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Atlas of Wildlife Migration: Wyoming's Ungulates
No full textFor thousands of years ungulates have migrated between seasonal ranges in the vast and beautiful landscapes of Wyoming. From mule deer and pronghorn that travel across the Red Desert to the wilderness journeys of elk and moose in the Greater Yellowstone Ecosystem, Wyoming boasts some of the longest and most spectacular migrations in North America. These epic, terrestrial migrations are to many a symbol of Wyomingâs vast intact landscapes. And although these migrations are part of the regionâs cultural heritage, they are poorly understood and threatened by rapidly changing landscapes. Recent research at the University of Wyoming has broken new ground in our understanding of Wyomingâs ungulat
Open problems in the theory of completely positive and copositive matrices
No full textWe describe the main open problems which are currently of interest in the theory of copositive and completely positive matrices. We give motivation as to why these questions are relevant and provide a brief description of the state of the art in each open problem
Decompositions into products of idempotents
No full textThe purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (â) that each right (left) singular element is a product of idempotents, and (2) to consider the question: âwhen is a singular nonnegative square matrix a product of nonnegative idempotent matrices?â The importance of the class of quasi- Euclidean rings in connection with the property (â) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154â170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118â122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309â314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices
Minimum ranks of sign patterns via sign vectors and duality
No full textA sign pattern matrix is a matrix whose entries are from the set {+,â,0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any mÃn sign pattern A with minimum rank n â 2, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer n ⥠9, there exists a nonnegative integer m such that there exists an m à n sign pattern matrix with minimum rank n â 3 for which rational realization is not possible. A characterization of m à n sign patterns A with minimum rank n â 1 is given (which solves an open problem in Brualdi et al. [R. Brualdi, S. Fallat, L. Hogben, B. Shader, and P. van den Driessche. Final report: Workshop on Theory and Applications of Matrices Described by Patterns. Banff International Research Station, Jan. 31 â Feb. 5, 2010.]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. Several related open problems are stated along the way
Combinatorial properties of generalized M-matrices
No full textAn M_â¨-matrix has the form A = sI â B with s ⥠Ï(B) and B^k is entrywise nonnegative for all sufficiently large integers k. In this paper, the existence of a preferred basis for a singular M_â¨- matrix A = sI â B with index(B) ⤠1 is proven. Some equivalent conditions for the equality of the height and level characteristics of A are studied. The well structured property of the reduced graph of A is discussed. Also the possibility of the existence of preferred basis for another generalization of M-matrices, known as GM-matrices, is studied
Nonnegative generalized doubly stochastic matrices with prescribed elementary divisors
No full textThis paper provides sufficient conditions for the existence of nonnegative generalized doubly stochastic matrices with prescribed elementary divisors. These results improve previous results and the constructive nature of their proofs allows for the computation of a solution matrix. In particular, this paper shows how to transform a generalized stochastic matrix into a nonnegative generalized doubly stochastic matrix, at the expense of increasing the Perron eigenvalue, but keeping other elementary divisors unchanged. Under certain restrictions, nonnegative generalized doubly stochastic matrices can be constructed, with spectrum \Lambda = {\lambda_1,\lambda_2 2, . . . , \lambda_n} for each Jordan canonical form associated with \Lambd
Higher numerical ranges of quaternion matrices
No full textLet n and k be two positive integers and k n. In this paper, the notion of kânumerical range of nâsquare quaternion matrices is introduced. Some algebraic and geometrical properties are investigated. In particular, a necessary and sufficient condition for the convexity of the kânumerical range of a quaternion matrix is given. Moreover, a new description of 1ânumerical range of normal quaternion matrices is also stated
Application of an identity for subtrees with a given eigenvalue
No full textFor an Hermitian matrix whose graph is a tree and for a given eigenvalue having Parter vertices, the possibilities for the multiplicity are considered. If V = {v_1, . . . , v_k} is a fragmenting Parter set in a tree relative to the eigenvalue , and T_{i+1} is the component of Tâ{v_1, v_2, . . . , v_i} in which v_{i+1} lies, it is shown that \sum_{i}^K N_i=m_A(\lambda)+2kâ1, in which N_i is the number of components of T_iâv_i in which lambda is an eigenvalue. This identity is applied to make several observations, including about when a set of strong Parter vertices leaves only 3 components with \lambda and about multiplicities in binary trees. Furthermore, it is shown that one can construct an Hermitian matrix whose graph is a tree that has a strong Parter set V such that |V | = k for each k in
On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results
No full textGiven a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A(G) and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in A(G)âv goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalizeour results to weighted graphs, or equivalently to symmetric matrices