This paper gives simple proofs of two theorems of Steck concerning the distribution of sample distribution functions.

The purpose of this paper is to introduce a nonlinear scalarisation function for solving a class of implicit vector equilibrium problems. We prove a scalarisation lemma to show the relation between the implicit vector equilibrium problem and the nonlinear scalarisation function. Then we derive some new existence theorems for solutions of implicit vector equilibrium problems, using the scalarisation lemma and the FKKM theorem.

Let Cφ be a composition operator on L2(λ), where λ is a σ-finite measure on a set X. Conditions under which Cφ is invertible, unitary, and normal are investigated in this paper.

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

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.

Rolewicz' drop property is a modification of a concept underlying Daneš' drop theorem. We characterise the drop property by the upper semicontinuity and compact valued property of the duality mapping for the dual. The characterisation suggests that we define a weak drop property which we show characterises the reflexivity of the space.

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

For an ideal I in a commutative ring R we consider the ideal In = ({in | i ∈I}). We show that if n! is a unit in R, then In = In. We give an example of a doubly generated ideal I with Is not finitely generated.

Let $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

We discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

In this article we show that finite loops with nilpotent inner mapping groups are centrally nilpotent.

Necessary conditions of the Fritz John type are given for a class of nonlinear programming problems over polyhedral cones in finite dimensional complex space.

Consider the problem to

where S is a polyhedral cone in, and Cm and f: C2n → C, g : C2n → Cm are differentiable functions. A necessary condition for a feasible point z0 to be optimal is that there exist τ≥0, ν ∈ S*, (τ, ν) ≠ 0, such that

and

We prove an existence theorem for differential inclusions in Banach spaces. Here {A (t): t ∈ [0,T]} is a family of linear operators generating a continuous evolution operator K (t, s). We concentrate on maps F with F (t,·) weakly sequentially hemi-continuous.

Moreover, we show a compactness of the set of all integral solutions of the above problem. These results are also applied to a semilinear optimal control problem. Some corollaries, important in the theory of optimal control, are given too. We extend in several ways theorems existing in the literature.

We show the existence of a convex representation of a maximal monotone operator by a convex function which is invariant with respect to the Fenchel conjugacy (up to an interchange of variables). We use the framework of generalized convexity.

Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight $n$. He also proved that the generating function for $f(n), $ the number of 1-shell totally symmetric plane partitions of weight $n$, is given by

$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$

$f(n)$

In this note some results concerning the equidistant set E(−x, x) and the kernel Mθ of the metric projection PM, where M is a Chebyshev subspace of a normed linear space X, have been obtained. In particular, when X = lp (1 < p < ∞), it has been proved that every equidistant set is closed in the bw-topology of the space. In c0 no equidistant set has this property.

It is proved that an orthocomplemented poset P is an orthomodular lattice if and only if it admits a suitable defined set of order preserving maps. These maps are called projections. They are, in fact, Just the projections of the Baer *-semigroup associated with the orthomodular lattice.

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.

If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.

Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

In this paper we investigate identities related to centralisers in rings and algebras. We prove, for example, the following result. Let A be a semisimple H* -algebra and let T: A → A be an additive mapping satisfying the relation T(xm+n+1) = xmT(x)xn for all x ∈ A and some integers m ≥ 1, n ≥ 1. In this case T is a left and a right centraliser.

Necessary and sufficient conditions on an arbitrary Gabriel filter of left ideals of a ring R are determined in order that the ring of quotients of R with respect to the filter be semi-simple artinian. Special instances include generalizations of earlier work on classical rings of quotients and maximal rings of quotients.