A simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

A linear mapping D of the algebra of polynomial functions P[0, 1] into the algebra of all continuous complex-valued functions C[0,1] is called a derivation provided D(fg) = fD(g) + gD(f) for all polynomials f and g. The derivations of P[0, 1] into C[0,1] are easily seen to be all mappings of the form Dw where w is a continuous function on [0, 1] and Dw (f) = wf' (f' denotes the ordinary derivative of f). In fact, w = D(x) where x is the coordinate function. Let Dw be such a derivation, and let ∥ · ∥ denote the supremum norm on C[0,1]. Then Dw gives rise to an algebra norm ∥ · ∥w on P[0,1] denned by .

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R) → W(R), then f is dispersive, i.e., given any v0 ∈ V(Q) and ε > 0, the image set f[D(v0, ε)], where D(v0, ε) = [v: v ∈ V(Q), ¦v – v0¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q) → W(Q) can be explosive in the sense that for any v0 ∈ V(Q) and ε > 0, the set f[D[v0, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a) → Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.

We sometimes think of medial (that is, infinite Dedekind-finite) sets as being “small” infinite sets. Medial cardinals can be defined as those cardinals that are incomparable to ℵℴ; hence we tend to think of them as being spread out on a plane “just above” the natural numbers, which seems to lend support to the view expressed above that medial sets are “small”.

Let S(α) denote the class of functions regular and analytic in the unit disc E = {z¦z¦< 1¦} and satisfying the condition, .

The close connection between certain types of matroids or combinatorial geometries and block designs is well known. The relationships previously discussed have centred on the loose analogy between the blocks of a design and the hyperplanesor flats ot the matroid or geometry. The matroids which arise in this way have had in the main a very tight regular structure. Here we show that theclass of matroids whose bases are the blocks of a design ismuch wider — indeed from Theorem 6 below we obatain a metroid in a canonical way from any balanced incomplete block design in which no pair of blocks differ by exactly one element.

Let F map [0, 1] into a Banach space B and let R(F) denote the set of all limits of Riemann sums of F. The set R(F) need not be convex in general (Nakamura and Amemiya (1966)) but is always convex when B is finite dimensional as first shown by Hartman (1947). A proof of Hartman's result, based on a description of R(F) when the range of F is finite, was given in Ellis (1959). In this note this description is refined, the extreme points of R(F) are determined and the following complete characterization of R(F) is obtained (where Nn = {1,2, …, n}).

A subset Y of a metric space (X, p) is called rigid if all the distances p(y1, y2) between points y1, y2 ∈ Y in Y are mutually different. The main purpose of this paper is to prove the existence of dense rigid subsets of cardinality c in Euclidean spaces En and in the separable Hilbert space l2. Some applications to abstract point set geometries are given and the connection with the theory of dimension is discussed.

In this note it is shown in the most frequently encountered spaces of test functions in the theory of generalized functions that the customary definitions of convergence are equivalent to apparently much weaker definitions. For example, in the space g the condition of uniform convergence of the functions together with all derivatives (which appears in the definition of convergence) is equivalent to the condition of pointwise convergence of the functions alone. Thus verification of convergence is simplified somewhat.

We let X be an arbitrary non-empty set throughout. Many papers have been written describing the automorphisms of various transformation semigroups defined on X: total (Lyapin (1955), Magill (1967), Malcev (1952), Schreier (1936)), partial (Gluskin (1959), Magill (1967)), partial and 1–1 (Liber (1953)), partial and shifting at most a finite number of elements (Subov (1961a)). In all these cases the automoprhisms are shown to be “inner”, and using the fact this authors deduce that the automorphism group of the given transformation semigroup is isomorphic to the group gx of all permutations defined on X.

Denote by P the Euclidean plane with a rectangular Cartesian coordinate system where the x-axis is horizontal and the y-axis is vertical. An arc in P shall mean a simple continuous curve Λ:{t: 0 ≦ t < 1} P having the properties that limitt→1 Λ(t) exists and limitt→1Λ(t) ≠ Λ(t0) for 0 ≦t0 < 1. An arc at a pointζ in P shall be an arc Λ where limt→1 Λ(t) = ζ. If S is an arbitrary subset of the plane, ζ is termed an ambiguous point relative to S provided there are arcs Λ and Γ at ζ with Λ ⊆ S and Γ ⊆ P–S; such arcs are referred to as arcs of ambiguity at ζ. If A is a set of arcs we say a point ζ in P is accessible via A provided there is an arc at ζ which is an element of A. If B is also a collection of arcs, then A and B are said to be pointwise disjoint if whenever α∈A and β ∈ B, α ∩ β = Ø. The collections A and B are said to be terminally arcwise disjoint if whenever α ∈ A and β ∈ B and both α and β are arcs at a point ζ in P, then a ∩ β contains no arc at ζ. If S is a planar set, we let A(S) denote the set of all arcs contained in S. Note that if S ∩ T = Ø then A(S) and A(T) are pointwise disjoint collections of arcs.

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).

In this paper, some methods are developed for obtaining explicitly a basis for the integral closure of a class of coordinate rings of algebraic space curves.

The investigation of this problem was motivated by a need for examples of integrally closed rings with specified subrings with a view toward examining questions of unique factorization in them. The principal result, giving the elements to be adjoined to a ring of the form k[x1, …,xn] to obtain its integral closure, is limited to the rather special case of the coordinate ring of a space curve all of whose singularities are normal. But in numerous examples where the curve has nonnormal singularities, the same method, which is essentially a modification of the method of locally quadratic transformations, also gives the integral closure.

Denote by Nn(α, β) the number of distinct fractions p/q, where 1 ≦ q ≦ n and α < p/q < β. Let .