✦ LIBER ✦

On the Metric Mahler Measure

✍ Scribed by A Dubickas; C.J Smyth

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 176 KB
Volume: 86
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2000.2579

No coin nor oath required. For personal study only.

✦ Synopsis

We define a new height function on the group of non-zero algebraic numbers :, the height of : being the infimum over all products of Mahler measures of algebraic numbers whose product is :. We call this height the metric Mahler measure, since its logarithm defines a metric in the factor group of the non-zero algebraic numbers modulo the group of all roots of unity. This metric induces the discrete topology on this factor group if and only if Lehmer's conjecture is true. Sharp upper and lower bounds are obtained for the metric Mahler measure of an algebraic number in terms of its Mahler measure, degree and house. The metric Mahler measure is computed for a class of numbers that includes Salem and Pisot numbers, and for roots of rationals. We also show that the set of all ratios of the logarithms of the metric and classical Mahler measures of algebraic numbers having a fixed degree is everywhere dense in the interval given by these bounds. 2001 Academic Press 1. INTRODUCTION For : in the multiplicative group Q * of non-zero algebraic numbers, its Mahler measure M(:) is defined as M(:)=A(:) d j=1 max[1, |: ( j) | ].

📜 SIMILAR VOLUMES

Remarks On Mahler′s Transcendence Measur

Remarks On Mahler′s Transcendence Measure for e

✍ M. Hata 📂 Article 📅 1995 🏛 Elsevier Science 🌐 English ⚖ 291 KB

A sightly improved classical transcendence measure for \(e\) will be given, by showing that the absolute constant in Mahler's measure can be taken to be 1 . We also give an improved linear independence measure for the system \(1, e, \ldots, e^{n}\). fr 1995 Academic Press. Inc

Jump processes and nonlinear fractional

Jump processes and nonlinear fractional heat equations on metric measure spaces

✍ Jiaxin Hu; Martina Zähle 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 174 KB

## Abstract Jump processes on metric‐measure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a __σ__ ‐stable type process decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form associated

Measurable selectors for the metric proj

Measurable selectors for the metric projection

✍ B. Cascales; M. Raja 📂 Article 📅 2003 🏛 John Wiley and Sons 🌐 English ⚖ 136 KB

## Abstract Let __Y__ and __Z__ be two topological spaces and __F__ : __Y__ × __Z__ → ℝ a function that is upper semi–continuous in the first variable and lower semi–continuous in the second variable. If __Z__ is Polish and for every __y__ ∈ __Y__ there is a point __z__ ∈ __Z__ with __F__(__y, z__)

A Metrical Result on Transcendence Measu

A Metrical Result on Transcendence Measures in Certain Fields

✍ Masaaki Amou 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 339 KB

We give transcendence measures for almost all elements of F q ((1ÂT )), the field of formal power series in 1ÂT over a finite field F q with q elements. 1996 Academic Press, Inc. where log q x means the logarithmic function with base q. Then our main result is stated as follows.

Remarks on the Atiyah-Hitchin Metric

✍ Ioannis Bakas 📂 Article 📅 2000 🏛 John Wiley and Sons 🌐 English ⚖ 70 KB 👁 2 views

We outline the construction of the Atiyah-Hitchin metric on the moduli space of SU (2) BPS monopoles with charge 2, first as an algebraic curve in C 3 following Donaldson and then as a solution of the Toda field equations in the continual large N limit. We adopt twistor methods to solve the underlyi

On the Norm of the Metric Projections

✍ Fernando Mazzone 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 87 KB

Let X be a Banach space. Given M a subspace of X we denote with P M the metric projection onto M. We define ?(X ) :=sup [&P M &: M a proximinal subspace of X]. In this paper we give a bound for ?(X ). In particular, when X=L p , we obtain the inequality &P M & 2 |2Â p&1| , for every subspace M of L