Elimination For Systems Of Algebraic Differential Equations, 2017 The Graduate Center, City University of New York

#### Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of ...

The Recognition Problem For Table Algebras And Reality-Based Algebras, 2017 University of Regina

#### The Recognition Problem For Table Algebras And Reality-Based Algebras, Allen Herman, Mikhail Muzychuk, Bangteng Xu

*EKU Faculty and Staff Scholarship*

Given a finite-dimensional noncommutative semisimple algebra A over C with involution, we show that A always has a basis B for which ( A , B ) is a reality-based algebra. For algebras that have a one-dimensional representation δ , we show that there always exists an RBA-basis for which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensional noncommutative semisimple algebra for which the algebra has a positive degree map, and give examples of RBA-bases of C ⊕ M n ( C ) for which the RBA has a positive degree map, for all n ≥ 2

Application Of Symplectic Integration On A Dynamical System, 2017 East Tennessee State University

#### Application Of Symplectic Integration On A Dynamical System, William Frazier

*Electronic Theses and Dissertations*

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic ...

Six Septembers: Mathematics For The Humanist, 2017 Duquesne University

#### Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

*Zea E-Books*

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics ...

A Game Of Monovariants On A Checkerboard, 2017 Lynchburg College

#### A Game Of Monovariants On A Checkerboard, Linwood Reynolds

*Student Scholar Showcase*

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot ...

Laurent Series Obtained By Long Division, 2017 Iowa State University

#### Laurent Series Obtained By Long Division, A. Abian, Leslie Hogben, Elgin H. Johnston

*Leslie Hogben*

Let r1,...,rn be the n root-moduli of the polynomial azn+bzm+c, where n>m>0 are integers and a,b,c are nonzero complex numbers. We give a necessary and sufficient condition in order that the long division of .1 by bzm+azn+c (where contrary to traditional long division, the divisor is ordered neither in the ascending nor in the descending powers of z) yield the Laurent series of 1/(azn+bzm+c) valid in the annulus rk< IzI k+1 for some root-modulus rk. Our method gives an effective way of obtaining Laurent series of 1 ...

Fractional Zero Forcing Via Three-Color Forcing Games, 2017 Iowa State University

#### Fractional Zero Forcing Via Three-Color Forcing Games, Leslie Hogben, Kevin F. Palmowski, David E. Roberson, Michael Young

*Leslie Hogben*

An r-fold analogue of the positive semidefinite zero forcing process that is carried out on the r-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach ...

Minimum Rank Of Skew-Symmetric Matrices Described By A Graph, 2017 University of Wyoming

#### Minimum Rank Of Skew-Symmetric Matrices Described By A Graph, Mary Allison, Elizabeth Bodine, Luz Maria Dealba, Joyati Debnath, Laura Deloss, Colin Garnett, Jason Grout, Leslie Hogben, Bokhee Im, Hana Kim, Reshmi Nair, Olga Pryporova, Kendrick Savage, Bryan Shader, Amy Wangsness Wehe

*Leslie Hogben*

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply ...

The Copositive Completion Problem: Unspecified Diagonal Entries, 2017 Iowa State University

#### The Copositive Completion Problem: Unspecified Diagonal Entries, Leslie Hogben

*Leslie Hogben*

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207–211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix.

The Symmetric M-Matrix And Symmetric Inverse M-Matrix Completion Problems, 2017 Iowa State University

#### The Symmetric M-Matrix And Symmetric Inverse M-Matrix Completion Problems, Leslie Hogben

*Leslie Hogben*

The symmetric M-matrix and symmetric M0-matrix completion problems are solved and results of Johnson and Smith [Linear Algebra Appl. 290 (1999) 193] are extended to solve the symmetric inverse M-matrix completion problem: (1) A pattern (i.e., a list of positions in an n×n matrix) has symmetric M-completion (i.e., every partial symmetric M-matrix specifying the pattern can be completed to a symmetric M-matrix) if and only if the principal subpattern R determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete, or, in graph theoretic terms, if and only ...

Potentially Eventually Exponentially Positive Sign Patterns, 2017 Columbia College

#### Potentially Eventually Exponentially Positive Sign Patterns, Marie Archer, Minerva Catral, Craig Erickson, Rana Haber, Leslie Hogben, Xavier Martinez-Rivera, Antonio Ochoa

*Leslie Hogben*

We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP sign patterns. A characterization of all 2×2 and 3×3 PEEP sign patterns is given.

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, 2017 Iowa State University

#### Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

*Leslie Hogben*

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above ...

Spectrally Arbitrary Patterns: Reducibility And The 2n Conjecture For N = 5, 2017 Drake University

#### Spectrally Arbitrary Patterns: Reducibility And The 2n Conjecture For N = 5, Luz M. Dealba, Irvin R. Hentzel, Leslie Hogben, Judith Mcdonald, Rana Mikkelson, Olga Pryporova, Bryan Shader, Kevin N. Vander Meulen

*Leslie Hogben*

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [J.H. Drew, C.R. Johnson, D.D. Olesky, P. van den Driessche, Spectrally arbitrary patterns, Linear Algebra Appl. 308 (2000) 121–137], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured ...

The Minimum Rank Of Symmetric Matrices Described By A Graph: A Survey, 2017 University of Regina

#### The Minimum Rank Of Symmetric Matrices Described By A Graph: A Survey, Shaun M. Fallat, Leslie Hogben

*Leslie Hogben*

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.

Orthogonal Representations, Minimum Rank, And Graph Complements, 2017 Iowa State University

#### Orthogonal Representations, Minimum Rank, And Graph Complements, Leslie Hogben

*Leslie Hogben*

Orthogonal representations are used to show that complements of certain sparse graphs have (positive semidefinite) minimum rank at most 4. This bound applies to the complement of a 2-tree and to the complement of a unicyclic graph. Hence for such graphs, the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph. The minimum rank of the complement of a 2-tree is determined exactly.

Minimum Rank Problems, 2017 Iowa State University

#### Minimum Rank Problems, Leslie Hogben

*Leslie Hogben*

A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal ...

On The Difference Between The Maximum Multiplicity And Path Cover Number For Tree-Like Graphs, 2017 Carleton University

#### On The Difference Between The Maximum Multiplicity And Path Cover Number For Tree-Like Graphs, Francesco Barioli, Shaun Fallat, Leslie Hogben

*Leslie Hogben*

For a given undirected graph G, the maximum multiplicity of G is defined to be the largest multiplicity of an eigenvalue over all real symmetric matrices A whose (i, j)th entry is non-zero whenever i ≠ j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertex-disjoint paths occurring as induced subgraphs of G that cover all the vertices of G. We derive a formula for the path cover number of a vertex-sum of graphs, and use it to prove that the vertex-sum of so-called non-deficient graphs is non-deficient. For ...

Path Cover Number, Maximum Nullity, And Zero Forcing Number Of Oriented Graphs And Other Simple Digraphs, 2017 Saint Olaf College

#### Path Cover Number, Maximum Nullity, And Zero Forcing Number Of Oriented Graphs And Other Simple Digraphs, Adam Berliner, Cora Brown, Joshua Carlson, Nathanael Cox, Leslie Hogben, Jason Hu, Katrina Jacobs, Kathryn Manternach, Travis Peters, Nathan Warnberg, Michael Young

*Leslie Hogben*

An oriented graph is a simple digraph obtained from a simple graph by choosing exactly one of the two arcs (u,v)(u,v) or (v,u)(v,u) to replace each edge {u,v}{u,v}. A simple digraph describes the zero-nonzero pattern of off-diagonal entries of a family of (not necessarily symmetric) matrices. The minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number and path cover number are related parameters. We establish bounds on the range of possible values of all ...

Rational Realization Of Maximum Eigenvalue Multiplicity Of Symmetric Tree Sign Patterns, 2017 Princeton University

#### Rational Realization Of Maximum Eigenvalue Multiplicity Of Symmetric Tree Sign Patterns, Atoshi Chowdhury, Leslie Hogben, Jude Melancon, Rana Mikkelson

*Leslie Hogben*

A sign pattern is a matrix whose entries are elements of {+, −, 0}; it describes the set of real matrices whose entries have the signs in the pattern. A graph (that allows loops but not multiple edges) describes the set of symmetric matrices having a zero-nonzero pattern of entries determined by the absence or presence of edges in the graph. DeAlba et al. [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness, Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl., in press, doi:10.1016/j.laa.2006.02.018.] gave ...

On Completion Problems For Various Classes Of P-Matrices, 2017 Grinnell College

#### On Completion Problems For Various Classes Of P-Matrices, John Bowers, Job Evers, Leslie Hogben, Steve Shaner, Karyn Snider, Amy Wangsness

*Leslie Hogben*

A P-matrix is a real square matrix having every principal minor positive, and a Fischer matrix is a P-matrix that satisfies Fischer’s inequality for all principal submatrices. In this paper, all patterns of positions for n × n matrices, n ⩽ 4, are classified as to whether or not every partial Π-matrix can be completed to a Π-matrix for Π any of the classes positive P-, nonnegative P-, or Fischer matrices. Also, all symmetric patterns for 5 × 5 matrices are classified as to completion of partial Fischer matrices, and all but two such patterns are classified as to positive P- or ...