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Generalizations of the Cauchy and Fujiwara Bounds for Products of Zeros of a Polynomial
No full textThe Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of zeros of the polynomial
The phiS polar decomposition when the cosquare of S is nonderogatory
No full textFor , define by . A matrix is \textit{orthogonal} if ; is \textit{symmetric} if ; has a \textit{polar decomposition} if for some orthogonal and symmetric . If has a polar decomposition, then commutes with the cosquare . Conditions under which the converse implication holds for the case where is nonderogatory, are obtained
Nostalgia for the 30-Year ‘Century of the Common Man’
No full textJack Metzgar grew up in a steelworking family during the best 30 years in U.S. history for common people, what the French call the Glorious 30 (trente glorieuses) from 1945 to 1975. It was a time of extraordinary economic prosperity that was widely shared. Average real incomes rose faster than ever before or since, with the bottom income quintiles advancing faster and stronger than the middle or top. This unprecedented shared prosperity did not lead to complacency and mindlessconsumerism, as was feared at the time, but rather to a golden age of collective action and a string of liberatory movements beginning with the black civil rights struggle and followed by the beginnings of the women’s and gay liberation movements, among many others. The following is an excerpt from an auto-ethnography Jack is writing about his experience of working-class and professional middle-class cultures from those times to today
Biderivations and linear commuting maps on simple generalized Witt algebras over a field
No full textLet be a simple generalizedWitt algebras over a field of characteristic zero. In this paper, weprove that each anti-symmetric biderivation of isinner. As an application of biderivations, we show that a linear map on is commuting if and only if is ascalar multiplication map on . The commutingautomorphisms and derivations of are determined
Odd Cycle Zero Forcing Parameters and the Minimum Rank of Graph Blowups
No full textThe minimum rank problem for a simple graph G and a given field F is to determine the smallest possible rank among symmetric matrices over F whose i, j-entry, i â j, is nonzero whenever i is adjacent to j, and zero otherwise; the diagonal entries can be any element in F. In contrast, loop graphs \mathscr{G} go one step further to restrict the diagonal i, i-entries as nonzero whenever i has a loop, and zero otherwise. When char F â 2, the odd cycle zero forcing number and the enhanced odd cycle zero forcing number are introduced as bounds for loop graphs and simple graphs, respectively. A relation between loop graphs and simple graphs through graph blowups is developed, so that the minimum rank problem of some families of simple graphs can be reduced to that of much smaller loop graphs
A Singular Value Inequality Related to a Linear Map
No full textIf is positive semidefinite with each block , we prove that where \Phi: X\mapsto X+(\tr X)I and means the -th largest singular value. This confirms a conjecture of the author in [Linear Algebra Appl. 459 (2014) 404-410]
Cyclic Refinements of the Different Versions of Operator Jensen's Inequality
No full textRefinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented
Stochastic forms of non-negative matrices and Perron-regularity
No full textGiven a real non-negative square matrix A, the problem of determining when two distinct constructions of stochastic matrices associated to A coincide is studied. All the constructions (or stochastic forms) that are considered are diagonal forms, i.e., the transformations act like A ô°â αD^(r)AD^(c), where D^(r) and D^(c) are diagonal matrices with positive diagonals and α > 0, all depending on A
On 1-sum flows in undirected graphs
No full textLet be a simple undirected graph. For a given set , a function is called an -flow. Given a vector , is a --flow if for each , the sum of the values on the edges incident to is . If , for all , then the --flow is called a -sum -flow. In this paper, the existence of --flows for various choices of sets of real numbers is studied, with an emphasis on 1-sum flows. Let be a subset of real numbers containing and denote . Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of -sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a -sum -flow or a -sum -flow are characterized. It is also shown that every -regular graph, with either odd or congruent to 2 modulo 4, admits a -sum -flow
