Extremal and probabilistic results for regular graphs

In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness. We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory. We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs. Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles

Geometristen muotojen reaaliaikainen tunnistus

Kynä- ja kosketuskäyttöliittymät vaativat toimiakseen tehokasta ja tarkkaa hahmontunnistusta. Tässä työssä esitellään reaaliaikaisen hahmontunnistuksen käsitteistöä, yleisiä menetelmiä ja aikaisempaa tutkimusta. Lyhyesti käsitellään eri tutkimusryhmien esittämiä hahmontunnistusjärjestelmiä. Lisäksi esitellään geometrisiin piirteisiin perustuva hahmontunnistusjärjestelmä. Työ antaa yksityiskohtaiset kuvaukset piirtoviivan esiprosessointi- ja piirteenirrotusalgoritmeista sekä hahmoluokittelumenetelmästä. Lisäksi kuvaillaan hahmontunnistusheuristiikka kahdelle yksinkertaiselle muodolle (nuoli ja tähti). Joukko koehenkilöitä käytti työssä toteutettua graa_sta käyttöliittymää, minkä tuloksena saatiin realistiset tulokset järjestelmän laskennallisesta suorituskyvystä ja tarkkuudesta: toteutettu järjestelmä on laskennallisesti nopea mutta tunnistustarkkuus monitulkintainen. Lopuksi pohditaan valitun lähestymistavan ongelmia ja rajoitteita.Effective sketch recognition is the basis for pen and touch-based human-computer interfaces. In this thesis the concepts, common methods and earlier work in the research area of online symbol recognition are presented. A set of shape recognition approaches proposed in the past by various research teams are briefly introduced. An online shape recognizer using global geometric features is described. The preprocessing and feature extraction algorithms as well as the shape classification method are described in detail. Recognition heuristics for two simple shapes (arrow and star) are suggested. A graphical user interface was implemented and a group of subjects employed to obtain realistic results of the computational performance and recognition accuracy of the system: the implemented system performs fast but the results on the recognition accuracy were ambiguous. Finally, the problems and restrictions of the approach are discussed