The object of this paper is to establish some relations between two generalized Nörlund methods and also between two absolute generalized Nörlund methods. Our theorems obtained here generalize many known results, including McFadden's Theorems which state the inclusion relations between two absolute Nörlund methods, and results of Ikuko Kayashima.

Spaces with homogeneous norms are closely related to the Beppo Levi spaces of Deny and Lions, to spaces of Riesz potentials, and to Sobolev spaces. In this paper we survey the literature on them, give a broad extension of their definition, and present their basic theory. Many of the properties of Sobolev spaces have their analogues. In fact, the two families are locally equivalent. Spaces with homogeneous norms are especially suited to the study of boundary value problems on for homogeneous elliptic operators with constant coefficients. We will use them extensively in a forthcoming paper to study elliptic partial differential equations with mixed boundary conditions on a smoothly bounded domain.

Expanding maps of the interval with unique turning points have periodic points of period 2n · 3 for some n and therefore are chaotic.

Generating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

We study linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$ when $q$ is an odd prime power. When $q^{\unicode[STIX]{x1D6FF}}+1$ is divisible by $n$, we obtain an exact count of linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$. For certain values of $n$ and $q$ and assuming Artin’s conjecture for primitive roots, we show that the relative distance of these codes satisfies a modified Gilbert–Varshamov bound.

By using the method of differential subordinations, we derive, among other results, some criteria for univalency in the unit disc.

Sufficient conditions are derived for all bounded solutions of a class of linear systems of differential difference equations to be oscillatory.

A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\setminus X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, Mrówka and Tsai [‘On local topological properties. II’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.19 (1971), 1035–1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.

In this paper we discuss the simplicity criteria of (−1,−1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε,δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1,δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.

The solution of the linear boundary layer problem is presented as a sum of the reduced solution and the layer function, which is approximated by a truncated orthogonal series. The layer subinterval is determined through the numerical layer length, which depends on the perturbation parameter and the degree of the spectral approximation. Modified Hermite polynomials are used as an orthogonal basis. Systematic methods are presented to obtain the recurrence relations for the coefficients in modified Hermite series solution of linear differential equations with polynomial coefficients. The upper bound for the error function is constructed and the numerical example is included.

Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k) with k > 1, . For a = 1 this equation reduces to the Erdös-Moser equation and the result to a result of Moser. Our method can also be used to deal with variants of the equation of the title, and two examples will be given. For one of them there are no integer solutions with

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form

where x ∈ Rn, ε is a small parameter,

Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

This paper gives a characterization of nonexpansive mappings from the unit sphere of ℓβ (Γ) onto the unit sphere of ℓβ (Δ) where 0<β≤1. By this result, we prove that such mappings are in fact isometries and give an affirmative answer to Tingley’s problem in ℓβ (Γ) spaces. We also show that the same result holds for expansive mappings between unit spheres of ℓβ (Γ) spaces without the surjectivity assumption.

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

In this paper we consider the situation that a group G has a subgroup H which is a dihedral 2-group and with connected transversals A and B in G. We show that G is then solvable and moreover, if G is generated by the set A ∪ B, then H is subnormal in G. We apply these results to loop theory and it follows that if the inner mapping group of a loop Q is a dihedral 2-group then Q is centrally nilpotent.

Two systems of queues are said to be isomorphic in the restricted sense if at least one of their output elements, for example waiting time, queue size, idle time, inter-departure time, and so on, has exactly or approximately the same probability distribution function, irrespective of differences in their input elements, for example, inter-arrival time, service time, and so on. This property has been used in deriving approximate solutions to systems which do not have exact solutions. The paper highlights the systems approach in queueing problems and its practical usefulness.

Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such family of presentations proposed by B.H. Neumann. We show that the problems are much easier than they first appear, albeit at the expense of considerable preliminary ‘experimentation’. This demonstrates how far the range of applicability of coset enumeration has improved.

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

Let p denote a prime, G a. finite soluble p'-group and A an elementary abelian p-group of operators on G. Suppose that A has order p4 and that if ω ∈ A# then CG(ω) has nilpotent derived group. Then G has nilpotent derived group.

We present a generalisation of the quasilinearisation method to obtain a monotone sequence of approximate solutions that converges quadratically to a solution of a nonlinear ordinary differential equation of order n ≥ 1.