graphical models for statistical inference and data assimilation in data assimilation for a system which evolves in time one combines past and current observations with a model of the dynamics of the system in order to improve the simulation of the system as well as any future predictions about it from a statistical point of view this process can be regarded as estimating many random variables which are related both spatially and temporally given observations of some of these variables typically corresponding to times past we require estimates of several others typically corresponding to future times graphical models have emerged as an efiective formalism for assisting in these types of inference tasks particularly for large numbers of random variables graphical models provide a means of representing dependency structure among the variables and can provide both intuition and e ciency in estimation and other inference computations we provide an overview and introduction to graphical models and describe how they can be used to represent statistical dependency and how the resulting structure can be used to organize computation the relation between statistical inference using graphical models and optimal sequential estimation algorithms such as kalman flltering is discussed we then give several additional examples of how graphical models can be applied to climate dynamics speciflcally estimation using multi resolution models of large scale data sets such as satellite imagery and learning hidden markov models to capture rainfall patterns in space and time

on a theorem of muckenhoupt and wheeden and a weighted inequality related to schrodinger operators we extend in several directions a theorem of b muckenhoupt and r wheeden relating the l p norms of riesz potentials and fractional maximal operators we apply these results to give a simple proof and sharpen a weighted inequality for schrodinger operators of chang wilson and wolff

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

spectral properties of schrodinger operators with irregular magnetic potentials for a spin 12 particle the two dimensional schrodinger operatorh a for a spin 12 particle is considered the magnetic fieldbgenerated byadoes not grow in some directions and stabilizes to a positively homogeneous function it is shown that the spectrum h a consists of disc h a and 0 the latter being an isolated eigenvalue of infinite multiplicity the former accumulating to only the principal term of the asymptotics of disc h a and of h a v wherebandvdo not grow in some directions is computed

the form boundedness criterion for the laplacian operator abstract in this paper we characterize the class of measurable functions or more generally real or complex valued distributions f such that the commutator operator c f f acting from h 1 r d to h 1 r d in other words we give a complete solution to the problem for the boundedness of the laplacian which is fundamental to quantum mechanics

multiscale smoothing error models a class of multiscale stochastic models based on scale recursive dynamics on trees has recently been introduced these models are interesting because they can be used to represent a broad class of physical phenomena and because they lead to efficient algorithms for estimation and likelihood calculation in this paper we provide a complete statistical characterization of the error associated with smoothed estimates of the multiscale stochastic processes described by these models in particular we show that the smoothing error is itself a multiscale stochastic process with parameters that can be explicitly calculated

the eigenvalue distribution of degenerate dirichlet forms we obtain a positive lower bound to the spectrum of certain second order elliptic operators h on l2 where rn has a lipschitz boundary and the coefficients of h become singular as one approaches the boundary we also find a general formula for the order of the asymptotic eigenvalue distribution of h in some situations where the classical limit formula of weyl courant titchmarsh and others is not applicable

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

on the spectrum of the dirichlet laplacian for horn shaped regions in rn with infinite volume abstract an inequality for trace et d is proven where d is the dirichlet laplacian for horn shaped regions d in rn the results of rozenbljum and simon for the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 0 are easily recovered an example of a horn shaped region in r2 where that asymptotics is exponential is given

trace properties of the dirichlet laplacian on obtient des bornes superieures et inferieures pour tr exp ht en termes de la fonction r x

nonclassical eigenvalue asymptotics for operators of schrodinger type on considere des operateurs de la forme a v x sur r n ou la metrique ij x 0 et le potentiel v x 0 on etudie la fonction de denombrement des valeurs propres par un calcul pseudo differentiel avec des symboles a valeur d operateurs

a simple separable hamiltonian having bound states in the continuum the author discusses a simple separable two freedom hamiltonian h describing the motion of a point particle in a plane region v 0 between two infinite parallel hard walls v infinity and with a rectangular finite potential well v v0 the quantal spectrum of h is discrete below the continuum limit ionisation energy different from the classical escape energy e 0 by the zero point energy of the transverse motion and has infinitely many bound states embedded in the continuum the author shows that this spectrum is fragile against perturbations of h if the hard walls are made diverging the bound states in the continuum disappear while if they are converging the continuum disappears and the spectrum is pure point

on some sharp spectral inequalities for schrodinger operators on semiaxis in this paper we obtain sharp lieb thirring inequalities for a schrodinger operator on semiaxis with a matrix potential and show how they can be used to other related problems among them are spectral inequalities on star graphs and spectral inequalities for schrodinger operators on half spaces with robin boundary conditions

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

image segmentation with a unified graphical model we propose a unified graphical model that can represent both the causal and noncausal relationships among random variables and apply it to the image segmentation problem specifically we first propose to employ conditional random field crf to model the spatial relationships among image superpixel regions and their measurements we then introduce a multilayer bayesian network bn to model the causal dependencies that naturally exist among different image entities including image regions edges and vertices the crf model and the bn model are then systematically and seamlessly combined through the theories of factor graph to form a unified probabilistic graphical model that captures the complex relationships among different image entities using the unified graphical model image segmentation can be performed through a principled probabilistic inference experimental results on the weizmann horse data set on the voc2006 cow data set and on the msrc2 multiclass data set demonstrate that our approach achieves favorable results compared to state of the art approaches as well as those that use either the bn model or crf model alone

form boundedness of the general second order differential operator we give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator i j ln aij i j j 1nbj j c with real or complex valued distributional coefficients aij bj and c acting from the sobolev space w1 2 n to its dual w 1 2 n this enables us to obtain analytic criteria for the fundamental notions of relative form boundedness compactness and infinitesimal form boundedness of c with respect to the laplacian on l 2 n in particular we establish a complete characterization of the form boundedness of the schrodinger operator i a 2 q with magnetic vector potential a lloc2 n n and q d n 2006 wiley periodicals inc

schrodinger operators with many bound states consider the schrodinger operators h d2 dx2 v x we present a method for estimating the potential in terms of the negative eigenvalues of these operators among the applications are inverse lieb thirring inequalities and several sharp results concerning the spectral properties of h

dirichlet neumann bracketing for horn shaped regions we use dirichlet neumann bracketing to obtain sharp upper and lower bounds for the spectral counting function of the dirichlet laplacian for a horn shaped region in rm the first and second term and an estimate for the remainder in the asymptotic expansion of the spectral counting function are obtained for a region in r2 given by x1 x2 x1er x2e x1 x2 1 2 12 212

approximate diagonalization of variable coefficient differential operators through similarity transformations approaches to approximate diagonalization of variable coefficient differential operators using similarity transformations are presented these diagonalization techniques are inspired by the interpretation of the uncertainty principle by fefferman known as the sak principle that suggests the location of eigenfunctions of self adjoint differential operators in phase space the similarity transformations are constructed using canonical transformations of symbols and anti differential operators for making lower order corrections numerical results indicate that the symbols of transformed operators can be made to closely resemble those of constant coefficient operators and that approximate eigenfunctions can readily be obtained

on the number of negative eigenvalues for a schrodinger operator with magnetic field we consider the schrodinger operator with magnetic field h sum limits j 1 n left frac 1 i frac partial partial x j a j right 2 vin mathbb r n under certain conditions on the magnetic fieldb curla we generalize the fefferman phong estimates bull a m s 9 129 206 1983 on the number of negative eigenvalues for v to the operatorh upper and lower bounds are established our estimates incorporate the contribution from the magnetic field the conditions onb in particular are satisfied if the magnetic potentialsaj x are polynomials

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

multiscale recursive estimation data fusion and regularization we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and or noisy measurements which may also be at several scales in particular multiscale signal representations lead naturally to pyramidal or tree like data structures in which each level in the tree corresponds to a particular scale of representation a class of multiscale dynamic models evolving on dyadic trees is introduced the main focus of this paper is on the description analysis and application of an extremely efficient optimal estimation algorithm for this class of models this algorithm consists of a fine to coarse filtering sweep followed by a coarse to fine smoothing step corresponding to the dyadic tree generalization of kalman filtering and rauch tung striebel smoothing the kalman filtering sweep consists of the recursive application of three steps a measurement update step a fine to coarse prediction step and a fusion step we illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of fractal regularizations of ill posed signal and image processing problems encountered

the spectrum of the schrodinger operator on decrit le spectre negatif de l operateur de schrodinger avec un potentiel singulier on determine la valeur exacte du fond du spectre et on etablit une double estimation

likelihood calculation for a class of multiscale stochastic models with application to texture discrimination a class of multiscale stochastic models based on scale recursive dynamics on trees has previously been introduced theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale e g 1 f processes as well as 1d markov processes and 2d markov random fields moreover efficient optimal estimation algorithms have been developed for these models by exploiting their scale recursive structure the authors exploit this structure in order to develop a computationally efficient and parallelizable algorithm for likelihood calculation they illustrate one possible application to texture discrimination and demonstrate that likelihood based methods using the algorithm achieve performance comparable to that of gaussian markov random field based techniques which in general are prohibitively complex computationally

a sharper uncertainty principle abstract the work strengthens the result established by l cohen on uncertainty principle involving phase derivative we propose stronger uncertainty principles not only in the classical setting for fourier transform but also for self adjoint operators we also deduce the conditions that give rise to the equal relation of the uncertainty principle examples are provided to show that the new uncertainty principle is truly sharper than the existing ones in literature

eigenvalue asymptotics of the neumann laplacian of regions and manifolds with cusps abstract we study the eigenvalue asymptotics of a neumann laplacian n in unbounded regions of r2 with cusps at infinity a typical example is x y r 2 x 1 y x 2 and prove that n e n n e h v e 2 vol where hv is the canonical one dimensional schrodinger operator associated to the problem we establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a dirichlet laplacian d for a class of cusp type regions of infinite volume

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

generalizations of the classical weyl and colin de verdiere s formulas and the orbit method the classical weyl formula expresses the leading term of the asymptotics of the counting function n h of the spectrum of a self adjoint operator h in an invariant form one can hear the volume of the subset of the cotangent bundle where the symbol of the operator h is less than in particular it is applicable to schrodinger operators with electric potentials growing at infinity the weyl formula is formulated in an invariant form however it gives for magnetic schrodinger operators with magnetic tensors growing at infinity for these operators colin de verdiere s formula is known but the form of the latter is not invariant in this article we suggest an invariant generalization of both weyl s and colin de verdiere s formulas for wide classes of schrodinger operators with polynomial electric and magnetic fields the construction is based on the orbit method due to kirillov and it allows one to hear the geometry of coadjoint orbits

multiscale methods for the segmentation and reconstruction of signals and images this paper addresses the problem of both segmenting and reconstructing a noisy signal or image the work is motivated by large problems arising in certain scientific applications such as medical imaging two objectives for a segmentation and denoising algorithm are laid out it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations the starting point for the development of a suitable algorithm is a variational approach to segmentation shah 1992 this paper then develops a precise statistical interpretation of a one dimensional 1 d version of this variational approach to segmentation the 1 d algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics a straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous gaussian markov random fields such procedures require an unacceptably large number of operations to meet the objective of developing a computationally efficient algorithm the use of previously developed multiscale statistical methods is investigated this results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics as desired

time frequency transfer function calculus symbolic calculus of linear time varying systems linear operators based on a generalized underspread theory we introduce a generalized concept of underspread linear time varying systems linear operators which contains a previous definition as a special case we show that an existing approximate transfer function calculus symbolic calculus can be extended to this wider and practically more relevant class of underspread operators as a mathematical underpinning of this calculus we establish explicit bounds on various error quantities associated with it the transfer function calculus provides a theoretical basis for various methods recently proposed for nonstationary signal processing and it has important implications in the theory of time varying power spectra

dynamic analysis of multi stepped distributed parameter rotor bearing systems exact solutions for a distributed parameter system are of great use for the physical understanding of the system or the sensitivity analysis and design of the system however exact or closed form solutions for multi stepped rotor bearing systems with distributed parameters have been rarely investigated the present paper proposes a modelling and analysis method to obtain exact solutions for multi stepped rotor bearing systems with distributed parameters to this end a modelling procedure to obtain an exact dynamic matrix for a timoshenko shaft element is presented by using a spatial state equation and the laplace transformation the assembling procedure for constructing the global matrix can be accomplished in the same manner as the finite element method the proposed method can readily provide exact eigensolutions frequency responses and unbalance responses for multi stepped rotor bearing systems with distributed parameters three numerical examples are also presented for validating or illustrating the proposed method the numerical study shows that the proposed method is very useful for the analysis of rotor bearing systems

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

the trace of the heat kernel in lipschitz domains we establish the existence of an asymptotic expansion as t 0 for the trace of the heat kernel for the neumann laplacian in a bounded lip schitz domain the proof of an asymptotic expansion for the heat kernel for the dirichlet laplacian is also sketched the treatment of the dirichlet lapla cian extends work of brossard and carmona who obtained the same result in c1 domains a classical question in analysis asks what is the relationship between the spectrum of the laplacian on a domain and the geometry of the domain this question has been studied by many authors including 1 3 7 11 12 14 15 17 19 one approach to this problem proceeds by considering the trace of the heat semigroup on d tr 0 i g t x x dx te jd where g is the heat kernel on d and 0 ai a2 a3 is the sequence of eigenvalues for a on d one attempts to give an asymptotic expansion for tx g near t 0 whose coefficients are quantities associated with the geometry of d thus one can deduce geometrical information about d from knowledge of the spectrum in this paper we will address this question under minimal smoothness as sumptions on the boundary of d in particular we will obtain asymptotic ex pansions for tx g when the order of smoothness of the boundary corresponds exactly to the order of accuracy of the asymptotic expansion in order to state our main theorem we need one definition by a lipschitz domain we mean a bounded connected open set whose boundary may be locally described as the graph of a lipschitz function our main result is main theorem suppose that d c r is a lipschitz domain then the heat kernel for the laplacian with dirichlet boundary conditions on 3d satisfies tx g t 4ntyn 2 d n x dd o txl2

modal analysis of rotating shafts a body fixed axis formulation approach abstract a spinning timoshenko beam subjected to a constant moving load is analyzed using a modal expansion technique the formulation is based on a body fixed axis reference system and thus should be applicable to handle any general cross sections using this alternative approach dynamic quantities such as natural frequencies mode shapes and system responses are easily computable it is shown that simply supported spinning timoshenko beams posses two pairs of natural frequencies corresponding to each mode shape through appropriate simplifications the coupled eighth order differential equations governing the problem are degenerated into a set of uncoupled fourth order equations with this simplified theory closed form expressions for natural frequencies and system transient response are derived however one pair of the natural frequencies is lost in the process but this is acceptable as the frequencies of this pair are much higher than those of the retained pair in addition a linearized approximation of the natural frequency similar to the one given for euler bernoulli beams is also proposed for a timoshenko shaft numerical simulations are performed to demonstrate the characteristics of the response

sobolev multipliers maximal functions and parabolic equations with a quadratic nonlinearity abstract we develop a general framework to describe global mild solutions to a cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity this class includes the navier stokes equations the subcritical dissipative quasi geostrophic equation and the parabolic elliptic keller segel system

spectral asymptotics for the schrodinger operator with potential which steadies at infinity we consider the discrete spectrum of the selfadjoint schrodinger operatora h h 2 v defined inl2 m with potentialv which steadies at infinity i e v x g x f 1 o 1 as x for 0 and some homogeneous functionsg andf of order zero let h 0 be the total multiplicity of the eigenvalues ofa h smaller thanm m being the minimum value ofg over the unit spheresm 1 hence m coincides with the lower bound of the essential spectrum ofa h we study the asymptotic behaviour of 1 as 0 or of h ash 0 the number 0 being fixed we find that these asymptotics depend essentially on the structure of the submanifold ofsm 1 where the functiong takes the valuem and generically are nonclassical i e even as a first approximation 2 m h differs from the volume of the set x 2m h2 2 v x m

on the spectral counting function for the dirichlet laplacian abstract we obtain upper and lower bounds for the spectral counting function associated to the dirichlet laplacian for an open set d in euclidean space satisfying a geometric and a potential theoretic condition

dixmier traces and residues on weak operator ideals we develop the theory of modulated operators in general principal ideals of compact operators for laplacian modulated operators we establish connes trace formula in its local euclidean model and a global version thereof it expresses dixmier traces in terms of the vector valued wodzicki residue we demonstrate the applicability of our main results in the context of log classical pseudo differential operators studied by lesch and a class of operators naturally appearing in noncommutative geometry c 2020 elsevier inc all rights reserved

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

discrete markov image modeling and inference on the quadtree noncasual markov or energy based models are widely used in early vision applications for the representation of images in high dimensional inverse problems due to their noncausal nature these models generally lead to iterative inference algorithms that are computationally demanding in this paper we consider a special class of nonlinear markov models which allow one to circumvent this drawback these models are defined as discrete markov random fields mrf attached to the nodes of a quadtree the quadtree induces causality properties which enable the design of exact noniterative inference algorithms similar to those used in the context of markov chain models we first introduce an extension of the viterbi algorithm which enables exact maximum a posteriori map estimation on the quadtree two other algorithms related to the mpm criterion and to bouman and shapiro s 1994 sequential map smap estimator are derived on the same hierarchical structure the estimation of the model hyper parameters is also addressed two expectation maximization em type algorithms allowing unsupervised inference with these models are defined the practical relevance of the different models and inference algorithms is investigated in the context of image classification problem on both synthetic and natural images

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

eigenvalue asymptotics and exponential decay of eigenfunctions for schrodinger operators with magnetic fields we consider the schr6dinger operator with magnetic field h va x 2 v x in rj assuming that v 0 and icurl a i v 1 is locally in certain reverse holder class we study the eigenvalue asymptotics and exponential decay of eigenfunctions introduction consider the schrodinger operator with magnetic field 0 1 h h a v jv a x v x inlr n 3 where i 1 v rn r is the electric scalar potential and a rn rn is the magnetic vector potential under some natural conditions on a and v h admits a self adjoint realization which is still denoted by h in l2 ir1n in this paper we shall study its eigenvalue asymptotics and exponential decay of its eigenfunctions let n a h denote the number of eigenvalues counting multiplicity of h smaller than a or the dimension of the spectral projection for h corresponding to x a in the case a x 0 the following phase space volume estimate due to cwickel lieb rosenbljum sil theorem 9 3 p 95 is classical 0 2 n a a v cnl i x e r n x r n 1h12 v x l where cn is a constant depending on n on the other hand there exist some simple degenerate potentials v x i e potentials which do not tend to infinity in all direction as lxi oo e g v x v xi x2 xn x2x n xn for which the right hand side of 0 2 is infinite and nevertheless a v x has a discrete spectrum furthermore in the case where a x 0 and v x 0 the phase space volume of the set where the symbol of h is less than a n i x e nr x r n e ijaj x l2 l j 1 received by the editors may 19 1995 1991 mathematics subject classification primary 35p20 35j10

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

modal analysis of continuous rotor bearing systems abstract modal analysis is applied to continuous rotor systems with various boundary conditions which include isotropic and anisotropic natural boundary conditions the rotor includes the effects of rotary inertia and gyroscopic moment in particular the whirl speeds and mode shapes backward and forward of a rotating shaft are obtained as spin speed and boundary conditions vary and the unbalance responses are calculated by using modal analysis the effects of asymmetry in boundary conditions on the system dynamic characteristics are also investigated

characterization of the central campanato space via the commutator operator of hardy type abstract let 1 p and 1 p 1 n then a function f l loc p r n is said to belong to the central campanato space c p r n if f c p r n sup r 0 1 b 0 r 1 b 0 r b 0 r f f b 0 r p d x 1 p where b 0 r denotes the ball centered at 0 with radius r 0 and f b 0 r 1 b 0 r b 0 r f y d y there are many known characterizations of c p r n for 0 1 n the aim of this paper is to introduce some characterizations of c p r n for 1 p 0 via the boundedness of commutator operators of hardy type some more explicit decompositions of these operators and the space c p r n which are different from that of 0 are here proposed to overcome the difficulties caused by 0 moreover some further interesting boundedness for the hardy type operators is also obtained

some estimates for commutators of riesz transforms associated with schrodinger operators abstract we consider the schrodinger operator l v on r n where the nonnegative potential v belongs to the reverse holder class b q 1 for some q 1 n 2 let q 2 1 when q 1 n and 1 q 2 1 1 q 1 1 n when n 2 q 1 n set min 1 2 n q 1 let h l p r n be the hardy space related to the schrodinger operator l for n n p 1 the commutator b r is generated by a function b where is a function space which is larger than the classical companato space and the riesz transform r v 1 2 we show that the commutator b r is bounded from l p r n into l q r n for 1 p q 2 where 1 q 1 p n and bounded from h l p r n into l q r n for n n p 1 where 1 q 1 p n moreover we prove that the commutator b r maps h l n n r n continuously into weak l 1 r n at last we give a characterization for the boundedness of the commutator b r in an extreme case

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

adventures of the coupled yang mills oscillators i semiclassical expansion we study the quantum mechanical motion in the x2y2 potentials with n 2 3 which arise in the spatially homogeneous limit of the yang mills ym equations these systems show strong stochasticity in the classical limit 0 and exhibit a quantum mechanical confinement feature we calculate the partition function z t going beyond the thomas fermi tf approximation by means of the semiclassical expansion using the wigner kirkwood wk method we derive a novel compact form of the differential equation for the wk function after separating the motion in the channels of the equipotential surface from the motion in the central region we show that the leading higher order corrections to the tf term vanish up to eighth order in if we treat the quantum motion in the hyperbolic channels correctly by adiabatic separation of the degrees of freedom finally we obtain an asymptotic expansion of the partition function in terms of the parameter g24t3

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

classical and non classical eigenvalue asymptotics for magnetic schrodinger operators abstract in this paper we consider schrodinger operators with magnetic fields and study which of the magnetic field and the scalar potential contributes to the leading term of the asymptotic distribution of eigenvalues by studying the asymptotic behavior of the trace of the heat kernels we see that if the norm of the magnetic field diverges at infinity faster than the scalar potential only the magnetic field contributes to the leading asymptotic and that only the scalar potential does if the opposite under mild conditions

dynamic trees for unsupervised segmentation and matching of image regions we present a probabilistic framework namely multiscale generative models known as dynamic trees dt for unsupervised image segmentation and subsequent matching of segmented regions in a given set of images beyond these novel applications of dts we propose important additions for this modeling paradigm first we introduce a novel dt architecture where multilayered observable data are incorporated at all scales of the model second we derive a novel probabilistic inference algorithm for dts structured variational approximation sva which explicitly accounts for the statistical dependence of node positions and model structure in the approximate posterior distribution thereby relaxing poorly justified independence assumptions in previous work finally we propose a similarity measure for matching dynamic tree models representing segmented image regions across images our results for several data sets show that dts are capable of capturing important component subcomponent relationships among objects and their parts and that dts perform well in segmenting images into plausible pixel clusters we demonstrate the significantly improved properties of the sva algorithm both in terms of substantially faster convergence rates and larger approximate posteriors for the inferred models when compared with competing inference algorithms furthermore results on unsupervised object recognition demonstrate the viability of the proposed similarity measure for matching dynamic structure statistical models

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

an overlapping tree approach to multiscale stochastic modeling and estimation recently a class of multiscale stochastic models has been introduced in which random processes and fields are described by scale recursive dynamic trees a major advantage of this framework is that it leads to an extremely efficient statistically optimal algorithm for least squares estimation in certain applications however estimates based on the types of multiscale models previously proposed may not be adequate as they have tended to exhibit a visually distracting blockiness we eliminate this blockiness by discarding the standard assumption that distinct nodes on a given level of the multiscale process correspond to disjoint portions of the image domain instead we allow a correspondence to overlapping portions of the image domain we use these so called overlapping tree models for both modeling and estimation in particular we develop an efficient multiscale algorithm for generating sample paths of a random field whose second order statistics match a prespecified covariance structure to any desired degree of fidelity furthermore we demonstrate that under easily satisfied conditions we can lift a random field estimation problem to one defined on an overlapped tree resulting in an estimation algorithm that is computationally efficient directly produces estimation error covariances and eliminates blockiness in the reconstructed imagery without any sacrifice in the resolution of fine scale detail

the partition function and level density for yang mills higgs quantum mechanics we calculate the partition function z t and the asymptotic integrated level density n e for yang mills higgs quantum mechanics for two and three dimensions n 2 3 due to the infinite volume of the phase space on energy shell for n 2 it is not possible to completely disentangle the coupled oscillators x2 y2 model from the higgs sector the situation is different for n 3 for which is finite the transition from order to chaos in these systems is expressed by the corresponding transitions in z t and n e analogous to the transitions in adjacent level spacing distribution from poisson distribution to wigner dyson distribution we also discuss a related system with quartic coupled oscillators and two dimensional quartic free oscillators for which contrary to ymhqm both coupling constants are dimensionless

a canonical correlations approach to multiscale stochastic realization we develop a realization theory for a class of multiscale stochastic processes having white noise driven scale recursive dynamics that are indexed by the nodes of a tree given the correlation structure of a 1 d or 2 d random process our methods provide a systematic way to realize the given correlation as the finest scale of a multiscale process motivated by akaike s use of canonical correlation analysis to develop both exact and reduced order models for time series we too harness this tool to develop multiscale models we apply our realization scheme to build reduced order multiscale models for two applications namely linear least squares estimation and generation of random field sample paths for the numerical examples considered least squares estimates are obtained having nearly optimal mean square errors even with multiscale models of low order although both field estimates and field sample paths exhibit a visually distracting blockiness this blockiness is not an important issue in many applications for such applications our approach to multiscale stochastic realization holds promise as a valuable general tool

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

logical reformulation of quantum mechanics i foundations the basic rules of quantum mechanics are reformulated they deal primarily with individual systems and do not assume that every ket may represent a physical state the customary kinematic and dynamic rules then allow to construct consistent boolean logics describing the history of a system following essentially griffiths proposal logical implication is defined within these logics the multiplicity of which reflects the complementary principle only one interpretative rule of quantum mechanics is necessary in such a framework it states that these logics providebona fide foundations for the description of a quantum system and for reasoning about it one attempts to build up classical physics including classical logic on these quantum foundations the resulting theory of measurement needs not to statea priori that the eigenvalues of an observable have to be the results of individual measurements nor to assume wave packet reduction both these properties can be obtained as consequences of the basic rules one also needs not to postulate that every observable is measurable even in principle a proposition calculus is obtained allowing in principle the replacement of the discussion of problems concerned with the practical interpretation of experiments by due calculations

characterization of multiplier spaces by wavelets and logarithmic morrey spaces in this paper we consider multipliers from sobolev spaces to lebesgue spaces we establish some wavelet characterization of multiplier spaces without using capacity further we give a sharp logarithmic morrey space condition for multipliers which lessens fefferman s morrey space condition to the logarithm level and generalizes lemarie s counter example to non integer cases and expresses his results in a more precise way

refined asymptotic expansion for the partition function of unbounded quantum billiards this paper presents a refined asymptotic expansion for the partition function t tr et of quantum billiards in the unbounded regions 0 x 0 yx 1 0 and 0 ye x 1 r2 where is the dirichlet laplacian simon ann phys 146 209 1983 j funct anal 53 84 1983 determined the leading divergence of the trace of the heat kernel for the first class of systems standard techniques are combined for the evaluation of for bounded region billiards with results by van den berg j funct anal 71 279 1987 for horn shaped regions using an optimized way of dividing the region into narrow and wide parts to determine the first three terms in the asymptotic expansion of results are also stated for bounded regions with cusps that can be obtained by the same method as an application the spectral staircase of the strongly chaotic billiard system defined in the region 0 xy 2 x 0 which has been discussed in connection with the riemann function and the search for quantum chaos is c

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

localization in infinite billiards a comparison between quantum and classical ergodicity consider the non compact billiard in the first quandrant bounded by the positive x semiaxis the positive y semiaxis and the graph of f x x 1 1 2 although the schnirelman theorem holds the quantum average of the position xis finite on any eigenstate while classical ergodicity entails that the classical time average of xis unbounded

refined asymptotics for eigenvalues on domains of infinite measure in this work we study the asymptotic distribution of eigenvalues in one dimensional open sets the method of proof is rather elementary based on the dirichlet lattice points problem which enable us to consider sets with infinite measure also we derive some estimates for the spectral counting function of the laplace operator on unbounded two dimensional domains