general synthesis procedures for fir lossless transfer matrices for perfect reconstruction multirate filter bank applications a recently reported procedure see ibid vol assp 35 p 476 492 apr 1987 and proc ieee int conf acoust speech sig processing apr 1987 p 2169 2172 for the design of m channel perfect reconstruction quadrature mirror filter qmf bands uses the concept of lossless alias component matrices the synthesis of such qmf banks centers around the generation of a lossless m m finite impulse response fir transfer matrix e z recent results for such generation have been somewhat ad hoc i e not sufficiently general a general procedure is outlined for the generation of such transfer matrices the procedure is based on a cascaded lattice structure derived from a state space viewpoint the structure is such that it generates only m m lossless fir transfer matrices regardless of parameter values and conversely any m m lossless fir transfer matrix can be obtained by a suitable choice of parameters these parameters turn out to be angles theta sub k and the structure is such that the number of angles is minimal a design example is presented to demonstrate the main results

the index of an algebraic variety let k be the field of fractions of a henselian discrete valuation ring mathcal o k let xk k be a smooth proper geometrically connected scheme admitting a regular model x mathcal o k we show that the index xk k of xk k can be explicitly computed using data pertaining only to the special fiber xk k of the model x

tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

linkage and sums of ideals it is shown under mild conditions that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection for a sum of geometrically linked cohen macaulay ideals we compute the depths of the conormal module arld the first koszul homology as applications we construct general examples of ideals which are strongly cohenmacaulay strongly nonobstructed but not in the linkage class of a complete intersection and gorenstein ideals which are strongly nonobstructed but not syzygetic

perfect linkage of cohen macaulay modules over cohen macaulay rings abstract in this paper we introduce and study the notion of linkage by perfect modules which we call perfect linkage for cohen macaulay modules over cohen macaulay local rings we explore perfect linkage in connection with syzygies maximal cohen macaulay approximations and yoshino isogawa linkage we recover a theorem of yoshino and isogawa and analyze the structure of double perfect linkage moreover we establish a criterion for two cohen macaulay modules of codimension one to be perfectly linked and apply it to the classical linkage theory for ideals we also construct various examples of linkage of modules and ideals

heights of ideals of minors we prove new height inequalities for determinantal ideals in a regular local ring or more generally in a local ring of given embedding codimension our theorems extend and sharpen results of faltings and bruns

deformation and linkage of gorenstein algebras general double linkage of gorenstein algebras is defined rigidity genericity and regularity up to codimension six all pass across general double linkage rigid strongly unobstructed codimension four gorenstein algebras which lie in different herzog classes are produced introduction we begin by reviewing the classical structure theorems let p k xi x 41 a p i and f be the minimal resolution of a by free p modules f 1 if a has codimension 1 and i is unmixed then f is given by 0 p p for some fin p 2 hilbert burch 7 schaps 28 if a has codimension 2 and is cohen macaulay then f is given by x x 0 p n i ppnp where xis an n x n 1 matrix xi l 1l i x and a i x is the determinant formed by omitting the ith row from x 3 buchsbaum and eisenbud 4 if a has codimension 3 and its gorenstein then f is given by xt x x o p pn pn p where xis an n x n alternating matrix xi l i lpfi x and pfi x denotes the pfaffian of the alternating submatrix of x formed by omitting row and column i in these structure theorems as the codimension of a increases so does the number of conditions on a it is reasonable to suspect that a structure theorem for codimension four gorenstein algebras a will require that an additional hypothesis be imposed on a we observe that in each case a there is a structure theorem b all deformations are unobstructed see herzog 13 and c each algebra a is in the linkage class of a complete intersection see peskine and szpiro 27 and watanabe 30 historically these three facts were found independently with hindsight we now see that b follows from c and a follows in turn from b in his thesis received by the editors june 10 1982 and in revised form march 29 1983 some of the contents of this paper were presented to the special session on commutative algebra at the american mathematical society meeting in cincinnati on january 13 1982 1980 mathematics subject classification primary 13c05 13d25 13h10 secondary 14b07 14d115 14m05 16ao3 both authors were supported in part by the national science foundation 1984 american mathematical society 0002 9947 84 1 00 25 per page

classification of the tor algebras of codimension four almost complete intersections let r m k be a local ring in which 2 is a unit assume that every element of k has a square root in k we classify the algebras tor r r j k as j varies over all grade four almost complete intersection ideals in r the analogous classification has already been found when j varies over all grade four gorenstein ideals 21 and when j varies over all ideals of grade at most three 5 30 the present paper makes use of the classification in 21 of the tor algebras of codimension four gorenstein rings as well as the usually nonminimal dg algebra resolution of a codimension four almost complete intersection which is produced in 25 and 26

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

periodic boundary value problems for second order impulsive differential systems on etudie le probleme aux valeurs limites periodiques pour un systeme d equations differentielles d ordre 2 faiblement couplees avec impulsion

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

maximally decimated perfect reconstruction fir filter banks with pairwise mirror image analysis and synthesis frequency responses structures are presented for the perfect reconstruction quadrature mirror filter bank that are based on lossless building blocks these structures ensure that the frequency responses of the analysis and synthesis filters have pairwise symmetry with respect to pi 2 and require fewer parameters than recently reported structures also based on lossless building blocks the design time for the proposed structures is correspondingly much less than for the earlier methods which did not incorporate such symmetry

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

betti numbers of modules of exponent two over regular local rings abstract let r m k be a regular local ring of dimension n and let m be a finite length module over r this paper gives an affirmative answer to horrocks questions when m 2 m 0 that is in this case the rank of the i th syzygy of m is at least formula and the i th betti number of m is at least formula

existence of periodic solutions for first order differential equations in this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities namely the method of upper and lower solutions and the alternative method for nonlinear problems at resonance the results obtained are in terms of the behavior of the nonlinear part at infinity

polyphase networks block digital filtering lptv systems and alias free qmf banks a unified approach based on pseudocirculants the relationship between block digital filtering and quadrature mirror filter qmf banks is explored necessary and sufficient conditions for alias cancellation in qmf banks are expressed in terms of an associated matrix derived from the polyphase components of the analysis and synthesis filters these conditions called the pseudocirculant conditions make it possible to unite qmf banks with the framework of block digital filtering directly absence of amplitude distortion in an alias free qmf bank translates into the losslessness property of the pseudocirculant matrix involved

periodic solutions of some lienard and duffing equations on etudie certains aspects du probleme periodique u f u u g t u 0 u 0 u 2 u 0 u 2 ou f est une fonction continue sur r et g t u est une fonction de caratheodory soumise a certaines restrictions de croissances on donne deux resultats d existence

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

new quadrature mirror filter structures this paper introduces new quadrature mirror filter qmf structures for the frequency domain analysis and synthesis of digital signals the conventional qmf technique is first extended to cover complex quadrature mirror filters cqmf in which a digital signal is split into n adjacent complex subbands where the real and imaginary parts are subsampled by 1 2n with respect to the original signal the computational complexity of qmf banks is then analyzed and a new scheme which reduces the computational complexity by about a factor of two over conventional qmf implementations is proposed finally the filter design tradeoffs are discussed and the microprogramed implementation of qmf banks is evaluated

symbolic squares in regular local rings in this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal our techniques come from residual intersections integral closures of ideals and differentials

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

an application of liaison theory to the eisenbud green harris conjecture abstract in this paper we apply liaison theory to the eisenbud green harris conjecture and prove that the conjecture holds for a certain subclass of homogeneous ideals in the linkage class of a complete intersection ideal in the case of three variables we prove that the conjecture holds for gorenstein ideals

finite morphisms to projective space and capacity theory we study conditions on a commutative ring r which are equivalent to the following requirement whenever x is a projective scheme over s spec r of fiber dimension leq d for some integer d geq 0 there is a finite morphism from x to p d s over s such that the pullbacks of coordinate hyperplanes give prescribed subschemes of x provided these subschemes satisfy certain natural conditions we use our results to define a new kind of capacity for subsets of the archimedean points of projective flat schemes x over the ring of integers of a number field this capacity can be used to generalize the converse part of the fekete szeg h o theorem

bertini theorems over finite fields let x be a smooth quasiprojective subscheme of pn of dimension m i 0 over fq then there exist homogeneous polynomials f over fq for which the intersection of x and the hypersurface f 0 is smooth in fact the set of such f has a positive density equal to aex m 1 1 where aex s zx q s is the zeta function of x an analogue for regular quasiprojective schemes over z is proved assuming the abc conjecture and another conjecture

on the projective dimension and the unmixed part of three cubics let r be a polynomial ring over a field in an unspecified number of variables we prove that if j r is an ideal generated by three cubic forms and the unmixed part of j contains a quadric then the projective dimension of r j is at most 4 to this end we show that if k r is a three generated ideal of height two and l r an ideal linked to the unmixed part of k then the projective dimension of r k is bounded above by the projective dimension of r l plus one

globally generated vector bundles on p1 p1 p1 with low first chern classes abstract we classify globally generated vector bundles on p 1 p 1 p 1 with small first chern class i e c 1 a 1 a 2 a 3 a i 2 our main method is to investigate the associated smooth curves to globally generated vector bundles via the hartshorne serre correspondence

a deformation theory approach to betti numbers of finite length modules let r k x xd be a polynomial ring in d variables with d at least 2 over a field k and let a4 be a finite length r module buchsbaum and eisenbud b e conjecture that the ith betti number pi should be at least f in their work on linkage huneke and ulrich h u prove this if m r i where i is a licci ideal by observing on pages 317 318 that after deforming and localizing they can assume that z is a complete intersection this paper extends this remark to ideals generated by monomials the technique works because deformation preserves the betti numbers exactly while localization can only make them go down in this paper we consider a sequence of localizations and deformations until we arrive at a module whose betti numbers we can compute in c one of us showed that if r kcx i xd and z was an ideal generated by monomials in the xi such that r z co and z is not an r sequence then the pi satisfy stronger inequality that pi 2 f ii in this paper we give a deformation theoretic proof of this by showing theorem l that if z is a finite colength ideal generated by monomials and is not a complete intersection then after a sequence of deformations and localizations we arrive at a new monomial ideal generated by d 1 elements consisting of powers of each of the xi and one other monomial

singularities of generic linkage of algebraic varieties let y be a generic link of a subvariety x of a nonsingular variety a we give a description of the grauert riemenschneider canonical sheaf of y in terms of the multiplier ideal sheaves associ ated to x and use it to study the singularities of y as the first application we give a criterion when y has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage as another application we give a quick and simple liaison method to generalize the results of de fernex ein and chardin ulrich on the castelnuovo mumford regularity bound for a projective variety

ideals having the expected reduction number 1 introduction let r be a noetherian local ring with infinite residue field k and let be an r ideal when studying algebraic properties of the blow up of spec along v l it is often important to have good upper bounds for the reduction number of the ideal recall that an ideal c is a reduction of if the extension of rees algebras r jt c r it is module finite or equivalently if ir l j r for some r 0 32 the least such r is denoted by rj i a reduction is minimal if it is minimal with respect to inclusion and the reduction number r l of is defined as min o j a minimal reduction of finally the analytic spread i of is the krull dimension of the fiber ring r lt r k or equivalently the minimal number of generators jbi j of any minimal reduction j of 1 32 philosophically speaking j is a simplification of with the reduction number r l measuring how closely the two ideals are related of the numerous recent results about rees algebras and associated graded rings of ideals having small reduction number 18 19 38 35 12 13 14 4 3

tight closure and linkage classes in gorenstein rings we study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class

on 2 4 complete intersection threefolds that contain an enriques surface we study nodal complete intersection threefolds of type 2 4 in mathbb p 5 which contain an enriques surface in its fano embedding we completely determine calabi yau birational models of a generic such threefold these models have hodge numbers h 11 2 h 12 32 we also describe calabi yau varieties with hodge numbers h 11 h 12 equal to 2 26 23 5 and 31 1 the last two pairs of hodge numbers are to the best of our knowledge new

links of symbolic powers of prime ideals in this paper we prove the following let r frak m be a cohen macaulay local ring of dimension d 2 suppose that either r is not regular or if r is regular assume that d 3 let t 2 be a positive integer if alpha 1 dots alpha d is a regular sequence contained in frak m t then alpha 1 dots alpha d frak m t subseteq frak m t this result gives an affirmative answer to a conjecture raised by polini and ulrich

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

hypersurfaces in projective schemes and a moving lemma let x s be a quasi projective morphism over an affine base we develop in this article a technique for proving the existence of closed subschemes h s of x s with various favorable properties we offer several applications of this technique including the existence of finite quasi sections in certain projective morphisms and the existence of hypersurfaces in x s containing a given closed subscheme c and intersecting properly a closed set f assume now that the base s is the spectrum of a ring r such that for any finite morphism z s pic z is a torsion group this condition is satisfied if r is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field we prove in this context a moving lemma pertaining to horizontal 1 cycles on a regular scheme x quasi projective and flat over s we also show the existence of a finite surjective s morphism to psd for any scheme x projective over s when x s has all its fibers of a fixed dimension d

numerical invariants of liaison classes 302 c huneke intersection what properties are then enjoyed by v for instance buchweitz was able to show that all such v are strongly nonobstructed see 5 for details the two questions above have been completely answered if x spec r with r a regular local ring and if codim v 2 in this case

sums of linked ideals it is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection conversely every gorenstein ideal of height at least two in the linkage class of a complete intersection can be obtained as a generalized localization of a sum of two geometrically linked ideals in the linkage class of a complete intersection we also investigate sums of doubly linked gorenstein ideals as an application we construct a perfect prime ideal which is strongly nonobstructed but not strongly cohen macaulay and a perfect prime ideal which is not strongly nonobstructed but whose entire linkage class is strongly cohen macaulay

subband decomposition an lms based algorithm to approximate the perfect reconstruction bank in the general case an algorithm based on least mean squares lms is described given an arbitrary invertible decomposition decimation process the algorithm will find the finite impulse response reconstruction filters which best approximate the perfect reconstruction ones by allowing the reconstruction filters impulse responses to be sufficiently long the quality of the approximation can be made as good as required two examples are presented for the implementation of this algorithm one in the case of a decomposition by a filter bank of galand 1977 where the reconstruction bank is already known the other in the situation of a two subband decomposition where one of the subbands covers two thirds of the frequency space and the other covers the remaining one third