Acyclicity criteria for complexes associated with an alternating map

When is a Gorenstein ideal of grade in a local ring , results of Boffi and Sánchez, and of Kustin and Ulrich show that for each one can construct in a canonical way a finite free complex that is ``approximately" a resolution for the ideal . Kustin and Ulrich also provide a sufficient condition that is acyclic, and a sufficient condition that is a resolution of . We complete these two acyclicity criteria by showing that the corresponding sufficient conditions are also necessary.

     

Gorenstein derived functors

Over any associative ring it is standard to derive using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product using Gorenstein flat modules.

     

Flat covers and cotorsion envelopes of sheaves

In this paper we prove that any sheaf of modules over any topological space (in fact, any -module where is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( -modules) does not have in general enough projectives.

     

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s 2n points in general linear position in P ⁿ , and the Koszul property of the generic Artinian Gorenstein algebra of socle degree 3.     

Heuristic Spectral Techniques for the Reduction of Bandwidth and Work-Bound of Sparse Matrices

In this paper, we present heuristic techniques for the reduction of the bandwidth of a sparse matrix as well as for the reduction of the cost of the associated Cholesky factorization. Our algorithms are inspired by the spectral method of Barnard, Pothen and Simon (1995), which derives a permutation for reducing the envelope-size of a sparse matrix by computing the second eigenvector of the associated Laplacian matrix. Two main modifications of that method are proposed and tested. The first is based on the experimental observation that it is often preferable to perform only few iterations of an iterative method converging to the second eigenvector; the second is the introduction of a weighted Laplacian. These simple ideas allow us to obtain a family of spectral methods that have been carefully tested on a set of matrices whose size ranges from few hundred to one million.     

n-Strongly Gorenstein Projective,Injective and Flat Modules

In this article, we study the relation between m-strongly Gorenstein projective (resp., injective) modules and n-strongly Gorenstein projective (resp., injective) modules whenever m ≠ n, and the homological behavior of n-strongly Gorenstein projective (resp., injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp., injective) modules.     

Envelope Theorems in Dynamic Programming

The envelope theorem is a statement about derivatives along an optimal trajectory. In dynamic programming the envelope theorem can be used to characterize and compute the optimal value function from its derivatives. We illustrate this here for the linear-quadratic control problem, the resource allocation problem, and the inverse problem of dynamic programming.     

超短脉冲光束传输缓变包络近似理论的失效和空间奇异性的形成与消除

研究了单周期或更短的超短脉冲光束在真空中的传输.结果显示,在此情况下缓变包络近似理论已不再有效,其失效的原因是,该理论导致了脉冲光束解的空间奇异性,并使脉冲光束不再具有符合物理意义的光束行为,而通过复解析信号的分析方法可以消除这种空间奇异性.给出了时间波形为Gauss,Hyperbolic Secant和Lorentz的脉冲光束奇异性的计算模拟,并给出了解决这些奇异性的复解析信号解.同样,会发现在脉冲光束超过一定的长度后两种解是一致的.     

Envelope Periodic Solutions to Coupled Nonlinear Equations

The envelope periodic solutions to some nonlinear coupled equations are obtained by means of the Jacobielliptic function expansion method. And these envelope periodic solutions obtained by this method can degenerate tothe envelope shock wave solutions and/or the envelope solitary wave solutions.     

An approximation of the first passage probability of systems under nonstationary random excitation

An approximate method is presented for obtaining analytical solutions for the conditional first passage probability of systems under modulated white noise excitation. As the method is based on VanMarcke＇s approximation, with normalization of the response introduced, the expected decay rates can be evaluated from the second-moment statistics instead of the correlation functions or spectrum density functions of the response of considered structures. Explicit solutions to the second-moment statistics of the response are given. Accuracy, efficiency and usage of the proposed method are demonstrated by the first passage analysis of single-degree-of-freedom （SDOF） linear systems under two special types of modulated white noise excitations.