There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

This is the first in a series of four articles 

Across all disciplines, at all levels, and throughout the world, health care is becoming more complex. Just 30 years ago the typical general practitioner in the United Kingdom practised from privately owned premises with a minimum of support staff, subscribed to a single journal, phoned up a specialist whenever he or she needed advice, and did around an hour's paperwork per week. The specialist worked in a hospital, focused explicitly on a particular system of the body, was undisputed leader of his or her “firm,” and generally left administration to the administrators. These individuals often worked long hours, but most of their problems could be described in biomedical terms and tackled using the knowledge and skills they had acquired at medical school.

You used to go to the doctor when you felt ill, to find out what was wrong with you and get some medicine that would make you better. These days you are as likely to be there because the doctor (or the nurse, the care coordinator, or even the computer) has sent for you. Your treatment will now be dictated by the evidence—but this may well be imprecise, equivocal, or conflicting. Your declared values and preferences may be used, formally or informally, in a shared management decision about your illness. The solution to your problem is unlikely to come in a bottle and may well involve a multidisciplinary team.

Not so long ago public health was the science of controlling infectious diseases by identifying the “cause” (an alien organism) and taking steps to remove or contain it. Today's epidemics have fuzzier boundaries (one is even known as “syndrome X”1): they are the result of the interplay of genetic predisposition, environmental context, and lifestyle choices.

The experience of …

https://www.bmj.com/content/bmj/323/7313/625.full.pdf

The challenge of complexity in health care

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

心臓病診療プラクティス 16. 心不全を癒す

Introduction General group actions: Topological dynamics Dynamical systems on Lebesgue spaces Ergodicity and mixing properties Invariant measures on topological systems Spectral theory Joinings Some applications of joinings Quasifactors Isometric and weakly mixing extensions The Furstenberg-Zimmer structure theorem Host's theorem Simple systems and their self-joinings Kazhdan's property and the geometry of $M_{\Gamma}(\mathbf{X})$ Entropy theory for $\mathbb{Z}$-systems: Entropy Symbolic representations Constructions The relation between measure and topological entropy The Pinsker algebra, CPE and zero entropy systems Entropy pairs Krieger's and Ornstein's theorems Prerequisite background and theorems Bibliography Index of symbols Index of terms.

http://www.gbv.de/dms/goettingen/357980980.pdf

Ergodic Theory via Joinings

The Li–Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand sufficient conditions for Li–Yorke chaos in a topological dynamical system are given. We solve a long–standing open question by proving that positive entropy implies Li–Yorke chaos. On the other hand properties of dynamical systems without Li–Yorke pairs are investigated; in addition to having entropy 0, they are minimal when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally it is proved that minimal systems without Li–Yorke pairs are disjoint from scattering systems.

On li-yorke pairs

It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity, minimality and positive topological entropy. In light of these investigations and several examples which we exhibit, we propose a natural family of dynamical systems- chi -systems-as a better abstract framework for a general theory of chaotic dynamics.

/pdf/sensitive-dependence-on-initial-conditions-a0upkk7udm.pdf

Sensitive dependence on initial conditions

This paper is a commentary on the recent work [4]. It has two goals: the first is to eliminate the C*-algebra machinery from the proofs of the results of [4]; the second, to provide a characterization of weak orbit equivalence of Cantor minimal systems in terms of their dimension groups.

Eli Glasner

Papers

Ergodic Theory via Joinings

On li-yorke pairs

Sensitive dependence on initial conditions

Weak orbit equivalence of cantor minimal systems

On minimal actions of Polish groups