We prove in this note that if two convex curves are internally tangent (lie on the same side of the common tangent), then in most cases they also cut each other infinitely many times. It follows that, in a certain sense, if two convex curves have an odd number of common points, then in general they are externally tangent (lie on different sides of the common tangent). The sense of the expressions most cases and in general remains to be precised.

In this paper we study the transcendence degree of fields generated over Q by the numbers associated with values of one-parameter subgroups of commutative algebraic groups. We show that in many instances these fields have a large transcendence degree when measured in terms of the available data.

Our method deals with points which are “well distributed” (in a sense which is made precise) among certain algebraic subgroups of the algebraic group under consideration. We verify that these results apply in many classical situations.

Following four important papers on Birkhoff interpolation by Turán and his associates ([2], [3], [4], [14]), Kis ([8], [19]) proved the following theorems.

Let T be a totally ordered set, PT the semigroup of partial transformations on T, and A(T) the l-group of order-preserving permutations of T. We show that PT is a regular left l-semigroup. Let be the set of α ∈ PT such that α is order-preserving and the domain of α is a final segment of T. Then is an l-semigroup, and we prove that it is the largest transitive l-subsemigroup of PT which contains A(T). When T is Dedekind complete, we characterize the largest regular l-semigroup of . When A(T) is also 0 − 2 transitive we show that there can be no l-subsemigroup of properly containing A(T) which is either inverse or a union of groups.

It is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.

In many respects the theory of semi-prime rings (i.e. rings without proper nilpotents) is similar to that for lattice-ordered groups. In this paper semi-prime rings are faithfully represented as subrings of continuous global sections of sheaves of integral domains with Boolean base spaces. This representation allows a simple description of a particular extension of a semi-prime ring as the corresponding ring of all continuous global sections. The ideals in a semi-prime ring R that give rise to the stalks in the sheaf representation are then characterized when R is projectable. Finally equivalent conditions are given for a semi-prime ring R to satisfy a condition, that in the case of lattice-groups, was termed “weak projectability” by Spirason and Strzelecki [8]. Some of the results that are common to semi-prime rings and lattice-groups (and semi-prime semigroups) have been extended to certain universal algebras by Davey [3].

A construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

Recently K. Mahler [1] introduced a set of φ(2n) matrices of n rows and columns which form under multiplication the abelian group of the residue classes prime to 2n modulo 2n. These remarkable matrices whose elements 0, 1 and −1, have latent roots and determinants which can be given explicitly. Thus we have new examples of matrices with given elements whose powers, roots, inverses and determinants can be written down precisely. Such matrices are often useful in testing the efficacy of methods for finding these functions for a general matrix.

A topological approach to the theory of equilibrium phase transitions in statistical physics is based on the topological hypothesis, which claims that phase transitions are due to changes of the topology of suitable submanifolds in the configuration space. In this paper we examine in detail the antiferromagnetic mean-field XY model and study the topology of the subenergy manifolds. The latter can be interpreted mechanically as the configuration space of a linkage with one telescopic leg. We use methods of Morse theory to describe explicitly the Betti numbers of this configuration space. We apply these results to the antiferromagnetic mean-field XY model and compute the exponential growth rate of the total Betti number. Previous authors studied the Euler characteristic rather than the total Betti number. We show that in the presence of an external magnetic field the model undergoes a single ‘total Betti number phase transition’.

It is shown that a so-called shortly connected combinatorial inverse semigroup is strongly lattice-determined “modulo semilattices”. One of the consequences of this theorem is the known fact that a simple inverse semigroup with modular lattice of full inverse subsemigroups is strongly lattice-determined [7]. The partial automorphism semigroup of an inverse semigroup S consists of all isomorphisms between inverse subsemigroups of S. It is proved that if S is a shortly connected combinatorial inverse semigroup, T an inverse semigroup and the partial automorphism semigroups of S and T are isomorphic, then either S and T are isomorphic or they are dually isomorphic chains (with respect to the natural partial order); moreover, any isomorphism between the partial automorphism semigroups of S and T is induced either by an isomorphism or, if S and T are dually isomorphic chains, by a dual isomorphism between S and T. Counter-examples are constructed to demonstrate that the assumptions about S being shortly connected and combinatorial are essential.

A queue at which arrivals occur randomly in batches of fixed size r and for which the service times are independent negative exponential variates is considered. Expressions are obtained for the moments of the transient waiting time distribution and the distribution of the number of customers in the system just before the nth batch arrives. The distribution of the number of customers served in a busy period is also determined.

Let A be a complex sequentially complete, locally convex (not necessarily commutative) topological algebra with the defining family {pα}α∈D of seminorms in which (*): for each sequence xn → 0 there exists xm ∈ {xn} such that xmk → 0 as k → ∞. Then each multiplicative linear functional on a Fréchet algebra satisfying the above condition (*) is Continuous.

These results answer open questions (1) and (2) (Mem. Amer. Math. Soc. 11, 1953) in the affirmative for Fréchet algebras in which (*) holds. It is also shown that a positive linear functional on such algebras with identity and continuous involution is continuous, thus partially generalizing Shah's result (1959).

We show that every one-codimensional closed two-sided ideal in a boundedly approximately contractible Banach algebra has a bounded approximate identity. We use this to give a complete characterization of bounded approximate contractibility of Beurling algebras associated to symmetric weights. We give a slight modification of a criterion for bounded approximate contractibility. We use our criterion to show that, for the quasi-SIN groups, in the presence of a certain growth condition on a weight, the associated Beurling algebra is boundedly approximately amenable if and only if it is boundedly approximately contractible. We show that approximate amenability of a Beurling algebra on an IN group necessitates the amenability of the group. Finally, we show that, for every locally compact abelian group, in the presence of a growth condition on the weight, 2n-weak amenability of the associated Beurling algebra is equivalent to every point-derivation vanishing at the augmentation character.

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.

We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.

We show that the problem of settling the existence of an n × n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph K n, n, such that T contains at least one circuit of each copy of K2,3 in Kn, n.

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

We prove the following statements about bounded linear operators on a complex separable infinite dimensional Hilbert space. (1) Let A and B* be subnormal operators. If A2X = XB2 and A3X = XB3 for some operator X, then AX = XB. (2) Let A and B* be subnormal operators. If A2X – XB2 ∈ Cp and A3X – XB3 ∈ Cp for some operator X, then AX − XB ∈ C8p. (3) Let T be an operator such that 1 − T*T ∈ Cp for some p ≥1. If T2X − XT2 ∈ Cp and T3X – XT3 ∈ Cp for some operator X, then TX − XT ∈ Cp. (4) Let T be a semi-Fredholm operator with ind T < 0. If T2X − XT2 ∈ C2 and T3X − XT3 ∈ C2 for some operator X, then TX − XT ∈ C2.

Let X be a closed, oriented, smooth 4-manifold with a finite fundamental group and with a non-vanishing Seiberg-Witten invariant. Let G be a finite group. If G acts smoothly and freely on X, then the quotient X/G cannot be decomposed as X1#X2 with (Xi) > 0, i = 1, 2. In addition let X be symplectic and c1(X)2 > 0 and b+2(X) > 3. If σ is a free anti-symplectic involution on X then the Seiberg-Witten invariants on X/σ vanish for all spinc structures on X/σ, and if η is a free symplectic involution on X then the quotients X/σ and X/η are not diffeomorphic to each other.