almost sure stabilizability and riccati s equation of linear systems with random parameters linear systems with stationary time varying parameters are considered the notions of almost sure stabilizability and almost sure detectability are explored under these properties the asymptotic behavior of the associated riccati equation is described in particular the stability of the kalman filter and the convergence of linear observers are proved

the cayley hilbert metric and positive operators abstract the cayley hilbert metric is defined for a real banach space containing a closed cone by restricting the domain of a particular type of positive nonlinear operator the banach contraction mapping theorem is used to prove the existence of a unique fixed point of the operator with explicit upper and lower bounds applications to quasilinear elliptic partial differential equations and to matrix theory are considered

contractive mappings and existence of cycle times for a monotone and homogeneous function abstract given a monotone and homogeneous self mapping f of the n dimensional positive cone a family of contractive mappings is used to define an equivalence relation in the index set as well as a total order among the equivalence classes then it is shown i that the cycle times are well defined at each index belonging to the maximal and minimal classes and ii that the cycle times of f exist at every index whenever a weak convexity condition is satisfied

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

a study of problems encountered in granger causality analysis from a neuroscience perspective granger causality methods were developed to analyze the flow of information between time series these methods have become more widely applied in neuroscience frequency domain causality measures such as those of geweke as well as multivariate methods have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings despite its widespread application in many fields there are ongoing concerns regarding the applicability of granger causality methods in neuroscience when are these methods appropriate how reliably do they recover the system structure underlying the observed data what do frequency domain causality measures tell us about the functional properties of oscillatory neural systems in this paper we analyze fundamental properties of granger geweke gg causality both computational and conceptual specifically we show that i gg causality estimates can be either severely biased or of high variance both leading to spurious results ii even if estimated correctly gg causality estimates alone are not interpretable without examining the component behaviors of the system model and iii gg causality ignores critical components of a system s dynamics based on this analysis we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations leading to highly counterintuitive and potentially misleading results through the analysis of these problems we provide important conceptual clarification of gg causality with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole

granger geweke causality estimation and interpretation abstract in a recent pnas article1 stokes and purdon performed numerical simulations to argue that granger geweke causality ggc estimation is severely biased or of high variance and ggc application to neuroscience is problematic because the ggc measure is independent of receiver dynamics here we use the same simulation examples to show that ggc measures when properly estimated either via the spectral factorization enabled nonparametric approach or the var model based parametric approach do not have the claimed bias and high variance problems further the receiver independence property of ggc does not present a problem for neuroscience applications when the nature and context of experimental measurements are taken into consideration ggc along with other spectral quantities yield neurophysiologically interpretable results

poisson equations associated with a homogeneous and monotone function necessary and sufficient conditions for a solution in a weakly convex case abstract given a monotone and homogeneous self mapping f of the n dimensional positive cone a communication matrix m f is introduced and a family f of functions is obtained by multiplying each component of f by arbitrary positive numbers assuming that f satisfies a weak form of convexity a necessary and sufficient criterion on the structure of m f is given so that each function in f has a nonlinear eigenvalue an alternative necessary and sufficient condition in terms of the recession function of f is also provided

the short cut test abstract the short cut test detects existence and uniqueness of laplacians on finitely ramified graph directed fractals previous results by sabot nussbaum and the author are improved and extended it opens up the way for further studies because it combines well established spectral dynamical and analytic techniques its algorithmic and recursive structure is designed to provide computable and flexible criteria

kalman filtering with intermittent observations weak convergence to a stationary distribution the paper studies the asymptotic behavior of discrete time random riccati equations rre arising in kalman filtering when the arrival of the observations is described by a bernoulli independent and identically distributed i i d process we model the rre as an order preserving strongly sublinear random dynamical system rds under a sufficient condition stochastic boundedness and using a limit set dichotomy result for order preserving strongly sublinear rds we establish the asymptotic properties of the rre the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution whose support exhibits fractal behavior for stabilizable and detectable systems stochastic boundedness and hence weak convergence holds for any nonzero observation packet arrival probability and in particular we can establish weak convergence at operating arrival rates well below the critical probability for mean stability the resulting invariant measure in that situation does not possess a first moment we apply the weak feller property of the markov process governing the rre to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points which in general is not dense in the set of positive semi definite matrices we use the explicit characterization of the support of the invariant distribution and the almost sure a s ergodicity of the sample paths to easily compute statistics of the invariant distribution a one dimensional example illustrates that the support is a fractured subset of the non negative reals with self similarity properties

multivariate granger causality and generalized variance granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables a shortcoming of the standard framework for granger causality is that it only allows for examination of interactions between single univariate variables within a system perhaps conditioned on other variables however interactions do not necessarily take place between single variables but may occur among groups or ensembles of variables in this study we establish a principled framework for granger causality in the context of causal interactions among two or more multivariate sets of variables building on geweke s seminal 1982 work we offer additional justifications for one particular form of multivariate granger causality based on the generalized variances of residual errors taken together our results support a comprehensive and theoretically consistent extension of granger causality to the multivariate case treated individually they highlight several specific advantages of the generalized variance measure which we illustrate using applications in neuroscience as an example we further show how the measure can be used to define partial granger causality in the multivariate context and we also motivate reformulations of causal density and granger autonomy our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems neural and otherwise

state space analysis of granger geweke causality measures with application to fmri the recent interest in the dynamics of networks and the advent across a range of applications of measuring modalities that operate on different temporal scales have put the spotlight on some significant gaps in the theory of multivariate time series fundamental to the description of network dynamics is the direction of interaction between nodes accompanied by a measure of the strength of such interactions granger causality and its associated frequency domain strength measures gems due to geweke provide a framework for the formulation and analysis of these issues in pursuing this setup three significant unresolved issues emerge first computing gems involves computing submodels of vector time series models for which reliable methods do not exist second the impact of filtering on gems has never been definitively established third the impact of downsampling on gems has never been established in this work using state space methods we resolve all these issues and illustrate the results with some simulations our analysis is motivated by some problems in fmri brain imaging to which we apply it but it is of general applicability

measures of conditional linear dependence and feedback between time series abstract measures of linear dependence and feedback for two multiple time series conditional on a third are defined the measure of conditional linear dependence is the sum of linear feedback from the first to the second conditional on the third linear feedback from the second to the first conditional on the third and instantaneous linear feedback between the first and second series conditional on the third the measures are non negative and may be expressed in terms of measures of unconditional feedback between various combinations of the three series the measures of conditional linear feedback can be additively decomposed by frequency estimates of these measures are straightforward to compute and their distribution can be routinely approximated by bootstrap methods an empirical example involving real output money and interest rates is presented

backscatter communications over correlated nakagami m fading channels in this paper we consider a radio frequency identification rfid backscatter system with two scenarios single input multiple output simo and multiple input single output miso we investigate the impact of channel correlation between the forward and backscatter links on the symbol error rate ser and we compare the performance of simo and miso rfid systems with maximum ratio combining reception over arbitrarily correlated nakagami m fading channels in order to gain an insight for the system design closed form expressions for the asymptotic ser and an upper bound are derived for m ary phase shift keying and quadrature amplitude modulation the diversity order in the fully correlated cases revealed from these expressions is half of that in the partially correlated and uncorrelated cases finally simulations are performed to validate the theoretical analysis

kalman filtering with random coefficients and contractions the riccati transformation of linear filtering control theory is shown to be a contraction on the space of positive symmetric matrices this is used to describe the asymptotic behavior of the filter for systems with stochastic stationary parameters

concave perron frobenius theory and applications many important results of classical perron frobenius theory can be extended from linear selfmappings of the standard cone in finite dimensional real space to concave selfmappings of this cone this is in particular true for minima of linear mappings albeit the spectrum of these special concave mappings is more intricate than that for linear mappings as classical perron frobenius theory has numerous applications there are many new applications for its concave extension 1991 mathematical subject classification primary 39a11 15a48 secondary 47h07 47h12

on estimation error outage for scalar gauss markov signals sent over fading channels measurements of a scalar linear gauss markov process are sent over a fading channel the fading channel is modeled as independent and identically distributed random variables with known realization at the receiver the optimal estimator at the receiver is the kalman filter in contrast to the classical kalman filter theory given a random channel the kalman gain and the error covariance become random then the probability distribution function of expected estimation error and its outage probability can be chosen for estimation quality assessment in this paper and in order to get the estimation error outage we provide means to characterize the stationary probability density function of the random expected estimation error furthermore and for the particular case of the i i d rayleigh fading channels upper and lower bounds for the outage probability are derived which provide insight and simpler means for design purposes we also show that the bounds are tight for the high snr regime and that the outage probability decreases linearly with the inverse of the average channel snr

from max plus algebra to nonexpansive mappings a nonlinear theory for discrete event systems discrete event systems provide a useful abstraction for modelling a wide variety of systems digital circuits communication networks manufacturing plants etc their dynamics stability equilibrium states cyclical behaviour asymptotic average delays are of vital importance to system designers however in marked contrast to continuous dynamical systems there has been little systematic mathematical theory that designers can draw upon in this paper we survey the development of such a theory based on the dynamics of maps which are nonexpansive in the l norm this has its origins in linear algebra over the max plus semiring but extends to a nonlinear theory that encompasses a variety of problems arising in other mathematical disciplines we concentrate on the mathematical aspects and set out several open problems

power allocation for outage minimization in state estimation over fading channels this paper studies the outage probability minimization problem for state estimation of linear dynamical systems using multiple sensors where an estimation outage is defined as an event when the state estimation error exceeds a pre determined threshold the sensors amplify and forward their measurements using uncoded analog transmission to a remote fusion center over wireless fading channels for stable systems the resulting infinite horizon problem can be formulated as a constrained average cost markov decision process mdp control problem a suboptimal power allocation that is less computationally intensive is proposed and numerical results demonstrate very close performance to the power allocation obtained from the solution of the mdp based average cost optimality equation motivated by practical considerations assuming that sensors can transmit with only a finite number of power levels optimization of the values of these levels is also considered using a stochastic approximation technique in the case of unstable systems a finite horizon formulation of the estimation outage minization problem is presented and solved an extension to the problem of minimization of the expected error covariance is also studied

distortion outage analysis for joint space time coding and kalman filtering in this paper we consider the scenario of transmitting a first order gauss markov vector signal over a multi input multi output mimo rayleigh nonfrequency selective fading channel the signal is reconstructed at the receiver side with the help of a kalman filter in order to minimize the mean squared error orthogonal space time codes are utilized in order to increase the quality of estimation and mitigate the destructing effects of the fading channel as a criterion for estimation quality assessment we use the distortion outage probability we first obtain upper and lower bounds for the outage probability as a function of system parameters we then perform high snr analysis of the bounds through which we prove the achievability of the maximum diversity order for a n times k mimo fading channel in addition we obtain upper and lower bounds for the coding gain of the distortion outage probability in the high snr regime and outline the relation between system parameters and the coding gain

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

entropy minimization dad problems and doubly stochastic kernels abstract the classical dad problem asks for a square matrix a with nonnegative entries when it is possible to find positive diagonal matrices d 1 and d 2 with d 1 ad 2 doubly stochastic we consider various continuous and measurable generalizations of this problem through a fusion of variational and fixed point techniques we obtain strong analogues of the classical results our extensions appear inaccessible by either technique separately

the whitney embedding theorem for tropical torsion modules classification of tropical modules abstract we prove here a tropical version of the well known whitney embedding theorem 32 stating that a smooth connected m dimensional compact differential manifold can be embedded into r 2 m 1 the tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module r p where p equals to 2 for m 2 and 3 p m m 1 otherwise is the number of rows supporting the torsion when the generators are given by the independent columns of a matrix of size n m as a corollary we get that tropical m dimensional torsion modules are classified by a m 1 m m 1 1 parameter family

the perron frobenius theorem for homogeneous monotone functions if a is a nonnegative matrix whose associated directed graph is strongly connected the perron frobenius theorem asserts that a has an eigenvector in the positive cone r n we associate a directed graph to any homogeneous monotone function f r n r n and show that if the graph is strongly connected then f has a nonlinear eigenvector in r n several results in the literature emerge as corollaries our methods show that the perron frobenius theorem is really about the boundedness of invariant subsets in the hilbert projective metric they lead to further existence results and open problems

measurement of linear dependence and feedback between multiple time series abstract measures of linear dependence and feedback for multiple time series are defined the measure of linear dependence is the sum of the measure of linear feedback from the first series to the second linear feedback from the second to the first and instantaneous linear feedback the measures are nonnegative and zero only when feedback causality of the relevant type is absent the measures of linear feedback from one series to another can be additively decomposed by frequency a readily usable theory of inference for all of these measures and their decompositions is described the computations involved are modest

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

nonlinear models of diffusion on a finite space the basic convergence theorems for finite state markov chains are extended to the nonlinear case an operatort inl1 of a finite space with counting measure is called nonexpansive if tf tg 1 f g 1 holds for allf g it is shown that for anyf there exists an integer p 1 such thattpnf converges sufficient conditions forp 1 are given in the case of continuous parameter nonexpansive semigrous tt t 0 ttf converges fort

computing the smallest fixed point of order preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs the problem of computing the smallest fixed point of an order preserving map arises in the study of zero sum positive stochastic games it also arises in static analysis of programs by abstract interpretation in this context the discount rate may be negative we characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential we apply this characterization to design a policy iteration algorithm which applies to the case of finite state and action spaces the algorithm returns a locally minimal fixed point which turns out to be globally minimal when the discount rate is nonnegative

spectral theorem for convex monotone homogeneous maps and ergodic control we consider convex maps f rn rn that are monotone i e that preserve the product ordering of rn and nonexpansive for the sup norm this includes convex monotone maps that are additively homogeneous i e that commute with the addition of constants we show that the fixed point set of f when it is nonempty is isomorphic to a convex inf subsemilattice of rn whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f this yields in particular an uniqueness result for the bias vector of ergodic control problems this generalizes results obtained previously by lanery romanovsky and schweitzer and federgruen for ergodic control problems with finite state and action spaces which correspond to the special case of piecewise affine maps f we also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters

uniqueness of the fixed point of nonexpansive semidifferentiable maps we consider semidifferentiable possibly nonsmooth maps acting on a subset of a banach space that are nonexpansive either in the norm of the space or in the hilbert s or thompson s metric inherited from a convex cone we show that the global uniqueness of the fixed point of the map as well as the geometric convergence of every orbit to this fixed point can be inferred from the semidifferential of the map at this point in particular we show that the geometric convergence rate of the orbits to the fixed point can be bounded in terms of bonsall s non linear spectral radius of the semidifferential we derive similar results concerning the uniqueness of the eigenline and the geometric convergence of the orbits to it in the case of positively homogeneous maps acting on the interior of a cone or of additively homogeneous maps acting on an am space with unit this is motivated in particular by the analysis of dynamic programming operators shapley operators of zero sum stochastic games

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

n ast nakagami a novel stochastic model for cascaded fading channels a generic and novel distribution referred to as nakagami constructed as the product of n statistically independent but not necessarily identically distributed nakagami m random variables rvs is introduced and analyzed the proposed distribution turns out to be a very convenient tool for modelling cascaded nakagami m fading channels and analyzing the performance of digital communications systems operating over such channels the moments generating probability density cumulative distribution and moments functions of the n nakagami distribution are developed in closed form using the meijer s g function using these formulas generic closed form expressions for the outage probability amount of fading and average error probabilities for several binary and multilevel modulation signals of digital communication systems operating over the n nakagami fading and the additive white gaussian noise channel are presented complementary numerical and computer simulation performance evaluation results verify the correctness of the proposed formulation the suitability of the n nakagami fading distribution to approximate the lognormal distribution is also being investigated using kolmogorov smirnov tests the rate of convergence of the central limit theorem as pertaining to the multiplication of nakagami m rvs is quantified

equivalence of communication and projective boundedness properties for monotone and homogeneous functions abstract this work concerns the eigenvalue problem for a monotone and homogeneous self mapping f of a finite dimensional positive cone a communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f a property that yields the existence of a nonlinear eigenvalue using the idea of dual function a similar result is obtained for lower eigenspaces

a classification of invariant distributions and convergence of imprecise markov chains abstract we analyse the structure of imprecise markov chains and study their convergence by means of accessibility relations we first identify the sets of states so called minimal permanent classes that are the minimal sets capable of containing and preserving the whole probability mass of the chain these classes generalise the essential classes known from the classical theory we then define a class of extremal imprecise invariant distributions and show that they are uniquely determined by the values of the upper probability on minimal permanent classes moreover we give conditions for unique convergence to these extremal invariant distributions

sum and difference of two squared correlated nakagami variates in connection with the mckay distribution general formulas for the probability density function of the sum and the difference of two correlated not necessarily identically distributed squared nakagami variates or equivalently gamma variates are derived these expressions are shown to be in the form of the mckay bessel function distributions in addition formulas for the moments of these distributions in terms of the gauss hypergeometric function are provided an application of these new results relevant to the calculation of outage probability in the presence of self interference is discussed

performance of mrc diversity systems for the detection of signals with nakagami fading bit error rate ber is analyzed theoretically for diversity reception in nakagami fading environment using an m branch maximal ratio combiner mrc coherent and incoherent reception of frequency shift keying fsk are considered using the multiple branch diversity system for both identical and different diversity branch fading parameters the effect of correlation is also considered for the dual diversity case the results are extended to include coherent phase shift keying cpsk and differential phase shift keying dpsk

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

granger causality for state space models granger causality has long been a prominent method for inferring causal interactions between stochastic variables for a broad range of complex physical systems however it has been recognized that a moving average ma component in the data presents a serious confound to granger causal analysis as routinely performed via autoregressive ar modeling we solve this problem by demonstrating that granger causality may be calculated simply and efficiently from the parameters of a state space ss model since ss models are equivalent to autoregressive moving average models granger causality estimated in this fashion is not degraded by the presence of a ma component this is of particular significance when the data has been filtered downsampled observed with noise or is a subprocess of a higher dimensional process since all of these operations commonplace in application domains as diverse as climate science econometrics and the neurosciences induce a ma component we show how granger causality conditional and unconditional in both time and frequency domains may be calculated directly from ss model parameters via solution of a discrete algebraic riccati equation numerical simulations demonstrate that granger causality estimators thus derived have greater statistical power and smaller bias than ar estimators we also discuss how the ss approach facilitates relaxation of the assumptions of linearity stationarity and homoscedasticity underlying current ar methods thus opening up potentially significant new areas of research in granger causal analysis

convexity and log convexity for the spectral radius abstract the starting point of this paper is a theorem by j f c kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix a related result of j cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements the first section of this paper gives a new unified proof of these results and also analyzes exactly when one has strict convexity the second section gives some very simple proofs of results of friedland and karlin concerning min max characterizations of the spectral radius of nonnegative matrices these arguments also yield as will be shown in another paper min max characterizations of the principal eigenvalue of second order elliptic boundary value problems on bounded domains the third section considers the cone k of nonnegative vectors in r n and continuous maps f k k which are homogeneous of degree one and preserve the partial order induced by k the cone spectral radius of such maps is defined and a direct generalization of kingman s theorem to a subclass of such nonlinear maps is given the final section of this paper treats a problem that arises in population biology if k 0 denotes the interior of k and f is as above when can one say that f has a unique eigenvector to within normalization in k 0 a subtle point to be noted is that f may have other eigenvectors in the boundary of k if u k 0 is an eigenvector of f u 1 and g x f x f x when can one say that for any x k 0 g p x the p th iterate of g acting on x converges geometrically to u the fourth section provides answers to these questions that are adequate for many of the population biology problems