Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type

We establish the uniqueness of the positive solution for equations of the form in , . The special feature is to consider nonlinearities whose variation at infinity is not regular (e.g., , , , , , , or ) and functions in vanishing on . The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the nonregular variation of at infinity with the blow-up rate of the solution near .

     

Torsion freeness of symmetric powers of ideals

Let be an ideal in a Noetherian commutative ring with unit, let be an integer, and let be the canonical surjective -module homomorphism from the th symmetric power of to the th power of . When or when is a perfect Gorenstein ideal of grade , we provide a necessary and sufficient condition for to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of . When is a maximal ideal of we show that is an isomorphism if and only if is a regular local ring. In all three cases for our results yield that if is an isomorphism, then is also an isomorphism for each .

     

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain in with smooth boundary, the cut locus is the closure of the set of nondifferentiability points of the distance from the boundary of . The normal distance to the cut locus, , is the map which measures the length of the line segment joining to the cut locus along the normal direction , whenever . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of is of class . Our main result is the global Hölder regularity of in the case of a domain with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain . The above regularity result for is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

     

Analyticity on translates of a Jordan curve

Let be a domain in which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate vertically to get where is such that . We prove that if is a continuous function on such that for each , the function has a continuous extension to which is holomorphic on , then is holomorphic on .

     

On -adic intermediate Jacobians

For an algebraic variety of dimension with totally degenerate reduction over a -adic field (definition recalled below) and an integer with , we define a rigid analytic torus together with an Abel-Jacobi mapping to it from the Chow group of codimension algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over . We compare and contrast the complex and -adic theories. Finally, we examine a special case of a -adic analogue of the Generalized Hodge Conjecture.

     

On homeomorphic Bernoulli measures on the Cantor space

Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability' and show that, if and are refinable and binomially equivalent, then and are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers and such that and are refinable, so the corresponding measures are topologically equivalent.

     

Analytic contractions, nontangential limits, and the index of invariant subspaces

Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of .

We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0.

It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent:

(1) for some ,

(2) for all , ,

(3) has nonzero Lebesgue measure,

(4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .

     

Compatible valuations and generalized Milnor -theory

Given a field and a subgroup of there is a minimal group for which there exists an -compatible valuation whose units are contained in . Assuming that has finite index in and contains for prime, we describe in computable -theoretic terms.

     

The expansion factors of an outer automorphism and its inverse

A fully irreducible outer automorphism of the free group of rank  has an expansion factor which often differs from the expansion factor of . Nevertheless, we prove that the ratio between the logarithms of the expansion factors of and is bounded above by a constant depending only on the rank . We also prove a more general theorem applying to an arbitrary outer automorphism of and its inverse and their two spectrums of expansion factors.

     

Complex symmetric operators and applications II

A bounded linear operator on a complex Hilbert space is called complex symmetric if , where is a conjugation (an isometric, antilinear involution of ). We prove that , where is an auxiliary conjugation commuting with . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition also extends to the class of unbounded -selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

Heegner points and Mordell-Weil groups of elliptic curves over large fields

Let be an elliptic curve defined over of conductor and let be the absolute Galois group of an algebraic closure of . For an automorphism , we let be the fixed subfield of under . We prove that for every , the Mordell-Weil group of over the maximal Galois extension of contained in has infinite rank, so the rank of is infinite. Our approach uses the modularity of and a collection of algebraic points on - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer and for a prime prime to , the rank of over all the ring class fields of a conductor of the form is unbounded, as goes to infinity.

     

Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group . We investigate -invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group . We use an important invariant of a flag manifold , its -root system, to give a simple necessary condition that admits a non-standard -invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds of a simple Lie group , only the manifold of complex structures in , and the complex projective space admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only -invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra of ). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

     

Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.

     

Sharp Sobolev inequalities in the presence of a twist

Let be a smooth compact Riemannian manifold of dimension . Let also be a smooth symmetrical positive -tensor field in . By the Sobolev embedding theorem, we can write that there exist such that for any ,

where is the standard Sobolev space of functions in with one derivative in . We investigate in this paper the value of the sharp in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.

     

The McMullen domain: Rings around the boundary

In this paper we show that there are infinitely many rings , around the McMullen domain in the parameter plane for the family of complex rational maps of the form where and . These rings converge to the boundary of the McMullen domain as . The rings contain parameter values that lie at the center of Sierpinski holes. That is, these parameters lie at the center of an open set in the parameter plane in which all of the corresponding maps have Julia sets that are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic of period either or .

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Parageometric outer automorphisms of free groups

We study those fully irreducible outer automorphisms of a finite rank free group which are parageometric, meaning that the attracting fixed point of  in the boundary of outer space is a geometric -tree with respect to the action of , but  itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of is strictly larger than the expansion factor of . As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric -trees.