This expository article on Shape Theory contains the main concepts of this theory with a formulation of the most important results of this theory and also with some open problems. The proofs are omitted, however the article gives references to the hooks and papers, in which the reader can find the proofs.

For simplicity, we formulate several theorems only in the form which clearly gives their geometric sense, even if there are known more general results with rather complicated formulations. The reader more interested in details, can find them in the original papers quoted in the list of references.

This note shows that the variety generated by Murskii's 3-element algebra contains an infinite descending chain of subvarieties, thus falsifying a conjecture made previously by the author and M.R. Vaughan-Lee.

It is shown that the functional fails to possess a variational derivative, contrary to what is claimed by Gelfand and Fomin. A modified definition is given with respect to which the functional does possess a variational derivative.

In various papers on some of the theoretical aspects of computing, the reader is confronted with a mathematical entity called a “multiset”, and is told that a multiset is “an unordered collection of elements that may have repeated occurrences of identical elements” (see, for example, Nachum Dershowitz and Zohar Manna, “Proving termination with multiset orderings”, Lecture Notes in Computer Science, 71). These entities are then manipulated in accordance with the classical laws of set algebra. The purpose of this note is to present a formal foundation for this concept of multiset, and to show by means of examples that if this foundation is adopted then, although certain sections of classical set algebra can be applied to multisets, it certainly cannot be applied in total.

In this paper, smoothness properties of solutions of the initial-Dirichlet problem for parabolic equations in regions with edges are considered. We obtain bounds for solutions and derivatives, and prove the Hölder continuity of the first derivatives and of the second derivatives multiplied by a suitable power of the distance from the edges.

The object of this paper is to establish some inclusion relations between two generalized Nörlund methods of summability. Our results are generalizations of Das's Theorem and one of them is also a generalization of a theorem of the author. They include the well-known fact that the Cesàro method (C, α) is weaker than (C, β) for β > α > 0.

Let S denote the class of functions f analytic and univalent in the open disc {z: |z| < 1} and normalized by f(0) = 0 = f′(0) − 1, and S*(α) denote the set of starlike functions of order α (0 ≤ α ≤ 1) in S. In this paper, the results of William M. Causey and William L. White [J. Math. Anal. Appl. 64 (1978), 458–466] are generalized by the following:

THEOREM 1. Let f, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and let F be defined by

where a, c ∈ N. Then F belongs to S*(β) for |z| less than a suitably chosen number.

THEOREM 2. Let F, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and f be given by **. Then f belongs to S*(β) , for |z| less than a suitably chosen number.

The relative Schoenflies theorem says that a locally flat embedding e: Sn−1 → Rn for which e−1(Rk) = Sk−1 extends to homeomorphism of the pair (Rn, Rk) provided the local collars respect Rk. In this note it is shown that the proviso is essential, at least when k = 3.

In the present paper we give a criterion for a function of Wiener's class to belong to the class of generalized absolute continuity, in terms of Fourier-Young coefficients {Ck}. More precisely, we prove the following theorem.

THEOREM. Let Λ = (λn,k) be a normal almost periodic matrix of real numbers such that λn,k ≥ λn,k+1 for all n and k. Then for any function f of Wiener's class Vν (1 < ν < 2) to be of class of generalized absolute continuity Ap (1 < p < ∞) it is necessary and sufficient that {|Ck|2} is summable Λ to zero.

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords bisected by 0, it is shown that A(K) ≤ 4. Also, if K has three such chords, A(K) ≤ 4.5. The results are generalised to any lattice in the plane.

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords through 0 of one of the following types, it is shown that A(K) ≤ 4, while if K has three of one type, A(K) ≤ 4.5. The types of chords considered are chords which partition K into two regions of equal area, chords which lie midway between parallel supporting lines of K, and chords such that K is invariant under reflection in them. The results are generalised to any lattice in the plane.

It is proved that certain rings satisfying generalized-commutator constraints of the form [xm, yn, yn, …, yn] = 0 must have nil commutator ideal.

Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X. If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M. Examples are presented to show that some of our hypotheses are needed.