Die eindeutige Primfaktorzerlegung

Zusammenfassung: Die Frage der Primfaktorzerlegung in Unterringen der komplexen Zahlen und der unmittelbar damit zusammenhängenden Sätze wird in der heutigen Algebra ohne grossen Aufwand und fast nebenbei behandelt: Studierende haben damit auch kaum Schwierigkeiten. In der Geschichte allerdings verlief die Entwicklung alles andere als gradlinig. Ein genauerer Blick auf die historischen Einzelheiten erlaubt interessante und in vielerlei Hinsicht überraschende Einsichten in die vertrackte Art und Weise, wie sich Mathematik manchmal entwickelt. Hier soll diese Geschichte erzählt werden, wie sie sich aus den neueren mathematikhistorischen Forschungen von H.M. Edwards, R. Bölling, O. Neumann und F. Lemmermeyer ergibt, und zwar auf einem Niveau, das einem Mathematikstudierenden nach einer Algebra-Vorlesung zugänglich is

Ein neuer Beweis eines Satzes von Magnus

The author provides a new and very simple proof of the following theorem of Magnus (2): Theorem. Let the group G have both a presentation with n + r generators and r relations also a presentation with n generators. Then G is the free group on n generator

Ein Zwischenfall, dem Heinz Hopf 1939 in Karlsruhe ausgesetzt war

Zusammenfassung: Am 9. Januar 1939 wurde Heinz Hopf, damals ordentlicher Professor für Mathematik an der ETH in Zürich, auf der Rückreise aus Berlin in Karlsruhe von der Gestapo verhaftet und in Untersuchungshaft genommen. Die dramatischen Umstände dieser gefährlichen Verwicklung lassen sich dank der heute im Archiv der Bibliothek der ETH vorhandenen Unterlagen in großen Zügen rekonstruieren. Darunter befinden sich auch Unterlagen, die dem Archiv erst vor kurzer Zeit von Dr. Klaus Völlm zur Verfügung gestellt worden sind. Es ergibt sich daraus ein beklemmendes Bild der Umstände, in denen Personen und Institutionen damals Entscheidungen haben fällen müsse

Two remarks on the homology of group extensions

In this note we apply a particular technique to obtain information on the homology homomorphism ε*: H*(G; A) → H* (Q; A) associated with a group extension and a Q-module A. The technique consists of using ε itself to pull-back (0.1); that is, we construct the pull-back extension induced from (0.1) by ε. This, however, is nothing but the semidirect product, N G, of N and G, with G operating on the left on N by conjugation. Thus we obtain from (0.1) the commutative diagram where ε1 is the projection and et is the multiplication ε1(n, x) = nx,nN,xG. We now apply the Lyndon-Hochschild-Serre spectral sequence functor to (0.2) and carry out computations in dimensions 2 and

Early Caregiving and Child Executive Function: Examining Preschool-aged Children with Prior Maltreatment

This thesis comprising three individual studies, examined the relationship between maltreatment, emotion-related parenting behaviours and executive function in a sample of maltreated and non-maltreated preschool children. Children (N = 107; ages 4-5) and caregivers were recruited from three cohorts: foster care agencies (serious maltreatment); child protection early intervention programs (low to moderate maltreatment); and community preschools (no maltreatment). Executive function was assessed using performance-based measures (Happy–Sad Stroop Task, Tapping Test and Dimensional Change Card Sort) and a caregiver rating scale (Behaviour Rating Inventory of Executive Function). Parenting was assessed with a self-report measure (Coping with Children’s Negative Emotions Scale) and maltreatment types were coded based on files and reports. Study 1 demonstrated that the profile of BRIEF-P scores for maltreated children was more elevated (indicative of worse executive function) relative to the non-maltreated group. Caregiver ratings were weakly correlated with performance-based executive function measures and maltreatment risk had a partial moderating effect on this association. Study 2 provided evidence that maltreatment risk uniquely predicted executive function, independent of parental depression and child ethnicity. Specifically, supportive parental reactions (emotion focused, expressive encouragement) buffered harmful effects of maltreatment on executive function, while unsupportive reactions (punitive, minimising) amplified the risk. Study 3 provided preliminary support for the common adverse effects of maltreatment on both pre-schoolers’ emotion and cognitive regulation. Findings highlighted that early maltreatment negatively impacts the development of executive function in preschool-aged children. Potential avenues for improving cognitive outcomes in maltreated young children include early assessment of executive function, evidenced-based parenting programs and early education programs