nonexistence and uniqueness of positive solutions of yamabe type equations on nonpositively curved manifolds we prove nonexistence and uniqueness of positive c2 solutions of the elliptic equation u a x u k x u 0 1 on a nonpositively curved complete manifold m g 0 introduction let m g be a complete noncompact riemannian manifold of dimension m 3 and let a x k x be assigned continuous functions on m the aim of this paper is to determine conditions involving the asymptotic behavior of a k and the geometry of m g in order to guarantee nonexistence and eventually uniqueness of positive c solutions of the elliptic equation u a x u k x u 0 0 1 on m for some constant 1 this problem has geometrical roots indeed up to an inessential multiplicative positive constant in front of the choice m 2 m 2 in 0 1 produces the well known yamabe equation relating the scalar curvature a x of g to that k x of the conformally deformed metric gu u 4 m 2 g u 0 in this case a fairly accurate general existence result satisfying the further request of completeness of gu has been given in rrv1 for negative scalar curvatures in the present investigation as naturally expected on a riemannian manifold the geometry is controlled via curvature conditions to be more precise we have fixed our attention on the two most typical cases i riem m g 0 and ii riem m g b2 for some constant b 0 here riem m g denotes the sectional curvature thus r with its canonical metric pertains to case i and in this special setting under the further assumption a x 0 a number of results have been obtained for instance by ni n lin l and more recently by cheng and lin c l however our methods beside having a geometrical flavor differ from those of the above authors mainly because while in r standard symmetrization techniques are at hand on a general manifold such an approach is obviously not feasible furthermore the appearance of the linear term in 0 1 plays a role in connection with k x which to our knowledge has not been thoroughly analysed even in euclidean space as received by the editors march 6 1995 1991 mathematics subject classification primary 53c21 58g03

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

an explicit formula for eigenvalues of bethe trees and upper bounds on the largest eigenvalue of any tree a bethe tree bd k is a rooted unweighted of k levels in which the root vertex has degree equal to d the vertices at level j 2 j k 1 have degree equal to d 1 and the vertices at level k are the pendant vertices in this paper we first derive an explicit formula for the eigenvalues of the adjacency matrix of bd k moreover we give the corresponding multiplicities next we derive an explicit formula for the simple nonzero eigenvalues among them the largest eigenvalue of the laplacian matrix of bd k finally we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any tree t these upper bounds are given in terms of the largest vertex degree and the radius of t and they are attained if and only if t is a bethe tree

poisson equation poincare lelong equation and curvature decay on complete kahler manifolds in the first part of this work the poisson equation on complete noncompact manifolds with nonnegative ricci curvature is studied sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained sharp estimates on the solutions are also derived in the second part these results are applied to the study of curvature decay on complete kahler manifolds in particular the poincare lelong equation on complete noncompact kahler manifolds with nonnegative holomorphic bisectional curvature is studied several applications are then derived which include the steinness of the complete kahler manifolds with nonnegative curvature and the flatness of a class of complete kahler manifolds satisfying a curvature pinching condition liouville type results for plurisubharmonic functions are also obtained 0 introduction in this paper we will discuss the poisson equation on complete noncompact manifolds and derive some applications on kahler manifolds let mm be a complete noncompact kahler manifold where m 2 is the complex dimension assume m has nonnegative holomorphic bisectional curvature and has maximal volume growth such that the scalar curvature decays like r 2 where r is the distance from a fixed point then it was proved in 20 by mok siu yau that one can solve the following poincare lelong equation 0 1 1 u received november 25 2000 and in revised form june 28 2001 the first author was partially supported by nsf grant dms9970284 usa the second author was partially supported by nsf of china project 10001001 and the third author was partially supported by earmarked grant of hong kong cuhk4217 99p

periodicity of grover walks on generalized bethe trees abstract we focus on the periodicity of the grover walk on the generalized bethe tree which is a rooted tree such that in each level the vertices have the same degree since the grover walk is induced by the underlying graph its properties depend on the graph in this paper we say that the graph induces periodic grover walks if and only if there exists k n such that the k th power of the time evolution operator becomes the identity operator our aim is to characterize such graphs we give the perfect characterizations of the generalized bethe trees which induce periodic grover walks

on a diffusive logistic equation we study a reaction diffusion version on all of rnof the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values our results which are obtained by the construction of sub and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady state distributions of population

a unified formulation of gaussian versus sparse stochastic processes part i continuous domain theory we introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous time stochastic processes the cornerstone of our approach is an innovation model that is driven by some generalized white noise process which may be gaussian or not e g laplace impulsive poisson or alpha stable this allows for a conceptual decoupling between the correlation properties of the process which are imposed by the whitening operator l and its sparsity pattern which is determined by the type of noise excitation the latter is fully specified by a levy measure we show that the range of admissible innovation behavior varies between the purely gaussian and super sparse extremes we prove that the corresponding generalized stochastic processes are well defined mathematically provided that the adjoint inverse of the whitening operator satisfies some lp bound for p 1 we present a novel operator based method that yields an explicit characterization of all levy driven processes that are solutions of constant coefficient stochastic differential equations when the underlying system is stable we recover the family of stationary continuous time autoregressive moving average processes carma including the gaussian ones the approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the levy processes which are sparse and non stationary finally we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform domain statistics

stochastic models for sparse and piecewise smooth signals we introduce an extended family of continuous domain stochastic models for sparse piecewise smooth signals these are specified as solutions of stochastic differential equations or equivalently in terms of a suitable innovation model the latter is analogous conceptually to the classical interpretation of a gaussian stationary process as filtered white noise the two specific features of our approach are 1 signal generation is driven by a random stream of dirac impulses poisson noise instead of gaussian white noise and 2 the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes we provide a complete characterization of these finite rate of innovation signals within gelfand s framework of generalized stochastic processes we then focus on the class of scale invariant whitening operators which correspond to unstable systems we show that these can be solved by introducing proper boundary conditions which leads to the specification of random spline type signals that are piecewise smooth these processes are the poisson counterpart of fractional brownian motion they are nonstationary and have the same 1 type spectral signature we prove that the generalized poisson processes have a sparse representation in a wavelet like basis subject to some mild matching condition we also present a limit example of sparse process that yields a map signal estimator that is equivalent to the popular tv denoising algorithm

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

positive solutions of an elliptic partial differential equation on rn abstract in this paper we discuss the existence nonexistence and uniqueness of positive solutions of a one parameter family of elliptic partial differential equations on r n n 2 these equations are of interests in mathematical biology and riemannian geometry our approach are based on variational arguments and comparison principles

existence and non existence results for a logistic type equation on manifolds we study the steady state solutions of a generalized logistic type equation on a complete riemannian manifold we provide sufficient conditions for existence respectively non existence of positive solutions which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold which is mostly taken into account by means of conditions on the volume growth of geodesic balls

localization of the grover walks on spidernets and free meixner laws a spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete time quantum walks on graphs we introduce the grover walk on a spidernet and its one dimensional reduction we derive an integral representation of the n step transition amplitude in terms of the free meixner law which appears as the spectral distribution as an application we determine the class of spidernets which exhibit localization our method is based on quantum probabilistic spectral analysis of graphs

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

a priori estimates uniqueness and existence of positive solutions of yamabe type equations on complete manifolds abstract we study the asymptotic behaviour of non negative solutions of yamabe type equations on a complete riemannian manifold then we provide a comparison result based on a form of the weak maximum principle at infinity which together with the a priori estimates previously obtained yields uniqueness under very general ricci assumptions the paper ends with an existence result and an application to the non compact yamabe problem

sparse stochastic processes and discretization of linear inverse problems we present a novel statistically based discretization paradigm and derive a class of maximum a posteriori map estimators for solving ill conditioned linear inverse problems we are guided by the theory of sparse stochastic processes which specifies continuous domain signals as solutions of linear stochastic differential equations accordingly we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions our estimators not only cover the well studied methods of tikhonov and l1 type regularizations as particular cases but also open the door to a broader class of sparsity promoting regularization schemes that are typically nonconvex we provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution magnetic resonance imaging and x ray tomographic reconstruction problems finally we compare the performance of estimators associated with models of increasing sparsity

on the elliptic equation δ 0 on complete riemannian manifolds and their geometric applications we study the elliptic equation u ku kup 0 on complete noncompact riemannian manifolds with k nonnegative three fundamental theorems for this equation are proved in this paper complete analyses of this equation on the euclidean space rn and the hyperbolic space hn are carried out when k is a constant its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

spectral radius index estimates for schrodinger operators and geometric applications in this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation vz avz 0 where a v are functions arising from geometry in particular we introduce a new technique to estimate the distance between two consecutive zeros these results are applied in the setting of complete riemannian manifolds in particular we prove index bounds for certain schrodinger operators and an estimate of the growth of the spectral radius of the laplacian outside compact sets when the volume growth is faster than exponential applications to the geometry of complete minimal hypersurfaces of euclidean space to minimal surfaces and to the yamabe problem are discussed

periodicity of grover walks on complete graphs with self loops abstract we investigate a periodicity of grover walk on complete graphs with a self loop at each vertex we study an evolution matrix which steps forward a state of probability amplitude vector by using algebraic method then we find a periodic behaviour that the probability amplitude vector at each vertex gets back to initial state after some steps it is shown that grover walk on complete graphs on n vertices with a self loop at each vertex is periodic with period 2n

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

a quantum walk induced by hoffman graphs and its periodicity abstract in order to define the discrete time quantum walks we divide the vertex set of a graph by two partitions and give a time evolution operator by a product of two local unitary operators on each partition in this paper we make such partitions from a hoffman graph which gives generalization of line graphs and study a staggered walk we first show that the square of the time evolution operator of the grover walk is unitarily equivalent to that of the staggered walk on the generalized line graph induced from a hoffman graph furthermore we focus on the periodicity of quantum walks and show that the staggered walk on the generalized line graph is periodic whenever the grover walk on the original graph is periodic by using the unitary equivalence this work is the first trial to develop the relation between quantum walks and hoffman graphs

mmse estimation of sparse levy processes we investigate a stochastic signal processing framework for signals with sparse derivatives where the samples of a levy process are corrupted by noise the proposed signal model covers the well known brownian motion and piecewise constant poisson process moreover the levy family also contains other interesting members exhibiting heavy tail statistics that fulfill the requirements of compressibility we characterize the maximum a posteriori probability map and minimum mean square error mmse estimators for such signals interestingly some of the map estimators for the levy model coincide with popular signal denoising algorithms e g total variation tv regularization we propose a novel non iterative implementation of the mmse estimator based on the belief propagation bp algorithm performed in the fourier domain our algorithm takes advantage of the fact that the joint statistics of general levy processes are much easier to describe by their characteristic function as the probability densities do not always admit closed form expressions we then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradient sparse signals

bayesian estimation for continuous time sparse stochastic processes we consider continuous time sparse stochastic processes from which we have only a finite number of noisy noiseless samples our goal is to estimate the noiseless samples denoising and the signal in between interpolation problem by relying on tools from the theory of splines we derive the joint a priori distribution of the samples and show how this probability density function can be factorized the factorization enables us to tractably implement the maximum a posteriori and minimum mean square error mmse criteria as two statistical approaches for estimating the unknowns we compare the derived statistical methods with well known techniques for the recovery of sparse signals such as the l1 norm and log l1 l0 relaxation regularization methods the simulation results show that under certain conditions the performance of the regularization techniques can be very close to that of the mmse estimator

the compact support principle for differential inequalities with gradient terms abstract we give some sufficient conditions to guarantee the validity of the compact support principle for solutions of some singular elliptic differential inequalities on complete non compact manifolds our results apply in particular to the p laplace and the mean curvature operators we also show that our results are sharp in many important cases

principal eigenvalues with indefinite weight functions both existence and non existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved the existence of a continuous family of principal eigenvalues is demonstrated

a unified formulation of gaussian versus sparse stochastic processes part ii discrete domain theory this paper is devoted to the characterization of an extended family of continuous time autoregressive moving average carma processes that are solutions of stochastic differential equations driven by white levy innovations these are completely specified by 1 a set of poles and zeros that fixes their correlation structure and 2 a canonical infinitely divisible probability distribution that controls their degree of sparsity with the gaussian model corresponding to the least sparse scenario the generalized carma processes are either stationary or nonstationary depending on the location of the poles in the complex plane the most basic nonstationary representatives with a single pole at the origin are the levy processes which are the non gaussian counterparts of brownian motion we focus on the general analog to discrete conversion problem and introduce a novel spline based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes we also rely on the concept of generalized increment process which suppresses all long range dependencies to specify an equivalent discrete domain innovation model a crucial ingredient is the existence of a minimally supported function associated with the whitening operator l this b spline which is fundamental to our formulation appears in most of our formulas both at the level of the correlation and the characteristic function we make use of these discrete domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous domain model

on the compact support principle on complete manifolds abstract in this paper we study the compact support principle for singular elliptic inequalities on complete manifolds with a significant dependence on the spatial variable in the process we underline the influence in our results both of the spatial variable and of the geometry of the ambient space

on the linearity of bayesian interpolators for non gaussian continuous time ar 1 processes bayesian estimation problems involving gaussian distributions often result in linear estimation techniques nevertheless there are no general statements as to whether the linearity of the bayesian estimator is restricted to the gaussian case the two common strategies for non gaussian models are either finding the best linear estimator or numerically evaluating the bayesian estimator by monte carlo methods in this paper we focus on bayesian interpolation of non gaussian first order autoregressive ar processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the levy khintchine representation theorem we redefine the bayesian estimation problem in the fourier domain with the help of characteristic forms by providing analytic expressions we show that the optimal interpolator is linear for all symmetric stable distributions the bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines we also show that the limiting case of levy type ar 1 processes the system of which has a pole at the origin always corresponds to a linear bayesian interpolator made of a piecewise linear spline irrespective of the innovation distribution finally we show the two mentioned cases to be the only ones within the family for which the bayesian interpolator is linear