fixed points of uniformly lipschitzian mappings abstract two fixed point theorems for uniformly lipschitzian mappings in metric spaces due respectively to e lifsic and to t c lim and h k xu are compared within the framework of the so called cat 0 spaces it is shown that both results apply in this setting and that lifsic s theorem gives a sharper result also a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces a class which includes those possessing property p of lim and xu

a normalized frequency domain lms adaptive algorithm a scheme is presented for obtaining an input power estimate for setting the algorithm gain parameter separately in each frequency bin in the frequency domain lms adaptive algorithm this is particularly important if the input has large spectral variations and a single feedback parameter set on the broad band power could result in instability in the adaptive filters in some frequency bins the estimate is incorporated directly into the algorithm as a data dependent time varying stochastic n using a gaussian data model and sample to sample data independence first order linear difference equations are derived and solved for the mean and misadjustment errors the performance of the scheme is compared to the case for which the input power level is known a priori for the same transient response only about ten samples need be averaged to yield the same misadjustment error

the existence of generating families for the cohomology ring of a graph abstract let be a finite d valent graph and g an n dimensional torus an action of g on is defined by a map that assigns to each oriented edge e of a 1 dimensional representation of g or alternatively a weight e in the weight lattice of g for the assignment e e to be a schematic description of a g action these weights have to satisfy certain compatibility conditions we attach to an equivariant cohomology ring h by definition this ring contains the equivariant cohomology ring of a point h g pt s g as a subring and in this paper we use graphical versions of standard morse theoretical techniques to analyze the structure of h as an s g module

h vectors of gorenstein simplicial posets abstract as is well known h vectors of simplicial convex polytopes are characterized those h vectors satisfy dehn sommerville equations and some inequalities conjectured by p mcmullen and first proved by r stanley using toric geometry the boundary of a simplicial convex polytope determines a gorenstein simplicial poset but there are many gorenstein simplicial posets which do not arise this way however it is known that h vectors of gorenstein simplicial posets still satisfy dehn sommerville equations and that every component in the h vectors is non negative in this paper we prove that h vectors of gorenstein simplicial posets must satisfy one more subtle condition conjectured by r stanley and complete the characterization of h vectors of gorenstein simplicial posets our proof is purely algebraic but the idea of the proof stems from topology

steady state solution of the deficient length constrained fblms algorithm in almost all analyses of the frequency domain block least mean square fblms algorithm the length of the adaptive filter is assumed to be equal to that of the unknown system impulse response however in many practical situations where the length of the system impulse response is extremely long the adaptive filter usually works in an under modeling situation the analysis results for the sufficient length fblms algorithm are not applicable to the deficient length case in this correspondence the steady state solution of the deficient length constrained fblms algorithm is analyzed simulations illustrate the accuracy of the theoretical results

unconstrained frequency domain adaptive filter an unconstrained frequency domain least mean squares uflms algorithm is presented the algorithm is based on the overlap save technique used in frequency domain filtering the proposed algorithm converges to the wiener solution without the time domain constraint on the impulse response as proposed by ferrara for a large number of taps geq32 the uflms algorithm offers a significant reduction in computation another advantage is its fast convergence for highly correlated input signals the performance of the algorithm is illustrated by a simulation

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

lim s theorems for multivalued mappings in cat 0 spaces let x be a complete cat 0 space we prove that if e is a nonempty bounded closed convex subset of x and t e k x a nonexpansive mapping satisfying the weakly inward condition i e there exists p e such that p 1 tx ie x x e 0 1 then t has a fixed point in banach spaces this is a result of lim on asymptotic centers and fixed points of nonexpansive mappings canad j math 32 1980 421 430 the related result for unbounded r trees is given

fixed point theory for almost convex functions traditionally metric fixed point theory has sought classes of spaces in which a given type of mapping nonexpansive assymptotically or generalized nonexpansive uniformly lipschitz etc from a nonempty weakly compact convex set into itself always has a fixed point in some situations the class of space is determined by the application while there is some degree of freedom in constructing the map to be used with this in mind we seek to relax the conditions on the space by considering more restrictive types of mappings

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

a unified approach to time and frequency domain realization of fir adaptive digital filters specific implementations of the finite impulse response fir block adaptive filter in the frequency domain are presented and some of their important properties are discussed the time domain block adaptive filter implemented in the frequency domain is shown to be equivalent to the frequency domain adaptive filter derived in the frequency domain provided data sectioning is done properly all of the known time and frequency domain adaptive filters 1 12 16 18 are contained in the set of possible block adaptive filter structures thus the block adaptive filter is generic and its formulation unifies the current theory of time and frequency domain fir adaptive filter structures a detailed analysis of overlap save and overlap add implementations shows that the former is to be preferred for adaptive applications because it requires less computation a fact that is not true for fixed coefficient filters

on the steady state performance of frequency domain lms algorithms the use of the fast fourier transform fft in the implementation of the least mean square lms algorithm in the frequency manuscript received may 24 1990 revised may 4 1992 a feuer is with the department of electrical engineering technionisrael institute of technology haifa 32000 israel r cristi is with the department of electrical and computer engineering naval postgraduate school monterey ca 93943 ieee log number 9203376 1053 587x 93 03 00 1993 ieee 420 ieee transactions on signal processing vol 41 no i january 1993 domain results in several types of algorithms two of which can be classified as constrained and unconstrained in this correspondence we point out that in general especially with correlated data the unconstrained algorithm may have a significant performance advantage in steady state if block mean square error is the criterion furthermore we point out here that in the constrained algorithms the choice of different step sizes in different frequency bins as is commonly done will very likely result in a deterioration of steady state performance this does not happen in the unconstrained algorithm

stability and convergence analysis of transform domain lms adaptive filters with second order autoregressive process in this paper the stability and convergence properties of the class of transform domain least mean square lms adaptive filters with second order autoregressive ar process are investigated it is well known that this class of adaptive filters improve convergence property of the standard lms adaptive filters by applying the fixed data independent orthogonal transforms and power normalization however the convergence performance of this class of adaptive filters can be quite different for various input processes and it has not been fully explored in this paper we first discuss the mean square stability and steady state performance of this class of adaptive filters we then analyze the effects of the transforms and power normalization performed in the various adaptive filters for both first order and second order ar processes we derive the input asymptotic eigenvalue distributions and make comparisons on their convergence performance finally computer simulations on ar process as well as moving average ma process and autoregressive moving average arma process are demonstrated for the support of the analytical results

frequency domain adaptation of causal digital filters the adaptation of causal fir digital filters in the discrete frequency domain is considered and it is shown how the bin normalized form of the lms algorithm can converge to a biased solution for problems such as linear prediction a discrete frequency domain version of newton s algorithm is derived and it is demonstrated how this can converge to the optimal causal solution even for linear prediction problems the algorithm employs a spectral factorization of the estimated power spectral density of the reference signal the entirely noncausal part of which is used before the causality constraint in the adaptation algorithm and the entirely causal part is applied after the causality constraint the spectral factors can be calculated online from a recursive estimate of the power spectral density without too great a loss of convergence speed the extension of the algorithm to the adaptation of feedforward controllers is also described in which case the spectral factors of the reference signals filtered by the plant response are required and these are shown to be equal to the spectral factors of the reference signal multiplied by the minimum phase part or the plant frequency response

invariant approximations for commuting mappings in cat 0 and hyperconvex spaces abstract in this paper for a commuting pair consisting of a point valued nonexpansive self mapping t and a set valued nonexpansive self mapping t of a hyperconvex metric space or a cat 0 space x we look for a solution to the problem of existence of z e x such that d z y d y e and z t z t z

torus graphs and simplicial posets for several important classes of manifolds acted on by the torus the information about the action can be encoded combinatorially by a regular n valent graph with vector labels on its edges which we refer to as the torus graph by analogy with the gkm graphs we introduce the notion of equivariant cohomology of a torus graph and show that it is isomorphic to the face ring of the associated simplicial poset this extends a series of previous results on the equivariant cohomology of torus manifolds as a primary combinatorial application we show that a simplicial poset is cohen macaulay if its face ring is cohen macaulay this completes the algebraic characterisation of cohen macaulay posets initiated by stanley we also study blow ups of torus graphs and manifolds from both the algebraic and the topological points of view