On Keller's conjecture in dimension seven

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. It is known that if $|L(T,x,i)|\leq 2$ for some $x\in \mathbb{R}^7$ and every $i\in [7]$ or $|L(T,x,i)|\geq 6$ for some $x\in \mathbb{R}^7$ and $i\in [7]$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $|L(T,x,i)|=5$ for some $x\in \mathbb{R}^7$ and $i\in [7]$. Thus, if there is a counterexample to Keller's conjecture in dimension seven, then $|L(T,x,i)|\in \{3,4\}$ for some $x\in \mathbb{R}^7$ and $i\in [7]$.Comment: 37 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1304.163

James Ralph Scales : a case study of sixteen years of university leadership

The purpose of this research was to identify factors in the leadership of James Ralph Scales during his presidency at Wake Forest University, 1967-1983. The identification of these factors was made through a historical and biographical case study. A second purpose was to compare Scales' leadership factors with those of five selected leadership frameworks. The case study analysis identified Scales' leadership factors as (1) constancy of "fit" between his style, values, and personal history and the style, values, and history of the institution; (2) an unmistakable commitment to the faculty as central to academic excellence; (3) a persistent articulation of the core values of an intellectual community; (4) a tolerance for situations requiring the management of ambiguity; (5) a spirit of magnanimity; (6) an active promotion of a climate of "possibility" through debate and personal initiative; (7) a sense of humor and an attractive physical presence; (8) a habit of person centered communication; and (9) a willingness to take risks because of a trust in the institution's resources

Studies in German Literature of the Nineteenth and Twentieth Centuries

Twenty-one distinguished American Germanists pay tribute to F. E. Coenen, previous longtime editor (1952-1968) of UNC Press' Studies in Germanic Languages and Literatures series. Their essays—reflecting a variety of approaches—deal with many major (Goethe, Kleist, Droste-Hülshoff, Keller, Nietsche, Rilke, Kafka, Hesse, Brecht, Thomas Mann, Musil) and some minor figures who have influenced the literary scene after 1800 and add significantly to both scholarship in and interpretation of modern German literature

Studies in German Literature of the Nineteenth and Twentieth Centuries

Twenty-one distinguished American Germanists pay tribute to F. E. Coenen, previous longtime editor (1952-1968) of UNC Press' Studies in Germanic Languages and Literatures series. Their essays—reflecting a variety of approaches—deal with many major (Goethe, Kleist, Droste-Hülshoff, Keller, Nietsche, Rilke, Kafka, Hesse, Brecht, Thomas Mann, Musil) and some minor figures who have influenced the literary scene after 1800 and add significantly to both scholarship in and interpretation of modern German literature

Validation of an observational instrument for measuring role behavior in social work groups as one aspect of maturity

Thesis (M.S.)--Boston Universit

Studies in German Literature of the Nineteenth and Twentieth Centuries: Festschrift for Frederic E. Coenen

Twenty-one distinguished American Germanists pay tribute to F. E. Coenen, previous longtime editor (1952-1968) of UNC Press' Studies in Germanic Languages and Literatures series. Their essays—reflecting a variety of approaches—deal with many major (Goethe, Kleist, Droste-Hülshoff, Keller, Nietsche, Rilke, Kafka, Hesse, Brecht, Thomas Mann, Musil) and some minor figures who have influenced the literary scene after 1800 and add significantly to both scholarship in and interpretation of modern German literature

Vagueness in mathematics talk

The Cockcroft Report claimed that "mathematics provides a means of communication which is powerful, concise and unambiguous". Such precision in language may be a conventional aim of mathematics, particularly when communicated in writing. Nonetheless, as this thesis demonstrates, vagueness is commonplace when people talk about mathematics. In this thesis, I examine the circumstances in which vagueness arises in mathematics talk, and consider the practical purposes which speakers achieve by means of vague utterances in this context. The empirical database, which is considered in Chapters 4 to 7, consists almost entirely of transcripts of mathematical conversations between adult interviewers (including myself) and one or two children. The data were collected from clinical interviews focused on a small number of tasks, and from fragments of teaching. For the most part, the pupils involved in the study were aged between 9 and 12, although the age-range in Chapter 7 extends from 4 to 25. I draw on a number of approaches to discourse associated with 'pragmatics' -a field of linguistics - to analyse the motives and communicative effectiveness of speakers who deploy vagueness in mathematics talk. I claim that, for these speakers, vagueness fulfills a number of purposes, especially 'shielding', i. e. self-protection against accusation of being wrong. Another purpose is to give approximate information; sometimes to achieve shielding, but also to provide the level of detail that is deemed to be appropriate in a given situation. A different purpose, associated with a particular form of vagueness (of reference), is to compensate for lexical gaps in pursuit of effective communication of concepts and ideas. I show, in particular, how speakers use the pronouns 'it' and 'you' in mathematics talk to communicate concepts and generalisations. Some consideration is given to the intentions of 'expert speakers of mathematics when they deploy vague language. Their purposes include some of those identified for novices. Teachers also use vagueness as a means of indirectness in addressing pupils; this strategy is associated with the redress of 'face threatening acts'. My thesis is that vagueness can be viewed and presented, not as a disabling feature of language, but as a subtle and versatile device which speakers can and do deploy to make mathematical assertions with as much precision, accuracy or as much confidence as they judge is warranted by both the content and the circumstances of their utterances. I report on the validation and generalisation of my findings by an Informal Research Group of school teachers, who transcribed and analysed their own classroom interactions using the methods I had developed

Review on higher homotopies in the theory of H-spaces

Higher homotopy in the theory of H-spaces started from the works by Sugawara in the 1950th. In this paper we review the development of the theory of H-spaces associated with it. Mainly there are two types of higher homotopies, homotopy associativity and homotopy commutativity. We give explanations of the polytopes used as the parameter spaces of those higher forms