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Brooks, R. (2022) Class Interruptions: Inequality and Division in African Diasporic Women’s Fiction. University of North Carolina Press.
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Maker Activities and Academic Writing in a Middle School Science Class
No full textAcademic language is an important focus area in middle school. However, academic writing in science classes is challenging for middle school students. Maker activities can contribute to students’ academic language development through artifact creation with tangible resources (Own, 2018). This makerspace and academic writing project, codesigned by a science teacher and a technology integration specialist, invited middle school science students to work on academic writing through maker activities
On the structure of isometrically embeddable metric spaces
No full textSince its popularization in the 1970s, the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work, we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space
Majorization inequalities via convex functions
No full textConvex functions have been well studied in the literature for scalars and matrices. However, other types of convex functions have not received the same attention given to the usual convex functions. The main goal of this article is to present matrix inequalities for many types of convex functions, including log-convex, harmonically convex, geometrically convex, and others. The results extend many known results in the literature in this direction. For example, it is shown that if are positive definite matrices and is a continuous -convex function on an interval containing the spectra of , then\begin{align*}\lambda^\downarrow (f(A\sigma B))\prec_w\lambda^\downarrow \left(f(A)\tau f(B)\right),\end{align*}for the matrix means and . Further, if , then\begin{align*} \lambda^\downarrow \left(f\left(e^{A\nabla_{\alpha}B}\right)\right)\prec_w\lambda^\downarrow \left(f(e^A)\tau f(e^B))\right).\end{align*}Similar inequalities will be presented for two-variable functions too
The Hamiltonian extended Krylov subspace method
No full textAn algorithm for constructing a -orthogonal basis of the extended Krylov subspace where is a large (and sparse) Hamiltonian matrix is derived (for or ). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting onto the subspace yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate where is a function which maps the Hamiltonian matrix to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods
W-weighted GDMP inverse for rectangular matrices
No full textIn this article, we introduce two new generalized inverses for rectangular matrices called -weighted generalized-Drazin--Moore--Penrose (GDMP) and -weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of -weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of -weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse- and forward-order laws for GD, -weighted GD, GDMP, and -weighted GDMP generalized inverses
Sign patterns associated with some graphs that allow or require diagonalizability
No full textThe problems of characterizing sign pattern matrices that allow or require diagonalizability are mostly open. In this paper, we introduce the concept of essential index for a tree sign pattern matrix and use it to investigate the allow problem on diagonalizability for sign pattern matrices having their graphs as trees. We characterize sign pattern matrices allowing diagonalizability, whose graphs are star or path. We also give a sufficient condition for sign pattern matrices whose graphs are trees to allow diagonalizability. Further, we give a necessary condition for a sign pattern matrix to require diagonalizability and characterize all star sign pattern matrices that require diagonalizability
Positivity of Hadamard powers of a few band matrices
No full textLet and be the sets of positive semidefinite and positive definite matrices of order , respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph with vertices. It is proved that for a connected simple graph of order , the set of powers preserving positive semidefiniteness on is precisely the same as the set of powers preserving positive definiteness on . In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely . The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices
Spectral properties of certain sequences of products of two real matrices
No full textThe aim of this paper is to analyze the asymptotic behavior of the eigenvalues and eigenvectors of particular sequences of products involving two square real matrices and , namely of the form , as . This analysis represents a detailed deepening of a particular case within a general theory on finite families of real square matrices already available in the literature. The Bachmann-Landau symbols and related results are largely used and are presented in a systematic way in the final Appendix