Topic
Jordan measure
About: Jordan measure is a research topic. Over the lifetime, 25 publications have been published within this topic receiving 328 citations.
Papers
Book•
1 Jan 2011TL;DR: In this paper, a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis, is presented, with a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text.
Abstract: This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Radamacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasised. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.
353 citations
26 citations
TL;DR: Weyl and Cassels as discussed by the authors gave a simplified version of Minkowski's original proof, and a different proof has been given by Davenport [1, 2].
Abstract: Let K be a bounded, open convex set in euclidean n-space R n , symmetric in the origin 0. Further let L be a lattice in R n containing 0 and put m 4 =infimum u i i=1,2,.....,n; extended over all positive real numbers u i for which u i K contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that m 1 m 2 ...m n V(K)≦ 2 n d(L) Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].
20 citations
Journal Article•
TL;DR: In this article, the Airy/Jordan criterion for the optimal conformal polynomial projection is defined as minimum of weighted mean of Airy and Jordan measure of distortion in points, while the weights are equal to areas of corresponding ellipsoidal trapezoids.
Abstract: The paper describes optimal conformal polynomial projections for Croatia according to the Airy/Jordan criterion. A brief introduction of history and theory of conformal mapping is followed by descriptions of conformal polynomial projections and their current application. The paper considers polynomials of degrees 1 to 10. Since there are conditions in which the 1st degree polynomial becomes the famous Mercator projection, it was not considered specifically for Croatian territory. The area of Croatia was defined as a union of national territory and the continental shelf. Area definition data were taken from the Euro Global Map 1:1 000 000 for Croatia, as well as from two maritime delimitation treaties. Such an irregular area was approximated with a regular grid consisting of 11 934 ellipsoidal trapezoids 2' large. The Airy/Jordan criterion for the optimal projection is defined as minimum of weighted mean of Airy/Jordan measure of distortion in points. The value of the Airy/Jordan criterion is calculated from all 11 934 centres of ellipsoidal trapezoids, while the weights are equal to areas of corresponding ellipsoidal trapezoids. The minimum is obtained by Nelder and Mead’s method, as implemented in the fminsearch function of the MATLAB package. Maps of Croatia representing the distribution of distortions are given for polynomial degrees 2 to 6 and 10. Increasing the polynomial degree results in better projections considering the criterion, and the 6th degree polynomial provides a good ratio of formula complexity and criterion value.
16 citations
Posted Content•
TL;DR: In this paper, the existence, uniqueness, and minimality properties of the Jordan decomposition for a finite signed measure were proved for a signed measure with respect to the Jordan measure decomposition.
Abstract: This is a note of purely didactical purpose as the proof of the Jordan measure decomposition is often omitted in the related literature. Elementary proofs are provided for the existence, the uniqueness, and the minimality property of the Jordan decomposition for a finite signed measure.
14 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2015 | 1 |
| 2014 | 2 |
| 2012 | 1 |
| 2011 | 3 |
| 2009 | 1 |
| 2007 | 1 |