commutators and linear spans of projections in certain finite c algebras assume that a is a unital separable simple c algebra with real rank zero stable rank one strict comparison of projections and that its tracial simplex t a has a finite number of extremal points we prove that every self adjoint element a in a with a 0 for all t a is the sum of two commutators in a and that every positive element of a is a linear combination of projections with positive coefficients assume that a is as above but unital and not necessarily unital then an element resp a positive element a of a is a linear combination resp a linear combination with positive coefficients of projections if and only if ra for every t a where denotes the extension of to a tracial weight on a and ra a denotes the range projection of a assume that a is unital and as above but t a has infinitely many extremal points then a is not the linear span of its projections this result settles two open problems of marcoux in 32

lattice preserving maps on lattices of continuous functions abstract lattice preserving maps between sublattices of continuous functions on locally compact spaces e g the functions with compact support are studied these maps though not necessarily linear are related to continuous functions between the underlying locally compact spaces consequences for first countable spaces including an automatic continuity for these lattice maps are provided

orthogonality preservers in c algebras jb algebras and jb triples abstract we study orthogonality preserving operators between c algebras jb algebras and jb triples let t a e be an orthogonality preserving bounded linear operator from a c algebra to a jb triple satisfying that t 1 d is a von neumann regular element then t a e 2 r d every element in t a and d operator commute in the jb algebra e 2 r d and there exists a triple homomorphism s a e 2 r d such that t l d r d s where r d denotes the range tripotent of d in e an analogous result for a being a jb algebra is also obtained when t a b is an operator between two c algebras we show that denoting h t 1 then t orthogonality preserving if and only if there exists a triple homomorphism s a b satisfying h s z s z h h s z s z h and t z l h r h s z 1 2 h r h s z s z r h h this allows us to prove that a bounded linear operator between two c algebras is orthogonality preserving if and only if it preserves zero triple products

frame theory based analysis and design of oversampled filter banks direct computational method this paper studies the frames corresponding to oversampled filter banks fbs for this class of fb frames we present a state space parameterization of perfect reconstruction fb frames and explicit and numerically efficient formulas to compute the tightest frame bounds to obtain the dual fb frame and to construct a tight paraunitary fb frame from a given untight nonparaunitary fb frame the derivation uses well developed techniques from modern control theory which results in the unified formulas for generic infinite impulse response iir and finite impulse response fir fbs these formulas involve only algebraic manipulations of real matrices and can be computed efficiently reliably and exactly without the approximation required in the existing methods for generic fbs the results provide a unified framework for frame theory based analysis and systematic design of oversampled filter banks

biseparating maps between operator algebras we prove that a biseparating map between spaces b e and some other banach algebras is automatically continuous and a multiple of an algebra isomorphism

biseparating maps and homeomorphic real compactifications abstract if x and y are realcompact and t c x c y is a ring isomorphism then two things happen 1 there is a homeomorphism h of x onto y and 2 t is a composition map for all in c x t h we strengthen that result in propositions 2 and 3 we discard the multiplicative property of t and assume only that t is biseparating in the sense that g 0 t tg 0 it remains true that x is homeomorphic to y but this time if t is also linear it is a weighted composition map for some weight function a in c y t a h

weighted composition operators ofc0 x s abstract in this paper we prove that into isometries and disjointness preserving linear maps from c 0 x into c 0 y are essentially weighted composition operators tf h f for some continuous map and some continuous scalar valued function h

constructive controllability algorithms for motion planning and optimization presents algorithms for planning feasible and minimum energy paths the algorithms rely on series expansions to characterize planning problems for polynomial control systems the resulting inversion problem is solved through an iterative contraction or a power series inversion while the design methodology is general our focus is on a class of polynomial control systems for which we provide explicit convergence guarantees we demonstrate performance and numerical characteristics using one dimensional and six dimensional systems

frame analysis for biorthogonal cosine modulated filterbanks this paper addresses the efficient computation of frame bounds for cosine modulated filterbanks we derive explicit expressions for the eigenvalues of the frame operator that can be easily computed from the prototype s polyphase components the number of channels and the downsampling factor may be even or odd and the oversampling factor is supposed to be an integer the analysis of low delay biorthogonal filterbanks shows that prototypes solely designed to minimize the stopband energy may lead to wide open frames and thus to an undesirable numerical behavior because the computational cost of determining the frame bounds with the proposed method is very low we can directly use the bounds during prototype optimization and obtain prototypes with minimum stopband energy under the condition of fixed frame bounds various design examples are presented

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

stabilization of relative equilibria for underactuated systems on riemannian manifolds steady motions of mechanical systems with symmetry are stabilized via a potential shaping and damping control a general methodology based on riemannian geometry tools provides exponential stabilization over the full phase space

an approximate nonlinear observer with polynomial coordinate transformation maps the state equivalence condition to the nonlinear observer form is stated in terms of the exactness of one form using the definition of the approximate closedness we relax the equivalence condition thereby obtaining a necessary and sufficient condition for the approximate observer error linearizability this approximate observer theory extends the realm of applications another great advantage of this approximation method is that the coordinate transformation is directly obtainable as vectors of polynomials while it is in general not possible with the existing method

optimal design of oversampled synthesis fbs with lattice structure constraints oversampled filter banks fbs with lattice structure allow fast implementation and guarantee the linear phase and perfect reconstruction property for such fbs the synthesis fb is not unique this nonuniqueness provides extra design freedom for optimal reduction of subband noise required in many applications however the existing design methods of synthesis fb for optimal reduction of subband noise cannot guarantee the linear phase and lattice structure property this paper studies the design of oversampled synthesis fbs with guaranteed lattice structure for optimal reduction of subband noise based on a state space parameterization of all synthesis fbs with lattice structure the explicit formulae are derived to find the optimal solution for noise reduction numerical examples are presented to demonstrate the effectiveness of the new design methods obtained

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

zero product preserving linear maps of ccr c algebras with hausdorff spectrum in this paper we try to attack a conjecture of araujo and jarosz that every bijective linear map between c algebras with both and its inverse 1 preserving zero products arises from an algebra isomorphism followed by a central multiplier we show it is true for ccr c algebras with hausdorff spectrum and in general some special c algebras associated to continuous fields of c algebras

a homotopy algorithm for approximating geometric distributions by integrable systems in the geometric theory of nonlinear control systems the notion of a distribution and the dual notion of codistribution play a central role many results in nonlinear control theory require certain distributions to be integrable distributions and codistributions are not generically integrable and moreover the integrability property is not likely to persist under small perturbations of the system therefore it is natural to consider the problem of approximating a given codistribution by an integrable codistribution and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory in this note we concentrate on the purely mathematical problem of approximating a given codistribution by an integrable codistribution we present an algorithm for approximating an m dimensional nonintegrable codistribution by an integrable one using a homotopy approach the method yields an approximating codistribution that agrees with the original codistribution on an m dimensional submanifold e 0 of r n

tight weyl heisenberg frames in l sup 2 z tight weyl heisenberg frames in l sup 2 z are the tool for short time fourier analysis in discrete time they are closely related to paraunitary modulated filter banks and are studied here using techniques of the filter bank theory good resolution of short time fourier analysis in the joint time frequency plane is not attainable unless some redundancy is introduced that is the reason for considering overcomplete weyl heisenberg expansions the main result of this correspondence is a complete parameterization of finite length tight weyl heisenberg frames in l sup 2 z with arbitrary rational oversampling ratios this parameterization follows from a factorization of polyphase matrices of paraunitary modulated filter banks which is introduced first

isomorphisms of ordered structures of abelian c subalgebras of c algebras abstract the aim of this note is to study the interplay between the jordan structure of c algebra and the structure of its abelian c subalgebras let abel a be a system of unital c subalgebras of a unital c algebra a ordered by set theoretic inclusion we show that any order isomorphism abel a abel b can be uniquely written in the form c c s a i c s a where is a partially linear jordan isomorphism between self adjoint parts of unital c algebras a and b as a corollary we obtain that for certain class of c algebras including von neumann algebras ordered structure of abelian subalgebras completely determines the jordan structure the results extend hitherto known results for abelian c algebras and may be relevant to foundations of quantum theory

a study of two channel complex valued filterbanks and wavelets with orthogonality and symmetry properties we investigate two channel complex valued filterbanks and wavelets that simultaneously have orthogonality and symmetry properties first the conditions for the filterbank to be orthogonal symmetric and regular for generating smooth wavelets are presented then a complete and minimal lattice structure is developed which enables a general design approach for filterbanks and wavelets with arbitrary length and arbitrary order of regularity finally two integer implementation methods that preserve the perfect reconstruction property of the filterbank are proposed their performances are evaluated via experimental results

time domain oversampled lapped transforms theory structure and application in image coding this paper generalizes time domain lapped transforms tdlts proposed by tran et al to oversampled systems thus leading to time domain oversampled lapped transforms tdolts these new transforms correspond to a subclass of oversampled linear phase perfect reconstruction filterbanks olpprfbs which can be implemented by adding a prefilter before the discrete cosine transform dct and a post filter after the inverse discrete cosine transform idct structures of the pre and post filters are developed and the frame theoretic properties of tdolts are analyzed a new parameterization of lattice matrices through the givens qr factorization is proposed for unconstrained optimization comparisons with other parameterization methods are also included several design examples along with some image coding results are presented to demonstrate the validity of the theory and the potential of tdolts in image coding especially in error resilient coding

poincare type inequalities for the homotopy operator with lφ ω norms abstract we obtain poincare type inequalities for the homotopy operator applied to differential forms in l m averaging domains which can be used to estimate the norms of operators and the integral properties of differential forms

linear disjointness preservers of w algebras in this paper we give a complete description of the structure of zero product and orthogonality preserving linear maps between w algebras in particular two w algebras are isomorphic if and only if there is a bijective linear map between them preserving their zero product or orthogonality structure in two directions it is also the case when they have equivalent linear and left right ideal structures

orthogonality and disjointness preserving linear maps between fourier and fourier stieltjes algebras of locally compact groups this paper is devoted to the study of orthogonality and disjointness preserving linear maps between fourier and fourier stieltjes algebras of locally compact groups we show that a linear bijection a g1 a g2 resp b g1 b g2 between two fourier algebras resp fourier stieltjes algebras of locally compact groups will induce a topological group isomorphism between g1 and g2 provided that preserves both disjointness and some kind of orthogonality this improves earlier results of j j font and m s monfared where amenability of the groups or continuity of the linear maps are assumed we also study the structure of bounded and unbounded disjointness preserving linear functionals of fourier algebras in the development general results about disjointness and orthogonality preserving linear maps between c algebras w algebras and their preduals are obtained

bound ratio minimization of filter bank frames this paper investigates and solves the problem of frame bound ratio minimization for oversampled perfect reconstruction pr filter banks fbs for a given analysis prfb a finite dimensional convex optimization algorithm is derived to redesign the subband gain of each channel the redesign minimizes the frame bound ratio of the fb while maintaining its original properties and performance the obtained solution is precise without involving frequency domain approximation and can be applied to many practical problems in signal processing the optimal solution is applied to subband noise suppression and tree structured fb gain optimization resulting in deeper insights and novel solutions to these two general classes of problems and considerable performance improvement effectiveness of the optimal solution is demonstrated by extensive numerical examples

approximate feedback linearization a homotopy operator approach in this paper we present an approach for finding feedback linearizable systems that approximate a given single input nonlinear system on a given compact region of the state space first we show that if the system is close to being involutive then it is also close to being linearizable rather than working directly with the characteristic distribution of the system we work with characteristic one forms i e with the one forms annihilating the characteristic distribution we show that homotopy operators can be used to decompose a given characteristic one form into an exact and an antiexact part the exact part is used to define a change of coordinates to a normal form that looks like a linearizable part plus nonlinear perturbation terms the nonlinear terms in this normal form depend continuously on the antiexact part and they vanish whenever the antiexact part does thus the antiexact part of a given characteristic one form is a measure of nonlinearizability of the system if the nonlinear terms are small by neglecting them we obtain a linearizable system approximating the original system one can design control for the original system by designing it for the approximating linearizable system and applying it to the original one we apply this approach for design of locally stabilizing feedback laws for nonlinear systems that are close to being linearizable

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

automatic continuity of biorthogonality preservers between compact c algebras and von neumann algebras abstract we prove that every biorthogonality preserving linear surjection between two dual or compact c algebras or between two von neumann algebras is automatically continuous

efficient computation of frame bounds using lmi based optimization this correspondence presents a simple and effective method for computing the optimal frame bounds of oversampled perfect reconstruction pr filter banks fbs it first shows that computation of the optimal frame bounds for complex valued oversampled pr fbs can be formulated as a convex optimization subject to complex valued linear matrix inequality lmi constraints and solved by effective interior point algorithms it then deals with discrete time weyl heisenberg wh frames to compute bounds on the wh frames by real valued lmi optimization the wh frames are closely related to modulated fbs and have complex coefficients four examples are given to illustrate the generality and effectiveness of the proposed method

non linear observer design by approximate error linearization abstract for a given non linear system an observer that provides exactly linear error dynamics can be computed by solving the so called generalized characteristic equation gce unfortunately the existence of a solution to the gce is not a generic property for unforced scalar output systems we show how spline functions may be used to construct approximate solutions that minimize a norm of the non linear terms obstructing linearization of the error dynamics the resulting error dynamics are shown to be locally exponentially stable a numerical example illustrates the design and performance of the observer

optimal two sided diagonal scaling for filter bank frames two sided diagonal scaling for transfer matrices is formulated an efficient algorithm is proposed to compute globally optimal solutions using the iterated bi section and linear matrix inequality lmi solver it is shown that the two sided scaling of filter bank fb frames can be implemented by the adjustment of channel gains and the periodic precoding of source signal and that the frame bound ratio of fb frames can be effectively improved by such scaling explicit formulas are established for both uniform and nonuniform fb frames including detail formulas for discrete weyl heisenberg frames and tree structured fbs discrete wavelets different examples show the effectiveness of the obtained results

optimal noise reduction in oversampled pr filter banks this paper studies the optimal noise reduction problem for oversampled filter banks fbs with perfect reconstruction pr constraint both the optimal design and worst case design are considered where the former caters for the noise with known power spectral density psd and the latter for the noise with unknown psd state space based explicit formulae involving only algebraic riccati equation and matrix manipulations are provided for the general iir or fir oversampled pr fbs and the relations between different cases are analyzed and revealed extensive numerical examples are provided to illustrate the proposed design methods and to show their effectiveness

survey approximate linearization via feedback an overview fostered by a growing interest in nonlinear control theory and catalyzed by the discovery in the early 1980s of the exact conditions under which a nonlinear plant can be linearized by static state feedback and coordinate transformation in the last decades there has been a rapid increase of interest in the search for approximate solutions to the problem of linearizing nonlinear systems by state or output feedback main reason for that is the limited applicability of the rigorous methods and the complexity sensitivity and design difficulties of the exact linearizing compensators if any in the present paper the literature on the subject is reviewed and organized in what is believed to be a new and consistent perspective recent works especially in the area of data based techniques are in fact described and related whenever possible to fundamental results previously obtained by model based differential geometric methods this is expected to bring modern system linearization methods closer to the needs of practicing control engineers and to stimulate further research eventually able to fill visible gaps in this direction

disjointness preservers of aw algebras abstract in this paper we study bijective linear maps a b between a w algebras preserving either zero products or range orthogonality such a map is automatically continuous and provides an algebra or a algebra isomorphism 1 1 from a onto b our results extend previous works for the case of w algebras and also cover such maps between c algebras satisfying an extra assumption on preserving abelian c subalgebras