maximum likelihood estimation from fuzzy data using the em algorithm a method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample this method is based on maximizing the observed data likelihood defined as the probability of the fuzzy data it is shown that the em algorithm may be used for that purpose which makes it possible to solve a wide range of statistical problems involving fuzzy data this approach called the fuzzy em fem method is illustrated using three classical problems normal mean and variance estimation from a fuzzy sample multiple linear regression with crisp inputs and fuzzy outputs and univariate finite normal mixture estimation from fuzzy data

statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size abstract usually the parameters of statistical tests are determined by crisp constraints on the probabilities and of the errors of type i and of type ii respectively and by the additional condition that the sample size n is to be selected as small as possible in this paper this approach is modified here statistical tests are proposed to be evaluated by the degree to which certain fuzzy requirements on and n are met aggregating these requirements by the minimum operator a procedure is presented how to determine a test maximizing this degree of satisfaction statistical tests with a continuously distributed test statistic and in particular two sided gau tests are discussed in more detail

possibilistic decisions and fuzzy statistical tests the paper deals with the problem of the interpretation of the results of statistical tests in terms of the theory of possibility the well known concept in statistics of the observed test size p also known as p value and significance has been given a new possibilistic interpretation and generalised for the case of imprecisely defined statistical hypotheses and vague statistical data the proposed approach allows a practitioner to evaluate the test results using intuitive concepts of possibility necessity or indifference

testing fuzzy hypotheses with crisp data abstract in this paper an approach is presented how statistical tests originally constructed to examine crisp hypotheses can also be applied to fuzzily formulated hypotheses in particular criterions and are proposed generalizing the probabilities of the errors of type i and type ii respectively the general approach is applied to one and two sided gaus tests here diagrams are given to determine the critical values in the most popular cases of 0 01 and 0 05 if in addition the value of is fixed in advance the sample size of a one or two sided gaus test can be obtained using supplementary graphs

normalization of modules we introduce techniques to derive estimates for the degrees of the generators of the integral closure of several classes of rees algebras of modules and to bound the length of normalization processes in the case of regular base rings the bounds are expressed in terms of buchsbaum rim multiplicities and a module version of briancon skoda numbers

development of fuzzy and control charts using α cuts statistical process control spc is an approach that uses statistical techniques to monitor the process shewhart introduced the control charts that are one of the most important techniques of quality control to detect if assignable causes exist the widely used control charts are and these are called traditional variable control charts in the traditional variable control charts center line upper control limit and lower control limit are represented by numeric values a process is either in control or out of control depending on numeric observation values for many problems control limits could not be so precise uncertainty comes from the measurement system including operators and gauges and environmental conditions in this context fuzzy set theory is a useful tool to handle this uncertainty numeric control limits can be transformed to fuzzy control limits by using membership functions if a sample mean is too close to the control limits and the used measurement system is not so sensitive the decision may be faulty fuzzy control limits provide a more accurate and flexible evaluation this study constructs the fuzzy and control charts with a cuts an application is presented for fuzzy control charts by using fuzzy and control charts the flexibility of traditional control limits is increased

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

the fuzzy decision problem an approach to the problem of testing statistical hypotheses with fuzzy information abstract this paper is devoted to the problems of testing statistical hypotheses about an experiment when the available information from its sampling is vague when the information supplied by the experimental sampling is exact the problems of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem in addition decision procedures may be used in problems of testing hypotheses in a similar manner the problem of testing statistical hypotheses about an experiment when the available sample information is vague is approached in this paper as a particular fuzzy decision problem as defined by tanaka okuda and asai this approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information whereas the present information about it can be expressed by means of fuzzy information the preceding framework allows us to extend the notion of risk function and some nonfuzzy decision procedures to the fuzzy case and particularize them to the problem of testing finally several illustrative examples are presented

the structure of the core of ideals the core of an r ideal i is the intersection of all reductions of i this object was introduced by d rees and j sally and later studied by c huneke and i swanson who showed in particular its connection to j lipman s notion of adjoint of an ideal being an a priori infinite intersection of ideals the core is difficult to describe explicitly we prove in a broad setting that core i is a finite intersection of minimal reductions core i is a finite intersection of general minimal reductions core i is the contraction to r of a universal ideal core i behaves well under flat extensions the proofs are based on general multiplicity estimates for certain modules

interval random sets and entropy abstract the problem of proper choice from the class of interval random sets generated by some fuzzy set is considered this problem is determined for the maximum entropy criterion the class of interval random sets is studied

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

construction of equilibrated singular basis functions without a priori knowledge of analytical singularity order in this paper a novel method is presented to construct singular basis functions for solving harmonic and bi harmonic problems with weak singularities such bases are found without any knowledge of the singularity order the singular bases are constructed by choosing a series as a tensor product of chebyshev polynomials and trigonometrical functions in radial and angular directions respectively and applying weak form of the governing equation with such features the singular bases are categorized as the equilibrated basis functions the constructed singular functions can be utilized as a complementary part to the smooth part of the approximation in the solution of problems with singularities to demonstrate the efficiency of employing such singular bases they are used in a boundary node method through the solution of some examples selected from the well known literature the capability of the method is shown it will be demonstrated that the main function and its derivatives are excellently approximated at very close neighborhood of the singular point the method may especially be found useful for those who research on the extended finite element method or similar ideas

fuzzy hypothesis testing with hybrid data abstract the formulation of binary hypothesis testing where the available data under one hypothesis is a superposition of a random and a fuzzy component is addressed in this formulation the likelihood ratio which is normally utilized in such types of decision making problems is fuzzified and compared to a threshold this involves the application of principles of ordering fuzzy sets which have been developed elsewhere by the authors decision rules are described which are independent of the shape of the membership function representing the fuzzy data these decision rules are illustrated for the case where the random component of the data is normally distributed probability of error performance curves using different ordering criteria are obtained and compared to the performance of non fuzzy hypothesis testing

fuzzy hypothesis testing for a population proportion based on set valued information abstract in this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one tailed hypergeometric test with fuzzy non complementary hypotheses and incorporate the conventional hypergeometric test with crisp hypotheses into its framework in particular we formulate hypotheses as ontic or epistemic fuzzy sets since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading further we suggest modeling of membership functions in terms of cost or frequency due to considerations of user s priorities or incomplete knowledge in relation to hypotheses formulation considering a hypothesis as union of its crisp and fuzzy areas we illustrate via real life examples that in contrast to classical test theory fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non complementary hypotheses the test concept is introduced for both test methods a test of significance and an alternative test and leads to a crisp test decision considering further cost aspects like the sample size the generalized error criteria are derived and interpreted in relation to both hypotheses additionally a corresponding sensitivity analysis is presented for one of the most promising test types

inner and outer fuzzy approximations of confidence intervals we extend the notion of confidence region to fuzzy data by defining a pair of fuzzy inner and outer confidence regions we show the connection with previous proposals as well as with recent studies on hypothesis testing with low quality data

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

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

exponential basis functions in solution of incompressible fluid problems with moving free surfaces in this paper a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries a lagrangian formulation established on pressure as a potential equation is employed in this method the approximate solution is expressed by a linear combination of exponential basis functions ebfs with complex valued exponents satisfying the governing equation constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique the numerical solution is performed in a time marching approach using an implicit algorithm in each time step the governing equation is solved at the beginning and the end of the step with the aid of an intermediate geometry the use of ebfs helps to find boundary velocities with high accuracy leading to a precise geometry updating the developed lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems non linear sloshing fluids in rigid rectangular two dimensional basins are particularly addressed

fuzzy hypotheses testing in the framework of fuzzy logic abstract testing hypotheses about the probability distribution underlying the available empirical data is one of the fundamental data analytic tasks in any application domain basically it consists in checking the null hypothesis that the probability distribution a priori assumed to belong to a certain set of distributions actually belongs to some its narrow subset which must be precisely delimited in advance however sometimes there are not enough clues for such a precise delimitation especially if the purpose of the data analysis is explorative a situation encountered increasingly often due to the growing amount of routinely collected data and the increasing importance of data mining that is why generalizations of statistical hypotheses testing to vague hypotheses have been investigated for more than a decade so far based on the most straightforward approach to replace the set defining the null hypothesis by a fuzzy set in this paper a principally different approach is presented motivated by the observational logic and its success in automated knowledge discovery its key idea is to view statistical testing of a fuzzy hypothesis as the application of an appropriate generalized quantifier of a fuzzy predicate calculus to predicates describing the data the theoretical principles of the approach are explained and its first implementations are briefly sketched

the largest non vanishing degree of graded local cohomology modules abstract this paper gives characterizations for the largest non vanishing degree of the local cohomology modules of a graded ring s in terms of a reduction of s and of the associated graded ring of an ideal i in terms of any reduction of i as a consequence this invariant can be computed explicitly for the associated graded ring of any ideal generated by a d sequence and of ideals having small analytic deviation

rees algebras of truncations of complete intersections abstract in this paper we describe the defining equations of the rees algebra and the special fiber ring of a truncation i of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m to describe explicitly the rees algebra r i in terms of generators and relations we map another rees ring r m onto it where m is the direct sum of powers of m we compute a grobner basis of the ideal defining r m it turns out that the normal domain r m is a koszul algebra and from this we deduce that in many instances r i is a koszul algebra as well

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

testing statistical hypotheses with vague data a definition of fuzzy test for testing statistical hypotheses with vague data is proposed then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one sided or two sided alternative hypotheses is shown this fuzzy test contrary to the classical approach leads not to the binary decision to reject or to accept given null hypothesis but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis respectively however it is a natural generalization of the traditional test i e if the data are precise not vague we get a classical statistical test with the binary decision a measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed

exponential basis functions in solution of static and time harmonic elastic problems in a meshless style in this paper exponential basis functions ebfs are used in a boundary collocation style to solve engineering problems whose governing partial differential equations pdes are of constant coefficient type complex valued exponents are considered for the ebfs two dimensional elasto static and time harmonic elasto dynamic problems are chosen in this paper the solution procedure begins with first finding a set of appropriate ebfs and then considering the solution as a summation of such ebfs with unknown coefficients the unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique the basis and various forms of the transformation have been addressed and discussed we shall propose several strategies for selection of ebfs with the aid of the basis explained for the transformation while using the transformation the number of ebfs should not necessarily be equal to or less than the number of boundary information data a library of ebfs has also been presented for further use the effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of ebfs a number of sample problems are solved to demonstrate the capabilities of the method it has been shown that the time harmonic problems with high wave number can be solved without much effort the method categorized in meshless methods can be applied to many other problems in engineering mechanics and general physics since ebfs can easily be found for almost all problems with constant coefficient pdes

on the depth of the rees algebra of an ideal module we study the rees algebra r e s e r s e of an ideal module e g re we use the technique of bourbaki ideals introduced by simis ulrich and vasconcelos 2003 22 to relate the rees algebra of a module e to the rees algebra of an ideal i i e r where r is a nagata extension of r we shall prove that depthr e depthr i e 1 and use it to deduce formulae for the depth of r e for ideal modules having small reduction number

the dimension and components of symmetric algebras let r be a commutative noetherian ring and let a4 be any finitely generated r module in this paper we compute the dimension of s m the symmetric algebra of m and identify many of the components of s m while the computation of the dimension of s m is not difficult the answer is intriguing and in view of the recent interest in symmetric algebras we hope the computation will be useful the dimension is given by the for mula dim s m max dim r p p m

kendall s rank correlation on quantized data an interval valued approach kendall s rank correlation coefficient also called kendall s is an efficient and robust way for identifying monotone relationships between two data sequences however when applied to digital data the high number of ties yields inconsistent results due to quantization here we propose an extension of kendall s that considers an epistemic view of a sequence of quantized data each sample is supposed to be the quantized version of an original value that is a real number we come up with an imprecise defined as the interval containing all values that could have been computed on sequences of original values before quantization we propose a very simple and straightforward algorithm to compute this interval valued we prove the exactness of the bounds and propose an experiment that illustrates the need for such an extension

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

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

statistical tests for fuzzy data abstract in this paper we investigate the impacts of vague and imprecise data on the statistical task of hypotheses testing test results are evaluated with respect to this ambiguity the issue of how to aggregate fuzzy data to a fuzzy sample vector is discussed and different eventualities are considered their consequences on the test results are compared the domain of the used test statistics is extended to fuzzy sample vectors by the means provided in the first part of this paper different methods for the defuzzification of the obtained fuzzy test results are discussed and the fuzzy nature of the classical verification task for statistical hypotheses is studied in this context a way of how to overcome the general insufficiency of statistical tests of being conclusive only in one direction is proposed by admitting fuzzy results

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

fuzzy hypotheses for guha implications abstract this paper presents a fuzzy generalization of a sophisticated approach to exploratory data analysis the general unary hypotheses automaton guha the guha paradigm to automatically generate sentences of an observational calculus which are supported by given data has been attracting attention for nearly 30 years most of the procedures for generating observational sentences encountered in the existing implementations of the approach are based on statistical hypotheses testing however there is an inner contradiction inherent to using common statistical tests to this end those tests always require a precise formulation of the tested conditions on the random variables characterizing the underlying real phenomena on the other hand due to its explorative purpose guha is predominantly used in situations in which the user has only a rather vague knowledge of those random variables thus being unable to precisely state the conditions to be tested therefore a fuzzy generalization of the approach is proposed it is based on a method of fuzzy hypotheses testing which was elaborated for the statistical tests used in guha and integrated into the context of an observational calculus due to space limitations the paper concentrates on a particular kind of generated observational sentences guha implications the fuzzy counterparts of two key concepts pertaining to guha implications are defined and a number of their properties are established the proposed approach is illustrated on two commonly used guha implications

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

on the depth of the associated graded ring of an ideal abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g fails to be cohen macaulay we assume that i has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers the main theorem unifies and generalizes several known results we also give conditions that imply the serre properties of the blow up rings

edge singularities in anisotropic composites abstract the stress singularity at the vertex of an anistropic wedge has the form r f r as r 0 where 0 f is a real function of the polar coordinates r in many cases f is independent of r the explicit form of f r depends on the eigenvalues of the elasticity constants called p here and on the order of singularity k when k is real k if k is complex is the real part of k the p s are all complex and consist of 3 pairs of complex conjugates which reduce to i when the material is isotropic the function f depends not only on p and k it also depends on whether p and k are distinct roots of the corresponding determinants in this paper we present the function f r in terms of p and k for the cases when p and k are single roots as well as when they are multiple roots the relationship between the complex variable z introduced in the analysis and the polar coordinates r is interpreted geometrically after presenting the form of f for individual cases a general form of f is given in eqn 74 we also show that the stress singularity at the crack tip of general anisotropic materials has the order of singularity 1 2 which is at least a multiple root of order 3 the implication of this on the form f r is discussed

a fictitious domain method using equilibrated basis functions for harmonic and bi harmonic problems in physics abstract in this paper we present a new meshless method to solve well known problems in physics and engineering with either constant or variable material properties in 2d space harmonic and bi harmonic problems are considered in this paper the method constructs a set of basis functions called equilibrated basis functions eqbfs through a weighted residual integration over a fictitious domain embedding the main one the bases can satisfy the governing partial differential equations approximately chebyshev polynomials of the first kind are employed to construct the eqbfs and exponential functions are used as the weights in the integrals the parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1d ones over a normalized intervals this reduces the computational efforts significantly either the eqbfs or the results of the integration process may be stored for further use the validity of the results is examined through an extended patch test a set of physical sample problems with variable non variable material properties as the potential flow over a cylinder steady state heat conduction problems in an anisotropic inhomogeneous functionally graded material potential and stokes flow through a channel of finite width obstructed by periodic cylinders and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method as a preliminary study we show that the method may effectively be used in a domain decomposition approach

similarity of personal preferences theoretical foundations and empirical analysis we study the problem of defining similarity measures on preferences from a decision theoretic point of view we propose a similarity measure called probabilistic distance that originates from the kendall s tau function a well known concept in the statistical literature we compare this measure to other existing similarity measures on preferences the key advantage of this measure is its extensibility to accommodate partial preferences and uncertainty we develop efficient methods to compute this measure exactly or approximately under all circumstances these methods make use of recent advances in the area of markov chain monte carlo simulation we discuss two applications of the probabilistic distance in the construction of the decision theoretic video advisor diva and in robustness analysis of a theory refinement technique for preference elicitation

fuzzy sets and statistical data abstract specific features of probability and possibility theories are discussed with emphasis on semantical aspects instead of putting forward the acknowledged usefulness of possibility theory for the non statistical modelling of subjective categories we try to figure out how statistical data and possibility theory could be matched as a result procedures for constructing weak possibilistic substitutes of probability measures and for processing imprecise statistical data are outlined they provide new insights on relationship between fuzzy sets and probability theories

on the canonical module of the rees algebra and the associated graded ring of an ideal the investigations of this paper originate in the following question suppose i is an ideal of a local gorenstein ring r m whose associated graded ring gr r is cohenmacaulay under which extra conditions is gr r actually a gorenstein ring hochster shows in 151 that gr r is a gorenstein ring if r is factorial and gr r is a domain hochster s arguments work as well if one only requires r to be a gorenstein ring however the condition that gr r is a domain cannot be weakened much in fact we given an example of a local complete intersection r m whose associated graded ring gr r is a reduced cohen macaulay ring but not a gorenstein ring the question when gr r is a gorenstein ring has a more satisfying answer for ideals i generated by a d sequence for which r i is a cohen macaulay ring d sequences were introduced and studied by huneke their relevance in the study of powers of ideals was shown by huneke in 161 for further investigations on d sequences the reader is referred to 111 now given an ideal i generated by a d sequence and such that r i is cohenmacaulay we show that gr r is a gorenstein ring if and only if i is strongly cohen macaulay again the notion strongly cohen macaulay was introduced by huneke it means that the koszul homology of a system of generators of i is cohen macaulay huneke showed in 191 that any ideal in the linkage class of a complete intersection is strongly cohen macaulay this result provides us with plenty of interesting examples the main subject of the paper however is the study of the canonical module of gr r and of the rees algebra r if r zr 12t2 sr t in 14 the canonical class os of the rees algebra s r zr was determined under the assumption that i is

rees algebras of ideals having small analytic deviation in this article we identify two large families of ideals of a cohen macaulay sometimes gorenstein local ring whose rees algebras are cohen macaulay our main results imply for example that if r m is a regular local ring and p is a prime ideal of r such that p n is unmixed for all n 1 then the rees algebra r pt is cohen macaulay if either dim r p 2 o dim r p 3 r p is cohen macaulay and r p is integrally closed

a fuzzy statistical test of fuzzy hypotheses abstract a fuzzy hypothesis is introduced and a testing method of a fuzzy hypothesis is proposed for random data the proposed test produces a conclusion which is also fuzzy some applications of the fuzzy statistical test are demonstrated particularly the test of the difference of two population means is considered in detail and the fuzzy hypothesis that two population means are nearly equal is tested by the proposed procedure

extension of fuzzy hypothesis testing with hybrid data abstract the problem of binary hypothesis testing where the available data under both hypotheses is a superposition of a random and a fuzzy component is addressed a rule that compares two fuzzy likelihood functions is used in the decision making process this rule is proved to result from the application of a fuzzified version of the bayes criterion in order that a crisp decision is ultimately obtained use is made of a criterion for ordering fuzzy sets over the real line which was developed elsewhere by the author a specific case which assumes normally distributed random component of the data is considered probability of error performance curves for fuzzy hypothesis testing are plotted and compared with the curves that correspond to the minimax probability of error criterion and the generalized likelihood ratio test when non fuzzy hypothesis testing is concerned

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

nonparametric rank based statistics and significance tests for fuzzy data nonparametric rank based statistics depending only on linear orderings of the observations are extended to fuzzy data the approach relies on the definition of a fuzzy partial order based on the necessity index of strict dominance between fuzzy numbers which is shown to contain in a well defined sense all the ordinal information present in the original data a concept of fuzzy set of linear extensions of a fuzzy partial order is introduced allowing the approximate computation of fuzzy statistics alpha cutwise using a markov chain monte carlo simulation approach the usual notions underlying significance tests are also extended leading to the concepts of fuzzy p value and graded rejection of the null hypothesis quantified by a degree of possibility and a degree of necessity at a given significance level this general approach is demonstrated in two special cases kendall s rank correlation coefficient and wilcoxon s two sample rank sum statistic

a bayesian approach to fuzzy hypotheses testing in statistical decisions we may come across imprecise fuzzy concepts one important case is a situation where we are interested in testing the hypotheses that are fuzzy rather than crisp in this paper we consider and analyze the testing of fuzzy hypotheses on the basis of a bayesian approach we illustrate our method by some examples and compare it with previous works on this topic

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

nonparametric kernel estimation based on fuzzy random variables this paper extends the classical nonparametric curve fitting methods for fuzzy random variables the study was conducted in two parts 1 fuzzy density estimation and 2 nonparametric regression the classical nonparametric density estimation was first developed based on the kernel method for a given fuzzy random sample at crisp and or fuzzy point based on crisp or fuzzy bandwidth the classical bandwidth selection was also extended when the underlying population was normal with known or unknown fuzzy variance moreover the classical nonparametric kernel based regression model with crisp and or input or output is extended whenever a given bandwidth is a crisp or fuzzy number the cross validation procedure for selecting the optimal value of the crisp smoothing parameter is also extended to fit the proposed nonparametric regression model the large sample properties of the proposed fuzzy estimators were also investigated by some theorems several numerical examples including that of real life data are used to illustrate the proposed methods in curve fitting estimation with crisp and or fuzzy information moreover the proposed methods were examined in comparison with some other existing methods and their effectiveness were clarified via some numerical examples and simulation studies both theatrical and numerical results indicated that the nonparametric curve fitting methods significantly reduced the sum of square errors as well as the spreads of the fuzzy curve fitting estimation

enrichment functions for weak singularities in 2d elastic problems with isotropic and orthotropic materials abstract in this paper a new enrichment technique is presented for weak singularities in isotropic and orthotropic 2d linear elasticity problems with absolutely no information of the singularity order in a novel numerical approach the enriching bases are constructed through a weighted residual imposition of the partial differential equation conceptually the bases can be categorized as equilibrated basis functions as another advantage of the proposed technique it will be shown that the enriched solution may be extracted for either isotropic or orthotropic materials in a similar manner noting that singularity effect in orthotropic materials has not been as widely referred to as for isotropic materials in the literature the presented examples selected from the well known literature reveal the accuracy and applicability of the enrichment in the framework of a boundary type method the method may also be found useful for the researchers focusing on the extended finite element method or other similar mesh based or mesh less methods

two sample hypothesis tests of means of a fuzzy random variable abstract in this paper we will consider some two sample hypothesis tests for means concerning a fuzzy random variable in two populations for this purpose we will make use of a generalized metric for fuzzy numbers and we will develop an exact study for the case of normal fuzzy random variables and an asymptotic study for the case of simple general fuzzy random variables

singular functions for heterogeneous composites with cracks and notches the use of equilibrated singular basis functions abstract in this paper a novel method is presented to solve problems with weak singularities in two dimensional heterogeneous media using equilibrated singular basis functions this especially includes crack problems in composites of functionally graded material types the method is mainly presented in a boundary formulation although it may be found quite useful in other mesh based or mesh less approaches as the extended finite element method xfem the present paper considers harmonic and elasticity problems the most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem to this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach after developing the formulation in a mapped polar co ordination some primary basis functions generated from chebyshev polynomials and trigonometric functions along with corresponding weight functions are employed the numerical examples either selected from the well known literature or solved by well established techniques will demonstrate the capability of the method in problems related to composite materials

combined rank tests for randomly censored paired data abstract many authors have dealt with the problem of extending ordinary two sample rank tests to cases where censoring occurs albers and akritas 1987 proposed simple tests for this purpose based on the idea of making separate rankings for uncensored and censored observations and subsequently combining the resulting two rank statistics similarly some papers appeared that studied the problem of censoring for the paired data case rather than the two sample case this article indicates how the approach of albers and akritas can be adapted to the paired data case the tests involved are not based on the ranks of the differences but the differences of the ranks in the pooled sample in this way use is made of interblock information the asymptotic distribution of the new statistic is obtained under the null hypothesis and contiguous location alternatives by way of example wilcoxon and savage type scores are introduced that are optimal for logistic location and exponential scale alternatives respecti

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

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

artin nagata properties minimal multiplicities and depth of fiber cones abstract minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras in this paper we introduce the notion of goto minimal j multiplicity for ideals of maximal analytic spread in a cohen macaulay ring inspired by the work of s goto a jayanthan t puthenpurakal and j verma we study the interplay among this new notion the notion of minimal j multiplicity introduced by c polini and y xie and the cohen macaulayness of the fiber cone of ideals satisfying certain residual assumptions we are also able to provide a bound on the reduction number of ideals of goto minimal j multiplicity having either cohen macaulay associated graded algebra or linear decay in the depth of their powers

the strong law of large numbers for fuzzy random variables abstract sequences of independent and identically distributed fuzzy random variables are considered it is shown that the strong law of large numbers holds also for fuzzy random variables this result is used to give a consistent estimator for the expectation of a fuzzy random variable

a two sample test with interval censored data via multiple imputation interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be recorded as having occurred between two examination times in this paper we consider the problem of comparing two interval censored samples we propose to impute exact failure times from interval censored observations to obtain right censored data then apply existing techniques such as harrington and fleming s g rho tests to imputed right censored data to appropriately account for variability a multiple imputation algorithm based on the approximate bayesian bootstrap abb is discussed through simulation studies we find that it performs well the advantage of our proposal is its simplicity to implement and adaptability to incorporate many existing two sample comparison techniques for right censored data the method is illustrated by reanalysing the breast cosmesis study data set

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