In this paper we examine the class G of simple undirected, connected graphs of diameter d > 1, girth 2d, and for any g ɛ G, if a pair of nodes are at distance d from each other, then that pair of nodes is connected by t distinct paths of length d, t > 1. (The girth of g is the length of the smallest circuit in g.)

We establish, in § 2, that for all g ɛ G, g is regular.

We establish necessary conditions for the existence of elements of G. If g ɛ G, we adopt the notation g = g(d, t, v, n), where v is the valence of g and n is the number of nodes. It is of course possible for g, h ɛ G, g ≠ h, and for given d, t, v, n to have both g(d, t, v, n) and h(d, t, v, n).

In particular, we show that if d = 2, t ≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular given t value.

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).

Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).

We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),

where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results is

where f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),

where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved that

where r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).

In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.

Let P be the class of all finite groups G whose powers GI have no countably infinite factor groups. Neumann and Yamamuro (1) proved that if G is a finite non-Abelian simple group, then G ∈ P. We generalize this result by proving the following theorem.

THEOREM. A finite group G ∈ P if and only if G is perfect

2. Inheritance properties ofP.

P1. If G ∈ P and N is normal in G, then G/N ∈ P.

Proof. Since (G/N)I is isomorphic to GI/NI it is clear that factor groups of (G/N)I are isomorphic to factor groups of GI, and hence finite or uncountable.

P2. If G = HK, where H ∈ P and K ∈ P, then G ∈ P.

Proof. We show that homomorphic images of GI are either finite or uncountable. Let ϕ be a homomorphism of GI. Then GIϕ = (HK)Iϕ = (HI/KI)ϕ = (HIϕ) (KIϕ). Since HIϕ and KIϕ must be finite or uncountable, the conclusion follows.

We shall use the following notation:

R = Dedekind domain;

K = quotient field of R;

Rp = ring of p-adic integers in K, p being a prime ideal in R;

A = finite-dimensional separable k-algebra;

G – R-order in A (for the definition cf. (3)).

All modules that occur are assumed to be finitely generated unitary left modules, unless otherwise specified. By a G-lattice we mean a G-module which is torsion-free as R-module. A G-lattice is called irreducible if it does not contain a proper G-submodule of smaller R-rank. If p is a prime ideal in R we shall write Gp = Rp ⊗RG; Mp = RP⊗RM for a G-lattice M, and KM = K ⊗R M. Two G-lattices M and N are said to lie in the same genus (notation M ∨ N) if Mp ≌ Np for every prime ideal p in R.

For any A-module L, let S(L) be the collection of G-lattices M, for which KM ≌ L. Suppose that S(L) splits into rg(L) genera, and into ri(L) classes under G-isomorphism. Maranda (6) has shown: If L is an absolutely irreducible A-module, then

Soient f(z) et g (z) deux fonctions univalentes normalisées convexes dans |z|. Nous avons démontré récemment (3) que leur moyenne géométrique F = (fg)1/2 applique le disque sur un domaine convexe et l'estimation est précise.

Nous voulons ici considérer le même problème pour la moyenne arithmétique des deux fonctions f(z) et g (z).

THÉORÈME. Soient f(z) et g(z) deux fonctions univalentes normalisées convexes dans |z|. Si K représente la plus petite racine positive de l'équation

1

alors la moyenne arithmétique applique |z| < κ sur un domaine convexe.

Il nous faut, pour la preuve du théorème, le lemme suivant.

LEMME. Si la function

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Then

is the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

A 2-sphere S in E3 is said to be locally spherical if for each point p in S and each ∈ > 0 there is a 2-sphere S' such that p Ç Int S”, S' ᴖ S is a continuum, and Diam S' < ∈. It is not known whether locally spherical spheres are tame; however, there are several partial results. Burgess (2) showed that S is tame if S' C\ S is a simple closed curve and Loveland (3) proved that S is tame if S can be side approximated missing the continuum S ᴖ S'. In this paper we demonstrate that S is tame if the continuum S P\ S' is irreducible with respect to separating S. This result is stated more precisely in Theorem 3. Theorem 2, which is used in the proof of Theorem 3, is a generalization of a theorem recently proved by Loveland (4).

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

The function μ(z, β, α) defined by

where

plays an important role in Volterra's theory of convolution-logarithms, and also in the Paley-Wiener inversion formula for the Laplace transformation. These and other properties of pi are briefly described in (3).

Our aim in this paper is to find the asymptotic behaviour of μ as |z| < ∞, a result which, as far as we are aware, has not been obtained. It is of interest to note that the procedure to be used also gives, with some minor modification, the asymptotic behaviour of μ as z → 0, a result that is well known (3, p. 219). When this behaviour becomes known, it becomes possible to write a significant generalization of Watson's Lemma.

In 1906, Barnes (1) published a significant paper containing many asymptotic results of major importance. It is of passing interest to note that this paper contains a general theorem from which the result of Watson's Lemma, published in 1918, can easily be obtained. It would seem that this latter result is misnamed in mathematical literature, and might well be called Barnes’ Lemma.

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.

In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.

Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.

Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:

(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.

We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.

We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.

A set of points (edges) of a graph is independent if no two distinct members of the set are adjacent. Gallai (1) observed that, if A0 (B0) is the minimum number of points (edges) of a finite graph covering all the edges (points) and A1 (B1) is the maximum number of independent points (edges), then:

holds, where m is the number of points of the graph.

The concepts of independence and covering are generalized in various ways for n-graphs. In this paper we establish certain connections between the corresponding extreme numbers analogous to the above result of Gallai.

Ray-Chaudhuri considered (2) independence and covering problems in n-graphs and determined algorithms for finding the minimal cover and some associated numbers.

Soit Eun espace vectoriel topologique localement convexe séparé et soit K un sous-ensemble convexe compact de E, le problème que nous voulons considérer est le suivant : nous partons d'une famille A de transformations affines de Kdans lui-même, pour chaque point xde K, nous considérons la fermeture de l'enveloppe convexe de {A(x)| A∊ A}, ce que nous appelions A(x). Si nous voulons que l'application x → A(x) vérifie certaines propriétés, quelles conditions devons-nous imposer à A ou à K? La solution de ce problème est reliée à plusieurs théorèmes de point fixe (en particulier au théorème de Markov-Kakutani (2) et de Ryll-Nardzewski (3)), ainsi qu'au lemme de Choquet-Deny sur les solutions bornées de l'équation de convolution μ * σ = μ. Le principal résultat de cet article est que l'application identité est un élément extrême de l'ensemble des applications affines d'un convexe borné en lui-même.

André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.

If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):