The divisor problem for d 4 (n) in short intervals

In this paper we establish an asymptotic formula for the sum

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.      

Some divisibilities amongst the terms of linear recurrences

Let (u_n)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu _n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u _n , where τ(n) is the number of the positive integer divisors of n, provided thatu _n satisfies some mild technical conditions.     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.     

On values of n 2 + 1 free of large prime factors

Let P ⁺(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation are given. Applications for the trapezoid and mid-point inequalities are also provided. Received: 19 May 2008     

The complex Busemann-Petty problem for arbitrary measures

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if and negative if . In this article we show that the answer remains the same if the volume is replaced by an “almost” arbitrary measure. This result is the complex analogue of Zvavitch’s generalization to arbitrary measures of the original real Busemann-Petty problem. Received: 6 May 2008     

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that , for θ > 1/2, with the assumption of an heuristic hypothesis weaker than the Lindel?f hypothesis. Received: 8 October 2007, Revised: 14 April 2008     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

Avoidable structures, I: Finite ordered sets, semilattices and lattices

We find all finite unavoidable ordered sets, finite unavoidable semilattices and finite unavoidable lattices. While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

On the Orbit of an m-Isometry

A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if . We prove some properties concerning the behaviour of the orbit of an m-isometry. For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N-supercyclic, that is, there does not exist an N-dimensional subspace E_N such that the orbit of T at E_N is dense in H.

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To Professor José Rodríguez Expósito on his 60th birthday     

From Lucid to TransLucid: Iteration, Dataflow, Intensional and Cartesian Programming

We present the development of the Lucid language from the Original Lucid of the mid-1970s to the TransLucid of today. Each successive version of the language has been a generalisation of previous languages, but with a further understanding of the problems at hand. The Original Lucid (1976), originally designed for purposes of formal verification, was used to formalise the iteration in while-loop programs. The pLucid language (1982) was used to describe dataflow networks. Indexical Lucid (1987) was introduced for intensional programming, in which the semantics of a variable was understood as a function from a universe of possible worlds to ordinary values. With TransLucid, and the use of contexts as firstclass values, programming can be understood in a Cartesian framework.      

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

On the \varpi_n-related elements in the stable homotopy groups of spheres

Let X = {x ₁, …, x _n } be a set of n points in the d-dimensional Euclidean space , with unit diameter. In this work we give the complete proof of a conjecture by Witsenhausen, stating that the maximum M(d, n) of in dimension d is attained if and only if the points are distributed as evenly as possible among the vertices of a regular d-dimensional simplex of edge-length 1. Received: 5 July 2007, Revised: 22 October 2007     

On Gale and braxial polytopes

Cyclic polytopes are characterized as simplicial polytopes satisfying Gale’s evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy Gale’s evenness condition. The faces of these polytopes are braxtopes, a certain class of nonsimplicial polytopes studied by the authors. In this paper we prove that the periodically-cyclic Gale polytopes of Bisztriczky are exactly the polytopes that satisfy Gale’s evenness condition and are braxial (all faces are braxtopes). The existence of other periodically-cyclic Gale polytopes is open. Supported in part by a grant from the University of Kansas General Research Fund and by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. Received: 20 September 2006     

The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices