Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords bisected by 0, it is shown that A(K) ≤ 4. Also, if K has three such chords, A(K) ≤ 4.5. The results are generalised to any lattice in the plane.

We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$ .

In this paper we extend some estimates of the right-hand side of a Hermite–Hadamard type inequality for functions whose derivatives’ absolute values are P-convex. Applications to the trapezoidal formula and special means are introduced.

In this paper we study optimality conditions for an efficient solution in various senses of a general multiobjective optimisation problem in abstract spaces. We utilise properties of the Clarke's generalised differential and properties of a conesubconvexlike function to derive a few necessary and/or sufficient conditions for a feasible solution to be a weak minimum (a minimum, a strong minimum or a proper minimum) of the vector optimisation problem. The results in this paper are extensions and refinements of some known results in vector optimisation.

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

Drensky and Lakatos (Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181–188) have established a convenient property of certain ideals in polynomial quotient rings, which can now be used to determine error-correcting capabilities of combined multiple classifiers following a standard approach explained in the well-known monograph by Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005)). We strengthen and generalise the result of Drensky and Lakatos by demonstrating that the corresponding nice property remains valid in a much larger variety of constructions and applies to more general types of ideals. Examples show that our theorems do not extend to larger classes of ring constructions and cannot be simplified or generalised.

For an abstract mathematical programming problem involving quasidifferentiable cone-constraints we obtain necessary (and sufficient) optimality conditions of the Kuhn-Tucker type without recourse to a constraint qualification. This extends the known results to the non-differentiable setting. To obtain these results we derive several simple conditions connecting various concepts in generalized convexity not requiring differentiability of the functions involved.

In this paper we discuss existence of positive solutions to a general nonlinear elliptic system of reaction-diffusion equations representing a predator-prey or competition model of interaction between two species, in a heterogeneous environment. We also consider homogeneous Dirichlet and/or Robin boundary conditions. In the predator-prey case we give necessary and sufficient conditions for the existence of positive solutions, while in the competition case we give sufficient conditions. We use index theory in a positive cone to attack our problem and characterise our results by the sign of the first eigenvalues of certain Schrodinger type operators.

Some structure theorems for totally disconnected groups are described. These theorems produce a certain positive integer valued function, called the scale function, on each totally disconnected group. The scale function has properties connected to the structure of the group and can be used to prove some conjectures of Hofmann and Mukherjea.

This paper discusses rational edged tetrahedra, in 3, 4 and n dimensions, with rational volume. The main results are (i) a proof of the existence of infinitely many tetrahedra with rational edge-lengths, face-areas and volume and (ii) a proof that there exist dimensions for which all regular hypertetrahedra with rational edge-lengths have rational hypervolume.

There are two distinct approaches in the literature to framing a version of the law of the iterated logarithm for martingales. One involves norming by constants, using the martingale variance and the other involves norming by random variables, using the sums of conditional variances of the increments, given their past. In this paper a portmanteau approach is provided, still based on the Skorokhod representation of the martingale, but involving normalization by more general random variables. This extends the functional forms of all the previously existing results.

We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra.

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then for any positive integer k.

Approximate methods are developed for modelling the growth and collapse of clouds of cavitation bubbles near an infinite and semi-infinite rigid boundary, a cylinder, between two flat plates and in corners and near edges formed by planar boundaries. Where appropriate, comparisons are made between this approximate method and the more accurate boundary integral methods used in earlier calculations. It is found that the influence of nearby bubbles can be more important than the presence of boundaries. In confined geometries, such as a cylinder, or a cloud of bubbles, the effect of the volume change due to growth or collapse of the bubble can be important at much larger distances. The method provides valuable insight into bubble cloud phenomena.

Let G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix, and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.

The basic elementary results about convex sets are derived successively from various properties of segments. The complete set of properties is shown to form a natural set of axioms characterising the convex sets in a real vector space.