Faculty PublicationsCopyright (c) 2020 University of South Florida St. Petersburg All rights reserved.
https://digital.stpetersburg.usf.edu/fac_publications
Recent documents in Faculty Publicationsen-usSat, 05 Dec 2020 01:48:10 PST3600A Student Typology Based on Instructional Method Preferences
https://digital.stpetersburg.usf.edu/fac_publications/4041
https://digital.stpetersburg.usf.edu/fac_publications/4041Fri, 04 Dec 2020 05:36:04 PSTPhilip J. Trocchia et al.Panama Papers and the Abuse of Shell Entities
https://digital.stpetersburg.usf.edu/fac_publications/4040
https://digital.stpetersburg.usf.edu/fac_publications/4040Fri, 04 Dec 2020 05:35:26 PSTCarl J. Pacini et al.Teaching About Religion with Conversations and Multicultural Literature in K-6 Classrooms
https://digital.stpetersburg.usf.edu/fac_publications/4039
https://digital.stpetersburg.usf.edu/fac_publications/4039Fri, 04 Dec 2020 05:35:03 PST
After reading Lailah's Lunchbox, a book about a young girl celebrating Ramadan, a circle of teachers discussed Ramadan through the perspective of the main character, Lailah. 1 Not only did the teachers in this graduate-level course learn about Muslim beliefs and practices during Ramadan (the ninth month of the Muslim year), they also gained a new perspective on what young Muslim students might be thinking and feeling as they attend school in the United States. One of the teachers shared dismay, recalling that she had sent a Muslim fourth grade student to the cafeteria during lunch period. "I had no idea it was Ramadan. The student asked to stay in the classroom and help me, and I just didn't put two and two together." The teacher realized that her student was fasting, not eating or drinking from dawn to sunset, for one month. Fasting during Ramadan is in the Islamic holy scriptures, the Koran. It is one of the Five Pillars of Islam, which are central practices of faithful Muslims.
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AnnMarie Alberton Gunn et al.Propagation Tree Decompositions and Linearly Independent Vertices
https://digital.stpetersburg.usf.edu/fac_publications/4038
https://digital.stpetersburg.usf.edu/fac_publications/4038Wed, 04 Nov 2020 11:19:06 PST
e explore the relationship between propagation tree decompositions of a graph of a given size and the OS-sets of the same size, showing that each can generate the other. We give a short constructive proof that the OS-sets are in bijective correspondence with a subset of the propagation tree decompositions.
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Lon MitchellOn the Minimum Rank among Positive Semidefinite Matrices with a Given Graph
https://digital.stpetersburg.usf.edu/fac_publications/4037
https://digital.stpetersburg.usf.edu/fac_publications/4037Wed, 04 Nov 2020 09:02:36 PST
Let P(G) be the set of all positive semidefinite matrices whose graph is G, and msr(G) be the minimum rank of all matrices in P(G). Upper and lower bounds for msr(G) are given and used to determine msr(G) for some well-known graphs, including chordal graphs, and for all simple graphs on less than seven vertices.
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Mathew Booth et al.Unitary Matrix Digraphs and Minimum Semidefinite Rank
https://digital.stpetersburg.usf.edu/fac_publications/4036
https://digital.stpetersburg.usf.edu/fac_publications/4036Wed, 04 Nov 2020 09:02:16 PST
For an undirected simple graph G, the minimum rank among all positive semidefinite matrices with graph G is called the minimum semidefinite rank (msr) of G. In this paper, we show that the msr of a given graph may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results to give a lower bound for the msr of certain bipartite graphs.
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Yunjiang Jiang et al.Maximal Total Absolute Displacement of a Permutation
https://digital.stpetersburg.usf.edu/fac_publications/4035
https://digital.stpetersburg.usf.edu/fac_publications/4035Wed, 04 Nov 2020 09:01:55 PST
We determine those permutations that have maximal absolute total displacement on a finite subset of real numbers, and give some corollaries.
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Lon MitchellLinearly Independent Vertices and Minimum Semidefinite Rank
https://digital.stpetersburg.usf.edu/fac_publications/4034
https://digital.stpetersburg.usf.edu/fac_publications/4034Wed, 04 Nov 2020 08:35:10 PST
We study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs. We present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph, and conjecture that the resulting graph invariant is equal to minimum semidefinite rank. Rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs. We also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex.
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Philip Hackney et al.Simplicity of C*-algebras using unique eigenstates
https://digital.stpetersburg.usf.edu/fac_publications/4033
https://digital.stpetersburg.usf.edu/fac_publications/4033Wed, 04 Nov 2020 08:10:18 PST
We consider a one-parameter family of operators that are constructed from a pair of isometries on Hilbert space with orthogonal ranges. For special values of the parameter, the operator plays a role in the representation theory of free groups and in free probability theory. For each parameter value, we identify the irreducible ∗-representations of the pair of isometries in which the operator has an eigenvalue. This yields a new technique for showing that certain C ∗ -algebras, including the C ∗ -algebra generated by the operator, are simple. We establish several other fundamental properties of this C ∗ -algebra and its generator.
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Lon Mitchell et al.Lower Bounds in Minimum Rank Problems
https://digital.stpetersburg.usf.edu/fac_publications/4032
https://digital.stpetersburg.usf.edu/fac_publications/4032Wed, 04 Nov 2020 08:09:53 PST
The minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph. The minimum semidefinite rank of a graph is the minimum rank among Hermitian positive semidefinite matrices with the given graph. We explore connections between OS-sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large.
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Lon Mitchell et al.Operators on the L∞-spaces of Bourgain and Delbaen
https://digital.stpetersburg.usf.edu/fac_publications/4031
https://digital.stpetersburg.usf.edu/fac_publications/4031Tue, 03 Nov 2020 14:28:11 PST
In 1980, J. Bourgain and F. Delbaen constructed two classes and of ∞-spaces each exhibiting many surprising properties. In particular, the members of each possess the Schur property and the members of are somewhat reflexive. In this paper, for a Banach space in either of these classes we define a bounded shift-type operator which is an isometry when restricted to a certain hyperplane.
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Kevin Beanland et al.On the Minimum Vector Rank of Multigraphs
https://digital.stpetersburg.usf.edu/fac_publications/4030
https://digital.stpetersburg.usf.edu/fac_publications/4030Tue, 03 Nov 2020 14:27:49 PST
The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in Fd. For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking the msr of chordal graphs to clique cover number also holds for the mvr of multigraphs. We study the difference between msr and mvr in the removal of duplicate vertices in multigraphs, and relate mvr to certain coloring problems.
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Lon Mitchell et al.Orthogonal Vector Coloring
https://digital.stpetersburg.usf.edu/fac_publications/4029
https://digital.stpetersburg.usf.edu/fac_publications/4029Tue, 03 Nov 2020 14:10:44 PST
A vector coloring of a graph is an assignment of a vector to each vertex where the presence or absence of an edge between two vertices dictates the value of the inner product of the corresponding vectors. In this paper, we obtain results on orthogonal vector coloring, where adjacent vertices must be assigned orthogonal vectors. We introduce two vector analogues of list coloring along with their chromatic numbers and characterize all graphs that have (vector) chromatic number two in each case.
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Gerald Haynes et al.Minimum Semidefinite Rank of Outerplanar Graphs and the Tree Cover Number
https://digital.stpetersburg.usf.edu/fac_publications/4028
https://digital.stpetersburg.usf.edu/fac_publications/4028Tue, 03 Nov 2020 09:32:03 PST
Let G= (V, E) be a multigraph with no loops on the vertex setV={1,2, . . . , n}. DefineS+(G) as the set of symmetric positive semidefinite matricesA= [aij] with aij6= 0,i6=j,ifij∈E(G) is a single edge and aij= 0,i6=j, if ij /∈E(G). Let M+(G) denote the maximum multiplicity of zero as an eigenvalue of A∈S+(G) and mr+(G) =|G|−M+(G) denote the minimum semidefinite rank ofG. The tree cover number of a multigraphG, denotedT(G), is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of G that cover all of the vertices of G. The authors present some results on this new graph parameterT(G). In particular, they show that for any outerplanar multigraphG,M+(G) =T(G).
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Francesco Barioli et al.On the Graph Complement Conjecture for Minimum Semidefinite Rank
https://digital.stpetersburg.usf.edu/fac_publications/4027
https://digital.stpetersburg.usf.edu/fac_publications/4027Tue, 03 Nov 2020 09:23:01 PST
In this note, we combine a number of recent ideas to give new results on the graph complement conjecture for minimum semidefinite rank.
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Lon MitchellOn the Minimum Semidefinite Rank of a Simple Graph
https://digital.stpetersburg.usf.edu/fac_publications/4026
https://digital.stpetersburg.usf.edu/fac_publications/4026Tue, 03 Nov 2020 09:22:29 PST
The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/ nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.
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Mathew Booth et al.Induced Trees, Minimum Semidefinite Rank, and Zero Forcing
https://digital.stpetersburg.usf.edu/fac_publications/4025
https://digital.stpetersburg.usf.edu/fac_publications/4025Tue, 03 Nov 2020 09:03:36 PST
We prove that the ordered subgraph number of a connected graph that has no duplicate vertices is at most three if and only if the complement does not contain a cycle on four vertices. The duality between zero forcing and ordered subgraphs then provides a complementary characterization for positive semidefinite zero forcing. We also provide some necessary conditions for when the minimum semidefinite rank can be computed using tree size.
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Rachel Cranfill et al.Lower Bounds for Minimum Semidefinite Rank from Orthogonal Removal and Chordal Supergraphs
https://digital.stpetersburg.usf.edu/fac_publications/4024
https://digital.stpetersburg.usf.edu/fac_publications/4024Tue, 03 Nov 2020 09:03:11 PST
The minimum semidefinite rank (msr) of a graph is the minimum rank among positive semidefinite matrices with the given graph. The OS-number is a useful lower bound for msr, which arises by considering ordered vertex sets with some connectivity properties. In this paper, we develop two new interpretations of the OS-number. We first show that OS-number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions. Our second interpretation of the OS-number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation-preserving. These interpretations not only give insight into the OS-number, but also allow us to prove some new results. For example we show that msr(G)=|G|−2 if and only if OS(G)=|Gzsfnc−2.
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Lon Mitchell et al.Bounds for Minimum Semidefinite Rank from Superpositions and Cutsets
https://digital.stpetersburg.usf.edu/fac_publications/4023
https://digital.stpetersburg.usf.edu/fac_publications/4023Mon, 02 Nov 2020 14:22:40 PST
The real (complex) minimum semidefinite rank of a graph is the minimum rank among all real symmetric (complex Hermitian) positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper we give an upper bound on the minimum semidefinite rank of a graph when the graph is modified from the superposition of two graphs by cancelling some number of edges. We also provide a lower bound for the minimum semidefinite rank of a graph determined by a given cutset. When the complement of the cutset is a star forest these lower and upper bounds coincide and we can compute the minimum semidefinite rank in terms of smaller graphs. This result encompasses the previously known case in which the cut set has order two or smaller. Next we provide results for when the cut set has order three. Using these results we provide an example where the positive semidefinite zero forcing number is strictly greater than the maximum positive semidefinite nullity.
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Jonathan Beagley et al.Fractals & Mysterious Triangles
https://digital.stpetersburg.usf.edu/fac_publications/4022
https://digital.stpetersburg.usf.edu/fac_publications/4022Mon, 02 Nov 2020 11:38:29 PSTMichael A. Jones et al.