Showing papers in "Mathematika in 1992"
TL;DR: In this article, it was shown that quasi self-similar fractals of equal Hausdorff dimensions that are homeomorphic to Cantor sets are equivalent under Holder bijections of exponents arbitrarily close to 1.
Abstract: We show that under certain circumstances quasi self-similar fractals of equal Hausdorff dimensions that are homeomorphic to Cantor sets are equivalent under Holder bijections of exponents arbitrarily close to 1. By setting up algebraic invariants for strictly self-similar sets, we show that such sets are not, in general, equivalent under Lipschitz bijections.
128 citations
TL;DR: In this article, it was shown that the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.
Abstract: Let A and B be two compact, convex sets in ℝ n , each symmetric with respect to the origin 0. L is any ( n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see [6]) raised the question: Does vol ( A ⌒ L ) B ⌒ L ) for every L imply vol ( A ) B )? The answer in case n = 2 is affirmative in a trivial way. Also in 1953 H. Busemann (see [4]) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in [5], the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.
88 citations
TL;DR: In this article, the expectation of vol K - vol K(n) when K is a smooth convex body was shown to be upper bounded by the probability of a random polytope approximating K with high probability.
Abstract: Let K subset-of R(d) be a convex body and choose points x1, x2,..., x(n) randomly, independently, and uniformly from K. Then K(n) = conv {x1,..., x(n)} is a random polytope that approximates K (as n --> infinity) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K - vol K(n) when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).
82 citations
TL;DR: In this paper, the authors obtained improvements in Vinogradov's mean value theorem widely applicable in additive number theory and showed that it is possible to obtain fairly strong bounds for Js,k(P), where P denotes the number of solutions of the simultaneous diophantine equations with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s.
Abstract: The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equationswith 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writingin which e(α) denotes e2πiα, we observe thatwhere Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).
70 citations
TL;DR: In this article, the authors consider a Hausdorff space X and a metric ρ and show that X is σ-fragmented by ρ if, for each e>0, the sets Xi, i ≥ 1, in (1.1) can be taken from any family of subsets of X, using sets from, where each set Xi has the property that each non-empty subset of Xi has a relatively open subset of ρ-diameter less than e.
Abstract: §1. Introduction. Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X. The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ-diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each e>0, it is possible to writewhere each set Xi, i≥1, has the property that each non-empty subset of Xi, has a non-empty relatively open subset of ρ-diameter less than e. If is any family of subsets of X, we say that X is σ-fragmented by the metric ρ, using sets from, if, for each e>0, the sets Xi, i ≥ 1, in (1.1) can be taken from
56 citations
TL;DR: In this paper, it was conjectured that minimal pairs in an equivalence class of the Hormander-Radstrom lattice are unique up to translation, and it was shown that this is the case for the two-dimensional case.
Abstract: In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hormander-Radstrom lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝn is contained in another convex body. This generalizes a theorem of Weil (cf. [10]).
55 citations
TL;DR: In this paper, it was shown that β = M(P0) = 1 38135, where P0 is the polynomial P0(x, y) =1 + x + y + y and δ = m(P1) = 2 79162, where M denotes Mahler's measure.
Abstract: Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.
34 citations
TL;DR: In this paper, the authors consider a convex body in Euclidean space R d, d ≥ 2, with volume V ( K ) = 1, and n ≥ d + 1, where n is a natural number.
Abstract: Let K be a convex body in Euclidean space R d , d ≥2, with volume V ( K ) = 1, and n ≥ d +1 be a natural number. We select n independent random points y 1 , y 2 , …, y n from K (we assume they all have the uniform distribution in K ). Their convex hull co { y 1 , y 2 , …, y n } is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope It is easy to see that if U : R d → R d is a volume preserving affine transformation, then for every convex body K with V ( K ) = 1, m ( K, n ) = m ( U ( K ), n ).
32 citations
TL;DR: The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation.
Abstract: §1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.
32 citations
TL;DR: The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body as mentioned in this paper, and corresponding inequalities for arbitrary convex bodies, where the successive minimum values are replaced by certain successive diameters and successive widths.
Abstract: The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths. We further give some applications of these results to successive radii, intrinsic volumes and the lattice point enumerator of a convex body.
30 citations
TL;DR: In this paper, the Opial's inequality was generalized to the case of functions of n variables, m, n ≥ 1, where m is the number of variables in the function.
Abstract: In this paper, the Opial's inequality, which has a wide range of applications in the study of differential and integral equations, is generalized to the case involving m functions of n variables, m, n ≥ 1.
TL;DR: Given ξ in [0, 1] and given ξ = [ 0, a1, a2…] denote a simple continued fraction expansion of ξ, we have shown that with the conventions p1 = q0 = 1 and q−1 = p0 = 0 we have
Abstract: Given ξ in [0, 1] let ξ = [0, a1, a2…] denote a simple continued fraction expansion of ξ Given the expansion ξ = [0, a1, a2, …] letThus with the conventions p1 = q0 = 1 and q−1 = p0 = 0 we have
TL;DR: In this paper, the authors restate a result for systems of additive equations over algebraic number fields, which they restate in a more specific setting, which is the case of additive systems of number fields.
Abstract: Dorner [8] has recently proved a result for systems of additive equations over algebraic number fields, which we restate in a more specific setting.
TL;DR: In this article, a new proof of a theorem of Bombieri and Davenport is given, where t(n) = t(k) be real, where e(u) = e2πiu.
Abstract: §1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define
TL;DR: In this paper, it was shown that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.
Abstract: Let k ≤ 2 be an integer, and setEk (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.
TL;DR: Theorem 7.12 as discussed by the authors states that every interval can be mapped, using a piecewise contractive map, onto a longer interval, assuming that the number of pieces in the partition of A is given.
Abstract: §1. Introduction and main results. A map f: A → R (A xs2282 R) is called piecewise contractive if there is a finite partition A = A1xs222A … xs222A An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).
TL;DR: Kertas ini akan membandingkan rumus matematik kaedah bayaran bulanan mengikut pendekatan konvensional dengan Kaedah yang diizinkan oleh Islam.
Abstract: Kertas ini akan membandingkan rumus matematik kaedah bayaran bulanan mengikut pendekatan konvensional dengan kaedah yang diizinkan oleh Islam.
Katakunci: Bayaran bulanan; konvensional; islam; pinjaman; pembayaran
TL;DR: An upper bound for the "sausage catastrophe" of dense sphere packings in 4-space is given in this paper, where the authors consider the problem of finding the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed.
Abstract: An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given.A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed ([5]). For d = 2 this problem was solved by Groemer ([6]).
TL;DR: In this paper, the authors considered the k-half factorial domain (HFD) property in Dedekind domains with small class group and proved the following Theorem: if D is a HFD for some integer k > 1, if, and only if, D is HFD.
Abstract: Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α 1 , …, α n , β 1 , …, β m of D the equality α 1 … α n = β 1 … β m implies that n = m . In [5] the present authors define D to be a k -half factorial domain ( k -HFD) if the equality above along with the fact that n or m ≤ k implies that n = m . In this paper we consider the k -HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k -HFD for some integer k > 1, if, and only if, D is HFD.
TL;DR: In this paper, the structure of the Kac modules V (Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2) is considered and the number of composition factors of V (λ) for such Λ is in exact agreement with the conjectures of [HKV].
Abstract: . We consider the structure of the Kac modules V (Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V (Λ) are constructed and it is shown that the number of composition factors of V (Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1( m/n ), and conjectured to be valid for all finite dimensional irreducible representations of sl( m/n ) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).
TL;DR: In 1985, Sarkozy as discussed by the authors proved a conjecture of Erdos by showing that the greatest square factor s n 2 of the binomial coefficient satisfies for arbitrary e > 0 and sufficiently large n where
Abstract: §1. Introduction . In 1985, Sarkozy [11] proved a conjecture of Erdos [2] by showing that the greatest square factor s ( n ) 2 of the “middle” binomial coefficient satisfies for arbitrary e > 0 and sufficiently large n Where
TL;DR: In this paper, the envelope of the family of planes which bisect the volume of a tetrahedron is studied, and the envelopes of these planes are studied in terms of the number of planes bisecting the volume.
Abstract: We study the envelope of the family of planes which bisect the volume of a tetrahedron
TL;DR: Theorem 2.2.3 as mentioned in this paper shows that minimal length functions correspond to minimal actions on R-trees by following Chiswell's construction of actions on trees from length functions, given in [4].
Abstract: In Theorem 7.13 of [1], Proposition 3.1 of [5], and Theorem 1 of [10], minimal group actions on R-trees are considered. If a group G acts on a tree T , then a Lyndon length function l u is associated with each point u ∈ T . Abstract minimal length functions are defined in Section 2 of this paper by a simple reduction process, where lengths of elements are reduced by a fixed amount (except that any length must remain non-negative). It is shown in Theorem 2.3 that minimal length functions correspond to minimal actions by following Chiswell's construction of actions on trees from length functions, given in [4]. A parallel result to Theorem 1 of [10] is given for minimal length functions in Theorem 2.2. One outcome of these results is that to determine which length functions can arise from an action of a group on the same tree, it suffices to consider only minimal length functions. Section 1 is concerned with some preparatory properties on lengths of products of elements. These lead in Proposition 1.6 to an alternative description of the maximal trivializable subgroup associated with a length function, defined in [3].