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 .

     

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 families of Gorenstein algebras

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring . Let be the scheme parametrizing graded quotients of with Hilbert function . We prove there is a close relationship between the irreducible components of , whose general member is a Gorenstein codimension quotient, and the irreducible components of , whose general member is a codimension Cohen-Macaulay algebra of Hilbert function related to . If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of , this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay' components of to ``Gorenstein' components of , in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay' component and a sum of two invariants of . Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.

     

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.

     

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.

     

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the -dimensional Euclidean space is said to be uniformly dense in a surface of codimension if, for every small the volume of the intersection of with a ball of radius and center does not depend on for

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary , and we show that the principal curvatures of satisfy certain identities.

The case in which the surface coincides with is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if , it must be either a circle or a straight line and (ii) if it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

     

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.

     

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.

     

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.

     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

On almost one-to-one maps

A continuous map of topological spaces is said to be almost -to- if the set of the points such that is dense in ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

Approximation and regularization of Lipschitz functions: Convergence of the gradients

We examine the possible extensions to the Lipschitzian setting of the classical result on -convergence: first (approximation), if a sequence of functions of class from to converges uniformly to a function of class , then the gradient of is a limit of gradients of in the sense that ; second (regularization), the functions can be chosen to be of class and -converging to in the sense that . In other words, the space of functions is dense in the space of functions endowed with the pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if is a locally Lipschitz function, we build a sequence of smooth functions such that

In other words, the space of functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality , which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

     

Partitioning -large sets: Some lower bounds

Let denote the property: if is an -large set of natural numbers and is partitioned into parts, then there exists a -large subset of which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for in terms of for some specific .

     

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

Functional distribution of with real characters and denseness of quadratic class numbers

We investigate the functional distribution of -functions with real primitive characters on the region as varies over fundamental discriminants. Actually we establish the so-called universality theorem for in the -aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed with and positive integers , there exist infinitely many such that for every the -th derivative has at least zeros on the interval in the real axis. We also study the value distribution of for fixed with and variable , and obtain the denseness result concerning class numbers of quadratic fields.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

Frequently hypercyclic operators

We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators on separable complex -spaces: is frequently hypercyclic if there exists a vector such that for every nonempty open subset of , the set of integers such that belongs to has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.

     

An infinitary extension of the Graham-Rothschild Parameter Sets Theorem

The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of , the Stone-Cech compactification of a discrete semigroup, we derive an infinitary extension of the Graham-Rothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extension of the original. The original theorem says that whenever in and the -parameter words are colored with finitely many colors, there exist a color and an -parameter word with the property that whenever a -parameter word of length is substituted in , the result is in the specified color. The ``simplest finite instance' referred to above is that, given finite colorings of the -parameter words for each , there is one -parameter word which works for each . Some additional Ramsey Theoretic consequences are derived.

We also observe that, unlike any other Ramsey Theoretic result of which we are aware, central sets are not necessarily good enough for even the and version of the Graham-Rothschild Parameter Sets Theorem.