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Recent documents in Wyoming Scholars Repositoryen-usThu, 14 Nov 2019 01:48:10 PST3600AN EIGENVALUE APPROACH FOR ESTIMATING THE GENERALIZED CROSS VALIDATION FUNCTION FOR CORRELATED MATRICES
https://repository.uwyo.edu/ela/vol35/iss1/34
https://repository.uwyo.edu/ela/vol35/iss1/34Thu, 07 Nov 2019 12:06:58 PST
This work proposes a fast estimate for the generalized cross-validation function when the design matrix of an experiment has correlated columns. The eigenvalue structure of this matrix is used to derive probability bounds satisfied by an appropriate index of proximity, which provides a simple and accurate estimate for the numerator of the generalized cross-validation function. The denominator of the function is evaluated by an analytical formula. Several simulation tests performed in statistical models having correlated design matrix with intercept confirm the reliability of the proposed probabilistic bounds and indicate the applicability of the proposed estimate for these models.
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Christos Koukouvinos et al.The $A_{\alpha}$- spectrum of graph product
https://repository.uwyo.edu/ela/vol35/iss1/33
https://repository.uwyo.edu/ela/vol35/iss1/33Thu, 07 Nov 2019 12:06:52 PST
Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, a complete characterization of the $A_{\alpha}$-spectrum of $G\square H$ for arbitrary graphs $G$ and $H$, and $G[H]$ for arbitrary graph $G$ and regular graph $H$ is given. Furthermore, $A_{\alpha}$-spectrum of the generalized lexicographic product $G[H_1,H_2,\ldots,H_n]$ for $n$-vertex graph $G$ and regular graphs $H_i$'s is considered. At last, the spectral radii of $A_{\alpha}(G\times H)$ and $A_{\alpha}(G\oplus H)$ for arbitrary graph $G$ and regular graph $H$ are given.
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Shuchao Li et al.A NOTE ON VARIANTS OF ZERO FORCING
https://repository.uwyo.edu/ela/vol35/iss1/32
https://repository.uwyo.edu/ela/vol35/iss1/32Thu, 07 Nov 2019 12:06:46 PST
A small improvement is made to the zero-forcing variants defined by Butler, Grout, and Hall (2015) for matrices with a given number of negative eigenvalues, resulting in a better value for the Barioli-Fallat tree and one negative eigenvalue.
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Lon MitchellThe sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs
https://repository.uwyo.edu/ela/vol35/iss1/31
https://repository.uwyo.edu/ela/vol35/iss1/31Sun, 27 Oct 2019 10:39:58 PDT
Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.
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Zhibin DuOn a Refined Operator Version of Young's Inequality and Its Reverse
https://repository.uwyo.edu/ela/vol35/iss1/30
https://repository.uwyo.edu/ela/vol35/iss1/30Sun, 27 Oct 2019 10:39:53 PDT
In this note, some refinements of Young's inequality and its reverse for positive numbers are proved, and using these inequalities, some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
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Alemeh Sheikhhosseini et al.On the Perron-Frobenius Theory of $M_v-$matrices and Eventually Exponentially Nonnegative Matrices
https://repository.uwyo.edu/ela/vol35/iss1/29
https://repository.uwyo.edu/ela/vol35/iss1/29Fri, 11 Oct 2019 10:07:38 PDT
${M_v}-$matrix is a matrix of the form $A = sI-B$, where $ 0 \le \rho (B) \le s$ and $B$ is an eventually nonnegative matrix. In this paper, $M_v-$matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an $M_v-$matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if $index_{0} B \leq 1$ are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or $M_v-$matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and $M_v-$matrices are presented. Various numerical examples are given to support our theoretical findings.
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Thaniporn Chaysri et al.Inequalities for sector matrices and positive linear maps
https://repository.uwyo.edu/ela/vol35/iss1/28
https://repository.uwyo.edu/ela/vol35/iss1/28Fri, 11 Oct 2019 10:07:16 PDT
Ando proved that if $A, B$ are positive definite, then for any positive linear map $\Phi$, it holds \begin{eqnarray*} \Phi(A\sharp_\lambda B)\le \Phi(A)\sharp_\lambda \Phi(B), \end{eqnarray*} where $A\sharp_\lambda B$, $0\le\lambda\le 1$, means the weighted geometric mean of $A, B$. Using the recently defined geometric mean for accretive matrices, Ando's result is extended to sector matrices. Some norm inequalities are considered as well.
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Fuping Tan et al.Minimal Estrada index of the trees without perfect matchings
https://repository.uwyo.edu/ela/vol35/iss1/27
https://repository.uwyo.edu/ela/vol35/iss1/27Tue, 24 Sep 2019 08:06:48 PDT
Trees possessing no Kekul ́e structures (i.e., perfect matching) with the minimal Estrada index are considered. Let T_n be the set of the trees having no perfect matchings with n vertices. When n is odd and n ≥ 5, the trees with the smallest and the second smallest Estrada indices among T_n are obtained. When n is even and n ≥ 6, the tree with the smallest Estrada index in T_n is deduced.
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Wen-Huan Wang et al.Consistency 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 Division