some comments on the behavior of sign sign adaptive identifiers abstract the sign sign variant of the lms algorithm with and without leakage is investigated three basic questions are considered through simplistic examples the need to redefine input excitations for parameter convergence the influence of designer selected variables on offset due to leakage and the effect of measurement noise in underexcited systems

minimal characteristic bisets for fusion systems we show that every saturated fusion system f has a unique minimal f characteristic biset f we examine the relationship of f with other concepts in p local finite group theory in the case of a constrained fusion system the model for the fusion system is the minimal f characteristic biset and more generally any centric linking system can be identified with the f centric part of f as bisets we explore the grouplike properties of f and conjecture an identification of normalizer subsystems of f with subbisets of f

normal subsystems of fusion systems the notion of a fusion system was first defined and explored by puig in the context of modular representation theory later broto levi and oliver significantly extended the theory of fusion systems as a tool in homotopy theory in this paper we begin a program to establish a local theory of fusion systems similar to the local theory of finite groups in particular we define the notion of a normal subsystem of a saturated fusion system and prove some basic results about normal subsystems and factor systems

subcentric linking systems for part of this research the author was supported by the danish national research foundation through the centre for symmetry and deformation dnrf92

stratification and duality for homotopical groups abstract we generalize quillen s f isomorphism theorem quillen s stratification theorem the stable transfer and the finite generation of cohomology rings from finite groups to homotopical groups as a consequence we show that the category of module spectra over c b g f p is stratified and costratified for a large class of p local compact groups g including compact lie groups connected p compact groups and p local finite groups thereby giving a support theoretic classification of all localizing and colocalizing subcategories of this category moreover we prove that p compact groups admit a homotopical form of gorenstein duality

asymptotically efficient parameter estimation using quantized output observations this paper studies identification of systems in which only quantized output observations are available an identification algorithm for system gains is introduced that employs empirical measures from multiple sensor thresholds and optimizes their convex combinations strong convergence of the algorithm is first derived the algorithm is then extended to a scenario of system identification with communication constraints in which the sensor output is transmitted through a noisy communication channel and observed after transmission the main results of this paper demonstrate that these algorithms achieve the cramer rao lower bounds asymptotically and hence are asymptotically efficient algorithms furthermore under some mild regularity conditions these optimal algorithms achieve error bounds that approach optimal error bounds of linear sensors when the number of thresholds becomes large these results are further extended to finite impulse response and rational transfer function models when the inputs are designed to be periodic and full rank

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

centralizers of normal subsystems revisited abstract in this paper we revisit two concepts which were originally introduced by aschbacher and are crucial in the theory of saturated fusion systems firstly we give a new approach to defining the centra lizer of a normal subsystem secondly we revisit the construction of the product of two normal subsystems which centralize each other

recursion operators of some equations of hydrodynamic type we give a general method for constructing recursion operators for some equations of hydrodynamic type admitting a nonstandard lax representation we give several examples for n 2 and n 3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators we also discuss a reduction of n 1 systems to n systems of some new equations of hydrodynamic type

existence and uniqueness of linking systems chermak s proof via obstruction theory we present a version of a proof by andy chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems this proof differs from the one in ch2 in that it is based on the computation of derived functors of certain inverse limits this leads to a much shorter proof but one which is aimed mostly at researchers familiar with homological algebra

control of fixed points and existence and uniqueness of centric linking systems a chermak has recently proved that to each saturated fusion system over a finite p group there is a unique associated centric linking system b oliver extended chermak s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish both proofs indirectly assume the classification of finite simple groups we show how to remove this assumption thereby giving a classification free proof of the martino priddy conjecture concerning the p completed classifying spaces of finite groups our main tool is a 1971 result of the first author on control of fixed points by p local subgroups this result is directly applicable for odd primes and we show how a slight variation of it allows applications for p 2 in the presence of offenders

transient and tracking performance bounds of the sign sign algorithm the paper provides a rigorous tracking analysis of the sign sign algorithm when used in the identification of a time varying plant with a white gaussian input the plant parameters vary according to a random walk model the assumptions allow nonstationarity of the plant input plant noise and increments of the plant parameters upper bounds are derived for the long term averages of the mean of the weight misalignment norm mean absolute error mean square weight misalignment and mean square error these bounds hold for all values of the algorithm step size all initial filter weight settings and all degrees of nonstationarity of the plant input plant noise and plant parameter increments lower bounds of the mean square weight misalignment and mean square error are also derived the step sizes that minimize the above bounds are derived a transient analysis of the algorithm is done in the case of a time invariant plant a tight lower bound of the convergence time is derived the above analytical results are supported by computer simulations

convergence analysis of an adaptive filter equipped with the sign sign algorithm convergence with probability 1 of the sign sign algorithm with a decreasing step size is proven when it is used in adaptive plant identification the plant input and plant noise are assumed stationary and m dependent the assumptions do not pose any upper bound on the memory m of the plant input and plant noise and hence they allow arbitrarily strong correlation in these signals a main interest of the paper is that it provides a condition of persistent excitation of the algorithm examples satisfying this condition are provided

control of fixed points over discrete p toral groups and existence and uniqueness of linking systems abstract the existence and uniqueness of linking systems associated to saturated fusion systems over discrete p toral groups were proved by levi and libman their proof makes indirect use of the classification of the finite simple groups here we extend some results and arguments of glauberman and lynd to show how to remove this assumption

quiver diagrams and signed adaptive filters a graphical technique for exploring the behavior of deterministic discrete time adaptive algorithms excited by periodic inputs is examined the constraint to periodic inputs is convenient the underlying concepts used also apply to stationary inputs drawn from finite alphabets the resulting graph is based on plotting single period parameter trajectories in the parameter error space and is essentially a discrete version of flow diagrams associated with continuous time systems for the least mean square lms adaptive fir filter with the algorithm step size sufficiently small the change of adapted parameters from a particular initial value is given by the total update over the input period with the parameterization frozen at its initial value this results in the familiar elliptical hyperparaboloid surface steepest descent interpretation this method can be extended to the lms adaptive infinite input response iir filter with suitable approximations based on a small step size

convergence of the signed output error adaptive identifier this paper considers an adaptive output error identifier with a signum function in its update kernel it is shown that the classical strict positive real spr condition required for the convergence of traditional adaptive identifiers does not suffice for the convergence of this signed identifier instead what is needed is a stronger operator condition called the strict dominant passive sdp condition we give an analog of the kalman yakubovic popov lemma for the sdp conditions and using it give a convergence proof under the assumptions of persistent excitation and sdp

persistence of excitation conditions and the convergence of adaptive schemes in the study of the behavior of adaptive filtering algorithms persistence of excitation of the input process arises as a sufficient condition for convergence and perhaps more importantly for convergence rate of the parameter estimates in this paper the underlying nature of the persistence requirement is presented and discussed relating its normal specification in terms of moment conditions with covariance decays etc to sample path properties deterministic and stochastic persistence conditions and persistence measures are treated as well as persistence conditions for output error equation error and adaptive control schemes

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

adaptive filtering with binary reinforcement recently there has been increased interest in high speed adaptive filtering where the usual stochastic gradient or least mean square lms algorithm is replaced with the simpler algorithm where adaptation is guided only by the polarity of the error signal in this paper the convergence of this binary reinforcement br algorithm is proved under the usual independence assumption and the surprising observation is made that unlike the lms algorithm convergence occurs for any positive value of the step size parameter while the stochastic gradient algorithm attempts to minimize a mean square error cost function the binary reinforcement algorithm in fact attempts to minimize a mean absolute error cost function it is proved for the binary reinforcement algorithm that the tap weight vector converges in distribution to a random vector that is suitably concentrated about the optimal value based on a least mean absolute error cost function for a sufficiently small step size the expected cost of the asymptotic weight vector can be made as close as desired to thc minimum cost attain ed by the optimum weight vector

characterizing persistent excitation for the sign sign equation error identifier abstract the sign sign ss algorithm is a computationally efficient adaptive identifier often used in signal processing tasks it is obtained by introducing signum functions on both the regressor and the prediction error multiplicands in the update kernel of the well known lms algorithm this paper gives a deterministic persistent excitation condition on the regressor sequence which guarantees ss convergence it also gives conditions under which ss may diverge and discusses how these persistent excitation conditions may be verified through a finite amount of computations

the general ff module theorem let p be a prime m a finite group with op m 1 v a faithful fpm module and j the subgroup of m generated by the best offenders on v in this paper we determine structure of j and the action of j on v

a note on oliver s p group conjecture abstract let s be a finite p group for an odd prime p oliver proposed the conjecture that the thompson subgroup j s is always contained in the oliver subgroup x s that means he conjectured that j s x s x s 1 let x 1 s be a subgroup of s such that x 1 s x s is the center of s x s in this short note we prove that j s x s if and only if j s x 1 s as an easy application we prove that j s x s x s p

fusion systems and localities we introduce objective partial groups of which the linking systems and p local finite groups of broto levi and oliver the transporter systems of oliver and ventura and the mathcal f localities of puig are examples as are groups in the ordinary sense as an application we show that if mathcal f is a saturated fusion system over a finite p group then there exists a centric linking system mathcal l having mathcal f as its fusion system and that mathcal l is unique up to isomorphism the proof relies on the classification of the finite simple groups in an indirect and for that reason perhaps ultimately removable way

tracking analysis of the sign sign algorithm for nonstationary adaptive filtering with gaussian data this correspondence is concerned with the analysis of the sign sign algorithm ssa when used to track a plant with randomly time varying parameters the input of the plant and the plant noise are assumed stationary and gaussian the paper derives expressions of the steady state excess mean square error spl xi the steady state mean square weight misalignment spl eta and the step sizes that minimize each one of them the paper also presents a comparison among the tracking properties of the ssa the sign algorithm sa and the signed regressor algorithm sra it is found that the three algorithms share the features that 1 spl xi does not depend on the eigenvalue spread of the input covariance matrix 2 spl eta depends on the eigenvalue spread and 3 the step size that minimizes spl xi is different from the one that minimizes spl eta the minimum values of spl xi attained by the sa and the sra are equal to each other and they are 1 db less than the one attained by the ssa the ratio of the minimum value of spl eta of the ssa to the one of the sa is found to be dependent on the input eigenvalue spread for equal eigenvalues this ratio is equal to 1 db the minimum value of spl eta of the ssa is found to be 1 db higher than the one of the sra independently of the input eigenvalue spread it is found that an advantage of the ssa with respect to both the sa and the sra is that the two optimum step sizes of the ssa are independent of the mean square plant input and the mean square plant noise

sharf convergence properties a class of stable algorithms for adapting infinite impulse response iir digital filters based on the concepts of nonlinear stability theory prominent in the control literature is emerging while this class of adaptive filters offers much promise in practical applications little has been done toward providing a characterization that would guide selection of design parameters such as adaptation constants and error smoothing coefficients this paper focuses on the simplest well behaved member of this class of adaptive recursive filters sharf progression from a local linearization of the nonlinear parameter estimate convergence behavior through an idealized eigenvalue eigenvector analysis of the parameter estimate time varying recursion to lyapunov function establishment for the full output and parameter error system reveals the exponential local nongradient descent convergence character of sharf and provides initial insight into the effects of adaptation constants and error smoothing coefficients on these characteristics

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

joint identification of plant rational models and noise distribution functions using binary valued observations system identification of plants with binary valued output observations is of importance in understanding modeling capability and limitations for systems with limited sensor information establishing relationships between communication resource limitations and identification complexity and studying sensor networks this paper resolves two issues arising in such system identification problems first regression structures for identifying a rational model contain non smooth nonlinearities leading to a difficult nonlinear filtering problem by introducing a two step identification procedure that employs periodic signals empirical measures and identifiability features rational models can be identified without resorting to complicated nonlinear searching algorithms second by formulating a joint identification problem we are able to accommodate scenarios in which noise distribution functions are unknown convergence of parameter estimates is established recursive algorithms for joint identification and their key properties are further developed

identification input design for consistent parameter estimation of linear systems with binary valued output observations input design is of essential importance in system identification for providing sufficient probing capabilities to guarantee convergence of parameter estimates to their true values this paper presents conditions on input signals that characterize their probing richness for strongly consistent parameter estimation of linear systems with binary valued output observations necessary and sufficient conditions on periodic signals are derived for sufficient richness these conditions are further studied under different system configurations including open loop and feedback systems and different scenarios of noises including actuator noise input measurement noise and output measurement noise in addition to system parameter estimation essential properties of identifiability and input conditions are also derived when sensor thresholds or noise distribution functions are unknown the findings of this paper provide a foundation to study identification of systems that either use binary valued or quantized sensors or involve communication channels which mandate quantization of signals

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated