On the characteristic polynomial of the almost Mathieu operator

Let be the rotation C*-algebra for angle . For with and relatively prime, is the sub-C*-algebra of generated by a pair of unitaries and satisfying . Let

be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of is .

     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

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.

     

On the Bochner theorem on orthogonal operators

We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation

where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

     

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 .

     

Semiclassical analysis for highly degenerate potentials

This paper characterizes the semi-classical limit of the fundamental energy,

and ground state of the Schrödinger operator in a bounded domain , in the highly degenerate case when and consists of two components, say and . The main result establishes that

and that approximates in the ground state of in if

     

Some remarks on spreading models and mixed Tsirelson spaces

We prove that if a Banach space with a bimonotone shrinking basis does not contain spreading models but every block sequence of the basis contains a further block sequence which is a spreading model for every , then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space , such that , does not contain spreading models.

     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Essential numerical range of elementary operators

Let and denote two -tuples of operators with and Let denote the elementary operators defined on the Hilbert-Schmidt class by We show that

Here is the essential numerical range, is the joint numerical range and is the joint essential numerical range.

     

Random Menshov spectra

We show that the spectra of frequencies obtained by random perturbations of the integers allows one to represent any measurable function on by an almost everywhere converging sum of harmonics:

     

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If is Borel measurable, define for -finite positive Borel measures on the bilinear integral expression

We give conditions on such that there is a constant , independent of and , with

Our results apply to a much larger class of functions than known before.

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

Signed sums of polynomial values

We give a generalization of Bleicher's result on signed sums of th powers. Let be an integral-valued polynomial of degree satisfying the necessary condition that there exists no integer 1$"> dividing the values for all integers . Then, for every positive integer and every integer , there are infinitely many integers and choices of such that

     

Global existence and nonexistence for degenerate parabolic systems

The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system

in the cylinder , where is bounded and are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by the first Dirichlet eigenvalue for the Laplacian on . We prove that there exists a global solution iff .

     

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.

     

On the best possible character of the norm in some a priori estimates for non-divergence form equations in Carnot groups

Let be a group of Heisenberg type with homogeneous dimension . For every we construct a non-divergence form operator and a non-trivial solution to the Dirichlet problem: in , on . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as

where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

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.