Cremer fixed points and small cycles

Let , , and let denote the sequence of convergents to the regular continued fraction of . Let be a function holomorphic at the origin, with a power series of the form . We assume that for infinitely many we simultaneously have (i) , (ii) the coefficients stay outside two small disks, and (iii) the series is lacunary, with for . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

     

Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators with unbounded coefficients in . Using a Bernstein approach we provide several uniform estimates for the semigroup generated by the realization of the operator in the space of all bounded and continuous or Hölder continuous functions in . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation (0$">) and the nonhomogeneous Dirichlet Cauchy problem . Then, we prove two different kinds of pointwise estimates of that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup in weighted -spaces related to the invariant measure associated with the semigroup.

     

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let be a bigraded ideal in the bigraded polynomial ring . Assume that has codimension 2. Then is a finite set of points. We prove that if is a local complete intersection, then any syzygy of the vanishing at , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

     

On Ore's conjecture and its developments

The -component of the index of a number field , , depends only on the completions of at the primes over . More precisely, equals the index of the -algebra . If is normal, then for some normal over and some , and we write for its index. In this paper we describe an effective procedure to compute for all and all normal and tamely ramified extensions of , hence to determine for all Galois number fields that are tamely ramified at . Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of .

     

Greedy wavelet projections are bounded on BV

Let be the space of functions of bounded variation on with . Let , , be a wavelet system of compactly supported functions normalized in , i.e., , . Each has a unique wavelet expansion with convergence in . If is the set of indicies for which are largest (with ties handled in an arbitrary way), then is called a greedy approximation to . It is shown that with a constant independent of . This answers in the affirmative a conjecture of Meyer (2001).

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

Weierstrass functions with random phases

Consider the function

where 1$">, , and is a non-constant 1-periodic Lipschitz function. The phases are chosen independently with respect to the uniform probability measure on . We prove that with probability one, we can choose a sequence of scales such that for every interval of length , the oscillation of satisfies . Moreover, the inequality is almost surely true at every scale. When is a transcendental number, these results can be improved: the minoration is true for every choice of the phases and at every scale.

     

Harmonic maps and 2-cycles, realizing the Thurston norm

Let be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, , of a harmonic map with Morse-type singularities delivers the Thurston norm of its homology class .

In particular, for a map with connected fibers and any well-positioned oriented surface in the homology class of a fiber, we show that the Thurston number satisfies an inequality

Here the variation is can be expressed in terms of the -invariants of the fiber components, and the twist measures the complexity of the intersection of with a particular set of ``bad" fiber components. This complexity is tightly linked with the optimal ``-height" of , being lifted to the -induced cyclic cover .

Based on these invariants, for any Morse map , we introduce the notion of its twist . We prove that, for a harmonic , if and only if .

     

Dimension of families of determinantal schemes

A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Mansfield's imprimitivity theorem for full crossed products

For any maximal coaction and any closed normal subgroup of , there exists an imprimitivity bimodule between the full crossed product and , together with compatible coaction of . The assignment implements a natural equivalence between the crossed-product functors `` ' and `` ', in the category whose objects are maximal coactions of and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of .

     

Existence and asymptotic behavior for a singular parabolic equation

We prove global existence of nonnegative solutions to the singular parabolic equation 0 \} } ( -u^{-\beta} + \lambda f(u) )=0$"> in a smooth bounded domain with zero Dirichlet boundary condition and initial condition , . In some cases we are also able to treat . Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called ``quenching'. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.

     

Limits of interpolatory processes

Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem

with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

     

Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action of a group on an associative algebra , we consider the crossed product . Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of obtained in the context of -algebras. In particular, we prove that is associative, provided that is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Generalized Ahlfors functions

Let be a bordered Riemann surface with genus and boundary components. Let be a smooth family of smooth Jordan curves in which all contain the point 0 in their interior. Let and let be the family of all bounded holomorphic functions on such that and for almost every . Then there exists a smooth up to the boundary holomorphic function with at most zeros on so that for every and such that for every . If, in addition, all the curves are strictly convex, then is unique among all the functions from the family .