Let SΓ be a vector space graph. A graphic subspace of SΓ need not be a direct summand with a graphic complement. A necessary and sufficient condition for the existence of a graphic complement is given. Also, it is shown that every graphic subspace possesses an o-special basis which extends to an o-special basis of SΓ.

A Radon–Nikodým theorem for operator-valued measures is applied to deduce the existence and uniqueness of conditional expectations in certain cases.

A chain rule is given for differentiating a multivariate function of a multivariate function. In the univariate case this chain rule reduces to Faa de Bruno's formula.

Using this, a simple procedure is given to obtain the rth order multivariate Hermite polynomial from the rth order univariate Hermite polynomial.

We give examples of finite groups of odd prime power order in which the commutators lying in the centre do not generate the intersection of the centre and the commutator subgroup.

Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = < X, Y, Z | Xk = Yl = Zm = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

We show that a certain local pinching condition on a compact minimal submanifold in a sphere leads to a condition on the codimension of the immersion.

A theorem on Nemitsky operators between Sobolev spaces, contained in a 1973 paper of Marcus and Mizel, is extended to Orlicz-Sobolev spaces.

In this paper we give necessary and sufficient conditions for exponential dichotomy of a general class of strongly continuous semigroups of operators defined on a Banach space. As a particular case we obtain a Datko theorem for exponential stability of a strongly continuous semigroup of class C0 defined on a Banach space.

Sufficient conditions are derived for the solutions of a system of linear delay-differential equations to behave asymptotically as those of the same system without the argument delays. A generalisation of the result to integrodifferential systems is indicated.

A survey over three different approaches to model hysteresis phenomena is given. The starting point for mathematical models are

(a) memory,

(b) population, or

(c) spatial distribution.

In any case the hysteresis effect is interpreted using an averaging procedure over the inner state of the system in consideration.

If R is (n(n−1)/2)-torsion free ring with 1 and satisfies the identity (xy)n = xnyn, then R is commutative provided that n = 4k.

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauder's techniques and coincidence degree.

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

Let Z(K) denote the center of a ring K. A ring R is compressible if Z(eRe) = eZ(R) for each idempotent e of R. In response to a question of S. Berberian, G. Bergman has constructed a (non-commutative) integral domain, satisfying a polynomial identity, for which the 2×2 matrix ring over the domain is not compressible. In contrast to Bergman's example, we show that the ring of nxn matrices over any commutative ring is always compressible.

The Schwarz-Pick lemma,

for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:

where Φ(z, r) is a quantity determined by the non-Euclidean area of the image of

and ψ(z, r) is that determined by the non-Euclidean length of the image of the boundary of D(z, r). The multiplicities in both images by f are not counted.

Let α be an algebraic integer of degree n > 1 with conjugates α = α1, α2, …, αn. Let D(α) denote the diameter of {α1, …, αn} and diam . In 1929 Favard showed that diam and that, when n = 2 or 3, the minimal values of diam (α) are less than 2. The author shows that diam and He also finds all algebraic integers a with diam (α) ≤ 2 when n ≤ 5. Similar results are found for D(α) and, in particular, D(α) > 2 when n = 5.

We consider a certain real hypersurface M of a complex projective space. The purpose of this paper is to characterize M in terms of Ricci curvatures.

Criteria for an analytic function f defined in |z| < 1 to belong to B0, the class of Bloch functions satisfying , and criteria for a meromorphic function g defined in |z| < 1 to belong to N0, namely, to satisfy are obtained in terms of the area and the length of the images of hyperbolic disks and hyperbolic circles, respectively.

A result by the author on the elimination of randomization (or relaxation) for variational problems is partially extended and then used to obtain a very general result on the denseness of the set of original control functions in the set of relaxed control functions. Also, a slight extension of Aumann's theorem on the integrals of multifunctions is shown to follow directly from the elimination result.

Let A ⊂ C(S) be a function algebra. In this paper we prove that is a weak peak set for A if and only if for any open neighbourhood U of S0 there is an f in A such that ∥f∥ ≤ 2, |f(x) − l| ≤ 1/3 on S0 and |f(x)| ≤ 1/3 on S\U.