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Linear preservers of Hadamard majorization
No full textLet be the set of all realmatrices. A matrix with nonnegative entries is called doubly stochastic if for all . For we say that is Hadamard-majorized by , denoted by , if there exists an doubly stochastic matrix such that .In this paper, some properties of on are first obtained, and then the (strong) linear preservers of on are characterized. For , it is shown that the strong linear preservers of Hadamard majorization on are precisely the invertible linear maps on which preserve the set of matrices of term rank 1.An interesting graph theoretic connection to the linear preservers of Hadamard majorization is exhibited. A number of examples are also provided in the paper
A unique “icy seep” aquatic habitat in the high Teton Range: Potential refuge for biological assemblages imperiled by climate change
No full textAlpine streams are threatened as meltwater sources diminish. We established baseline monitoring efforts of alpine streams in the Teton Range during 2015 by describing biotic and abiotic conditions of surface glacier and snowmelt streams. Our results indicated a third alpine stream type, icy seeps. Icy seeps are fed by subterranean ice melt, extremely cold but stable water temperature (summer mean <2°C), moderately high streambed stability (Pfankuch Index ~18-25), and relatively high specific conductivity (>50 μS cm-1). In 2016, we documented several icy seeps in the Teton massif, and our data suggest that they have the potential to serve as climate refugia for organisms and processes associated with extremely cold meltwater, due to the subterranean ice sources being more insulated from atmospheric conditions than surface glaciers and snowpack. Our 2016 work focused on locating icy seeps and documenting physical conditions, invertebrates, diatoms, and microbes in these alpine streams. We are processing the biological data, but microbial assemblages in icy seeps are similar to those found in streams fed by dwindling surface glacier ice. We are preparing to submit a paper about the microbial patterns, including how they compare to alpine streams representing comparable hydrological sources in Glacier National Park.
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Block imbedding and interlacing results for normal matrices
No full textA pair of matrices is said to be imbeddable precisely when one is an isometric projection of the other on a suitable subspace. The concept of imbedding has been the subject of extensive study. Particular emphasis has been placed on relating the spectra of the matrices involved, especially when both matrices are Hermitian or normal. In this paper, the notion of block imbedding is introduced and shown to be intimately connected to an extension of interlacing for eigenvalues of normal matrices. Thus, a generalization of a classic Theorem of K. Fan and G. Pall is obtained, which is then applied to yield bounds on the number of eigenvalues of a block imbeddable pair in a closed, convex set. Moreover, a wide class of normal matrices, for which block imbedding applies, is indicated. Finally, comments and links on the necessary imbedding conditions of D. Carlson and E.M. de Sa, and J.P. Queiro and A.L. Duarte are provided
Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters
No full textThis paper provides some properties of Brauer symmetry classes of tensors. A dimension formula is derived for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups whose non-linear Brauer characters have support being a cyclic group. Using the derived formula, necessary and sufficient condition are investigated for the existence of an o-basis of dicyclic groups, semi-dihedral groups, and also those things are reinvestigated on dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included
The distance spectral radius of graphs with given number of odd vertices
No full textThe graphs with smallest, respectively largest, distance spectral radius among the connected graphs, respectively trees with a given number of odd vertices, are determined. Also, the graphs with the largest distance spectral radius among the trees with a given number of vertices of degree 3, respectively given number of vertices of degree at least 3, are determined. Finally, the graphs with the second and third largest distance spectral radius among the trees with all odd vertices are determined
Homeomorphic Images of Orthogonal Bases
No full textNecessary and sufficient conditions are obtained for a sequence in a Hilbert space to be, up to the elimination of a finite subset of , the linear homeomorphic image of an orthogonal basis of some Hilbert space . This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques
The inverse along an element in rings
No full textSeveral properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered
Controllability and nonsingular solutions of Sylvester equations
No full textThe singularity problem of the solutions of some particular Sylvester equations is studied. As a consequence of this study, a good choice of a Sylvester equation which is associated to a linear continuous time system can be made such that its solution is nonsingular. This solution is then used to solve an eigenstructure assignment problem for this system. From a practical point view, this study can also be applied to automatic control when the system is subject to input constraints
An Expansion Property of Boolean Linear Maps
No full textGiven a finite set , a Boolean linear map on is a map from the set of all subsets of into itself with such that holds for all . For fixed subsets of , to predict if is reachable from in the dynamical system driven by , one can assume the existence of nonnegative integers with , find an upper bound for the minimum of all such assumed integers , and test if really appears in . In order to get such an upper bound estimate, this paper establishes an expansion property for the Boolean linear map . Namely, the authors find a lower bound on the size of for any nonnegative integer . Besides presenting several direct applications of the derived expansion property, this paper collects some related problems on Boolean linear dynamical systems, including problems on primitive multilinear maps and inhomogeneous topological Markov chains
On the characterization and parametrization of strong linearizations of polynomial matrices
No full textIn the present note, a new characterization of strong linearizations, corresponding to a given regular polynomial matrix, is presented. A linearization of a regular polynomial matrix is a matrix pencil which captures the finite spectral structure of the original matrix, while a strong linearization is one incorporating its structure at infinity along with the finite one. In this respect, linearizations serve as a tool for the study of spectral problems where polynomial matrices are involved. In view of their applications, many linearization techniques have been developed by several authors in the recent years. In this note, a unifying approach is proposed for the construction of strong linearizations aiming to serve as a bridge between approaches already known in the literature