On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

On graphs whose star sets are (co-)cliques

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G=K_1,2 or K_2,…,2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets.     

On semicomplete multipartite digraphs whose king sets are semicomplete digraphs

Reid [Every vertex a king, Discrete Math. 38 (1982) 93-98] showed that a non-trivial tournament H is contained in a tournament whose 2-kings are exactly the vertices of H if and only if H contains no transmitter. Let T be a semicomplete multipartite digraph with no transmitters and let K_r(T) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K₄(T). Very recently, Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] proved that Q contains no transmitters and gave an example to show that the direct extension of Reid's result to semicomplete multipartite digraphs with 2-kings replaced by 4-kings is not true. In this paper, we (1) characterize all semicomplete digraphs D which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of D. While it is trivial that K₄(Q)⊆K₄(T), Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] showed that K₃(Q)⊆K₃(T) and K₂(Q)=K₂(T). Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] also provided an example to show that K₃(Q) need not be the same as K₃(T) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K₃(Q)=K₃(T). In the course of proving our result (1), we (2) show that K₃(Q)=K₃(T) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph.     

Arrangements of n Points whose Incident-Line-Numbers are at most n/2

We consider a set X of n noncollinear points in the Euclidean plane, and the set of lines spanned by X, where n is an integer with n ≥ 3. Let t(X) be the maximum number of lines incident with a point of X. We consider the problem of finding a set X of n noncollinear points in the Euclidean plane with t(X) ￡ ?n/2 ?{t(X) \le \lfloor n/2 \rfloor}, for every integer n ≥ 8. In this paper, we settle the problem for every integer n except n = 12k + 11 (k ≥ 4). The latter case remains open.     

Semigroups whose Julia sets are Cantor targets

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2.     

On Banach spaces whose norm-open sets are -sets in the weak topology

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.     

Sperner systems containing at most k sets of every cardinality

We prove using a direct construction that one can choose n − 2 subsets of an n-element set with different cardinality such that none of them contains any other. As a generalization, we prove that if for any j we can have at most k subsets containing exactly j elements (k> 1), then for n 5 we can choose at most k(n − 3) subsets from an n-element set such that they form a Sperner system. Moreover, we prove that this can be achieved if n is large enough, and give a construction for n 8k − 4.     

Non-measurable sets of reals whose measurable subsets are countable

In this paper we generalize Sierpinski's concept of sets of typeS and give a characterization of such sets in terms of a partition of the reals. We also give a similar characterization of Lusin sets.     

Restricted Lie algebras all whose elements are semisimple

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra.     

On separating systems of graphs

Given a finite loopless graph G (resp. digraph D), let σ(G), ?(G) and ψ(D) denote the minimal cardinalities of a completely separating system of G, a separating system of G and a separating system of D, respectively. The main results of this paper are:

$\text{δ(G) =}\text{min}\text{m}{}^{\mathrm{m}}{}_{\mathrm{?m/2?}}\text{?γ(G)}\text{and}\text{?(G) = ?}\text{log}{}_2\text{γ(G)?}\text{where}\text{γ(G)}$

denotes the chromatic number of G. (ii) All the problems of determining σ(G), ?(G) and ψ(D) are NP-complete.     

Group rings whose symmetric elements are Lie nilpotent

Let be the group ring of a group over a field , with characteristic different from . Let denote the natural involution on sending each group element to its inverse. Denote by the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided has no -elements, if is Lie nilpotent, then so is . In this paper, we determine when is Lie nilpotent, if does contain -elements.

     

A class of functions whose sublevel sets are absolute retracts

In this paper, we prove a result of which the following is a corollary: If X is a Banach space and J:X→R is a contraction, then the nonempty sublevel sets of the function x→‖x‖+J(x) are absolute retracts.     

On systems of bilinear forms whose minimal division-free algorithms are all bilinear

In this paper we prove that for a certain class of systems of bilinear forms, all minimal division-free algorithms are essentially bilinear. This class includes systems for computing products in finite algebraic extension fields, and systems for computing the products of Toeplitz and Hankel matrices with vectors. Our results, together with the classification theorems of 10., 12., 169–180) completely describe all minimal division-free algorithms for computing these systems. We also prove, as an immediate consequence of our results, that the multiplicative complexity of the quaternion product over a real field is 8.     

Convex polytopes whose projection bodies and difference sets are polars

In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d≥2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inR ^d (d≥3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.     

On polynomial sets of classS k,and sets of rankk

Summary Al-Salam and Verma introduced the class S^k of polynomial sets, and characterized such sets by their generating functions. Here we define a class V^k of sets of rank k by means of recurrence relations, and show that S^k ⊂ V^k; and we characterize those sets of S^k, and of V^k, that are orthogonal. Entrata in Redazione il 24 giugno 1977.