CONFORMAL MEASURES ASSOCIATED TO ENDS OF HYPERBOLIC n-MANIFOLDS

Let be a non-elementary Kleinian group acting on the closedn-dimensional unit ball and assume that its Poincaréseries converges at the exponent . Let M be the -quotient ofthe open unit ball. We consider certain families = {E₁, ...,E_p} of open subsets of M such that is compact. The sets E_i are the ends of M and is a completecollection of ends for M. We associate to each end E an -conformalmeasure such that the measures corresponding to different endsare mutually singular if non-trivial. Additionally, each -conformalmeasure for on the limit set () of can be written as a sumof such conformal measures associated to ends E . In dimension3, our results overlap with some results of Bishop and Jones(The law of the iterated logarithm for Kleinian groups, Cont.Math. 211 (1997), 17–50.).     

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant of quasiprojective -varieties X with values ina commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closedin X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We firstdefine a unique extension ' of to finite type Artin -stacks, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -groupacting on X, and [X/G] is the quotient stack. This only worksif (G) is invertible in for all special -groups G, which excludes = as (_m) = 0. But we can extend the construction to get roundthis. Then we develop the theory of stack functions on Artin stacks.These are a universal generalization of constructible functionson Artin stacks. There are several versions of the construction:the basic one , and variants ‘twisted’ by motivic invariants. We associate a -vector space or a -module to each Artin stack , with functorial operations of multiplication, pullbacks * and pushforwards _*under 1-morphisms ;, and so on. They will be important tools in the author's series on ‘Configurationsin abelian categories’.     

LABELLING THE CHARACTER TABLES OF SYMMETRIC AND ALTERNATING GROUPS

Let X be a character table of the symmetric group S_n. It isshown that unless n = 4 or n = 6, there is a unique way to assignpartitions of n to the rows and columns of X so that for all and , X is equal to (), the value of the irreducible characterof S_n labelled by on elements of cycle type . Analogous resultsare proved for alternating groups, and for the Brauer charactertables of symmetric and alternating groups.     

DISTRIBUTION OF ANGLES BETWEEN GEODESIC RAYS ASSOCIATED WITH HYPERBOLIC LATTICE POINTS

For every two points z₀, z₁ in the upper-half plane , considerall elements in the principal congruence group (N), actingon by fractional linear transformations, such that the hyperbolicdistance between z₁ and z₀ is at most R > 0. We study thedistribution of angles between the geodesic rays [z₁, z₀] asR , proving that the limiting distribution exists independentlyof N and explicitly computing it. When z₁ = z₀, this is foundto be the uniform distribution on the interval [–/2, /2].     

Decomposability of Reflexive Cycle Algebras

We give, for each n 3, an example of a reflexive operator algebra_n with the following properties: (i) each finite rank operatorwith rank less than n – 1 is the sum of rank-one operatorsin _n, and (ii) there is an operator of rank n – 1 in _nwhich is not the sum of rank-one operators in _n. The invariantsubspace lattice of _n is finite and distributive with 2n join-irreducibleelements. We show also that the indecomposability of _n is relatedto the existence of a chordless cycle in a bipartite graph associatedwith _n.     

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Let ₁, ...,_r R be ‘not very well approximable’,for example, Q-linearly independent real algebraic numbers.Then there are infinitely many positive square-free integersn such that ||n_i|| << n^–(2/3r)+(1 i r), where||·|| denotes distance to the nearest integer.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.     

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C² (M) such that dµ_V e^V d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that Ric_M – Hess_V –Kand imply the log-Sobolevinequality for the Dirichlet form µ_V(| f |²).     

ON A BRESAR-SEMRL CONJECTURE AND DERIVATIONS OF BANACH ALGEBRAS

In this paper, we answer a question on derivations of densealgebras of linear operators posed by Brear and emrl. Our theoremimplies the following result: let be a complex Banach algebra,and let d and g be continuous derivations of . If dg(x) is quasi-nilpotentfor every x , then dg(x)³ lies in the radical of for everyx . This result was proved by Brear and emrl with the additionalassumption gd = dg.     

COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN-BILU THEOREM

We note a link between combinatorial results of Bollobásand Leader concerning sumsets in the grid, the Brunn–Minkowskitheorem and a result of Freiman and Bilu concerning the structureof sets A with small doubling. Our main result is the following. If > 0 and if A is a finitenon-empty subset of a torsion-free abelian group with |A + A| K|A|, then A may be covered by e^KO(1) progressions of dimension log ₂ K + and size at most |A|.     

The multiplier algebra of a nuclear quasidiagonal C*-algebra

If is a unital separable simple nuclear quasidiagonal C*-algebra,then ( ) has the AF-property in the strict topology; that is,there is a unital AF-subalgebra ( ) such that is strictlydense in ( ). We also give a multiplier algebra characterizationof nuclearity and quasidiagonality for a unital separable simpleC*-algebra.     

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.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

In this paper, we consider the following semilinear ellipticproblem: where f(x, t) tends to p(x) and q(x) L(^N), respectively, ast 0 and t +. We prove that there exist two numbers l and Lwith L < l such that problem (P) has at least one positivesolution for (– l, –L) and has no positive solutionfor all [–l,–L]. The existence and non-existenceof positive solutions for problem (P) at = –l and =–L are also discussed.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

Multiplicative decomposability of shifted sets

The following two problems are open.

1. Do two sets of positiveintegers and exist, with at leasttwo elements each, suchthat + coincides with the set of primes for sufficiently largeelements?
2. Let ={6, 12, 18}. Is there an infinite set of positiveintegerssuch that +1? A positive answer would imply that thereare infinitelymany Carmichael numbers with three prime factors.

In this paper we prove the multiplicative analogue of the firstproblem, namely that there are no two sets and , with at leasttwo elements each, such that the product coincides with anyadditively shifted copy +c of the set of primes for sufficientlylarge elements. We also prove that shifted copies of sets ofintegers that are generated by certain subsets of the primescannot be multiplicatively decomposed.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM

Let M be an orientable connected and compact real hypersurfaceof the complex space form C^{(n + 1)/2}. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequalityn²² (n² – 1) + f², where is the characteristic vectorfield, A is the shape operator and (n – 1) is the infimumof the Ricci curvatures of hypersurface M, then it is shownthat is a constant and M is the sphere Sⁿ(²) in C^{(n + 1)/2}.     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .