The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. The small Davenport constant $\mathsf d (G)$ is the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ which has no nontrivial, product-one subsequence. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence---this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that $\mathsf d(G)+1\leq \mathsf D(G)$, and if $G$ is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Now suppose $G$ has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that $\mathsf d(G)=\frac12|G|$ if $G$ is non-cyclic, and $\mathsf d(G)=|G|-1$ if $G$ is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that $\mathsf D(G)=\mathsf d(G)+|G'|$, where $G'=[G,G]\leq G$ is the commutator subgroup of $G$

Representation of Finite Abelian Group Elements by Subsequence Sums

Let $G\cong C_{n_1}\oplus ... \oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1... w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\geq |W|+|G|-1\geq |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma(W)\equiv 0\mod |G|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\sum_{i=1}^{n}w_is_i$, with $s_1... s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\geq {1/2}|G|$. A theorem of Gao, generalizing an older result of Olson, says that if $G$ is a finite abelian group, and $S$ is a sequence over $G$ with $|S|\geq |G|+D(G)-1$, then either every element of $G$ can be represented as a $|G|$-term subsequence sum from $S$, or there exists a coset $g+H$ such that all but at most $|G/H|-2$ terms of $S$ are from $g+H$. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis $|S|\geq |G|+D(G)-1$ can be relaxed to $|S|\geq |G|+d^*(G)$, where d^*(G)=\Sum_{i=1}^{r}(n_i-1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least $d^*(G)$ of the $w_i$ are relatively prime to $|G|$

On the arithmetic of Krull monoids with infinite cyclic class group

Let $H$ be a Krull monoid with infinite cyclic class group $G$ and let $G_P \subset G$ denote the set of classes containing prime divisors. We study under which conditions on $G_P$ some of the main finiteness properties of factorization theory--such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the existence of monotone and of near monotone chains of factorizations--hold in $H$. In many cases, we derive explicit characterizations

Inverse Additive Problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{\mu_d(A)}{M}+\frac{\mu_d(B)}{N}),$$ where $M=\sup\{\mu_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$ and $N=\sup\{\mu_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}$. Standard compression arguments show that the above bound also holds when $M=\mu_{d-1}(\pi(A))$ and $N=\mu_{d-1}(\pi(B))$, where $\pi$ denotes a projection of $\mathbb R^d$ onto $H$, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this later bound, showing that equality holds if and only if $A$ and $B$ are obtained from a pair of homothetic convex bodies by `stretching' along the direction of the projection, which is made formal in the paper. When $d=2$, we characterize the case of equality in the former bound as well

The endogenous caspase-8 inhibitor c-FLIPL regulates ER morphology and crosstalk with mitochondria

Components of the death receptors-mediated pathways like caspase-8 have been identified in complexes at intracellular membranes to spatially restrict the processing of local targets. In this study, we report that the long isoform of the cellular FLICE-inhibitory protein (c-FLIPL), a well- known inhibitor of the extrinsic cell death initiator caspase-8, localizes at the endoplasmic reticulum (ER) and mitochondria-associated membranes (MAMs). ER morphology was disrupted and ER Ca2+-release as well as ER-mitochondria tethering were decreased in c-FLIP-/- mouse embryonic fibroblasts (MEFs). Mechanistically, c-FLIP ablation resulted in enhanced basal caspase-8 activation and in caspase-mediated processing of the ER-shaping protein reticulon-4 (RTN4) that was corrected by re-introduction of c-FLIPL and caspase inhibition, resulting in the recovery of a normal ER morphology and ER-mitochondria juxtaposition. Thus, the caspase-8 inhibitor c-FLIPL emerges as a component of the MAMs signaling platforms, where caspases appear to regulate ER morphology and ER-mitochondria crosstalk by impinging on ER-shaping proteins like the RTN4

Development of fluorescent probes for bioimaging applications

Fluorescent probes, which allow visualization of cations such as Ca2+, Zn2+ etc., small biomolecules such as nitric oxide (NO) or enzyme activities in living cells by means of fluorescence microscopy, have become indispensable tools for clarifying functions in biological systems. This review deals with the general principles for the design of bioimaging fluorescent probes by modulating the fluorescence properties of fluorophores, employing mechanisms such as acceptor-excited Photoinduced electron Transfer (a-PeT), donor-excited Photoinduced electron Transfer (d-PeT), and spirocyclization, which have been established by our group. The a-PeT and d-PeT mechanisms are widely applicable for the design of bioimaging probes based on many fluorophores and the spirocyclization process is also expected to be useful as a fluorescence off/on switching mechanism. Fluorescence modulation mechanisms are essential for the rational design of novel fluorescence probes for target molecules. Based on these mechanisms, we have developed more than fifty bioimaging probes, of which fourteen are commercially available. The review also describes some applications of the probes developed by our group to in vitro and in vivo systems

Opera and poison : a secret and enjoyable approach to teaching and learning chemistry

The storyline of operas, with historical or fictional characters, often include potions and poisons. This has prompted a study of the chemistry behind some operatic plots. The results were originally presented as a lecture given at the University of Minho in Portugal, within the context of the International Year of Chemistry. The same lecture was subsequently repeated at other universities as an invited lecture for science students and in public theaters for wider audiences. The lecture included a multimedia and interactive content that allowed the audience to listen to arias and to watch video clips with selected scenes extracted from operas. The present article, based on the lecture, demonstrates how chemistry and opera can be related and may also serve as a source of motivation and inspiration for chemistry teachers looking for alternative pedagogical approaches. Moreover, the lecture constitutes a vehicle that transports chemistry knowledge to wider audiences through examples of everyday molecules, with particular emphasis on natural products.The author is pleased to express his gratitude to Jorge Calado and Michael John Smith for useful discussions. The author also thanks the reviewers of the manuscript for their helpful comments and suggestions. Thanks are due to the Foundation for Science and Technology (FCT,Portugal), QREN and FEDER/EU for financial support through the research centers, CQ/UM PEst-C/QUI/UI0686/2011. Ciencia Viva, Portugal, is also acknowledged for financial support of the activities organized by the University of Minho during the International Year of Chemistry. The author also expresses his gratitude to Ana Paula Ferreira and Andre Cunha Leal from RTP Antena 2 who contributed immensely to the popularization of the lecture on which this paper is based on

Double-Stranded RNA Attenuates the Barrier Function of Human Pulmonary Artery Endothelial Cells

Circulating RNA may result from excessive cell damage or acute viral infection and can interact with vascular endothelial cells. Despite the obvious clinical implications associated with the presence of circulating RNA, its pathological effects on endothelial cells and the governing molecular mechanisms are still not fully elucidated. We analyzed the effects of double stranded RNA on primary human pulmonary artery endothelial cells (hPAECs). The effect of natural and synthetic double-stranded RNA (dsRNA) on hPAECs was investigated using trans-endothelial electric resistance, molecule trafficking, calcium (Ca2+) homeostasis, gene expression and proliferation studies. Furthermore, the morphology and mechanical changes of the cells caused by synthetic dsRNA was followed by in-situ atomic force microscopy, by vascular-endothelial cadherin and F-actin staining. Our results indicated that exposure of hPAECs to synthetic dsRNA led to functional deficits. This was reflected by morphological and mechanical changes and an increase in the permeability of the endothelial monolayer. hPAECs treated with synthetic dsRNA accumulated in the G1 phase of the cell cycle. Additionally, the proliferation rate of the cells in the presence of synthetic dsRNA was significantly decreased. Furthermore, we found that natural and synthetic dsRNA modulated Ca2+ signaling in hPAECs by inhibiting the sarco-endoplasmic Ca2+-ATPase (SERCA) which is involved in the regulation of the intracellular Ca2+ homeostasis and thus cell growth. Even upon synthetic dsRNA stimulation silencing of SERCA3 preserved the endothelial monolayer integrity. Our data identify novel mechanisms by which dsRNA can disrupt endothelial barrier function and these may be relevant in inflammatory processes

Ca2+-Mg2+-dependent ATP-ase activity in hemodialyzed children. Effect of a hemodialysis session

In the course of chronic kidney disease (CKD) the intracellular erythrocyte calcium (Cai2+) level increases along with the progression of the disease. The decreased activity of Ca2+-Mg2+-dependent ATP-ase (PMCA) and its endogenous modulators calmodulin (CALM), calpain (CANP), and calpastatin (CAST) are all responsible for disturbed calcium metabolism. The aim of the study was to analyze the activity of PMCA, CALM, and the CANP-CAST system in the red blood cells (RBCs) of hemodialyzed (HD) children and to estimate the impact of a single HD session on the aforementioned disturbances. Eighteen patients on maintenance HD and 30 healthy subjects were included in the study. CALM, Cai2+ levels and basal PMCA (bPMCA), PMCA, CANP, and CAST activities were determined in RBCs before HD, after HD, and before the next HD session. Prior to the HD session, the level of Cai2+ and the CAST activity were significantly higher, whereas bPMCA, PMCA, and CANP activities and the CALM level were significantly lower than in controls. After the HD session, the Cai2+ concentration and the CAST activity significantly decreased compared with the basal values, whereas the other parameters significantly increased, although they did not reach the levels of healthy children. The values observed prior to both HD sessions were similar. Cai2+ homeostasis is severely disturbed in HD children, which may be caused by the reduction in the PMCA activity, CALM deficiency, and CANP-CAST system disturbances. A single HD session improved these disturbances but the effect is transient

A comparison of in vitro properties of resting SOD1 transgenic microglia reveals evidence of reduced neuroprotective function

<p>Abstract</p> <p>Background</p> <p>Overexpression of mutant copper/zinc superoxide dismutase (<it>SOD1</it>) in rodents has provided useful models for studying the pathogenesis of amyotrophic lateral sclerosis (ALS). Microglia have been shown to contribute to ALS disease progression in these models, although the mechanism of this contribution remains to be elucidated. Here, we present the first evidence of the effects of overexpression of mutant (TG G93A) and wild type (TG WT) human <it>SOD1 </it>transgenes on a set of functional properties of microglia relevant to ALS progression, including expression of integrin β-1, spreading and migration, phagocytosis of apoptotic neuronal cell debris, and intracellular calcium changes in response to an inflammatory stimulus.</p> <p>Results</p> <p>TG SOD1 G93A but not TG SOD1 WT microglia had lower expression levels of the cell adhesion molecule subunit integrin β-1 than their NTG control cells [NTG (G93A) and NTG (WT), respectively, 92.8 ± 2.8% on TG G93A, 92.0 ± 6.6% on TG WT, 100.0 ± 1.6% on NTG (G93A), and 100.0 ± 2.7% on NTG (WT) cells], resulting in decreased spreading ability, with no effect on ability to migrate. Both TG G93A and TG WT microglia had reduced capacity to phagocytose apoptotic neuronal cell debris (13.0 ± 1.3% for TG G93A, 16.5 ± 1.9% for TG WT, 28.6 ± 1.8% for NTG (G93A), and 26.9 ± 2.8% for NTG (WT) cells). Extracellular stimulation of microglia with ATP resulted in smaller increase in intracellular free calcium in TG G93A and TG WT microglia relative to NTG controls (0.28 ± 0.02 μM for TG G93A, 0.24 ± 0.03 μM for TG WT, 0.39 ± 0.03 μM for NTG (G93A), and 0.37 ± 0.05 μM for NTG (WT) microglia).</p> <p>Conclusions</p> <p>These findings indicate that, under resting conditions, microglia from mutant <it>SOD1 </it>transgenic mice have a reduced capacity to elicit physiological responses following tissue disturbances and that higher levels of stimulatory signals, and/or prolonged stimulation may be necessary to initiate these responses. Overall, resting mutant <it>SOD1</it>-overexpressing microglia may have reduced capacity to function as sensors of disturbed tissue/cellular homeostasis in the CNS and thus have reduced neuroprotective function.</p