Weierstrass functions in Zygmund's class

Consider the function

where 1$"> and is an almost periodic function. It is well known that the function lives in the so-called Zygmund class. We prove that is generically nowhere differentiable. This is the case in particular if the elementary condition is satisfied. We also give a sufficient condition on the Fourier coefficients of which ensures that is nowhere differentiable.

     

Joins of projective varieties and multisecant spaces

Let , , be integral varieties. For any integers 0$">, , and set and . Let be the set of all linear -spaces contained in a linear -space spanned by points of , points of , ..., points of . Here we study some cases where has the expected dimension. The case was recently considered by Chiantini and Coppens and we follow their ideas. The two main results of the paper consider cases where each is a surface, more particularly:

or

     

A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations

In 1992, P. Polácik showed that one could linearly imbed any vector field into a scalar semi-linear parabolic equation on with Neumann boundary condition provided that there exists a smooth vector field on such that

In this short paper, we give a classification of all the domains on which one may find such a type of vector field.

     

A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

In this paper, we prove the following general result. Let be a real Hilbert space and a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that

Then, for each for which the set is not convex and for each convex set dense in , there exist and 0$"> such that the equation

has at least three solutions.

     

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

We study the conformal scalar curvature problem

where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .

     

Almost automorphic solutions of semilinear evolution equations

We are concerned with the semilinear differential equation in a Banach space ,

where generates an exponentially stable -semigroup and is a function of the form . Under appropriate conditions on and , and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.

     

Morita equivalence for quantum Heisenberg manifolds

We discuss Morita equivalence within the family 0, \mu,\nu\in\mathbb{R}\}$"> of quantum Heisenberg manifolds. Morita equivalence classes are described in terms of the parameters , and the rank of the free abelian group associated to the -algebra .

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Knot signature functions are independent

A Seifert matrix is a square integral matrix satisfying

To such a matrix and unit complex number there corresponds a signature,

Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.

     

Proofs of two conjectures of Gray involving the double suspension

In proving that the fiber of the double suspension has a classifying space, Gray constructed fibrations

and

He conjectured that is homotopic to the -power map on when is an odd prime. Harper proved this is true when looped once. We remove the loop when . Gray also conjectured that at odd primes  factors through a map

We show that this is true as well when .

     

On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .

     

Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where 0$"> and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

     

On a class of sublinear quasilinear elliptic problems

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem

where is a bounded domain in with smooth boundary , is continuous and p-sublinear at and is a large parameter.

     

Degree of a holomorphic map between unit balls from to

Let be a rational proper holomorphic map between the unit ball in and the unit ball in Write

where and are holomorphic polynomials, with Recall that the degree of is defined by

   deg

In this paper, we give a bound estimate for the degree of improving the bound given by Forstneric (1989).

     

Unprovability of sharp versions of Friedman's sine-principle

For every and every function of one argument, we introduce the statement : ``for all , there is such that for any set of rational numbers, there is of size such that for any two -element subsets and in , we have

We prove that for and any function eventually dominated by , the principle is not provable in . In particular, the statement is not provable in Peano Arithmetic. In dimension 2, the result is: does not prove , where and is the inverse of the Ackermann function.

     

Relaxation and convexity of functionals with pointwise nonlocality

It is shown that the relaxation of the integral functional involving argument deviations

in weak topology of a Lebesgue space (where and are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions : satisfies the special condition, called unifiability, which can be regarded as collective nonergodicity or commensurability property, and is automatically satisfied only if . If, however, either 1$"> or 1$">, then it is shown that as opposed to the classical case without argument deviations, for nonunifiable function matrix one can always construct an integrand so that the functional itself is already weakly lower semicontinuous but not convex.

     

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.

     

A ``deformation estimate" for the Toeplitz operators on harmonic Bergman spaces

Let denote the open unit ball in for and the Lebesgue volume measure on . For , the (weighted) harmonic Bergman space is the space of all harmonic functions which are in . For , the Toeplitz operator is defined on by , where is the orthogonal projection of onto . In this note, we prove that for radial, .

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .