University of Wyoming Open Journals
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Digital Escape Room: Agency, Personalized Instruction, and Flow Theory
No full textScheduled during the final two weeks of a practicum internship, this asynchronous online lesson for preservice teachers focuses on Flow Theory and provides first-hand experience with agentic instruction to deepen students’ pedagogical understandings and promote future implementation of such practices. Applying principles of inquiry-based learning and digital escape room design, students are tasked with completing the digital escape room to rescue their instructor who has been trapped in a digital world by his jealous colleagues. Delivered through the Canvas learning management system, students seemingly enjoy the activity, despite occasionally struggling, and some students plan their own digital escape rooms as a result of this activity
Minimal rank weighted weak Drazin inverses
No full textThe concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted -WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained
Faces of the signed Birkhoff polytopes
No full textWe study faces of the signed Birkhoff polytopes, denoted by . We describe its nonempty faces, -dimensional faces, -dimensional faces, and facets. Moreover, we study the diameter and Hamiltonian connectivity of the graph of . In the end, we show that the reduced Gröbner basis of the toric ideal of the signed Birkhoff polytope with respect to the graded reverse lexicographic order induced by rank orders has square-free initial monomials of degree
Principal eigenvectors in hypergraph Turán problems
No full textFor a general class of hypergraph Turán problems with uniformity , we investigate the principal eigenvector for the -spectral radius (in the sense of Keevash-Lenz-Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to ). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the -spectral radius of an arbitrary -graph in terms of the degrees of the graph. This builds on earlier work of Cardoso-Trevisan, Li-Zhou-Bu, Cioabă-Gregory, and Zhang. The case 1 < p < r of our results leads to some subtleties connected to Nikiforov's notion of -tightness, arising from the Perron-Frobenius theory for the -spectral radius. We raise a conjecture about these issues and provide some preliminary evidence for our conjecture
Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces
No full textWe propose and justify a matrix reduction method for calculating the optimal approximation of an observed matrix by a sum of matrix products where each and is known and where the unknown matrix kernels are determined by minimizing the Frobenius norm of the error. The sum can be represented as a bounded linear mapping with unknown kernel from a prescribed subspace onto a prescribed subspace defined, respectively, by the collective domains and ranges of the given matrices and . We show that the optimal kernel is and that the optimal approximation is the projection of the observed mapping onto a mapping from to . If is large, and may also be large and direct calculation of and becomes unwieldy and inefficient. The proposed method avoids this difficulty by reducing the solution process to finding the pseudo-inverses of a collection of much smaller matrices. This significantly reduces the computational burden
Eigenvalue characterization of some structured matrix pencils under linear perturbation
No full textWe study the effect of linear perturbations on three families of matrix pencils. The matrix pairs of the first two families are Hermitian/skew-Hermitian with special block cases appeared in continuous-time control, and the matrix pairs of the third family are special non-Hermitian block matrices appeared in discrete-time control. For the first family of matrix pencils and more general cases of the second family of matrix pencils, based on the properties of the involved matrices, we obtain some upper or lower bounds on the set of eigenvalues of linearly perturbed matrix pencils which are on the imaginary axis. Studying a special block matrix pencil, which is associated with continuous-time control, leads us to some linear perturbation that do not preserve (properly) the structure of the matrices. This, in turn, leads to a numerical technique for finding the nearest Hermitian/skew-Hermitian matrix pencil which can satisfy conditions such that, for some nonzero real perturbation parameter, some or all of its eigenvalues lie on the imaginary axis. We also study the linearly perturbed matrix pencils, associated with discrete-time control, using an one-to-one equivalence between the matrix pencil of continuous-time problem and the matrix pencil of discrete-time problem
tensors and -tensors
No full textStrong -tensors play an important role in the fields of science and engineering. In this paper, we first propose a new subclass of strong -tensors that we call the class of tensors. We also prove that if a tensor is an tensor, it is a strong -tensor. As an application, a sufficient condition for an tensor to certify positive definiteness of even-order real symmetric tensors is proposed. Furthermore, we propose a new class of tensors by means of tensors, naming it -tensors, and show that -tensors are a subclass of it. Meanwhile, some properties of -tensor were introduced. The numerical examples demonstrate the validity of our results
An extension of the Perron-Frobenius theory to arbitrary matrices and cones
No full textWe develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and do not preserve a cone. We introduce a new concept called a "quasi-eigenvalue of a matrix," which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts
The Man with a Million Names: A Personal Essay on Transit Work
No full textThis essay is a scholarly personal narrative about transit work, especially the operation of omnibuses, horse cars, trolleys, and trams in New York City in the nineteenth century. The culminating event is the trolley strike of 1895, the longest in New York history, and the theme is the need for solidarity between transit workers and the riding public, and thus for what is now is called union “Bargaining for the Public Good.” In this essay, the author speaks as both a transit worker and an historian. 
