On the Auslander-Reiten Periodicity of Self-Injective Algebras

This paper investigates how to relate the syzygy periodicityof a self-injective algebra A to its Auslander–Reitenperiodicity. Moreover, a characterization is provided of theAuslander–Reiten bounded A–A-bimodules that areperiodic. 2000 Mathematics Subject Classification 16G70, 16E40(primary), 16G20 (secondary).     

The Accuracy of Error Estimates for Systems of Linear Algebraic Equations

Satisfactory error estimates are obtained from iterative refinementof the solution using M^–l, an approximation to the inverseof A and involving ||I–M^–1A||.     

The Marica-Schonheim Inequality in Lattices

The Marica-Schönheim Inequality says that if A is a finitefamily of sets, then |A–||A| where A–A=[A₁\A₂:A₁,A₂A]. For a finite lattice L and AL, we define a–b=(J_a\J_b)where J_a=[jL:ja and j is join-irreducible], and if AL then welet A–A=[a₁–a₂: a₁, a₂A]. Then the analogue of theMarica-Schöonheim Inequality is |A–A|A| for all AL.We prove that this is true if L is distributive or complementedand modular or L is a partition lattice.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

Minimal bases of matrix pencils and coprime matrix fraction descriptions

Two new methods for the construction of coprime MFDs are proposedstarting from a minimal state space realization of a transferfunction matrix. The approaches are based on the constructionof minimal bases for the kernels of the state-space-based pencils[s I – A, – B], [s I – A^t, –C^t]. Thelatter problem is explicitly solved using the Toeplitz matrixconstruction of minimal bases of matrix pencils. The numericalalgorithms of the methods are presented and some examples demonstratingthe implementation of the algorithms are also given. Received 18 May 2000. Revised 5 September 2000. Accepted 5 September 2000.     

Fields of Definition for Division Algebras

Let A be a finite-dimensional division algebra containing abase field k in its center F. A is defined over a subfield F₀if there exists an F₀-algebra A₀ such that . The following are shown. (i) In many cases A canbe defined over a rational extension of k. (ii) If A has odddegree n 5, then A is defined over a field F₀ of transcendencedegree 1/2(n–1)(n–2) over k. (iii) If A is a Z/mx Z/2-crossed product for some m 2 (and in particular, if Ais any algebra of degree 4) then A is Brauer equivalent to atensor product of two symbol algebras. Consequently, M_m(A) canbe defined over a field F₀ such that trdeg_k(F₀) 4. (iv) IfA has degree 4 then the trace form of A can be defined overa field F₀ of transcendence degree 4. (In (i), (iii) and (iv)it is assumed that the center of A contains certain roots ofunity.)     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

A WKB analysis of the buckling condition for a cylindrical shell of arbitrary thickness subjected to an external pressure

We consider the plane-strain buckling of a cylindrical shellof arbitrary thickness which is made of a Varga material andis subjected to an external hydrostatic pressure on its outersurface. The WKB method is used to solve the eigenvalue problemthat results from the linear bifurcation analysis. We show thatthe circular cross-section buckles into a non-circular shapeat a value of µ₁ which depends on A₁/A₂ and a mode number,where A₁ and A₂ are the undeformed inner and outer radii, andµ₁ is the ratio of the deformed inner radius to A₁. Inthe large mode number limit, we find that the dependence ofµ₁ on A₁/A₂ has a boundary layer structure: it is constantover almost the entire region of 0 < A₁/A₂ < 1 and decreasessharply from this constant value to unity as A₁/A₂ tends tounity. Our asymptotic results for A₁ – 1 = O(1) and A₁– 1 = O(1/n) are shown to agree with the numerical resultsobtained by using the compound matrix method.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

Topological Entropy for the Canonical Endomorphism of Cuntz-Krieger Algebras

It is shown that Voiculescu's topological entropy for the canonicalendomorphism of a simple Cuntz–Krieger algebra O_A equalsthe logarithm of the spectral radius of A. 1991 MathematicsSubject Classification 46L55.     

Euler Characteristics in Relative K-Groups

Suppose that M is a finite module under the Galois group ofa local or global field. Ever since Tate's papers [17, 18],we have had a simple and explicit formula for the Euler–Poincarécharacteristic of the cohomology of M. In this note we are interestedin a refinement of this formula when M also carries an actionof some algebra A, commuting with the Galois action (see Proposition5.2 and Theorem 5.1 below). This refinement naturally takesthe shape of an identity in a relative K-group attached to A(see Section 2). We shall deduce such an identity whenever wehave a formula for the ordinary Euler characteristic, the keystep in the proof being the representability of certain functorsby perfect complexes (see Section 3). This representabilitymay be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over Aimplies the ‘isogeny invariance’ of the equivariantconjectures on special values of the L-function put forwardin [3], and this was our motivation to write this note. Incidentally,isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer)was also a motivation for Tate's original paper [18]. I am verygrateful to J-P. Serre for illuminating discussions on the subjectof this note, in particular for suggesting that I consider representability.I should also like to thank D. Burns for insisting on a mostgeneral version of the results in this paper. 2000 MathematicsSubject Classification 19A99, 18G99, 11R34.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

On the Number of Sums and Products

A new lower bound on max{|A + A|, |A · A|} is given,where A is a finite set of complex numbers. 2000 MathematicsSubject Classification 11B75 (primary), 52C10 (secondary).     

Q-Curves and Abelian Varieties of GL2-Type

The relation between Q-curves and certain abelian varietiesof GL₂-type was established by Ribet (‘Abelian varietiesover Q and modular forms’, Proceedings of the KAIST MathematicsWorkshop (1992) 53–79) and generalized to building blocks,the higher-dimensional analogues of Q-curves, by Pyle in herPhD Thesis (University of California at Berkeley, 1995). Inthis paper we investigate some aspects of Q-curves with no complexmultiplication and the corresponding abelian varieties of GL₂-type,for which we mainly use the ideas and techniques introducedby Ribet (op. cit. and ‘Fields of definition of abelianvarieties with real multiplication’, Contemp.\ Math. 174(1994) 107–118). After the Introduction, in Sections 2and 3 we obtain a characterization of the fields where a Q-curveand all the isogenies between its Galois conjugates can be definedup to isogeny, and we apply it to certain fields of type (2,...,2).In Section 4 we determine the endomorphism algebras of all theabelian varieties of GL₂-type having as a quotient a given Q-curvein easily computable terms. Section 5 is devoted to a particularcase of Weil's restriction of scalars functor applied to a Q-curve,in which the resulting abelian variety factors over Q up toisogeny as a product of abelian varieties of GL₂-type. Finally,Section 6 contains examples: we parametrize the Q-curves comingfrom rational points of the modular curves X*N having genuszero, and we apply the results of Sections 2–5 to someof the curves obtained. We also give results concerning theexistence of quadratic Q-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

Restricted addition in Z/nZ and an Application to the Erdos-Ginzburg-Ziv Problem

Some elementary results are given on restricted sums 2A and3A of a subset AZ/nZ in the case when the cardinality of A,is close to n/2. In the last section, some of these resultsare used to derive some new values of a function related tothe Erds–Ginzburg–Ziv problem.     

On Atkin-Lehner Quotients of Curves

We study the erednik–Drinfeld p-adic uniformization ofcertain Atkin–Lehner quotients of Shimura curves overQ. We use it to determine over which local fields they haverational points and divisors of a given degree. Using a criterionof Poonen and Stoll, we show that the Shafarevich–Tategroup of their jacobians is not of square order for infinitelymany cases. 1991 Mathematics Subject Classification 11G18, 11G20,14G20, 14G35, 14H40.     

A Characterization of Strongly Continuous Groups of Linear Operators on a Hilbert Space

It is proved that the infinitesimal generator A of a stronglycontinuous semigroup of linear operators on a Hilbert spacealso generates a strongly continuous group if and only if theresolvent of –A, ( + A)^–1, is also a bounded functionon some right-hand-side half plane of complex numbers, and convergesstrongly to zero as the real part of tends to infinity. Anapplication to a partial differential equation is given. 1991Mathematics Subject Classification 47D03.