Positive definite spherical functions on Olshanskii domains

Let be a simply connected complex Lie group with Lie algebra , a real form of , and the analytic subgroup of corresponding to . The symmetric space together with a -invariant partial order is referred to as an Olshanskii space. In a previous paper we constructed a family of integral spherical functions on the positive domain of . In this paper we determine all of those spherical functions on which are positive definite in a certain sense.

     

Characterizations of spectra with -injective cohomology which satisfy the Brown-Gitler property

We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum satisfies the Brown-Gitler property if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum standard if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is reduced if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

     

Embeddings in generalized manifolds

We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

The spectrum of infinite regular line graphs

Let be an infinite -regular graph and its line graph. We consider discrete Laplacians on and , and show the exact relation between the spectrum of and that of . Our method is also applicable to -semiregular graphs, subdivision graphs and para-line graphs.

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

A construction of homologically area minimizing hypersurfaces with higher dimensional singular sets

We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if is smooth, compact dimensional manifold, , and if is an embedded, orientable submanifold of dimension , then we construct metrics on such that the homologically area minimizing hypersurface , homologous to , has a singular set equal to a prescribed number of spheres and tori of codimension less than . Near each component of the singular set, is isometric to a product , where is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.

     

Banach spaces with the Daugavet property

A Banach space is said to have the Daugavet property if every operator of rank satisfies . We show that then every weakly compact operator satisfies this equation as well and that contains a copy of . However, need not contain a copy of . We also study pairs of spaces and operators satisfying , where is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with is as small as possible and give characterisations in terms of a smoothness condition.

     

Möbius-like groups of homeomorphisms of the circle

An orientation preserving homeomorphism of is Möbius-like if it is conjugate in to a Möbius transformation. Our main result is: given a (noncyclic) group whose every element is Möbius-like, if has at least one global fixed point, then the whole group is conjugate in to a Möbius group if and only if the limit set of is all of . Moreover, we prove that if the limit set of is not all of , then after identifying some closed subintervals of to points, the induced action of is conjugate to an action of a Möbius group. Said differently, is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case is isomorphic, as a group, to a Möbius group.

This result has another interpretation. Namely, we prove that a group of orientation preserving homeomorphisms of whose every element can be conjugated to an affine map (i.e., a map of the form ) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of is the one of an affine group.

     

Examples of Möbius-like groups which are not Möbius groups

In this paper we give two basic constructions of groups with the following properties:

(a)

, i.e., the group is acting by orientation preserving homeomorphisms on ;

(b)

every element of is Möbius-like;

(c)

, where denotes the limit set of ;

(d)

is discrete;

(e)

is not a conjugate of a Möbius group.

Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group (of a certain type) and then we change the underlying circle upon which acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by . Now we form a new group which is generated by all of and an additional element whose existence is enabled by the inserted intervals. This group has all the properties (a) through (e).

     

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

     

Simple and semisimple Lie algebras and codimension growth

We study the exponential growth of the codimensions of a finite dimensional Lie algebra over a field of characteristic zero. In the case when is semisimple we show that exists and, when is algebraically closed, is equal to the dimension of the largest simple summand of . As a result we characterize central-simplicity: is central simple if and only if .

     

Hankel Operators on Bounded Analytic Functions

For a domain in the complex plane and a bounded measurable function on , the generalized Hankel operator on is the operator of multiplication by followed by projection into . Under certain conditions on we show that either is compact or there is an embedded on which is bicontinuous. We characterize those 's for which is compact in the case that is a Behrens roadrunner domain.

     

Overgroups of irreducible linear groups, II

Determining the subgroup structure of algebraic groups (over an algebraically closed field of arbitrary characteristic) often requires an understanding of those instances when a group and a closed subgroup both act irreducibly on some module , which is rational for and . In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26-69), we give a classification of all such triples when is a non-connected algebraic group with simple identity component , is an irreducible -module with restricted -high weight(s), and is a simple algebraic group of classical type over sitting strictly between and .

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

Random intersections of thick Cantor sets

Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .

     

Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave connecting and for the scalar viscous conservation law in two space dimensions. We assume that the initial data tends to constant states as , respectively. Then, the convergence rate to of the solution is investigated without the smallness conditions of and the initial disturbance. The proof is given by elementary -energy method.