It is shown that for any almost periodic linear differential system asymptotic stability and uniform stability together imply uniform asymptotic stability.

Let G be a finite group. It is shown that G is 2-closed if and only if

(a) every 2-block of G has full defect, and

(b) every Sylow 2-intersection is centralized by a Sylow 2-subgroup of G.

As a consequence it is shown that G is a TI-group if and only if every 2-block of G has either full defect or defect zero and (b) holds. This result and a theorem of Kwok yield complete characterizations of finite groups with certain relations being satisfied by every nonprincipal irreducible character.

Oscillation criteria are obtained, for the nonlinear delay differential inequality u(u″+f(t, u(t), u(g(t)))) ≤ 0. The main theorems give sufficient conditions (and in some cases sufficient and. necessary conditions) for all solutions u(t) to have arbitrary large zeros. Generalizations to more general cases are discussed.

In this paper, we study the oscillation of nth-order differential equations. Recently, Atkinson and the present authors studied (separately) the comparison properties of differential inequalities. Kartsatos treated the nth-order ordinary case and proposed several open problems.

The purpose of this paper is to answer one of them in the affirmative concerning more general functional differential equations. The result is that if under several conditions, the equation

is oscillatory for n even or a solution x(t) of (1) is oscillatory or for n odd, then this is also the case for the equation

Let G be a doubly transitive permutation group on a finite set Ω, and for α in Ω suppose that Gα has a set Σ of non-trivial blocks of imprimitivity in Ω - {α}. If Gα is 2-transitive but not faithful on Σ, when is it true that the stabiliser in Gα of a block of Σ does not act faithfully on that block (that is, there is a nontrivial element in Gα which fixes every point of the block)? In a previous paper this question was answered when is the alternating or symmetric group, or a Mathieu group in its usual representation. In this paper we answer the question when , permuting the q + 1 points of the projective line, for some prime power q. We show that the only groups which arise satisfy either

(i) PSL(3, q) ≤ G ≤ PΓL(3, q) in its natural representation, or

(ii) G is a group of collineations of an affine translation plane of order q, and contains the translation group.

Suslev's constraint multipliers are used to derive the equations of motion of dynamical systems (holonomic or nonholonomic) in the form of Poincaré-Četaev equations and in the canonical form. For holonomic systems defined by redundant variables, the constraint multipliers occuring in the canonical equations are determined and a modification of the Hamilton-Jacobi Theorem for integrating the canonical equations is presented.

We give some results on the number of G-orbits (G = GL(n, q)) on groups isogenous (in the algebraic-geometric sense) to SL(n, q). The conjectured isogeny invariance of this number is contradicted by computer calculations.

Using homological methods (and in particular the 5-term sequence of Hochschild-Serre in the homology of groups) a generalization of the theorem of John G. Thompson (J. Austral. Math. Soc. 16 (1973), 16–17) referred to in the title is proved.

A method of construction of designs with parameters v1 = r2 = p2, r1 = v2 = p + 1, b = p(p+1), k = p which may be used for the two-way elimination of heterogeneity is discussed. These designs were first studied in connection with estimating tobacco mosaic virus. Our designs have the advantage that every treatment occurs at most once in a row or column. We give the designs explicitly for p = 3, 4, 5.

Suppose T is the natural transformation

where the variables are abelian groups. We find conditions on A, B, and C that guarantee that T is injective.

A recent improvement by H. Guggenheimer of the lower bound for the product of the volumes of a pair of polar reciprocal convex bodies is used to replace one of the results of Kurt Mahler, “Polar analogues of two theorems by Minkowski”, Bull. Austral. Math. Soc. 11 (1974), 121–129 by a best possible one. A similar improvement can be made for the other theorems in my note.

The semigroup of all onto mappings of a set to itself and the semigroup of all one-to-one mappings of a set to itself are shown to have the property that every automorphism is inner.

In this note, Lp estimators for the parameters in the linear model y = Xβ are considered. In particular, it is shown that these estimators are invariant under scale transformations on the dependent variable; that is, if is an Lp estimator for β, then for any nonzero scalar a. It is shown that this result does not extend to more general transformations on y, and the invariance property does not hold for general nonlinear models.

The subclasses S*(α, β) and C*(α, β) of T, the class of analytic and univalent functions of the form have been considered. Sharp results concerning coefficients, distortion of functions belonging to S*(α, β) and C*(α, β) are determined along with a representation formula for the functions in S*(α, β). Furthermore, it is shown that the classes S*(α, β) and C*(α,.β) are closed under arithmetic mean and convex linear combinations.

The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such that

where S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S.

If a finite group G has a ‘central’ involution t whose centralizer in G is (t) × H then, under certain conditions on H, G cannot be simple.

A strengthened form of the Kantorovich convergence theorem for the modified Newton's method is proved. The result is compared with previously known results.

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.

Let p, q, u, and v be any four positive integers, and let δ be a number in the interval 0 < δ ≤ 2. In one of his papers, Kurt Mahler, Bull. Austral. Math. Soc. 10 (1974), 325–335, proved that if q satisfies the inequalities

then

In this note, by a slightly different treatment of some inequalities in Mahler's paper, we easily obtain the same result with q only restricted by the first condition.

In this note the matrix equation A + WB + BTW + WCW = 0 is considered. A monotoneity result and an inertia theorem on the location of the eigenvalues of W and B + CW are proved.