This paper centres around the variety 0 of distributive Ockham algebras, and those subvarieties of 0 which are generated by a single finite subdirectly irreducible algebra A. We use H.A. Priestley's duality for bounded distributive lattices throughout. First, intrinsic descriptions of the duals of certain finite subdirectly irreducibles are given; these are later used to determine projectives in the dual categories. Next, left adjoints to the forgetful functors from 0 and Var(A) into bounded distributive lattices are obtained, thereby allowing us to describe all free algebras and coproducts of arbitrary algebras. Finally, by applying the duality, we characterize injectivity in Var(A) for each finite subdirectly irreducible algebra A.

We show that the space of smooth maps from a compact smooth manifold into another smooth manifold can be endowed with the structure of a smooth manifold if we use the Γ-differentiation of Yamamuro. We then generalise the Smale Density Theorem to mappings between these manifolds.

A new proof is given of the following result: Let m and d be positive integers, and let a set of md + m − d points be given in d-dimensional space. Then the set can be partitioned into m sets such that the m convex polytopes spanned by the sets have a non-empty intersection.

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

The purpose of this paper is to give a general theorem on the absolute Riesz summability factor of Fourier series which implies Matsumoto's Theorem [Tôhoku Math. J. 8 (1956), 114–124] and to deduce some results from the theorem.

Some methods are given for constructing regular r-valent r-connected non-hamiltonian graphs; often the graphs are also non-r-edge-colorable. The extent of the class of such graphs constructible from these methods and previous methods is discussed.

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

The motion of a slender body parallel and very close to a flat interface which separates two immiscible liquids of differing density and viscosity is considered for very small Reynolds numbers. Approximate analytical expressions are obtained for the distribution of forces acting on the slender body. The limiting case of a rigid plane wall yields results obtained previously.

Arbitrary Prandtl number convection in a layer with nearly insulating boundaries is investigated. For B1/3 > 3.95P (P is the Prandtl number and B being the ratio between the thermal conductivities of the boundary and of the fluid) two-dimensional rolls are stable. The heat transported by the stable rolls reaches its peak at a critical P = B1/3/5.77 beyond which the heat flux decreases with increasing P. For B1/3 = 3.95P rolls become unstable to disturbances in the form of rolls oriented at a right angle to the original rolls. For B1/3 > 3.95P square pattern convection represents the preferred stable convection. The stable square pattern transports the maximum amount of heat at a critical P = B1/3/3.7.

We shall show necessary and sufficient conditions for which closed set X in the complex plane is a spectral set of an operator T on a complex Hilbert space.

The following theorem proved in this paper is a generalization of a result of Jakob Nielsen. Suppose G is a group of linear fractional transformations acting on the unit disc D in the complex plane; suppose also that each element of G, except the identity, is either a hyperbolic or a parabolic transformation. Then any homeomorphism h of the open disc onto itself which satisfies the functional equation hg = g′h, for some automorphism g → g′ of G, has a unique extension to a homeomorphism of D onto itself.

In a natural way, associated with each rooted tree there exists a pair of two-person games, each game possessing the root as initial position. When a rooted tree is selected at random from the set of all rooted trees which possess {1, 2, …, n} as vertex set, the number of winning moves available to the first player to move in each of the associated games is a random variable. For fixed n, we determine the distribution of this random variable. As an immediate consequence, we find the probability that the first player to move has no winning move at all. The “saddlepoint method” is applied to a certain contour integral to obtain the asymptotic distribution of the number of winning moves as n → ∞.

There are several geometric characterizations of inner product spaces amongst the normed linear spaces. Mahlon M. Day's refinement “rhombi suffice as well as parallelograms”, of P. Jordan and J. von Neumann parallelogram law is well known. There are some characterizations which employ various notions of orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality then the space is an inner product space; geometrically it means that if the diagonals of a rectangle, with sides perpendicular in Birkhoff-James sense, are equal then the space is an inner product space. In the main result of this note we improve upon this characterization and show that here unit squares suffice as well as rectangles.

We generalise the standard notion of transversality to maps between manifolds modelled on locally convex spaces and prove that all standard transversal theorems remain valid for this generalised notion.

As first applications, we use our generalised Transversal Density Theorem to prove several simple “generic” results for local smooth maps which parallel the usual ones.

If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

We study the set of points at which two algebraically independent meromorphic functions have algebraic coefficients in their Laurent expansions. After a survey of the present knowledge in this field, we obtain two general transcendence criteria which sharpen previous results of Straus, Schneider and Lang. As a corollary, we give a new proof, based on Gel'fond's method, of some of Siegel's results on E-functions.

It is shown that, under certain conditions, a transformation group is pointwise equicontinuous if and only if it is P-equicontinuous for some replete semigroup P in the phase group.

An operator T on a Hilbert space is in the class of k-quasihyponormal operators Q(k), if T*k(T*T−TT*)Tk ≥ 0. It is shown that if T is in Q(k) and S is normal such that TX = XS, where X is one to one with dense range, then T is normal; and is unitarily equivalent to S. It is proved that S can be replaced by a cohyponormal operator, if T in Q(1) is one to one. It is also shown that two quasisimilar operators in Q(k) have equal spectra, and every reductive operator quasisimilar to a normal operator is normal.

A class of groups is said to have the small embedding property if every amalgam of two groups from amalgamating a normal subgroup embeds into a group not generating the variety of all groups. Very few large classes have the small embedding property. For example, the minimal non-abelian varieties containing non-abelian finite groups do not; neither does any class of groups containing ; but the set of finitely generated groups in a nilpotent variety not containing does have the small embedding property.

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.