The number of rational numbers determined by large sets of integers

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha \beta)^{1+\epsilon}XY$ for any $\epsilon > 0$, where the implied constant depends on $\epsilon$ alone. We then construct examples that show that this bound cannot in general be improved to $\gg \alpha \beta XY$. We also resolve the natural generalisation of our problem to arbitrary subsets $C$ of the integer points in $[1,X] \times [1,Y]$. Finally, we apply our results to answer a question of S\'ark\"ozy concerning the differences of consecutive terms of the product sequence of a given integer sequence.Comment: 11 page

An explicit upper bound for L(1,χ)L(1, \chi) when χ\chi is quadratic

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq 2\cdot 10^{23}$ and $|L(1, \chi)|\leq \frac{9}{20}\log q$ for all $q\geq 5\cdot 10^{50}$.Comment: 17 page

Accountability of school stakeholders in ensuring orphaned children’s school attendance

BACKGROUND : In recent years, there appears to have been more interest than ever in the education of orphaned children, especially in terms of their school attendance. Although some studies have reported on the efforts of caregivers, teachers, government, Non-Governmental Organisations and others in providing the educational needs of orphaned learners, little is known about accountability in terms of their school attendance. AIM : The aim of this study was to explore the perspectives of school stakeholders on managing the school attendance of orphaned learners. SETTING : This research study was conducted in three primary schools with large numbers of orphaned children and high rates of absenteeism in a district in the Limpopo province, South Africa. METHODS : A qualitative approach was adopted, using semi-structured interviews to collect data from nine participants who were identified through purposive sampling. The participants were class teachers, life orientation teachers and teachers responsible for orphaned learners. RESULTS : The findings reflect that there is a lack of material resources and emotional support, physical and sexual abuse, little moral responsibility, few family values and inadequate use of resources provided by government and Non-governmental organizations to support the education of orphaned children. CONCLUSION : In this article, the researchers argue that the provision of material resources and emotional needs alone do not fully address the problem of the school absenteeism of orphaned learners and suggest a combined accountability of, and consequential measures for, school stakeholders in ensuring the school attendance of orphaned children.http://www.sajce.co.zaam2019Education Management and Policy Studie

Utilisation of Mucin Glycans by the Human Gut Symbiont Ruminococcus gnavus Is Strain-Dependent

Commensal bacteria often have an especially rich source of glycan-degrading enzymes which allow them to utilize undigested carbohydrates from the food or the host. The species Ruminococcus gnavus is present in the digestive tract of ≥90% of humans and has been implicated in gut-related diseases such as inflammatory bowel diseases (IBD). Here we analysed the ability of two R. gnavus human strains, E1 and ATCC 29149, to utilize host glycans. We showed that although both strains could assimilate mucin monosaccharides, only R. gnavus ATCC 29149 was able to grow on mucin as a sole carbon source. Comparative genomic analysis of the two R. gnavus strains highlighted potential clusters and glycoside hydrolases (GHs) responsible for the breakdown and utilization of mucin-derived glycans. Transcriptomic and functional activity assays confirmed the importance of specific GH33 sialidase, and GH29 and GH95 fucosidases in the mucin utilisation pathway. Notably, we uncovered a novel pathway by which R. gnavus ATCC 29149 utilises sialic acid from sialylated substrates. Our results also demonstrated the ability of R. gnavus ATCC 29149 to produce propanol and propionate as the end products of metabolism when grown on mucin and fucosylated glycans. These new findings provide molecular insights into the strain-specificity of R. gnavus adaptation to the gut environment advancing our understanding of the role of gut commensals in health and disease

Contribution au probleme de Goldbach: tout entier superieur a 1 somme d'au plus 13 nombres premiers

SIGLEINIST T 75102 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Chowla’s Conjecture: From the Liouville Function to the Moebius Function

International audienceThis aim of this short note is to try to clarify the links between different version of the “Chowla conjecture”

How teachers manage absenteeism of orphaned learners in primary schools

The study was based on the increasing number of orphaned learners and child headed families in our schools and our community. Some of the orphaned learners are unable to attend school regularly. A review of literature reveals that parent or other care givers are held responsible for learners` school attendance and in the case of orphaned there is no parent figure to ensure that learners attend school. The purpose of the study was to explore how teachers manage the absenteeism of orphaned learners in selected primary schools in Limpopo Province. The research approach used in this study was qualitative. I choose qualitative methods because as a researcher, I intended to gain holistic overview of how teachers manage absenteeism of orphaned learners at school. As a researcher I anticipated that teachers shared with me the challenges they experienced when learners are absent from school. I chose qualitative research approach because of its interactive nature. Research design chosen for the study was multiple case study involving three schools within one circuit in Limpopo province. Purposive sampling was used. The participants selected for the study were teachers who are responsible for orphaned learners, Life Orientation teachers and class teachers. A total number of ten (10) teachers were selected from the three schools. The school principals in the respective schools were involved in identifying the participants. Data were collected by using interviews and doing document analysis. The interviews were semi structured with pre-set questions. Learner attendance policy and attendance registers were analysed. The research findings showed that childheaded families are less likely to attend school than non-orphans. The recommendations in the study are made in terms of the findings and within the context of this study.Dissertation (MEd)--University of Pretoria, 2016.tm2016Education Management and Policy StudiesMEdUnrestricte

On the Missing Log Factor

International audienceThis is a written account of a talk with the same title given during the main Chaire Morlet conference. Its guideline is the elementarily proven bound |\sum _{n\leqslant x}\mu (n)/n|\leqslant 1. The trivial bound for the implied summation is \log x+ \operatorname {\mathrm {\mathcal {O}}}(1), while the Prime Number Theorem tells us that it is o(1). Our starting estimate thus lies in-between, a fact that we explore under different lights

Séries de Dirichlet à deux variables et distribution des valeurs de fonctions arithmétiques

Nous traitons deux problèmes liés aux séries de Dirichlet. Nous étudions d'abord le prolongement analytique d'une certaine classe de séries de Dirichlet à deux variables: g(s_1,s_2,a,r) = somme_d=1 r(d) / a(d)s1ds2, où a(d) est une fonction multiplicative strictement positive et r(d) est une fonction multiplicative. Nous démontrons, sous certaines hypothèses, un théorème général qui permet d'approcher cette série de Dirichlet par une série connue, modulo une autre série pour laquelle nous obtenons des majorations très précises. Nous utilisons ensuite cet outil pour obtenir des résultats quantitatifs sur la distribution des valeurs de fonctions arithmétiques. Sous certaines hypothèses sur les fonctions a(d) et r(d), nous déterminons lim_x?8 1/X somme_d<x_a(d)<z r(d) (0<z=X) et mesurons la vitesse de convergence vers la loi limite. La classe de fonctions a(d) est beaucoup plus large que celle considérée jusqu'à maintenant. L'introduction de r(d) semble nouvelle.We deal with two problems related to Dirichlet series. First we study the analytic continuation of a class of Dirichlet series with two variables: g(s_1,s_2,a,r) = sum_d=1 r(d) / a(d)s1ds2, where a(d) is a positive multiplicative function and r(d) is a multiplicative function. We prove, under suitable hypotheses, a general Theorem which allows us to approach this Dirichlet series by a known series, up to another series for which we get very precise upper bounds. Then we use this tool to get quantitative results on the distribution of values of arithmetical functions. Under suitable hypotheses on the functions a(d) and r(d), we determine lim_x?8 1/X sum_d<x_a(d)<z r(d) (0<z=X) and estimate the rate of convergence towards the limit distribution. The class of functions a(d) is much wider than that considered so far. The introduction of r(d) seems to be new.LILLE1-Bib. Electronique (590099901) / SudocSudocFranceF

What are the effects of a cashless society on VAT evasion? : A study on Denmark, Finland &amp; Sweden

The title of this essay is "What are the effects of a cashless society on VAT evasion –A study on Denmark, Finland &amp; Sweden". Due to an increasingly digitalized world there will be different effects on the economy. We are getting closer to a cashless society every day, but we do not know the consequences that this will have. VAT evasion has long been a problem and has been easy to go through with, due to all the payments made with cash. It would be interesting to see if there is going to be a change in VAT evasion now as we go towards a cashless society. The aim of the study was therefore to answer the question: what are the effects of a cashless society on VAT evasion. To answer this question we focused on collecting data from three Scandinavian countries: Denmark, Finland and Sweden. Many articles were read on the subject before interesting data was collected to be analyzed. The data, mostly gathered from the European Central Bank, included the VATgap, number of payment terminals, number of ATMs, percentage of total payments made with cards, GDP and the Consumer Price Index for each of the three chosen countries. In the theoretical framework the theory around VAT is presented as well as a short discussion about the underlying factors on VAT evasion. There is also a section on how we have chosen to represent the cashless society and how this will be measured in the paper. In order with previous research the hypothesis of the study was formed to be that the VAT gap will decrease as we gotowards a cashless society. Multiple regressions were made on the data collected and the result analyzed. There was no significant relationship found between the VAT gap and any of the three explanatory variables. Instead the VAT gap seemed to be connected to what country that was studied. The coefficients of the variables seemed to indicate that there might be a positive relationship between the VAT gap divided by GDP and the number of payment terminals. The reason for this relationship was discussed to possibly be blamed on the fact that card usage increases the total transactions due to the speed and simplicity of card payments. In order to make the results more reliable it was suggested that the study would be enlarged to include more countries and specifically countries that are less digitalized and perceived to be more corrupt