Numerology

2018 ◽  
pp. 69-74
Author(s):  
Christophe Reutenauer
Keyword(s):  
Quadratic Forms ◽  
Triple P ◽  
Mathematical Objects ◽  
Markoff Numbers

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.

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.

Prävention von oppositionellen und aggressiven Verhaltensstörungen bei Kindern: Triple P Die Triple P-Materialien und Informationen zur Triple P-Ausbildung sind zu beziehen über: Verlag für Psychotherapie, PAG Institut für Psychologie, Hoyastraße 1a, 48147 Münster, Tel.: 0251/518941. - ein Programm zu einer positiven Erziehung

2000 ◽  
Vol 9 (1) ◽  
pp. 20-29 ◽  
Author(s):  
Annett Kuschel ◽  
Yvonne Miller ◽  
Evi Köppe ◽  
Anne Lübke ◽  
Kurt Hahlweg ◽  
...  
Keyword(s):  
Public Health ◽  
Positive Parenting ◽  
Parenting Program ◽  
Triple P ◽  
Positive Parenting Program ◽  
Verhaltensstörungen Bei Kindern

Zusammenfassung. Es wird ein Überblick zum Stand der Forschung in den Bereichen Epidemiologie, Ätiologie und Behandlung von aggressivem und oppositionellem Problemverhalten bei Kindern gegeben. Triple P (Positive Parenting Program) - ein mehrstufiges präventives Programm zu positiver Erziehung - wird als ein erfolgreiches Beispiel für neue präventive Ansätze im Public Health Bereich vorgestellt. Dazu werden die theoretischen Grundlagen beschrieben, die verschiedenen Interventionsebenen und Materialien vorgestellt sowie Ergebnisse zur Effektivität berichtet. Abschließend wird über den Stand der Einführung von Triple P in Deutschland informiert.

Veränderungsmechanismen in Elterntrainings am Beispiel von Workplace Triple P

2010 ◽  
Vol 19 (2) ◽  
pp. 102-108 ◽  
Author(s):  
Doreen Hartung ◽  
Franziska Lups ◽  
Kurt Hahlweg
Keyword(s):  
Triple P

Elterntrainings haben sich als eine geeignete Interventionsform erwiesen, um dysfunktionales Erziehungsverhalten zu reduzieren und elterliche Selbstwirksamkeit zu steigern. Wie diese Veränderungen erzielt werden ist jedoch größtenteils unerforscht. Am Beispiel von Workplace Triple P wurde überprüft, ob Veränderungen in der elterlichen Selbstwirksamkeit zu Veränderungen im Erziehungsverhalten führen oder ob Veränderungen im Erziehungsverhalten zu Veränderungen in der elterlichen Selbstwirksamkeit führen. Als Stichprobe dienten berufstätige Eltern einer Experimental- und Wartelistenkontrollgruppe (N = 97 Eltern; 74 Mütter, 23 Väter; n [EG] = 42; n [KG] = 55). Regressionsbasierte Mediatoranalysen belegen, dass die Effekte des Trainings auf die elterliche Selbstwirksamkeit durch die Reduktion des dysfunktionalen Erziehungsverhaltens mediiert werden. Umgekehrt konnte jedoch nicht gezeigt werden, dass die Effekte des Trainings auf das dysfunktionale Erziehungsverhalten der Eltern durch Verbesserungen der elterlichen Selbstwirksamkeit mediiert werden. Diese Ergebnisse unterstützen die Bedeutung von Elterntrainings, in denen konkrete Erziehungs- und Stressmanagementfertigkeiten geübt und im Alltag umgesetzt werden, um positive Veränderungen im Familienleben zu erzielen.

Exploration-Stage Implementation Variation

2014 ◽  
Vol 222 (1) ◽  
pp. 37-48 ◽  
Author(s):  
Stephanie Romney ◽  
Nathaniel Israel ◽  
Danijela Zlatevski
Keyword(s):  
Parent Training ◽  
Average Cost ◽  
Cost Savings ◽  
Training Group ◽  
Group Differences ◽  
Triple P ◽  
Community Based ◽  
Parent Training Program ◽  
A Site ◽  
The Cost

The present study examines the effect of agency-level implementation variation on the cost-effectiveness of an evidence-based parent training program (Positive Parenting Program: “Triple P”). Staff from six community-based agencies participated in a five-day training to prepare them to deliver a 12-week Triple P parent training group to caregivers. Prior to the training, administrators and staff from four of the agencies completed a site readiness process intended to prepare them for the implementation demands of successfully delivering the group, while the other two agencies did not complete the process. Following the delivery of each agency’s first Triple P group, the graduation rate and average cost per class graduate were calculated. The average cost-per-graduate was over seven times higher for the two agencies that had not completed the readiness process than for the four completing agencies ($7,811 vs. $1,052). The contrast in costs was due to high participant attrition in the Triple P groups delivered by the two agencies that did not complete the readiness process. The odds of Triple P participants graduating were 12.2 times greater for those in groups run by sites that had completed the readiness process. This differential attrition was not accounted for by between-group differences in participant characteristics at pretest. While the natural design of this study limits the ability to empirically test all alternative explanations, these findings indicate a striking cost savings for sites completing the readiness process and support the thoughtful application of readiness procedures in the early stages of an implementation initiative.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Sign in / Sign up

Export Citation Format

Share Document