检索结果-内蒙古大学图书馆

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 2020年第2238期476卷 20200038页

作者： Chistikov, Dmitry Goulko, Olga Kent, Adrian Paterson, Mike Univ Warwick Ctr Discrete Math & Applicat DIMAP Coventry CV4 7AL W Midlands England Univ Warwick Dept Comp Sci Coventry CV4 7AL W Midlands England Boise State Univ Dept Phys 1910 Univ Dr Boise ID 83725 USA Univ Cambridge Ctr Math Sci DAMTP Ctr Quantum Informat & Fdn Wilberforce Rd Cambridge CB3 0WA England Perimeter Inst Theoret Phys 31 Caroline St North Waterloo ON N2L 2Y5 Canada

We consider versions of the grasshopper problem (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference 2 pi, we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper's jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length pi, we show this is true except when the jump length phi is of the form pi(p/q) with p,q coprime and p odd. For these jump lengths, we show the optimal probability is 1 - 1/q and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is 1 - 1/q for p,q coprime, p odd and q even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent & Pitalda-Garcia 2014 Phys. Rev. A 90, 062124) that if phi = pi/q, where q is an element of N, then the optimal retention probability of 1 - 1/q for the grasshopper's jump is provided by a hemispherical lawn. We show that in all other cases where 0 < phi < pi/2, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent & Pitalua-Garcia 2014 Phys. Rev. A 90, 062124). We discuss the implications for Bell experiments and related cryptographic tests.

关键词： Bell inequalities geometric combinatorics grasshopper problem

来源：评论

学校读者我要写书评

暂无评论

CONGRUENCE CLASSES OF LARGE CONFIGURATIONS IN VECTOR SPACES OVER FINITE FIELDS

引用

FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI 2020年第1期62卷 131-141页

作者： McDonald, Alex Univ Rochester UR Math 915 Hylan BldgRC Box 270138 Rochester NY 14627 USA

In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent s d case, fixing all pairs of distnaces leads to an overdetermined system, so q((k+1 2)) is no longer the correct number of congruence classe... 详细信息

In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent s < d such that any set E subset of F-q(d) with vertical bar E vertical bar greater than or similar to q(s) determines greater than or similar to q((k+1) (2)) congruence classes of (k + 1)-point configurations for k <= d. Because congruence classes can be identified with tuples of distances between distinct points when k <= d, and because there are (k+1 2) such pairs, this means any such E determines a positive proportion of all congruence classes. In the k > d case, fixing all pairs of distnaces leads to an overdetermined system, so q((k+1 2)) is no longer the correct number of congruence classes. We determine the correct number, and prove that vertical bar E vertical bar greater than or similar to q(s) still determines a positive proportion of all congruence classes, for the same s as in the k <= d case.

关键词： Erdos combinatorics geometric combinatorics

来源：评论

学校读者我要写书评

暂无评论

Sets in R^d determining k taxicab distances

引用

INVOLVE, A JOURNAL OF MATHEMATICS 2020年第3期13卷 487-509页

作者： Balaji, Vajresh Edwards, Olivia Loftin, Anne Marie Mcharo, Solomon Phillips, Lo Rice, Alex Tsegaye, Bineyam Millsaps Coll Dept Math Jackson MS 39210 USA

We address an analog of a problem introduced by Erdos and Fishburn, itself an inverse formulation of the famous Erdos distance problem, in which the usual Euclidean distance is replaced with the metric induced by the l(1)-norm, commonly referred to as the taxicab metric. Specifically, we investigate the following question: given d, k is an element of N, what is the maximum size of a subset of R-d that determines at most k distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension d = 2, as well as the k = 1 case in dimension d = 3, and we also provide a full resolution in the general case under an additional hypothesis.

关键词： Erdos distance problem taxicab metric discrete geometry geometric combinatorics

来源：评论

学校读者我要写书评

暂无评论

ON KAKEYA-NIKODYM TYPE MAXIMAL INEQUALITIES

引用

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2017年第9期369卷 6351-6372页

作者： Xi, Yakun Johns Hopkins Univ Dept Math Baltimore MD 21218 USA

We show that for any dimension d >= 3, one can obtain Wolff's L(d+2)/2 bound on Kakeya-Nikodym maximal function in R-d for d >= 3 without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional L-2 estimate with an auxiliary maximal function. We also prove that the same L(d+2)/2 bound holds for Nikodym maximal function for any manifold ( M-d, g) with constant curvature, which generalizes Sogge's results for d = 3 to any d >= 3. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function.

关键词： Kakeya maximal function Nikodym maximal function geometric combinatorics

来源：评论

学校读者我要写书评

暂无评论

geometric combinatorics in discrete settings

Geometric combinatorics in discrete settings

引用

作者： Covert, David University of Missouri - Columbia

学位级别：Ph.D.

This thesis is a compilation of work in which the author studies geometric configurations in finite fields and the integers modulo q. The results of this dissertation are threefold. First, we prove a finite field analog of the Furstenberg-Katznelson-Weiss theorem on triangles in [special characters omitted]. Second, we study volume sets in [special characters omitted] and discuss some applications to sum-product problems. Finally, we study geometric combinatorics in [special characters omitted]. We generalize a result of Hart and Iosevich [27] which has applications to sum-product problems. Finally, we show that the [special characters omitted] analogue of a sphere with unital radius is qd –1-dimensional.

关键词： geometric combinatorics finite fields sum-product problem

来源：评论

学校读者我要写书评

暂无评论

Results on the Erdos-Falconer distance problem in Z_q^d for odd q

引用

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2015年第2期426卷 727-733页

作者： Covert, David J. Univ Missouri St Louis MO 63121 USA

The Eras Falconer distance problem in Z(q)(d) asks one to show that if E subset of Z(q)(d) is of sufficiently large cardinality, then the set of distances determined by E satisfies Delta(E) = Z(q). Previous results were known only in the case q = p(e), where p is an odd prime, and as such only showed that all units were obtained in the distance set. We give the first such result over rings Z(q), where q is no longer confined to be a prime power, and despite this, we show that the distance set of E contains all of Z(q), whenever E is of sufficiently large cardinality. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Erdos-Falconer distance problem Discrete mathematics geometric combinatorics

来源：评论

学校读者我要写书评

暂无评论

ON TRIANGLES DETERMINED BY SUBSETS OF THE EUCLIDEAN PLANE, THE ASSOCIATED BILINEAR OPERATORS AND APPLICATIONS TO DISCRETE GEOMETRY

引用

ANALYSIS & PDE 2012年第2期5卷 397-409页

作者： Greenleaf, Allan Iosevich, Alex Univ Rochester Dept Math Rochester NY 14627 USA

We prove that if the Hausdorff dimension of a compact set E subset of R-2 is greater than 7/4, then the set of three-point configurations determined by E has positive three-dimensional measure. We establish this by showing that a natural measure on the set of such configurations has Radon-Nikodym derivative in L-infinity if dim(H)(E) > 7/4, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, B(f, g)(x) = integral integral f(x - u) g (x - v) dK (u, v), where K is surface measure on the set {(u, v) Sigma R-2 x R-2 : vertical bar u vertical bar = vertical bar v vertical bar = vertical bar u - v vertical bar = 1}, and we prove a scale of estimates that includes B : L--1/2(2)(R-2) x L-2(R-2) -> L-1 (R-2) on positive functions. As an application of our main result, it follows that for finite sets of cardinality n and belonging to a natural class of discrete sets in the plane, the maximum number of times a given three-point configuration arises is O(n(9/7+epsilon)) (up to congruence), improving upon the known bound of O(n(4/3)) in this context.

关键词： Falconer-Erdos distance problem distance set geometric combinatorics multilinear operators triangles

来源：评论

学校读者我要写书评

暂无评论

INDUCING POLYGONS OF LINE ARRANGEMENTS

引用

INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS 2011年第3期21卷 351-368页

作者： Scharf, Ludmila Scherfenberg, Marc Free Univ Berlin Inst Comp Sci D-14195 Berlin Germany

We show that anarrangement A of n lines in general position in the plane has an inducing polygon of size O(n). Additionally, we present a simple algorithm for finding an inducing n-path for A in O(n log n) time and an... 详细信息

关键词： geometric combinatorics combinatorial algorithms line arrangements inducing polygons

来源：评论

学校读者我要写书评

暂无评论

Convex hull realizations of the multiplihedra

引用

TOPOLOGY AND ITS APPLICATIONS 2008年第2期156卷 326-347页

作者： Forcey, Stefan Tennessee State Univ Dept Math & Phys Nashville TN 37209 USA

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A(n)-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A(infinity)-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph-associahedra. (C) 2008 Elsevier B.V. All rights reserved.

关键词： n-category morphisms A-infinity maps Multiplihedron Homotopy geometric combinatorics

来源：评论

学校读者我要写书评

暂无评论

Equipartitions of measures in R⁴

引用

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2008年第1期360卷 153-169页

作者： Zivaljevic, Rade T. SANU Math Inst Belgrade 11001 Serbia

A well-known problem of B. Grunbaum (1960) asks whether for every continuous mass distribution (measure) d mu = f dm on R-n there exist n hyperplanes dividing R-n into 2(n) parts of equal measure. It is known that the answer is positive in dimension n = 3 (see H. Hadwiger (1966)) and negative for n >= 5 (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grunbaum's problem in the critical dimension n = 4 by proving that each measure mu in R-4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional affine subspace L of R-4. Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence ( 1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4;see G. C. Tootill (1956) and D. E. Knuth (2001).

关键词： geometric combinatorics partitions of masses Gray codes

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：