weak singularity and absence of metastability in random ising ferromagnets in the above models there exists a weak singularity of the magnetisation in addition to its jump on the first order transition line it corresponds to the griffiths 1969 singularity above the temperature of the phase transition analytic continuation from positive to negative magnetic fields yielding a metastable state is impossible

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field

adaptive estimation of eigensubspace in a recent work we recast the problem of estimating the minimum eigenvector eigenvector corresponding to the minimum eigenvalue of a symmetric positive definite matrix into a neural network framework we now extend this work using an inflation technique to estimate all or some of the orthogonal eigenvectors of the given matrix based on these results we form a cost function for the finite data case and derive a newton based adaptive algorithm the inflation technique leads to a highly modular and parallel structure for implementation the computational requirement of the algorithm is o n sup 2 n being the size of the covariance matrix we also present a rigorous convergence analysis of this adaptive algorithm the algorithm is locally convergent and the undesired stationary points are unstable computer simulation results are provided to compare its performance with that of two adaptive subspace estimation methods proposed by yang and kaveh 1988 and an improved version of one of them for stationary and nonstationary signal scenarios the results show that the proposed approach performs identically to one of them and is significantly superior to the remaining two

improved perturbation expansion for disordered systems beating griffiths singularities we introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge due to the presence of griffiths type singularities it is organized inductively over an infinite sequence of increasing distance scales in each induction step one redefines what is means by the unperturbed system a procedure somewhat reminiscent of k a m theory our techniques may be useful in dealing with the so called large field problem in real space renormalization group schemes

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

some numerical results on the block spin transformation for the 2d ising model at the critical point we study the block spin transformation for the 2d ising model at the critical temperaturetc we consider the model with the constraint that the total spin in each block is zero an old argument by cassandro and gallavotti strongly supports the gibbsianness of the transformed measure provided that such model has a critical temperaturet c lower thantc after describing a possible rigorous approach to the problem we present numerical evidence that indeedt c tc and study the dobrushin shlosman uniqueness condition

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

on the inflation method in adaptive noise subspace estimator it is shown that yang and kaveh s 1987 1988 inflation method is not necessary to accomplish an adaptive noise subspace estimator without the inflation method an alternative noise subspace estimator involving less computation and simpler parallel implementation is developed we also prove that if the initial noise subspace basis consists of a set of mutually orthonormal vectors the proposed estimator has a better performance than that with the inflation method both in the mean sense and in the mean square sense computer simulations are provided to substantiate the superiority of the estimator without the inflation method

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

neural network learning algorithms for tracking minor subspace in high dimensional data stream a novel random gradient based algorithm is developed for online tracking the minor component mc associated with the smallest eigenvalue of the autocorrelation matrix of the input vector sequence the five available learning algorithms for tracking one mc are extended to those for tracking multiple mcs or the minor subspace ms in order to overcome the dynamical divergence properties of some available random gradient based algorithms we propose a modification of the oja type algorithms called ojam which can work satisfactorily the averaging differential equation and the energy function associated with the ojam are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set the corresponding energy or lyapunov functions exhibit a unique global minimum attained if and only if its state matrices span the ms of the autocorrelation matrix of a vector data stream the other stationary points are saddle unstable points the globally convergence of ojam is also studied the ojam provides an efficient online learning for tracking the ms it can track an orthonormal basis of the ms while the other five available algorithms cannot track any orthonormal basis of the ms the performances of the relative algorithms are shown via computer simulations

cluster expansion for d dimensional lattice systems and finite volume factorization properties we consider classical lattice systems with finite range interactions ind dimensions by means of a block decimation procedure we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite volume partition functions holds in this way we extend a result of olivieri

non gibbsian states and their gibbs description the driving principle behind this paper is the following thesis every physically reasonable random field has to be a gibbs random field in this paper the so called non gibbsian random fields are considered the usual definition of the gibbs field is generalized in such a way so as to include some of the discovered non gibbsian fields the new definition is then used to show that the projection of the two dimensional ising model onto the one dimensional sublattice 1 falls into the class of the generalized gibbs fields

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

a minor subspace analysis algorithm this paper proposes a learning algorithm which extracts adaptively the minor subspace spanned by the eigenvectors corresponding to the smallest eigenvalues of the autocorrelation matrix of an input signal we show both analytically and by simulation results that the weight vectors provided by the proposed algorithm are guaranteed to converge to the minor subspace of the input signal for wider applications we also present the complex valued version of the proposed minor subspace analysis algorithm

perturbative analysis of disordered ising models close to criticality we consider a two dimensional ising model with random i i d nearest neighbor ferromagnetic couplings and no external magnetic field we show that if the probability of supercritical couplings is small enough the system admits a convergent cluster expansion with probability one the associated polymers are defined on a sequence of increasing scales in particular the convergence of the above expansion is compatible with the infinite differentiability of the free energy but does not imply its analyticity the basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

a note on finite time blowup for dissipative klein gordon equation abstract this is a continuation of the study on an arbitrarily positive initial energy finite time blowup result of the solution to the initial value problem of strongly damped klein gordon equation based on an adapted concavity method both the conclusions and arguments of the present paper are not bounded by the coefficients of dissipative damping terms

least squares type algorithm for adaptive implementation of pisarenko s harmonic retrieval method pisarenko s harmonic retrieval method involves determining the minimum eigenvalue and the corresponding eigenvector of the covariance matrix of the observed random process recently thompson 9 suggested a constrained gradient search procedure for obtaining an adaptive version of pisarenko s method and his simulations have verified that the frequency estimates provided by his procedure were unbiased however the main cost of this technique was that the initial convergence rate could be very slow for certain poor initial conditions restating the constrained minimization as an unconstrained nonlinear problem we derived an alternative gauss newton type recursive algorithm which also used the second derivative matrix or hessian this algorithm may also be viewed as an approximate least squares algorithm simulations have been performed to compare this algorithm to a slight variant of thompson s original algorithm the most important conclusions are that the least squares type algorithm has faster convergence in the beginning while its convergence rate close to the true parameters depends on the signal to noise ratio of the input signal the approximate least squares algorithm resolves the sinusoids much faster than the gradient version

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

absence of renormalization group pathologies near the critical temperature two examples we consider real space renormalization group transformations for ising type systems which are formally defined by exp left h sigma right sum limits sigma t sigma sigma exp left h sigma right wheret is a probability kernel i e t 1 for every configuration for each choice of the block spin configuration let let be the measure on spin configurations which is formally given by taking the probability of to be proportional tot exp h we give a condition which is sufficient to imply that the renormalized hamiltonianh is defined roughly speaking the condition is that the collection of measures is in the high temperature phase uniformly in the block spin configuration the proof of this result uses methods of olivieri and picco we use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples decimation with spacingb 2 on the square lattice with 1 36 c and the kadanoff transformation with parameterp on the trian gular lattice in a subset of the p plane that includes values of greater than c

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

performance analysis of adaptive eigenanalysis algorithms we present a rigorous analysis of several popular forms of short memory adaptive eigenanalysis algorithms using a stochastic averaging method a first order analysis shows that the algorithms do not have any equilibrium points despite published claims to the contrary through averaging analysis we show that they hover around an appropriate eigenvector a second order analysis is also given without the gaussian noise assumption and our results greatly outperform an earlier approximation in the literature the second order analysis has been of much interest in the offline study but in the dynamic adaptive case is uncommon

a combinatorial proof of tree decay of semi invariants we consider finite range gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi invariants supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions this hypothesis holds for completely analytical gibbs fields in this context the tree decay of semi invariants has been proven via analyticity arguments however the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called griffiths phase when analyticity arguments fail

two connections between random systems and non gibbsian measures in this contribution we discuss the role disordered or random systems have played in the study of non gibbsian measures this role has two main aspects the distinction between which has not always been fully clear 1 from disordered systems disordered systems can be used as a tool analogies with as well as results and methods from the study of random systems can be employed to investigate non gibbsian properties of a variety of measures of physical and mathematical interest 2 of disordered systems non gibbsianness is a property of various joint measures describing quenched disordered systems we discuss and review this distinction and a number of results related to these issues moreover we discuss the mean field version of the non gibbsian property and present some ideas how a kac limit approach might connect the finite range and the mean field non gibbsian properties

rotational search methods for adaptive pisarenko harmonic retrieval two adaptation algorithms for adaptive pisarenko harmonic retrieval are described they are derived by considering the associated minimum eigenvalue problem as an optimization problem which seeks the minimum of a quadratic cost function given a hyperspherical constraint an iterative search procedure is used in which each search path is constrained to lie on the unit hypersphere computational complexity per iteration is approximately one third that of previous adaptive phr algorithms simulations reveal that at low snr the trial eigenvector can converge to the true minimum eigenvector of the sample covariance matrix long before this matrix is a good estimate of the true covariance matrix

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

against the convergence of the minor component analysis neurons we investigate the dynamical behavior of some linear neurons devised for the minor component analysis contrary to their convergence theorems our analysis reveals the existence of a divergence in a finite time

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

percolation path large deviations and weakly gibbs states we present a unified approach to establishing the gibbsian character of a wide class of non gibbsian states arising in the renormalisation group theory inside the realm of the pirogov sinai theory for lattice spin systems we prove that rg transformations applied to low temperature phases give rise to weakly gibbsian measures in other words we show that the griffiths pearce israel scenario of rg pathologies is carried by atypical configurations the renormalized measures are described by an effective interaction with relative energies well defined on a full measure set of configurations in this way we complete the first part of the dobrushin restoration program to give a gibbsian description to non gibbsian states a disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed the percolation is controlled via a novel type of pathwise large deviation theory

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

the mca exin neuron for the minor component analysis the minor component analysis mca deals with the recovery of the eigenvector associated to the smallest eigenvalue of the autocorrelation matrix of the input data and is a very important tool for signal processing and data analysis it is almost exclusively solved by linear neurons this paper presents a linear neuron endowed with a novel learning law called mca exinn and analyzes its features the neural literature about mca is very poor in the sense that both a little theoretical basis is given almost always focusing on the ode asymptotic approximation and only experiments on toy problems at most four dimensional problems are presented without any numerical analysis this work addresses these problems and lays sound theoretical foundations for the neural mca theory in particular it classifies the mca neurons according to the riemannian metric and justifies from the analysis of the degeneracy of the error cost the different behavior in approaching convergence the cost landscape is studied and used as a basis for the analysis of the asymptotic behavior all the phases of the dynamics of the mca algorithms are investigated in detail and together with the numerical analysis lead to the identification of three possible kinds of divergence here called sudden dynamic and numerical the importance of the choice of low initial conditions is also explained a lot of importance is given to the experimental part where simulations on high dimensional problems are presented and analyzed the orthogonal regression or total least squares tls technique is also presented together with a real world application on the identification of the parameters of an electrical machine it can be concluded that mca exin is the best mca neuron in terms of stability no finite time divergence speed and accuracy

renormalization group at criticality and complete analyticity of constrained models a numerical study we study the majority rule transformation applied to the gibbs measure for the 2d ising model at the critical point the aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is gibbsian we analyze the validity of dobrushin shlosman uniqueness dsu finite size condition for the constrained models corresponding to different configurations of the image system it is known that dsu implies in our 2d case complete analyticity from which as recently shown by haller and kennedy gibbsianness follows we introduce a monte carlo algorithm to compute an upper bound to vasserstein distance appearing in dsu between finite volume gibbs measures with different boundary conditions we get strong numerical evidence that indeed the dsu condition is verified for a large enough volumev for all constrained models

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed