ETD Collection

Permanent URI for this collectionhttps://wiredspace.wits.ac.za/handle/10539/104


Please note: Digitised content is made available at the best possible quality range, taking into consideration file size and the condition of the original item. These restrictions may sometimes affect the quality of the final published item. For queries regarding content of ETD collection please contact IR specialists by email : IR specialists or Tel : 011 717 4652 / 1954

Follow the link below for important information about Electronic Theses and Dissertations (ETD)

Library Guide about ETD

Browse

Filters

2008 - 200912010 - 20171

Settings

Has files: YesSubject: search.filters.subject.Arithmetic

Search Results

Now showing 1 - 2 of 2
  • No Thumbnail Available
    Item
    Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations
    (2017) Alanazi, Abdulaziz Mohammed
    In this thesis, various partition functions with respect to `-regular overpartitions, a special partition inequality and partition con gurations are studied. We explore new combinatorial properties of overpartitions which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition and `-regular partition functions. We provide both generating function and bijective proofs. We then establish an in nite set of Ramanujan-type congruences for the `-regular overpartitions. This signi cantly extends the recent work of Shen which focused solely on 3{regular overpartitions and 4{regular overpartitions. We also prove some of the congruences for `-regular overpartition functions combinatorially. We then provide a combinatorial proof of the inequality p(a)p(b) > p(a+b), where p(n) is the partition function and a; b are positive integers satisfying a+b > 9, a > 1 and b > 1. This problem was posed by Bessenrodt and Ono who used the inequality to study a maximal multiplicative property of an extended partition function. Finally, we consider partition con gurations introduced recently by Andrews and Deutsch in connection with the Stanley-Elder theorems. Using a variation of Stanley's original technique, we give a combinatorial proof of the equality of the number of times an integer k appears in all partitions and the number of partition con- gurations of length k. Then we establish new generalizations of the Elder and con guration theorems. We also consider a related result asserting the equality of the number of 2k's in partitions and the number of unrepeated multiples of k, providing a new proof and a generalization.
  • No Thumbnail Available
    Item
    The transition across the cognitive gap - the case for long division - : Cognitive architecture for division : base ten decomposition as an algorithm for long division
    (2008-11-04T12:53:01Z) Du Plessis, Jacques Desmond
    This is an action research study which focuses on a didactical model founded on base ten decomposition as an algorithm for performing division on naturals. Base ten decomposition is used to enhance the algebraic structure of division on naturals in an attempt to cross the cognitive divide that currently exists between arithmetic long division on naturals and algebraic long division on polynomials. The didactical model that is proposed and implemented comprises three different phases and was implemented over five one hour lessons. Learners’ work and responses which were monitored over a fiveday period is discussed in this report. The structure of the arithmetic long division on naturals formed the conceptual basis from which shorter methods of algebraic long division on polynomials were introduced. These methods were discussed in class and reported on in this study.