title: Decomposing dense bipartite graphs into 4-cycles

authors: Cavenagh, Nicholas J.

abstract: Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles.

text: Decomposing dense bipartite graphs into 4-cycles Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles.

year: 2015-02-25

title: Constellations and their relationship with categories

authors: Gould, Victoria

subjects: mathematics

abstract: Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category (Formula presented.), its canonical extension, in a simplest possible way, and that P is a quotient of (Formula presented.) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences (Formula presented.) on a given category (Formula presented.) (those for which (Formula presented.), and show that the category of constellations is equivalent to the category of (Formula presented.)-categories, that is, categories equipped with distinguished canonical congruence (Formula presented.). The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are (Formula presented.)-categories) may be subsumed by constellation theory.

text: Constellations and their relationship with categories Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category (Formula presented.), its canonical extension, in a simplest possible way, and that P is a quotient of (Formula presented.) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences (Formula presented.) on a given category (Formula presented.) (those for which (Formula presented.), and show that the category of constellations is equivalent to the category of (Formula presented.)-categories, that is, categories equipped with distinguished canonical congruence (Formula presented.). The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are (Formula presented.)-categories) may be subsumed by constellation theory.

year: 2017

title: An evaluation of Te Rau Puawai workforce 100: Academic mentor's perspectives

authors: Nikora, Linda Waimarie

subjects: Maori students

abstract: To evaluate the Te Rau Puawai programme, the Ministry of Health commissioned the
Maori and Psychology Research Unit of the University of Waikato in July 2001. The
overall aim of the evaluation was to provide the Ministry with a clearer understanding
of the programme including: the perceived critical success factors, the barriers if any
regarding Te Rau Puawai, the impact of the programme, the extent to which the
programme may be transferable, gaps in the programme, and suggested
improvements.
As an integral component of the Te Rau Puawai programme and as a specialised
source of support, academic mentors were identified as an important group to receive
comment from. This report documents the evaluative exchange that occurred by way
of email questionnaire, that we had with 18 out of 46 academic mentors of Te Rau
Puawai bursars in the College of Humanities and Social Sciences responded to our
questionnaire.

text: An evaluation of Te Rau Puawai workforce 100: Academic mentor's perspectives To evaluate the Te Rau Puawai programme, the Ministry of Health commissioned the
Maori and Psychology Research Unit of the University of Waikato in July 2001. The
overall aim of the evaluation was to provide the Ministry with a clearer understanding
of the programme including: the perceived critical success factors, the barriers if any
regarding Te Rau Puawai, the impact of the programme, the extent to which the
programme may be transferable, gaps in the programme, and suggested
improvements.
As an integral component of the Te Rau Puawai programme and as a specialised
source of support, academic mentors were identified as an important group to receive
comment from. This report documents the evaluative exchange that occurred by way
of email questionnaire, that we had with 18 out of 46 academic mentors of Te Rau
Puawai bursars in the College of Humanities and Social Sciences responded to our
questionnaire.

year: 2002-05-01