Stable geometric dimension of vector bundles over even-dimensional real projective spaces

In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order over if is even and sufficiently large and . In this paper, we use the Bendersky-Davis computation of to show that the 1981 result extends to all (still provided that is sufficiently large). If , the result is often different due to anomalies in the formula for when , but we also determine the stable geometric dimension in these cases.

     

On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Covering a compact set in a Banach space by an operator range of a Banach space with basis

A Banach space has the approximation property if and only if every compact set in is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for spaces and quotients of spaces in terms of covering compact sets in by operator ranges from spaces. A Banach space is a space if and only if every compact set in is contained in the closed convex symmetric hull of a basic sequence which converges to zero.

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

     

On the Cohen-Macaulay property of multiplicative invariants

We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group . By definition, these are -actions on Laurent polynomial algebras that stabilize the multiplicative group consisting of all monomials in the variables . For the most part, we concentrate on the case where the base ring is . Our main result states that if acts non-trivially and the invariant ring is Cohen-Macaulay, then the abelianized isotropy groups of all monomials are generated by the bireflections in and at least one is non-trivial. As an application, we prove the multiplicative version of Kemper's -copies conjecture.

     

The fourth power moment of automorphic -functions for over a short interval

In this paper we will prove bounds for the fourth power moment in the aspect over a short interval of automorphic -functions for on the central critical line Re. Here is a fixed holomorphic or Maass Hecke eigenform for the modular group , or in certain cases, for the Hecke congruence subgroup with . The short interval is from a large to . The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg -function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

     

A moment approach to analyze zeros of triangular polynomial sets

Let be a zero-dimensional ideal of such that its associated set of polynomial equations for all is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in . We also provide a necessary and sufficient (numerical) condition for all the zeros of to be in a given set , without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the 's for to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set . In the proof technique, we use a deep result of Curto and Fialkow (2000) on the -moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the 's via the Newton sums of . In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.

     

Sharp dimension estimates of holomorphic functions and rigidity

Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that

for all , with equality for some positive integer if and only if is holomorphically isometric to . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

     

On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface over a curve of genus with the numerical invariant . Let    Pic be a line bundle in the numerical class of . We prove that for , satisfies property if and , and for , satisfies property if and . By using these facts, we obtain Mukai-type results. For ample line bundles , we show that satisfies property when and or when and . Therefore we prove Mukai's conjecture for ruled surface with . We also prove that when is an elliptic ruled surface with , satisfies property if and only if and .

     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

-manifolds with planar presentations and the width of satellite knots

We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

Corollary.

The width of a satellite knot is no less than the width of its pattern knot and so

.

     

Open loci of graded modules

Let be an excellent homogeneous Noetherian graded ring and let be a finitely generated graded -module. We consider as a module over and show that the -loci of are open in . In particular, the Cohen-Macaulay locus    is Cohen-Macaulay is an open subset of . We also show that the -loci on the homogeneous parts of are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module over an excellent ring and for an ideal which is not contained in any minimal prime of , the -loci for the modules are eventually stable.

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

The braid index is not additive for the connected sum of 2-knots

Any -dimensional knot can be presented in a braid form, and its braid index, , is defined. For the connected sum of -knots and , it is easily seen that holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of -dimensional knots; the equality holds for -knots. We prove that the equality does not hold for -knots unless or is a trivial -knot. We also prove that the -knot obtained from a granny knot by Artin's spinning is of braid index , and there are infinitely many -knots of braid index .

     

Besov spaces with non-doubling measures

Suppose that is a Radon measure on which may be non-doubling. The only condition on is the growth condition, namely, there is a constant 0$"> such that for all and 0,$">

where In this paper, the authors establish a theory of Besov spaces for and , where 0$"> is a real number which depends on the non-doubling measure , , and . The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function on the plane and a non-zero define the Hilbert transform of in the direction to be

p.v.

Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

     

On polynomial-factorial diophantine equations

We study equations of the form and show that for some classes of polynomials the equation has only finitely many solutions. This is the case, say, if is irreducible (of degree greater than 1) or has an irreducible factor of ``relatively large" degree. This is also the case if the factorization of contains some ``large" power(s) of irreducible(s). For example, we can show that the equation has only finitely many solutions for , but not that this is the case for (although it undoubtedly should be). We also study the equation , where is one of several other ``highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.