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Welcome to the Department of Mathematics

What's Happening This Week In Mathematics....

CSC Weekly Seminar
Friday, November 28, 2014
2:30 pm
TASC-2, Rm 8500

Speaker: Paul Tupper (Mathematics, SFU)


Title: Exemplar Dynamics and Sound Merger in Language

Abstract:
We develop a model of phonological contrast in natural language. Specifically, the model describes the maintenance of contrast between different words in a language, and the elimination of such contrast when sounds in the words merge. An example of such a contrast is that provided by the two vowel sounds "i" and "e", which distinguish pairs of words such as "pin" and "pen" in most dialects of English. We model language users' knowledge of the pronunciation of a word as consisting of collections of labeled exemplars stored in memory. Each exemplar is a detailed memory of a particular utterance of the word in question. In our model an exemplar is represented one or two phonetic variables along with a weight indicating how strong the memory of the utterance is. Starting from an exemplar-level model we derive integro-differential equations for the evolution of exemplar density fields in phonetic space. Using these latter equations we investigate under what conditions two sounds merge causing words to no longer contrast. Our main conclusion is that for the preservation of phonological contrast, it is necessary that anomalous utterances of a given word are discarded, and not merely stored in memory as an exemplar of another word.

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Wei Chen, M.Sc. Thesis Defence, Mathematics   Room: PIMS 8500 TASC II

Title:
Enumeration of Set Partitions Refined by Crossing and Nesting Numbers

Abstract:
The standard representation of set partitions gives rise to two natural statistics: a crossing number and a nesting number. Chen, Deng, Du, Stanley, and Yan (2007) proved, via a non-trivial bijection involving sequences of Young tableaux, that these statistics have a symmetric joint distribution. Recent results by Marberg (2013) has lead to algorithmic tools for the enumeration of set partitions with fixed crossing number and fixed nesting number. In this thesis we further consider set partitions refined by these two statistics. These sub-classes can be recognized by finite automata, and consequently have rational generating functions. Our main contribution is an investigation into the structure of the automata, the corresponding adjacency matrices, and the generating functions/

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Event: 56th annual meeting of the Canadian Operational Research Society (CORS) in Ottawa.

As in the past two years, we encouraged the undergraduates in Math 402W Operations Research Clinic to submit their projects to the CORS undergraduate student paper competition.  And, once again, they won both prizes.  This year, all 3 projects submitted were chosen as finalists, and presented their work at the meeting. (The fourth finalist was from the University of Alberta; nine entries were received.)  Congratulations go to:

Winners:
Kishley Bhalla, Craig Mathews, W. Brett Robinson and Katie Sclater "Selecting Optimal Tolling Levels: A Case Study for the Fraser River in the Greater Vancouver Area"

Honourable Mention:
Nicole Mo, Alborz Namazi, Joyce Tai and Eric Yuen "Optimal Locations of Telecommunication Equipment: A Case Study for the City of Richmond, British Columbia, Canada"

Finalists:
Kingsley Cheang, Feiqi He, Sarah Lin and Ashlie Neufelt "The Community Mailbox Location-Routing Problem"

Additionally, Second Prize in the CORS Practice Competition went to Daniel Karapetyan (SFU Math postdoc 2011-13) and Abraham Punnen, for their paper "Operational Research Models and Algorithms for Fleet Size Planning and Schedule Optimisation for the British Columbia Ferry Services Inc."  Finalists for the practice competition included teams from the University of Toronto and IBM, first prize went to UOIT.

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Upcoming Events

  • CSC Weekly Seminar - Paul Tupper (SFU)
    2:30 PM - 3:30 PM
    November 28, 2014
    CSC Weekly Seminar Friday, November 28, 2014 2:30 pm TASC-2, Rm 8500 Speaker: Paul Tupper (Mathematics, SFU) Title: Exemplar Dynamics and Sound Merger in Language Abstract: We develop a model of phonological contrast in natural language. Specifically, the model describes the maintenance of contrast between different words in a language, and the elimination of such contrast when sounds in the words merge. An example of such a contrast is that provided by the two vowel sounds "i" and "e", which distinguish pairs of words such as "pin" and "pen" in most dialects of English. We model language users' knowledge of the pronunciation of a word as consisting of collections of labeled exemplars stored in memory. Each exemplar is a detailed memory of a particular utterance of the word in question. In our model an exemplar is represented one or two phonetic variables along with a weight indicating how strong the memory of the utterance is. Starting from an exemplar-level model we derive integro-differential equations for the evolution of exemplar density fields in phonetic space. Using these latter equations we investigate under what conditions two sounds merge causing words to no longer contrast. Our main conclusion is that for the preservation of phonological contrast, it is necessary that anomalous utterances of a given word are discarded, and not merely stored in memory as an exemplar of another word.
  • Wei Chen, M.Sc. Thesis Defence, Mathematics Room: PIMS 8500 TASC II
    10:00 AM - 12:00 PM
    December 2, 2014
    (Sr. Supervisor: Marni Mishna) (Co-Supervisor: Lily Yen) Title: Enumeration of Set Partitions Refined by Crossing and Nesting Numbers Abstract: The standard representation of set partitions gives rise to two natural statistics: a crossing number and a nesting number. Chen, Deng, Du, Stanley, and Yan (2007) proved, via a non-trivial bijection involving sequences of Young tableaux, that these statistics have a symmetric joint distribution. Recent results by Marberg (2013) has lead to algorithmic tools for the enumeration of set partitions with fixed crossing number and fixed nesting number. In this thesis we further consider set partitions refined by these two statistics. These sub-classes can be recognized by finite automata, and consequently have rational generating functions. Our main contribution is an investigation into the structure of the automata, the corresponding adjacency matrices, and the generating functions.
  • Yian Xu, M.Sc. Thesis Defence, Mathematics K9509
    3:00 PM - 5:00 PM
    December 9, 2014
    (Sr. Supervisor: Luis Goddyn) Title: Generalized Thrackles and Graph Embeddings Abstract Let G be a graph drawn on a surface X such that every two distinct edges meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that e(G) ? n(G) for every thrackle on sphere. The best known bound is e(G) ? 1:428n(G). By using discharging rules we show that m ? 1:392n, provided that four particular graphs have no thrackle drawing. Furthermore we show that the following are equivalent: ? G has a drawing on X where every two edges meet an odd number of times (a generalized thrackle) ? G has a drawing on X where every two edges meet exactly once [at a common point] (a 1􀀀thrackle) ? G has a special embedding on a surface whose genus differs from the genus of X by at most 1. This theorem, when applied to embedded signed graphs, implies recent results [European J. Combin., 30, 1704-1717 (2009)] regarding the crossing number.
  • Daryl Funk, Ph.D. Thesis Defence, Mathematics Room K9509 Burnaby Campus
    10:00 AM - 12:30 PM
    January 8, 2015
    (Sr. Supervisor: Matt DeVos) (Co-Supervisor: Luis Goddyn) Title: On Excluded Minors and Biased Graph Representations of Frame Matroids Abstract: A biased graph is a graph in which every cycle has been given a bias, either balanced or unbalanced. Biased graphs provide representations for an important class of matroids, the frame matroids. As with graphs, we may take minors of biased graphs and of matroids, and a family of biased graphs or matroids is minor-closed if it contains every minor of every member of the family. For any such class, we may ask for the set of those objects that are minimal with respect to minors subject to not belonging to the class — i.e., we may ask for the set of excluded minors for the class. A frame matroid need not be uniquely represented by a biased graph. This creates complications for the study of excluded minors. Hence this thesis has two main intertwining lines of investigation: (1) excluded minors for classes of frame matroids, and (2) biased graph representations of frame matroids. Trying to determine the biased graphs representing a given frame matroid leads to the necessity of determining the biased graphs representing a given graphic matroid. We do this in Chapter 3. Determining all possible biased graph representations of non-graphic frame matroids is more di cult. In Chapter 5 we determine all biased graphs representations of frame matroids having a biased graph representation of a certain form, subject to an additional connectivity condition. Perhaps the canonical examples of biased graphs are group-labelled graphs. Not all biased graphs are group-labellable. In Chapter 2 we give two characterisations of those biased graphs that are group labellable, one topological in nature and the other in terms of the existence of a sequence of closed walks in the graph. In contrast to graphs, which are well-quasi-ordered by the minor relation, this characterisation enables us to construct infinite antichains of biased graphs, even with each member on a fixed number of vertices. These constructions are then used to exhibit infinite antichains of frame matroids, each of whose members are of a fixed rank. In Chapter 4, we begin an investigation of excluded minors for the class of frame matroids by seeking to determine those excluded minors that are not 3-connected. We come close, determining a set E of 18 particular excluded minors and drastically narrowing the search for any remaining such excluded minors. Keywords: Frame matroid; biased graph; excluded minors; representations; group-labelling; gain graph; well-quasi-ordering; lift matroid; graphic matroid
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