the tight integral closure of a set of ideals abstract this paper celebrates the enormous contributions of david buchsbaum to commutative algebra homological algebra and representation theory with grateful appreciation for the inspiration he has provided for myself and so many others

arithmetic of normal rees algebras two reasons drive this study of the arithmetical properties of blow up like algebras first one still lacks effective criteria for the rees algebra of an ideal to be normal and second there are several new phenomena related to torsionfree symmetric algebras of modules notably bundlesthat need clarification as yet another related question there is the description of the canonical modules of such algebras a necessary step in determining their gorenstein loci the main thread for these questions is the understanding of the divisor class group essentially defined by a pair made up of a morphism and a free subgroup of finite rank the behaviour of the mapping has puzzling

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

a fully discrete bem fem for the exterior stokes problem in the plane we reformulate the johnson nedelec approach for the exterior two dimensional stokes problem taking advantage of the parameterization of the artificial boundary the main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem this one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals we give a complete error analysis of both the galerkin and fully discrete galerkin methods

a new bem fem coupling strategy for two dimensional fluid solid interaction problems we present a numerical method to solve a fluid solid interaction problem posed in the plane in this scheme we use a finite element method to approximate the solid vibrations and the near wave field the far field effects are taken into account by means of boundary integral equations posed on an artificial interface that contains the obstacle the boundary unknown involved in our formulation is approximated by a spectral method we obtain a fully discrete galerkin procedure whose main advantage is the simplicity of the quadratures used to approximate the weakly singular boundary integrals we provide numerical results that illustrate the accuracy of our method and the stability of the algorithm used to solve the linear systems of equations that arise from this discretization technique

exploring isospectral spring mass systems with firefly algorithm this paper investigates in line spring mass systems an fixed at one end and free at the other with n degrees of freedom d f the objective is to find feasible in line systems b n that are isospectral to a given system the spring mass systems a n and b n are represented by jacobi matrices an error function is developed with the help of the jacobi matrices a n and b n the problem of finding the isospectral systems is posed as an optimization problem with the aim of minimizing the error function the approach for creating isospectral systems uses the fact that the trace of two isospectral jacobi matrices a n and b n should be identical a modification is made to the diagonal elements of the given jacobi matrix a n to create the isospectral systems the optimization problem is solved using the firefly algorithm augmented by a local search procedure numerical results are obtained and resulting isospectral systems are shown for 4 d f and 10 d f systems

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

f purity and rational singularity in graded complete intersection rings a simple criterion is given for determining almost completely whether the positively graded complete intersection ring r k x1 xn t g of dimension n has an f pure type singularity at m x1 x specifically if deg x a 0 for 1 0 for 1 0 the characterization given by this theorem is particularly effective if the singularity of r at m is isolated in that case 8 0 so that only the condition t 1d y lta 0 is not solved by the above result in particular it follows from work of kei ichi watanabe that if r has an isolated rational singularity then r has f pure type the converse is also almost true with the only exception being the case where t 1d e i ta 0 in proving this criterion a weak but more stable form of f purity called f contractedness is defined and explored r is f contracted in characteristic p 0 if every system of parameters for m is contracted with respect to the frobenius map f r r just as for f purity the notion of f contracted type is defined in characteristic 0 by reduction to characteristic p the two notions of f pure type and f contracted type coincide when r is gorenstein whence in particular when r is a complete intersection ring 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m r m is exact a notion of f pure type can then be defined in characteristic 0 by reduction to characteristic p the concept of f purity was suggested originally by the hochster roberts proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 4 the same two authors later demonstrated that f purity measures nice singularities in the sense that it implies much simplification and vanishing in the computation of local cohomology 5 the problem of identifying those rings which are f pure type is quite difficult the following special cases are known 1 the invariant ring of a reductive linear group acting on a regular ring is f pure type 5 received by the editors october 8 1985 1980 mathematics subject classification 1985 revision primary 13h99 the author wishes to thank the research council of the university of missouri for a research grant which partially supported this project 1987 american mathematical society 0002 9947 87 1 00 25 per page

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

inverse vibration problem for un damped 3 dimensional multi story shear building models various researchers have contributed to the identification of the mass and stiffness matrices of two dimensional 2 d shear building structural models for a given set of vibratory frequencies the suggested methods are based on the specific characteristics of the jacobi matrices i e symmetric tri diagonal and semi positive definite matrices however in case of three dimensional 3 d structural models those methods are no longer applicable since their stiffness matrices are not tri diagonal in this paper the inverse problem for a special class of vibratory structural systems i e 3 d shear building models is investigated a practical algorithm is proposed for solving the inverse eigenvalue problem for un damped 3 d shear buildings the problem is addressed in two steps first using the target frequencies a so called normalized eigenvector matrix which is a banded matrix containing the information related to the frequencies and mode shapes of the target structural system is determined in this regard similar to the solution of inverse problem for 2 d shear building structural models in which an auxiliary structure is constructed by adding constraints or springs to the original system three auxiliary structures are proposed to solve the problem for 3 d cases in the second step the normalized eigenvector matrix is utilized to obtain the normalized stiffness matrix in turn this matrix is decomposed into the stiffness and mass matrices of the system finally a numerical example is presented to demonstrate the efficiency of the proposed algorithm in determining the mass and stiffness matrices of a 3 d structural model for a given set of target vibrational frequencies

general isospectral flows for linear dynamic systems the matrix a a0 a1 2a2 kak lal with matrix coefficients a0 a1 a2 al cm n defines a linear dynamic system of dimension m n when m n and when det a 0 for some values of the eigenvalues of this system are well defined a one parameter trajectory of such a system a0 a1 a2 al is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values ir this paper presents the most general form for isospectral flows of linear dynamic systems of orders l 2 3 4 and the forms for isospectral flows for even higher order systems are evident from the patterns emerging based on the definition of a class of coordinate transformations called structure preserving transformations the concept of isospectrality and the associated flows is seen to extend to cases where m n

an inverse problem for a functionally graded elliptical plate with large deflection and slightly disturbed boundary this paper deals with the inverse problem of a functionally graded material fgm elliptical plate with large deflection and disturbed boundary under uniform load the properties of functionally graded material are assumed to vary continuously through the thickness of the plate and obey a simple power law expression based on the volume fraction of the constituents based on the classical nonlinear von karman plate theory the governing equations of a thin plate with large deflection were derived in order to solve this non classical problem a perturbation technique was employed on displacement terms in conjunction with taylor series expansion of the disturbed boundary conditions the displacements of in plane and transverse are obtained in a non dimensional series expansion form with respect to center deflection of the plate the approximate solutions of displacements are solved for the first three terms and the corresponding internal stresses can also be obtained

inverse spectral theory of finite jacobi matrices we solve the following physically motivated problem to determine all finite jacobi matrices j and corresponding indices i j such that the green s function is proportional to an arbitrary prescribed function f z our approach is via probability distributions and orthogonal polynomials we introduce what we call the auxiliary polynomial of a solution in order to factor the map j i j where square brackets denote the equivalence class consisting of scalar multiples this enables us to construct the solution set as a fibration over a connected semi algebraic coordinate base the end result is a wealth of explicit constructions for jacobi matrices these reveal precise geometric information about the solution set and provide the basis for new existence theorems

a variant of wangʼs theorem in this paper we give a new formula of j j for any parameter ideal j in a gorenstein local ring r of positive characteristic in terms of test ideals j j j jd 1 where jd 1 denotes the jd 1 test ideal of r as an application we give a variant of wang s theorem namely we prove that if j is a parameter ideal in a cohen macaulay local ring r m of dimension d 2 with j ms then j m d 1 s 1 resp j m d 1 s 1 1 is integral over j resp if r is not regular moreover we prove that after reduction to characteristic p 0 a similar assertion holds true for cohen macaulay q gorenstein normal local domain essentially of finite type over a field of characteristic zero under some extra assumption

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

wavelet galerkin algorithms for boundary integral equations the implementation of a fast wavelet based galerkin discretization of second kind integral equations on piecewise smooth surfaces gamma subset ir 3 is described it allows meshes consisting of triangles as well as quadrilaterals the algorithm generates a sparse approximate stiffness matrix with cn o n log n 2 nonvanishing entries in o n log n 4 operations where n is the number of degrees of freedom on the boundary while essentially retaining the asymptotic convergence rate of the full galerkin scheme a new proof of the matrix compression estimates is given based on derivative free kernel estimates the condition number of the sparse stiffness matrices is bounded independently of the meshwidth the data structure containing the compressed stiffness matrix is described in detail it requires o cn memory and can be set up in o cn operations numerical experiments show that the asymptotic performance estimates apply for moderate n problems with n 106 degrees of freedom were computed in core on a workstation the impact of various parameters in the compression scheme on the performance and the accuracy of the algorithm is studied

inverse structural modification using constraints in a structural modification problem the mass and stiffness matrices are modified to obtain a desired spectrum in this paper this is done by imposing constraints on the structure the undamped natural vibrations of a constrained linear structure are calculated by solving a generalized eigenvalue problem derived from the equations of motion for the constrained system involving lagrangian multipliers the coefficients of the constraint matrix are taken as design variables and a set of equations defining the inverse structural modification problem is formulated this modification problem requires an iterative method for its solution an algorithm based on newton s method is employed each iteration step involves the calculation of a rectangular jacobian and the solving of an associated underdetermined system of linear equations the system can be solved by using the moore penrose inverse the method is demonstrated in some numerical examples

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

the artificial boundary conditions for incompressible materials on an unbounded domain in this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in frak r 2 a series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given then the original problem is reduced to a problem on a bounded domain which be solved numerically by a mixed finite element method the numerical example shows that our artificial boundary conditions are very effective

an optimal iterative process for the johnson nedelec method of coupling boundary and finite elements we reformulate the discretization of the johnson nedelec method 11 of coupling boundary elements and finite elements for an exterior bidimensional laplacian this new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix we show that if the parameter of discretization is sufficiently small the fully discrete scheme is well posed and the error estimates remain unaltered the rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems

on the reconstruction of a damped vibrating system from two complex spectra part 1 theory abstract the paper concerns an n degree of freedom damped vibrating system consisting ofn 1 masses connected in parallel by springs and dampers to an n th mass the paper analyzes the construction of such a system from the given complex eigenvalue data the analysis has two parts the establishment of the conditions on the eigenvalues which ensure that they correspond to an actual system the derivation of the system parameters from the eigenvalues

a dual dual formulation for the coupling of mixed fem and bem in hyperelasticity we combine a dual mixed finite element method with the boundary integral equation method to study the solvability and galerkin approximations of a linear nonlinear transmission problem arising in plane elastostatics our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain and it leads to what we call a dual dual mixed formulation since the corresponding operator equation becomes of a twofold saddle point structure we derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis in particular we extend the finite element space for plane elasticity given by the peers element and define an explicit finite element boundary element subspace satisfying all the required compatibility conditions most of our analysis makes use of an extension of the classical babuska brezzi theory to a class of nonlinear saddle point problems

an inverse eigenvalue problem for the symmetric tridiagonal quadratic pencil with application to damped oscillatory systems a method is presented which constructs an n by n tridiagonal symmetric quadratic pencil which has its 2n eigenvalues and the 2n 2 of its n 1 dimensional leading principal subpencil prescribed it is shown that if the given eigenvalues are distinct there are at most 2 n 2n 3 n 2 different solutions in the degenerate case where some of the given eigenvalues are common there are an infinite number of solutions apart from finding the roots of certain polynomials the problem is solved in a finite number of steps where the problem has only a finite number of solutions they can all be found in a systematic manner the method is demonstrated with a simple example and its use is illustrated with a practical engineering application in vibrations

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

inverse finite element vibration problems the paper concerns the reconstruction of a consistent fem model of an in line system of 2 dof elements fixed at one end and free at the other such a system has tridiagonal stiffness and mass matrices k m because each element has one rigid body mode khas negative codiagonal and is constrained to have a particular form mhas positive codiagonal it is shown how to construct an infinite family of such models so that each has a specified undamped frequency response at the free end and how to construct a system with a damper at the free end so that the system has specified complex eigenvalues

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

multi symbolic rees algebras and strong f regularity let i be a divisorial ideal of a strongly f regular ring a k i watanabe raised the question whether the symbolic rees algebra mathcal r s i oplus n ge 0 i n is cohen macaulay whenever it is noetherian we develop the notion of multi symbolic rees algebras and use this to show that mathcal r s i is indeed cohen macaulay whenever a certain auxiliary ring is finitely generated over a

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

two new solvable potentials we succeeded in finding two new solvable potentials by working in a complete square integrable basis that carries a tridiagonal jacobi matrix representation for the wave operator the conventional methods do not lead to exact solutions for these potentials hence a new approach has been adopted for these potentials first we obtain the potential parameter spectrum the set of values of the potential parameters that lead to an exact solution at a given energy then the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum for a given potential parameter this procedure has been applied in 3d to obtain the energy spectrum for a special screened coulomb potential with a barrier and in 1d for a single wave potential

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

new implementation techniques for the exterior stokes problem in the plane abstract we present a method to solve numerically two dimensional stokes problems on exterior domains our scheme is based on the fully discrete bem fem formulation proposed in 21 whose main advantage is that only elemental quadratures are used to approximate the weakly singular boundary integrals we show in this article that it is possible to maintain this important property without using curved triangles in the discretization process this modification makes the method easier to implement and the numerical experiments reveal that it still keeps the optimal order of convergence of the original scheme we also introduce in this paper a new iterative method to solve the complicated linear systems of equations that arises from this type of bem fem discretizations

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings