or the relation e0 is borel reducible to permutative equivalence between sequences of normalized disjoint blocks of x or of x if en n n is unconditional then either x is isomorphic to l2 or x contains 2 subspaces or 2 quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

non linear dynamics of a suspended travelling cable subject to transverse fluid excitation starting with the analysis of the fluid drag and lift on a suspended travelling cable subjected to transverse fluid excitation the paper presents the expression of forces on the cable and then a set of partial differential equations of in plane and out of plane motions of the cable in the case of small ratio of sag to span the in plane and out of plane modes of the first order dominate the motions of cable thus the partial differential equations of cable are reduced to two ordinary differential equations of the second order by means of the galerkin approach because the stiffness terms disappear in the ordinary differential equations when the cable is at equilibrium position the co ordinate transform proposed by pilipchuk is used to describe the stretch and rotation of mid span section of cable from the equilibrium position so that the transformed differential equations include linear stiffness terms afterwards the differential equations are simplified by using the perturbation approach of two variables under the assumption that the young s module of cable is not very small as a result the approximate cable dynamics yields a two dimensional autonomous system and does not exhibit any chaotic motions according to the approximated model two equilibrium positions of cable are determined and their stability is analyzed finally the influences of travelling velocity and cable density on the cable dynamics are discussed on the basis of numerical computations the case studies show that the travelling velocity should be limited when a very light cable is laid under water in order to avoid harmful and dangerous swings

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

robust stability analysis of linear time delay systems by lambert w function some extreme point results robust stability conditions are derived for linear time delay systems using lambert w function the characteristic quasi polynomials of the systems are assumed to be factorized it is proven that if uncertainties in the coefficients of the quasi polynomial are set in appropriate regions in the complex plane we can enjoy extreme point results finite number of stability checks at some points of the boarder of the regions suffice the strength of lambert w function approach lies in the fact that the function is implemented on some standard software packages such as mathematica maple or matlab which afford to compute the function value very easily the above two points make the stability test for the class of uncertain time delay systems quite practical

the smooth ideal we give a classical proof of the generalization of the characterization of smoothness to quotients of polish spaces by borel equivalence relations as an application we describe the extent to which any given borel equivalence relation on a polish space is encoded by the corresponding ideal generated by the family of borel sets on which it is smooth

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

parameter dependent lyapunov functions for robust stability of time varying linear systems under point delays abstract this paper investigates the stabilization dependent on the delays of in general time varying linear systems with multiple constant point time delays the matrices describing the state space dynamics are parameterized by in general time varying function matrices governed by coefficient functions in compact sets the closed loop sufficiency type stability conditions are obtained from lyapunov s stability theory by constructing parameter dependent candidates which satisfy in the most general case a riccati matrix differential inequality

sets of filter convergence of sequences of continuous functions abstract we consider sets of filter convergence and divergence to infinity of sequences of continuous functions for borel filters we characterize the sets of filter convergence of sequences of continuous functions on the real line for borel filters we also give sufficient and necessary conditions for results involving sets of filter divergence to infinity in particular we give the full description of such sets for the statistical convergence for metric spaces

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

robust hurwitz stability test for linear systems with uncertain commensurate time delays the note deals with the robust hurwitz stability of a class of linear systems with uncertain commensurate time delays which is formulated as the robust hurwitz stability of a nonpolytopic family of quasi polynomials a np hard problem from the viewpoint of computational complexity on the basis of the edge theorem and sylvester resultant elimination a necessary and sufficient condition is developed for the robust stability of the whole family and it yields an effective testing procedure an illustrative example is given to show the effectiveness of the method

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

stabilization of vibration systems via delayed state difference feedback this paper presents a study of the stabilization problem via delayed state feedback of pyragas type for unstable vibration systems that have even number of characteristic roots with positive real parts on the basis of stability switches with respect to the time delay a new stabilization criterion is established and an effective procedure for determining the admissible values of the feedback gains and the delay is given two examples are given in detail to demonstrate the efficiency of the theory

forced vibrations of a beam on a tensionless foundation abstract this paper is concerned with the response of an elastic beam resting on a tensionless winkler foundation subjected to a concentrated load a uniformly distributed load and a concentrated load the external load are assumed to be time dependent and the non linear governing differential equation of forced vibrations is derived by employing the galerkin method the free vibration mode functions of the completely free beam are adopted as the co ordinate functions of the displacement function of the beam the results which represent the static and dynamic responses of the beam are presented in the figures it is concluded that the presence of the lift off affects the response of the beam markedly by changing the shape of the displacement curve as well as the foundation reactions

one to one selections marriage theorems let a be a lebesgue measurable subset of 0 1 x 0 1 such that each vertical and each horizontal section of a has positive

the conjugacy problem in ergodic theory all common probability preserving transformations can be represented as elements of mpt the group of measure preserving transformations of the unit interval with lebesgue measure this group has a natural polish topology and the induced topology on the set of ergodic transformations is also polish our main result is that the set of ergodic elements t in mpt that are isomorphic to their inverse is a complete analytic set this has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not borel this is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is borel this result explains perhaps why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable the construction that we use is general enough to show that the set of ergodic t s with nontrivial centralizer is also complete analytic on the positive side we show that the isomorphism relation is borel when restricted to the rank one transformations which form a generic subset of mpt it remains an open problem to nd a good explicit method of checking when two rank one transformations are isomorphic in memoriam prior to the

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

travelling wave solutions for the 2 1 dimensional nizhnik novikov veselov equation abstract using an auxiliary ordinary differential equation method we have obtained many exact travelling wave solutions for the 2 1 dimensional nizhnik novikov veselov equation which is considered as a model for an incompressible fluid the key of this method is to construct appropriate travelling wave functions in terms of solving a system of nonlinear algebraic equations the method is also extended to the case of other nonlinear partial differential equations with solitary wave solutions triangular periodic wave solutions and elliptic function solutions et al

bifurcations of a generalized van der pol oscillator with strong nonlinearity a generalized van der pol oscillator with parametric excitation is studied for its bifurcation on the basis of the mlp method we enable a strongly nonlinear system to be transformed into a small parameter system the bifurcation response equation of a 1 2 subharmonic resonance system is determined by the multiple scales method according to the singularity theory the bifurcation of equilibrium points is analyzed the stability of the zero solution is researched by the eigenvalues of the variational matrix and the bifurcation sets are constructed in various regions of the parameter plane

robust stability of perturbed systems with time delays abstract motivated by dynamical system considerations a number of new results on robust stability of perturbed polynomials have been recently obtained in the literature under consideration the perturbations are known only within given bounds and enter affine linearly into the polynomial coefficients in this paper this formulation is expanded to include time delays under a properness assumption a necessary and sufficient condition for robust stability is developed an example is given to illustrate application of the main result

a finite element method for computation of structural intensity by the normal mode approach abstract a method for numerical computation of structural intensity in thin walled structures is presented the method is based on structural finite elements beam plate and shell type enabling computation of real eigenvalues and eigenvectors of the undamped structure which then serve in evaluation of complex response the distributed structural damping is taken into account by using the modal damping concept while any localized damping is treated as an external loading determined by use of impedance matching conditions and eigenproperties of the structure emphasis is given to aspects of accuracy of the results and efficiency of the numerical procedures used high requirements on accuracy of the structural response displacements and stresses needed in intensity applications are satisfied by employing the swept static solution which effectively takes into account the influence of higher modes otherwise inaccessible to numerical computation a comparison is made between the results obtained by using analytical methods and the proposed numerical procedure to demonstrate the validity of the method presented

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

uniformisations partielles et criteres a la hurewicz dans le plan we give characterizations of the borel sets potentially in some wadge class among the borel sets with countable vertical sections of a product of two polish spaces to do this we use some partial uniformization results

measurement based determination of the irrotational part of the structural intensity by means of test functional series expansion the irrotational part of the structural intensity for flexural wave motion in plate like structures is calculated from measurement data of kinematic quantities such as out of plane velocities by a new method that makes use of a test functional series expansion the computation of the structural intensity and its irrotational part which allows to effectively assess vibrational sources and sinks is less sensitive to measurement noise as compared to standard approaches

robust stability of time delay systems with an uncertain time delay constant frequency domain methods are described for testing the robust stability of a class of time delay systems with an uncertain time constant both parametric and non parametric uncertainties are considered our technique determines without a sweep of time delay the maximum time delay constant which preserves the robust stability

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

delay independent stability of retarded dynamic systems of multiple degrees of freedom on the basis of generalized sturm theory this paper presents a simple but systematic approach to the delay independent stability analysis of the linear dynamic systems involving multiple degrees of freedom and possibly two time delays the approach enables one to complete the stability analysis in a much simpler way than before through the use of a maple routine given in the paper to demonstrate the approach the paper gives a detailed analysis the corresponding sufficient and necessary conditions as well as stable regions in parameter space of concern for the delay independent stability of a vibrating system with time delays in state feedback an active tendon for a tall structure and an active suspension of a quarter car model

non linear vibrations of a beam on an elastic foundation the non linear vibrations of an elastic beam resting on a non linear tensionless winkler foundation subjected to a concentrated load at the centre is presented in this paper since the foundation is assumed to be tensionless the beam may lift off the foundation and there exists different regions namely contact and non contact regions since the contact regions are not known in advance the problem appears as a non linear one even though there is no non linear term in the foundation model in this case the calculation of the roots of a non linear equation is needed to obtain contact lengths the perturbation technique is used to solve the non linear governing equation associated with the problem using this technique the non linear problem is reduced to the solution of a set of linearized partial differential equations the lift off points and the displacements are obtained in linear and non linear cases and the variation of these points with respect to various parameters are presented it is concluded that the contact length varies with the magnitude of the load because of the non linearity

robust stability of time delay systems there are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space the edge theorem and the theory based on the concept of convex directions many known results can be explained with these two results this paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions the main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families one of the primary sources of motivation for the class of problems considered in this paper is derived from process control a typical problem formulation almost always includes a delay element in each subsystem process block when we interconnect a number of such blocks in a feedback system the study of robust stability involves quasipolynomials of the sort considered in this paper

robust stability for time delay systems the edge theorem and graphical tests the robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope p of quasi polynomials i e polynomials in one complex variable and exponential powers of the variable given a set d in the complex plane the goal is to find a constructive technique to verify whether all roots of every quasi polynomial in p belong to d that is to verify the d stability of p first it is demonstrated by counterexample that kharitonov s theorem does not hold for general delay systems next it is shown that under a mild assumption on the set d a polytope of quasi polynomials is d stable if and only if the edges of the polytope are d stable this extends the edge theorem for the d stability of a polytope of polynomials the third result gives a constructive graphical test for checking the d stability of a polytope of quasi polynomials which is especially simple when the set d is the open left half plane an application is given to demonstrate the power of the results

stability criteria for linear time invariant systems with point delays based on one dimensional routh hurwitz tests this brief deals with the asymptotic stability of a class of linear time invariant systems subject to point constant uncommensurate delays results are obtained dependent on and independent of the delays which may be tested be performing a finite set of 1 d routh hurwitz tests on a corresponding set of auxiliary delay free linear time invariant systems plus some supplementary conditions related to either commutation pairwise of the associate dynamics matrices or their norm closeness or some simple testable conditions if the matrices have particular forms and there is only one single delay the obtained results are applicable to the case when a finite set of matrices describing the dynamics of a set of delay free auxiliary dynamical systems are hurwitz and either commute pair wise or their norms are sufficiently close to each other some extensions are given to robust stability of point time delay time invariant systems with commensurate delays parametrized by an uncertain parameter vector whose components are in real intervals of known limits

counting roots of the characteristic equation for linear delay differential systems abstract a formula is given that counts the number of roots in the positive half plane of the characteristic equation for general real constant coefficient linear delay differential systems the formula is used to establish necessary and sufficient conditions for asymptotic stability of the zero solution of linear delay differential systems the formula is potentially useful in verifying stability hypotheses that arise in bifurcation analysis of autonomous delay differential systems application of the formula to hopf bifurcation theory for delay differential systems is discussed and an example application to an equation with two delays is given

regularization method for measurement of structural intensity using nearfield acoustical holography the regularization method for measurement of structural intensity using nearfield acoustical holography is proposed spatial derivatives of normal displacement are necessary to obtain the structural intensity the derivative operations amplify high wave number components of measurement noise therefore the estimation of an appropriate wave number filter is crucial for implementation of the measurement of structural intensity in conventional methods this wave number filter is determined from the flexural wavelength and the same wave number filter is applied to obtain all spatial derivatives as a result structural intensity obtained from the pressure hologram whose signal to noise ratio is low is seriously contaminated by the noise to overcome this difficulty regularization theory is applied to determine the appropriate wave number filter for each order of derivatives the effectiveness of the proposed method is demonstrated by experiments

robust stabilization of a class of polytopic linear time varying continuous systems under point delays and saturating controls abstract this paper investigates the stabilization dependent on the delays of in general time varying linear systems with multiple constant point time delays under saturating state feedback controls the matrices describing the state space dynamics and control belong to polytopes also the controller gain matrix is characterized as belonging to another polytope whose vertices are computed from the knowledge of a closed bounded ball containing a set of values of the state norm and also the component wise saturating gains and saturation parameterizations this knowledge defines the polytope vertices through scaling diagonal matrices being associated with the various operation modes in the linear and saturated zones of each input component in these conditions the resulting closed loop system is of polytopic nature whose whole number of vertices is at most equal to the product of both numbers of vertices of the above two polytopes characterizing the plant parameterization and the saturation the closed loop sufficiency type stability conditions are obtained from lyapunov s stability theory by constructing candidates for each vertex of the polytopic closed loop system each satisfying in the most general case a riccati matrix differential inequality some conditions guaranteeing the stability conditions are obtained from a general kalman yakubovitch popov kyp lemma and some weaker stability conditions are also obtained for the time invariant case from a set of linear matrix inequalities associated with the set of vertices