kalman decomposition for implicit linear systems the kalman decomposition for a general class of implicit systems is introduced and studied the properties of strong reachability observability acceptance of all inputs and outer sequences output uniqueness uniqueness and trajectory observability of the original system are related to the properties of its subsystems obtained via the kalman decomposition the problems of transfer relation regularizability by feedback output feedback and output injection are studied by means of this decomposition

intrinsic approximation for fractals defined by rational iterated function systems mahler s research suggestion following k mahler s suggestion for further research on intrinsic approxi mation on the cantor ternary set we obtain a dirichlet type theorem for the limit sets of rational iterated function systems we further investigate the behavior of these ap proximation functions under random translations we connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a khinchin type theorem finally using a result of s ramanujan we prove a theorem motivating a conjecture regarding the distribution of rationals in reduced form on the cantor ternary set

dynamic equilibria for linear systems and quadratic costs abstract this paper considers the computation of dynamic equilibria from a systems viewpoint attention is focused on an economic system governed by a linear production system quadratic production costs and a linear demand function the resulting necessary conditions for a dynamic equilibrium take the form of a linear two point boundary value problem similar in structure to the conditions of optimal control however the conditions are not equivalent to those of optimal control and the standard riccati equation approach cannot generally be applied a new procedure based on descriptor variable theory provides a general solution which is a generalization of the riccati equation approach

reformulation of the stochastic bem to improve the computational efficiency in the prediction of the vibro acoustic behaviour of structures with uncertainties abstract non determinism plays a crucial role in the prediction of the vibro acoustic behaviour of structures the stochastic boundary element method sbem allows computing the average response of systems with geometrical uncertainties the method consists in including such perturbations in the standard bem formulation and adding a set of equations representative of the energetic behaviour due to the presence of unknown cross products the addition of a set of auxiliary equation is necessary to solve the problem as a consequence the system dimension becomes very large and the application of the methodology is limited to small models this paper deals with a reformulation of the sbem approach such that the system dimension is drastically decreased and the methodology can be applied to more complex cases moreover the approach is extended to predict the variance of the field variable at no additional computational cost after a theoretical introduction the sbem is applied to two dimensional examples with pressure and velocity boundary conditions the comparison with monte carlo simulations shows a good agreement of the results and a reduction in the computational load is achieved

a closed formula for the determination of the impulsive solutions of linear homogeneous matrix differential equations a closed formula is given which allows the determination of the impulsive solutions of linear homogeneous matrix differential equations lhmde directly in terms of finite and infinite spectral data of the associated polynomial matrix specifically the notions of finite and infinite jordan pairs for a general polynomial matrix are defined and it is pointed out the strong relationship among them and the impulsive solutions of lhmde

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

quantitative recurrence properties in conformal iterated function systems abstract we consider a quantitative version of poincare s recurrence theorem in a conformal iterated function system let i i be a conformal iterated function system on 0 1 d with a countable index set denote by j the attractor of let f 0 1 d r be a positive function s n f x be the sum f x f w 1 1 x f w 1 w n 1 1 x analogous to an ergodic sum and consider the set of points for which the inequality x w 1 w n 0 1 d x w 1 w n 1 x e s n f x is realized for infinitely many n n with w 1 w n n the set above contains the points x whose orbit returns very close to x infinitely many times with the quality of approximation depending on the time n and the point x this can be viewed as a quantitative version of poincare s recurrence theorem it is shown that its hausdorff dimension is the solution to some pressure function the setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b adic expansions continued fraction expansion as well as some dynamical systems defined on fractal sets applying the main result to diophantine approximation gives a partial answer to a question by k mahler

wave propagation in one dimensional waveguides with slowly varying random spatially correlated variability this paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation more specifically it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium this variability plays a significant role in the so called mid frequency region for dynamics and vibration but wave based methods are typically only applicable to homogeneous and uniform waveguides the wkb approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion an alternative wave formulation approximation with piecewise constant properties is also derived and included so that the internal reflections are taken into account but this requires a discretization of the waveguide moreover a fourier like series the karhunen loeve expansion is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam it is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the wkb approach is given

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

a review of 2 d implicit systems abstract the one dimensional notion of causality has no counterpart in the two dimensional plane where at best partial orderings of i j may be defined two dimensional implicit systems unlike the familiar state space 2 d models do not require any notion of causality or recursibility instead they require the milder notion of regularity thus the implicit models are more suited to the description of naturally occurring two dimensional systems such as are described by the hyperbolic equation and the heat equation as well as for applications in image processing this paper provides a brief summary of the current state of the theory of 2 d implicit systems

analysis of descriptor systems using numerical algorithms in this paper we analyze numerical methods for the solution of the large scale dynamical system e dot y t ay t g t y t 0 y 0 where e and a are matrices possibly singular systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables special cases of such systems are algebraic differential systems we discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure detect nonsolvable systems resolve initial value consistency problems and are applicable to stiff descriptor systems we also present an algorithm for the control of the local truncation error on only the state variables

sets of exact approximation order by rational numbers for a non increasing function let exact be the set of real numbers that are approximable by rational numbers to order but to no or der c with 0 c 1 in a previous paper we determined the hausdorfi dimension of the set exact when x 7 x2 x is non increasing and the sum p x 1 x x converges in the present note we complement this result by estab lishing that exact has full hausdorfi dimension for a large class of functions that do not decrease too slowly and are such that the sum p x 1 x x diverges furthermore we discuss the case where x 7 x 2 x is a constant function

diophantine approximation and cantor sets we provide an explicit construction of elements of the middle third cantor set with any prescribed irrationality exponent this answers a question posed by kurt mahler

passive vibration suppression of flexible space structures via optimal geometric redesign acomputationalframework is presented for the design of large flexible space structures with nonperiodicgeometries to achieve passive vibration suppression the present system combines an approximationmodel management framework ammf developed for evolutionary optimization algorithms eas with reduced basis approximate dynamic reanalysis techniques a coevolutionary genetic search strategy is employed to ensure that design changes during the optimization iterations lead to low rank perturbations of the structural system matrices for which thereduced basis methods give high quality approximations the k means algorithm is employed for cluster analysis of the population of designs to determine design points at which exact analysis should be carried out the fitness of the designs in an ea generation is then approximated using reduced basis models constructed around the points where exact analysis is carried out results are presented for the optimal design of a two dimensional cantilevered space structure to achieve passive vibration suppression it is shown that significant vibration isolation of the order of 50 db over a 100 hz bandwidth can be achieved further it is demonstrated that the ammf can potentially arrive at a better design compared to conventional approaches when a constraint is imposed on the computational budget available for optimization

kalman type decompostion for implicit linear discrete time systems and its applications a kalman type decomposition for implicit linear discrete time systems is studied in detail it is shown that such a decomposition can be used to study the problem of regularizability

diophantine approximation on rational quadrics we compute the hausdorff dimension of sets of very well approximable vectors on rational quadrics we use ubiquitous systems and the geometry of locally symmetric spaces as a byproduct we obtain the hausdorff dimension of the set of rays with a fixed maximal singular direction which move away into one end of a locally symmetric space at linear depth infinitely many times

energy flow variability in a pair of coupled stochastic rods abstract random variations in the transfer functions describing energy flows in a pair of coupled axially vibrating stochastic rods are considered the material properties of the rods are modelled either as random variables or as random processes and the consequent effects on the nature of the power receptance functions are studied by using monte carlo simulation techniques the relevance of these results is discussed in the context of variability in average estimates obtained when using statistical energy analysis sea formalisms the response statistics are shown to vary significantly with the details of the statistical modelling of the system especially in the low and medium frequency ranges moreover the confidence limits show no tendency to converge toward the mean with increasing frequency critical systems the responses of which deviate maximally from the average values are identified and attempts made to understand their nature

a dynamical dimension transference principle for dynamical diophantine approximation diophantine approximation in dynamical systems concerns the diophantine properties of the orbits in classic diophantine approximation the powerful mass transference principle established by beresnevich and velani provides a general principle to the dimension for a limsup set in this paper we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system more precisely let x t be a topological dynamical system where x is a compact metric space and t x rightarrow x is an expanding continuous transformation given y o in x we consider the following limsup set mathcal w t f driven by the dynamical system x t begin aligned big x in x x in b z e s n f log t z text for some z in t n y o text with infinitely many n in mathbb n big end aligned where log t is a function reflecting the local conformality of the transformation t f is a non negative continuous function over x and s n f log t z denotes the ergodic sum f log t z cdots f log t t n 1 z by proposing a dynamical ubiquity property assumed on the system x t we obtain that the dimensions of x and mathcal w t f are both related to the bowen manning mccluskey formulae namely the solution to the pressure functions begin aligned texttt p t log t 0 text and texttt p t log t f 0 text respectively end aligned we call this phenomenon a dynamical dimension transference principle because of its partial analogy with the mass transference principle this general principle unifies and extends some known results which were considered only separatedly before these include the b adic expansions expanding rational maps over julia sets inhomogeneous diophantine approximation on the triadic cantor set and finite conformal iterated function systems

descriptor systems expanded descriptor equation and markov parameters generalized concepts of solvability and conditionability are presented for descriptor systems and tests are given for these generalized properties an expression is derived for descriptor system markov parameters the notion of expanded descriptor equation is presented

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

wave propagation in semi infinite bar with random imperfections of density and elasticity module abstract mathematical modeling and properties of a linear longitudinal wave propagating in a slender bar with random imperfections of material density and young modulus of elasticity is discussed fluctuation components of material properties are considered as continuous stochastic functions of the length coordinate two types of fluctuation and their influence on response properties have been investigated in particular the delta correlated and a diffusion type processes investigation itself is based on markov processes and corresponding fokker planck kolmogorov equation the stochastic moments closure as a solution method has been used many effects due to the stochastic nature of the problem have been detected along the bar a drop of the mean value of the response with the simultaneous increase of the response variance have been observed this effect does not represent any conventional damping but a gradual drop of the deterministic and an increase of the stochastic components of the overall response the rate of the response indeterminacy increases with the increase of the length coordinate increasing values of material imperfection variances and the rising excitation frequency can lead to a critical state when the length of the propagating wave is comparable with the correlation length of imperfections this state will manifest itself as a radical change of the response character the problem will pass beyond the boundaries of stochastic mechanics and lose its physical meaning similar effects can be observed in the fem analysis where there is also a certain permissible upper boundary of the excitation frequency corresponding with the size and type of the element used

diophantine approximation and badly approximable sets let x d be a metric space and d a compact subspace of x which supports a non atomic finite measure m we consider natural classes of badly approximable subsets of loosely speaking these consist of points in which stay clear of some given set of points in x the classical set bad of badly approximable numbers in the theory of diophantine approximation falls within our framework as do the sets bad i j of simultaneously badly approximable numbers under various natural conditions we prove that the badly approximable subsets of have full hausdorff dimension applications of our general framework include those from number theory classical complex p adic and formal power series and dynamical systems iterated function schemes rational maps and kleinian groups

an integral formulation with random parameters adapted to the study of the vibrational behaviour of structures in the middle and high frequency field the fundamentals of a new boundary element formulation applied to the study of structural dynamics on a wide frequency range is proposed in this paper random geometrical parameters are introduced to the integral representations in order to model the increasing sensitivity of the harmonic vibrational responses to any parameter perturbation when the frequency increases it is shown that the significant unknowns of the formulation are the expectations of the square boundary kinematic unknowns indeed these variables are able to describe accurately the responses of the structures in the low frequency field and deliver a smooth response in the middle and high frequency field corresponding to the global trend of the structural dynamic response in this frequency domain the formulation is established for isolated and assembled one two and three dimensional structures numerical applications illustrate the effectiveness of the method

on sums of degrees of the partial quotients in continued fraction expansions of laurent series abstract for any formal laurent series x n v c n z n with coefficients c n lying in some given finite field let x a 0 x a 1 x a 2 x be its continued fraction expansion it is known that with respect to the haar measure almost surely the sum of degrees of partial quotients deg a 1 x deg a n x grows linearly in this note we quantify the exceptional sets of points with faster growth orders than linear ones by their hausdorff dimension which covers an earlier result by j wu

sets of exact logarithmic order in the theory of diophantine approximation for each real number alpha let e alpha denote the set of real numbers with exact order alpha a theorem of guting states that for alpha geq 2 the hausdorff dimension of e alpha is equal to 2 alpha in this note we introduce the notion of exact t logarithmic order which refines the usual definition of exact order our main result for the associated refined sets generalizes guting s result to linear forms and moreover determines the hausdorff measure at the critical exponent in fact the sets are shown to satisfy delicate zero infinity laws with respect to lebesgue and hausdorff measures these laws are reminiscent of those satisfied by the classical set of well approximable real numbers for example as demonstrated by khintchine s theorem

lyapunov like equations and reachability observabiliy gramians for descriptor systems reachability and observability gramians are defined for the descriptor systems ex k 1 ax k bu k y k cx k and e dot x t ax t bu t y t cx t the pencils ze a and se a are assumed to be regular if e is singular these gramians do not in general satisfy the expected lyapunov like sylvester equations for example epa t ape t bb t if p is the continuous time teachability gramian this is shown by expressing the solution of the descriptor system in terms of the parameters of the laurent expansion of ze a 1 or se a 1 we derive the lyapunov like equations which these gramians do satisfy

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

a mass transference principle and the duffin schaeffer conjecture for hausdorff measures a hausdorff measure version of the duffin schaeffer conjecture in metric number theory is introduced and discussed the general conjecture is established modulo the original conjecture the key result is a mass transference principle which allows us to transfer lebesgue measure theoretic statements for lim sup subsets of r k to hausdorff measure theoretic statements in view of this the lebesgue theory of lim sup sets is shown to underpin the general hausdorff theory this is rather surprising since the latter theory is viewed to be a subtle refinement of the former

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

descriptor systems decomposition into forward and backward subsystems it is shown that a decomposition of the descriptor space with certain properties induces a decomposition of the descriptor system into forward and backward subsystems different decompositions correspond to different apportionments of the required boundary conditions between initial and final times applications include several different but equivalent methods for solving the descriptor system and a clarification of the relations between some previously known methods of solution

asymptotic analysis of wave propagation along weakly non uniform repetitive systems we consider monochromatic wave propagation along a long finite one dimensional slightly non uniform waveguide whose ends are connected to uniform semi infinite waveguides the non uniformity in the system parameters which is assumed slowly varying and deterministic can be tuned to produce a desired scattered wave field or reflection transmission properties for a broad range of incident wave fields with this objective in mind we obtain an analytic solution for wave propagation along repetitive systems asymptotic in the slowness of the variation of the system parameters we consider systems governed by a second order finite difference equation and apply the wkb method allowing the index variable to be complex this allows complex turning points to be considered the coefficients of the difference equation are represented by their discrete fourier modes for complex turning points we obtain exponentially small reflection a new result in the context of difference equations the asymptotic solution besides revealing how the non uniformity in the parameters affects wave propagation furnishes an analytic expression for the system scattering matrix as a function of the system parameters it also sheds light on the mechanism of localization phenomena for this class of repetitive systems we also compare the asymptotic results with numerical experiments for large finite one dimensional non uniform chains of coupled pendula

almost no points on a cantor set are very well approximable we prove that almost no numbers in cantor9s middle thirds set are very well approximable by rationals more generally we discuss diophantine properties of almost every point where almost every is understood relative to any measure on the real line satisfying a decay condition introduced in the work of veech

on a problem of k mahler diophantine approximation and cantor sets let k denote the middle third cantor set and mathcal a 3 n n 0 1 2 ldots given a real positive function let w mathcal a psi denote the set of real numbers x in the unit interval for which there exist infinitely many p q in mathbb z times mathcal a such that x p q q the analogue of the hausdorff measure version of the duffin schaeffer conjecture is established for w mathcal a psi cap k one of the consequences of this is that there exist very well approximable numbers other than liouville numbers in k an assertion attributed to k mahler explicit examples of irrational numbers satisfying mahler s assertion are also given

on input function observers for generalized state space systems abstract linear control systems of the form e dx dt ax bu y cx du are considered where e and a are not necessarily square matrices and no assumption is made on the rank of the matrix pencil se a the problem to be solved is this find a system of the form edz dt z bcirc u z dcirc where is some linear combination of y the output of the original system and its derivatives and u is the input of the original system for the solution of this problem a method based on the use of matrix generalized inverses is developed that can be applied both to time invariant and time varying systems a numerical procedure is elaborated for the time invariant case that either yields a system e dz dt z bcirc u z dcirc in which the coefficient matrices are of the smallest possible dimensions or it shows that there is no such system this procedure makes use of a recent paper of wilkinson on singular systems of linear differential equations in the special case where e