On Rings of Invariants for Cyclic p-Groups

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p \u3e0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular. We also study the a-invariant for a graded ring S, that is, the maximal graded degree of the top local cohomology module of S. We give an upper bound for the a-invariant of R^G and we show for any subgroup H\leq G, we can bound the a-invariant of R^G by the a-invariant of R^H. We extend this result to more general modular rings of invariants where R^{G\u27} is quasi-Gorenstein and G\u27 has a normal, cyclic, p-Sylow subgroup. Given a subgroup H \leq G we consider the natural action of H on R and the associated ring of invariants R^H. When G acts in a particular way, we determine the representation underlying the action of H on R. Building on work of Watanabe and Yoshida, we estimate the Hilbert-Kunz multiplicity of R^G in a way that does not depend on finding explicit generators for R^G. %, that is, we give an upper bound for the asymptotic growth rate of the colength of Frobenius powers of the homogeneous maximal ideal. We extend this result to modular rings of invariants for groups G\u27 which have a normal, cyclic, p-Sylow subgroup. Finally, we also consider computations of the norm of x_4 for a cyclic modular action on k[x_1, \ldots, x_n]. This builds on and extends work of Sezer and Shank

Dominance Over N

Abstract.This paper provides an overview of the b-dominance order over the natural numbers, N, using the base b expansion of natural numbers. The b-dominance order is an accessible partially-ordered set that is less complex than the divisor relation but more complex than ≤; thus, it supplies a good medium through which an undergraduate can be exposed to the subject of order theory. Here we discuss many ideas in order theory, including the Poincaré polynomial and the Möbius function. Acknowledgements: The authors thank the M.J. Murdock Charitable Trust and the Pacific Lutheran University Division of Natural Sciences for their generous support. They would also like to extend their thanks to Dr. Tom Edgar for the project idea and all his help throughout their summer program. Page 24 RHIT Undergrad. Math. J., Vol. 14, no. 2

Dominance Over ℵ

This paper provides an overview of the b-dominance order over the natural numbers, ℵ, using the base b expansion of natural numbers. The b-dominance order is an accessible partially-ordered set that is less complex than the divisor relation but more complex than ≤; thus, it supplies a good medium through which an undergraduate can be exposed to the subject of order theory. Here we discuss many ideas in order theory, including the Poincare polynomial and the Mobius function

LibQuest: A Web App for Exploring the IC Library

LibQuest is a web app that is designed to help students explore the physical space of the library and can be used in place of a library scavenger hunt or tour. Questions encourage students to interact with service points and physical collections, and are randomized to discourage copying. Text-input questions are included to increase difficulty, as are questions that can only be answered by visiting the building

Managing Radiation Degradation of CCDs on the Chandra X-Ray Observatory--III

The CCDs on the Chandra X-ray Observatory are vulnerable to radiation damage from low-energy protons scattered off the telescope's mirrors onto the focal plane. Following unexpected damage incurred early in the mission, the Chandra team developed, implemented, and maintains a radiation-protection program. This program--involving scheduled radiation safing during radiation-belt passes, intervention based upon real-time space-weather conditions and radiation-environment modeling, and on-board radiation monitoring with autonomous radiation safing--has successfully managed the radiation damage to the CCDs. Since implementing the program, the charge-transfer inefficiency (CTI) has increased at an average annual rate of only 3.2x 10(exp -6) (2.3 percent) for the front-illuminated CCDs and 1.0x10(exp -6) (6.7 percent) for the back-illuminated CCDs. This paper describes the current status of the Chandra radiation-management program, emphasizing enhancements implemented since the previous papers

SARS-CoV-2 infection in patients with multiple myeloma: survey in 23 centers across Europe and USA

Introduction: Despite several studies, the impact of coronavirus disease 2019 on patients with multiple myeloma remains uncertain. Material and methods: We performed a survey that covered the period of the first and second waves of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic in 23 centers inseven countries. Out of 352 patients with myeloma and SARS-CoV-2, 23% died. Results/Conclusions: Logistic regression showed a lower risk of death among patients treated with proteasome inhibitor and a higher risk of death for those who had a severe or a very severe course of disease