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The purpose of this paper is to investigate from two viewpoints an order-induced topology on a set X.

This paper will be concerned with some fundamental properties of a line. In particular, fundamental ordering properties of a line segment are covered.

Whereas benchmarking experiments are very frequently used to investigate the performance of statistical or machine learning algorithms for supervised and unsupervised learning tasks, overall analyses of such experiments are typically only carried out on a heuristic basis, if at all. We suggest to determine winners, and more generally, to derive a consensus ranking of the algorithms, as the linear order on the algorithms which minimizes average symmetric distance (Kemeny-Snell distance) to the performance relations on the individual benchmark data sets. This leads to binary programming problems which can typically be solved reasonably efficiently. We apply the approach to a medium-scale benchmarking experiment to assess the performance of Support Vector Machines in regression and classification problems, and compare the obtained consensus ranking with rankings obtained by simple scoring and Bradley-Terry modeling. / Series: Research Report Series / Department of Statistics and Mathematics

<p>Cilj ove teze je da se ispitaju&nbsp; lanci u parcijalnim uredjenjima (P(X), &sub;),&nbsp;pri čemu je P(X) skup domena izomorfnih podstruktura relacijske strukture&nbsp;X. Po&scaron;to se svaki lanac u parcijalnom uredjenju može produžiti do maksimalnog lanca, dovoljno je ispitati maksimalne lance u P(X). Dokazano je da, ako je X ultrahomogena relacijska struktura koja ima netrivijalne izomorfne&nbsp;podstrukture, onda je svaki maksimalan lanac u (P(X) &cup; {&empty;}&nbsp; , &sub;) kompletno&nbsp;linearno uredjenje koje se utapa u R i ima neizolovan minimum. Ako &nbsp;je X&nbsp;relacijska struktura, dat je dovoljan uslov da za svako kompletno linearno uredjenje L koje se utapa&nbsp; u R i ima neizolovan minimum, postoji maksimalan lanac u (P(X) &cup; {&empty;}&nbsp; , &sub;) izomorfan L.&nbsp; Dokazano je i da ako je&nbsp;X neka od sledećih relacijskih struktura: Rado graf, Hensonov graf, random poset, ultrahomogeni&nbsp; poset Bn&nbsp; ili&nbsp; ultrahomogeni&nbsp; poset Cn; onda je&nbsp;L izomorfno maksimalnom lancu u (P(X) &cup; {&empty;}&nbsp; , &sub;) ako i samo ako je &nbsp;L&nbsp;kompletno,&nbsp; utapa se u R i ima neizolovan minimum. Ako je X prebrojiv&nbsp;antilanac ili disjunktna unija &micro; kompletnih&nbsp; grafova sa &nu; tačaka za &micro;&nu; = &omega;, onda je L izomorfno maksimalnom lancu u (P(X) &cup; {&empty;}&nbsp; , &sub;) ako i samo ako&nbsp;je bulovsko,&nbsp; utapa se u R i ima neizolovan minimum.</p> / <p>The purpose of this thesis is to investigate chains in partial orders (P(X), &sub;), where P(X) is the set of domains of isomorphic substructures of a relational structure X. Since each chain in a partial&nbsp; order can be extended to a maximal one, it is enough to describe maximal chains in P(X). It is proved that, if X is an ultrahomogeneous relational structure with non-trivial isomorphic substructures, then each maximal&nbsp; chain in (P(X)&cup; {&empty;}&nbsp; , &sub;) is a complete, R-embeddable linear order with minimum&nbsp; non-isolated. If X is a relational structure, a condition is given for X, which is sufficient&nbsp; for (P(X) &cup; {&empty;}&nbsp; , &sub;) to embed each complete,&nbsp; R-embeddable&nbsp; linear order with minimum non-isolated as a maximal&nbsp; chain.&nbsp; It is also proved that if X is one of the follow- ing relational structures: Rado graph, Henson graph, random poset, ultrahomogeneous poset Bn or ultrahomogeneous poset Cn; then L is isomorphic to a maximal&nbsp; chain in (P(X) &cup; {&empty;}&nbsp; , &sub;) if and only if L is complete, R-embeddable with minimum non-isolated. If X is a countable&nbsp; antichain&nbsp; or disjoint union of &micro; complete graphs with &nu; points where &micro;&nu; = &omega;, then L is isomorphic to a maximal&nbsp; chain&nbsp; in (P(X) &cup; {&empty;}&nbsp; , &sub;) if and only if L is Boolean, R-embeddable with minimum non-isolated.</p>