Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Asymptotics for minimal discrete energy on the sphere

We investigate the energy of arrangements of points on the surface of the unit sphere in that interact through a power law potential where and is Euclidean distance. With denoting the minimal energy for such -point arrangements we obtain bounds (valid for all ) for in the cases when and . For , we determine the precise asymptotic behavior of as . As a corollary, lower bounds are given for the separation of any pair of points in an -point minimal energy configuration, when . For the unit sphere in , we present two conjectures concerning the asymptotic expansion of that relate to the zeta function for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of when (the divergent case).

     

Complicated dynamics of parabolic equations with simple gradient dependence

Let be a smooth bounded domain. Given positive integers , and , , ..., , consider the semilinear parabolic equation

where and are smooth functions. By refining and extending previous results of Polácik we show that arbitrary -jets of vector fields in can be realized in equations of the form (E). In particular, taking we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on .

     

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let denote the classical equilibrium distribution (of total charge ) on a convex or -smooth conductor in with nonempty interior. Also, let be any th order ``Fekete equilibrium distribution' on , defined by point charges at th order ``Fekete points'. (By definition such a distribution minimizes the energy for -tuples of point charges on .) We measure the approximation to by for by estimating the differences in potentials and fields,

both inside and outside the conductor . For dimension we obtain uniform estimates at distance from the outer boundary of . Observe that throughout the interior of (Faraday cage phenomenon of electrostatics), hence on the compact subsets of . For the exterior of the precise results are obtained by comparison of potentials and energies. Admissible sets have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions of equal point charges on . In that context there is an important open problem for the sphere which is discussed at the end of the paper.

     

Rumely's local global principle for algebraic PC fields over rings

Let be a finite set of rational primes. We denote the maximal Galois extension of in which all totally decompose by . We also denote the fixed field in of elements in the absolute Galois group of by . We denote the ring of integers of a given algebraic extension of by . We also denote the set of all valuations of (resp., which lie over ) by (resp., ). If , then denotes the ring of integers of a Henselization of with respect to . We prove that for almost all , the field satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that for each and for each . Then . We also prove two approximation theorems for .

     

Boundary slopes of punctured tori in 3-manifolds

Let be an irreducible 3-manifold with a torus boundary component , and suppose that are the boundary slopes on of essential punctured tori in , with their boundaries on . We show that the intersection number of and is at most . Moreover, apart from exactly four explicit manifolds , which contain pairs of essential punctured tori realizing and 6 respectively, we have . It follows immediately that if is atoroidal, while the manifolds obtained by - and -Dehn filling on are toroidal, then , and unless is one of the four examples mentioned above.

Let be the class of 3-manifolds such that is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that , where has a torus component , and . Let be slopes on such that . Then . The exterior of the Whitehead sister link shows that this bound is best possible.

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Trace on the boundary for solutions of nonlinear differential equations

Let be a second order elliptic differential operator in with no zero order terms and let be a bounded domain in with smooth boundary . We say that a function is -harmonic if in . Every positive -harmonic function has a unique representation

where is the Poisson kernel for and is a finite measure on . We call the trace of on . Our objective is to investigate positive solutions of a nonlinear equation

for [the restriction is imposed because our main tool is the -superdiffusion which is not defined for ]. We associate with every solution a pair , where is a closed subset of and is a Radon measure on . We call the trace of on . is empty if and only if is dominated by an -harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.

     

Small subalgebras of Steenrod and Morava stabilizer algebras

Let (resp. be the subalgebra of the Steenrod algebra (resp. th Morava stabilizer algebra) generated by reduced powers , (resp. , . In this paper we identify the dual of (resp. , for with some Frobenius kernel (resp. -points) of a unipotent subgroup of the general linear algebraic group . Using these facts, we get the additive structure of for odd primes.

     

A new degree bound for vector invariants of symmetric groups

Let be a commutative ring, a finitely generated free -module and a finite group acting naturally on the graded symmetric algebra . Let denote the minimal number , such that the ring of invariants can be generated by finitely many elements of degree at most .

For and , the -fold direct sum of the natural permutation module, one knows that , provided that is invertible in . This was used by E. Noether to prove if .

In this paper we prove for arbitrary commutative rings and show equality for a prime power and or any ring with . Our results imply

for any ring with .

     

Generalized Hestenes' Lemma and extension of functions

Suppose we have an -jet field on which is a Whitney field on the nonsingular part of . We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on , if the field is flat enough at the singular part , then it is a Whitney field on (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when is subanalytic. In Section II, we show that a function on can be extended to one on if the differential goes to faster than the order of divergence of the principal curvatures of and if the first covariant derivative of is sufficiently flat. For the general case of functions with , we give a similar result for in Section III.

     

A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems

We study two-point Lagrange problems for integrands :

 

Under very weak regularity hypotheses [ is Hölder continuous and locally elliptic on each compact subset of ] we obtain, when is of superlinear growth in , a characterization of problems in which the minimizers of (P) are -regular for all boundary data. This characterization involves the behavior of the value function : defined by . Namely, all minimizers for (P) are -regular in neighborhoods of and if and only if is Lipschitz continuous at . Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where is nearly autonomous, nearly independent of , or jointly convex in are presented.

     

A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.

     

Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Generalized Witt algebras, over a field of characteristic , were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras over , where the ingredients are an abelian group , a vector space over , and a map which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra
, with the right kernel of being , are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.

     

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with and nonbirational bicanonical map. Our main result is that, if is such a surface and if is minimal with no pencil of curves of genus , then is the symmetric product of a curve of genus , and therefore and . Furthermore we obtain some results towards the classification of minimal surfaces with . Such surfaces have , and we show that if and only if is the symmetric product of a curve of genus . We also classify the minimal surfaces with with a pencil of curves of genus , proving in particular that for those one has .

     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

     

Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines

In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries , with the integral taken over the domain D. We prove that the solution is a biharmonic function in except on the interior boundaries , and satisfies some matching conditions on . There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of a positive integer, then the solution is a polyharmonic function of order for , satisfying matching conditions on , and is called a polyspline of order . Uniqueness and existence for polysplines of order , provided that the interior boundaries are sufficiently smooth surfaces and , is proved. Three examples of data sets possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional splines.

     

On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.

     

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasi-diagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.