Copositivity and Complete Positivity

A real matrix $A$ is called copositive if $x^TAx \ge 0$ holds for all $x \in \mathbb R^n_+$. A matrix $A$ is called completely positive if it can be factorized as $A = BB^T$ , where $B$ is an entrywise nonnegative matrix. The concept of copositivity can be traced back to Theodore Motzkin in 1952, and that of complete positivity to Marshal Hall Jr. in 1958. The two classes are related, and both have received considerable attention in the linear algebra community and in the last two decades also in the mathematical optimization community. These matrix classes have important applications in various fields, in which they arise naturally, including mathematical modeling, optimization, dynamical systems and statistics. More applications constantly arise. The workshop brought together people working in various disciplines related to copositivity and complete positivity, in order to discuss these concepts from different viewpoints and to join forces to better understand these difficult but fascinating classes of matrices

Optimal Rates of Statistical Seriation

Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.Comment: V2 corrects an error in Lemma A.1, v3 corrects appendix F on unimodal regression where the bounds now hold with polynomial probability rather than exponentia

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

A course in mathematics appreciation

The purpose of this study is to create a mini-course in mathematics appreciation at the senior high school level. The mathematics appreciation course would be offered as an elective to students in the 11th or 12th grade, who are concurrently enrolled in trigonometry or calculus. The topics covered in the mathematics appreciation course include: systems of numeration, congruences, Diophantine equations, Fibonacci sequences, the golden section, imaginary numbers, the exponential function, pi, perfect numbers, numbers with shape, ciphers, magic squares, and root extraction techniques. In this study, the student is exposed to mathematical proofs, where appropriate, and is encouraged to create practice problems for other members of the class to solve. Also, areas for research are suggested so that the student may explore, even more deeply, areas which hold a particular interest for that student. These topics are treated with a three-pronged approach – historical, recreational, and practical. It is the author\u27s contention, supported by research, that this approach, along with the choice of topics, will assist in developing and enhancing the mathematics potential of the student to the highest possible extent

Western Bushido: The American Invention of Asian Martial Arts

Prior to the Second World War, very few Americans were aware that martial arts existed outside of the Olympic institutions (e.g. boxing and wrestling) and it wasn’t until the 1960s and 1970s that Asian martial culture became mainstream in the English-speaking world. This changed when a group of dedicated, unorthodox Westerners applied themselves to the study and dissemination of East Asian martial arts, soon raising their popularity to its current level. This project explores the social and cultural process whereby the martial arts were imbued with a violent nationalist rhetoric in Japan before World War II and came to be a part of daily life in the United States in the decades that followed. Central ideas in this process are the creation of an imagined and exotic Asia and discourses of masculinity as they are negotiated within the larger framework of transforming American society. Source material for this contextual cultural analysis includes archival and interview data as well as popular publications, films, and other multimedia in addition to standard library research. By merging these three methods, it is possible to develop a well-rounded picture of trends in society over time and, in particular, how the folk history of any one group has influenced the broader zeitgeist. In this case, the invented traditions of prewar Japanese martial arts can be seen to travel across the Pacific via American servicemen and undergo radical transformations over time depending on the needs of practitioners and spectators in any given period

Knighthood and anti-heroic behaviour in the figures of Falstaff and Don Quixote

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French Chivalry

Chivalry denotes the ideals and practices considered suitable for a noble. The word itself is reminiscent of the aristocratic society of medieval France dominated by mounted warriors. As early as the eleventh century, several different views of chivalric standards and behavior had appeared. During the next four hundred years, these conceptions of the ideal nobleman were developed by and for the feudal ruling class. Sidney Painter studies chivalry from the perspectives of both social history and the history of ideas. The first chapter provides readers unfamiliar with medieval history the background required for understanding the chapters on chivalry

Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details