On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action

Let be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring (associated to and the Frobenius homomorphism , in the indeterminate ) that is both -torsion and Artinian over .

     

Capital Investment — An Optimal Control Perspective

Many managers appear to have a mental model of how investment decisions should be carried out. This paper attempts to identify some of the characteristics of such a mental model by constructing an optimal control model of the process of authorizing an investment project. The key activities in the model are design, support generation and authorization. It is assumed that the effort that can be expended on each is limited. The objective function is the net present value of the cash flows associated with the project. The solution to the model is readily interpreted qualitatively. For viable projects, four different patterns of decision-making are found, each of which is optimum under appropriate circumstances. It is argued that in many actual investment decisions it should be possible for managers to approximate reasonably closely the optimal behaviour, and that therefore the optimal control model may shed light on managers' mental model.     

Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y^′=f(t,y)$y^{\prime }=f(t,y)$ with initial conditions y(_t0)=_y0$y\left(t_0\right)=y_0$. Its formula uses y^′$y^{\prime }$, y^″$y^{″}$ and y^?$y^{?}$ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.     

A conservative formulation for plasticity

In this paper we propose a fully conservative form for the continuum equations governing rate-dependent and rate-independent plastic flow in metals. The conservation laws are valid for discontinuous as well as smooth solutions. In the rate-dependent case, the evolution equations are in divergence form, with the plastic strain being passively convected and augmented by source terms. In the rate-independent case, the conservation laws involve a Lagrange multiplier that is determined by a set of constraints; we show that Riemann problems for this system admit scale-invariant solutions.     

Chaos,transport and mesh convergence for fluid mixing

Chaotic mixing of distinct fluids produces a convoluted structure to the interface separating these fluids. For miscible fluids （as considered here）, this interface is defined as a 50% mass concentration isosurface. For shock wave induced （Richtmyer-Meshkov） instabilities, we find the interface to be increasingly complex as the computational mesh is refined. This interfacial chaos is cut off by viscosity, or by the computational mesh if the Kolmogorov scale is small relative to the mesh. In a regime of converged interface statistics, we then examine mixing, i.e. concentration statistics, regularized by mass diffusion. For Schmidt numbers significantly larger than unity, typical of a liquid or dense plasma, additional mesh refinement is normally needed to overcome numerical mass diffusion and to achieve a converged solution of the mixing problem. However, with the benefit of front tracking and with an algorithm that allows limited interface diffusion, we can assure convergence uniformly in the Schmidt number. We show that different solutions result from variation of the Schmidt number. We propose subgrid viscosity and mass diffusion parameterizations which might allow converged solutions at realistic grid levels.     

Associated primes of graded components of local cohomology modules

The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all > 0$">. This paper is concerned with the asymptotic behaviour of as .

The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when f$">. They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

On the photoisomerization of the benzisothiazole portion of ziprasidone

Photostability challenge of ziprasidone in solution shows that the benzisothiazole moiety undergoes isomerization to the corresponding benzthiazole. A model compound, 3-piperazinyl-1,2-benzisothiazole, also undergoes this photoisomerization. Identification of the products has been confirmed by synthesis of the proposed molecules.