Chapter 31 - An update on the elusive fixed-point property

Paper 25: Jonathan M. Borwein, “I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly,” American Mathematical Monthly, vol. 122 (2015), p. 195–216. Copyright 2015 Mathematical Association of America. All Rights Reserved.

"100 Years Ago This Month in The American Mathematical Monthly." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1

"100 Years Ago This Month in The American Mathematical Monthly." The American Mathematical Monthly, 130(2), p. 181

(1939). Index to Volume 46, 1939 the American Mathematical Monthly. The American Mathematical Monthly: Vol. 46, No. 10, pp. 665-674.

(1917). Mathematical Problems in Relation to the History of Economics and Commerce. The American Mathematical Monthly: Vol. 24, No. 5, pp. 221-223.

In celebration of both a special “big” π Day (3/14/15) and the 2015 centennial of the Mathematical Association of America, we review the illustrious history of the constant π in the pages of the American Mathematical Monthly.

The following pages show the output of the Maple system applied to some selected problems from the Problem Section of the American Mathematical Monthly. The purpose of assembling these examples is to show some of the facilities and commands of the Maple system. These examples were run on our Vax 11/785 computer, under the Unix operating system. This is a rather random sample, and does not represent a systematic search of problems in the AMM.

Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework. - The American Mathematical Monthly is an excellent volume which will be a valuable companion both for those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students. - Mathematical Reviews An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features: The origin of Ito's correction formulae for Brownian motion and the Poisson process can be traced to communication relations or, equivalently, the uncertainty principle. Quantum stochastic interpretation enables the possibility of seeing new relationships between fermion and boson fields. Quantum dynamical semigroups as well as classical Markov semigroups are realized through unitary operator evolutions. The text is almost self-contained and requires only an elementary knowledge of operator theory and probability theory at the graduate level.

Previous article Next article Random Partitions of SetsV. N. SachkovV. N. Sachkovhttps://doi.org/10.1137/1119019PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Leo Moser and , Max Wyman, An asymptotic formula for the Bell numbers, Trans. Roy. Soc. Canada. Sect. III. (3), 49 (1955), 49–54 MR0078489 (17,1201c) 0066.31001 Google Scholar[2] V. N. Sachkov, Distribution of the number of distinct elements of a symmetric basis in a random $mA$-sample, Theory Prob. Applications, 16 (1971), 494–505 10.1137/1116053 0263.60002 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Combinatorial Models of the Distribution of Prime Numbers27 May 2021 | Mathematics, Vol. 9, No. 11 Cross Ref On the maximal multiplicity of block sizes in a random set partitionRandom Structures & Algorithms, Vol. 56, No. 3 Cross Ref Random partitions of a set with given number of blocksDiscrete Mathematics and Applications, Vol. 13, No. 3 Cross Ref Random Set Partitions: Asymptotics of Subset CountsJournal of Combinatorial Theory, Series A, Vol. 79, No. 2 Cross Ref Some Probabilistic Aspects of Set Partitions10 April 2018 | The American Mathematical Monthly, Vol. 104, No. 3 Cross Ref Random Set PartitionsWilliam M. Y. Goh and Eric Schmutz17 February 2012 | SIAM Journal on Discrete Mathematics, Vol. 7, No. 3AbstractPDF (1107 KB)On sets of integers with prescribed gapsMonatshefte f�r Mathematik, Vol. 116, No. 2 Cross Ref Gap-Free Set Partitions11 October 2006 | Random Structures & Algorithms, Vol. 3, No. 1 Cross Ref Generation of a random partition of a finite set by an urn modelJournal of Combinatorial Theory, Series A, Vol. 35, No. 2 Cross Ref Counting subsets of the random partition and the 'Brownian Bridge' processStochastic Processes and their Applications, Vol. 15, No. 2 Cross Ref Volume 19, Issue 1| 1974Theory of Probability & Its Applications History Submitted:14 February 1973Published online:28 July 2006 InformationCopyright © 1974 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1119019Article page range:pp. 184-190ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

In 1979, Goebel and Kuczumow showed that very large class of non-weak* compact, closed, bounded and convex subsets of \(\ell ^1\) has the fixed point property (FPP) for nonexpansive mappings. Later, in 2008, Lin proved that \(\ell ^1\) can be renormed to have FPP for nonexpansive mappings. \(c_0\)-analogue of Lin’s result is still open. However, renorming \(c_0\), we prove that Goebel and Kuczumow analogy can be proved under affinity condition. That is, we prove that there exist a renorming of \(c_0\) and a very large class of non-weakly compact, closed, bounded and convex subsets of \(c_0\) with FPP for affine nonexpansive mappings whereas Dowling, Lennard and Turett proved in 2004 that weak compactness is equivalent to FPP for nonexpansive mappings when \(c_0\) is considered with its usual norm instead of any equivalent norm.

In this paper the preservation of the fixed point property under UV decompositions is studied. It is shown that if K is an n-dimensional complex with the fixed point property and G is U V n − 1 U{V^{n - 1}} decomposition of K, then K/G also will have the fixed point property. Furthermore, if X is a compact metric space with the fixed point property, and G is a U V n U{V^n} decomposition of X such that X/G may be embedded in a suitably small Euclidian space, R m {R^m} , then X/G retains the fixed point property.

In this note we answer three questions raised by Richard K. Guy in his column Unsolved Problems in The American Mathematical Monthly of December 1984 [1]. Guy discusses tiling a three-dimensional cuboid using (at least two) incomparable cuboids with integer sides. Two cuboids are incomparable if neither will fit inside the other with sides parallel. (To compare two cuboids, write their dimensions in increasing order: a1 K a2 ?. a3, b1 b1.) Bill Sands found a tiling of a 3 x 4 x 15 cuboid (volume 180) using the six pieces 1 x 1 x 15, 1 X 2 X 11, 1 X 3 X 10, 2 X 3 X 6, 2 x 4 x 4, 3 x 3 x 5. (See FIGURE 1).

(1922). Two New Constructions of the Strophoid. The American Mathematical Monthly: Vol. 29, No. 2, pp. 55-58.

(1932). Questions and Discussions. The American Mathematical Monthly: Vol. 39, No. 4, pp. 229-231.

(1928). The Use of the Discriminant in Finding the Singular Solution of f(x, y, p) = 0. The American Mathematical Monthly: Vol. 35, No. 10, pp. 529-535.