On the denominators of harmonic numbers

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers $n$ such that $v_n$ is divisible by the least common multiple of $1, 2, \cdots, \lfloor {n^{1/4}}\rfloor$ has density one. In particular, for any positive integer $m$, the set of positive integers $n$ such that $v_n$ is divisible by $m$ has density one.Comment: 6 page

Effects of gut hormones, glucagon-like peptide-1 and desacyl ghrelin, on eating behaviour in obesity and ex-smokers

Introduction: Unhealthy eating behaviour is more prevalent in obesity and contributes to weight regain after dieting. Smoking cessation weight gain, a common reason for relapse to cigarettes, also has adverse health consequences. Gut hormones, such as GLP-1 and desacyl ghrelin (DAG), reduce appetite and weight in obesity and Prader-Willi syndrome. GLP-1 and ghrelin signalling systems modulate central reward networks for food and nicotine in preclinical and human studies. However, the impact of GLP-1 and DAG on neurocircuitry involved in eating behaviour in obesity and ex-smokers remains unclear therefore further insight is needed to guide clinical use of gut hormones in prevention of weight gain during dieting and smoking cessation. Aims: Here, the effects of acute administration of GLP-1 analogue, Exenatide or DAG was explored in dieting adults with obesity, or in abstinent nicotine-dependence (double blind randomised placebo controlled cross-over design), on food cue responsivity using fMRI in reward-processing regions, food intake, food reward and appetite. Results: In dieting group with obesity, both Exenatide and DAG increased BOLD signal to high-energy (HE) food pictures in prefrontal cortex regions, implicated in inhibitory control. In contrast, in ex-smokers, both Exenatide and DAG decreased BOLD signal to HE food pictures in the mesolimbic reward-processing regions and prefrontal cortex, suggesting a reduction in anticipatory food reward with a concomitant decrease in executive control. With Exenatide, there was also a reduction in HE food appeal, food intake and appetite ratings in both groups. With DAG, there was no overall effect on HE food appeal, food intake or appetite ratings in both groups. Conclusion: These findings are in accord with the possibility that Exenatide and DAG could be used in prevention of smoking cessation weight gain. This experimental medicine study has provided pilot data for a larger clinical study to trial these gut hormones as potential therapies in smoking cessation.Open Acces

Enhanced spin injection efficiency in a four-terminal double quantum dot system

Within the scheme of quantum rate equations, we investigate the spin-resolved transport through a double quantum dot system with four ferromagnetic terminals. It is found that the injection efficiency of spin-polarized electrons can be significantly improved compared with single dot case. When the magnetization in one of four ferromagnetic terminals is antiparallel with the other three, the polarization rate of the current through one dot can be greatly enhanced, accompanied by the drastic decrease of the current polarization rate through the other one. The mechanism is the exchange interaction between electrons in the two quantum dots, which can be a promising candidate for the improvement of the spin injection efficiency.Comment: 10 pages and 5 figure

Waiver of employer’s termination right by doctrine of election

A contract creates a legal obligation upon the contracting parties. Generally, a contract may be terminated before completion at common law or by the exercise of express rights set out in the contract itself. Contracts can be brought to an end in a variety of ways, either by performance, agreement, frustration or by breach. Under common law, the innocent party can terminate the contract by the operation of law when a party intimates by words or conducts that he does not intend to honour his obligations or when the guilty party commits breach so serious that evinces its intention not to perform. However, a repudiation of contract does not automatically terminate the innocent party’s obligations under the contract. The innocent party has choices between the right to continue the contract or to accept the repudiation of the guilty party as terminating the contract. In legal terminology the choice is known as “election”. In order for the doctrine of election to operate effectively, there are certain essential elements and conditions to be fulfilled. If an election is not done correctly due to certain circumstances, it can be considered as an ineffective election. The employer’s decision to affirm the contract or treat the contract as an end would be challenged. In situation where the employer elects to terminate the contract, it would become a waiver to the employer’s termination right. Therefore, this study focused on the circumstances that are considered as waiver of the employer’s termination right by election. From the result of this research, it can be concluded that the circumstances that may lead the employer to have waive their termination right by election are delay in termination, unequivocal conduct to affirm and affirmation of contract