A remark on Littlewood-Paley theory for the distorted Fourier transform

We consider the classical theorems of Mikhlin and Littlewood-Paley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator . We show that for such operators which display a zero energy resonance the full range in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to .

     

Finite time blow up for a Navier-Stokes like equation

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so-called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space . We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

     

On boundary blow-up problems for the complex Monge-Ampère equation

We prove the regularity for some complex Monge-Ampère equations with boundary data equal to .

     

Classification of escaping exponential maps

We give a complete classification of the set of parameters for which the singular value of escapes to under iteration. In particular, we show that every path-connected component of this set is a curve to infinity.

     

An addition to the -theorem for subharmonic and entire functions of zero lower order

We obtain a sharp asymptotic relation between the infimum and the maximum on a circle of a subharmonic function of zero lower order. An example is constructed, which shows the sharpness of the relation in the class of entire functions of zero order such that , where as .

     

Global behavior of the branch of positive solutions to a logistic equation of population dynamics

We consider the nonlinear problem arising in population dynamics:

where is a constant and is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions in -framework as , where ( ). Especially, for the original logistic equation, namely the case where and , we obtain not only the asymptotic expansion formula for but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.

     

On the stable module category of a self-injective algebra

Let be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable -modules which belong to Auslander-Reiten components of the form or . The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.     

Characterizing indecomposable plane continua from their complements

We show that a plane continuum is indecomposable iff has a sequence of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence of open arcs, with and , there is a sequence of shadows , where each is a shadow of , such that . Such an open arc divides into disjoint subdomains and , and a shadow (of ) is one of the sets .

     

Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

     

Random polynomials with prescribed Newton polytope

The Newton polytope of a polynomial is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in of a system of polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in for these distributions.

Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree in complex variables with its usual SU-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope . We then determine the asymptotics of the conditional expectation of simultaneous zeros of polynomials with Newton polytope as . When , the unit simplex, it is clear that the expected zero distributions are uniform relative to the Fubini-Study form. For a convex polytope , we show that there is an allowed region on which is asymptotically uniform as the scaling factor . However, the zeros have an exotic distribution in the complementary forbidden region and when (the case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as .

     

Cesàro function spaces fail the fixed point property

The Cesàro sequence spaces , are reflexive but they have the fixed point property. In this paper we prove that in contrast to these sequence spaces the corresponding Cesàro function spaces on both and for are not reflexive and they fail to have the fixed point property.

     

Infinite partition regular matrices: solutions in central sets

A finite or infinite matrix is image partition regular provided that whenever is finitely colored, there must be some with entries from such that all entries of are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that must have images in every central subset of . We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever is finitely colored, there must exist injective sequences and in with all sums of the forms and with in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.

     

On possible non-homeomorphic substructures of the real line

We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength:

Theorem 1. The following are equiconsistent:

(i) a Jónsson cardinal;

(ii) a sufficiently elementary submodel of the universe of sets with not homeomorphic to

The reverse direction is a corollary to:

Theorem 2. is Jónsson hereditarily separable, hereditarily Lindelöf, with .

We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.

     

Semi-continuity of metric projections in -direct sums

Let be a proximinal subspace of finite codimension of . We show that is proximinal in and the metric projection from onto is Hausdorff metric continuous. In particular, this implies that the metric projection from onto is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

     

Infinite dimensional universal subspaces generated by Blaschke products

Let be the Banach algebra of all bounded analytic functions in the unit disk . A function is said to be universal with respect to the sequence of noneuclidian translates, if the set is locally uniformly dense in the set of all holomorphic functions bounded by . We show that for any sequence of points in tending to the boundary there exists a closed subspace of , topologically generated by Blaschke products, and linear isometric to , such that all of its elements are universal with respect to noneuclidian translates. The proof is based on certain interpolation problems in the corona of . Results on cyclicity of composition operators in are deduced.

     

Singular solutions of parabolic -Laplacian with absorption

We consider, for and , the -Laplacian evolution equation with absorption

We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in , and satisfy for all . We prove the following:

(i)

When , there does not exist any such singular solution.

(ii)

When , there exists, for every , a unique singular solution that satisfies as .

Also, as , where is a singular solution that satisfies as .

Furthermore, any singular solution is either or for some finite positive .

     

Stabilization of Tsirelson-type norms on spaces

We consider classical Tsirelson-type norms of and their modified versions on spaces, . We show that the modified Tsirelson-type norms do not distort any of the subspaces of the spaces. We prove that Tsirelson-type norms, being equivalent to their modified versions, may at most 2-distort spaces.

     

Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define . In particular, let be such that , as for some , is o-minimal, and the expansion of by the set does not define . Then there exist 0$"> and such that as .

     

Canonical divisors in weighted Bergman spaces

Canonical divisors in Bergman spaces can be found as solutions of extremal problems. We derive a formula for certain extremal functions in the weighted Bergman spaces for -1$"> and . This leads to a study of the zeros of a specific family of hypergeometric functions.