Showing papers in "Journal of The London Mathematical Society-second Series in 1972"

Uniform Distribution of Polynomials Over Finite Fields

Abstract: 1. Let k be the finite field of order q = p. A polynomial Q(x) in M(n), the set of polynomials in k[x] of degree n, is defined to have factor pattern X = \...n" if Q factorises in k[x] into a product of irreducible polynomials, there being precisely ad prime factors of degree d (for each d = 1, ..., n) in such a decomposition. The number of polynomials with factor pattern X in a subset A of M(«) will be denoted by TX(A). Thus, it was noted in [3] that, for fixed n,

Transitive Permutation Groups of Prime Degree

Abstract: This paper is intended as a survey of what is now known about transitive permutation groups of prime degree. The topic arises from early work on the theory of equations. Over 200 years ago Lagrange was led to an interest in irreducible polynomial equations of prime degree by showing1 that if every such equation were soluble in terms of root-extraction then polynomial equations of arbitrary degree would be. Even after Abel and Galois had shown that such solutions are impossible in general, Galois still devoted a good proportion of his work to equations of prime degree. It was of course he who emphasised the groups involved. Several 19th century mathematicians, notably Mathieu and Jordan, continued the work of Galois and provided foundations for the rich material that has been published since 1900. At present the problems concerning groups of prime degree remain near the centre of permutation group theory, retaining their interest partly as tests of the power and scope of techniques of finite group theory, partly as being typical of a range of similar problems concerning groups of degrees kp (with k < p) and p m where p is prime.

An Interpolation Series Associated with the Bessel–Hankel Transform

Abstract: [5]; they form a complete orthonormal set in the Hilbert function space known as the Paley-Wiener functions, and wn(m) = 5nm (Kronecker's Symbol) for all integers n and m. This means that the cardinal series is not only an orthogonal expansion for the Paley-Wiener functions, but it also provides a process for interpolation at the integers since the series reduces formally to am when x is an integer m. In the present note we shall consider further sets of this type and in particular a set involving Bessel functions. Cardinal series interpolation has important applications in information theory, where it was introduced by C. E. Shannon [11]. It is, for example, important for the electrical engineer to know that a certain type of transmitted signal, a function of time, lies in a subspace (the Paley-Wiener functions) of L(-oo, oo), and that this subspace possesses an orthogonal basis with respect to which the \" coordinates \" of the signal are actually values taken by the signal at certain instants of time. It was with this application in mind that H. P. Kramer introduced a generalisation of the cardinal series in a lemma which we adopt as the starting point for the present discussion. LEMMA 1 (Kramer [7]). Let (a, b) be a finite interval of U (the real numbers). Let K(x, t) e L(a, b)for each xeU and suppose that the sequence of real numbers {xn} (where n runs over some indexing set of integers) is such that {K(xn, t)} forms a complete orthogonal set (COS) in L(a, b). If

On the Numerical Range of Compact Operators on Hilbert Spaces

Abstract: Proof. If I is a cluster point of W(T), then there exists a sequence {(Tx,, x,)}, where Ix,ll = 1 for all n, converging to A. Since the unit ball in a Hilbert space is weakly sequentially compact, there exists a subsequence {s,,} which is weakly convergent to an x where 11x1 < 1. Since T is a compact operator, {Tx,,} is strongly convergent to Tx. However, I(Tx,,, x,,) (Tx, x)l < I( Tx,,, x,,) (Tx7 x,,>l + l (Tx, x,,) (Tx, x)l < Ilx,,I I I Tx,,Txll + I(x,,, Tx) ( 4 Tx)l. Therefore {(Tx,,, x,,)} converges to (Tx, x) and so (Tx, x) = A. If I # 0, clearly .x # 0, SO

The Point Partition Numbers of Closed 2-Manifolds

Abstract: Let M be a closed 2-manifold. The chromatic number of M is defined to be the maximum chromatic number of all graphs which can be imbedded in M. The famous Four Colour Conjecture states that the chromatic number of the sphere is four. One of the oddities of mathematics is that the chromatic number of the familiar sphere is still unknown, although the chromatic number of every other closed 2-manifold, whether oiientable or non-orientable, is known. In 1959, Ringel published in [6] his proof that the chromatic number of the closed non-orientable 2-manifold of genus y (y > 0) is [|(7+V(l+24y))] if y # 2, and the chromatic number of the Klein bottle (y = 2) is six. More recently, Ringel and Youngs in [7], 1968, announced their solution to the long-standing Heawood Map-Colouring Conjecture: The chromatic number of the closed orientable 2-manifold of genus y (y > 0) is [i(7 +V(l + 48y))]. A colouring number for graphs closely related to the chromatic number is the point-arboricity (see [2]). The point-arboricity of the closed 2-manifold M is defined to be the maximum point-arboricity of all graphs which can be imbedded in M. In 1969, Kronk showed in [4] that the point-arboricity of the closed orientable 2-manifold of genus y (y > 0) is [i(9+VO+48y))]. Chartrand and Kronk, also in 1969, proved in [1] that the point-arboricity of the sphere is three. The similarity of the three results mentioned above suggested the generalization treated in this paper.