A note on Gabor orthonormal bases

The study of Gabor bases of the form for has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer , is an orthonormal basis for if and only if is a tiling of . Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such for , , . Inspired by their work, this paper addresses Gabor orthonormal bases of the form for and some other related problems, where is as above. For a fixed , the generating function of a Gabor orthonormal basis for corresponding to the above is characterized explicitly provided that , which is new even if ; a Shannon type sampling theorem about such is derived when , ; for an arbitrary positive integer , an explicit expression of the with being an orthonormal basis for is obtained under the condition that .

     

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 .

     

Irreducibility of the -classes on smooth rational surfaces

We give a characterization for a -divisor on a smooth rational surface to be irreducible under the assumption that an anticanonical divisor of is nef. Here is nef means for every effective divisor on , and a -divisor is a divisor such that the two numerical conditions hold.

As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of -curves. A -curve is a smooth rational curve of self-intersection .

     

A compact group which is not Valdivia compact

A compact space is Valdivia compact if it can be embedded in a Tikhonov cube in such a way that the intersection is dense in , where is the sigma-product ( the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

     

Remarks concerning linear characters of reflection groups

Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

     

-algebras whose torsion part is not cyclic

We consider algebras over a Dedekind domain with the property and generalize Schultz' structure theory of the case to Dedekind domains. We construct examples of mixed -algebras, which are non-split extensions of the submodule of elements infinitely divisible by the relevant prime ideals. This is also new in the case .

     

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let be a Noetherian commutative ring, a finitely generated -algebra, and a pure -subalgebra of . Then is finitely generated over .

     

Positive scalar curvature and minimal hypersurfaces

We show that the minimal hypersurface method of Schoen and Yau can be used for the ``quantitative' study of positive scalar curvature. More precisely, we show that if a manifold admits a metric with or , where is the scalar curvature of , any 2-tensor on and the Weyl tensor of , then any closed orientable stable minimal (totally geodesic in the second case) hypersurface also admits a metric with the corresponding positivity of scalar curvature. A corollary pertaining to the topology of such hypersurfaces is proved in a special situation.

     

Bounds for the index of the centre in capable groups

A group is called capable if it is isomorphic to for some group . Let be a capable group. I. M. Isaacs (2001) showed that if is finite, then the index of the centre is bounded above by some function of . We show that if , then with some constant and this bound is essentially best possible. We complete a result of Isaacs, showing that if is a cyclic group, then .

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Fixed points of nonexpansive mappings in spaces of continuous functions

Let be a compact metrizable space and let be the Banach space of all real continuous functions defined on with the maximum norm. It is known that fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when contains a perfect set. However the space , where and is the first infinite ordinal number, enjoys the w-FPP, and so also satisfies this property if . It is unknown if has the w-FPP when is a scattered set such that . In this paper we prove that certain subspaces of , with , satisfy the w-FPP. To prove this result we introduce the notion of -almost weak orthogonality and we prove that an -almost weakly orthogonal closed subspace of enjoys the w-FPP. We show an example of an -almost weakly orthogonal subspace of which is not contained in for any .

     

Relations approximated by continuous functions

Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

     

Semi-continuity of metric projections in -direct sums

Let be a proximinal subspace of finite codimension of . We show that is proximinal in and the metric projection from onto is Hausdorff metric continuous. In particular, this implies that the metric projection from onto is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

     

The Rokhlin lemma for homeomorphisms of a Cantor set

For a Cantor set , let denote the group of all homeomorphisms of . The main result of this note is the following theorem. Let be an aperiodic homeomorphism, let be Borel probability measures on , and let 0$"> and . Then there exists a clopen set such that the sets are disjoint and 1 - \varepsilon, i= 1,\ldots,k$">. Several corollaries of this result are given. In particular, it is proved that for any aperiodic the set of all homeomorphisms conjugate to is dense in the set of aperiodic homeomorphisms.

     

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence whose moment matrix is positive semidefinite and has finite rank is the sequence of moments of an -atomic nonnegative measure on . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure having degree at most if the principal submatrix of (indexed by all monomials of degree ) has full rank . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set , where and . If is positive semidefinite and has a flat extension such that all localizing matrices are positive semidefinite, then has an atomic representing measure supported by . We also review an application of this result to the problem of minimizing a polynomial over the set .

     

The content of a Gaussian polynomial is invertible

Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.

     

Planar algebras and the Ocneanu-Szymanski theorem

We give a very simple `planar algebra' proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible -subfactor , the relative commutants and admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that acts on with invariants .

     

D-resultant and subresultants

We establish a connection between the D-resultant of two polynomials and and the subresultant sequence of and . This connection is used to decide in a more explicit way whether or . We also show how to extract a faithful parametrization from a given one.

     

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.

     

More on convexity numbers of closed sets in

The convexity number of a set is the least size of a family of convex sets with . is countably convex if its convexity number is countable. Otherwise is uncountably convex.

Uncountably convex closed sets in have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all , it is consistent that there is an uncountably convex closed set whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of .

Moreover, we construct a closed set whose convexity number is and that has no uncountable -clique for any 1$">. Here is a -clique if the convex hull of no -element subset of is included in . Our example shows that the main result of the above-named authors, a closed set either has a perfect -clique or the convexity number of is in some forcing extension of the universe, cannot be extended to higher dimensions.