projective fourier analysis for patterns abstract identifying psl 2 c as a projective group for patterns in the conformal camera model the projective harmonic analysis on its double covering group sl 2 c is presented in the noncompact and compact pictures the pictures used to study different aspects of irreducible unitary representations of semisimple lie groups bypassing technicalities of representation theory but stressing the motivation and similarities with euclidean fourier analysis each constructed picture of the projective fourier analysis includes the fourier transform plancherel s theorem and convolution property projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern after removing conformal distortions by using only one projective fourier transform of the original pattern what is demonstrated in a computer simulation the convolution properties in both pictures must by used to develop algorithms for projectively invariant matching of patterns work in progress on fast algorithms for computing with projective fourier transforms and for rendering image projective transformations is discussed efficient computations of the convolutions would follow from the both fast projective fourier transforms and their inverses

the revised fundamental theorem of moment invariants a revised fundamental theorem of moment invariants for pattern recognition which corrects the fundamental theorem proposed by m k hu 1962 is presented the correction affects neither similitude scale nor rotation invariants derived using the original theorem but it does affect features invariant to general linear transformations four of the latter invariants were presented originally by hu these are revised to take the correction to the fundamental theorem into account furthermore these four invariants are combined to yield three new invariants which are additionally invariant to changes in the illumination of an image

the canonical coordinates method for pattern recognition ii isomorphisms with affine transformations abstract the canonical coordinates method for pattern recognition is extended conditions are examined for the existence of an isomorphism between a given deformation group and some affine transformation new invariants for many non linear deformations are derived several examples with real images are given

the canonical coordinates method for pattern deformation theoretical and computational considerations a method for the analysis of deformed patterns is presented and analyzed the image is transformed into a new set of coordinates in which the deformation has a particular simple form a number of deformations are considered the practical implementation of the method is discussed similar aspects of biological vision are also considered

continuity properties of balanced realizations in this paper topological and geometrical properties of pre balanced and balanced realizations are considered it is shown that analytic pre balancing coordinate transformations do exist and that the set of pre balanced realizations forms an analytic submanifold explicit formulas for the tangent spaces and their dimension are derived continuity properties of balanced realizations are studied for systems with more than two inputs and two outputs explicit points of discontinuity of balancing coordinate transformations are described a certain class of balancing coordinate transformations is shown to be discontinuous even for distinct and fixed singular values

an exactly solvable model of a multidimensional incommensurate structure the paper considers the class of schrodinger multidimensional discrete operators with quasi periodic unbounded potential for which essentially complete spectral analysis may be carried out in the case of sufficiently high incommensurability of almost periods the spectrum of such operators is found to be pure point and simple the eigenfunctions exponentially localized and the low frequency conductivity exponentially small in the one dimensional case for any incommensurability the spectrum does not contain the absolutely continuous component while for small incommensurability the spectrum is singular continuous

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

n dimensional moment invariants and conceptual mathematical theory of recognition n dimensional solids the proof of the generalized fundamental theorem of moment invariants gftmi is presented for n dimensional pattern recognition on the basis of gftmi the moment invariants of affine transformation and subgroups of affine transformation are constructed using these invariants the conceptual mathematical theory of recognition of geometric figures solids and their n dimensional generalizations is worked out by means of this theory it is possible for the first time to analyze scenes consisting not only of polygons and polyhedra but also scenes consisting of geometric figures and solids with curved contours and surfaces respectively in general it is the author s opinion that this theory is a useful step toward the essential development of robot vision and toward creating machine intelligence to make machines able to think by means of geometric concepts of different generalities and dimensions and by associations of these concepts

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

anderson localization for generic deterministic operators abstract we consider a class of ensembles of lattice schrodinger operators with deterministic potentials including quasi periodic potentials with diophantine frequencies depending upon an infinite number of parameters in an auxiliary measurable space using a variant of the multi scale analysis we prove anderson localization for generic ensembles in the strong disorder regime and establish an analog of minami type bounds for spectral spacings

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

geometric fourier analysis of the conformal camera for active vision suppose one intends to design an active vision system that should perform some artificial intelligence functions for instance it should recognize a planar object or a three dimensional object containing a piece of a planar surface in a dynamic scene ideally such a system should be built upon some data model representing visual inputs and algorithms storing processing and analyzing visual information that are well adapted to image transformations produced by different perspectives between planar objects and the imaging system in spite of its importance this problem remained unsolved until recently in this article building on the author s work projective fourier analysis for patterns is constructed in the framework of geometric fourier analysis on groups and homogeneous spaces it is done by identifying in the conformal camera the group mathbf sl 2 mathbb c which gives image projective transformations by acting through linear fractional mappings on the image plane homogeneous under th

recognition of distorted patterns by invariance kernels abstract a novel method for the recognition of deformed patterns is presented it is shown that for planar deformations generated by lie transformation groups there exist kernels such that the associated integral transforms are invariant under the deformations many examples of deformations and the corresponding kernels are given

a global parametrization of symptotically stable linear systems abstract in this paper we construct a homeomorphism from the set of p m transfer functions of mcmillan degree n onto the open subspace of asymptotically stable linear systems this homeomorphism yields a one to one correspondence between 1 i canonical forms for state space equivalence of minimal systems and 2 ii canonical forms for state space equivalence of asymptotically stable minimal systems implications for the topology of various sets of asymptotically stable systems are given

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

anderson localization for the 1 d discrete schrodinger operator with two frequency potential we prove the complete exponential localization of eigenfunctions for the 1 d discrete schrodinger operators with quasi periodic potentials having two basic frequencies it is shown also that for such operators there is no forbidden zones in the spectrum unlike the operators with one basic frequency

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

anderson localization for schrodinger operators on ℤ with strongly mixing potentials in this paper we show that for a e x 0 2 the operators defined on as and with dirichlet condition 1 0 have pure point spectrum in with exponentially decaying eigenfunctions where 0 and are small as it is a simple consequence of known techniques that for small one has 2 2 spectrum h x for a e x 0 2 we thus established anderson localization on the spectrum up to the edges and the center more general potentials than cosine can be treated but only those energies with nonzero spectral density are allowed finally we prove the same result for operators on the whole line with potential where a 2 2 is a hyperbolic toral automorphism f c1 2 f 0 and small the basis for our analysis is an asymptotic formula for the lyapunov exponent for 0 by figotin pastur and generalized by chulaevski spencer we combine this asymptotic expansion with certain martingale large deviation estimates in order to apply the methods developed by bourgain and goldstein in the quasi periodic case

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

uniform anderson localization unimodal eigenstates and simple spectra in a class of haarsh deterministic potentials abstract we consider a particular class of families of multi dimensional lattice schrodinger operators with deterministic e g quasi periodic potentials generated by the hull given by an orthogonal series over the haar wavelet basis on the torus of arbitrary dimension with expansion coefficients considered as independent parameters in the strong disorder regime we prove anderson localization for generic operator families and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers we also prove a variant of the minami estimate for deterministic potentials and simplicity of the spectrum

from fixed energy localization analysis to dynamical localization an elementary path we review several techniques initiated by a remarkable work written by spencer j stat phys 51 1009 1019 1988 a quarter of century ago used and further developed in numerous subsequent researches we also describe a fairly general elementary derivation of spectral and strong dynamical anderson localization from the fixed energy analysis of the green s functions on locally finite graphs of polynomial growth obtained either by the multi scale analysis or by the fractional moment method this derivation goes in the same direction as the simon wolf method commun pure appl math 39 75 90 1986 but provides more quantitative estimates can be adapted to multi particle models and combined with a simplified variant of the germinet klein argument commun math phys 222 415 448 2001 results in an elementary proof of strong dynamical localization on arbitrary graphs of polynomial growth

localization in v dimensional incommensurate structures we exhibit a class of quasi periodic unbounded potential in thev dimensional discrete schrodinger equation for which the spectrum is only pure point with exponentially localized states and a dense set of eigenvalues in

correlation estimates in the anderson model we give a new proof of correlation estimates for arbitrary moments of the resolvent of random schrodinger operators on the lattice that generalizes and extends the correlation estimate of minami for the second moment we apply this moment bound to obtain a new n level wegner type estimate that measures eigenvalue correlations through an upper bound on the probability that a local hamiltonian has at least n eigenvalues in a given energy interval another consequence of the correlation estimates is that the results on the poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation invariant selfadjoint lattice operators with decaying off diagonal terms and random potentials

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions