Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics

In this thesis, we consider several combinatorial topics, belonging to the areas appearing in the thesis title. Given a non-empty complete metric space $(X,d)$, a family of $n$ continuous maps $f_1,f_2,\dots,f_n\colon X\to X$ is a \emph{contractive family} if there exists $\lambda0$ and $k\in\mathbb{N}$, we construct a subset $A\subset\mathbb{Z}_N$ for some $N$, such that $|A^2+kA|\leq\epsilon N$, while $A-A=\mathbb{Z}_N$. (Here $A-A=\{a_1-a_2:a_1,a_2\in A\}$ and $A^2+kA=\{a_1a_2+a'_1+a'_2+\dots+a'_k:a_1,a_2,a'_1,a'_2,\dots,a'_k\in A\}$.) We also prove some extensions of this result. Among other ingredients, the proof also includes an application of a quantitative equidistribution result for polynomials. In the final part, we consider the Graham-Pollak problem for hypergraphs. Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. We disprove a conjecture that $f_4(n)\geq (1+o(1))\binom{n}{2}$, by showing that $f_4(n)\leq\frac{14}{15}(1+o(1))\binom{n}{2}$. The proof is based on the relationship between this problem and a problem about decomposing products of complete graphs, and understanding how the Graham-Pollak theorem (for graphs) affects what can happen here.I would like to thank Trinity College and Department of Pure Mathematics and Mathematical Statistics for their generous financial support and hospitality during PhD studies

Impact of a recruitment campaign on students’ applications to medical school

Objective. Medical schools might benefit if they have information about the activities that may help them to increase the number of applicants with superior academic qualifications. Methods. The study was carried out at the Split University School of Medicine, Croatia. Medical school applicants were surveyed in 2007 and 2009. Promotional activities consisted of presentations on medical education covering six general high-schools in 2008, and a presentation on regional television, aired in 2008 and 2009 shortly before the admission term. Results. The survey response rate was 79% (299/379). The number of applicants in 2009 increased by 37% and the percentage of applicants from general high schools increased by 10%, in relation to 2007. The percentage of students with the best grades in all four years of high school was 42% in the both surveys. Presumed quality was a major influencing factor for choosing Split University Medical School. However, the medical school in the Croatian capital remained the first choice for students. Only a few applicants stated that the medical schools’ promotional campaigns influenced their decision about where to study; 9.7% and 1.5%, respectively. Conclusion. Presentations in elected general high schools and a television campaign should be further explored as possible activities that regional schools may use to recruit potential applicants

The Significance of COVID-19 Diseases in Lipid Metabolism Pregnancy Women and Newborns

Coronavirus disease (COVID-19) is an infectious disease caused by SARS-CoV-2. Elderly people, people with immunodeficiency, autoimmune and malignant diseases, as well as people with chronic diseases have a higher risk of developing more severe forms of the disease. Pregnant women and children can becomesick, although more often they are only the carriers of the virus. Recent studies have indicated that infants can also be infected by SARS-CoV-2 and develop a severe form of the disease with a fatal outcome. Acute Respiratory Distress Syndrome (ARDS) ina pregnant woman can affect the supply of oxygen to the fetus and initiate the mechanism of metabolic disorders of the fetus and newborn caused by asphyxia. The initial metabolic response of the newborn to the lack of oxygen in the tissues is the activation of anaerobic glycolysis in the tissues and an increase in the concentration of lactate and ketones. Lipid peroxidation, especially in nerve cells, is catalyzed by iron released from hemoglobin, transferrin and ferritin, whose release is induced by tissue acidosis and free oxygen radicals. Ferroptosis-inducing factors can directly or indirectly affect glutathione peroxidase through various pathways, resulting in a decrease in the antioxidant capacity and accumulation of lipid reactive oxygen species (ROS) in the cells, ultimately leading to oxidative cell stress, and finally, death. Conclusion: damage to the mitochondria as a result of lipid peroxidation caused by the COVID-19 disease can cause the death of a newborn and pregnant women as well as short time and long-time sequelae