scholarly journals MINIMAL DUVAL EXTENSIONS

2004 ◽  
Vol 15 (02) ◽  
pp. 349-354 ◽  
Author(s):  
TERO HARJU ◽  
DIRK NOWOTKA
Keyword(s):  
Research Topic ◽  
Maximum Length ◽  
Short Survey ◽  
Sturmian Words ◽  
Lyndon Words

A word v=wu is a (nontrivial) Duval extension of the unbordered word w, if (u is not a prefix of v and) w is an unbordered factor of v of maximum length. After a short survey of the research topic related to Duval extensions, we show that, if wu is a minimal Duval extension, then u is a factor of w. We also show that finite, unbordered factors of Sturmian words are Lyndon words.

scholarly journals Sturmian words, Lyndon words and trees

1997 ◽  
Vol 178 (1-2) ◽  
pp. 171-203 ◽  
Author(s):  
Jean Berstel ◽  
Aldo de Luca
Keyword(s):  
Sturmian Words ◽  
Lyndon Words

Lyndon Words and Christoffel Words

2018 ◽  
pp. 87-94
Author(s):  
Christophe Reutenauer
Keyword(s):  
Lexicographical Order ◽  
Sturmian Words ◽  
Lyndon Words

This chapter covers the lexicographical ordering of lower Christoffel words, which is equivalent to the ordering by their slopes (Borel and Laubie). Lower Christoffel words are particular Lyndon words. They are maximum for the lexicographical order among Lyndon words of a given slope (Borel and Laubie). They are, together with the upper Christoffel words, the only unbordered finite Sturmian words (Chuan). They are exactly the Lyndon words which are Sturmian words (Berstel and de Luca). The standard factorization of a lower Christoffel word is obtained by cutting before the smallest lexicographical suffix. Finally, they are exactly the Lyndon words which are equilibrated (Melançon).

From Christoffel Words to Markoff Numbers

2018 ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Special Class ◽  
Rational Numbers ◽  
Real Numbers ◽  
Inner Automorphisms ◽  
Sturmian Words ◽  
Markoff Numbers ◽  
Lyndon Words

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

Experimental Studies Of Multilayer Glass Structures On Compression, Compression With Bending And Simple Bending

2019 ◽  
pp. 22-30
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Applied Load ◽  
Experimental Studies ◽  
Strength Properties ◽  
Research Topic ◽  
Normative Values ◽  
Laminated Glass ◽  
The Cross ◽  
Experimental Values

The work of multilayer glass structures for central and eccentric compression and bending are considered. The substantiation of the chosen research topic is made. The description and features of laminated glass for the structures investigated, their characteristics are presented. The analysis of the results obtained when testing for compression, compression with bending, simple bending of models of columns, beams, samples of laminated glass was made. Overview of the types and nature of destruction of the models are presented, diagrams of material operation are constructed, average values of the resistance of the cross-sections of samples are obtained, the table of destructive loads is generated. The need for development of a set of rules and guidelines for the design of glass structures, including laminated glass, for bearing elements, as well as standards for testing, rules for assessing the strength, stiffness, crack resistance and methods for determining the strength of control samples is emphasized. It is established that the strength properties of glass depend on the type of applied load and vary widely, and significantly lower than the corresponding normative values of the strength of heat-strengthened glass. The effect of the connecting polymeric material and manufacturing technology of laminated glass on the strength of the structure is also shown. The experimental values of the elastic modulus are different in different directions of the cross section and in the direction perpendicular to the glass layers are two times less than along the glass layers.

Unexpectedness

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
John-Carlos Perea ◽  
Jacob E. Perea
Keyword(s):  
American Indian ◽  
San Francisco ◽  
Interdisciplinary Research ◽  
Ethnic Studies ◽  
College Of Education ◽  
Present Moment ◽  
State University ◽  
Research Projects ◽  
Short Survey ◽  
Wide Range

The concepts of expectation, anomaly, and unexpectedness that Philip J. Deloria developed in Indians in Unexpected Places (2004) have shaped a wide range of interdisciplinary research projects. In the process, those terms have changed the ways it is possible to think about American Indian representation, cosmopolitanism, and agency. This article revisits my own work in this area and provides a short survey of related scholarship in order to reassess the concept of unexpectedness in the present moment and to consider the ways my deployment of it might change in order to better meet the needs of my students. To begin a process of engaging intergenerational perspectives on this subject, the article concludes with an interview with Dr. Jacob E. Perea, dean emeritus of the Graduate College of Education at San Francisco State University and a veteran of the 1969 student strikes that founded the College of Ethnic Studies at San Francisco State University.

Research Methods for Accounting and Finance

2016 ◽  
Keyword(s):  
Research Methods ◽  
Research Process ◽  
Research Topic ◽  
First Time

An essential text for accounting and finance students undertaking research for the first time. It demystifies the research process by providing the novice researcher with a must-have guide through all of the stages of the research process, from identifying a research topic to the finished project.

ARCHITECTURE IN EFFECT: A Glance at Critical Historiography

2014 ◽  
Vol 8 (1) ◽  
pp. 184
Author(s):  
Navid Gohardani
Keyword(s):  
Subject Matter ◽  
Research Topic ◽  
Modern Architecture ◽  
Multiple Levels ◽  
Multidisciplinary Perspective ◽  
Critical Historiography ◽  
Research Domain ◽  
Critical Aspects

Historiography marks a relatively unexplored research domain in architecture. Despite the obscure nature of this subject matter, architectural historiography equally illuminates a hidden pathway to the historical interaction of architecture with art or literature. Critical historiography adds another dimension to this emerging research topic that further encapsulates multiple levels of criticism. In recognition of a growing interest for historiography, it can be argued that the critical aspects of historiography may serve as crucial instruments for an enhanced understanding of architectural historiography. In this article, the realm of architectural historiography is investigated through a multidisciplinary perspective that revisits architectural criticism, critical historiography, modern architecture, phenomenology, and a number of aspects of architectural historiography in the Swedish Million Homes Program.

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