Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Isolated invariant curves of a foliation

We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation of in terms of the Milnor number of . This result gives a bound for the degree of an algebraic invariant curve of a foliation of in terms of the degree of , provided that all the branches of are isolated separatrices.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

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.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define . In particular, let be such that , as for some , is o-minimal, and the expansion of by the set does not define . Then there exist 0$"> and such that as .

     

-algebras and the tail-equivalence relation on Bratteli diagrams

We show that the -algebra associated to the tail-equivalence relation on a Bratteli diagram, according to a procedure recently introduced by the first-named author and A. Lopes, is isomorphic to the -algebra of the diagram. More generally we consider an approximately proper equivalence relation on a compact space for which the quotient maps are local homeomorphisms. We show that the algebra associated to under the above-mentioned procedure is isomorphic to the groupoid -algebra .

     

-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups of size continuum such that every countable Borel -cover of contains a -cover of .

     

Cube-approximating bounded wavelet sets in

We prove that for any real expansive matrix , there exists a bounded -dilation wavelet set in the frequency domain (the inverse Fourier transform of whose characteristic function is a band-limited single wavelet in the time domain ). Moreover these wavelet sets can approximate a cube in arbitrarily. This result improves Dai, Larson and Speegle's result about the existence of (basically unbounded) wavelet sets for real expansive matrices.

     

A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs

Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

     

Maximality theorems for Fréchet algebras

The algebra of unbounded holomorphic functions that is contained in the algebra is studied. For in but not in , we show that the algebra generated by and is dense in for all .

     

Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

Entire pluricomplex Green functions and Lelong numbers of projective currents

Let be a positive closed current of bidimension (1,1) and unit mass on the complex projective space . We prove that the set of points where has Lelong number larger than is contained in a complex line if , and for some complex line if . We also prove that in dimension 2 and if , then for some conic .

     

On homeomorphisms and quasi-isometries of the real line

We show that the group of piecewise-linear homeomorphisms of having bounded slopes surjects onto the group of all quasi-isometries of . We prove that the following groups can be imbedded in : the group of compactly supported piecewise-linear homeomorphisms of , the Richard Thompson group , and the free group of continuous rank.

     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

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.

     

Nikolskii-type inequalities for shift invariant function spaces

Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

Theorem. Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then

for every and

     

A theorem on reflexive large rank operator spaces

If every nonzero operator in an -dimensional operator space has rank , then is reflexive.