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Recent documents in Wyoming Scholars Repositoryen-usSun, 22 Sep 2019 02:34:29 PDT3600Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$
https://repository.uwyo.edu/ela/vol35/iss1/26
https://repository.uwyo.edu/ela/vol35/iss1/26Thu, 12 Sep 2019 13:10:05 PDT
For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the conjugate transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-conjugate (transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.
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Xin Liu et al.The cone of Z-transformations on Lorentz cone
https://repository.uwyo.edu/ela/vol35/iss1/25
https://repository.uwyo.edu/ela/vol35/iss1/25Thu, 12 Sep 2019 13:09:49 PDT
In this paper, the structural properties of the cone of $\calz$-transformations on the Lorentz cone are described in terms of the semidefinite cone and copositive/completely positive cones induced by the Lorentz cone and its boundary. In particular, its dual is described as a slice of the semidefinite cone as well as a slice of the completely positive cone of the Lorentz cone. This provides an example of an instance where a conic linear program on a completely positive cone is reduced to a problem on the semidefinite cone.
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Sandor Zoltan Nemeth et al.SPN Graphs
https://repository.uwyo.edu/ela/vol35/iss1/24
https://repository.uwyo.edu/ela/vol35/iss1/24Thu, 29 Aug 2019 09:14:58 PDT
A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized.
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Leslie Hogben et al.Alpha Adjacency: A generalization of adjacency matrices
https://repository.uwyo.edu/ela/vol35/iss1/23
https://repository.uwyo.edu/ela/vol35/iss1/23Thu, 29 Aug 2019 09:14:40 PDT
B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.
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Matt Hudelson et al.Maps Preserving Norms of Generalized Weighted Quasi-arithmetic Means of Invertible Positive Operators
https://repository.uwyo.edu/ela/vol35/iss1/22
https://repository.uwyo.edu/ela/vol35/iss1/22Sun, 25 Aug 2019 08:56:40 PDT
In this paper, the problem of describing the structure of transformations leaving norms of generalized weighted quasi-arithmetic means of invertible positive operators invariant is discussed. In a former result of the authors, this problem was solved for weighted quasi-arithmetic means, and here the corresponding result is generalized by establishing its solution under certain mild conditions. It is proved that in a quite general setting, generalized weighted quasi-arithmetic means on self-adjoint operators are not monotone in their variables which is an interesting property. Moreover, the relation of these means with the Kubo-Ando means is investigated and it is shown that the common members of the classes of these types of means are weighted arithmetic means.
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Gergő Nagy et al.On Sign Pattern Matrices that Allow or Require Algebraic Positivity
https://repository.uwyo.edu/ela/vol35/iss1/21
https://repository.uwyo.edu/ela/vol35/iss1/21Sun, 25 Aug 2019 08:56:27 PDT
A square matrix M with real entries is algebraically positive (AP) if there exists a real polynomial p such that all entries of the matrix p(M) are positive. A square sign pattern matrix S allows algebraic positivity if there is an algebraically positive matrix M whose sign pattern is S. On the other hand, S requires algebraic positivity if matrix M, having sign pattern S, is algebraically positive. Motivated by open problems raised in a work of Kirkland, Qiao, and Zhan (2016) on AP matrices, all nonequivalent irreducible 3 by 3 sign pattern matrices are listed and classify into three groups (i) those that require AP, (ii) those that allow but not require AP, or (iii) those that do not allow AP. A necessary condition for an irreducible n by n sign pattern to allow algebraic positivity is also provided.
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Diane Christine Pelejo et al.The inverse eigenvalue problem for Leslie matrices
https://repository.uwyo.edu/ela/vol35/iss1/20
https://repository.uwyo.edu/ela/vol35/iss1/20Mon, 19 Aug 2019 17:03:07 PDT
The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n > 2$ real numbers.
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Luca BenvenutiOn Orthogonal Matrices with Zero Diagonal
https://repository.uwyo.edu/ela/vol35/iss1/19
https://repository.uwyo.edu/ela/vol35/iss1/19Mon, 19 Aug 2019 17:02:52 PDT
This paper considers real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as $\OMZD(n)$. It is shown that there exists an $\OMZD(n)$ if and only if $n\neq 1,\ 3$, and that a symmetric $\OMZD(n)$ exists if and only if $n$ is even and $n\neq 4$. Also, a construction of $\OMZD(n)$ obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered.
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Robert F. Bailey et al.On the Block Structure and Frobenius Normal Form of Powers of Matrices
https://repository.uwyo.edu/ela/vol35/iss1/18
https://repository.uwyo.edu/ela/vol35/iss1/18Mon, 19 Aug 2019 17:02:38 PDT
The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix $A$ and the power $q$ are provided so that for any invertible matrix $S$, if $S^{-1}A^qS$ is block upper triangular, then so is $S^{-1}AS$ when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. The article concludes by applying the block upper triangular powers result to the cone Frobenius normal form of powers of a eventually cone nonnegative matrix.
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Mashael M. Al Baidani et al.Extending Standing to Non-Clients Moving to Disqualify Opposing Counsel in Wyoming
https://repository.uwyo.edu/wlr/vol19/iss2/9
https://repository.uwyo.edu/wlr/vol19/iss2/9Tue, 13 Aug 2019 14:31:24 PDTJohn M. Burman et al.WY Law DivisionUnoccupied: How a Single Word Affects Wyoming’s Ability to Regulate Tribal Hunting Through a Federal Treaty; Herrera v. Wyoming
https://repository.uwyo.edu/wlr/vol19/iss2/8
https://repository.uwyo.edu/wlr/vol19/iss2/8Tue, 13 Aug 2019 14:31:16 PDTJason MitchellWY Law DivisionRightful Compensation for a Wrongful Conviction: In Defense of a Compensation Statute in the State of Wyoming
https://repository.uwyo.edu/wlr/vol19/iss2/7
https://repository.uwyo.edu/wlr/vol19/iss2/7Tue, 13 Aug 2019 14:31:10 PDTMeridith J. HeneageWY Law DivisionThe Wild, Wild West: The Mechanics and Potential Uses of Trust Decanting
https://repository.uwyo.edu/wlr/vol19/iss2/6
https://repository.uwyo.edu/wlr/vol19/iss2/6Tue, 13 Aug 2019 14:31:03 PDTJohn FritzWY Law DivisionWyoming, Take Another Look at Unions: How Unions Can Increase Equality for Women in the Workplace
https://repository.uwyo.edu/wlr/vol19/iss2/5
https://repository.uwyo.edu/wlr/vol19/iss2/5Tue, 13 Aug 2019 14:30:56 PDTHallie GuidryWY Law DivisionWyoming is More Likely Than Not Behind in Guardianship Proceedings: The Unconstitutional Standard for Guardianship Under Wyoming Statute § 3-2-104
https://repository.uwyo.edu/wlr/vol19/iss2/4
https://repository.uwyo.edu/wlr/vol19/iss2/4Tue, 13 Aug 2019 14:30:48 PDTKasey J. BenishWY Law DivisionWhat Fane Lozman Can Teach Us About Free Speech
https://repository.uwyo.edu/wlr/vol19/iss2/3
https://repository.uwyo.edu/wlr/vol19/iss2/3Tue, 13 Aug 2019 14:30:40 PDTJesse D. H. SnyderGeneral Law DivisionSexual Consent as a Common Law Doctrine
https://repository.uwyo.edu/wlr/vol19/iss2/2
https://repository.uwyo.edu/wlr/vol19/iss2/2Tue, 13 Aug 2019 14:30:34 PDTColin ColtGeneral Law DivisionRape is Not a Contract: Recognizing the Fundamental Difficulties in Applying Economic Theories of Jurisprudence to Criminal Sexual Assault
https://repository.uwyo.edu/wlr/vol19/iss2/1
https://repository.uwyo.edu/wlr/vol19/iss2/1Tue, 13 Aug 2019 14:30:27 PDTTori R. A. KrickenGeneral Law DivisionGeneralization of real interval matrices to other fields
https://repository.uwyo.edu/ela/vol35/iss1/17
https://repository.uwyo.edu/ela/vol35/iss1/17Mon, 12 Aug 2019 01:19:40 PDT
An interval matrix is a matrix whose entries are intervals in $\R$. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in $\Q$. It is proved that a (real) interval $p \times q$ matrix with the endpoints of all its entries in $\Q$ contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than $\min\{p,q\}$ if and only if it contains a rational matrix with rank smaller than $\min\{p,q\}$; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field $K$ and a matrix $\al$ whose entries are subsets of $K$, a criterion to find the maximal rank of a matrix contained in $\al$ is described.
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Elena RubeiPure PSVD approach to Sylvester-type quaternion matrix equations
https://repository.uwyo.edu/ela/vol35/iss1/16
https://repository.uwyo.edu/ela/vol35/iss1/16Thu, 25 Jul 2019 11:36:33 PDT
In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type quaternion matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ is considered by using the PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given quaternion matrices of compatible sizes $(i=1,2,3,4)$. Some necessary and sufficient conditions for the existence of a solution to this system are derived. Moreover, the general solution to this system is presented when it is solvable.
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Zhuo-Heng He