On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

Arc-analytic roots of analytic functions are Lipschitz

Let be an arc-analytic function (i.e., analytic on every analytic arc) and assume that for some integer the function is real analytic. We prove that is locally Lipschitz; even if is less than the multiplicity of . We show that the result fails if is only a , arc-analytic function (even blow-analytic), . We also give an example of a non-Lipschitz arc-analytic solution of a polynomial equation , where are real analytic functions.

     

Weak properties of weighted convolution algebras

Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, converges to 0 in norm.

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

Local existence of -sets, projective tensor products, and Arens regularity for

Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.

     

Double covering of curves

Let be a smooth projective algebraic curve of genus and an integer with . For all integers we prove the existence of a double covering with a smooth curve of genus and the existence of a degree morphism that does not factor through . By the Castelnuovo-Severi inequality, the result is sharp (except perhaps the bound ).

     

On two problems of Erdos and Hechler: New methods in singular madness

For an infinite cardinal , denotes the set of all cardinalities of nontrivial maximal almost disjoint families over .

Erdos and Hechler proved in 1973 the consistency of for a singular cardinal and asked if it was ever possible for a singular that , and also whether for every singular cardinal .

We introduce a new method for controlling for a singular and, among other new results about the structure of for singular , settle both problems affirmatively.

     

Cohomology of symplectic reductions of generic coadjoint orbits

Let be a generic coadjoint orbit of a compact semi-simple Lie group . Weight varieties are the symplectic reductions of by the maximal torus in . We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of , and it makes explicit the dependence on and a parameter in .

     

Lattice polygons and Green's theorem

Associated to an -dimensional integral convex polytope is a toric variety and divisor , such that the integral points of represent . We study the free resolution of the homogeneous coordinate ring as a module over . It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface. In particular, for a planar polytope , satisfies Green's condition if contains at least lattice points.

     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

Distinguished representations and poles of twisted tensor -functions

Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

     

Transference for amenable actions

Suppose acts amenably on a measure space with quasi-invariant -finite measure . Let be an isometric representation of on and a finite Radon measure on . We show that the operator has -operator norm not exceeding the -operator norm of the convolution operator defined by . We shall also prove an analogous result for the maximal function associated to a countable family of Radon measures .

     

On the norm of an idempotent Schur multiplier on the Schatten class

We show that if the norm of an idempotent Schur multiplier on the Schatten class lies sufficiently close to , then it is necessarily equal to . We also give a simple characterization of those idempotent Schur multipliers on whose norm is .

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

The growth of valuations on rational function fields in two variables

Given a valuation on the function field , we examine the set of images of nonzero elements of the underlying polynomial ring under this valuation. For an arbitrary field , a Noetherian power series is a map that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on . Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let denote the images under the valuation of all nonzero polynomials of at most degree in the variable . We construct a bound for the growth of with respect to for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

     

Hecke algebras for the basic characters of the unitriangular group

Let denote the unitriangular group of degree over the finite field with elements. In a previous paper we obtained a decomposition of the regular character of as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character of . We prove that is induced from a linear character of an algebra subgroup of , and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of as characters induced from an algebra subgroup of . Finally, we identify a special irreducible constituent of , which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption where is the characteristic of the field) that gives a necessary and sufficient condition for to have a unique irreducible constituent.

     

Uncountable categoricity for gross models

A model is said to be gross if all infinite definable sets in have the same cardinality (as ). We prove that if for some uncountable , has a unique gross model of cardinality , then for any uncountable , has a unique gross model of cardinality .

     

The duality theory of a finite dimensional discrete quantum group

Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

     

Stability of -algebras associated to graphs

We characterize stability of graph -algebras by giving five conditions equivalent to their stability. We also show that if is a graph with no sources, then is stable if and only if each vertex in can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph -algebra. Specifically, if is a graph and is the graph formed by adding a head to each vertex of , then is the stabilization of ; that is, .