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Full-Text Articles in Physical Sciences and Mathematics
Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang
Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang
Department of Mathematics: Faculty Publications
We study sequences of Betti numbers (βRi (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I☐ generated by the quadratic relations of the associated graded ring of R satisfies height I☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, …
An Application Of The Spectral Theorem To The Laplacian On A Riemannian Manifold, Paul Bracken
An Application Of The Spectral Theorem To The Laplacian On A Riemannian Manifold, Paul Bracken
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
There continues to be great interest in the study of the heat equation on Riemannian manifolds. This may be due to the remarkable more recent work of Patodi [1]. It may also be due in part to the asymptotic expansion of Minakshisundaram and Pleijel. The heat equation involves a parabolic partial differential equation that describes the distribution of heat in a given region over time. This equation has also appears in probability theory to describe random walks. The heat equation is also of importance in Riemannian geometry, topology and applied mathematics.
Lower Bounds For The Maximum Genus Of 4-Regular Graphs, Ding Zhou, Rongxia Hao, Weili He
Lower Bounds For The Maximum Genus Of 4-Regular Graphs, Ding Zhou, Rongxia Hao, Weili He
Turkish Journal of Mathematics
This paper investigates the maximum genus and upper embeddability of connected 4-regular graphs. We obtain lower bounds on the maximum genus of connected 4-regular simple graphs and connected 4-regular graphs without loops in terms of the Betti number. The definition of the Betti number is referred to [Gross and Tucker, Topological Graph Theory, New York, 1987]. Furthermore, we give examples that show that these lower bounds are tight.
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Department of Mathematics: Dissertations, Theses, and Student Research
This work is primarily concerned with the study of artinian modules over commutative noetherian rings.
We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.
In the last chapter we prove that if the Betti …