Partitions into Primes

We investigate the asymptotic behavior of the partition function defined by , where denotes the von Mangoldt function. Improving a result of Richmond, we show that , where is a positive constant and denotes the times iterated logarithm. We also show that the error term can be improved to if and only if the Riemann Hypothesis holds.

     

The space for nondoubling measures in terms of a grand maximal operator

Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.

     

Dihedral coverings of algebraic surfaces and their application

In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following:

Let be a reduced plane curve of even degree having only nodes and cusps. If 2d^2 - 6d + 6$">, then is non-abelian.

Note that Nori's result implies that is abelian, provided that .

     

Holomorphic extensions from open families of circles

For a circle write . A continuous function on extends holomorphically from (into the disc bounded by ) if and only if the function defined on has a bounded holomorphic extension into . In the paper we consider open connected families of circles , write , and assume that a continuous function on extends holomorphically from each . We show that this happens if and only if the function defined on has a bounded holomorphic extension into the domain for each open family compactly contained in . This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.

     

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 .

     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

Periodic solutions of conservation laws constructed through Glimm scheme

We present a periodic version of the Glimm scheme applicable to special classes of systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in , for Bakhvalov's class, and in , in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in with , for some constant which is for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, , are uniformly bounded in , for all 0$">, where is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the -systems with , , , which, for , model isentropic gas dynamics in Lagrangian coordinates.

     

Representation formulae and inequalities for solutions of a class of second order partial differential equations

Let be a possibly degenerate second order differential operator and let be its fundamental solution at ; here is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of on to satisfy the representation formula

We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition

then (R) holds with no growth assumptions on .

     

Maximal degree subsheaves of torsion free sheaves on singular projective curves

Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.

     

Prescribing analytic singularities for solutions of a class of vector fields on the torus

We consider the operator acting on distributions on the two-torus where and are real-valued, real analytic functions defined on the unit circle We prove, among other things, that when changes sign, given any subset of the set of the local extrema of the local primitives of there exists a singular solution of such that the projection of its analytic singular support is furthermore, for any and any closed subset of there exists such that and We also provide a microlocal result concerning the trace of at

     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .

     

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 .

     

Convergence of double Fourier series and -classes

The double Fourier series of functions of the generalized bounded variation class are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class defined here, have quadrant limits at every point and, for there exist at most countable sets and such that, for and is continuous at . It is shown that the previously studied class contains essentially discontinuous functions unless the sequence satisfies a strong condition.

     

Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence

Let be a connected, linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent -orbits in and the nilpotent -orbits in . We show that this correspondence associates each spherical nilpotent -orbit to a nilpotent -orbit that is multiplicity free as a Hamiltonian -space. The converse also holds.

     

Schauder estimates for equationswith fractional derivatives

The equation where and are fractional derivatives of order and is studied. It is shown that if , , and are Hölder-continuous and , then there is a solution such that and are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (). Finally the solution of () with is studied.     

Limits in the uniform ultrafilters

Let be the space of uniform ultrafilters on . If is regular, then there is an which is not an accumulation point of any subset of of size or less. is also good, in the sense of Keisler.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

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 .