Random Interval Graphs

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform distribution on $[0,1]$. We then say that the $i$th of the $n$ vertices is the interval $[X_{i},Y_{i}]$ if $X_{i}<Y_{i}$ and the interval $[Y_{i},X_{i}]$ if $Y_{i}<X_{i}$. We then say that two vertices are adjacent if and only if the corresponding intervals intersect. We recall from our MA902 essay that fact that in such a graph, each edge arises with probability $2/3$, and use this fact to obtain estimates of the number of edges. Next, we turn to how these edges are spread out, seeing that (for example) the range of degrees for the vertices is much larger than classically, by use of an interesting geometrical lemma. We further investigate the maximum degree, showing it is always very close to the maximum possible value $(n-1)$, and the striking result that it is equal to $(n-1)$ with probability exactly $2/3$. We also recall a result on the minimum degree, and contrast all these results with the much narrower range of values obtained in the alternative \lq comparable\rq\, model $G(n,2/3)$ (defined later). We then study clique numbers, chromatic numbers and independence numbers in the Random Interval Graphs, presenting (for example) a result on independence numbers which is proved by considering the largest chain in the associated interval order. Last, we make some brief remarks about other ways to define random interval graphs, and extensions of random interval graphs, including random dot product graphs and other ways to define random interval graphs. We also discuss some areas these ideas should be usable in. We close with a summary and some comments

The Quicksort algorithm and related topics

Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort algorithm and its variants. Specifically, we study the time complexity of the algorithm and we provide a complete demonstration of the variance of the number of comparisons required, a known result but one whose detailed proof is not easy to read out of the literature. We also examine variants of Quicksort, where multiple pivots are chosen for the partitioning of the array. The rest of this work is dedicated to the analysis of finding the true order by further pairwise comparisons when a partial order compatible with the true order is given in advance. We discuss a number of cases where the partially ordered sets arise at random. To this end, we employ results from Graph and Information Theory. Finally, we obtain an alternative bound on the number of linear extensions when the partially ordered set arises from a random graph, and discuss the possible application of Shellsort in merging chains

Οι προσδιοριστικοί παράγοντες ανάπτυξης της οικονομίας, ως δείκτης ευημερίας και βιωσιμότητας των επιχειρήσεων.

Σκοπός της παρούσας έρευνας είναι να διερευνηθεί η σχέση μεταξύ των προσδιοριστικών παραγόντων της οικονομικής ανάπτυξης, της ευημερίας και της βιωσιμότητας των επιχειρήσεων. Το υπόβαθρο για την εκπόνησή της βασίζεται στο ότι στη διεθνή βιβλιογραφία διαπιστώνονται κοινοί προσδιοριστικοί παράγοντες της ανάπτυξης της οικονομίας και της ευημερίας και βιωσιμότητας των επιχειρήσεων. Για την εκπόνηση της έρευνας χρησιμοποιήθηκε η δευτερογενής ανάλυση (βιβλιογραφική επισκόπηση) καθώς και η πρωτογενής έρευνα με τη βοήθεια της ποσοτικής μεθόδου με ερωτηματολόγιο που συμπληρώθηκε από 119 εργαζομένους στην Αττική στην πλατφόρμα Google Forms. Από την έρευνα διαπιστώθηκε ότι οι παράγοντες εκείνοι που προσδιορίζουν τη βιωσιμότητα των επιχειρήσεων είναι: α) όσοι σχετίζονται με τον ίδιο τον επιχειρηματία, β) όσοι σχετίζονται με την οικονομική ανάπτυξη, γ) όσοι σχετίζονται με την επιχείρηση, δ) όσοι σχετίζονται με το ευρύτερο περιβάλλον. Συνολικά, επομένως, διαπιστώνεται πως παράγοντες που σχετίζονται με τη βιωσιμότητα των επιχειρήσεων είναι παράγοντες που αφορούν ως επί το πλείστον τα χαρακτηριστικά του επιχειρηματία, αλλά και την οικονομική, κοινωνική και περιβαλλοντική βιωσιμότητα. Ωστόσο, παρ’ όλο που προσδιοριστικοί παράγοντες της οικονομικής ανάπτυξης επηρεάζουν τη βιωσιμότητα των επιχειρήσεων, δε σχετίζονται μεταξύ τους σε στατιστικά σημαντικό βαθμό. Ως εκ τούτου θα πρέπει να δοθεί περισσότερη έμφαση από κυβερνητικούς φορείς στην ανάπτυξη αυτών των παραγόντων που θα μπορέσουν να συμβάλουν στη διασφάλιση της βιωσιμότητας των επιχειρήσεων. Επίσης, είναι αναγκαία η διεξαγωγή περαιτέρω ερευνών στα ίδια τα ενδιαφερόμενα μέρη, όπως επιχειρηματίες και εργαζομένους.The aim of this study is to examine the relationship between the determinants of economic growth, prosperity and business viability. The background for its conduct is based on the fact that in the international literature there are common determinants of the development of the economy and the prosperity and viability of enterprises. For the conduct of the research, the secondary analysis (bibliographic review) was used, as well as the primary research with the help of the quantitative method with a questionnaire completed by 119 employees in Attica. The questionnaire was uploaded on the Google Forms platform. The results indicated that the factors that determine the viability of businesses are the following: a) those related to the entrepreneur, b) those related to economic development, c) those related to the company, d) those related to the wider environment. Overall, it was found that factors related to business viability are factors that mostly concern the characteristics of the entrepreneur, as well as the economic, social and environmental sustainability. However, although determinants of economic growth affect business viability, they are not related to a statistically significant degree. Therefore, more emphasis should be placed by government agencies upon the development of these factors that can contribute to the viability of businesses. In addition it is necessary to conduct further research on stakeholders, such as entrepreneurs and employees

Profiling student smokers:a behavioral approach

The aim of the present study is to construct a coherent profile of student smokers in Greece, based on their behavioral and demographic characteristics. In this context, we collected data by administrating an anonymous self-completed questionnaire, which was answered by students of University and Technological Educational Institute (T.E.I.) of Patras. The final sample consists of 1,190 student smokers. For the purposes of the present study, principal component analysis was utilized to explore and detect the demographic and behavioral profiles of Greek student smokers. The factor solution identified 5 demographic factors and 14 behavioral factors. All factors were labeled, interpreted and discussed in the light of existing knowledge in order to understand better the consumer behavior of student smokers

Profiling student smokers:a behavioral approach

The aim of the present study is to construct a coherent profile of student smokers in Greece, based on their behavioral and demographic characteristics. In this context, we collected data by administrating an anonymous self-completed questionnaire, which was answered by students of University and Technological Educational Institute (T.E.I.) of Patras. The final sample consists of 1,190 student smokers. For the purposes of the present study, principal component analysis was utilized to explore and detect the demographic and behavioral profiles of Greek student smokers. The factor solution identified 5 demographic factors and 14 behavioral factors. All factors were labeled, interpreted and discussed in the light of existing knowledge in order to understand better the consumer behavior of student smokers