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LIGHTS OUT! on graph products over the ring of integers modulo k
No full textLIGHTS OUT! is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over , where the vertices are either lit or unlit, but the game can be generalized to , where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over . We extend this work to and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over
Decomposition of matrices into commutators of unipotent matrices of index 2
No full textLet be the complex field. Denote by the group of all complex matrices with determinant . It is proved that every matrix in can be decomposed into a product of two commutators of unipotent matrices of index . Moreover, two is the smallest such number
Laplacian integral subcubic signed graphs
No full textA (signed) graph is called Laplacian integral if all eigenvalues of its Laplacian matrix are integers. In this paper, we determine all connected Laplacian integral signed graphs of maximum degree 3; among these signed graphs,there are two classes of Laplacian integral signed graphs, one contains 4 infinite families of signed graphs and another contains 29 individual signed graphs
On -th roots of nilpotent matrices
No full textA new necessary and sufficient condition for the existence of an -th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the -th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an -th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an -th root is corrected and generalized. Moreover, for a singular matrix having an -th root with a pair of nilpotent Jordan blocks of sizes and , a new -th root is constructed by replacing that pair by another one of sizes and , for special . This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix over an arbitrary field that is a sum of two commuting matrices, several results for the existence of -th roots of are obtained
A Dozen Images Made in or Near Youngstown, Ohio, That Show Why People Need Both Jobs and Fish
No full textEconomy vs. ecology. That’s one way to frame the debate that once raged in Youngstown, Ohio, between those who focused on the health of the Mahoning River and those who gave priority to the health of the local economy and the jobs it provided. The latter point of view was often stated in terms of ‘Jobs, not fish!’ and its proponents asked: Compared to jobs in steel mills, which make it possible for workers to have homes and a decent way of life, what does it matter that fish can’t live in the river? Initially, the steel industry benefitted a surprisingly small number of people, mostly owners and investors who treated workers as a resource to be exploited, much like the air and water. But later, thanks to union struggles, workers lived well in the Mahoning Valley, and environmental problems, such as a dirty river, were viewed as a necessary evil. In fact, the foulness of the river assured residents that the mills were going strong and were a source of prosperity. In Youngstown today, deindustrialization has made economic insecurity a fact of life, and the Mahoning, once known as the dirtiest river in the United States, is home to many species of fish. The story of the changes that have taken place in the river landscape centers around the supposed incompatibility of having both jobs along the river’s banks and fish in its waters. Ideas from cultural geography can teach us how to view a landscape where so much conflict has played out.
When geographer James S. Duncan presented the idea of a landscape as texts which communicate and transmit information, he also argued that reading the landscape can reveal how power relations have played out in a given region. Sherry Lee Linkon and John Russo built on similar notions in Steeltown USA: Work and Memory in Youngstown as they showed how people's memories, experiences, and struggles are represented in the landscape. Linkon & Russo also noted that conflict and landscape have a reciprocal relationship. ‘Landscapes not only are constructed by economic and social conflict,’ they stated, ‘but also reinforce such divisions of power.’ ( Linkon & Russo, 2002, pp. 15-16). Such a reading of the Mahoning River landscape yields a complex story about the ways people transformed the natural world in order to benefit from it and then lived with the environmental consequences of that transformation. Though this story is very much about how power and class relations have played out there, in the twentieth century such conflict was often overshadowed by tensions between advocates for steel workers and advocates for the river. Recently, however, the growing understanding of the concept of environmental justice, which has been applied to working-class issues by, among others, Christina Robertson & Jennifer Westerman in their call for a working-class ecology (Robertson & Westerman, 2015) and Karen Bell in her agenda for a just transition to sustainability (Bell, 2020), lays the groundwork for alliances between environmentalists and working-class people that were not present when the Mahoning River was an ‘industrial stream.’ Cultural geographers have also shown us that depictions of a landscape contribute to its meaning(s). Building on such ideas, Linkon & Russo examined the landscape of Youngstown through the lens of images and stories, and this essay will view the more specific landscape of the Mahoning River by examining a dozen images created in or near Youngstown since the early twentieth century. Not all of these images depict the river itself, yet all help to clarify the way the conflict between economy and ecology has played out in the Mahoning Valley
Fazio, M., Launius, C., and Strangleman, T., eds. (2021) Routledge International Handbook of Working-Class Studies. Routledge
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Navigating Academia as a Working-Class Academic
No full textDespite an increasing focus on the impact of class in higher education, less has been said about the experiences of those working-class people who navigate from student to scholar. In the largest interview study to date, conducted in the United Kingdom, this paper draws upon extensive qualitative interview data with ninety working-class academics. This article highlights the hostile encounters faced by these academics but also illuminates the forms of capital and the assets they bring to academia. The article suggests how we can move forward before providing a reminder that the working class should not be viewed by their supposed deficits (real or imaginary)
Additive Maps of Rank k Bivectors
No full textLet and be linear spaces over fields and , respectively, such that Dim and . Let be the second exterior power of . Fixing an even integer satisfying , it is shown that a map satisfies for all rank bivectors if and only if is an additive map. Examples showing the indispensability of the assumption on are given
Nullities for a class of 0-1 symmetric Toeplitz band matrices
No full textLet denote the symmetric Toeplitz band matrix whose first superdiagonals and first subdiagonals have all entries , and whose remaining entries are all . For all n > k >0 with even, we give formulas for the nullity of . As an application, it is shown that over half of these matrices are nonsingular. For the purpose of rapid computation, we devise an algorithm that quickly computes the nullity of even for extremely large values of and , when is even. The algorithm is based on a connection between the nullspace vectors of and the cycles in a certain directed graph