On the excess of sets of complex exponentials

For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.

     

On the boundary of attractors with non-void interior

Let be a family of contractive mappings on such that the attractor has nonvoid interior. We show that if the 's are injective, have non-vanishing Jacobian on , and have zero Lebesgue measure for then the boundary of has measure zero. In addition if the 's are affine maps, then the conclusion can be strengthened to . These improve a result of Lagarias and Wang on self-affine tiles.

     

Integer-valued polynomials over Krull-type domains and Prüfer -multiplication domains

Let be a domain with quotient field . The ring of integer-valued polynomials over is . We characterize the Krull-type domains such that is a Prüfer -multiplication domain.

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

Maximal estimates for the -dimensional Walsh-Fourier series

The -dimensional dyadic martingale Hardy spaces are introduced and it is proved that the maximal operator of the means of a Walsh-Fourier series is bounded from to and is of weak type , provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the means of a function converge a.e. to the function in question. Moreover, we prove that the means are uniformly bounded on whenever . Thus, in case , the means converge to in norm. The same results are proved for the conjugate means, too.

     

Lomonosov's theorem cannot be extended to chains of four operators

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if is the operator without a non-trivial closed invariant subspace constructed by C. J. Read, then there are three operators , and (non-multiples of the identity) such that commutes with , commutes with , commutes with , and is compact. It is also shown that the commutant of contains only series of .

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Let be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector and the point . Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces (). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. . For known examples of 4-dimensional Osserman manifolds of signature we check also that . By the presentation of a class of examples we show that curvature homogeneity and do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.

     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

The Furuta inequality with negative powers

Let be bounded linear operators on a Hilbert space satisfying . Furuta showed the operator inequality
as long as positive real numbers satisfy and . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

     

On the divergence of the means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double means of the Walsh-Fourier series () for functions in ( is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function we have a function such as and does not converge to a.e. (in the Pringsheim sense).

     

Inequalities for the gamma function

We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

     

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

     

Schreier theorem on groups which split over free abelian groups

Let be either a free product with amalgamation or an HNN group where is isomorphic to a free abelian group of finite rank. Suppose that both and have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if contains a finitely generated normal subgroup which is neither contained in nor free, then the index of in is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of -manifolds and , the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of and has at least one nontorus boundary.

     

A remark on Kulesza's example

In this note we simplify the proof of some properties of Kulesza's metric space with ind and Ind.

     

The Hausdorff operator is bounded on the real Hardy space

We prove that the Hausdorff operator generated by a function is bounded on the real Hardy space . The proof is based on the closed graph theorem and on the fact that if a function in is such that its Fourier transform equals for (or for ), then .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

Remarks about Schlumprecht space

Let denote the Schlumprecht space. We prove that

(1) is finitely disjointly representable in ;

(2) contains an -spreading model;

(3) for any sequence of natural numbers, is isomorphic to the space .

     

Existence of solutions for first order singular problems

We develop the lower and upper solutions method for first order initial value problems as well as for first order periodic problems in case the nonlinearity presents singularities. More precisely we prove that if we have a lower solution and an upper solution of these problems, which are not necessarily continuous nor ordered, we have a solution wedged between and .