Shaun Fallat

Permanent URI for this collectionhttps://hdl.handle.net/10294/4838

Office: CW 307.2
E-mail: Shaun.Fallat@uregina.ca
Phone: 306-585-4107
Website: http://uregina.ca/~sfallat/

Current classes
Math 223 (Introduction to Abstract Algebra), Stat 252 (Statistical Inference)

Research interests
Matrix theory, discrete mathematics, graph theory, combinatorial matrix analysis

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Browsing Shaun Fallat by Author "van den Driessche, Pauline"

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    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.
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    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
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    Parameters Related to Tree-Width, Zero Forcing, and Maximum Nullity of a Graph
    (Wiley Periodicals, Inc., 2013) Barioli, Francesco; Barrett, Wayne; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; Shader, Bryan; van den Driessche, Pauline; van der Holst, Hein
    Tree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdi`ere type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d’arborescence, path-width, and proper path-width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.
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    The principal rank characteristic sequence over various fields
    (Elsevier, 2014) Barrett, Wayne; Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, Michael
    Given an n-by-n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0; 1,..., n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported.