Local rings of countable Cohen-Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore, we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

     

Promoting students’ graphical understanding of the calculus

Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were recorded on audio tape and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data, we were able to construct a picture of the students’ graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to ‘walk’ an associated displacement time graph, we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a ‘physical feel’.     

WEAK CONVERGENCE FOR NON-UNIFORM φ-MIXING RANDOM FIELDS

ＷＥＡＫＣＯＮＶＥＲＧＥＮＣＥＦＯＲＮＯＮＵＮＩＦＯＲＭφＭＩＸＩＮＧＲＡＮＤＯＭＦＩＥＬＤＳＬＵＣＨＵＡＮＲＯＮＧＡｂｓｔｒａｃｔＬｅｔ｛ξｔ，ｔ∈Ｚｄ｝ｂｅａｎｏｎｕｎｉｆｏｒｍφｍｉｘｉｎｇｓｔｒｉｃｔｌｙｓｔａｔｉｏｎａｒｙｒｅａ...     

One-step optical trinary signed-digit arithmetic using redundant bit representations

A simple one-step fully parallel trinary signed-digit arithmetic is proposed for parallel optical computing. This technique performs multidigit carry-free addition and borrow-free subtraction in constant time. The trinary signed-digit arithmetic operations are based on redundant bit representation of the digits. Optical implementation of the proposed arithmetic can be carried out using correlation or matrix multiplication based schemes. An efficient matrix multiplication based optical implementation that employs a fixed number of minterms for any operand length is developed. It is shown that only 30 minterms (less than recently reported techniques) are enough for implementing the one-step trinary addition and subtraction.     

A simple proof of the Uncertainty Principle for compact groups

We give a simple proof of the Uncertainty Principle for finite nonabelian groups, which generalizes directly to compact groups.     

HOW TO CONSTRUCT LAX REPRESENTATION FOR CONSTRAINED FLOWS OF THEBOUSSINESQ HIERARCHY VIA ADJOINT REPRESENTATIONS

1IntroductionTheBoussinesqequationarisesinseveralphysicalapplicationandhasbeenstudiedquiteextensivelyinthepast[1--3].Inarecentpaper[4],itwasfoundthattheBoussinesqhierarchycanbeobtainedfromthezero-curvatureconditionassociatedwiththegroupSL(3,R).ThisshowsadirectrelationshipbetweenthegroupSL(3,R)andtheW3algebraofZamolodchikov.Recentlytherehasbeenconsiderableinterestinthedecompositionofsolitonequationsviaconstraintsrelatingpotentialandeigenfunctions,becausethedecompositionprovidesaneffectiveme…     

新经典MHD效应与内扭曲模本征方程

本文从一简化的新经典ＭＨＤ方程组出发，利用气球模表示及多尺度近似等方法，在ｑ＝１磁面附近的过渡层内导出了包含新经典ＭＨＤ效应的内扭曲模本征方程。与从前相关工作的比较表明，新经典ＭＨＤ效应对通常采用的磁流体或电阻磁流体内扭曲模本征方程，均产生十分重要的修正。     

A MODEL FOR WHITE NOISE ANALYSIS IN P-ADIC NUMBER FIELDS

ＡＭＯＤＥＬＦＯＲＷＨＩＴＥＮＯＩＳＥＡＮＡＬＹＳＩＳＩＮＰ－ＡＤＩＣＮＵＭＢＥＲＦＩＥＬＤＳ（黄志远）￥ＡｎｄｒｅｗＫｈｒｅｎｎｉｋｏｖ（ＭｏｓｃｏｗＩｎｓｔ．ｏｆＥｌｅｃｔｒｏｎｉｃＥｎｇｉｎｅｅｒｉｎｇ，１０３４９８，Ｚｅｌｅｎｏｇｒａｄ，Ｋ－...     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature.

In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.

Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.

In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).