-actions with

Suppose that is a smooth -action on a closed smooth -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set vanish in positive dimension. This paper shows that if 2^k\dim F$"> and each -dimensional part possesses the linear independence property, then bounds equivariantly, and in particular, is the best possible upper bound of if is nonbounding.

     

A hereditarily subspace of without the Schur property

Let p_1 > p_2 > \cdots > 1$">. We construct an easily determined -symmetric basic sequence in , which spans a hereditarily subspace without the Schur property. An immediate consequence is the existence of hereditarily subspaces of without the Schur property.

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

-invariant subspaces of

In this note, we give a new proof of the characterization of the -invariant subspaces of for in using ideas from approximation theory.     

Gateaux derivative of norm

We prove that for Hilbert space operators and , it follows that

,\end{displaymath}">

where . Using the concept of -Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in , and to give an easy proof of the characterization of smooth points in .

     

Measures of concordance determined by -invariant measures on

A measure, , on is said to be -invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, , generated in a certain way by a measure, , on is shown to be a measure of concordance if and only if the generating measure is positive, regular, -invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist's beta as a special case.

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Examples of non-formal closed -connected manifolds of dimensions

We construct closed -connected manifolds of dimensions that possess non-trivial rational Massey triple products. We also construct examples of manifolds such that all the cup-products of elements of vanish, while the group is generated by Massey products: such examples are useful for the theory of systols.

     

The vanishing of implies that is regular

Let be an excellent local ring of positive prime characteristic. We show that if , then is regular. This improves a result of Schoutens, in which the additional hypothesis that was an isolated singularity was required for the proof.

     

``Beurling type' subspaces of and

In this note we extend the ``Beurling type' characterizations of subspaces of and to and , respectively.

     

On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of

     

Line arrangements in

If is a hyperbolic manifold and is a simple closed geodesic, then lifts to a collection of lines in acted upon by . In this paper we show that such a collection of lines cannot contain a particular type of subset (called a bad triple) unless has orientation-reversing elements. This fact allows us to extend certain lower bounds on hyperbolic volume to the non-orientable case.

     

Local isometries of

In this paper we study the structure of local isometries on . We show that when is first countable and is uniformly convex and the group of isometries of is algebraically reflexive, the range of a local isometry contains all compact operators. When is also uniformly smooth and the group of isometries of is algebraically reflexive, we show that a local isometry whose adjoint preserves extreme points is a -module map.

     

The distribution functions of and

Let be the sum of the positive divisors of . We show that the natural density of the set of integers satisfying is given by , where denotes Euler's constant. The same result holds when is replaced by , where is Euler's totient function.

     

The -local -homology of

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the -homology of a space. As an application, we compute the -homology of in a manner analogous to Mahowald and Milgram's computation of the -homology .

     

Real

The famous problem involves applying two maps: and to positive integers. If is even, one applies , if it is odd, one applies . The conjecture states that each trajectory of the system arrives to the periodic orbit . In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both and independently of . That is, we consider the action of the free semigroup with generators and on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by and .

     

The homological determinant of quantum groups of type

Let be a Hecke symmetry depending algebraically on a parameter . We show that the homology of the Koszul complex associated with is one-dimensional when is not a root of unity. A generator of this homology group then induces the homological determinant of the quantum group associated with .

     

Hopf algebras of dimension

Let be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If is not semisimple and for some odd integer , then or is not unimodular. Using this result, we prove that if for some odd prime , then is semisimple. This completes the classification of Hopf algebras of dimension .

     

Brangesian spaces in

In this note, we characterize certain algebraic subspaces of extending D. Singh's result.

     

Spectral pictures of and

The spectral pictures of products and of Banach space operators are compared; in particular when one of them is `of index zero'.