improved hardy and rellich inequalities on riemannian manifolds in this paper we establish improved hardy and rellich type inequalities on a riemannian manifold m furthermore we also obtain sharp constants for improved hardy and rellich type inequalities on the hyperbolic space n

the propagation of sound in ducts lined with circumferentially non uniform admittance of the form ηo ηq exp iqθ in this paper closed form solutions are obtained to a problem of sound propagation in a duct with circumferentially non uniform admittance traditionally solutions to duct problems have been sought in terms of modes which can be expressed as products of functions of space and time co ordinates this method fails when the boundary conditions are non uniform in the text of the paper non separable solutions are developed to obtain a closed form solution to the problem the non separable modes are a sum of sub modes which satisfy the wave equation but not the boundary conditions expressions are obtained for the sub modal distribution functions which ensure that together the sub modes shall form a wave packet which will satisfy the boundary conditions

hardy and uncertainty inequalities on stratified lie groups abstract we prove various hardy type and uncertainty inequalities on a stratified lie group g in particular we show that the operators t f l 2 f where is a homogeneous norm 0 q p and l is the sub laplacian are bounded on the lebesgue space l p g as consequences we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality we also deduce a general version of the heisenberg pauli weyl inequality relating the l p norm of a function f to the l q norm of f and the l r norm of l 2 f

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

on the optimization of non uniform acoustic liners on annular nozzles abstract the solution of the convected wave equation with uniform axial flow in cylindrical co ordinates is used together with non uniform impedance wall boundary conditions to specify the acoustic modes in a cylindrical or annular nozzle the radial eigenfunctions in this case are bessel functions and the method applies equally well to sheared and swirling mean flows provided that the appropriate eigenfunctions are used the eigenvalues or radial wavenumbers are determined by i the roots of a linear combination of bessel functions for a cylindrical nozzle with uniform wall impedance ii the roots of a 2 2 determinant whose terms are linear combinations of bessel and neumann functions for an annular nozzle with uniform but distinct impedances at each wall iii the roots of an infinite determinant for a cylindrical nozzle with circumferentially non uniform wall impedance iv the roots of the determinant of a 2 2 block of infinite matrices for an annular nozzle with distinct non uniform impedance distributions at the two walls the case of circumferentially non uniform but axially uniform wall impedance allows the existence of an axial wavenumber for each frequency and each eigenvalue or radial wavenumber the acoustic liner may be optimized by maximizing the decay of a particular wave mode e g the slowest decaying or a combination of them e g the total acoustic energy

on the acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution the present paper considers the propagation of sound in a cylindrical duct with a wall section of finite length covered by an acoustic liner whose impedance is an arbitrary function of position the cases of i uniform wall impedance and wall impedance varying along the ii circumference or iii axis of the duct or iv both simultaneously are explicitly considered it is shown that a nonuniform wall impedance couples modes with distinct azimuthal l or axial m wave numbers so that their radial wave numbers k can no longer be calculated separately for each pair m l the radial wave numbers are the roots of an infinite determinant in the case when the wall impedance varies either i circumferentially or ii radially if the wall impedance varies iv both radially and circumferentially then the radial wave numbers are the roots of a doubly infinite determinant i e an infinite determinant in which each term is an infinite determinant the infinite determinants specifying the radial wave numbers are written explicitly for sound in a cylindrical nozzle with a uniform axial flow in which case the radial eigenfunctions are bessel functions the method of calculation of the radial wave numbers applies equally well to a cylindrical nozzle with shear flow and or swirling flows with the bessel functions replaced by other eigenfunctions the radial wave numbers are calculated by truncation of the infinite determinants for several values of the aspect ratio defined as the ratio of length to diameter it is shown that a nonuniform wall impedance will give rise to additional modes compared with a uniform wall impedance the radial wave numbers specify the eigenfrequencies for the acoustic modes in the duct the imaginary parts of the eigenfrequencies specify the decay of the sound field with time and thus the effectiveness of the acoustic liner

sharp poincare hardy and poincare rellich inequalities on the hyperbolic space abstract we study hardy type inequalities associated to the quadratic form of the shifted laplacian h n n 1 2 4 on the hyperbolic space h n n 1 2 4 being as it is well known the bottom of the l 2 spectrum of h n we find the optimal constant in a resulting poincare hardy inequality which includes a further remainder term which makes it sharp also locally the resulting operator is in fact critical in the sense of 17 a related improved hardy inequality on more general manifolds under suitable curvature assumption and allowing for the curvature to be possibly unbounded below is also shown it involves an explicit curvature dependent and typically unbounded potential and is again optimal in a suitable sense furthermore with a different approach we prove rellich type inequalities associated with the shifted laplacian which are again sharp in suitable senses

on sound propagation in a linear shear flow abstract it is shown that the acoustic wave equation in a linear shear flow always has a critical layer where the doppler shifted frequency vanishes since this is the only singularity of the wave equation apart from the point at infinity the power series solution about the critical layer has infinite radius of convergence two linearly independent solutions are even and odd functions of distance from the critical layer their plots show that acoustic oscillations are suppressed near the critical layer a linear combination of these solutions specifies the general acoustic field the constants of integration are determined from boundary conditions of which several examples are given the total acoustic field is illustrated for rigid and impedance wall conditions the cases illustrated include both real and complex wave fields these wave fields are small amplitude perturbations of the acoustic wave equation in a linear shear flow it is shown in the appendix that the perturbations of vorticity and dilatation are coupled and thus combine features of acoustic and hydrodynamic modes

some trace hardy type inequalities and trace hardy sobolev maz ya type inequalities abstract we prove a trace hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of alvino et al and of tzirakis on the upper half space we also prove some trace hardy sobolev maz ya type inequalities which generalize the recent results of filippas et al in applications we derive some hardy type inequalities and hardy sobolev maz ya type inequalities for fractional laplacian finally we prove the logarithmic sobolev trace inequalities and logarithmic hardy trace inequalities on the upper half spaces the best constants in these inequalities are explicitly computed in the radial case

on some strong poincare inequalities on riemannian models and their improvements abstract we prove second and fourth order improved poincare type inequalities on the hyperbolic space involving hardy type remainder terms since theirs l h s only involve the radial part of the gradient or of the laplacian they can be seen as stronger versions of the classical poincare inequality we show that such inequalities hold true on model manifolds as well under suitable curvature assumptions and sharpness of some constants is also discussed

sharp constants in the hardy rellich inequalities optimal constants are found in hardy rellich inequalities containing derivatives of arbitrary not necessarily integer order l some new inequalities of this type are also obtained

optimizing improved hardy inequalities let be a bounded domain in rn n 3 containing the origin motivated by a question of brezis and vazquez we consider an improved hardy inequality with best constant b that we formally write as n 22 21 x 2 bv x we first give necessary conditions on the potential v under which the previous inequality can or cannot be further improved we show that the best constant b is never achieved in h01 and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in h01 our analysis reveals that the original inequality can be repeatedly improved by adding on the right hand side specific potentials this leads to an infinite series expansion of hardy s inequality the series obtained is in some sense optimal in establishing these results we derive various sharp improved hardy sobolev inequalities

remarks on the rellich inequality we study the rellich inequalities in the framework of equalities we present equalities which imply the rellich inequalities by dropping remainders this provides a simple and direct understanding of the rellich inequalities as well as the nonexistence of nontrivial extremisers

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

the hardy inequality and nonlinear parabolic equations on carnot groups abstract in this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation u t g p u v x u p 1 in 0 t 1 p 2 u x 0 u 0 x 0 in u x t 0 on 0 t where g p is the p sub laplacian on a carnot group g and v l loc 1

exact and approximate directivity patterns of the sound radiated from a cylindrical duct abstract the exact solution for radiation from a cylindrical pipe obtained by using the wiener hopf method has been available for over 50 years approximate solutions can be used to simplify the problem and their accuracy compared with the exact solution firstly the exact solution can be simplified by using the method of the steepest descent to cast the problem in the form of weinstein s u approximation a simpler technique involves the use of the kirchhoff approximation this vastly simplifies the problem by considering the radiation emanating from a circular aperture in a thin screen the various formulae are then compared in a parameter space related to the physical structure of a propagating model if one considers the behaviour of the approximations solely as one of the governing parameters is altered i e as k0a is increased the physical behaviour of the propagating mode would not linearly change in considering the parameters in terms of mode angles one can observe the accuracy of the approximations for various parameters while linearly changing the ray structure

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

frame wavelet sets in r d in this paper we discuss the characterization of frame wavelet sets we extend some results obtained earlier in the one dimensional case more specifically we completely characterize tight frame wavelet sets in higher dimensions and obtain some necessary conditions and sufficient conditions for a set e to be a frame wavelet set in rd several examples are presented and compared with those in the one dimensional case using our results one can easily construct various frame wavelet sets

hardy lieb thirring inequalities for fractional schrodinger operators we show that the lieb thirring inequalities on moments of negative eigenvalues of schrodinger like operators remain true with possibly different constants when the critical hardy weight c x 2 is subtracted from the laplace operator we do so by first establishing a sobolev inequality for such operators similar results are true for fractional powers of the laplacian and the hardy weight and in particular for relativistic schrodinger operators we also allow for the inclusion of magnetic vector potentials as an application we extend for the first time the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge z 2 for less than some critical value

sharp trace hardy sobolev maz ya inequalities and the fractional laplacian in this work we establish trace hardy and trace hardy sobolev maz ya inequalities with best hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity we then use them to produce fractional hardy sobolev maz ya inequalities with best hardy constants for various fractional laplacians in the case where the domain is the half space our results cover the full range of the exponent s in 0 1 of the fractional laplacians in particular we give a complete answer in the l2 setting to an open problem raised by frank and seiringer sharp fractional hardy inequalities in half spaces in around the research of vladimir maz ya international mathematical series 2010

eigensolutions for liners in uniform mean flow ducts the problem of sound attenuation in a rectangular acoustically lined duct containing uniform mean flow is investigated analytically by means of the generalized wiener hopf technique for lined sections of finite axial extent uniqueness of the solution is enforced by imposing edge conditions at the liner interfaces several possible edge conditions are considered including the kutta condition the corresponding solutions differ by eigensolutions and it is demonstrated that solution methods like the mode matching and singularity method imply differing edge conditions the power attenuation as well as the spl traces appear to be rather insensitive to the imposed edge conditions but striking differences are observed for the reflection coefficient

hardy and rellich inequalities identities and sharp remainders on homogeneous groups abstract we give sharp remainder terms of l p and weighted hardy and rellich inequalities on one of most general subclasses of nilpotent lie groups namely the class of homogeneous groups as consequences we obtain analogues of the generalised classical hardy and rellich inequalities and the uncertainty principle on homogeneous groups we also prove higher order inequalities of hardy rellich type all with sharp constants a number of identities are derived including weighted and higher order types

improved lp poincare inequalities on the hyperbolic space abstract we investigate the possibility of improving the p poincare inequality h n u p p p u p p on the hyperbolic space where p 1 and p n 1 p p is the best constant for which such inequality holds we prove several different and independent improved inequalities one of which is a poincare hardy inequality namely an improvement of the best p poincare inequality in terms of the hardy weight r p r being geodesic distance from a given pole certain hardy maz ya type inequalities in the euclidean half space are also obtained

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

new characterizations of wavelet frames in l2 r abstract in this paper the classical problem in frame theory i e characterizing the functions g l 2 r and parameters a 1 b 0 such that d a n t m b g generates a frame for l 2 r is discussed several new necessary conditions and sufficient conditions are established we also obtain some new necessary and sufficient conditions under certain assumptions in particular we introduce a type of special functions which are similar to wiener type functions and use them to characterize wavelet frames

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics