Zero-separating invariants for finite groups

We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G over k and every v ∈ V^G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) = 0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisble by p). We show that δ(G) = |P|. If N_G(P)/P is cyclic, we show σ(G) ≥ |N_G(P)|. If G is p-nilpotent and P is not normal in G, we show σ(G) ≤ |G|/l , where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case

Separating Invariants for the Basic G_a actions

Abstract. We explicitly construct a finite set of separating invariants for the basic G_a -actions. These are the finite dimensional indecomposable rational linear representations of the additive group G_a of a field of characteristic zero, and their invariants are the kernel of the Weitzenbock derivation

Zero-separating invariants for linear algebraic groups

Let G be linear algebraic group over an algebraically closed field k acting rationally on a G-module V , and N(G,V) its nullcone. Let δ(G, V ) and σ(G, V ) denote the minimal number d, such that for any v ∈ V^G \ N(G,V) and v ∈ V \ N(G,V) respectively, there exists a homogeneous invariant f of positive degree at most d such that f (v) = 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V . For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL 2 (k) which contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G 0 is unipotent. Our results also lead to a more elementary proof that β_sep(G) is finite if and only if G is finite

On separating a fixed point from zero by invariants

Assume a fixed point v in V^G can be separated from zero by a homogeneous invariant f ∈ k[V]^G of degree p^r d where p > 0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f . It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded sep- arating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separat- ing algebra implies the group is generated by bireflections. Ad- ditionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay

Improving the Mental Health Care Delivery System for Elderly Nursing Home Patients

It is well known that the mental health care delivery system for aged nursing home patients is inadequate. Based on information gained from face to face interviews and from a mail survey of nursing home personnel, the range and usefulness of the resources and services available for mental health care in nursing homes are identified. This information is then used to derive recommendations for the development of a more effective mental health care delivery package for nursing homes

On separating a fixed point from zero by invariants

Assume a fixed point v in V^G can be separated from zero by a homogeneous invariant f ∈ k[V]^G of degree p^r d where p > 0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f . It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power

Impact of Corrosion Inhibitors on Antibiotic Resistance, Metal Resistance, and Microbial Communities in Drinking Water

Corrosion inhibitors, including zinc orthophosphate, sodium orthophosphate, and sodium silicate, are commonly used to prevent the corrosion of drinking water infrastructure. Metals such as zinc are known stressors for antibiotic resistance selection, and phosphates can increase microbial growth in drinking water distribution systems (DWDS). Yet, the influence of corrosion inhibitor type on antimicrobial resistance in DWDS is unknown. Here, we show that sodium silicates can decrease antibiotic resistant bacteria (ARB) and antibiotic-resistance genes (ARGs), while zinc orthophosphate increases ARB and ARGs in source water microbial communities. Based on controlled bench-scale studies, zinc orthophosphate addition significantly increased the abundance of ARB resistant to ciprofloxacin, sulfonamides, trimethoprim, and vancomycin, as well as the genes sul1, qacEΔ1, an indication of resistance to quaternary ammonium compounds, and the integron-integrase gene intI1. In contrast, sodium silicate dosage at 10 mg/L resulted in decreased bacterial growth and antibiotic resistance selection compared to the other corrosion inhibitor additions. Source water collected from the drinking water treatment plant intake pipe resulted in less significant changes in ARB and ARG abundance due to corrosion inhibitor addition compared to source water collected from the pier at the recreational beach. In tandem with the antibiotic resistance shifts, significant microbial community composition changes also occurred. Overall, the corrosion inhibitor sodium silicate resulted in the least selection for antibiotic resistance, which suggests it is the preferred corrosion inhibitor option for minimizing antibiotic resistance proliferation in DWDS. However, the selection of an appropriate corrosion inhibitor must also be appropriate for the water chemistry of the system (e.g., pH, alkalinity) to minimize metal leaching first and foremost and to adhere to the lead and copper rule