orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

calculating laplace transforms in terms of the toda molecule a method for calculating a continued fraction expansion of laplace transforms is presented in terms of the tau function of the continuous time toda molecule equation and its integrable discretization the z transform and a discrete analogue of the laplace transform of a set of finite samples are also formulated

the poincare series of the module of derivations of affine monomial curves abstract let a be a graded k algebra and m be a finitely generated a module the poincare series p m a z is the formal power series i dim k tor i a k m z i we study the poincare series of der k k s the module of derivations of a numerical semigroup ring k s and we relate it to the poincare series of k over k s and to the type of s we then use this in order to determine the poincare series of der k k s or at least its rationality for some classes of examples we finally give an example of a non rational p der k k s k s z

differential operators on monomial curves abstract let k be an algebraically closed field of characteristic 0 let n 0 be a numerical semigroup and let a k be the corresponding semigroup algebra we give an explicit description of the ring d d a of k linear differential operators on a the associated graded ring grd a and the module of derivations derk a we also classify all graded d modules which are finitely generated torsion free a modules of rank 1 considered as modules over the sub ring a d a

compactly supported orthogonal and biorthogonal sqrt 5 refinement wavelets with 4 fold symmetry recently radic5 refinement hierarchical sampling has been studied and radic5 refinement has been used for surface subdivision compared with other refinements such as the dyadic or quincunx refinement radic5 refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates this special property has been shown to be very useful to represent adaptively and render complex and procedural geometry when radic5 refinement is used for multiresolution data processing radic5 refinement filter banks and wavelets are required while the construction of 2 d nonseparable bi orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers the construction of bi orthogonal wavelets with radic5 refinement has not been investigated the main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with radic5 refinement in this paper we obtain block structures of orthogonal and biorthogonal radic5 refinement fir filter banks with 4 fold rotational symmetry we construct compactly supported orthogonal and biorthogonal wavelets based on these block structures

detail enhanced exposure fusion in a typical processing chain of image enhancement an exposure fusion scheme can be used to synthesize a more detailed low dynamic range ldr image directly from a set of differently exposed ldr images without generation of an intermediate high dynamic range image in this brief we introduce a new quadratic optimization based method to extract fine details from a vector field the new method extracts fine details from a set of differently exposed ldr images simultaneously the extracted fine details are then added to an intermediate ldr image which is fused by simply using an existing exposure fusion scheme with this the proposed scheme can enhance fine details to produce sharper images

stieltjes continued fraction and qd algorithm scalar vector and matrix cases the definition in previous studies of vector stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals left unsolved the question of a direct definition of their coefficients in terms of the original data a vector of stieltjes series the subject was more undefined in the matrix case a new version of the qd algorithm for matrix problem allows to extend to the vector and matrix cases the result of stieltjes expansion of a scalar function in terms of a stieltjes continued fraction beside this connection it solves the inverse miura transform and gives interesting identities between general band matrix and sparse band matrix finally as a consequence we extend to some dynamical systems a method known for toda lattices

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

on the structure of free resolutions of length 3 the sources of structure theory of finite free resolutions go back to hilbert in his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal this theorem was then extended by burch to arbitrary commutative rings there are various motivations for extending this results one of them is a variety of applications of the hilbert burch theorem in deformation theory and algebraic geometry cf for example s another reason comes from serre s conjecture one would like to understand in terms of resolutions why the lifting property of grothendieck does not hold in 1974 buchsbaum and eisenbud in bel introduced so called structure theorems various factorisations which occur for acyclic complexes of arbitrary length then hochster in h generalized the hilbert burch theorem to arbitrary free complexes of length 2 his results were then improved in br pw still the structure of complexes of length bigger than 2 is not understood the first steps in this direction are the results of b and the construction of a generic ring for complexes of length 3 of type 1 rz 1 1 given in pw in this paper i construct a ring which i believe is the generic ring for complexes of length 3 the construction involves new structure theorems and a new idea the notion of defects of structure theorems the paper is organized as follows section 1 is introductory i give there the background from representation theory and the cohomology groups of a lie algebra section 2 contains the proof of the new structure theorems we need the construction of the generic ring r and of the defect lie algebra l in section 3 i show that genericity of r follows from vanising of the homology of a certain family of complexes over u l the enveloping algebra of l in section 4 i apply the results of section 2 to study multi

a generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system abstract a generalized eigenvalue algorithm for a certain class of tridiagonal matrix pencils is presented the algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil the convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system

multiresolution analysis haar bases and self similar tilings of r sup n orthonormal bases for l sup 2 r sup n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l sup 2 r for example each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions the construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings

hierarchical subsampling giving fractal regions recursive image subsampling which yields support areas approaching fractals is described and analyzed using iterated function systems the subsampling scheme is suitable in e g hierarchical image processing and image coding schemes for hexagonally sampled images a hierarchical subsampling structure is given which yields hexagon like regions with fractal borders

spectral transformation chains and some new biorthogonal rational functions a discrete time chain associated with the generalized eigenvalue problem for two jacobi matrices is derived various discrete and continuous symmetries of this integrable equation are revealed a class of its rational elementary and elliptic functions solutions appearing from a similarity reduction are constructed the latter lead to large families of biorthogonal rational functions based upon the very well poised balanced hypergeometric series of three types the standard hypergeometric series 9 f 8 basic series 10 9 and its elliptic analogue 10 e 9 for an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly

triangular 7 and quadrilateral 5 subdivision schemes regular case abstract this paper is devoted to the study of triangular 7 and quadrilateral 5 surface subdivisions both approximation and interpolatory subdivision schemes are considered with illustrative examples of both scalar valued and matrix valued 7 and 5 subdivision masks that satisfy the sum rule of sufficiently high orders in particular optimal sobolev smoothness is determined and holder smoothness estimates are presented

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

one dimensional local rings of maximal and almost maximal length let a m k denote a one dimensional cohen macaulay local ring with maximal ideal m and residue class field k we assume a is a reduced excellent local ring which has a canonical module a with no loss of generality we can also assume k is infinite let denote the integral closure of a in its total quotient ring q and let b denote the conductor of a in a has maximal length if la aa la ab t a here t a denotes the cohen macaulay type of a and la denotes the length of the a module a has almost maximal length if la aa la ab t a 1 the two main results of this paper are as follows a has maximal length if and only if either a is gorenstein or there exists an x m and a positive integer p such that m x b andba xpa let e a denote the multiplicity of a then a has almost maximal length and 1 t a e a if and only if there exists a transversal x of m such that m x b and la bxpa 1 here p min i xi b these two theorems generalize more specific results for semigroup rings obtained in brown and curtis numerical semigroups of maximal and almost maximal length semigroup forum vol 42 springer verlag berlin new york 1991 219 235

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

integrable discrete time chains for the frobenius stickelberger thiele polynomials the notion of frobenius stickelberger thiele fst polynomials is introduced spectral transformations for these polynomials analogous to the christoffel and geronimus transformations for orthogonal polynomials are constructed they yield an integrable discrete time chain the fst chain related to the generalized e algorithm relations of the fst polynomials to the pade interpolation problem and to general and symmetric biorthogonal rational functions are considered in detail

combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete toda molecules combinatorial expressions are presented of the solutions to initial value problems of the discrete and ultradiscrete toda molecules for the discrete toda molecule a subtraction free expression of the solution is derived in terms of non intersecting paths for which two results in combinatorics flajolet s interpretation of continued fractions and gessel viennot s lemma on determinants are applied by ultradiscretizing the subtraction free expression the solution to the ultradiscrete toda molecule is obtained it is finally shown that the initial value problem of the ultradiscrete toda molecule is exactly solved in terms of shortest paths on a specific graph the behavior of the solution is also investigated in comparison with the box ball system