Results are formulated about the image and the kernel of the kth iterate fk of a function f : A → A. In this way, an extremely general version of Fitting's classical lemma is obtained. Two applications are presented: the first is a characterization of strongly π-regular rings, while the second is a “lattice theoretical Fitting lemma”.

The cosine transforms of functions on the unit sphere play an important role in convex geometry, Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting mathematical object useful for applications. More general integral transforms are introduced that reveal distinctive features of higher-rank objects in full generality. These new transforms are called the composite cosine transforms, by taking into account that their kernels agree with the composite power function of the cone of positive definite symmetric matrices. It is shown that injectivity of the composite cosine transforms can be studied using standard tools of the Fourier analysis on matrix spaces. In the framework of this approach, associated generalized zeta integrals are introduced and new simple proofs given to the relevant functional relations. The technique is based on application of the higher-rank Radon transform on matrix spaces.

One of the main open problems in the theory of Asplund spaces is whether every Asplund space admits a Fréchet differentiable bump function. This problem is also open for C(K) Asplund spaces, where it is unknown even for C∞-Fréchet smooth bump (a general Asplund space does not always admit C2-Fréchet smooth bump – it suffices to consider ℓ3/2[DGZ2]).

Suppose that {tn} is the sequence of positive roots of ζ (½ + it) counted according to multiplicity and arranged in non-decreasing order; in my paper [6] I proved that

and my main objective here is to improve this bound.

Two results are proved involving the quantitative illumination parameter B(d) of the unit ball of a d-dimensional normed space introduced by Bezdek (1992). The first is that B(d) = O(2dd2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) that of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. Morgan (1992) conjectured that s(d) ≤ 2d, and Cieslik (1990) conjectured that v(d) ≤ 2(2d − 1). It is proved that s(d) ≤ v(d) ≤ B(d) which, combined with the above estimate of B(d), improves the previously best known upper bound v(d) < 3d.

For each integer n ≥ 2, let β(n) be the sum of the distinct prime divisors of n and let (x) stand for the set of composite integers n ≤ x such that n is a multiple of β(n). Upper and lower bounds are obtained for the cardinality of (x).

This is a study of relations between pure cubic fields and their normal closures. Explicit formula shows how the discriminant, regulator and class number of the normal closure can be expressed in terms of the cubic field.

In this paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem [5] is established for a single linear form with a “slowly decreasing” error function. To explain this in more detail, some notation is introduced. Throughout this paper, m, n are positive integers; i.e., m, n ∈ ℕ; x = (x1,…, xn) will denote a point or vector in ℝn, q = (q1,…, qn) will denote a non-zero vector in ℤn and

|x| := max{|x1|,…,|xn|} = ‖X‖∞

will denote the height of the vector x. Let Ψ : ℕ → (0, ∞) be a (non-zero) function which converges to 0 at ∞. The notion of a slowly decreasing functionΨ is defined in [3] as a function for which, given c ∈ (0, 1), there exists a K = K(c) > 1 such that Ψ(ck) ≤ KΨ(k). Of course, since Ψ is decreasing, Ψ(k) ≤ Ψ(ck). For any set X, |X| will denote the Lebesgue measure of X (there should be no confusion with the height of a vector).

The Ehrhart polynomials for the class of 0-symmetric convex lattice polytopes in Euclidean n-space ℝn are investigated. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima of such polytopes are closely related by their geometric and arithmetic means. It is also shown that the roots of the Ehrhart polynomials of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, the structure of the roots in the planar case is studied. Here it turns out that their distribution reflects basic properties of lattice polygons.

In this paper is considered the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over ℚ with ℚ-torsion group ℤ2 × ℤ2. The existence is shown of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

In this paper, an improvement of a large sieve type inequality in high dimensions is presented, and its implications on a related problem are discussed.

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein.

Let (X, ρ) be a complete metric space and let ℱ = {T1,…, Tn} be a finite family of self-maps of X. Suppose that there is a constant γ ∈ (0, 1) such that, for any x, y ∈ X, there exists T ∈ ℱ with ρ(T(x), T(y)) ≤ γρ(x, y). Then some composition of members of ℱ has a fixed point.

In this paper this conjecture is disproved, We also show that it does hold for a (continuous) commuting ℱ in the case n = 2. It is conjectured that it holds for commuting ℱ for any n.

Equifacetal simplices, all of whose codimension one faces are congruent to one another, are studied. It is shown that the isometry group of such a simplex acts transitively on its set of vertices and, as an application, equifacetal simplices are shown to have unique centres. It is conjectured that a simplex with a unique centre must be equifacetal. The notion of the combinatorial type of an equifacetal simplex is introduced and analysed, and all possible combinatorial types of equifacetal simplices are constructed in even dimensions.

Linear stability of an incompressible triple-deck flow over a wall roughness is considered for disturbances of high frequency. The wall roughness consists of two relatively short obstacles placed far apart on an otherwise flat surface. It is shown that the flow is unstable to feedback or global mode disturbances. The feedback loop is formed by algebraically decaying disturbances propagating upstream and weakly growing Tollmien-Schlichting waves travelling downstream and as such represents an interaction between modes from continuous and discrete spectra of the corresponding parallel-flow problem. An example of growth rate calculation for a specific roughness is considered.

This paper treats finite lattice packings Cn + K of n copies of some centrally symmetric convex body K in Ed for large n. Assume that Cn is a subset of a lattice Λ, and ϱ is at least the covering radius; namely, Λ + ϱK covers the space. The parametric density δ(Cn, ϱ) is defined by δ(Cn, ϱ) = n · V(K)/V(convCn + ϱK). It is shown that, if δ(Cn, ϱ) is minimal for n large, then the shape of conv Cn is approximately given by Wulff's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs–Curie energy. It is also proved that, in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wulff-shape associated to some densest packing lattice of K.

The dimension of a graph, that is, the dimension of its incidence poset, has become a major bridge between posets and graphs. Although allowing a nice characterization of planarity, this dimension behaves badly with respect to homomorphisms.

We introduce the universal dimension of a graph G as the maximum dimension of a graph having a homomorphism to G. The universal dimension, which is clearly homomorphism monotone, is related to the existence of some balanced bicolouration of the vertices with respect to some realizer.

Nontrivial new results related to the original graph dimension are subsequently deduced from our study of universal dimension, including chromatic properties, extremal properties and a disproof of two conjectures of Felsner and Trotter.

The question of the maximum number $\mbox{ex}(m,n,C_{2k})$ of edges in an m by n bipartite graph without a cycle of length 2k is addressed in this note. For each $k \geq 2$, it is shown that $\mbox{ex}(m,n,C_{2k}) \leq \begin{cases} (2k-3)\bigl[(mn)^{\frac{k+1}{2k}} + m + n\bigr] & \mbox{ if }k \mbox{ is odd,}\\[2pt] (2k-3)\bigl[m^{\frac{k+2}{2k}}\, n^{\frac{1}{2}} + m + n\bigr] & \mbox{ if }k \mbox{ is even.}\\ \end{cases}$

A solid diagram of volume n is a packing of n unit cubes into a corner so that the heights of vertical stacks of cubes do not increase in either of two horizontal directions away from the corner. An asymptotic distribution of the dimensions – heights, depths, and widths – of the diagram chosen uniformly at random among all such diagrams is studied. For each k, the planar base of k tallest stacks is shown to be Plancherel distributed in the limit $n\to\infty$.

Let $P(n)$ and $C(n)$ denote, respectively, the maximum possible numbers of Hamiltonian paths and Hamiltonian cycles in a tournament on n vertices. The study of $P(n)$ was suggested by Szele [14], who showed in an early application of the probabilistic method that $P(n) \geq n!2^{-n+1}$, and conjectured that $\lim ( {P(n)}/ {n!} )^{1/n}= 1/2.$ This was proved by Alon [2], who observed that the conjecture follows from a suitable bound on $C(n)$, and showed $C(n) <O(n^{3/2}(n-1)!2^{-n}).$ Here we improve this to $C(n)<O\big(n^{3/2-\xi}(n-1)!2^{-n}\big),$ with $\xi = 0.2507$… Our approach is mainly based on entropy considerations.