ALMOST COHEN–MACAULAY MODULES

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R _P , R _P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X ₁, …, X _n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.     

FINITENESS CONDITIONS FOR GENERALIZED EXPONENTS OF DIGRAPHS

1.IntroductionR.A.BrualdiandBolianLiuintroducedin199011]theconceptofgeneralizedexponentsforprimitivedigraphs.Thisconceptisageneralizationofthetraditionalconceptoftheexponentsforprimitivedigraphsandhasbackgroundsinmemorylesscommunicationsystemsassociatedwithdigraphs.Inthispaperweshowthatgeneralizedexponentscanalsobedefinedfordigraphswhicharenotnecessarilyprimitive.WewillgivenecessaryandsufficientconditionsforthefinitenessofthegeneralizedexponentsexpD(k),f(D,k)andF(D,k).Theseideasandresultss…     

GORENSTEIN MODULES,FINITE INDEX,AND FINITE COHEN–MACAULAY TYPE

ABSTRACT

A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.     

CERTAIN SUBCLASSES OF ANALYTIC p－VALENT FUNCTIONS FOR OPERATORS ON HILBERT SPACE

ＣＥＲＴＡＩＮＳＵＢＣＬＡＳＳＥＳＯＦＡＮＡＬＹＴＩＣｐ－ＶＡＬＥＮＴＦＵＮＣＴＩＯＮＳＦＯＲＯＰＥＲＡＴＯＲＳＯＮＨＩＬＢＥＲＴＳＰＡＣＥＪｉａｎＭｉｎｇ（蹇明）（Ｄｅｐｔ．ｏｆＭａｔｈ．ＷｕｈａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｗｕ...     

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let (S,≤) be an ordered set. Recall that (S,≤) is artinian if every strictly decreasingsequence of elements of S is ?nite, and that (S,≤) is narrow if every subset of pairwiseorder-incomparable elements of S is ?nite. Let S be a commutative monoid. Unl…     

THE GIBBS DERIVATIVES OF GENERALIZED BRIDGE FUNCTIONS

The logical derivatives of p-adic copy-shift and shift-copy generalized Bridge functions are given in this paper. When p equals 2, they are exactly the logical derivatives of natural ordering copy-shift and copy-shift Bridge functions.     

THE HILBERT BOUNDARY PROBLEM OF DOUBLY PERIODIC ANALYTIC FUNCTIONS

The doubly periodic Hilbert boundary value problem is discussed in this paper.First,certain kind of integral representations of doubly quasi-periodic analytic functions inmultiplication is established so that the Dirichlet problem of such functions is solved.Then,by the method of regularization,the Hilbert boundary value problem is transferredto such a problem,and it is reduced at length to some Fredholm integral equation.Thenumber of solutions and conditions of solvability as well as the form of the generalsolution are obtained.     

APPROXIMATION OF CONTINUOUS FUNCTIONS BY GENERALIZED KARAMATA MEANS

A sufficient condition for the order of approximation of a continuous 2π periodic function with a givenmajorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order isobtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums oftheir Fourier series to be of Jackson order. The error estimate is shown to be the best possible.     

GENERALIZED WEAK FUNCTIONS

In this paper the authors introduce some new ideas on generalized numbers and generalized weak functions. They prove that the product of any two weak functions is a generalized weak function. So in particular they solve the problem of the multiplication of two generalized functions.     

APPROXIMATION OF CONTINUOUS FUNCTIONS BY THE GENERALIZED BERNSTEIN-BEZIER POLYNOMIALS

§1. Introduction In [1], for any α>0, and a function φ defined on [0,1], Geng-Zhe Change defined the generalized Bernstein-Bezier polynomial ofφ as follows: B_(n, a)(φ, x) = sum from k=0 to n φ(k/n){f_(nk)~a(x)-f_(n,k+1)~a,(x)} (1.1)where f_(n, n+1) (x) =0 and f_(n, k)(x) = sum from j=k to n x~j(1-x)~(n-j) k=0,1,...,n. (1.2)are the Bezier base functions of degree n.Obviously, for any x ∈(0, 1), we have     

A NOTE ON LAGRANGE INTERPOLATION OF FUNCTIONS OF GENERALIZED BOUNDED VARIATION

In this paper we solve a remained problem in[2],whether the following estimate approximation for theclass f∈C[-1,1]∩BV by Lagrange interpolation based on the Jacobi abscissasi,L_n~(α,β)(f,x)-f(x)=Ο(1/n)holds,ifα≠β≥-1.     

THE GENERALIZED AIRY FUNCTIONS

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｏｗｎｇｓｅｃｏｎｄｏｒｄｅｒｌｉｎｅａｒｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｄ２ｙｄｘ２－ｘλｙ＝０，（１．１）ｗｈｅｒｅｔｈｅｃｏｎｓｔａｎｔλ≥０ｓａｔｉｓｆｉｅｓ（－１）λ＝±１．Ｗｈｅ...     

ASSESSMENT OF LOCAL INFLUENCE FOR GENERALIZED CORRELATION COEFFICIENTS

ＡＳＳＥＳＳＭＥＮＴＯＦＬＯＣＡＬＩＮＦＬＵＥＮＣＥＦＯＲＧＥＮＥＲＡＬＩＺＥＤＣＯＲＲＥＬＡＴＩＯＮＣＯＥＦＦＩＣＩＥＮＴＳ（陈建宝，陈涛）￥ＣｈｅｎＪｉａｎｂａｏＣｈｅｎＴａｏ（Ｄｅｐｔ．ｏｆＳｔａｔｉｓｔｉｃｓ，ＹｕｎｎａｎＵｎｉｖ，Ｋｕｎｍｉ...     

ON APPROXIMATION OF CLASSES OF FUNCTIONS FOR MODIFIED SZASZ OPERATORS

In this note,using some results of the probability theory we obtained some estimates ofthe approximation of classes or functions for modified Szasz operators with regard to two spe-ciai classes or functions.     

COMPARISON THEOREMS FOR GREEN FUNCTIONS OF MARKOV CHAINS

The author presents some straightforward proofs for two comparison theorems for Green functions of Markov chains, which slightly improve the previous results by Varopoulos, Durrett and Yan and Chen. A recent result by Rogers and Williams about instantaneous Markov chains is also improved by using the same idea.     

UNIQUENESS OF GENERALIZED SOLUTION FOR THE CAUCHY PROBLEM OF TRANSPORTATTION EQUATIONS

1IntroductionInthepresentpaper,weconsidertheuniquenessofsolutionforthetransportationsystemwithinitialdatawherepanduaredensityandvelocityrespectively.Thesystemofcollservationlaws(1.l)describesthefluidmechanicswithconstantpressure,whichisusedinplasmaphysicswhosepressureisassumedtobeconstant.TheCauchyproblemforthissystemisnotwell-posedintheclassofboundedmeasurablefunctions(see[31).In[8],Z.Wang,F.HuangandX.DinghaveprovedtheealstenceofsolutionsdefinedbyLebesgue--Stieltjesintegral.Beforedetaildi…     

PERTURBING FAMILIES OF GENERALIZED LYAPUNOV FUNCTIONS AND RELATIVE STABILITY IN TERMS OF TWO MEASURES

ＰＥＲＴＵＲＢＩＮＧＦＡＭＩＬＩＥＳＯＦＧＥＮＥＲＡＬＩＺＥＤＬＹＡＰＵＮＯＶ　ＦＵＮＣＴＩＯＮＳＡＮＤＲＥＬＡＴＩＶＥＳＴＡＢＩＬＩＴＹＩＮＴＥＲＭＳＯＦＴＷＯ　ＭＥＡＳＵＲＥＳＳｈａｎｇＬｉｊｕｎ（商立军）（ＴｈｅＦｏｕｒｔｈＭｉｌｉｔａｒｙＭｅ...     

TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted l-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series.The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic contin- uous functions spaces.Tractability is the minimal number of function samples required to solve the problem in polynomial in ε~(-1)and d.and the strong tractability is the pres- ence of only a polynomial dependence in ε.This problem has been recently studied for quasi-Monte Carlo quadrature rules.quadrature rules with non-negative coefficients. and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables.The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14]on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative.The arguments are not constructive.     

ON GUNDERSEN''''S QUESTION FOR THE UNICITY OF MEROMORPHIC FUNCTIONS

The uniqueness of meromorphic functions that share four values are investigated, some results are obtained to show that if two nonconstant entire functions share three finite values IM, then the functions necessarily share all three values CM.     

WEIGHT FUNCTIONS FOR SOME NEW CLASSES OF ORTHOGONAL POLYNOMIALS

This article continue the discussion of finding weight functions for orthogonal polynomials in thtee situations. The Gegenbauer polynoraials are shown to have a distributional weight function.The polynomials of Geronimus [6] which are orthogonal on[—1,1]ale explicitely calculated.An application to Pade approximations is made.Two negative situations are mentioned.