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On The Eneström-Kakeya Theorem  [PDF]
Gulshan Singh, Wali Mohammad Shah
Applied Mathematics (AM) , 2010, DOI: 10.4236/am.2010.16073
Abstract: In this paper, we prove some generalizations of results concerning the Eneström-Kakeya theorem. The results obtained considerably improve the bounds by relaxing the hypothesis in some cases.
随机算子值enestr?m-kakeya定理的一个注记  [PDF]
武汉理工大学学报 , 2001,
Abstract: ?关于代数方程和算子情形的enestr¨omkakeya定理已被讨论。我们在这篇注记中,利用作用在hilbert空间上的随机算子值的随机数值半径不等式证明了随机算子值enestr¨omkakeya定理
On the Location of Zeros of Polynomials  [PDF]
Gulshan Singh, Wali Mohammad Shah
American Journal of Computational Mathematics (AJCM) , 2011, DOI: 10.4236/ajcm.2011.11001
Abstract: In this paper, we prove some extensions and generalizations of the classical Eneström-Kakeya theorem.
On The Enestrm-Kakeya Theorem  [PDF]
A Liman,Tawheeda Rasool,WM Shah
BIBECHANA , 2014, DOI: 10.3126/bibechana.v10i0.8067
Abstract: In this paper we present some interesting generalizations of Enestr?m-Kakeya type results concerning the location of zeros of a polynomial in the complex plane. We relax the hypothesis and put less restrictive conditions on the coefficients of the polynomial, and thereby generalize some classical results. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.8067 ? BIBECHANA 10 (2014) 71-81
The endpoint multilinear Kakeya theorem via the Borsuk--Ulam theorem  [PDF]
Anthony Carbery,Stefan Ingi Valdimarsson
Mathematics , 2012,
Abstract: We give an essentially self-contained proof of Guth's recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk-Ulam theorem.
The Kakeya Problem  [PDF]
Rongchuan Tao, Yingzi Yang, Xiaoxiao Zou, Zifan Dong, Siran Chen
Advances in Pure Mathematics (APM) , 2019, DOI: 10.4236/apm.2019.92006
Abstract: This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.
Discrete analogues of Kakeya problems  [PDF]
Marina Iliopoulou
Mathematics , 2013,
Abstract: This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let $\mathfrak{L}$ be a collection of $L$ lines in $\mathbb{R}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, that is, the set of points each of which lies in at least three non-coplanar lines of $\mathfrak{L}$. It is known that $|J|=O(L^{3/2})$ (first proved by Guth and Katz). For each joint $x\in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $$\sum_{x\in J} N(x)^{1/2}=O(L^{3/2}), $$while we also extend this result to real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as to curves in $\mathbb{R}^3$ parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in $\mathbb{R}^3$; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions.
Recent progress on the Kakeya conjecture  [PDF]
Nets Katz,Terence Tao
Mathematics , 2000,
Abstract: The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).
Kakeya Sets and Directional Maximal Operators in the Plane  [PDF]
Michael Bateman
Mathematics , 2007,
Abstract: We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
New bounds on Kakeya problems  [PDF]
Nets Katz,Terence Tao
Mathematics , 2001,
Abstract: We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.
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