The maximum number of cliques in graphs without large matchings

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,T,F)$ and is called the Generalized Tur\'{a}n number. When $T=K_2$, it reduces to the classical Tur\'{a}n number $ex(n,F)$. In 1961, Erd\H{o}s and Gallai proved that $$ex(n,M_{k+1})=\max\left\{\binom{2k+1}{2}, \binom{k}{2}+k(n-k)\right\},$$ where $M_{k+1}$ is a matching of size $k+1$. In this note, we prove that for any $s\geq 2$, $$ex(n,K_s,M_{k+1})=\max\left\{\binom{2k+1}{s}, \binom{k}{s}+(n-k)\binom{k}{s-1}\right\}.$$Comment: Zoltan Furedi, Alexandr Kostochka and Ruth Luo's paper "A variation of a theorem by Posa" indicates an even shorter proo

Advocating the broad use of the decision tree method in education

Predictive studies have been widely undertaken in the field of education to provide strategic information about the extensive set of processes related to teaching and learning, as well as about what variables predict certain educational outcomes, such as academic achievement or dropout. As in any other area, there is a set of standard techniques that is usually used in predictive studies in the field education. Even though the Decision Tree Method is a well-known and standard approach in Data Mining and Machine Learning, and is broadly used in data science since the 1980's, this method is not part of the mainstream techniques used in predictive studies in the field of education. In this paper, we support a broad use of the Decision Tree Method in education. Instead of presenting formal algorithms or mathematical axioms to present the Decision Tree Method, we strictly present the method in practical terms, focusing on the rationale of the method, on how to interpret its results, and also, on the reasons why it should be broadly applied. We first show the modus operandi of the Decision Tree Method through a didactic example; afterwards, we apply the method in a classification task, in order to analyze specific educational data.(undefined)info:eu-repo/semantics/publishedVersio

Meta-analysis for bioequivalence studies : interchangeability of generic drugs and similar containing Hydrochlorothiazide is possible but not with Enalapril Maleate

INTRODUÇÃO: O programa de genéricos no Brasil propiciou maior acesso da população a medicamentos. Para garantir a intercambiabilidade entre medicamentos referência e genérico ou similar, é necessário que eles sejam bioequivalentes. Com o crescimento do número de medicamentos genéricos, é comum que pacientes o substituam por outro genérico ou similar. Contudo, essa troca pode não garantir a manutenção da bioequivalência. Para avaliar a segurança na intercambiabilidade entre diferentes genéricos e similares com Hidroclorotiazida e Maleato de Enalapril, foi realizada metanálise de vários estudos de bioequivalência que utilizaram esses medicamentos. MÉTODOS: Foram utilizados dados provenientes de estudos de bioequivalência de genéricos e similares registrados pela Agência Nacional de Vigilância Sanitária (Anvisa). A compatibilidade dos dados de cada um dos estudos foi analisada, e a determinação de um intervalo de confiança para as diferenças entre as médias dos parâmetros farmacocinéticos, área sob a curva (ASC) e concentração plasmática máxima (Cmáx ), foi feita para cada estudo por meio da metanálise. RESULTADOS: A intercambiabilidade entre as combinações dos três produtos com Hidroclorotiazida foi confirmada com base nos intervalos de confiança obtidos. Para os medicamentos com Maleato de Enalapril, a intercambiabilidade não foi confirmada em 50% das comparações estudadas dos produtos. CONCLUSÃO: A intercambiabilidade foi comprovada entre os três produtos com Hidroclorotiazida. No entanto, para o Maleato de Enalapril, metade dos produtos estudados não é intercambiável, uma vez que não contempla os intervalos preconizados pelos testes de bioequivalência; portanto, a resposta farmacocinética e, por conseguinte, a efetividade do medicamento podem ser alteradas.INTRODUCTION: The generic drugs program provided a better population's access to medicines. To ensure interchangeability between a brand-name and generic or similar drugs is necessary that they are bioequivalent. With the growing number of generic drugs, it is common for patients to replace a generic to another or one similar. However, this exchange can not guarantee the maintenance of bioequivalence. To evaluate the safety interchangeability between different generic and similar drugs with Hydrochlorothiazide and Enalapril Maleate, a meta-analysis was carried out with several bioequivalence studies with these drugs. METHODS: Data from bioequivalence of generic and similar drugs approved by the National Health Surveillance Agency (Anvisa) (drug regulatory agency in Brazil) were used. The compatibility of data from each study was analyzed and the determination of a confidence interval for the differences between the means of pharmacokinetic parameters, area under the curve (ASC0-t) and maximum plasma concentration (Cmax), was made for each study by meta-analysis. RESULTS: The interchangeability between the combinations of the three products with Hydrochlorothiazide was confirmed based on the obtained confidence intervals. For the drugs studied with Enalapril Maleate interchangeability has not been confirmed for 50% of the product comparisons. CONCLUSION: The exchange was established between the three products with hydrochlorothiazide. However, for the Enalapril Maleate half of the products studied are not interchangeable, considering they do not match the established intervals for bioequivalence tests, so the pharmacokinetics behavior and thus the effectiveness of the product may be changed

Moving Parallel and Transversal Lines with Touches on Smartphones: A Look through Screenrecording

This article illustrates how videos generated through capturing touches on screens (screenrecording) can assist in analyses related to manipulation and reasoning by primary education learners in solving geometry tasks. We are presenting two protocols as ways to elucidate the use of application AZ Screen Recorder in learning about parallel lines intersected by a transversal line. In the first protocol, the interactions and the ways adopted by the subjects when analyzing the relation between corresponding angles are highlighted. In the second protocol, the relevant aspects are related to the particularities of the atmosphere of dynamic geometry used and the strategies adopted by the learners in the study of corresponding angles and collateral internal angles. The screen recording allowed us to (re)view the students’ touches in two simultaneous sources (audio and interactions on screen), using the same application. The analyses point to the fact that screenrecording made it possible to identify interactions on screen, especially through synchronicity, which contributed to track learning processes and to reformulate a research planning