MacSphereThe MacSphere digital repository system captures, stores, indexes, preserves, and distributes digital research material.https://macsphere.mcmaster.ca:4432022-09-27T04:06:29Z2022-09-27T04:06:29ZHistory of the Earth Vol 11iSci Class of 2023http://hdl.handle.net/11375/278792022-09-27T03:28:59Z2021-01-01T00:00:00ZTitle: History of the Earth Vol 11
Authors: iSci Class of 20232021-01-01T00:00:00ZToric Ideals of Finite Simple GraphsKeiper, Grahamhttp://hdl.handle.net/11375/278782022-09-27T02:15:10Z2022-01-01T00:00:00ZTitle: Toric Ideals of Finite Simple Graphs
Authors: Keiper, Graham
Abstract: This thesis deals with toric ideals associated with finite simple graphs. In particular we
establish some results pertaining to the nature of the generators and syzygies of toric
ideals associated with finite simple graphs.
The first result dealt with in this thesis expands upon work by Favacchio, Hofscheier,
Keiper, and Van Tuyl which states that for G, a graph obtained by
"gluing" a graph H1 to a graph H2 along an induced subgraph, we can obtain the toric ideal associated to G from the toric ideals associated to H1 and H2 by taking their sum as ideals in the larger ring and saturating by a particular monomial f. Our contribution is to
sharpen the result and show that instead of a saturation by f, we need only examine the colon ideal with f^2.
The second result treated by this thesis pertains to graded Betti numbers of toric
ideals of complete bipartite graphs. We show that by counting specific subgraphs one
can explicitly compute a minimal set of generators for the corresponding toric ideals as well as minimal generating sets for the first two syzygy modules. Additionally we provide formulas for
some of the graded Betti numbers.
The final topic treated pertains to a relationship between the fundamental group
the finite simple graph G and the associated toric ideal to G. It was shown by
Villareal as well as Hibi and Ohsugi that the generators of a toric ideal associated to a finite simple graph correspond to the closed even walks of the graph G, thus linking algebraic properties to combinatorial ones. Therefore it is a natural question whether there is a relationship between the toric ideal associated to the graph G and the fundamental group of the graph G. We show, under the assumption that G is a bipartite graph with some additional assumptions, one can conceive of the set of binomials in the toric ideal with coprime terms, B(IG), as a group with an appropriately chosen operation ⋆ and establish a group isomorphism (B(IG), ⋆) ∼= π1(G)/H where H is a normal subgroup. We exploit this relationship further to obtain information about the generators of IG as well as bounds on the Betti numbers. We are also able to characterise all regular sequences and hence compute the depth of the toric ideal of G. We also use the framework to prove that IG = (⟨G⟩ : (e1 · · · em)^∞) where G is a set of binomials which correspond to a generating set of π1(G).2022-01-01T00:00:00ZMemory for temporally nonadjacent tonal centers mediated by musically salient featuresSpyra, Joannahttp://hdl.handle.net/11375/278772022-09-27T01:18:09Z2022-01-01T00:00:00ZTitle: Memory for temporally nonadjacent tonal centers mediated by musically salient features
Authors: Spyra, Joanna
Abstract: Research on memory often describes the remarkable longevity of music. However, memory for music is not uniform. Cook (1987) found that participants were not able to tell apart excerpts that modulated from those that did not when the excerpt was longer than 1 minute in length. This suggests that participants were no longer able to remember, and compare, musical keys after a relatively short period of time. Farbood (2016) and Woolhouse et al. (2016) further explored the limitations of memory for tonal structures finding that, in fact, harmonic memory only lasts up to 21 seconds after modulation. However, this research was done using homophonic stimuli—arpeggios or quarter-note chords—that may not be representative of the music participants would be listening to regularly. The focus of this project was to explore how the addition of certain musical features, such as melodic or rhythmic figurations, may influence harmonic memory. Observing these possible influences may provide us with insight into the processes responsible for auditory memory and how it differs from other domains, such as speech or vision. Chapter 1 explores prominent memory literature and music cognition experiments that support, or address concerns with, common memory models. Here, I introduce a cognitive system which reconciles music research with models by memory specialists such as Baddeley and Snyder. Chapter 2 presents a detailed account of background empirical literature, including Farbood (2016) and Woolhouse et al. (2016). Though fundamental to the exploration of temporally nonadjacent harmonic memory, this research is potentially limited in its generalizability due to the homophonic nature of the stimuli. Chapter 3 explores this limitation by testing the effects of adding surface features—melodic and rhythmic components often used for elaboration in composition—on memory for large-scale tonal structures. Results found that harmonic memory is, indeed, enhanced and prolonged by these elaborative components, lasting up to 33 seconds, well past the limit found in previous research. Farbood (2016) further claimed that harmonic memory is significantly interrupted by new, highly harmonic excerpts. However, results from Woolhouse et al. (2016), Spyra et al. (2021) and those from Chapter 3 all question this claim as they employed stimuli that was highly harmonic. Chapter 4 investigates the contradiction by testing whether functional diatonic, functional chromatic, or random sequences degraded harmonic memory for an original key. Functional diatonic intervening information resulted in increased harmonic memory, directly contradicting Farbood’s original findings. In Chapter 5, these results are explored in terms of prominent memory models in the field of cognition, supporting standard models of memory such as that by Baddeley and Hitch (1974) or Atkinson and Shiffrin (1968), as well as my proposed cognitive system. This is further elaborated by discussing the process of undergoing a musical judgement task from perception through to decision-making. In summary, this project suggests that more generalizable stimuli containing realistic musical features produce a significant boost in harmonic memory. Furthermore, this arguably calls into question standard practices in analysis that categorize surface features as hierarchically less important than ’deeper’ harmonic events, and thus, potentially less important from a cognitive perspective. Which is to say, this evidence suggests that these features may play a vital role in remembering nonadjacent harmonic
structures.2022-01-01T00:00:00ZRules of Customary Behaviour in the Mūlasarvāstivāda-vinayaAltenburg, Gerjanhttp://hdl.handle.net/11375/278762022-09-27T01:09:48Z2022-09-01T00:00:00ZTitle: Rules of Customary Behaviour in the Mūlasarvāstivāda-vinaya
Authors: Altenburg, Gerjan
Abstract: This dissertation is a study of the rules of customary behaviour (āsamudācārika-dharmas) found in a North Indian Buddhist monastic law code, the Mūlasarvāstivāda-vinaya. Other than Gregory Schopen, few scholars have noted the significance of these rules. Schopen points out that according to this vinaya, adherence to rules of customary behaviour is foundational for achieving nirvāṇa. Yet, these rules have been practically ignored in contemporary scholarship. Building on Schopen’s work, I approach this material with two main questions: 1) What are rules of customary behaviour? and 2) How do rules of customary behaviour function in the Mūlasarvāstivāda-vinaya?
In an attempt to answer these questions, I explore passages from the Mūlasarvāstivāda-vinaya in which the Buddha prescribes rules of customary behaviour for specific monastics in response to a variety of narrative situations. I organize this material into three thematic chapters. First, I discuss rules of customary behaviour related to the administration of the monastic community (saṃgha). Next, I explore rules of customary behaviour relevant only in specific environments. Finally, I investigate rules of customary behaviour prescribed in response to illnesses in the saṃgha.
Through the above exploration, I demonstrate three main points:
1) that there are three ways that rules of customary behaviour appear in the Mūlasarvāstivāda-vinaya;
2) the adoption of rules of customary behaviour prescribed in narratives in the Mūlasarvāstivāda-vinaya does not necessarily signal the creation of a new monastic office or official duty; and
3) In the vast majority of cases, these rules seem to be designed to protect the integrity of the saṃgha and accommodate monks or nuns who are experiencing temporary challenges to their ecclesiastical status.2022-09-01T00:00:00Z