It is shown that the problem of deciding whether a curve in a finite surface complex is isotopic to a point is NP-complete. This contrasts with the recursively enumerable (RE)-completeness of the corresponding homotopy problem, and exhibits surface complexes as a common framework for NP-complete and RE-complete algorithmic problems.

A simple formula for the inverse of a block matrix with non-zero blocks in the principal diagonal and the first sub-diagonal only is proved. The matrix had arisen in an investigation of a difference equation.

This paper furthers the foundation of the theory of quasigroups obeying the law x(xy) = yx by studying their algebraic properties. Much information is obtained by analysing the cycle decomposition of left translations regarded as permutations, and other results are obtained by representation in terms of abelian groups with an operation.

An abelian group G is said to be subdirectly irreducible if there exists a subdirectly irreducible ring R with additive group G. If G is subdirectly irreducible, and if every ring R with additive group G, and R2 ≠ 0, is subdirectly irreducible, then G is said to be strongly subdirectly irreducible. The torsion, and torsion free, subdirectly irreducible and strongly subdirectly irreducible groups are classified completely. Results are also obtained concerning mixed subdirectly irreducible and strongly subdirectly irreducible groups.

In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equation

with initial conditions

where

and |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).

These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0

This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).

This paper is a survey of the current state of knowledge on groups of exponent 8. It contains a report on a first stage of an attempt to answer the Burnside questions for these groups.

A theorem of Sharkovsky on the coexistence of cycles for one-dimensional difference equations is generalized to a class of difference equations of arbitary dimension. The mappings defining these difference equations are such that the ith component depends only on the first i independent variables.

This paper gives a detailed classification of the indecomposable representations of one non-trivial “commutative quiver”, gives brief details of certain generalisations, and uses methods likely to be more widely applicable.

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

The category of groups forms a full subcategory of the category of hypergroups. This larger category also contains other group theoretic objects, such as the conjugacy class hypergroup and character hypergroup of a finite group.

Definitions of hypergroups and hypergroup morphisms are given and related to double algebras (which are simultaneously algebras and cogebras, but not Hopf algebras in general). Quotient and orbit hypergroups are defined. Coefficients are allowed in a more or less arbitrary field.

These concepts provide a new language in which groups and their character tables can be fruitfully discussed.

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

Several fixed point theorems for nonexpansive self mappings in metric spaces and in uniform spaces are known. In this context the concept of orbital diameters in a metric space was introduced by Belluce and Kirk. The concept of normal structure was utilized earlier by Brodskiĭ and Mil'man. In the present paper, both these concepts have been extended to obtain definitions of β-orbital diameter and β-normal structure in a uniform space having β as base for the uniformity. The closed symmetric neighbourhoods of zero in a locally convex space determine a base β of a compatible uniformity. For 3-nonexpansive self mappings of a locally convex space, fixed point theorems have been obtained using the concepts of β-orbital diameter and β-normal structure. These theorems generalise certain theorems of Belluce and Kirk.

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.

Schneider (Math. Nachr. 60 (1974), 167–180) has established the following result. Consider the mixed type equation

in G ⊂ R2 which is a simply connected region, bounded for y > 0 by a piece-wise smooth curve Γ0 connecting the points A(0, 0) and B(1, 0), and for y < 0 by the solutions of k(y).(dy)2 + (dx)2 = 0 which meet at the point G(½, yc), such that for ,

S(x, y) = F(yy) + 8λ·(k/k′)2 > 0 in Ḡ ∩ {y < 0}, “Schneider's Condition”, where F(y) = 1 + 2(k/k′)′, and such that S = S(x, y) is integrable in G2, “Frankl's Condition”. Then the Tricomi Problem (T): L[u] = f with has a weak solution u ∈ L2(Ḡ) and the Adjoint Tricomi Problem (T+): L+[w] = f with has at most one semistrong solution.

In this present paper we get the above result of Schneider in a much more generalized way, so that here our uniqueness theorem and existence results include cases where S(x, y) may be negative in G2.

The paper presents a necessary and sufficient condition for a given compact totally disconnected space C to be the projective limit of a given directed cone of epimorphisms onto finite discrete quotients of C. This problem is related to the question of when a compact totally disconnected algebra is profinite and some observations in this direction are recorded.

A duality theorem, and a converse duality theorem, are proved for a nonlinear fractional program, where the numerator of the objective function involves a concave function, not necessarily differentiable, and also the support function of a convex set, and the denominator involves a convex function, and the support function of a convex set. Various known results are deduced as special cases.

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

It is shown that the inequality

holds for an infinity of integers p, q and that here the factor ½ may not be replaced by a smaller number. Corresponding best possible inequalities are given for the numbers e±2/t, where t is a positive integer.

Let S be a closed convex cone in Cn, S* the polar cone, g a continuous map from Cn into itself, and e a fixed vector in S*. In this paper we prove that there is a connected set T in S of stationary points of (Dr (e), g) where Dr (e) is the set of all x in S with re(e, x) ≤ r. This extends the results of Lemke and Eaves to the complex nonlinear case and arbitrary closed convex cones in Cn. We show that if g is strictly monotone on S, then T is both unique as well as arcwise connected. This partly solves the open problems raised by Eaves in this more general setting. We also show that if x is a stationary point of (Dr (e), g) and re(e, x) < r then x is a stationary point of (S, g).