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Deflating invariant subspaces for rank structured pencils
No full textIt is known that executing a perfect shifted step via the implicit algorithm may not result in a deflation of the perfect shift. Typically, several steps are required before deflation actually takes place. This deficiency can be remedied by determining the similarity transformation via the associated eigenvector. Similar techniques have been deduced for the algorithm and for the rational algorithm. In this paper, we present a similar approach for executing a perfect shifted step on a general rank structured pencil instead of a specific rank structured one, e.g., a Hessenberg-Hessenberg pencil. For this, we rely on the rank structures present in the transformed matrices. A theoretical framework is presented for dealing with general rank structured pencils and deflating subspaces. We present the corresponding algorithm allowing to deflate simultaneously a block of eigenvalues rather than a single one. We define the level- poles and show that these poles are maintained executing the deflating algorithm. Numerical experiments illustrate the robustness of the presented approach showing the importance of using the improved scaled residual approach
Inverse of the squared distance matrix of a complete multipartite graph
No full textLet be a connected graph on vertices and be the length of the shortest path between vertices and in . We set for every vertex in . The squared distance matrix of is the matrix with entry equal to if and equal to if . For a given complete -partite graph on vertices, under some condition we find the inverse as a rank-one perturbation of a symmetric Laplacian-like matrix with . We also investigate the inertia of
The power of bidiagonal matrices
No full textBidiagonal matrices are widespread in numerical linear algebra, not least because of their use in the standard algorithm for computing the singular value decomposition and their appearance as LU factors of tridiagonal matrices. We show that bidiagonal matrices have a number of interesting properties that make them powerful tools in a variety of problems, especially when they are multiplied together. We show that the inverse of a product of bidiagonal matrices is insensitive to small componentwise relative perturbations in the factors if the factors or their inverses are nonnegative. We derive componentwise rounding error bounds for the solution of a linear system , where or is a product of bidiagonal matrices, showing that strong results are obtained when the are nonnegative or have a checkerboard sign pattern. We show that given the factorization of an totally nonnegative matrix into the product of bidiagonal matrices, can be computed in flops and that in floating-point arithmetic the computed result has small relative error, no matter how large is. We also show how factorizations involving bidiagonal matrices of some special matrices, such as the Frank matrix and the Kac-Murdock-Szegö matrix, yield simple proofs of the total nonnegativity and other properties of these matrices
Diagonalizably realizable implies universally realizable
No full textA spectrum of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix . The spectrum is diagonalizably realizable () if the realizing matrix is diagonalizable, and is universally realizable () if it is realizable for each possible Jordan canonical form allowed by In 1981, Minc proved that if is the spectrum of a diagonalizable positive matrix, then is universally realizable. One of the main open questions about the problem of universal realizability of spectra iswhether implies . Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable
“A small step forward can be as important as a big one” – Parliamentary debate about the first abortion law in Sweden in 1938
No full text“A small step forward can be as important as a big one”. So says the social democrat Agda Östlund in the Second Chamber of the Swedish Parliament on Wednesday May 18, 1938, when she justifies her support for a new abortion law.[1] The law gives the right to abortion for women with faltering health and many children, but not for those who suffer financial hardship or social disgrace after becoming pregnant out of wedlock. The debate, and the bill, is characterized by a spirit of cooperation and willingness to compromise. An exception is the conflict between the female members. Voting takes place by standing up and only a few votes against the bill. Previous research has not considered class as central to this debate. However, as we shall see, the debate is about the working-class woman.
[1] Andra kammaren 1938:35, Ang. förslag till lag om avbrytande av havandeskap, p. 27
High S. (2022) Deindustrializing Montreal: Entangled Histories of Race, Residence, and Class. McGill-Queen’s University Press
No full textVisual Mnemonics and Gamification: A New Approach to Teaching Muscle Physiology
No full textThis learning representation is an innovative approach to teaching muscle physiology in a first-year medical school curriculum, utilizing assets from the Medimon game-based website. Medimon is a game designed to enhance students' preference and retention of medical concepts through interactive and visually engaging game contexts. The Medimon game experience allows students to (a) engage with characters representing various physiological components, (b) explore buildings designed to align with visual mnemonics, and (c) reinforce knowledge via game activities of muscle physiology, including muscle, cardiac, and smooth muscle structure and function. By leveraging the detailed visual mnemonics of the game, we implemented Medimon game assets into a presentation on muscle physiology to supplement existing presentation materials and promote long-term retention of muscle physiology
