Beurling and Lipschitz Algebras

It is well known that there exist infinite closed subsets Eof T such that A(E) = C(E) (see, for example, [3]). Such setsare called Helson sets. Let E be a closed subset of T, let 0< < 1, and let A(E) be the restriction of the Beurlingalgebra A(T). Then A(E) lipE. We shall show that A(E) = lipEif and only if E is finite. This answers a question raised byPedersen [5], where partial results were obtained. 1991 MathematicsSubject Classification 46J10.     

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

Let E/Q be an elliptic curve. For a prime p of good reduction,let E(F_p) be the set of rational points defined over the finitefield F_p. Denote by (#E(F_p)) the number of distinct prime divisorsof #E(F_p). For an elliptic curve with complex multiplication,the normal order of (#E(F_p)) is shown to be log log p. The normalorder of the number of distinct prime factors of the exponentof E(F_p) is also studied. 2000 Mathematics Subject Classification11N37, 11G20.     

On Towers Approximating Homological Localizations

Our object of study is the natural tower which, for any givenmap f:AB and each space X, starts with the localization of Xwith respect to f and converges to X itself. These towers canbe used to produce approximations to localization with respectto any generalized homology theory E_*, yielding, for example,an analogue of Quillen's plus-construction for E_*. We discussin detail the case of ordinary homology with coefficients inZ/p or Z[1/p]. Our main tool is a comparison theorem for nullificationfunctors (that is, localizations with respect to maps of theform f:Apt), which allows us, among other things, to generalizeNeisendorfer's observation that p-completion of simply-connectedspaces coincides with nullification with respect to a Moorespace M(Z[1/p], 1).     

The Fitting ideal problem

Let A be a Noetherian local ring and let E be a finitely generatedA-module having rank r. In this note one deals with the expectedinequality (^rE) height (Fitt_r(E)), where height (Fitt_r(E))is the codimension of the rth Fitting ideal of E, and (M) standsfor the analytic spread of a module M. One establishes caseswhere the inequality holds as well as where it fails. A specialcase where the inequality holds implies the celebrated Zak inequalityfor the dimension of the image of the Gauss map.     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

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.     

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.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Embeddings of Sz(32) in E8(5)

We show that the Suzuki group Sz(32) is a subgroup of E₈(5),and so is its automorphism group. Both are unique up to conjugacyin E₈(F) for any field F of characteristic 5, and the automorphismgroup Sz(32):5 is maximal in E₈(5). 1991 Mathematics SubjectClassification 20E28.     

The tower of K-theory of truncated polynomial algebras

Let A be a regular noetherian F_p-algebra. The relative K-groupsK_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluatedby the author and Ib Madsen in terms of the big de Rham–Wittgroups W_rA^q of the ring A. In this paper, we evaluate the mapsof relative K-groups and Nil-groups induced by the canonicalprojection f: A[x]/(x^m) A[x]/(xⁿ). The result depends stronglyon the prime p. It generalizes earlier work by Stienstra onthe groups in degrees 2 and 3. Received February 28, 2007.     

Measures of pseudorandomness for finite sequences: typical values

Mauduit and Sárközy introduced and studied certainnumerical parameters associated to finite binary sequences E_N {–1, 1}^N in order to measure their ‘level of randomness’.Those parameters, the normality measure (E_N), the well-distributionmeasure W(E_N), and the correlation measure C_k(E_N) of order k,focus on different combinatorial aspects of E_N. In their work,amongst others, Mauduit and Sárközy (i) investigatedthe relationship among those parameters and their minimal possiblevalue, (ii) estimated (E_N), W(E_N) and C_k(E_N) for certain explicitlyconstructed sequences E_N suggested to have a ‘pseudorandomnature’, and (iii) investigated the value of those parametersfor genuinely random sequences E_N. In this paper, we continue the work in the direction of (iii)above and determine the order of magnitude of (E_N), W(E_N) andC_k(E_N) for typical E_N. We prove that, for most E_N {–1,1}^N, both W(E_N) and (E_N) are of order N, while C_k(E_N) is oforder for any given 2 k N/4.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

Determination of Homomorphisms between Linear Groups of the same Degree over Division Rings

We determine all group homomorphisms SL_n(D) GL_n(E), where Dand E are division rings such that char D = char E and E isfinite-dimensional over its centre. Apart from the case wheren = 2 and D is F₂ or F_3, all non-trivial group homomorphismsSL_n(D) GL_n(E) are induced from homomorphisms and anti-homomorphismsD E.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.     

The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

On the Reflexivity of the Space {pi}p(E, F) of p-Absolutely Summing Operators, 1 <= p < + {infty}

We give a representation of the dual of the space _p(E,F) ofp-absolutely summing operators (1 p < + ) under certainconditions on E and F. One deduces that the space _p(E, F), 1 p < + , is reflexive if and only if E and F are reflexive.We improve results of Gordon, Lewis, Retherford and Saphar.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

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.     

Error Bounds for the Truncation of Infinite Linear Differential Systems

An infinite system of linear differential equations of the form = Ax + f, x(0) = y is considered, wherex, y and f are infinite column vectors in E, and A is a constantinfinite matrix defining a bounded operator on E, where E isl₁, (c₀), or l. Explicit error bounds are obtained for the approximationof the solution of the infinite system by the solutions of thefinite truncated systems.