quantum markov states on cayley trees abstract it is known that any locally faithful quantum markov state qms on one dimensional setting can be considered as a gibbs state associated with hamiltonian with commuting nearest neighbor interactions in our previous results we have investigated quantum markov states qms associated with ising type models with competing interactions which are expected to be qms but up to now there is no any characterization of qms over trees we notice that these qms do not have one dimensional analogues hence results of related to one dimensional qms are not applicable therefore the main aim of the present paper is to describe of qms over cayley trees namely we prove that any qms associated with localized conditional expectations can be realized as integral of product states w t r a gibbs measure moreover it is established that any locally faithful qms associated with localized conditional expectations can be considered as a gibbs state corresponding to hamiltonians on the cayley tree with commuting competing interactions

a constructive description of ground states and gibbs measures for ising model with two step interactions on cayley tree we consider the ising model with competing two step interactions and spin values 1 on a cayley tree of order k 1 we constructively describe ground states and verify the peierls condition for the model we define notion of a contour for the model on the cayley tree using a contour argument we show the existence of two different gibbs measures

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

integrability and algebraic entropy of k periodic non autonomous lyness recurrences abstract this work deals with non autonomous lyness type recurrences of the form x n 2 a n x n 1 x n where a n n is a k periodic sequence of complex numbers with minimal period k we treat such non autonomous recurrences via the autonomous dynamical system generated by the birational mapping f a k f a k 1 f a 1 where f a is defined by f a x y y a y x for the cases k 1 2 3 6 the corresponding mappings have a rational first integral by calculating the dynamical degree we show that for k 4 and for k 5 generically the dynamical system is no longer rationally integrable we also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters are k 1 2 3 6

adaptive subspace detectors we use the theory of generalized likelihood ratio tests glrts to adapt the matched subspace detectors msds of scharf 1991 and of scharf and frielander 1994 to unknown noise covariance matrices in so doing we produce adaptive msds that may be applied to signal detection for radar sonar and data communication we call the resulting detectors adaptive subspace detectors asds these include kelly s 1987 glrt and the adaptive cosine estimator ace of kaurt and scharh see ibid vol 47 p 2538 41 1999 and of scharf and mcwhorter see proc 30th asilomar conf signals syst comput pacific grove ca 1996 for scenarios in which the scaling of the test data may deviate from that of the training data we then present a unified analysis of the statistical behavior of the entire class of asds obtaining statistically identical decompositions in which each asd is simply decomposed into the nonadaptive matched filter the nonadaptive cosine or t statistic and three other statistically independent random variables that account for the performance degrading effects of limited training data

a contour method on cayley trees we consider finite range lattice models on cayley trees with two basic properties the existence of only a finite number of ground states and with a peierls type condition we define the notion of a contour for the model on the cayley tree by a contour argument we show the existence of s different where s is the number of ground states gibbs measures

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

on gibbs measures of models with competing ternary and binary interactions and corresponding von neumann algebras in the present paper the ising model with competing binary j and binary j1 interactions with spin values 1 on a cayley tree of order 2 is considered the structure of gibbs measures for the model is studied we completely describe the set of all periodic gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the cayley tree types of von neumann algebras generated by gns representation associated with diagonal states corresponding to the translation invariant gibbs measures are determined it is proved that the factors associated with minimal and maximal gibbs states are isomorphic and if they are of type iii then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively some concrete examples of factors are given too

exact solution of an ising model with competing interactions on a cayley tree the exact solution of an ising model with competing restricted interactions on the cayley tree and in the absence of an external field is presented a critical curve is defined where it is possible to get phase transitions above it and a single gibbs state is obtained elsewhere

bounds on maximum likelihood ratio part ii application to antenna array detection estimation with imperfect wavefront coherence the multiple hypothesis testing problem of the detection estimation of an unknown number of independent gaussian point sources is adequately addressed by likelihood ratio lr maximization over the set of admissible covariance matrix models we introduce nonasymptotic lower and upper bounds for the maximum lr since lr optimization is generally a nonconvex multiextremal problem any practical solution could now be tested against these bounds enabling a high probability of recognizing nonoptimal solutions we demonstrate that in many applications the lower bound is quite tight with approximate maximum likelihood ml techniques often unable to approach this bound the introduced lower bound analysis is shown to be very efficient in determining whether or not performance breakdown has occurred for subspace based direction of arrival doa estimation techniques we also demonstrate that by proper lr maximization we can extend the range of signal to noise ratio snr values and or number of data samples wherein accurate parameter estimates are produced yet when the snr and or sample size falls below a certain limit for a given scenario we show that ml estimation suffers from a discontinuity in the parameter estimates a phenomenon that cannot be eliminated within the ml paradigm

regularized covariance matrix estimation in complex elliptically symmetric distributions using the expected likelihood approach part 2 in the first part of these two papers we extended the expected likelihood approach originally developed in the gaussian case to the broader class of complex elliptically symmetric ces distribut

ising models on the lobachevsky plane we consider the ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane for large values of the inverse temperature we construct an uncountable number of mutually singular gibbs states

on contour arguments for the three state potts model with competing interactions on a semi infinite cayley tree we consider the potts model with competing two step interactions and spin values 1 2 3 on a semi infinite cayley tree of order 2 we describe ground states and verify the peierls condition for the model using a contour argument we show the existence of three different gibbs measures associated with translation invariant ground states

recursive estimation of the covariance matrix of a compound gaussian process and its application to adaptive cfar detection adaptive detection of signals embedded in gaussian or non gaussian noise is a problem of primary concern among radar engineers we propose a recursive algorithm to estimate the structure of the covariance matrix of either a set of gaussian vectors that share the spectral properties up to a multiplicative factor or a set of spherically invariant random vectors sirvs with the same covariance matrix and possibly correlated texture components we also assess the performance of an adaptive implementation of the normalized matched filter nmf relying on the newly introduced estimate in the presence of compound gaussian clutter dominated disturbance in particular it is shown that a proper initialization of the recursive procedure leads to an adaptive nmf with the constant false alarm rate cfar property and that it is very effective to operate in heterogeneous environments of relevant practical interest

phase diagrams of ising models on husimi trees i pure multisite interaction systems lattice spin systems with multisite interactions have rich and interesting phase diagrams we present some results for such systems involving ising spins 1 using a generalization of the bethe lattice approximation first we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems second a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made we exhibit a full bifurcation diagram chaos period 3 windows etc for the magnetization of the base site of this system

the cfar adaptive subspace detector is a scale invariant glrt the constant false alarm rate cfar matched subspace detector cfar msd is the uniformly most powerful invariant test and the generalized likelihood ratio test glrt for detecting a target signal in noise whose covariance structure is known but whose level is unknown previously the cfar adaptive subspace detector cfar asd or adaptive coherence estimator ace was proposed for detecting a target signal in noise whose covariance structure and level are both unknown and whose covariance structure is estimated with a sample covariance matrix based on training data we show here that the cfar asd is glrt when the test measurement is not constrained to have the same noise level as the training data as a consequence this glrt is invariant to a more general scaling condition on the test and training data than the well known glrt of kelly 1986

regularized covariance matrix estimation in complex elliptically symmetric distributions using the expected likelihood approach part 2 the under sampled case in abramovich bounds on maximum likelihood ratio part i application to antenna array detection estimation with perfect wavefront coherence ieee trans signal process vol 52 pp 1524 1536 june 2004 it was demonstrated for multivariate complex gaussian distribution that the probability density function p d f of the likelihood ratio lr for the unknown actual covariance matrix r0 does not depend on this matrix and is fully specified by the matrix dimension m and the number of independent training samples t this invariance property hence enables one to compare the lr of any derived covariance matrix estimate against this p d f and eventually get an estimate that is statistically as likely as r0 this expected likelihood quality assessment allowed significant improvement of music doa estimation performance in the so called threshold area and for diagonal loading and tvar model order selection in adaptive detectors recently the so called complex elliptically symmetric ces distributions have been introduced for description of highly in homogeneous clutter returns the aim of this series of two papers is to extend the el approach to this class of ces distributions as well as to a particularly important derivative namely the complex angular central distribution acg for both cases we demonstrate a similar invariance property for the lr associated with the true scatter matrix 0 furthermore we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology this first part is devoted to the conventional scenario t m while part ii deals with the undersampled scenario t m

statistical analysis for the angular central gaussian distribution on the sphere summary the angular central gaussian distribution is an alternative to the bingham distribution for modeling antipodal symmetric directional data in this paper the statistical theory for the angular central gaussian model is presented some topics treated are maximum likelihood estimation of the parameters testing for uniformity and circularity and principal components analysis comparisons to methods based upon the sample second moments are made via an example

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

non bernoullian quantum k systems we construct an uncountable family of pairwise non conjugate non bernoullian k systems of type iii1 with the same finite cnt entropy we also investigate clustering properties of multiple channel entropies for strong asymptotically abelian systems of type ii and iii we prove that a wide enough class of systems has the k property in particular such systems as the space translations of a one dimensional quantum lattice with the gibbs states of araki the space translations of the ccr algebra and the even part of the car algebra with the quasi free states of park and shin non commutative markov shifts in the accardi sense are entropic k systems

the effros marechal topology in the space of von neumann algebras ii we continue our study 9 of the effros marechal topology on the space vn h of all von neumann algebras acting on a fixed separable hilbert space h in particular we prove that factors of each of the types ii1 ii and for fixed 0 1 iii form a dense subset of vn h moreover the density of type i factors or equivalently of injective factors in vn h is proved to be equivalent to famous conjectures by a connes and e kirchberg

phase transitions for quantum markov chains associated with ising type models on a cayley tree the main aim of the present paper is to prove the existence of a phase transition in quantum markov chain qmc scheme for the ising type models on a cayley tree note that this kind of models do not have one dimensional analogous i e the considered model persists only on trees in this paper we provide a more general construction of forward qmc in that construction a qmc is defined as a weak limit of finite volume states with boundary conditions i e qmc depends on the boundary conditions our main result states the existence of a phase transition for the ising model with competing interactions on a cayley tree of order two by the phase transition we mean the existence of two distinct qmc which are not quasi equivalent and their supports do not overlap we also study some algebraic property of the disordered phase of the model which is a new phenomena even in a classical setting

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

unified framework to regularized covariance estimation in scaled gaussian models we consider regularized covariance estimation in scaled gaussian settings e g elliptical distributions compound gaussian processes and spherically invariant random vectors asymptotically in the number of samples the classical maximum likelihood ml estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex we propose a unified framework for regularizing this estimate in order to improve its finite sample performance our approach is based on the discovery of hidden convexity within the ml objective we begin by restricting the attention to diagonal covariance matrices using a simple change of variables we transform the problem into a convex optimization that can be efficiently solved we then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices we regularize the problem using appropriately convex penalties these allow for shrinkage towards the identity matrix shrinkage towards a diagonal matrix shrinkage towards a given positive definite matrix and regularization of the condition number we demonstrate the advantages of these estimators using numerical simulations

robust shrinkage estimation of high dimensional covariance matrices we address high dimensional covariance estimation for elliptical distributed samples which are also known as spherically invariant random vectors sirv or compound gaussian processes specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples large p small n we start from a classical robust covariance estimator tyler 1987 which is distribution free within the family of elliptical distribution but inapplicable when n p using a shrinkage coefficient we regularize tyler s fixed point iterations we prove that for all n and p the proposed fixed point iterations converge to a unique limit regardless of the initial condition next we propose a simple closed form and data dependent choice for the shrinkage coefficient which is based on a minimum mean squared error framework simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy tailed samples finally as a real world application we demonstrate the performance of the proposed technique in the context of activity intrusion detection using a wireless sensor network

notes on the non autonomous lyness equation abstract we prove that the solutions of the non autonomous lyness equation x n 1 x n b n x n 1 b n 0 x 0 0 x 1 0 are bounded away from zero and infinity if the sequence b n is monotonic we also give an example to show that the solutions need not be bounded even when the sequence b n attains only two values