Browsing by Author "Fallat, Shaun"

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Open Access
Actionable Three-Way Decisions
(Faculty of Graduate Studies and Research, University of Regina, 2018-09) Gao, Cong; Yao, Yiyu; Fallat, Shaun; Hamilton, Howard; Hepting, Daryl; Yang, Xue-Dong; Ciucci, Davide
In this thesis, we analyze both the trisecting and acting aspects of three-way decisions. In an evaluation based model of three-way decisions, there are two steps: trisecting and acting. The trisecting step constructs three regions based on an evaluation func- tion and a pair of thresholds. The acting step adopts proper strategies to deal with objects in these regions. For the trisecting step, this thesis examines statistical interpretations for the con- struction of three regions. The interpretations rely on an understanding that the middle region consists of normal or typical instances in a population, while two side regions consist of, abnormal or atypical instances. By using statistical information such as median, mean, percentiles, and standard deviation, two interpretations are discussed. One is based on non-numeric values and the other is based on numeric values. For non-numeric values, median and percentiles are used to construct three pair-wise disjoint regions. For numeric values, mean and standard deviation are used. The interpretations provide a solid statistical basis of three-way decisions for appli- cations. This thesis analyzes a chi-square statistic as a measure for searching for the optimal pair of thresholds for trisecting. An optimization based method for determining the pair of thresholds is to minimize or maximize an objective function that quanti es the quality, cost, or bene t of a trisection. We use the chi-square statistic to interpret and establish an objective function in the context of classi cation. The maximization of the chi-square statistic searches for a strong correlation between the trisection and the classi cation. For the acting step, this thesis introduces actionable strategies to three-way de- cision. We present a general framework of actionable three-way decisions with four change-based actionable models according to action bene t and action cost. Two of the four models provide the bounds of the cost and bene t and the other two models quantify the maximum bene t under limited cost and the minimum cost for a desired bene t, respectively. We design and analyze algorithms for these models. To reduce action cost and increase bene t, we introduce the R4 reduction frame- work for actionable three-way decision. The framework consists of reductions of attributes, attribute-value pairs, classi cation rules, and actions for creating more bene t and reducing cost. The rst three types of reductions are rede ned for the context of three-way decisions and the action reduction is proposed. Attribute reduc- tion removes some attributes from all classi cation rules to reduce the action cost. Attribute-value pair reduction shortens the left hand side of a rule to reduce the ac- tion cost without sacri cing any classi cation power or action bene t. Rule reduction and action reduction remove redundant classi cation rules and actions, respectively, to reduce computational cost. The Addition strategy for reduction is adapted and its correctness is proven. Based on this strategy, an algorithm for attribute and attribute-value pair reductions is designed. Finally, we report experimental results to support the proposed four actionable three-way decision models and the R4 reduction framework.
Open Access
Calculating and Preserving Star Sets and Star Complements of General Matrices
(Faculty of Graduate Studies and Research, University of Regina, 2017-08-18) Bergen, Ryan Paul; Fallat, Shaun; Meagher, Karen; Hoeber, Orland
This thesis presents several results relating to star sets and star complements of graphs. While a method for calculating star sets and star complements involving pro- jection matrices has been known since their introduction, a second method involving determinants is demonstrated and shown to be equivalent to the rst method. Some of the theory for star sets and star complements is expanded to general diagonalizable matrices, regardless of symmetry. The concept of preserving star sets between two general matrices is introduced and shown to be an equivalence relation, and attempts are made to classify what types of matrices can preserve the star sets of a general matrix. Finally, we determine some results for general matrices that occur when star set preservation overlaps with other equivalence relations.
Open Access
Cameron-Liebler Sets for 2-Transitive Groups
(Faculty of Graduate Studies and Research, University of Regina, 2020-11) Palmarin, Daniel Michael; Fallat, Shaun; Meagher, Karen; Herman, Allen; Butz, Cortney
This research was conducted on 2-transitive groups whose minimal normal subgroup is abelian. Suppose G is such a group and ΓG is its derangement graph. Any maximum coclique S of ΓG has a characteristic vector xS. Each xS is a boolean vector contained in a particular module, which is called the permutation module MP. This module has a dimension of 1 + (n − 1)2, where n = deg(G), and it is spanned by {xij | i,j∈{1,...,n}}, where each xij is the characteristic vector of Sij, the set of permutations that map i to j. Apart from the xij, which correspond to the stabilizers of G and their cosets, this research set out to find any other boolean vectors that are contained in Mp using linear programming. Henceforth, such boolean vectors are defined to be Cameron-Liebler sets for 2-transitive groups. In addition to finding Cameron-Liebler sets, analyses were performed on each group to determine: (1) whether the strict EKR property holds; (2) the number of maximum cocliques that are subgroups, cosets, or neither; (3) isomorphism classes and conjugacy classes of the maximum cocliques that are subgroups; (4) the dimension of C′, the maximum cliques that are subgroups (along with their right cosets), and C, all maximum cliques; and (5) the spectrum of ΓG and whether the ratio bound is satisfied with equality.
Open Access
Colin de Verdiere parameters of chordal graphs
(International Linear Algebra Society, 2013-01) Mitchell, Lon; Fallat, Shaun
The Colin de Verdi`ere parameters mu and nu are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For nu the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on computing μ for split graphs.
Open Access
Complexity Parameters for Learning Multi-Label Concept Classes,
(Faculty of Graduate Studies and Research, University of Regina, 2015-03) Samei, Rahim; Zilles, Sandra; Yang, Boting; Hamilton, Howard; Mouhoub, Malik; Fallat, Shaun; Pestov, Vladimir
In Computational Learning Theory, one way to model a concept is to consider it as a member of the Cartesian products of instances (sets), where each instance may correspond to a binary or multi-valued domain. A concept class is a set of concepts, and the goal of learning algorithms is to identify the target concept in a concept class from a small number of examples, i.e., labeled instances. This thesis studies multi-label concept classes and three important learning complexity parameters for these classes. The first parameter examined in this work is the Vapnik-Chervonenkis-dimension (VCD) and its previously studied analogues for multi-label concept classes such as the Graph-dimension, Pollard's pseudo-dimension and the Natarajan dimension. This thesis also extends the study of sample compression schemes to multi-label concept classes. A sample compression scheme (SCS) for a concept class C compresses every set S of labeled examples for some concept in C to a subset, which is decompressed to some concept that is consistent with S. The size of the SCS is the cardinality of its largest compressed set. The best possible size of an SCS is the second parameter studied below. This work formulates a sufficient condition, which we call the reduction property, for a notion of VC-dimension (VCD ) to yield labeled compression schemes for maximum classes of VCD d in which the compression sets have size at most d. This compression scheme is in fact an extension of Floyd and Warmuth's binary scheme to the multi-label case. The same condition also yields a so-called tight SCS, which we de ne to generalize Kuzmin and Warmuth's unlabeled binary scheme to the multi-label case. The Graph dimension satis es our su cient condition, while neither Pollard's pseudo-dimension nor the Natarajan dimension does. Moreover, this thesis shows that any class of Graph-dimension 1 has an SCS of size 1. The third parameter studied in this thesis is the recursive teaching dimension (RTD), a complexity parameter of the recently introduced recursive teaching model, which is of interest in the study of binary concept classes because of its connection to the VCD. In the recursive teaching model, the teacher and learner agree to start the learning process with the easiest concept in the class, i.e., a concept with the minimum teaching dimension (minimum number of labeled examples needed to identify a concept among all other concepts in the class). This concept is then removed from the concept class and a concept with the minimum teaching dimension in the remaining class is chosen to teach and remove, and the process continues until no concepts are left in the class. This procedure is called a recursive teaching plan and the recursive teaching dimension of the class is the largest minimum teaching dimension encountered in the plan. This thesis establishes a further connection between RTD and VCD in both the multi-label and binary cases, by providing an upper bound on the size of classes of a given RTD, analogous to Sauer's bound on the size of classes of a given VCD . Maximum (largest possible) classes of a given VCD are proven to be RTD-maximum as well. It is shown that for any VCD -maximum class C where VCD fulfills the reduction property, there is a teaching plan for C in which the recursive teaching sets coincide with the compression sets resulting from a tight compression scheme. Methodologically, an algebraic approach turns out to be useful, for example to prove an interesting graph-theoretic property of VCD -maximum classes.
Open Access
Convergence and Comparison Theorems for Various Splittings of Matrices Based on Generalized Inverses
(Faculty of Graduate Studies and Research, University of Regina, 2015-03) Agasthian, Vijayaparvathy; Guo, Chun-Hua; Argerami, Martin; Fallat, Shaun; Yang, Boting; Wei, Yi-Min
The convergence of iterative methods for numerically solving the linear systems of equations associated with different types of splittings has been well studied in the literature. In this dissertation, we define new types of splittings based on generalized inverses (Moore-Penrose inverse, Drazin inverse, Group inverse) and present convergence results based on those splittings, namely, Weak Nonnegative Proper Splitting, Proper Regular Splitting, Index Proper Regular Splitting, Weak Nonnegative Index Proper Splitting, Group Proper Regular Splitting, and Weak Nonnegative Group Proper Splitting. We also present some new comparison theorems between the spectral radii of matrices, which are useful in the analysis of the rate of convergence of iterative methods for the different types of splittings of matrices.
Open Access
Dietary Niche and Foraging Ecology of a Generalist Predator, Double-Crested Cormorant (Phalacrocorax Auritus): Insight Using Stable Isotopes
(Faculty of Graduate Studies and Research, University of Regina, 2012-04) Doucette, Jennifer Lee; Somers, Christopher; Brigham, Mark; Wissel, Bjoern; Fallat, Shaun; Hobson, Keith
The ability of predator populations to expand their ranges and adapt to new environments is often attributed to having a generalist dietary strategy, which is thought to be represented both at the population and individual level. Cormorants (Phalacrocorax spp.) are considered to be opportunistic generalists capable of using a wide variety of aquatic prey. This reputation is partially responsible for the global conflict between piscivorous cormorants and fish harvesters, which is one of the most widespread wildlife management issues in history. Despite the persistent belief that cormorants adversely affect economically important fish populations, relatively little is known about their trophic ecology and habitat use. Stable nitrogen and carbon isotopes are popular tools for studying food webs, and offer a comprehensive assessment of diet, trophic position, and ecological niche when combined with traditional diet analyses. However, the interpretation of isotope data may be confounded by variation in the lipid content of sample tissues. No validated lipid-normalization procedures are currently available for any cormorant species, or any fish-eating birds. As such, I first determined the effect of lipids on the stable carbon and nitrogen isotopes (δ13C and δ15N) values in cormorant tissues, and tested three published lipid-normalization models on stable isotope signatures in double-crested cormorant (P. auritus) muscle and liver tissues. The presence of lipids in cormorant muscle and liver altered the stable isotopes values, indicating corrections were required. However, the effects of lipids in cormorants were unpredictable and thus violated a major assumption of published lipid-normalization models. As a result, lipids must be chemically removed from cormorant muscle and liver tissue. I then examined the diet and trophic position of breeding populations of double-crested cormorants from three different lakes. The results revealed that cormorants generally occupied top-predator positions and relied heavily on pelagic prey in all food webs examined. The isotopic values of cormorants and pelagic predatory fish were sometimes similar, suggesting that dietary overlap is possible. To determine whether cormorants are true dietary generalists I studied double-crested cormorants from breeding colonies spanning three major ecoregions. Analyses of stomach contents revealed that at the population level cormorant diet varied widely by location, likely reflecting local food-web structure. However, within populations individuals were much more specialized than expected. Temporal shifts in δ13C and δ15N values in cormorant tissues with different turnover rates (muscle vs. liver) indicated that foraging varied among populations. The dietary niche occupied by cormorants will affect their interactions with fish, highlighting the importance of understanding their impacts to fish populations both at the population and individual level. Ultimately, my research has shown that cormorants do not consume prey indiscriminately, and instead may have more specific and uniform dietary niche requirements than previously considered. From a management perspective, cormorants should not be assumed to have negative effects on fish in all situations; however, further attention is required to determine the impacts of dietary overlap with sport fish. Ecologically, I have shown that generalist species can be much more consistent and specialized than previously considered. Further, individuals within generalist species may be highly specialized, which will change the overall effects of the population on other species in the food web.
Open Access
Distinguishing Linear Sets and Pattern Languages With Membership Examples
(Faculty of Graduate Studies and Research, University of Regina, 2017-09) Gao, Ziyuan; Zilles, Sandra; Yang, Boting; Yao, Yiyu; Fallat, Shaun; Reidenbach, Daniel
In Computational Learning Theory, a binary concept is often represented by a set of pairs (x; l), where x varies over a set of instances known as a universe or instance space, and l is a label, either "+" or "-", indicating whether or not x belongs to the concept. This thesis will consider a type of machine learning task known as supervised learning, where an algorithm M is fed with a set of labelled data for a target concept belonging to a given class of concepts, and M produces a hypothesis function that is used to accurately predict the correct labels of unseen instances. The learning process typically involves two agents, a teacher and a learner. The task of a teacher is to provide labelled examples for a target concept to a learner, who in turn makes a hypothesis based on the seen data. The present thesis focusses on the teacher's role; specifically, the question: given any concept class C and a learning algorithm M, what is the minimum number of labelled examples needed by M to exactly identify any given concept belonging to C? This quantity may be conceived as a measure of the information complexity or teaching complexity of C. Our work will study the teaching complexity of three related families of concept and the erasing pattern languages. Linear sets and pattern languages are mathematical objects that are closely connected to automata theory and formal languages. In this thesis, the teaching complexity of a concept class will be measured mainly by two combinatorial parameters, the teaching dimension (TD) and the recursive teaching dimension (RTD). The TD of a concept C with respect to a class C containing C is defined as the size of a smallest sample (called a teaching set for C with respect to C) that is labelled consistently with C but not with any C0 in C distinct from C; the TD of C is the worst-case TD over all C in C (or infinity if the set consisting of the teaching dimensions of all C in C with respect to C is unbounded). Concerning the TD, we are chiefly interested in two questions with respect to any given concept class C: first, is there a decidable characterization of the concepts in C that have finite TD with respect to C; second, given a representation for concepts in C, how does the TD of C with respect to C vary with the size of the representation of C? The second main parameter studied in this work, the recursive teaching dimension (RTD), arises from the recently introduced recursive teaching model, and it is of particular interest in learning theory due to its links to other learning models as well as to the Vapnik-Chervonenkis dimension { one of the most important parameters in Statistical Learning Theory { in the context of finite concept classes. Our main finding about the RTD in this thesis is that for many classes of linear sets (resp. pattern languages), recursive teaching is significantly more sample efficient than the classical teaching protocol.
Open Access
Eigenvalues of K-Uniform Hypergraphs
(Faculty of Graduate Studies and Research, University of Regina, 2017-08) Gorr, Adam Vernon; Meagher, Karen; Fallat, Shaun; Herman, Allen; Gosselin, Shonda; Butz, Cory
We de ne two separate attempts to generalize the de nition of eigenvalues to hypergraphs and show several results related to each. The rst approach is rooted in 2-dimensional matrices and allows for the generalization of many results from graph theory. The second approach covered is more sophisticated and may only be applied to k-uniform hypergraphs. We include the development of a sound algorithm using the resultant of polynomials that can be used for any k-uniform hypergraph. Speci c examples are provided to demonstrate the power of the algorithm. Further, we show that certain results hold for the eigenvalues and associated eigenvectors of k-uniform hypergraphs and those hypergraphs obtained from combinatorial designs such as Steiner triple systems.
Open Access
The enhanced principal rank characteristic sequence
(Elsevier, 2015) Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, Michael
The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n ×n matrix is a sequence from A,S, or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the kth entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence ofAs andSs ending inAis attainable, and any sequence ofAs andSs followed by one or moreNs is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications.
Open Access
The enhanced principal rank characteristic sequence for skew-symmetric matrices
(Elsevier, 2015-08-15) Fallat, Shaun; Olesky, Dale; van den Driessche, Pauline
The enhanced principal rank characteristic sequence (epr-sequence) was originally defined for an n ×n real symmetric matrix or an n ×n Hermitian matrix. Such a sequence is defined to be l1l2···ln where lk is A,S, or N depending on whether all, some, or none of the matrix principal minors of order k are nonzero. Here we give a complete characterization of the attainable epr-sequences for real skew-symmetric matrices. With the constraint that lk=0 if k is odd, we show that nearly all epr-sequences are attainable by skew-symmetric matrices, which is in contrast to the case of real symmetric or Hermitian matrices for which many epr-sequences are forbidden. ©2015
Open Access
Enriched model categories and the Dold-Kan correspondence
(Faculty of Graduate Studies and Research, University of Regina, 2024-10) Ngopnang Ngompe, Arnaud; Frankland, Martin; Stanley, Donald; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Ponto, Kate
The work we present in this thesis is an application of the monoidal properties of the Dold–Kan correspondence and is constituted of two main parts. In the first one, we observe that by a theorem of Christensen and Hovey, the category of nonnegatively graded chain complexes of left R-modules has a model structure, called the Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. Hence, the Dold–Kan correspondence induces a model structure on the category of simplicial left R-modules and some properties, notably it is monoidal. In the second part, we observe that changing the enrichment of an enriched, tensored and cotensored category along the Dold–Kan correspondence does not preserve the tensoring nor the cotensoring. Thus, we generalize this observation to any weak monoidal Quillen adjunction and we give an insight of which properties are preserved and which are weakened after changing the enrichment of an enriched model category along a right weak monoidal Quillen adjoint.
Open Access
The Erdős-Ko-Rado Theorem for intersecting families of permutations.
(Faculty of Graduate Studies and Research, University of Regina, 2010) Purdy, Alison May; Meagher, Karen; Fallat, Shaun; Zilles, Sandra
The Erdős-Ko-Rado Theorem is a fundamental result in extremal set theory. It describes the size and structure of the largest collection of subsets of size k from a set of size n having the property that any two subsets have at least t elements in common. Following the publication of the original theorem in 1961, many different proofs and extensions have appeared, culminating in the publication of the Complete Erdős-Ko-Rado Theorem by Ahlswede and Khachatrian in 1997. A number of similar results for families of permutations have appeared. These include proofs of the size and structure of the largest family of permutations having the property that any two permutations in the family agree on at least one element of the underlying set. In this thesis we apply techniques used in the proof of the Complete Erdős-Ko-Rado Theorem for set systems to prove a result for certain families of t-intersecting permutations. Specifically, we give the size and structure of a fixed t-intersecting family of permutations provided that n ≥2t + 1 and show that this lower bound on n is optimal.
Open Access
The Erdős-Ko-Rado Theorem for Transitive Permutation Groups
(Faculty of Graduate Studies and Research, University of Regina, 2022-03) Razafimahatratra, Andriaherimanana Sarobidy; Meagher, Karen; Fallat, Shaun; Herman, Allen; Yang, Boting; Mojallal, Seyed Ahmad; Bamberg, John
Given a transitive permutation group \(G \le Sym(\Omega)\), a subset \(F\) of \(G\) is \(\textit {intersecting}\) if any two elements of \(F\) agree on some elements of \(\Omega\). We are interested in the problem of finding the structure of the largest intersecting families of \(G\). This problem is the analogue of the \(\textit {Erdős-Ko-Rado (EKR) Theorem}\) for transitive permutation groups. We say that a transitive group \(G \le Sym(\Omega )\) has the \(\textit{EKR property}\) if any intersecting set of \(G\) has size at most the order of a stabilizer of a point of \(G\). Moreover, \(G\) has the \(\textit {strict-EKR property}\) if the largest intersecting sets in \(G\) are cosets of a stabilizer of a point of \(G\). In this thesis, we use various algebraic techniques to prove EKR-type results for finite transitive groups. In particular, we prove that the action of the symmetric group on the 2-tuples with distinct entries and 2-subsets of \([n]\) have the EKR property, and construct families of transitive groups that are as far away as possible from having the EKR property. Then, we show that any transitive subgroup of \(GL_2(q)\) acting on the non-zero vectors of \(\mathbb F^2_q\) has the EKR property. We also prove that for any odd primes \(p\), the size of the largest intersecting set in a transitive group of degree \(2p\) is at most twice the order of a point stabilizer. In addition, we show that if \(G\) is transitive of degree a product of two odd primes, then \(G\) has the EKR property whenever the socle of \(G\) admits an imprimitive subgroup.
Open Access
Extensions of the Erdős-Ko-Rado Theorem to Perfect Matchings
(Faculty of Graduate Studies and Research, University of Regina, 2022-03-31) Nasrollahi Shirazi, Mahsa; Meagher, Karen; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Nasserasr, Shahla; Guo, Krystal
One of the important results in extremal set theory is the Erdős-Ko-Rado (EKR) theorem which gives a tight upper bound on the size of intersecting sets. The focus of this thesis is on extensions of the EKR theorem to perfect matchings and uniform set partitions. Two perfect matchings are said to be t-intersecting if they have at least t edges in common. In 2017, Godsil and Meagher algebraically proved the EKR theorem for intersecting perfect matchings on the complete graph with 2k vertices. In 2017, Lindzey presented an asymptotic refinement of the EKR theorem on perfect matchings. In this thesis, we extend their results to 2-intersecting and also to set-wise 2-intersecting perfect matchings. These results are not asymptotic. A perfect matching is in fact a special case of a uniform set partition. Another focus of this thesis is on partially 2-intersecting uniform set partitions. We find the largest set of 2-intersecting uniform set partitions, when the number of parts is sufficiently large. The result on uniform set partitions is part of a joint research project with Karen Meagher and Brett Stevens.
Open Access
Fusions of association schemes
(Faculty of Graduate Studies and Research, University of Regina, 2023-01) Joshi, Neha; Allen, Herman; Meagher, Karen; Fallat, Shaun; Floricel, Remus; Fan, Lisa; Sankey, Alyssa
Since their introduction as symmetric coherent configurations by Bose and Mesner in 1959, association schemes have gained significant importance in algebraic combinatorics. An important breakthrough was achieved by Delsarte’s PhD thesis where he proved that many problems from coding theory, combinatorial design theory and statistics can be treated using the concept of association schemes [12]. Since its initial introduction, many algebraists and graph theorists have been studying the existence, construction and generalizations of various association schemes [1, 3, 7, 8, 17, 19, 28, 29, 30]. Because of their impressive construction, association schemes are useful to these subjects and there is always a search for new association schemes. One easy way to construct a new association scheme is by taking either direct or wreath products of two existing association schemes. One such example that we studied was given by Sankey in [32]. Another way to construct a new association scheme is by fusing specific relations of an existing association scheme. The resulting association scheme is known as a fusion (previously referred to as “subscheme” by mathematicians) [4]. The focus of this thesis is to examine a few important association schemes and classify them based on their fusions. It can be observed from literature that the nonexistence of nontrivial fusions is a rare phenomenon and this is the motivation behind this thesis. An association scheme, A, is said to be fusion-primitive if there does not exist any nontrivial fusions of A. In 1992, Muzychuk proved that there does not exist any nontrivial fusions of the Johnson scheme, J (n, k) for all n > 3k + 4 with k ≥ 4 [29]. In 1994, this result was further refined for all n > 3k + 1 by Uchida in [36]. In this thesis, we prove that the Johnson scheme does not have any nontrivial fusions for all n, n > 2k + 1 with k ≤ 20. In addition, we classify almost all of the multiplicity-free subgroups of the symmetric group based on their nontrivial fusions. The Hamming scheme, H(n, q) is an important example in coding theory [10]. The fusionprimitivity of the Hamming scheme has been discussed before (see [28]). In this thesis, we study the generalized Hamming scheme, H(n,A) and prove that it is fusion-imprimitive (that is, it always has a nontrivial fusion). We also classify all fusions of the generalized Hamming scheme, H(2,A) where A is the association scheme corresponding to a strongly-regular graph.
Open Access
Maximum Intersecting Families of Permutations
(Faculty of Graduate Studies and Research, University of Regina, 2013-07) Ahmadi, Bahman; Meagher, Karen; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Dukes, Peter
In extremal set theory, the Erd}os-Ko-Rado (EKR) theorem gives an upper bound on the size of intersecting k-subsets of the set {1; : : : ;n}. Furthemore, it classi es the maximum-sized families of intersecting k-subsets. It has been shown that similar theorems can be proved for other mathematical objects with a suitable notion of \intersection". Let G B Sym(n) be a permutation group with its permutation action on the set {1; : : : ;n}. The intersection for the elements of G is de ned as follows: two permutations ; > G are intersecting if (i) = (i) for some i > {1; : : : ;n}. A subset S of G is, then, intersecting if any pair of its elements is intersecting. We say G has the EKR property if the size of any intersecting subset of G is bounded above by the size of a point stabilizer in G. If, in addition, the only maximum-sized intersecting subsets are the cosets of the point-stabilizers in G, then G is said to have the strict EKR property. It was rst shown by Cameron and Ku [10] that the group G = Sym(n) has the strict EKR property. Then Godsil and Meagher presented an entirely di erent proof of this fact using some algebraic properties of the symmetric group. A similar method was employed to prove that the projective general linear group PGL(2; q), with its natural action on the projective line Pq, has the strict EKR property. The main objective in this thesis is to formally introduce this method, which we call the module method, and show that this provides a standard way to prove Erd}os-Ko-Rado theorems for other permutation groups. We then, along with proving Erd}os-Ko-Rado theorems for various groups, use this method to prove some permutation groups have the strict EKR property. We will also show that this method can be useful in characterizing the maximum independent sets of some Cayley graphs. To explain the module method, we need some facts from representation theory of groups, in particular, the symmetric group. We will provide the reader with a su cient level of background from representation theory as well as graph theory and linear algebraic facts about graphs.
Open Access
The maximum nullity of a complete edge subdivision graph is equal to it zero forcing number
(International Linear Algebra Society, 2014-06) Barrett, Wayne; Butler, Steve; Catral, Minnie; Hall, Tracy; Fallat, Shaun; Hogben, Leslie; Young, Michael
Barrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, °G) = Z(°G) by introducing the bridge tree of a connected graph. Since this equality is valid for all fields, °G has field independent minimum rank, and we also show that °G has a universally optimal matrix.
Open Access
Measurement of the Beam Asymmetry for the ETA and ETA Prime Mesons with the Gluex Experiment
(Faculty of Graduate Studies and Research, University of Regina, 2019-05) Beattie, Tegan Donald; Papandreou, Zisis; Huber, Garth; Lolos, George; Fallat, Shaun; Stevens, Justin; Voutier, Eric
The GlueX Experiment in Hall D at Jefferson Lab ultimately aims to provide evidence of hybrid mesons, quark-antiquark pairs with gluonic excitations, which are predicted by quantum chromodynamics calculations on the lattice. GlueX features a linearly-polarized photon beam incident on a liquid hydrogen target and a nearly-four-pi hermetic detector capable of measuring positions and momenta of both charged and neutral final state particles generated in the target. The short-term physics goals of GlueX are to measure observables and properties of the known particles, especially those that are likely decay products of hybrids, such as the eta and eta-prime mesons. An overview of the detector subsystems and the long-term physics goals of GlueX are presented, along with analyses on the Sigma beam asymmetry of the eta and eta-prime mesons. The measurements of Sigma are presented versus the four-momentum transfer squared using a photon beam energy of 8.2–8.8 GeV in the three main decay channels of the eta (eta → 2gamma, eta → pi+pi-pi0, and eta → 3pi0) and the largest branching fraction decay channel of the eta-prime (eta-prime → pi+pi-eta, with eta → 2gamma). These are the first results for the beam asymmetry of the eta-prime measured at this beam energy. The results, which are in agreement with past published measurements for the eta by GlueX and with modern theory predictions for the eta and eta-prime, indicate a dominance of natural parity exchange in the resonance production mechanism.
Open Access
Measurement of the pion exclusive electro-production cross-section in the E12-19-006 experiment in Hall-C at Jefferson Lab
(Faculty of Graduate Studies and Research, University of Regina, 2024-09) Kumar, Vijay; Huber, Garth; Mobed, Nader; Barbi, Mauricio; Mack, David; Fallat, Shaun; Ireland, David
One of the most effective methods for exploring the transition from hadronic de- grees of freedom to quark-gluon degrees of freedom in Quantum Chromodynamics (QCD) is through the investigation of \exclusive" pion and kaon electro-production reactions at various Q2 and -t values. The E12-19-006 experiment is conducted within the confines of experimental Hall C at the Thomas Jefferson National Accel- erator Facility, USA, for such studies. The primary aim of the experiment is to first enhance our comprehension of the pion electro-production cross-section and its form factor at Q2 = 0.38 and 0.42 GeV2. This is the first run period of the E12-19-006 experiment which ran in summer 2019. A more profound understanding of the pion electro-production reaction, 1H(e,e'π+)n, at low Q2 is deemed essential to employ this electro-production reaction (an indirect technique) for the high Q2 studies, thereby delving deeper into the realm of QCD. Consequently, this dissertation presents a thorough analysis of the experimental data acquired in the first run period of the E12-19-006 experiment. In pursuit of precision, a series of systematic studies (target boiling correction study, the elastic reaction cross-section measurements, study for determining vari- ous kinematics offsets, etc.) are conducted to discern the accuracy of the analyzed data, a prerequisite for the use of Rosenbluth separation technique to separate the pion electro-production cross-section terms in t bins. The separated pion electro- production cross-section through the Rosenbluth separation technique is then used to extract the pion electromagnetic form factor. In this dissertation, the pion electro-production cross-section is carefully dissected into its four constituent components: longitudinal (𝜎L), transverse (σT ), longitudinal- transverse (σLT ), and transverse-transverse (σTT ), using the full version of Rosenbluth separation technique for the Q2 = 0.38 GeV2. The technique is simultaneously fitted to the unseparated pion electro-production cross-sections at the three values of polar- ization of the virtual photon (ϵ), i.e., ϵ = 0.286, 0.629 and 0.781. An iterative process is applied to refine the parameters of the model cross-sections until the yield ratio of experimental and Monte Carlo simulation converges. In this study, 21 iterations are conducted to refine the model cross-section parameters. The final pion electro- production cross-section terms are then determined for 7 t bins using the optimized parameters of the model cross-sections.