erratum to on the complete intersection conjecture of murthy with an example in complete intersections j algebra 458 2016 156 170 abstract this is an erratum to 4 and as well to 3 this article provides examples to indicate inconsistencies in 3 4

generalized umbrella rings in this paper the notion of generalized umbrella rings is introduced we show that every generalized umbrella ring is a stably coherent ring as an application we show that every finitely generated projective module over r x is free if r is a generalized umbrella ring with weak global dimension two

orthogonal multiwavelets of multiplicity four abstract we consider solutions of a system of refinement equations with a 4 1 function vector and three nonzero 4 4 coefficient matrices we give explicit expressions of coefficient matrices such that the refinement function vector and the corresponding wavelet vector have properties of short support 0 2 symmetry or antisymmetry and orthogonality the properties of convergence of the subdivision scheme approximation order and smoothness of the refinement functions are also discussed

laplace approximations for sums of independent random vectors we consider expressions of the form zn e exp nf sn n where sn is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log zn supxf x h x where h is the so called entropy function in an earlier paper a more precise evaluation of zn is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

unimodular rows over monoid rings abstract for a commutative noetherian ring r of dimension d and a commutative cancellative monoid m the elementary action on unimodular n rows over the monoid ring r m is transitive for n max d 2 3 the starting point is the case of polynomial rings considered by a suslin in the 1970s the main result completes a project initiated in the early 1990s and suggests a new direction in the study of k theory of monoid rings

euler class group of a laurent polynomial ring local case abstract let r be a noetherian commutative ring of dimension n 2 and let a r t t 1 assume that the height of the jacobson radical of r is at least 2 let p be a projective a module of rank n dim a 1 with trivial determinant we define an abelian group called the euler class group of a denoted by e a let be an isomorphism from a to det p to the pair p we associate an element of e a called the euler class of p denoted by e p then we prove that a necessary and sufficient condition for p to have a unimodular element is the vanishing of e p in e a earlier bhatwadekar and raja sridharan have defined the euler class group of r denoted by e r and have proved similar results for projective r module of rank n later m k das defined the euler class group of the polynomial ring r t denoted by e r t and proved similar results for projective r t modules of rank n with trivial determinant

vanishing of euler class groups abstract in this paper we prove some theorems about vanishing of euler class groups for example suppose b a x f 1 where a x is a polynomial ring f a x is a non zero divisor and dim b dim a 1 3 then we prove that the euler class group e b l 0 for any rank one projective b module l

a note on projective modules over real affine algebras let a be an affine domain over r of dimension d let f a be an element not belonging to any real maximal ideal of a and let p be a projective a module of rank d 1 let a p af pf be a unimodular element and q af pf a p af if p is free then a result of ojanguren and parimala 6 theorem shows that q is extended from a a consequence of this result is that if d 3 then all projective modules over af are free where a r x1 x2 x3 see 6 for motivation in this paper we prove the following result 3 10 which is a generalization of the above result of ojanguren and parimala

injective stability for k 1 of classical modules in rao 1994 14 the second author and w van der kallen showed that the injective stabilization bound for k1 of the general linear group is d 1 over a regular affine algebra over a perfect c1 field where d is the krull dimension of the base ring which is finite and at least 2 in this article we prove that the injective stabilization bound for k1 of the symplectic group is d 1 over a geometrically regular ring containing a field where d is the dimension of the base ring which is finite and at least 2 using the local global principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d 1 for k1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above

on a question of quillen let r be a regular local ring and f a regular parameter of r quillen asked whether every projective r f module is free we settle this question when r is a regular local ring of an affine algebra over a field k further if r has infinite residue field we show that projective modules over laurent polynomial extensions of r f are also free

some properties of projective modules over discrete hodge algebras all rings in this paper are assumed to be commutative with 1 and all modules finitely generated let b be a discrete hodge algebra over a r ing a by definition this means that b r i where r a xi x and i is an ideal in r generated by monomials t vorst showed in 1 12 that thebehavior of projective modules over such b is very similar to the case of an ordinary polynomial ring in the following sense if all projective modules over any polynomial ring with ground ring a are extended from a then every projective module over a discrete hodge algebra over a is extended from a let b be a discrete hodge algebra over a noetherian ring a and p a projective module over b s mandal proved in 7 that if rank p dim b dim a 1 then

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

applications of stein s method for concentration inequalities stein s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and gibbs measures in this paper we provide some extensions of the theory and three applications 1 we obtain a concentration inequality for the magnetization in the curie weiss model at critical temperature where it obeys a nonstandard normalization and super gaussian concentration 2 we derive exact large deviation asymptotics for the number of triangles in the erd s renyi random graph g n p when p 0 31 similar results are derived also for general subgraph counts 3 we obtain some interesting concentration inequalities for the ising model on lattices that hold at all temperatures

stability results for projective modules over blowup rings let r be a normal affine domain of dimension n 3 over an algebraically closed field k suppose chark 0 or chark p n let g f1 fr be a r regular sequence and a r f1 g fr g let p be a stably free a module of rank n 1 then murthy proved that there exists a projective r module q such that q r a p and n 1 q r 9 theorem 2 10 as a consequence of murthy s result if f g c x1 xn with g 0 then all stably free modules over c x1 xn f g of rank n 1 are free 9 corollary 2 11 in this paper we prove the following result 3 5 which generalizes the above result of murthy

laplace approximations for sums of independent random vectors ii degenerate maxima and manifolds of maxima we consider expressions of the form z n e exp nf s n n where s n is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log z n sup x f x h x where h is the so called entropy function in an earlier paper a more precise evaluation of z n is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

matrix refinement type equations abstract taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the fourier transform of an l 1 solution of the following random matrix refinement type equation f x det l c f l x m p d where m r p is a random vector and l r p p c r q q are random matrices moreover the dimension of the space of all l 1 solutions of this refinement type equation is examined

approximation by translates of refinable functions the functions f 1 x dots f r x are refinable if they are combinations of the rescaled and translated functions f i 2x k this is very common in scientific computing on a regular mesh the space v 0 of approximating functions with meshwidth h 1 is a subspace of v 1 with meshwidth h 1 2 these refinable spaces have refinable basis functions the accuracy of the computations depends on p the order of approximation which is determined by the degree of polynomials 1 x dots x p 1 that lie in v 0 most refinable functions such as scaling functions in the theory of wavelets have no simple formulas the functions f i x are known only through the coefficients c k in the refinement equation scalars in the traditional case r times r matrices for multiwavelets the scalar sum rules that determine p are well known we find the conditions on the matrices c k that yield approximation of order p from v 0 these are equivalent to the strang fix conditions on the fourier transforms hat f i omega but for refinable functions they can be explicitly verified from the c k

gibbs phenomenon for nonnegative compactly supported scaling vectors abstract this paper considers gibbs phenomenon for scaling vectors in l 2 r we first show that a wide class of multiresolution analyses suffer from gibbs phenomenon to deal with this problem in contemp math 216 1998 63 79 walter and shen use an abel summation technique to construct a positive scaling function p r 0 r 1 from an orthonormal scaling function that generates v 0 a reproducing kernel can in turn be constructed using p r this kernel is also positive has unit integral and approximations utilizing it display no gibbs phenomenon these results were extended to scaling vectors and multiwavelets in proceedings of wavelet analysis and multiresolution methods 2000 pp 317 339 in both cases orthogonality and compact support were lost in the construction process in this paper we modify the approach given in proceedings of wavelet analysis and multiresolution methods 2000 pp 317 339 to construct compactly supported positive scaling vectors while the mapping into v 0 associated with this new positive scaling vector is not a projection the scaling vector does produce a riesz basis for v 0 and we conclude the paper by illustrating that expansions of functions via positive scaling vectors exhibit no gibbs phenomenon

a matrix valued wavelet kl like expansion for wide sense stationary random processes matrix valued wavelet series expansions for wide sense stationary processes are studied in this paper the expansion coefficients a are uncorrelated matrix random process which is a property similar to that of a matrix karhunen loe spl grave ve mkl expansion unlike the mkl expansion however the matrix wavelet expansion does not require the solution of the eigen equation this expansion also has advantages over the fourier series which is often used as an approximation to the mkl expansion in that it completely eliminates correlation the basis functions of this expansion can be obtained easily from wavelets of the matrix valued lemarie spl acute meyer type and the power spectral density of the process

multiwavelet density estimation accurate density estimation methodologies play an integral role in a variety of scientific disciplines with applications including simulation models decision support tools and exploratory data analysis in the past histograms and kernel density estimators have been the predominant tools of choice primarily due to their ease of use and mathematical simplicity more recently the use of wavelets for density estimation has gained in popularity due to their ability to approximate a large class of functions including those with localized abrupt variations however a well known attribute of wavelet bases is that they cannot be simultaneously symmetric orthogonal and compactly supported multiwavelets more general vector valued constructions of wavelets overcome this disadvantage making them natural choices for estimating density functions many of which exhibit local symmetries around features such as a mode we extend the methodology of wavelet density estimation to use multiwavelet bases and illustrate several empirical examples of multiwavelet density estimation

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

construction of biorthogonal multiwavelets abstract there are perfect formulas for the constructions of biorthogonal uniwavelets let x k z p k 2x k x k z p k 2x k be a pair of biorthogonal uniscaling functions then a pair of biorthogonal uniwavelet associated with the above biorthogonal uniscaling functions can be easily expressed as x k z 1 k 1 p 1 k 2x k x k z 1 k 1 p 1 k 2x k however it seems that there is not such a good formula of similar structure for biorthogonal multiwavelets in this paper we will give a procedure for constructing compactly supported biorthogonal multiwavelets which makes construction of biorthogonal multiwavelets easy like in the construction of biorthogonal uniwavelet our approach is also suitable for the case of compactly supported orthogonal multiwavelets four examples for constructing multiwavelets are given

vector valued wavelets and vector filter banks in this paper we introduce vector valued multiresolution analysis and vector valued wavelets for vector valued signal spaces we construct vector valued wavelets by using paraunitary vector filter bank theory in particular we construct vector valued meyer wavelets that are band limited we classify and construct vector valued wavelets with sampling property as an application of vector valued wavelets multiwavelets can be constructed from vector valued wavelets we show that certain linear combinations of known scalar valued wavelets may yield multiwavelets we then present discrete vector wavelet transforms for discrete time vector valued or blocked signals which can be thought of as a family of unitary vector transforms

lifting scheme for biorthogonal multiwavelets originated from hermite splines we present new multiwavelet transforms of multiplicity 2 for manipulation of discrete time signals the transforms are implemented in two phases 1 pre post processing which transforms the scalar signal into a vector signal and back and 2 wavelet transforms of the vector signal both phases are performed in a lifting manner we use the cubic interpolatory hermite splines as a predicting aggregate in the vector wavelet transform we present new pre post processing algorithms that do not degrade the approximation accuracy of the vector wavelet transforms we describe two types of vector wavelet transforms that are dual to each other but have similar properties and three pre post processing algorithms as a result we get fast biorthogonal algorithms to transform discrete time signals that are exact on sampled cubic polynomials the bases for the transform are symmetric and have short support

orthogonal multiwavelets with optimum time frequency resolution a procedure to design orthogonal multiwavelets with good time frequency resolution is introduced formulas to compute the time durations and the frequency bandwidths of scaling functions and multiwavelets are obtained parameter expressions for the matrix coefficients of the multifilter banks that generate symmetric antisymmetric scaling functions and multiwavelets supported in o n are presented for n 2 6 orthogonal multiwavelets with optimum time frequency resolution are constructed and some optimal multifilter banks are provided

multiwavelet frames in signal space originated from hermite splines we present a method for construction of multiwavelet frames for manipulation of discrete signals the frames are generated by three channel perfect reconstruction oversampled multifilter banks the design of the multifilter bankstarts from a pair of interpolatory multifilters we derive these interpolatory multifilters from the cubic hermite splines we use the original preprocessing algorithms which transform scalar signals into vector arrays that serve as inputs to the oversampled analysis multifilter banks these preprocessing algorithms do not degrade the approximation accuracy of the transforms of the vectors by multifilter banks the postprocessing algorithms convert the vector output of the synthesis multifilter banks into scalar signal the discrete framelets generated by the designed filter banks are symmetric and have short support the analysis framelets have four vanishing moments whereas the synthesis framelets converge to hermite splines supported on the interval 1 1

pic is a contracted functor we show that there is a natural decomposition pic a t t 1 cong pic a oplus npic a oplus npic a oplus h 1 a for any commutative ringa where pic a is the picard group of invertiblea modules andh1 a is the etale cohomology grouph1 spec a a similar decomposition of pic x t t 1 holds for any schemex this makes pic a contracted functor in the sense of bass h1 a is always a torsionfree group and is zero ifa is normal for pseudo geometric rings h1 a is an effectively computable finitely generated free abelian group we also show thath1 a t t 1 h1 a i e nh1 lh1 0 this yields the formula for group rings pic a t 1 t 1 1 t m t m 1 cong pic a oplus coprod limits i 1 m h 1 a oplus coprod limits k 1 m coprod limits i 1 2 k left k m right n k pic a

projective modules over overrings of polynomial rings abstract let a be a commutative noetherian ring of dimension d and let p be a projective r a x 1 x l y 1 y m 1 f 1 f m module of rank r max 2 dim a 1 where f i a y i then i the natural map r gl r r el r 1 r k 1 r is surjective 3 8 ii assume f i is a monic polynomial then r 1 is an isomorphism 3 8 iii el 1 r p acts transitively on um r p in particular p is cancellative 3 12 iv if a is an affine algebra over a field then p has a unimodular element 3 13 in the case of laurent polynomial ring i e f i y i i ii are due to suslin 1977 12 iii is due to lindel 1995 4 and iv is due to bhatwadekar lindel and rao 1985 2

a method of construction for biorthogonal multiwavelets system with 2r multiplicity biorthogonal multiwavelets systems are consisted of a pair of biorthogonal multiscaling functions and the corresponding a pair of multiwavelets a method for constructing biorthogonal multiwavelets systems with multiplicity 2r was derived in particular hermite interpolation mask was considered as primal masks a pair of primal and dual masks with high order of sum rules should be firstly designed in order to obtain a biorthogonal multiwavelet with high vanishing moment concretely starting from a short sequence primal masks possessing hermite interpolation properties a necessary and sufficient conditions such that primal masks satisfies the preassigned order of sum rules are established then a dual masks with short support symmetry and any preassigned order of sum rules is constructed it follows that two multiscaling functions are obtained and then multiwavelets and dual multiwavelets are derived from which a general design framework is obtained for constructing biorthogonal multiwavelets system with multiplicity 2r associated with hermite interpolation functions the biorthogonal multiwavelets constructed by our design framework has desirable properties such as symmetry short support high vanishing moments and simple structures with explicit expressions finally an example is given

refinement of reproducing kernels we continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel hilbert space associated with the refinement kernel contains that with the original kernel as a subspace to motivate this study we first develop a refinement kernel method for learning which gives an efficient algorithm for updating a learning predictor several characterizations of refinement kernels are then presented it is shown that a nontrivial refinement kernel for a given kernel always exists if the input space has an infinite cardinal number refinement kernels for translation invariant kernels and hilbert schmidt kernels are investigated various concrete examples are provided

the serre problem for discrete hodge algebras all rings in this paper will be commutative and noetherian let a b be a split injection of rings a finitely generated projective b module is called extended from a if there exists a finitely generated projective a module q such that p q b furthermore we call an c eglr b resp slr b extended from a if can be written as f i with fi glr a resp slr a and 7 e b the group of elementary matrices now let a be a ring in this paper we will consider a algebras of the following type a x1 xn i where i is an ideal generated by monomials such an a algebra is called a discrete hodge algebra over a and vice versa every discrete hodge algebra over a is of this type 5 for the proofs of most of our results we can restrict ourselves to the special class of square free discrete hodge algebras over a by which is meant that i can be generated by square free monomials the varieties over a corresponding to these square flee discrete hodge algebras are unions of intersections of coordinate hyperplanes h i x 1 xn anixi 0 i 1 n typical examples are a x 1 xn and a x 1 x 2 x i x 2 the main result of this paper is the following

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

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

multirate filter banks with block sampling multirate filter banks with block sampling were recently studied by khansari and leon garcia 1993 in this paper we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices respectively we show that multirate filter banks with block sampling studied by khansari and leon garcia are special vector filter banks where the transfer matrices are pseudocirculant we present some fundamental properties for the basic building blocks such as noble identities interchangeability of down up sampling polyphase representations of m channel vector filter banks and multirate filter banks with block sampling we then present necessary and sufficient conditions for the alias free property finite impulse response fir systems with fir inverses paraunitariness and lattice structures for paraunitary vector filter banks we also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling as an application of this theory we present all possible perfect reconstruction delay chain systems with block sampling we also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling in this paper we also present a connection between vector filter banks and vector transforms studied by li vector filter banks also play important roles in multiwavelet transforms and vector subband coding

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

image compression through embedded multiwavelet transform coding in this paper multiwavelets are considered in the context of image compression and two orthonormal multiwavelet bases are experimented each used in connection with its proper prefilter for evaluating the effectiveness of multiwavelet transform for coding images at low bit rates an efficient embedded coding of multiwavelet coefficients has been realized the performance of this multiwavelet based coder is compared with the results obtained for scalar wavelets

matrix extension and biorthogonal multiwavelet construction abstract suppose that p z and p z are two r n matrices over the laurent polynomial ring r z where r p z p z i r on the unit circle t we develop an algorithm that produces two n n matrices q z and q z over r z satisfying the identity q z q z i n on t such that the submatrices formed by the first r rows of q z and q z are p z and p z respectively our algorithm is used to construct compactly supported biorthogonal multiwavelets from multiresolutions generated by univariate compactly supported biorthogonal scaling functions with an arbitrary dilation parameter m z where m 1

a general approach for analysis and application of discrete multiwavelet transforms this paper proposes a general paradigm for the analysis and application of discrete multiwavelet transforms particularly to image compression first we establish the concept of an equivalent scalar wavelet filter bank system in which we present an equivalent and sufficient representation of a multiwavelet system of multiplicity r in terms of a set of r equivalent scalar filter banks this relationship motivates a new measure called the good multifilter properties gmps which define the desirable filter characteristics of the equivalent scalar filters we then relate the notion of gmps directly to the matrix filters as necessary eigenvector properties for the refinement masks of a given multiwavelet system second we propose a generalized efficient and nonredundant framework for multiwavelet initialization by designing appropriate preanalysis and post synthesis multirate filtering techniques finally our simulations verified that both orthogonal and biorthogonal multiwavelets that possess gmps and employ the proposed initialization technique can perform better than the popular scalar wavelets such as daubechies d8 wavelet and the d 9 7 wavelet and some of these multiwavelets achieved this with lower computational complexity

an algorithm for matrix extension and wavelet construction this paper gives a practical method of extending an n x r matrix p z r n with laurent polynomial entries in one complex variable z to a square matrix also with laurent polynomial entries if p z has orthonormal columns when z is restricted to the torus t it can be extended to a paraunitary matrix if p z has rank r for each z t it can be extended to a matrix with nonvanishing determinant on t the method is easily implemented in the computer it is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter

on automorphisms of modules over polynomial rings write f xfi bxf then f is in xp and f q 0 clearly 1 qf is in el p x and it is a lift of 1 pg in a similar way we can lift 1 pg when g is unimodular this completes the proof of 2 8 the following is a variant of a proposition of lindel l 2 71 see 4 3 for the statement like the proposition of lindel in his paper this proposi tion plays a key roll in the proof of our main theorems 3 1 and 3 2 we shall go into detailed discussions on lindel s proposition l 2 71 in our later sections sects 4 5 6 and now we shall state our proposition 2 9

stability theorems for overrings of polynomial rings let r be a polynomial ring over a field projective modules over rings of the type r ix y r x y where r is a non zero element of r have been considered by murthy swan weibel see mu and w these rings lie between r x and r x x 1 and even when they are regular projective modules over them need not be free mu example 6 2 this leads us to study stability properties of projective modules over rings a which lie between r x and r x x 1 when r is any commutative noetherian ring we prove stability theorems for projective modules over such rings a in w these results theorems 4 2 and 4 3 have been proved for a r x by plumstead p and for a r x x 1 by mandal ma in w we prove similar results for the allied class of rings d of the type r x y xy stability theorems for gl a and gl d are proved in w when a r x or r x x 1 our theorems 6 2 i and 6 4 i had already been proved by suslin s finally in w 7 we study pic r x y r xy when r is a pid theorem 7 1 extends a result of murthy mu corollary 5 3 it would be interesting to know whether theorems 4 1 and 4 2 can be extended to the polynomial ring a t 1 tm where a lies between r x and r x x 1 more precisely for a projective a t 1 t module p of rank dim a we would like to know if the following statements are true i p has a unimodular element ii p has the cancellation property we have been able to show that when a r x the statement i is true b r theorem 3 1

an algorithm for constructing symmetric orthogonal multiwavelets by matrix symmetric extension symmetric extensions of paraunitary matrix is studied starting from a given m band compactly supported orthogonal multiscaling function with some symmetric property the problem to construct a corresponding m band multiwavelet which have the symmetric property can be reduced to the symmetric paraunitary extension of a given matrix it can be shown that such matrix extension is always solvable and an algorithmic approach to solve the symmetric paraunitary extension problem is presented this leads to a practical method for the construction of compactly supported symmetric orthogonal multiwavelets from a given multiscaling functions with an arbitrary integer dilation factor m 1 as an application some examples are given in particular the well known chui lian multiwavelets is recovered via the method

on the existence of unimodular elements and cancellation of projective modules over noetherian and non noetherian rings let r be a commutative ring of dimension d s r x or r x 1 x and p a finitely generated projective s module of rank r then p is cancellative if p has a unimodular element and r d 1 moreover if r dim s then p has a unimodular element and therefore p is cancellative as an application we prove that if r is a ring of dimension d of finite type over a prufer domain and p is a projective r x or r x 1 x module of rank at least d 1 then p has a unimodular element and is cancellative

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform

the euler class group of a polynomial algebra in this paper we continue our study on the theory of the euler class group of a polynomial algebra a t where a is a commutative noetherian ring containing q of dimension n for such a ring a in 9 we defined the notion of the nth euler class group en a t of a t for simplicity let us call it e a t in 9 we also studied the relations between e a t and e a where e a is the nth euler class group of a for example there is canonical map e a e a t which is an injective group homomorphism and it is an isomorphism when a is a smooth affine domain 9 proposition 5 7 in general these two groups are not isomorphic see discussion preceding 9 proposition 5 7 in this context the following question is natural