An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87)for analysing the time discretization error and assuming moreregularity on the initial data, we improve on the error boundderived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16,257-287) for a fully practical piecewise linear finite elementapproximation with a backward Euler time discretization of amodel for phase separation of a multi-component alloy.     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

On graphs having domination number half their order

In this paper we present a characterization of connected graphs of order 2n with domination numbern. Using this class of graphs, we determine an infinite class of graphs with the property that the domination number of the product of any two is precisely the product of the domination numbers.     

Computing the boxicity of a graph by covering its complement by cointerval graphs

If $\text{F}$ is a family of sets, its intersection graph has the sets in $\text{F}$ as vertices and an edge between two sets if and only if they overlap. This paper investigates the concept of boxicity of a graph G, the smallest n such that G is the intersection graph of boxes in Euclidean n-space. The boxicity, b(G), was introduced by Roberts in 1969 and has since been studied by Cohen, Gabai, and Trotter. The concept has applications to niche overlap (competition) in ecology and to problems of fleet maintenance in operations research. These applications will be described briefly. While the problem of computing boxicity is in general a difficult problem (it is NP-complete), this paper develops techniques for computing boxicity which give useful bounds. They are based on the simple observation that b(G)≤k if and only if there is an edge covering of $\text{G}$ by spanning subgraphs of $\text{G}$, each of which is a cointerval graph, the complement of an interval graph (a graph of boxicity ≤1.).     

MEXIT: Maximal un-coupling times for stochastic processes

Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this “un-coupling” or “maximal agreement” construction as MEXIT, standing for “maximal exit”. After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of MEXIT for Brownian motions with two different constant drifts.     

A vector Chebyshev algorithm

We consider polynomials orthogonal relative to a sequence of vectors and derive their recurrence relations within the framework of Clifford algebras. We state sufficient conditions for the existence of a system of such polynomials. The coefficients in the above relations may be computed using a cross-rule which is linked to a vector version of the quotient-difference algorithm, both of which are proved here using designants. An alternative route is to employ a vector variant of the Chebyshev algorithm. This algorithm is established and an implementation presented which does not require general Clifford elements. Finally, we comment on the connection with vector Padé approximants. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability of Partially Implicit Langevin Schemes and Their MCMC Variants

A broad class of implicit or partially implicit time discretizations for the Langevin diffusion are considered and used as proposals for the Metropolis–Hastings algorithm. Ergodic properties of our proposed schemes are studied. We show that introducing implicitness in the discretization leads to a process that often inherits the convergence rate of the continuous time process. These contrast with the behavior of the naive or Euler–Maruyama discretization, which can behave badly even in simple cases. We also show that our proposed chains, when used as proposals for the Metropolis–Hastings algorithm, preserve geometric ergodicity of their implicit Langevin schemes and thus behave better than the local linearization of the Langevin diffusion. We illustrate the behavior of our proposed schemes with examples. Our results are described in detail in one dimension only, although extensions to higher dimensions are also described and illustrated.     

Shotgun reconstruction in the hypercube

Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the -dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for , almost every -coloring can be reconstructed from the multiset of colorings of 1-balls.     

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.