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Differential Opportunity for Men from Low-Income Backgrounds across Pennsylvania
No full textThis study examines the place-based differences in opportunity experienced by men from lowincome backgrounds across U.S. and Pennsylvania counties. Our quantitative findings suggest that U.S. and Pennsylvania counties are very unequal in terms of how men raised in low-income families fare in adulthood on measures of upward mobility, household income, college graduation, incarceration, and marriage. A variety of county-level measures of concentrated disadvantage were associated with these outcomes, including county household income, poverty rate, degree of racial segregation, college graduation rate, single parenthood rate, social capital rate, and job growth rate. Additionally, anonymous qualitative data from phone interviews with county commissioners from some of the Pennsylvania counties that struggled the most in our analysis helped to confirm our findings with valuable on-the-ground perspectives. We discuss these findings and their implications for equality of opportunity in the U.S. and the state of Pennsylvania
A note on majorization properties of the Lieb function
No full textIn this note, the Lieb function (A,B) \to \Phi (A,B) = \tr \exp ( A + \log B ) for an Hermitian matrix and a positive definite matrix is studied. It is shown that satisfies a majorization property of Sherman type induced by a doubly stochastic operator. The variant for commuting matrices is also considered. An interpretation is given for the case of the orthoprojection operator onto the space of block diagonal matrices
The maximal \alpha-index of trees with k pendent vertices and its computation
No full textLet be a graph with adjacency matrix and let be the diagonal matrix of the degrees of . The index of is the spectral radius of the matrix where . Let be the tree of order and pendent vertices obtained from a star and pendent paths of almost equal lengths attached to different pendent vertices of . It is shown that if and is a tree of order with pendent vertices then% with equality holding if and only if . This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix (). Let and , . It is also obtained that the spectrum of is the union of the spectra of two special symmetric tridiagonal matrices of order and when or the union of the spectra of three special symmetric tridiagonal matrices of order , and when . Thus the index of can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order if or order if
Extremal properties of the distance spectral radius of hypergraphs
No full textThe distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. The unique hypertrees with minimum distance spectral radii are determined in the class of hypertrees of given diameter, in the class of hypertrees of given matching number, and in the class of non-hyperstar-like hypertrees, respectively. The unique hypergraphs with minimum and second minimum distance spectral radii are determined in the class of unicylic hypergraphs. The unique hypertree with maximum distance spectral radius is determined in the class of -th power hypertrees of given matching number
GMPCP for one row and one column completion
No full textIn this paper, the possible Kronecker invariants of a matrix pencil obtained by simultaneous one row and one column completion of a given matrix pencil are characterized. This presents a new approach to the General Matrix Pencil Completion Problem (GMPCP), where simultaneous row and column completion is considered
Some Graphs Determined by their Signless Laplacian (Distance) Spectra
No full textIn literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph provided , or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph (n>l+m) which is obtained from the complete graph by removing all the edges of a complete bipartite subgraph is studied. It is shown that the graph with is determined by its signless Laplacian spectrum, and it is proved that the graph is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for . Here, this problem is completely solved for all positive integer . The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He
Strongly self-inverse weighted graphs
No full textLet G be a connected, bipartite graph. Let Gw denote the weighted graph obtained from G by assigning weights to its edges using the positive weight function w : E(G) ! (0;1). In this article we consider a class Hnmc of bipartite graphswith unique perfect matchings and the family WG of weight functions with weight 1 on the matching edges, and characterize all pairs G in Hnmc and w in WG such that Gw is strongly self-inverse
Parameterized Structure-Preserving Transformations of Matrix Polynomials
No full textThis paper examines the relationship between the companion forms of regular matrix polynomials with singular leading coefficients. When two such polynomials have the same underlying finite and infinite Jordan structures, it is shown that their companion forms are connected by a strict equivalence transformation that can be parameterized using the commutant of the companion forms' common Weierstrass canonical form. The process developed herein for generating such parameterized transformations is applied to the useful class of diagonalizable quadratic polynomials
Solving an Open Problem About the G-Drazin Partial Order
No full textG-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: If A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices