Showing papers in "Mathematika in 1984"
TL;DR: In this paper, the authors considered the problem of finding the set of n points on the sphere whose sum of all Euclidean distances between points is maximal and what is the maximum.
Abstract: This paper is concerned with the solution of the following interesting geometrical problem For what set of n points on the sphere is the sum of all Euclidean distances between points maximal, and what is the maximum? Our starting point is the following surprising “invariance principle” due to K B Stolarsky: The sum of the distances between points plus the quadratic average of a discrepancy type quantity is constant Thus the sum of distances is maximized by a well distributed set of points We now introduce some notation to make the statement more precise
98 citations
TL;DR: It is known that the Riemann hypothesis implies for all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]), but there is not even a conjectural value of ck for any other k as mentioned in this paper.
Abstract: LetIt is conjectured that for any k,for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis impliesfor all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).
60 citations
TL;DR: In this article, it was shown that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ (t) is defined for t ≥ 0 as the continuous solution of the equations.
Abstract: For x, y ≥ 1, let Ψ(x, y) denote the number of positive integers less than or equal to x and free of prime factors greater than y. The behaviour of the function Ψ(x, y) has been the object of numerous articles (see e.g. Norton's memoir [5] and the bibliography there). It turns out that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ(t) is defined for t ≥ 0 as the continuous solution of the equations
54 citations
TL;DR: In this article, it was shown that a natural way to arrange n = 2p, where 2p + 1 = 3q, with p, q primes, is to specify that P2 should be a prime q, which would equally allow one to replace P2 by a prime in Chen's p + 2 = P2 result.
Abstract: Rather more than thirty years ago Erdős and Mirsky [2] asked whether there exist infinitely many integers n for which d(n) = d(n + 1). At one time it seemed that this might be as hard to resolve as the twin prime problem, see Vaughan [6] and Halberstam and Richert [3, pp. 268, 338]. The reasoning was roughly as follows. A natural way to arrange that d(n) = d(n + l) is to take n = 2p, where 2p + 1 = 3q, with p, q primes. However sieve methods yield only 2p + 1 = 3P2 (by the method of Chen [1]). To specify that P2 should be a prime q entails resolving the “parity problem” of sieve theory. Doing this would equally allow one to replace P2 by a prime in Chen's p + 2 = P2 result.
51 citations
32 citations
TL;DR: In contrast to the Stokes paradox for flow past an isolated cylinder, if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity as discussed by the authors.
Abstract: A study is made of Stokes flows in which a line rotlet or stokeslet is in the presence of a circular cylinder in a viscous fluid. In contrast to the Stokes Paradox for flow past an isolated cylinder, it is shown that if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity. A similar phenomenon can occur when two equal cylinders rotate with equal and opposite angular velocities, and the flow pattern is then such that there is a closed streamline enclosing both cylinders.
29 citations
TL;DR: In this paper, Pach's number was used to obtain the number of forbidden subgraphs in the universal graph of infinite graphs, which was later used in the construction of the Universal Graph.
Abstract: Keywords: infinite graphs ; universal graph ; forbidden subgraphs Note: Professor Pach's number: [034] Reference DCG-ARTICLE-1984-005 Record created on 2008-11-14, modified on 2017-05-12
29 citations
TL;DR: In this article, the authors consider the second order linear differential equation where p and q are real-valued and p(t) > 0 for all t ≥ T. Their interest here is the oscillatory nature of solutions of (1.1).
Abstract: We consider the second order linear differential equationwhere p and q are real-valued and p(t) > 0 for all t ≥ T. Our interest here is the oscillatory nature of solutions of (1.1). More particularly we consider the following questions, (I), (II) and (III).
25 citations
TL;DR: In this paper, it was shown that if K is a number field and E/K an elliptic curve, then there exists a global minimal Weierstrass equation for E. The main goal in this paper is to prove the converse to this statement.
Abstract: Let K be a number field and E/K an elliptic curve. As is well known [3, 4,[ if K has class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.
19 citations
TL;DR: In this article, the Fourier-Stieltjes transform of a finite, complex-valued measure on R and its Fourier transform on R has been studied and defined.
Abstract: We state some definitions belonging to the two halves of the title, going far enough to state our main results.Fourier transforms. Let μ be a finite, complex-valued measure on R and its Fourier-Stieltjes transform. We define ℛ to be the set of μ with When μ ∈ ℛ and φ is of class (continuously differentiable of compact support), the identity shows that θ · μ ∈ ℛ.
TL;DR: In this paper, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are defined.
Abstract: © Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this article, it was shown that for almost all e-tuples σ = (σ 1, σ, σ e ) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the prime field, the following results hold.
Abstract: The following theorem is proved in [2[. Let K be a finitely generated field over its prime field. Then for almost all e-tuples σ = (σ 1 , …, σ e ) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the following results hold.
TL;DR: In this paper, it was shown that every compact convex set in with mean width equal to that of a line segment of length 2 and with Steiner point at the origin is contained in the unit ball.
Abstract: It is shown that every compact convex set in with mean width equal to that of a line segment of length 2 and with Steiner point at the origin is contained in the unit ball. As a consequence, the diameter with respect to the Hausdorff metric of the space of all such sets is 1. There also results a sharp bound for the Hausdorff distance between any two compact convex sets.
TL;DR: In this article, it was shown that the cardinality of a measure can be computed in terms of certain fundamental cardinals (Corollary 11 and Theorem 16) and that cardinality can be used to measure the additivity and co-finality of the measure.
Abstract: Cardinal functions of topologies have been extensively studied. Cardinal functions of measures have attracted less interest, perhaps because there are fewer straightforward results which are independent of special axioms. In this paper I consider the “additivity” and “cofinality” of a measure (Definition 1) and show that they can often be calculated in terms of certain fundamental cardinals (Corollary 11 and Theorem 16).
TL;DR: In this article, it has been shown that the temperature predicted by the coupled dynamic theory is approximated by a solution of the classical heat equation, which is sometimes possible to justify the uncoupled and quasi-static approximations which are commonly invoked to simplify the solution of initial and boundary value problems in the linear theory of thermoelasticity.
Abstract: It has been shown in [1, 2] that it is sometimes possible to justify the uncoupled and quasi-static approximations which are commonly invoked to simplify the solution of initial and boundary value problems in the linear theory of thermoelasticity. The justification involves showing, among other things, that the temperature predicted by the coupled dynamic theory is approximated by a solution of the classical heat equation.
TL;DR: In this article, it was shown that if G is a positive definite integral ternary quadratic lattice of discriminant d and c is a square free integer which is primitively represented by the genus of G, then G primitively represents all sufficiently large integers of the type ct 2, with g.c.d.
Abstract: It is proved here that, if G is a positive definite integral ternary quadratic lattice of discriminant d and c is a squarefree integer which is primitively represented by the genus of G , then G primitively represents all sufficiently large integers of the type ct 2 , with g.c.d. ( t , 2 d ) = 1, which are primitively represented by the spinor genus of G .
TL;DR: In this paper, the inferior limits of p n, q n, t n and t n were defined and the corresponding superior limits were defined, and it was immediately clear from these definitions and their assumptions about a n that the latter since na n → 0 by Olivier's theorem.
Abstract: Let a n be a non-increasing real sequence such that converges; then clearly a n ↓ 0. We shall ignore the trivial case where a n = 0 for all large n , and so we assume that a n > 0 for all n , from now onwards. In [1] J. B. Wilker introduced certain new sequences associated with the rate of convergence of , and obtained various relations between them, in order to investigate packing problems in convex geometry. Let us define We write P , Q and T respectively for the inferior limits of p n , q n and t n , and P 1 , Q 1 and T 1 for the corresponding superior limits. Further, we put It is immediately clear from these definitions and our assumptions about a n that the latter since na n → 0 by Olivier's theorem [2, p. 124].
TL;DR: In this article, the K-Lusin sets in a Hausdorff space X were characterized as the sets that can be obtained as the image of some paracompact Cech complete space G, under a continuous injective map that maps discrete families in G to discretely σ -decomposable families in X.
Abstract: In [4] we initiated a study of K -Lusin sets. We characterized the K -Lusin sets in a Hausdorff space X as the sets that can be obtained as the image of some paracompact Cech complete space G , under a continuous injective map that maps discrete families in G to discretely σ -decomposable families in X [4, Theorem 2, p. 195]. Unfortunately, we cannot substantiate a second characterization of K -Lusin sets in completely regular spaces, given in the second part of Theorem 14 of [4].
TL;DR: Gale transform can be used to determine all closed faces and Radon partitions of the α-polytope as discussed by the authors, and a byproduct is a characterization of closed faces using nets of functionals.
Abstract: Gale transforms are constructed for certain infinite dimensional α-polytopes. In a manner analogous to the finite dimensional case the Gale transform can be used to determine all closed faces and Radon partitions of the α-polytope. A by-product is a characterization of closed faces using nets of functionals.