A method of analysing the general linear regression model is described, for the case where the observations are correlated. For many applications the correlations are structured, with neighbouring observations being more strongly correlated than those some distance apart in time or space. Such correlation structures may often be assumed to belong to some class of models indexed by a small number of parameters. Estimation and inference procedures which are able to cope with a wide range of correlation models, are described and the methods are applied to problems which occur in biometry.

A powerful tool in the construction of orthogonal designs has been amicable orthogonal designs. Recent results in the construction of Hadamard matrices has led to the need to find amicable orthogonal designs A, B in order n and of types (u1, U2, …, u6) and (ν1, ν2, …, νr) respectively satisfying At = -A, Bt = B, and ABt = BAt with

For simplicity, we say A, B are amicable orthogonal designs of type (u1, u2, …, us; v1, v2, …, vr).

We completely answer the question in order 8 by showing (1, 2, 2, 2; 8), (1, 2, 4; 2, 2, 4), (2, 2, 3; 2, 6), (7, 1, 7) and those designs derived from the above are the only possible.

We use our results to obtain new orthogonal designs in order 32.

We find necessary and sufficient conditions on a topological space X so that S(X), the semigroup of all continuous selfmaps of X, is isomorphic to the multiplicative semigroup of a near-ring. The analogous problem is also considered for the semigroup of all continuous selfmaps which fix some point of X.

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

A new cardinal inequality, |K (X)| ≤ 2L∗(X) · psw (X), is proved in this paper. It strengthens the result of D.K. burke and R. Hodel the |K (X)| ≤ 2e (X) · psw (X).

A variety of groups has the amalgam embedding property if every amalgam of two -groups can be embedded in a -group. In this note the author proves that if is a variety of exponent O which satisfies a law W(x1, X2,…, xt) but not W(x1, x2,… xt) then does not have the amalgam embedding property.

We prove first that a logical fraction functor from a topos to a topos must be a filter-power functor, then we prove that such functors can have adjoints only when the filter is principal. Finally we refine this so that we are able to prove that the filter-power of a Grothendieck topos is Grothendieck if and only if the filter is principal.

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

The purpose of this paper is to study the trajectory multifunction Φ(·) determined by the indefinite set valued integral of a measurable Banach space valued multifunction F(·), that is for all t ∈ [0, T], , where the set valued integral is interpreted in the sense of Aumann. We study the topological and algebraic properties of SΦ equaling the set of selectors of Φ(·) whose primitive is an integrable selector of F(·). We also determine several useful properties that Φ(·) possesses and finally we present some convergence and stability results using the Kuratowski-Mosco convergence of sets.

We give another formulation of Theorem 1 of Ribenboim's paper, Bull. Austral. Math. Soc. 2 (1970), 81–88. Our condition on a module to be totally ordered such that this order extends a given strict order seems to be more useful for applications than condition (2) of Ribenboim. This is shown for example in Corollaries 3 and It of our paper. Corollary 4 is due to Ribenboim. Corollary 3 generalizes the corresponding well known result on abelian groups.

The problem of unicity in the uniform approximation of vector-valued functions is considered. A recent result of Cheney and Wulbert concerning unicity in the uniform approximation of real-valued functions is extended and some “point-wise” criteria for unicity are given.

ABS mthods are direct iteration methods for solving linear systems where the i-th iterate satisfies the first i equations, and therefore a system on m equations is solved in at most m ABS steps. In this paper, using a new rank two update of the Abaffian matrix, we introduce a class of ABS-type methods for solving full row rank linear equations, where the i-th iterate solves the first 2i equations. So, termination is achieved in at most ⌊(m + 1)/2⌋ steps. We also show how to decrease the dimension of the Abaffian matrix by choosing appropriate parameters.

We give a complete characterisation of univalent logharmonic mappings from the domain D of such that has countable many components onto where pj is a singleton in C.

Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

Let f be a real function defined on [0, 1] × R3 and let η ∈ (0, 1). Very recently, C.P. Gupta and V. Lakshimikantham, making use of the Leray-Schauder continuation theorem and Wirtinger-type inequalities, established an existence result for the problem

(Theorem 1 and Remark 4 of [Nonlinear Anal. 16 (1991), 949–957]).

The aim of the present paper is simply to point out how, bu means of a completely different approach, it is possible to improve that result not only by requiring much general on f, but also by giving a precise pointwise estimate on x″′

Let f(z) be an n–Valued algebroid function of finite lower order. In the present paper, we give a spread relation of f(z) and some applications of the spread relation.

Let Y = {z ∈ C: z3 ∈ [0, 1]} (equipped with subspace topology of the complex space C) and let f: Y → Y be a continuous map. We show that if f is pointwise chain recurrent (that is, every point of Y is chain recurrent under f), then either f12 is the identity map or f12 is turbulent. This result is a generalisation to Y of a result of Block and Coven for pointwise chain recurrent maps of the interval.

We prove large time Gaussian bounds for the semigroup kernels associated with complex, second-order, subelliptic operators on Lie groups of polynomial growth.

In this paper we study the existence of mild solutions for Cauchy problem

We derive conditions under which the mild solutions exist, and also get the relative compactness of the solution set, which extend and improve some existing results in this area.

The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.