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Recent documents in Wyoming Scholars Repositoryen-usMon, 22 Oct 2018 01:50:06 PDT3600On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices
https://repository.uwyo.edu/ela/vol33/iss1/8
https://repository.uwyo.edu/ela/vol33/iss1/8Fri, 12 Oct 2018 13:20:03 PDT
We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.
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Katarzyna Filipiak et al.Explicit Block-Structures for Block-Symmetric Fiedler-like pencils
https://repository.uwyo.edu/ela/vol34/iss1/36
https://repository.uwyo.edu/ela/vol34/iss1/36Tue, 09 Oct 2018 16:18:51 PDT
In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial $P(\lambda)$, regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular $P(\lambda)$ in an easy way, allowing the computation of the minimal indices of a singular $P(\lambda)$ in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were constructed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial $P(\lambda)$ is symmetric (Hermitian), it is convenient to use linearizations of $P(\lambda)$ that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of $P(\lambda)$, which are symmetric (Hermitian) when $P(\lambda)$ is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem. In particular, it was shown that all GFP and GFPR, after permuting some block-rows and block-columns, belong to the family of extended block Kronecker pencils, which are defined explicitly in terms of their block-structure. Unfortunately, those permutations that transform a GFP or a GFPR into an extended block Kronecker pencil do not preserve the block-symmetric structure. Thus, in this paper, the family of block-minimal bases pencils, which is closely related to the family of extended block Kronecker pencils, and whose pencils are also defined in terms of their block-structure, is considered as a source of canonical forms for block-symmetric pencils. More precisely, four families of block-symmetric pencils which, under some generic nonsingularity conditions are block minimal bases pencils and strong linearizations of a matrix polynomial, are presented. It is shown that the block-symmetric GFP and GFPR, after some row and column permutations, belong to the union of these four families. Furthermore, it is shown that, when $P(\lambda)$ is a complex matrix polynomial, any block-symmetric GFP and GFPR is permutationally congruent to a pencil in some of these four families. Hence, these four families of pencils provide an alternative but explicit approach to the block-symmetric Fiedler-like pencils existing in the literature.
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M. I. Bueno et al.On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs
https://repository.uwyo.edu/ela/vol34/iss1/35
https://repository.uwyo.edu/ela/vol34/iss1/35Tue, 09 Oct 2018 16:18:40 PDT
Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.
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Shuting Liu et al.Positive and Z-operators on Closed Convex Cones
https://repository.uwyo.edu/ela/vol34/iss1/34
https://repository.uwyo.edu/ela/vol34/iss1/34Sun, 07 Oct 2018 21:44:37 PDT
Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.
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Michael J. OrlitzkySpectral Bounds for the Connectivity of Regular Graphs with Given Order
https://repository.uwyo.edu/ela/vol34/iss1/33
https://repository.uwyo.edu/ela/vol34/iss1/33Sun, 07 Oct 2018 21:44:25 PDT
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, two upper bounds are presented for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.
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Aida Abiad et al.Bounded linear operators that preserve the weak supermajorization on $\ell^1(I)^+$
https://repository.uwyo.edu/ela/vol34/iss1/32
https://repository.uwyo.edu/ela/vol34/iss1/32Sun, 07 Oct 2018 21:44:14 PDT
Linear preservers of weak supermajorization which is defined on positive functions contained in the discrete Lebesgue space $\ell^1(I)$ are characterized. Two different classes of operators that preserve the weak supermajorization are formed. It is shown that every linear preserver may be decomposed as sum of two operators from the above classes, and conversely, the sum of two operators which satisfy an additional condition is a linear preserver. Necessary and sufficient conditions under which a bounded linear operator is a linear preserver of the weak supermajorization are given. It is concluded that positive linear preservers of the weak supermajorization coincide with preservers of weak majorization and standard majorization on $\ell^1(I)$.
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Martin Z. Ljubenović et al.BOUNDS ON THE SUM OF MINIMUM SEMIDEFINITE RANK OF A GRAPH AND ITS COMPLEMENT
https://repository.uwyo.edu/ela/vol34/iss1/31
https://repository.uwyo.edu/ela/vol34/iss1/31Wed, 26 Sep 2018 08:55:19 PDT
The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.
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Sivaram Narayan et al.Identifying combinatorially symmetric Hidden Markov Models
https://repository.uwyo.edu/ela/vol34/iss1/30
https://repository.uwyo.edu/ela/vol34/iss1/30Thu, 23 Aug 2018 17:51:34 PDT
A sufficient criterion for the unique parameter identification of combinatorially symmetric Hidden Markov Models, based on the structure of their transition matrix, is provided. If the observed states of the chain form a zero forcing set of the graph of the Markov model, then it is uniquely identifiable and an explicit reconstruction method is given.
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Daniel BurgarthThe Largest Eigenvalue and Some Hamiltonian Properties of Graphs
https://repository.uwyo.edu/ela/vol34/iss1/29
https://repository.uwyo.edu/ela/vol34/iss1/29Thu, 23 Aug 2018 17:51:26 PDT
In this note, sufficient conditions, based on the largest eigenvalue, are presented for some Hamiltonian properties of graphs.
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Rao LiPositive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^*_3=C_3, A_4XA^*_4=C_4$ in Hilbert $C^*$-modules
https://repository.uwyo.edu/ela/vol34/iss1/28
https://repository.uwyo.edu/ela/vol34/iss1/28Thu, 23 Aug 2018 17:51:16 PDT
Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is given to show that the proposed investigation in the setting of Hilbert $C^*$-modules is different from that of Hilbert spaces.
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Rasoul Eskandari et al.Proof of a Conjecture of Graham and Lovasz concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree
https://repository.uwyo.edu/ela/vol34/iss1/27
https://repository.uwyo.edu/ela/vol34/iss1/27Thu, 23 Aug 2018 11:00:37 PDT
The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.
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Ghodratollah Aalipour et al.Extremal octagonal chains with respect to the spectral radius
https://repository.uwyo.edu/ela/vol34/iss1/26
https://repository.uwyo.edu/ela/vol34/iss1/26Thu, 23 Aug 2018 11:00:22 PDT
Octagonal systems are tree-like graphs comprised of octagons that represent a class of polycyclic conjugated hydrocarbons. In this paper, a roll-attaching operation for the calculation of the characteristic polynomials of octagonal chain graphs is proposed. Based on these characteristic polynomials, the extremal octagonal chains with n octagons having the maximum and minimum spectral radii are identified.
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Xianya Geng et al.Decoding the “Testimonial” Tug of War: When a Cellphone Search Warrant and a Showing of Substantial Need and Undue Hardship Justify Cellphone Passcode Compulsion
https://repository.uwyo.edu/wlr/vol18/iss2/8
https://repository.uwyo.edu/wlr/vol18/iss2/8Tue, 21 Aug 2018 16:50:35 PDTZara S. MasonGeneral Law DivisionA Proposal for a National Tribally Owned Lien Filing System to Support Access to Capital in Indian Country
https://repository.uwyo.edu/wlr/vol18/iss2/7
https://repository.uwyo.edu/wlr/vol18/iss2/7Tue, 21 Aug 2018 16:50:29 PDTWilliam H. Henning et al.General Law DivisionIdentifying Inefficiencies: Exploring Ways to Write Briefs More Quickly within the Time Demands of Legal Practice
https://repository.uwyo.edu/wlr/vol18/iss2/6
https://repository.uwyo.edu/wlr/vol18/iss2/6Tue, 21 Aug 2018 16:50:22 PDTJacob M. CarpenterGeneral Law DivisionESTATE LAW—Balancing the Competing Interests of Efficiency, Finality, and Freedom of Disposition in Ancillary Administration Proceedings: Lon V. Smith Foundation v. Devon Energy Corp., 2017 WY 121, 403 P.3d 997 (Wyo. 2017)
https://repository.uwyo.edu/wlr/vol18/iss2/5
https://repository.uwyo.edu/wlr/vol18/iss2/5Tue, 21 Aug 2018 16:50:17 PDTKaylee HarmonWY Law DivisionJust How Liberal is Liberal?: Wyoming Courts’ Treatment of Civil Pro Se Pleadings
https://repository.uwyo.edu/wlr/vol18/iss2/4
https://repository.uwyo.edu/wlr/vol18/iss2/4Tue, 21 Aug 2018 16:50:10 PDTMichael J. KlepperichWY Law DivisionBeauty and the H-2Beast: How the Equality State Fails its Female Guest Workers
https://repository.uwyo.edu/wlr/vol18/iss2/3
https://repository.uwyo.edu/wlr/vol18/iss2/3Tue, 21 Aug 2018 16:50:05 PDTCatherine DiSantoWY Law DivisionThe Discovered Country: Wyoming’s Primacy as a Trust Situs Jurisdiction
https://repository.uwyo.edu/wlr/vol18/iss2/2
https://repository.uwyo.edu/wlr/vol18/iss2/2Tue, 21 Aug 2018 16:49:59 PDTAmy M. StaehrWY Law DivisionImproving Wyoming’s Attorney-Client Privilege
https://repository.uwyo.edu/wlr/vol18/iss2/1
https://repository.uwyo.edu/wlr/vol18/iss2/1Tue, 21 Aug 2018 16:49:52 PDTJohn M. Burman et al.WY Law Division