Effective Distribution Coefficient of Silicon Dopants During Magnetic Czochralski Growth

Silicon crystals have been grown under axial magnetic fieldsof up to 0.2 T. At 0.05 T the field starts to interact withthe melt and dopant incorporation becomes erratic. As the fieldis increased to 0.2 T the interaction becomes more controlledand it is seen that the effective distribution coefficient (k_eff)moves towards unity. A theory based on the analysis by Burton, Prim & Slichterof the dependence of the effective distribution coefficient(k_eff) on growth and crystal rotation rates in Czochralski growthis extended to include the effect of an imposed steady axialmagnetic field. The flow fields incorporated into the theoryare based on the analysis of the hydromagnetic flow at a rotatingdisc due to Kakutani (J. Phys. Soc. Jpn (1962) 17, 1496). Itis shown that k_eff approaches unity as the field increases.     

Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

Equilibrium attractivity of Runge-Kutta methods

For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable Runge–Kutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor Runge–Kutta solutions. This property, called equilibriumattractivity, means that the norm of the righthand side f neverincreases. It is a property dual to algebraic stability sinceneither is sufficient for the other, in general. We derive sufficientalgebraic conditions for Runge–Kutta methods and proveequilibrium attractivity of the high-order algebraically stableRadau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocationmethods (which are not algebraically stable). No smoothnessassumptions on f and no stepsize restrictions are required but,except for some simple cases, f has to satisfy certain additionalproperties which are generalizations of the simple one-sidedLipschitz condition using more than two argument points. Thesemultipoint conditions are discussed in detail.     

Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model

Theorem 4 is a characterization of Woodin cardinals in terms of Skolem hulls and Mostowski collapses. We define weakly hyper-Woodin cardinals and hyper-Woodin cardinals. Theorem 5 is a covering theorem for the Mitchell-Steel core model, which is constructed using total background extenders. Roughly, Theorem 5 states that this core model correctly computes successors of hyper-Woodin cardinals. Within the large cardinal hierarchy, in increasing order we have: measurable Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin, and superstrong cardinals. (The comparison of Shelah versus hyper-Woodin is due to James Cummings.)

     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

Solvability of symmetric word equations in positive definite letters

Let S(X, B) be a symmetric (‘palindromic’) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case.     

Bicriterion shortest hyperpaths in random time-dependent networks

In relevant application areas, such as transportation and telecommunications,there has recently been a growing focus on random time-dependentnetworks (RTDNs), where arc lengths are represented by time-dependentdiscrete random variables. In such networks, an optimal routingpolicy does not necessarily correspond to a path, but ratherto an adaptive strategy. Finding an optimal strategy reducesto a shortest hyperpath problem that can be solved quite efficiently. The bicriterion shortest path problem, i.e. the problem offinding the set of efficient paths, has been extensively studiedfor many years. Recently, extensions to RTDNs have been investigated.However, no attempt has been made to study bicriterion strategies.This is the aim of this paper. Here we model bicriterion strategy problems in terms of bicriterionshortest hyperpaths, and we devise an algorithm for enumeratingthe set of efficient hyperpaths. Since the computational effortrequired for a complete enumeration may be prohibitive, we proposesome heuristic methods to generate a subset of the efficientsolutions. Different criteria are considered, such as expectedor maximum travel time or cost; a computational experience isreported.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

Modelling the capillary locking of point defects in nematic liquid crystals

DEDICATED TO JERRY ERIKSEN: It has long been known that the equilibrium configuration withminimum energy of a nematic liquid crystal within a cylindersubject to homeotropic conditions on the lateral boundary isescaped along the axis of the cylinder, and there is no singularityof the orientation field. So problems of explaining the presenceof point defects in capillary tubes and of exploring their stabilityarise. There is enough evidence to believe that the menisciplay a central role in preventing the orientation field arounda point defect from unwinding towards the escaped configuration.We propose a variational model which describes how a point defectinteracts with a meniscus. This interaction fades away at afinite distance. When active, it is two-sided, being repulsiveat first and then attractive when the defect comes closer tothe centre of curvature of the meniscus. Thus, when a defectis enclosed between two menisci they can become antagonists,so that there is a metastable equilibrium position where thedefect could be locked in.     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

A Solution to the Invariant Subspace Problem

In this paper we exhibit a Banach space X, and a continuouslinear operator T: X X, such that X has no nontrivial closedT-invariant subspace.     

Orthogonal polynomials with a resolvent-type generating function

The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

Crystal Bases for Quantum Affine Algebras and Combinatorics of Young Walls

In this paper we give a realization of crystal bases for quantumaffine algebras using some new combinatorial objects which wecall the Young walls. The Young walls consist of colored blockswith various shapes that are built on a given ground-state walland can be viewed as generalizations of Young diagrams. Therules for building Young walls and the action of Kashiwara operatorsare given explicitly in terms of combinatorics of Young walls.The crystal graph of a basic representation is characterizedas the set of all reduced proper Young walls. The characterof a basic representation can be computed easily by countingthe number of colored blocks that have been added to the ground-statewall. 2000 Mathematical Subject Classification: 17B37, 17B65,81R50, 82B23.     

Finite Subsequences of Shift Register Sequences

In a stream cipher a cryptogram is produced from a binary datastream by modulo-2-adding it to a keystream sequence. The securityof the system relies on the inability of an interceptor to determinethis keystream sequence. One obvious requirement for such asystem is that there should be sufficiently many possibilitiesfor the keystream sequence that the interceptor cannot possiblytry them all. In this paper we consider the likelihood of an interceptor beingable to decipher the cryptogram correctly even though he maybe trying the wrong keystream sequence. This possibility arisesbecause the length of any particular message is likely to beconsiderably shorter than the period of the keystream sequence,and thus only a comparatively small section of the keystreamsequence is used. Hence, if the interceptor tries a sequencewhich intersects (i.e. agrees) with the keystream sequence inthe appropriate positions, he will deduce the message correctly. A number of the standard methods for generating keystream sequencesuse shift registers as ‘building blocks’. So welook in considerable detail at the number of intersections (ofvarious lengths) for sequences generated by two different shiftregisters. We also show that if a keystream sequence has linearequivalence n, then the local linear equivalence of any subsequenceof length at least 2n is n. This means that if the message haslength at least 2n and the keystream sequence has linear equivalencen, then there is no other sequence of linear equivalence lessthan n+1 which can be used to decipher correctly.     

Long Arithmetic Progressions in Sum-Sets and the Number x-Sum-Free Sets

In this paper we obtain optimal bounds for the length of thelongest arithmetic progression in various kinds of sum-sets.As an application, we derive a sharp estimate for the numberof sets A of residues modulo a prime n such that no subsum ofA equals x modulo n, where x is a fixed residue modulo n. 2000Mathematics Subject Classification 05A16, 11B25, 11P32.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

CARTAN DETERMINANTS OF CELLULAR ALGEBRAS

Let A be a finite-dimensional cellular algebra over a field.It is shown that the Cartan determinant of A is a positive integer;furthermore, the Cartan determinant of value 1 is equivalentto the condition that A is quasi-hereditary. In this article,we establish a formula for expressing the Cartan determinantsof cellular algebras. By applying this formula, we obtain anecessary and sufficient condition for a cellular algebra withCartan determinant 2 to be standardly stratified. Moreover,we provide a sufficient condition for an arbitrary cellularalgebra to be standardly stratified and an inductive constructionof standardly stratified cellular algebras.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)