coefficient ideals in and blowups of a commutative noetherian domain abstract the ratliff rush ideal associated to a nonzero ideal i in a commutative noetherian domain r with unity is n 1 in 1 rin is r s b i where b i r i a p a i 0 p spec r i a is the blowup of i we observe that certain ideals are minimal or even unique in the class of ideals having the same associated ratliff rush ideal if r m is local quasi unmixed and analytically unramified and if i is m primary then we show that the coefficient ideal i k of i i e the largest ideal containing i whose hilbert polynomial agrees with that of i in the highest k terms is also contracted from a blowup b i k which is obtained from b i by a process similar to s2 ification this allows us to generalise the notion of coefficient ideas we investigate these ideas in the specific context of a two dimentional regular local ring observing the interaction of these notions with the zariski theory of complete ideals

nonparametric probabilistic approach of uncertainties for elliptic boundary value problem the paper is devoted to elliptic boundary value problems with uncertainties such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media model uncertainties are induced by the mathematical physical process which allows the boundary value problem to be constructed from the design system if experiments are not available the bayesian approach cannot be used to take into account model uncertainties recently a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non linear dynamical systems nevertheless the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem for instance for a linear elastostatic problem relative to an elastic bounded domain we then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems the theory and its validation are presented

on krull gabriel dimension and galois coverings abstract assume that k is an algebraically closed field r a locally support finite locally bounded k category g a torsion free admissible group of k linear automorphisms of r and a r g we show that the krull gabriel dimension kg r of r equals the krull gabriel dimension kg a of a furthermore we determine the krull gabriel dimension of locally support finite repetitive k categories these results are applied to determine the krull gabriel dimension of standard selfinjective algebras of polynomial growth finally we show that there are no super decomposable pure injective modules over standard selfinjective algebras of domestic type

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

maps between tree and band modules the indecomposable modules over special biserial algebras are known to be of a specially simple form so called tree and band modules cf 6 2 crawley boevey has shown how to describe the homomorphisms between tree modules cf 3 and our objective in this paper is to extend this result to band modules tree and band modules occur as indecomposable representations of arbitrary algebras nevertheless we follow crawley boevey and consider zero relation algebras as the appropriate context given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k linear maps which are associated to these quiver homomorphisms it is interesting to note that in general the classification of maps between representations of a special biserial algebra a is a wild problem in fact the maps are equivalent to the representations of t a for instance take for a the path algebra of an a n 2 6 then t a is wild although the maps between indecomposable representations of a are easily to describe throughout k will be a fixed algebraically closed field maps are written on the left this paper results from a stay at the university of liverpool supported by the deutscher akademischer austauschdienst i am very grateful to sheila brenner for her suggestions

ringel s conjecture for domestic string algebras we classify indecomposable pure injective modules over domestic string algebras verifying ringel s conjecture on the structure of such modules

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

fuzzy behavior of mechanical systems with uncertain boundary conditions the problem of structures sensitivities to uncertain boundary conditions is presented in this paper and the effect of uncertain prescribed displacements on the static response of structures is discussed the imprecise or uncertain prescribed displacements are treated as fuzzy quantities with known membership functions for numerical computations a fuzzy quantity can be approximated by sets of closed intervals with respect to specified cuts this transforms the problem into a set of interval equations at each level a numerical method is developed to solve the system of fuzzy equations simple stress analysis is also used to establish and illustrate the method a numerical example is presented to demonstrate the capability of the method

a fuzzy finite element procedure for the calculation of uncertain frequency response functions of damped structures part 2 numerical case studies this work introduces a numerical algorithm to calculate frequency response functions of damped finite element models with fuzzy uncertain parameters part 1 of this paper focusses on the numerical procedure for the solution of the underlying interval finite element problem based on the undamped procedure and the principle of rayleigh damping part 2 of this paper illustrates the applicability of the methodology through four case studies the concepts of the interval and the fuzzy finite element frequency response function analysis are illustrated for different types of uncertainties the obtained results are compared with the results of monte carlo simulations

integrally closed and componentwise linear ideals in a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is cohen macaulay in higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do we restrict our attention to 0 dimensional homogeneous ideals in polynomial rings r of arbitrary dimension we identify a class of integrally closed ideals the goto class mathcal g which is closed under product and it has a suitable unique factorization property ideals in mathcal g have a cohen macaulay associated graded ring if either they are monomial or dim r 3 our approach is based on the study of the relationship between the notions of integrally closed contracted full and componentwise linear ideals

depth of fiber cones of ideals with almost minimal mixed multiplicity let r m be a cohen macaulay local ring of dimension d 0 i an m primary ideal with almost minimal mixed multiplicity such that depth g i d 1 we show that fm i has almost maximal depth i e depth fm i d 1

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

generalized hilbert coefficients and northcott s inequality abstract let r be a cohen macaulay local ring of dimension d with infinite residue field let i be an r ideal that has analytic spread l i d satisfies the g d condition and the weak artin nagata property a n d 2 we provide a formula relating the length i n 1 j i n to the difference p i n h i n where j is a general minimal reduction of i p i n and h i n are respectively the generalized hilbert samuel polynomial and the generalized hilbert samuel function we then use it to establish formulas to compute the generalized hilbert coefficients of i as an application we extend northcott s inequality to non m primary ideals furthermore when equality holds we prove that the ideal i enjoys nice properties indeed if this is the case then the reduction number of i is at most one and the associated graded ring of i is cohen macaulay we also recover results of g colome nin c polini b ulrich and y xie on the positivity of the generalized first hilbert coefficient j 1 i our work extends that of s huckaba c huneke and a ooishi to ideals that are not necessarily m primary

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

comparison of static response of structures using convex models and interval analysis method in this paper by combining the finite element analysis and non probabilistic convex models we present the numerical algorithm of non probabilistic convex models and interval analysis method for the static displacement of structures with uncertain but bounded parameters under the condition of the box or interval vector determined from the ellipsoid of the uncertain but bounded structural parameter vector by comparing the numerical algorithm of non probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non probabilistic convex models is tighter than those produced by the interval analysis method copyright 2003 john wiley sons ltd

a dual layer dimension wise fuzzy finite element method for structural analysis with epistemic uncertainties abstract epistemic uncertainty propagation is gaining momentum but remains a great challenge in the numerical analysis of a mechanical system a dual layer dimension wise analysis procedure for the fuzzy finite element method is proposed the core algorithm layer is to identify the minimal and maximal min max points of each slice of the response surface at the zero cut the data processing layer aims to filtrate the min max points of each response slice at each nonzero cut subsequently the min max input vectors at any alpha cut are dimension wisely assembled and the response interval at alpha cut is calculated by deterministic finite element analysis at the min max input vectors finally the fuzzy description of the system response is recomposed two examples are investigated to verify both the accuracy and efficiency of the proposed method

assessment of uncertainty on structural dynamic responses with the short transformation method abstract the predictive capability of finite element models is limited by their deterministic nature typically not all model parameters are exactly known while even small deviations may have significant effects on the predicted response parameter uncertainty should therefore be taken into account e g with fuzzy arithmetic the absence of fuzzy solvers led to interval arithmetic as a numerical alternative the transformation method tm presented by m hanss replaces interval arithmetic with a set of deterministic computations for each interval all parameter extrema are combined in every possible way in a design of experiments terminology the tm is a so called full factorial design the tm is applicable if the output is monotonic in the inputs unlike interval arithmetic it does not overestimate the response uncertainty as only parameter combinations are evaluated that actually occur in this paper the tm has been applied to visualise uncertain frequency response functions frfs obtained with modal superposition this yields accurate results when validated against monte carlo data but the computation time is rather high the short transformation method stm is proposed as an attractive alternative to the original tm a full set of deterministic computations combining all interval extrema is only performed at the lowest interval for higher levels a smaller set is evaluated this allows reconstructing the fuzzy frf from a much lower number of deterministic computations with only a small reduction in the accuracy of frfs both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters

exact bounds for the static response set of structures with uncertain but bounded parameters abstract the numerical estimation of the static displacement bounds of structures with uncertain but bounded parameters is considered in this paper by representing each uncertain but bounded parameter as an interval number or vector a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations in which the coefficient matrix and the right hand side term are respectively the interval matrix and the interval vector in this study we present two new simple mathematical proofs of the vertex solution theorem using cramer s rule for solving linear interval equations different from the other proof methods to find the upper and lower bounds on the set of solutions the first takes advantage of optimization theory while the second is based on interval extension by means of a typical example considered first by hansen it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the oettli prager criterion examples of a three stepped beam and a 10 bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method

σ pure injective modules for string algebras and linear relations abstract we prove that indecomposable pure injective modules for a string algebra are string or band modules the key step in our proof is a splitting result for infinite dimensional linear relations

krull gabriel dimension of domestic string algebras we classify the indecomposable pure injective modules over a wide class of 1 domestic string algebras and calculate the krull gabriel dimension of these algebras

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

a spectral approach for fuzzy uncertainty propagation in finite element analysis abstract uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge this paper proposes a polynomial chaos type spectral approach based on orthogonal function expansion a fuzzy variable is represented as a set of interval variables via the membership function the interval variables are further transformed into the standard interval 1 1 smooth nonlinear functions of standard interval variables are projected in the basis of legendre polynomials by exploiting its orthogonal properties over the interval 1 1 the coefficients associated with the basis functions are obtained by a galerkin type of error minimisation the method is first illustrated using scalar functions of multiple fuzzy variables later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables the response of such systems can be expressed in the complete basis of multivariate legendre polynomials the coefficients obtained by galerkin type of error minimisation can be calculated from the solution of an extended set of linear algebraic equations an eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way a numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results

hilbert coefficients of hilbert filtrations adic filtrations see 7 11 18 21 wx in the first section of this paper we extend most of the techniques of 4 w5 to the case of hilbert filtrations and by doing so we are also able toxoffer alternative simple proofs of some results of 7 wxin the second section thanks to the setting just described if

the component mode transformation method a fast implementation of fuzzy arithmetic for uncertainty management in structural dynamics abstract the study of mechanical systems with uncertain parameters is of increasing importance in the analysis and in the design of structures because an adequate description of these uncertainties is necessary to provide a reliable forecasting model in order to update or evaluate an uncertain model different methods have been proposed but many of these such as the monte carlo simulation or the transformation method require multiple sequential evaluations of the model moreover to cover an adequate portion of the uncertain parameter space the number of necessary evaluations of the model grows exponentially with the number of uncertain parameters leading often to a large number of runs the proposed methodology deals with uncertain parameters that are modeled by the use of fuzzy numbers and the transformation method is applied to characterize the dynamic behavior of the structure in order to reduce the order of magnitude of the computational time a reduction of the model dimension obtained through a component mode synthesis is performed advantages and drawbacks of the proposed method are discussed together with an application to the study of the dynamics of a beam

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

graded rings associated with contracted ideals the study of the ideals in a regular local ring r m of dimension 2 has a long and important tradition dating back to the fundamental work of zariski zs more recent contributions are due to several authors including cutkosky huneke lipman rees sally and teissier among others see c1 c2 h hs l lt r one of the main result in this setting is the unique factorization theorem for complete i e integrally closed ideals proved originally by zariski zs theorem 3 appendix 5 it asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way up to the order of the factors by definition an ideal is simple if it cannot be written as a product of two proper ideals another important property of a complete ideal i is that its reduction number is 1 which in turns implies that the associated graded ring gri r is cohen macaulay and its hilbert series is well understood this is due to lipman and teissier lt see also hs the class of contracted ideals plays an important role in the original work of zariski as well as in the work of huneke h an ideal i of r is contracted if i r ir m x for some x m m2 any complete ideal is contracted but not the other way round the associated graded ring gri r of a contracted ideal i need not be cohen macaulay and its hilbert series can be very complicated

a new uncertainty analysis for the transformation method in this paper a new uncertainty analysis for the transformation method tm is proposed as a practical implementation of fuzzy arithmetic the tm is a convenient tool for the simulation and analysis of systems with uncertain parameters that are expressed by fuzzy numbers the proposed uncertainty analysis and the sensitivity analysis of the tm complete each other in providing some quantification of the relationship between the uncertainties of the system input and the system output the computation of gain factors is proposed which allows the estimation of the absolute and relative measures of uncertainty these measures allow the quantification of the influence of the uncertainty of the input on the uncertainty of the output