traces of sobolev functions on fractal type sets and characterization of extension domains abstract we describe traces of sobolev functionsu w1 p r n 1 r nin terms of sobolev spaces on metric spaces 7 our results apply to smooth submanifolds fractal subsets as well as to open subsets of r n in particular if 0 r nis a john domain then we characterize thosew1 p functions which can be extended tow1 p r n if is uniform then this result implies jones extension theorem 14 in the case of traces on fractal subsets our results are related to those of jonsson and wallin 16

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

hilbert coefficients and the gorenstein property of the associated graded ring in this article we investigate the gorenstein property of the associated graded ring ga i of an ideal i in a gorenstein local ring a m of positive dimension we especially concentrate on the case where i is m primary assuming that the rees algebra of i is cohen macaulay we then give necessary and sufficient conditions for the gorensteiness of ga i in terms of the hilbert coefficients of i

decoupling elastic waves and its applications abstract in this paper we consider time harmonic elastic wave scattering governed by the lame system it is known that the elastic wave field can be decomposed into the shear and compressional parts namely the pressure and shear waves that are generally coexisting but propagating at different speeds we consider the third or fourth kind impenetrable scatterer and derive two geometric conditions respectively related to the mean and gaussian curvatures of the boundary surface of the scatterer that can ensure the decoupling of the shear and pressure waves the decoupling results are new to the literature and are of significant interest for their own sake as an interesting application we apply the decoupling results to the uniqueness and stability analysis for inverse elastic scattering problems in determining polyhedral scatterers by a minimal number of far field measurements

layer stripping for a transversely isotropic elastic medium in this paper we develop a layer stripping algorithm for transversely isotropic elastic materials in three dimensional space we first prove that by making displacement and traction measurements at the boundary of these elastic materials one can determine the elastic tensor and all its derivatives at the boundary the elastic measurements made at the boundary are encoded in the dirichlet to neumann dn map then we use the riccati equation developed in g nakamura and g uhlmann a layer stripping algorithm in elastic impedance tomography in inverse problems in wave propagation ima vol math appl 90 springer verlag new york 1997 pp 375 384 for the dn map in any anisotropic elastic medium to approximate the elastic tensor in the interior

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

on the depth of the fiber cone of filtrations preprint enviat per a la seva publicacio en una revista cientifica journal of algebra 15 december 1997 volume 198 issue 2 pages 428 445 https doi org 10 1006 jabr 1997 7147

on near cloak in acoustic scattering abstract invisibility cloaking in acoustic scattering via the approach of transformation optics is considered the near cloaks of both passive medium and active radiating object are investigated from a practical viewpoint we are especially interested in the cloaking of an arbitrary but regular content it is shown that one cannot achieve the near cloak unless some special mechanism is introduced into the construction a general lossy layer is incorporated into the construction of our near cloaking devices we derive very accurate estimates of the scattering amplitude in terms of the regularization parameter and the material parameters of the lossy layer in different settings

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium we consider an isotropic lame system for an elastic medium consisting of finitely many imperfections of small diameter embedded in a homogeneous reference medium the lame coefficients of the imperfections are different from those of the background medium first we establish an asymptotic formula for the displacement vector in terms of the reference lame coefficients the location of the imperfections and their geometry second we use this asymptotic expansion to establish continuous dependence estimates for certain characteristics of the imperfections in terms of the boundary data and to derive integral boundary formulae that yield to a very effective identification procedure

a characterization of two dimensional rational singularities via core of ideals abstract the notion of p g ideals for normal surface singularities has been proved to be very useful on the other hand the core of ideals has been proved to be very important concept and also very mysterious one however the computation of the core of an ideal seems to be given only for very special cases in this paper we will give an explicit description of the core of p g ideals of normal surface singularities as a consequence we give a characterization of rational singularities using the inclusion of the core of integrally closed ideals

uniqueness of the density in an inverse problem for isotropic elastodynamics we consider the unique determination of the density of a nonhomogeneous isotropic elastic object from measurements made at the surface we model the behavior of the bounded 3 dimensional object by the linear hyperbolic system of operators for isotropic elastodynamics the material properties of the object its density and elastic properties correspond to the smooth coefficients of these differential operators the data for this inverse problem in the form of the correspondence between applied surface tractions and resulting surface displacements is modeled by the dynamic dirichlet to neumann map on a finite time interval in an earlier paper we show that the speeds c p s of compressional and sheer wave propagation through the object are uniquely determined by the dirichlet to neumann map here we extend that result by showing that the density is also determined in the interior by the dirichlet to neumann map in the case for example that c p 2c s at only isolated points in the object we use techniques from microlocal analysis and integral geometry to solve this fully three dimensional problem

the almost gorenstein rees algebras of parameters abstract there is given a characterization for the rees algebras of parameters in a gorenstein local ring to be almost gorenstein graded rings a characterization is also given for the rees algebras of socle ideals of parameters the latter one shows almost gorenstein rees algebras rather rarely exist for socle ideals if the dimension of the base local ring is greater than two

the trace problem for vector fields satisfying hormander s condition trace theorems are proved for non isotropic sobolev and l p lipschitz spaces defined by vector fields satisfying hormander s bracket condition of order 2 it is shown that the loss of regularity by traces is the same as in the classical case

trace theorems for vector fields in the setting of carnot caratheodory spaces we prove some trace theorems for sobolev functions we consider the trace on a non characteristic surface for hormander vector fields of step r ge 1 and the trace on the boundary of a class of domains in the grushin plane

gorenstein graded rings associated to ideals let a be a noetherian local ring with the maximal ideal m and dim a d let i 6 a be an ideal in a and s hta i this article studies the question of when the associated graded ring g i i 0 i i is gorenstein to state our result we set up some notation let be an integer such that d and let j be a reduction of i generated by elements a1 a2 a we denote by rj i the reduction number of i with respect to j the analytic spread of i is i dim a m a g i then s i we always assume that the generating set a1 a2 a of j is a basic generating set for j in the sense of aberbach huneke and trung aht which means that jiaq is a reduction of iaq for all q v i with i hta q here let v i be a set of prime ideals in a containing i and ji a1 a2 ai for 0 i by aht 7 2 there always exists a basic generating set for j if the field a m is infinite let

on the gorenstein property of rees and form rings of power of ideals in this paper we determine the exponents n for which the rees ring r in and the form ring gra in are gorenstein rings where i is a strongly cohen macaulay ideal of linear type including complete and almost complete intersections or an m primary ideal in a local ring a with maximal ideal m given an ideal i in a local ring a m it is well known that the cohenmacaulayness of the rees algebra r i implies the cohen macaulayness of all rees algebras r in the same is true for the form rings gra i and gra in see 3 2 7 8 and 8 8 5 in this paper we show that in contrast to the cohenmacaulay property the gorenstein property of r in and gra in only holds for special exponents n if in particular gra i is gorenstein and r i is cohen macaulay it turns out that these special exponents are closely related to the a invariant of the form ring gra i mainly under this aspect we prove some results concerning the gorenstein property of rees and form rings of powers of i strongly cohen macaulay ideals of linear type including almost complete intersections in gorenstein rings ii m primary ideals in cohen macaulay rings and iii equimultiple prime ideals p m in a generalized cohen macaulay ring our investigations are essentially based on the explicit computation of the a invariants of form and rees rings see 2 for the above classes of idealsusing a structure theorem for the canonical module of r i in 7 we can determine in 3 the exponents n 1 for which r in and gra in are gorenstein rings see in particular theorem 3 5 a more geometrical interpretation of the results in 3 is the observation that in these situations the gorenstein property of proj r i can be deduced from the gorenstein property of a certain veronesean subring r in of r i it might be an interesting question when the gorenstein property of a blow up proj r i is inherited to an appropriate rees ring r j with proj r j proj r i over speca in 4 part i we characterize for cohen macaulay or gorenstein rings a the gorenstein property of r md i i 1 2 3 by conditions on the received by the editors june 30 1991 and in revised form april 10 1992 1991 mathematics subject classification primary 13d03 13h10 13h15 secondary 14b05 14b 1 5 1994 american mathematical society 0002 9947 94 1 00 25 per page

variational approach to shape derivatives for elasto acoustic coupled scattering fields and an application with random interfaces abstract we establish the theoretical results governed by helmholtz equation and lame system of shape derivatives of solutions to the elasto acoustic coupled scattering problem the primary techniques use the variational approach and the admissible perturbation characterized by the velocity method unlike perturbations of the boundary in the normal direction the velocity method is introduced to conduct sensitivity analysis for an arbitrary domain with the least smooth conditions on a geometric boundary in view of different boundary regularities shape derivatives are investigated only in suitable sobolev spaces as a further application of our results we derive the first order shape derivatives of solutions to stochastic elasto acoustic equations with random interfaces which can be used to obtain the approximation expectation variance and high order moments through taylor shape expansion

global uniqueness for an inverse boundary value problem arising in elasticity we prove that we can determine the lame parameters of an elastic isotropic inhomogeneous medium in dimensionsn 3 by making measurements of the displacements and corresponding stresses at the boundary of the medium

cohen macaulay graded rings associated to ideals let g i be the associated graded ring of an ideal i in a cohen macaulay local ring a we give a sufficient condition for g i to be a cohen macaulay ring it is described in terms of the depths of a i n for finitely many n the reduction numbers of i q for certain prime ideals q and the artin nagata property of i in the sense of ulrich contemp math vol 159 pp 373 400 the main theorem unifies several results which are already known in this aspect of the theory although the statement of the main theorem is rather complicated we restate the conditions under special situations so that they are practical and simple

core and residual intersections of ideals d rees and j sally defined the core of an r ideal i as the in tersection of all minimal reductions of i however it is not easy to give an explicit characterization of it in terms of data attached to the ideal until recently the only case in which a closed formula was known is the one of inte grally closed ideals in a two dimensional regular local ring due to c huneke and i swanson the main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local cohen macaulay ring we also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core

on a invariant formulas abstract let g i be the form ring of an ideal i of positive height in a local ring a in this work we will provide formulas for the a invariant of g i our main result will only need the assumption that a is cohen macaulay and that g i fulfills serre s condition sl where l is the analytic spread of i as a consequence of our formula we will prove upper bounds for the reduction exponent of i in the case that a is a regular local ring if g i fulfills serre s condition sl then this upper bound is l 1 and in the case that g i is even gorenstein it is l 2

ulrich ideals and 2 agl rings abstract the notion of 2 almost gorenstein local ring 2 agl ring for short is a generalization of the notion of almost gorenstein local ring from the point of view of sally modules of canonical ideals in this paper for further developments of the theory we discuss three different topics on 2 agl rings the first one is to clarify the structure of minimal presentations of canonical ideals and the second one is the study of the question of when certain fiber products so called amalgamated duplications are 2 agl rings we also explore ulrich ideals in 2 agl rings mainly two generated ones

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

reduction numbers and multiplicities of multigraded structures the aim of this paper is twofold first we want to relate the cohen macaulay property of r i i to the cohen macaulay propa 1 r erty of the ordinary rees rings r i j s 1 r then we determine a j the multiplicities of multi rees and form rings in particular we ask when minimal multiplicity occurs for i s s i s i the multi rees algebras r i s r i i 1 r a r a behave to some extent like the ordinary rees algebras r i in particua lar the cohen macaulay property of such multi rees algebras can be characterized in terms of conditions on the local cohomology of the form ring gr i including the condition a gr i 0 where a gr i denotes a a a w x the a invariant of gr i see hhr 1 the a invariant has in certain a w cases been related to the reduction exponent r i of the ideal i see t2 x gh moreover the cohen macaulay property of r i implies that of a w x r i for all r 1 the converse is not true in general see hhr1 but a r in certain applications the assumption r i is cohen macaulay can be a

recovering complex elastic scatterers by a single far field pattern abstract we consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded homogeneous and isotropic elastic medium the inverse problem is nonlinear and ill posed our study is conducted in an extremely general and practical setting the number of scatterers is unknown in advance and each scatterer could be either a rigid body or a cavity which is not required to be known in advance and moreover there might be components of multiscale sizes presented simultaneously we develop several locating schemes by making use of only a single far field pattern which is widely known to be challenging in the literature the inverse scattering schemes are of a totally direct nature without any inversion involved for the recovery of multiple small scatterers the nonlinear inverse problem is linearized and to that end we derive sharp asymptotic expansion of the elastic far field pattern in terms of the relative size of the cavities the asymptotic expansion is based on the boundary layer potential technique and the result obtained is of significant mathematical interest for its own sake the recovery of regular size extended scatterers is based on projecting the measured far field pattern into an admissible solution space with a local tuning technique we can further recover multiple multiscale elastic scatterers

gorenstein associated graded rings of analytic deviation two ideals abstract this paper studies the question of when the associated graded ring g i n 0in in 1 of a certain ideal i in a local ring is gorenstein the main result implies for example that if a is a regular local ring 708 is a prime ideal in a with dim a p 2 and a p is a complete intersection in codimension one then the associated graded ring g p is gorenstein if and only if the reduction number of p is at most 1