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.

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.

Roughly speaking, a system of linear differential equations has an exponential dichotomy if it has a subspace of solutions shrinking exponentially and a complementary subspace of solutions growing exponentially. In the case of constant coefficients, this happens if and only if the eigenvalues of the coefficient matrix have nonzero real parts. In the general case, Lazer has shown that if the coefficient matrix function is bounded and satisfies a diagonal dominance condition (which, in the constant case, is a sufficient but not necessary condition that the eigenvalues have nonzero real parts) then the system has an exponential dichotomy. In this paper we prove the same result with a weaker diagonal dominance condition, thus generalizing a theorem of Nakajima.

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.

The aim of the present note is to give the degree of approximation by Szász operators.

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

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.

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx. 5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$. We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

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.

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

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

$f(n)$

An integral representation formula in terms of the normal Gauss map for minimal surfaces in 3-dimensional solvable Lie groups with left invariant metric is obtained.

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.

We prove that (a) if R is a commutative coherent ring, the weak global dimension of R equals the supremum of the flat (or (FP–)injective) dimensions of the simple R-modules; (b) if R is right semi-artinian, the weak (respectively, the right) global dimension of R equals the supremum of the flat (respectively, projective) dimensions of the simple right R-modules; (c) if R is right semi-artinian and right coherent, the weak global dimension of R equals the supremum of the FP-injective dimensions of the simple right R-modules.

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

The aim of this paper is to give an inversion theorem for set-valued maps involving both some known results for functions and set-valued maps. To do this we introduce a notion of strict differentiability for set-valued maps and we use a Newton like method assuming the derivative to be surjective. Moreover we prove the pseudo-Lipschitz regularity of the inverse.

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.

Dedicated to George Szekeres on the occasion of his 90th Birthday

In 1990, new statistics on partitions (called cransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such that

and is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

Let ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

In this note we give external characterizations of absolutely topologicai functors U: A →X (where A may be a large category) as functors which are injective with respect to all functors based on the category X, or equivalently, which do not admit essential extensions over X. From this we deduce several characterizations and properties of the MacNeille completion of faithful functors. These results generalize the work of Horst Herriich [Math. Z. 150 (1976), 101–110] in case of small categories, where the crucial point of this generalization lies in the fact that we cannot make use of the existence of MacNeille completions and we therefore have to introduce a local-initial-completion-construction.