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Derivations on rank triangular matrices
No full textLet be an integer and let be the algebra of upper triangular matrices over a unital ring . In this paper, we characterize derivations on strictly upper triangular matrices, i.e., additive maps satisfying for all strictly upper triangular matrices . We then deduce this result a complete structural characterization of derivations on rank upper triangular matrices, where is a fixed integer and is a division ring
Decompositions of periodic matrices into a sum of special matrices
No full textWe study the problem of when a periodic square matrix of order over an arbitrary field is decomposable into the sum of a square-zero matrix and a torsion matrix and show that this decomposition can always be obtained for matrices of rank at least when is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when equals the field of the real numbers
Domination number and (signless Laplacian) spectral radius of cactus graphs
No full textA cactus graph is a connected graph whose block is either an edge or a cycle. A vertex set is said to be a dominating set of a graph if every vertex in is adjacent to a vertex in . There are several results on the (signless Laplacian) spectral radius and domination number in graph theory. In this paper, we determine the unique graph with the maximum adjacency spectral radius and signless Laplacian spectral radius among all cactus graphs with fixed domination number
Graph products that allow two distinct eigenvalues
No full textThe parameter of a graph is the minimum number of distinct eigenvalues of a symmetric matrix whose pattern is given by . We introduce a novel graph product by which we construct new infinite families of graphs that achieve . Several graph families for which it is already known that can also be thought of as arising from this new product
Artificial Intelligence-Enhanced Digital Storytelling: Empowering Young Creators in a Summer STEM Camp
No full textThis lesson engages students in grades 3–6 in creating digital stories enhanced by artificial intelligence (AI), encouraging logical thinking, creativity, and problem-solving. Using a structured, project-based format, students developed multimedia-rich narratives with support from AI tools for idea generation, writing refinement, and visual content creation. The students explored STEM role models, wrote story drafts, and edited videos using WeVideo. The project concluded with the production of AI-supported digital stories, assessed through a structured rubric
A note on products of finite-dimensional quadratic matrices
No full textLet be a field, be a positive integer, and , where and are two nonzero elements in . Denote by the ring of all matrices over . A matrix is called quadratic with respect to if . In this paper, we investigate the question of when a matrix in can be expressed as a product of quadratic matrices with respect to . First, we prove that if is a field with more than elements, is an integer, and has determinant , where are integers such that , then can be expressed as a product of quadratic matrices with respect to . In particular, if , for some integer , and has a determinant that is a power of , then can be expressed as a product of at most quadratic matrices with respect to . As a corollary, we derive results on the factorization of matrices as products of certain special quadratic matrices
Eigenvalues and component factors in graphs
No full textFor a set of connected graphs, an -factor of is a spanning subgraph of if each component of is isomorphic to an element of Kano, Lu and Yu [Electron. J. Combin. 26 (2019) P4.33] provided a good characterization based on an isolated vertex condition for the existence of a -factor in graphs. Motivated by the above elegant result, we in this paper focus on the existence of a -factor in graphs from perspective of eigenvalues. By adopting a crucial technique due to Tait [J. Combin. Theory Ser. A 166 (2019) 42-58] and combining typical spectral methods and structural analysis, we present tight sufficient conditions in terms of the spectral radius and the distance spectral radius for a graph to contain a -factor, respectively
“Workers without Borders:” Envisioning Sociality in Xiao Hai’s Poems
No full textNew worker poetry has emerged as a unique literary voice in contemporary China. This paper places Chinese new workers as the global working class and focuses on the poetics of their global vision. Through a close reading of poems written by Xiao Hai (1980-), one of the prolific worker poets, I argue that the new worker poet constructs global sociality at the levels of aesthetics, social critique, and cultural proposal. Aesthetically, Xiao Hai has borrowed inspiration from classical Chinese poetry, western counter-culture icons, and contemporary avant-garde spirit in his writings on laborers’ ordeals. Global sociality embodies a powerful critique of hierarchical global systems in which laborers are positioned at the bottom. It is also a cultural ideal rooted in revolutionary nostalgia and classical notions, a passionate call for connection among like-minded people, and an awareness of workers’ shared identity. Raising their voices in poetry, Xiao Hai, as well as other worker poets, actively explore opportunities to make their voices heard on a broader scal
Versal deformations: a tool of linear algebra
No full textA versal deformation of a matrix is a normal form to which all matrices , close to , can be reduced by similarity transformation smoothly depending on the entries of . In this paper, we discuss versal deformations and their use in codimension computations, in investigation of closure relations of orbits and bundles, in studying changes of canonical forms under perturbations, as well as in the reduction of unstructured perturbations to structured perturbations
Schur functions and immanantal identities
No full textLittlewood developed the theory of symmetric functions and immanants. It is known that some identities for immanants correspond to the ordinary products of Schur functions via the Littlewood-Richardson rule. We discuss the relations between immanants and plethysm, another type of products of Schur functions. We present immanantal identities corresponding to the most basic formula of plethysm. As an application, we show some inequalities for positive semidefinite Hermitian matrices