input output stabilization of linear systems on spl zopf a formal framework is set up for the discussion of generalized autoregressive with external input models of the form ay bu where a and b are linear operators with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers different notions of stability are explored and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context various problems inherent in using spl zopf as the time axis are analyzed in this operatorial framework

linear block codes over cyclic groups the main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of g sup i where g is a group generating a low dimensional signal constellation and i is an index set in this paper we study the properties of these subgroups when g is cyclic we exploit the fact that any cyclic group of q elements is isomorphic to the additive group of z sub q the ring of integers modulo q so that we can make use of concepts related to linearity our attention is focused mainly on indecomposable cyclic groups i e of prime power order since they are the elementary building blocks of any abelian group in analogy with the usual construction of linear codes over fields we define a generator matrix and a parity check matrix trellis construction and bounds on the minimum euclidean distance are also investigated some examples of coded modulation schemes based on this theory are also exhibited and their performance evaluated

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

worst case input output identification we consider worst case l1 identification of causal linear shift invariant systems from time series many results are given on general aspects of identification algorithm performance existence of optimal algorithms robust convergence and input experiment design the identification methodology studied here is compatible with the modelling requirements of modern robust control design

empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

structural identification of muitivariabie systems by the eigenvector method abstract the structural identification of linear muitivariabie systems can be performed on the basis of the predicted percentage of reconstruction errors ppcre associated to a sequence of increasing complexity models this paper extends some previous results describing the properties of the ppcre for canonical models whose parameters are estimated by means of the eigenvector method

some structural properties of convolutional codes over rings convolutional codes over rings have been motivated by phase modulated signals some structural properties of the generator matrices of such codes are presented successively stronger notions of the invertibility of generator matrices are studied and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by massey and mittelholzer 1990 it is shown that a generator matrix that can be decomposed into a direct sum is basic minimal and noncatastrophic if and only if all generator matrices for the constituent codes are basic minimal and noncatastrophic respectively it is also shown that if a systematic generator matrix can be decomposed into a direct sum then all generator matrices of the constituent codes are systematic but that the converse does not hold some results on convolutional codes over z p sup e are obtained

dynamical systems and convolutional codes over finite abelian groups polynomial algebraic techniques have always played a central role in linear systems theory and also in the theory of convolutional codes we show how such techniques can be generalized to study systems and codes defined over abelian groups the systems are considered from the behavioral point of view as developed by willems in the 1980s and some of our results can be seen as extensions of willems results to group systems we also address a certain number of coding oriented questions and we propose concrete methods based on these algebraic techniques for the synthesis of encoders inverters and syndrome formers for codes over finite abelian groups

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

guaranteed stability with subspace methods abstract we show how stability of models can be guaranteed when using the class of identification algorithms which have become known as subspace methods in many of these methods the a matrix is obtained or can be obtained as the product of a shifted matrix with a pseudo inverse we show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position then a stable model results the cost of this is some possibly large distortion of the results but in some applications that is outweighed by the advantage of guaranteed stability

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

extending the scope of empirical likelihood this article extends the scope of empirical likelihood methodology ill three directions to allow for plug in estimates of nuisance parameters in estimating equations slower than root n rates of convergence and settings in which there are a relatively large number of estimating equations compared to the sample size calibrating empirical likelihood confidence regions with plug in is sometimes intractable due to the complexity of the asymptotics so we introduce a bootstrap approximation that call be used in such situations we provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results

on the control of finite support behaviors abstract in the paper the main properties of finite support behaviors are introduced and characterized in terms of the polynomial matrices involved in their representation necessary and sufficient conditions for the existence of an input output possibly non anticipating description of a finite support behavior are given the control problem is also addressed and properly restated in the behavior theory context as an interconnection problem conditions for the existence of controllers that make the resulting connected system observable as well as a complete description of the class of characteristic polynomials that the resulting controlled system possibly exhibits are given finally the problem of controlling a dynamical system in input output from by means of a controller with the same structure is investigated and solved

canonical kernel representations for behaviors over finite abelian groups abstract in this paper the problem of constructing canonical kernel representations for systems whose alphabet set is an abelian group is addressed in standard linear system theory there exist two important canonical kernel representations minimal and row reduced kernel representations in this paper we propose an extension of such canonical kernel representations to the group case this topic has applications in the theory of convolutional codes over groups where kernel representations correspond to syndrome formers which are essential in decoding

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

from time series to linear system part ii exact modelling abstract in the second part of this paper the problem of finding an exact model for a q dimensional infinite time series is considered first a mathematical vocabulary for discussing exact modelling is developed it is then shown how the results of part i guarantee the existence of a most powerful ar model for an observed time series two algorithms for obtaining such an ar model are subsequently derived one of these algorithms gives a shortest lag input output model the problem of obtaining a minimal state space realization of the observed time series is also considered in order to do that realization theory based on the truncated behaviour is developed as an extensive example the classical situation with impulse response measurements is discussed

on minimal tree realizations of linear codes a tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree a tree decomposition can be extended to a tree realization i e a cycle free realization of the code on the underlying tree by specifying a state space at each edge of the tree and a local constraint code at each vertex of the tree the constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes a measure of the complexity of maximum likelihood ml decoding for a code is its treewidth which is the least constraint complexity of any of its tree realizations it is known that among all tree realizations of a linear code that extends a given tree decomposition there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree in this paper we give two new constructions of these minimal realizations as a by product of the first construction a generalization of the state merging procedure for trellis realizations we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree the second construction relies on certain code decomposition techniques that we develop we further observe that the treewidth of a code is related to a measure of graph complexity also called treewidth we exploit this connection to resolve a conjecture of forney s regarding the gap between the minimum trellis constraint complexity and the treewidth of a code we present a family of codes for which this gap can be arbitrarily large

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

on the choice of inputs in identification for robust control the thesis that noisy identification has close ties to the study of the singular value decomposition of perturbed matrices is investigated in particular by assuming an upper bound on the norm of the perturbation one can obtain a convex parametrization of an uncertain family of systems which contains the system generating the data in this approach the second smallest singular value s of an appropriately defined data matrix becomes a quantity of importance as it provides an upper bound for the size of the uncertain family this yields a new tool leading to the design of input functions which are optimal or persistently exciting from the point of view of identification for robust control

continuity of latent variable models the authors study the continuity of the behavior of dynamical systems as a function of the parameters in their behavioral equations a system is defined in terms of its behavior and continuity requires that this behavior be continuous in the limit the problem is demonstrated by means of an example involving an rlc circuit whose port behavior exhibits a surprising discontinuity as a function of the numerical values of the elements in the circuit the main result states that a system described by means of difference equations involving manifest external and latent internal variables will have a continuous behavior in the limit if the limit system is observable

codes on graphs recent progress abstract this paper surveys the field of codes on graphs from its origins in the work of gallager and tanner through its explosion in 1995 96 due to the rediscovery of ldpc codes and wiberg s thesis and up to the present

on dualizing trellis based app decoding algorithms the trellis of a finite abelian group code is locally i e trellis section by trellis section related to the trellis of the corresponding dual group code which allows one to express the basic operations of the a posteriori probability app decoding algorithm defined on a single trellis section of the primal trellis in terms of the corresponding dual trellis section using this local approach any algorithm employing the same type of operations as the app algorithm can thus be dualized even if the global dual code does not exist e g nongroup codes represented by a group trellis given this the complexity advantage of the dual approach for high rate codes can be generalized to a broader class of app decoding algorithms including suboptimum algorithms approximating the true app which may be more attractive in practical applications due to their reduced complexity moreover the local approach opens the way for mixed approaches where the operations of the app algorithm are not exclusively performed on the primal or dual trellis this is inevitable if the code does not possess a trellis consisting solely of group trellis sections as e g for certain terminated group or ring codes the complexity reduction offered by applying dualization is evaluated as examples we give a dual implementation of a suboptimum app decoding algorithm for tailbiting convolutional codes as well as dual implementations of app algorithms of the sliding window type moreover we evaluate their performance for decoding usual tailbiting codes or convolutional codes respectively as well as their performance as component decoders in iteratively decoded parallel concatenated schemes

computing components and projections of curves over finite fields this paper provides an algorithmic approach to some basic algebraic and combinatorial properties of algebraic curves over finite fields the number of points on a curve or a projection its number of absolutely irreducible components and the property of being exceptional

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

local irreducibility of tail biting trellises this paper investigates tail biting trellis realizations for linear block codes intrinsic trellis properties are used to characterize irreducibility on given intervals of the time axis it proves beneficial to always consider the trellis and its dual simultaneously a major role is played by trellis properties that amount to observability and controllability of trellis fragments of various lengths for fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to irreducibility for reducible trellises a constructive reduction procedure is presented the considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary atomic trellises

a general hankel norm approximation scheme for linear recursive filtering let x t and y t be stochastic processes which are weakly stationary and stationarily correlated we consider the problem of finding an approximate recursive low dimensional filter of x t based on the observation of the past of y t using hankel norm techniques several estimation problems have been investigated in the past using these techniques we present here a general framework which includes many of these approaches as special cases we also discuss some new applications the approximate filter so constructed allows for an a priori bound on the estimation error

confidence tubes for multiple quantile plots via empirical likelihood the nonparametric empirical likelihood approach is used to obtain simultaneous confidence tubes for multiple quantile plots based on k independent possibly right censored samples these tubes are asymptotically distribution free except when both k 3 and censoring is present pointwise versions of the confidence tubes however are asymptotically distribution free in all cases the various confidence tubes are valid under minimal conditions the proposed methods are applied in three real data examples 1 introduction the quantile quantile q q plot is a well known and attractive graphical method for comparing two distributions especially when confidence limits are added in this paper we develop q q plot methods for the comparison of two or more distributions from randomly censored data more specifically we consider the problem of finding simultaneous confidence tubes for multiple quantile plots for brevity multi q plots from k independent samples of possibly right censored survival times the multi q plot is defined to be the k dimensional curve q1 p qk p parameterized by 0 p 1 where qj is the quantile function of the jth distribution it specializes to the ordinary q q plot in the two sample case the comparison of quantile functions is particularly useful for the analysis of survival data in biomedical settings frail and strong individuals corresponding to low and high values of p often respond to different treatments in different ways so treatment effects can be hard to determine from comparison of mean or median survival times alone see for example doksum 1974 the approach developed here allows comparison of treatments simultaneously across all frailty levels our approach is based on the nonparametric empirical likelihood method this method was originally developed by thomas and grunkemeier 1975 and owen 1988 1990 as a way of improving upon wald type confidence regions there now exists a substantial literature on empirical likelihood indicating that it is widely viewed as a desirable and natural approach to

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

minimal and canonical rational generator matrices for convolutional codes a full rank k spl times n matrix g d over the rational functions f d generates a rate r k n convolutional code c g d is minimal if it can be realized with as few memory elements as any encoder for c and g d is canonical if it has a minimal realization in controller canonical form we show that g d is minimal if and only if for all rational input sequences u d the span of u d g d covers the span of u d alternatively g d is minimal if and only if g d is globally zero free or globally invertible we show that g d is canonical if and only if g d is minimal and also globally orthogonal in the valuation theoretic sense of monna 1970

the structure of the state representation of shift invariant controllable and observable group codes in this paper an investigation on the structure of the canonical trellis section of shift invariant l controllable and m observable group codes is carried out necessary and sufficient conditions for a set of group homomorphisms in order that they represent the trellis section of this class of codes are established

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

trellis group codes for the gaussian channel in this paper trellis group codes are introduced as an extension of slepian group codes to codes over sequence spaces a trellis group code is defined over r sup n as the orbit of a bi infinite seed sequence x sub 0 spl isin r sup n sup z under an infinite defining group of transformations this group of transformations is generated by a symbolic system the theory is developed by combining a nontrivial extension of the notion of an isometric labeling with results from the theory of symbolic dynamics over groups new results presented here include a useful characterization of uniform partitions and a symbolic dynamic classification of trellis group codes the theory is used to develop a class of rotationally invariant nonabelian trellis group codes for qam modulation it is also shown that the 8 state rotationally invariant trellis code designed by wei 1984 used in the v 32 and v 32 bis international modem standard belongs to this class

an efficient algorithm for constructing minimal trellises for codes over finite abelian groups we present an efficient algorithm for computing the minimal trellis for a group code over a finite abelian group given a generator matrix for the code we also show how to compute a succinct representation of the minimal trellis for such a code and present algorithms that use this information to compute efficiently local descriptions of the minimal trellis this extends the work of kschischang and sorokine see ibid vol 41 no 6 p 1926 37 1995 who treated the case of linear codes over fields an important application of our algorithms is to the construction of minimal trellises for lattices a key step in our work is handling codes over cyclic groups c sub p spl alpha where p is a prime such a code can be viewed as a module over the ring z sub p spl alpha because of the presence of zero divisors in the ring modules do not share the useful properties of vector spaces we get around this difficulty by restricting the notion of linear combination to a p linear combination and by introducing the notion of a p generator sequence which enjoys properties similar to those of a generator matrix for a vector space

system theory for system identification abstract prediction and control problems require for their solution mathematical models in the form of dynamic systems in system theory the approach to this problem is to investigate realization problems for system identification novel in this paper is the emphasis on the realization problem for observation based realizations and on the realization problem for control a major aspect of these realization problems is computation and complexity algebraic approaches to realization problems may lead to the distinction of subclasses of systems with required finiteness properties to decompositions and to abstractions for hierarchical systems

empirical likelihood for cumulative hazard ratio estimation with covariate adjustment summary in medical studies it is often of scientific interest to evaluate the treatment effect via the ratio of cumulative hazards especially when those hazards may be nonproportional to deal with nonproportionality in the cox regression model investigators usually assume that the treatment effect has some functional form however to do so may create a model misspecification problem because it is generally difficult to justify the specific parametric form chosen for the treatment effect in this article we employ empirical likelihood el to develop a nonparametric estimator of the cumulative hazard ratio with covariate adjustment under two nonproportional hazard models one that is stratified as well as a less restrictive framework involving group specific treatment adjustment the asymptotic properties of the el ratio statistic are derived in each situation and the finite sample properties of el based estimators are assessed via simulation studies simultaneous confidence bands for all values of the adjusted cumulative hazard ratio in a fixed interval of interest are also developed the proposed methods are illustrated using two different datasets concerning the survival experience of patients with non hodgkin s lymphoma or ovarian cancer

a note on the empirical likelihood confidence band for hazards ratio with covariate adjustment in medical studies comparing two treatments in the presence of censored data the stratified cox model is an important tool that has the ability to flexibly handle non proportional hazards while allowing parsimonious covariate adjustment in order to capture the cumulative treatment effect the ratio of the treatment specific cumulative baseline hazards is often used as a measure of the treatment effect pointwise and simultaneous confidence bands associated with the estimated ratio provide a global picture of how the treatment effect evolves over time recently dong and matthews 2012 biometrics 68 408 418 proposed to construct a pointwise confidence interval for the ratio using a plug in type empirical likelihood approach however their result on the limiting distribution of the empirical likelihood ratio is generally incorrect and the resulting confidence interval is asymptotically undercovering in this article we derive the correct limiting distribution for the likelihood ratio statistic we also present simulation studies to demonstrate the effectiveness of our approach

confidence bands for survival curves under the proportional hazards model summary in this paper we show how to construct simultaneous confidence bands for the subjectspecific survival curve under the cox proportional hazards model the idea is to approximate the distribution of the normalized cumulative hazard estimator by a zero mean gaussian process whose distribution can be easily generated through simulation numerical studies indicate that the proposed bands are appropriate for practical use a liver disease example is presented

on minimality of convolutional ring encoders convolutional codes are considered with code sequences modeled as semi infinite laurent series it is well known that a convolutional code c over a finite group g has a minimal trellis representation that can be derived from code sequences it is also well known that for the case that g is a finite field any polynomial encoder of c can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis in this paper we seek to extend this result to the finite ring case g bbzpr by introducing a so called ldquo p encoderrdquo we show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over bbzpr to produce a particular type of p encoder ldquominimal p encoderrdquo whose controller canonical realization is a minimal trellis with nonlinear features the minimum number of trellis states is then expressed as p gamma where gamma is the sum of the row degrees of the minimal p encoder in particular we show that any convolutional code over bbzpr admits a delay free p encoder which implies the novel result that delay freeness is not a property of the code but of the encoder just as in the field case we conjecture that a similar result holds with respect to catastrophicity i e any catastrophic convolutional code over bbzpr admits a noncatastrophic p encoder

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

information flow decomposition for network coding we propose a method to identify structural properties of multicast network configurations by decomposing networks into regions through which the same information flows this decomposition allows us to show that very different networks are equivalent from a coding point of view and offers a means to identify such equivalence classes it also allows us to divide the network coding problem into two almost independent tasks one of graph theory and the other of classical channel coding theory this approach to network coding enables us to derive the smallest code alphabet size sufficient to code any network configuration with two sources as a function of the number of receivers in the network but perhaps the most significant strength of our approach concerns future network coding practice namely we propose deterministic algorithms to specify the coding operations at network nodes without the knowledge of the overall network topology such decentralized designs facilitate the construction of codes that can easily accommodate future changes in the network e g addition of receivers and loss of links

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

robust input output stabilization on z for persistent signals robustness of feedback stabilization is studied for causal linear time invariant input output systems defined on the full time axis z the signal space setting used includes also certain unbounded signals this study has been possible due to the use of two operator plant models ay bu this avoids the intrinsic difficulties of unbounded causal input output plant mappings y gu on z the full time axis robust stabilization results obtained here are analogous to those on the semi infinite time axis n in the finite dimensional case however there are certain important differences between the full time axis results and the semi infinite time axis results for infinite dimensional systems even when two operator plant models are used

a structure theory for linear dynamic errors in variables models we deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise the case of uncorrelated noise components and the bounded noise case is considered if also the inputs may be contaminated by noise a number of additional complications in identification arise in particular the underlying system is not uniquely determined from the population second moments of the observations a description of classes of observationally equivalent systems is given continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

a behavioral approach to time varying linear systems part 1 general theory we develop a behavioral approach to linear time varying differential algebraic systems the analysis is almost everywhere in the sense that the statements hold on mathbb r setminus mathbb t where mathbb t is a discrete set controllability observability and autonomy are introduced and related to the behavior of the system classical results on the behavior of time invariant systems are studied in the context of time varying systems

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

estimating cumulative treatment effects in the presence of nonproportional hazards often in medical studies of time to an event the treatment effect is not constant over time in the context of cox regression modeling the most frequent solution is to apply a model that assumes the treatment effect is either piecewise constant or varies smoothly over time i e the cox nonproportional hazards model this approach has at least two major limitations first it is generally difficult to assess whether the parametric form chosen for the treatment effect is correct second in the presence of nonproportional hazards investigators are usually more interested in the cumulative than the instantaneous treatment effect e g determining if and when the survival functions cross therefore we propose an estimator for the aggregate treatment effect in the presence of nonproportional hazards our estimator is based on the treatment specific baseline cumulative hazards estimated under a stratified cox model no functional form for the nonproportionality need be assumed asymptotic properties of the proposed estimators are derived and the finite sample properties are assessed in simulation studies pointwise and simultaneous confidence bands of the estimator can be computed the proposed method is applied to data from a national organ failure registry

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

shifts on compact and discrete lie groups algebraic topological invariants and classification problems abstract this paper is devoted to an investigation of various dynamical concepts for group shift systems which are invariant by algebraic conjugacy i e topological conjugacy preserving the group structure the concept of controllability which is stronger than topological transitivity and the concept of limit dimension which is analogous to topological entropy are discussed at length particular attention is dedicated to the abelian case for which duality results are also established

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

constraint complexity of realizations of linear codes on arbitrary graphs a graphical realization of a linear code c consists of an assignment of the coordinates of c to the vertices of a graph along with a specification of linear state spaces and linear local constraint codes to be associated with the edges and vertices respectively of the graph the kappa complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes kappa complexity is a reasonable measure of the computational complexity of a sum product decoding algorithm specified by a graphical realization the main focus of this paper is on the following problem given a linear code c and a graph g how small can the kappa complexity of a realization of c on g be as useful tools for attacking this problem we introduce the vertex cut bound and the notion of vc treewidth for a graph which is closely related to the well known graph theoretic notion of treewidth using these tools we derive tight lower bounds on the kappa complexity of any realization of c on g our bounds enable us to conclude that good error correcting codes can have low complexity realizations only on graphs with large vc treewidth along the way we also prove the interesting result that the ratio of the kappa complexity of the best conventional trellis realization of a length n code c to the kappa complexity of the best cycle free realization of c grows at most logarithmically with code length n such a logarithmic growth rate is in fact achievable

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

codes on graphs observability controllability and local reducibility this paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility trimness and properness are dual properties of constraint codes a linear or group realization with a constraint code that is not both trim and proper is locally reducible a linear or group realization on a finite cycle free graph is minimal if and only if every local constraint code is trim and proper a realization is called observable if there is a one to one correspondence between codewords and configurations and controllable if it has independent constraints a linear or group realization is observable if and only if its dual is controllable a simple counting test for controllability is given an unobservable or uncontrollable realization is locally reducible parity check realizations are controllable if and only if they have independent parity checks in an uncontrollable tail biting trellis realization the behavior partitions into disconnected sub behaviors but this property does not hold for nontrellis realizations on a general graph the support of an unobservable configuration is a generalized cycle