Some Applications of the Boundary-Locus Method and the Method of D-Partitions

We summarize aspects of the method of D-partitions and its analogue,the boundary-locus technique. We apply these techniques to adiscussion of the stability of the delay equation where y(t)=(t), t [–, 0]. Some comparisons with resultsvalid for other functional equations are introduced. Of interestare the following features: (i) the results corresponding toour results but applicable to delay-differential equations arenot found in the literature, (ii) some unconventional parameterspaces are introduced, and (iii) the possibility of extendingresults which hold for a basic test equation to more generalequations is established.     

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.     

Morita Equivalent Twisted Actions and a New Version of the Packer-Raeburn Stabilization Trick

We show that every twisted action (, ) of a locally compactgroup G on a C*-algebra A is Morita equivalent to an ordinaryaction of G/N, where N is the domain of . This result allowsus to apply many results known for ordinary covariant systemsto the more general twisted case. Especially, this is true forresults which are obtained by the Mackey machine.     

Direct Sums of Operator Spaces

It is proved that if X and Y are operator spaces such that everycompletely bounded operator from X into Y is completely compactand Z is a completely complemented subspace of X Y, then thereexists a completely bounded automorphism : X Y X Y with completelybounded inverse such that Z = X₀ Y₀, where X₀ and Y₀ are completelycomplemented subspaces of X and Y, respectively. If X and Yare homogeneous, the existence is proved of such a under aweaker assumption that any operator from X to Y is strictlysingular. An upper estimate is obtained for ||||_cb||^–1||_cbif X and Y are separable homogeneous Hilbertian operator spaces.Also proved is the uniqueness of a ‘completely unconditional’basis in X Y if X and Y satisfy certain conditions.     

Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces

We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceX_n, of weight such that every n-dimensional complete metricspace of weight is embeddable in X_n, as a closed subset.     

Comparison Theorems in Delay Differential Equations in a Critical State and Applications

First, two comparison theorems are established between the firstorder delay differential equation x'(t)+p(t)x((t))=0, tt₀ (*) and two related second order ordinary differential equationsin the case when Next, by using these comparison theorems some new oscillationand non-oscillation criteria are given for (*) under condition(**). As a consequence, an open problem by Elbert and Stavroulakisis completely solved.     

An Efficient Enthalpy-type Method for the Stefan Problem

The paper presents a new finite-difference method for solvingthe one-dimensional two-phase Stefan problem. Under assumptionson the data which guarantee the temperature u and the movingboundary s to belong to and , respectively, we obtain L₂-errorestimates of order O(h + h^–?) provided the time step is chosen such that Numerical aspects are discussed.     

Dynamic crack-tip singular stress in a cylindrically orthotropic elastic solid

Anti-plane wave motion is induced in a cylindrically orthotropicelastic solid containing a semi-infinite stress-free crack,by a point impulsive body force. First, the static version ofthe problem is solved for the fracture stress _z. Here, a globalsolution is obtained and then examined at the crack tip in orderto determine the nature of the spatial singularity. Next, thedynamic problem is treated and it is found that the dominantspatial singularity for _z at the crack tip is the same as inthe static case. However, the dynamic part of the stress intensityfactor, T, may introduce a further singularity. Several equivalentexpressions are presented for T, one of which is examined insome detail.     

Inequalities for Oscillating p-Functions

Upper bounds on [ ) are derived for those p-functions such thatp() = m is minimum, and p' (+) = q(l-m).     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

Mathematical Models of Solid-Solid Phase Transitions in Steel

The austenite-pearlite phase transition in steel occurs overa large range of temperatures and gradually in time. Accordingto the ‘additivity rule’, for any prescribed temperatureevolution (?), at any time t, the fraction F(t) of transformedaustenite is characterized by the condition where is a prescribed positive function. This lawdefines a causal functional for any T > 0. The properties of F are here analysed. The cooling of a spatially distributed austenitic body is thenstudied. The enthalpy conservation equation is coupled withlimit conditions and with the constitutive relation F = F().Existence of a strong solution is then proven for a formulationin Sobolev spaces.     

Delay dependent stability regions of {Theta}-methods for delay differential equations

In this paper asymptotic stability properties of -methods fordelay differential equations (DDEs) are considered with respectto the test equation y'(t) = ay(t) + by(t - ), t > 0, y(t)= g(t), - t 0, where > 0. First we examine extensivelythe instance where a, b and g(t) is a continuous real-valuedfunction; then we investigate the more general case of a, b C and g(t) a continuous complex-valued function. The last decade has seen a relatively large number of papersdevoted to the study of the stability of -methods, using thetest equation (0.1). In those papers, conditions that are strongerthan necessary for the (asymptotic) stability of the zero solutionare assumed; for instance, [a]+¦b¦ < 0, thatis the set of complex pairs (a, b) such that the zero solutionof (0.1) is asymptotically stable for every > 0. In thispaper we study, instead, the stability properties of -methodsfor equation (0.1) with an arbitrary but fixed value of .     

Length Spectrums of Riemann Surfaces and the Teichmuller Metric

Suppose that T (S₀) is the Teichmüller space of a compactRiemann surface S₀ of genus g > 1. Let d_T(·, ·)be the Teichmüller metric of T(S₀) and let d_S(·,·) be a metric of T(S₀) defined by the length spectrumsof Riemann surfaces. The author showed in a previous paper thatd_T and d_S are topologically equivalent and , where C(₁) is a constant depending on ₁. In thispaper, it is shown that d_T and d_S are not metrically equivalent;that is, there is no constant C > 0 such that for all ₁ and ₂ in T(S₀). 2000 Mathematics SubjectClassification 32G15, 30C62, 30C75.     

The Martin Boundary and Ratio Limit Theorems for Killed Random Walks

It is shown that if S is an aperiodic random walk on the integers,S* is the Markov chain that arises when S is killed when itleaves the non-negative integers, and H⁺ is the renewal processof weak increasing ladder heights in S, then there is a 1:1correspondence between functions which are non-negative andsuperregular for S* and H⁺. This allows all the regular functionsfor S* to be described, and thus a result due to Spitzer tobe completed for the recurrent case. This result is then appliedto give a ratio limit theorem for P_x(*=n)/P₀{*=n}, where * isthe lifetime of S*, in the case when S drifts to –, andthe right-hand tail of its step distribution is ‘locallysub-exponential’.     

On Some Steady-state Moving Boundary Problems in the Linear Theory of Viscoelasticity

Three steady-state moving boundary problems in the linear theoryof viscoelasticity are considered when the inertia terms aretaken into account. Attention is directed to the dimensionlessparameter = V/L where V is the speed of the boundary, is therelaxation time of the medium, and L is a length associatedwith the geometry of the problem. When is small the problemsare recognized as having a singular perturbation character andthe method of matched asymptotic expansions is used. The problemsare (i) a semi-infinite crack moving steadily in a clamped viscoelasticstrip, (ii) a finite length crack moving in an infinite viscoelasticmedium, and (iii) a steady rolling cylinder on a visco-elastichalf-space.     

Optimal Stopping in the L log L-Inequality of Hardy and Littlewood

Let B = (B_t)_t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(^r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu_* = 1 + 1/e^c(c – 1) and = (c – 1)/c for c >1, with X_t = |B_t| and S_t = max_0rt|B_r|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(^r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v_* = c/e(c – 1) and =(c – 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog⁺ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

SEMISIMPLE BANACH ALGEBRAS GENERATED BY STRONGLY CONTINUOUS REPRESENTATIONS OF MOORE GROUPS

We prove that if is a strongly continuous bounded representationof a Moore group G on a Banach space X, and if the Arveson spectrumof is scattered, then the closure with respect to the weakoperator topology in L(X) of the algebra generated by the transforms _Gf(t) (t)d t with f L¹(G) is a semisimple Banach algebra.     

The Alternative Dunford-Pettis Property, Conjugations and Real Forms of C*-Algebras

Let be a conjugation, alias a conjugate linear isometry oforder 2, on a complex Banach space X and let X be the real formof X of -fixed points. In contrast to the Dunford–Pettisproperty, the alternative Dunford–Pettis property neednot lift from X to X. If X is a C^*-algebra it is shown thatX has the alternative Dunford–Pettis property if and onlyif X does and an analogous result is shown when X is the dualspace of a C^*-algebra. One consequence is that both Dunford–Pettisproperties coincide on all real forms of C^*-algebras.