Partitions into Primes

We investigate the asymptotic behavior of the partition function defined by , where denotes the von Mangoldt function. Improving a result of Richmond, we show that , where is a positive constant and denotes the times iterated logarithm. We also show that the error term can be improved to if and only if the Riemann Hypothesis holds.

     

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let denote the moduli space of Riemann spheres with ordered marked points. In this article we define the group of quasi-special symmetric outer automorphisms of the algebraic fundamental group for all to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of and commuting with the group of outer automorphisms of obtained by permuting the marked points. Our main result states that is isomorphic to the Grothendieck-Teichmüller group for all .

     

Semi-classical limit for random walks

Let be a discrete symmetric random walk on a compact Lie group with step distribution and let be the associated transition operator on . The irreducibles of the left regular representation of on are finite dimensional invariant subspaces for and the spectrum of is the union of the sub-spectra on the irreducibles, which consist of real eigenvalues . Our main result is an asymptotic expansion for the spectral measures

along rays of representations in a positive Weyl chamber , i.e. for sequences of representations , with . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on (for which is essentially a direct sum of Harper operators).

     

Specializations of Brauer classes over algebraic function fields

Let be either a number field or a field finitely generated of transcendence degree over a Hilbertian field of characteristic 0, let be the rational function field in one variable over , and let . It is known that there exist infinitely many such that the specialization induces a specialization , where has exponent equal to that of . Now let be a finite extension of and let . We give sufficient conditions on and for there to exist infinitely many such that the specialization has an extension to inducing a specialization , the residue field of , where has exponent equal to that of . We also give examples to show that, in general, such need not exist.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

Partial differential equations with matricial coefficients and generalized translation operators

Let be the Bessel operator with matricial coefficients defined on by

where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

     

Geometry of Banach spaces having shrinking approximations of the identity

Let and let be a compact set of scalars. We introduce property of Banach spaces by the requirement that whenever is a bounded net converging weak to in and . Using with 1$">, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to -, -, and -ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

     

Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex , to compute , where is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension , we define a double covering of to be an extension of degree , such that is Galois. We demonstrate that each class gives rise to a double covering of , by . When lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of is abelian and hence . The may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension

RÉSUMÉ. On considère dans un ouvert borné de , à bord régulier, le problème de Dirichlet

où , est positive et s'annule sur un ensemble fini de points de . On démontre alors sous certaines hypothèses sur et si est assez petit, que le problème (1) possède une solution convexe unique .

ABSTRACT. We consider in a bounded open set of , with regular boundary, the Dirichlet problem

where , is positive and vanishes on , a finite set of points in . We prove, under some hypothesis on and if is sufficiently small, that the problem (1) has a unique convex solution .

     

Conformal actions of on Lorentz manifolds

We prove that, for , a locally faithful action of or of by conformal transformations of a connected Lorentz manifold must be a proper action.

     

Rigidity of Coxeter groups

Let be a Coxeter group acting properly discontinuously and cocompactly on manifolds and such that the fixed point sets of finite subgroups are contractible. Let be a -homotopy equivalence which restricts to a -homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then is -homotopic to a -homeomorphism.     

The truncated complex -moment problem

Let denote a sequence of complex numbers ( 0, \gamma _{ij}=\bar{\gamma}_{ji}$">), and let denote a closed subset of the complex plane . The Truncated Complex -Moment Problem for entails determining whether there exists a positive Borel measure on such that ( ) and . For a semi-algebraic set determined by a collection of complex polynomials , we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix and the localizing matrices . We prove that there exists a -atomic representing measure for supported in if and only if and there is some rank-preserving extension for which , where or .

     

Principal curvatures of isoparametric hypersurfaces in

Let be an isoparametric hypersurface in , and the inverse image of under the Hopf map. By using the relationship between the eigenvalues of the shape operators of and , we prove that is homogeneous if and only if either or is constant, where is the number of distinct principal curvatures of and is the number of non-horizontal eigenspaces of the shape operator on .

     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

A classification of one dimensional almost periodic tilings arising from the projection method

For each irrational number , with continued fraction expansion , we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope . The invariant is a sequence of integers in the space and whenever modulo the equivalence relation generated by tail equivalence and . Each tile in a tiling , of slope , is coded by an integer . Using a composition operation, we produce a sequence of tilings . Each tile in gets absorbed into a tile in . A choice of a starting tile in will thus produce a sequence in . This is the invariant.

     

Orbit equivalence of global attractors of semilinear parabolic differential equations

We consider global attractors of dissipative parabolic equations

on the unit interval with Neumann boundary conditions. A permutation is defined by the two orderings of the set of (hyperbolic) equilibrium solutions according to their respective values at the two boundary points and We prove that two global attractors, and , are globally orbit equivalent, if their equilibrium permutations and coincide. In other words, some discrete information on the ordinary differential equation boundary value problem characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.

     

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that is a second countable locally compact transformation group. We let denote the set of Morita equivalence classes of separable dynamical systems where is a -algebra and is compatible with the given -action on . We prove that is a commutative semigroup with identity with respect to the binary operation for an appropriately defined balanced tensor product on -algebras. If and act freely and properly on the left and right of a space , then we prove that and are isomorphic as semigroups. If the isomorphism maps the class of to the class of , then is Morita equivalent to .

     

Projective manifolds with small pluridegrees

Let be a very ample line bundle on a connected complex projective manifold of dimension . Except for a short list of degenerate pairs , and there exists a morphism expressing as the blowup of a projective manifold at a finite set , with nef and big for the ample line bundle . The projective geometry of is largely controlled by the pluridegrees for , of . For example, , where is the genus of a curve section of , and is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of . In this article, a detailed analysis is made of the pluridegrees of . The restrictions found are used to give a new lower bound for the dimension of the space of sections of . The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to .

     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .