Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings

The formulation and development of instruments to measure field dependence-field independence using spatial and verbal modalities

The thesis gives a description of the formulation and development of an Embedded Shapes Test which was subsequently used as a method of measuring the construct/cognitive style of field dependence – field independence, using a spatial modality. Two additional methods of measuring field dependence – field independence were also formulated and developed, which used a verbal modality, in the form of a Sense Word Test (SWT) and Non-Sense Word Test (NSWT). Each of the above field dependence – field independence 'tests' were used in a series of pilot studies and studies, as part of their development and application. With two of the studies, of which there were four, additional ‘tests’ were used to measure cognitive attributes considered to have some association with those inherent in the measurement of field dependence – field independence. These included the Chronological Order Integration Test (COIT) and the Gestalt Picture Completion Test (GPCT) in Study Three; and the GPCT; British Ability Scales (BASI)/Short Form IQ; and the Cognitive Styles Analysis (CSA) in Study Four. The COIT in Study Three provided a framework to measure field dependence – field independence using a verbal modality through the medium of text. Therefore, this enabled comparisons to be made within a verbal modality through the medium of words and non-sense words by the SWT and NSWT, as well as a spatial modality by the EST. The inclusion of the GPCT, in Studies Three and Four enabled the process of measuring field dependence – field independence to be compared with a process that appeared to be opposite to it, i.e. an integration – synthesis process, or putting together pieces of information, instead of a segregated – analytical process, or taking apart a piece of information, respectively. The BASI/Short Form IQ was included in Study Four to provide a measure of cognitive ability. Since an argument directed at the concept and measurement of field dependence – field independence is that it is measuring cognitive ability rather than cognitive style, a comparison was made between field dependence – field independence and cognitive ability/intelligence as measured by the BASI/Short Form IQ. A further comparison with measurements of field dependence – field independence was made possible by the inclusion of the CSA in terms of a Wholistic or Analytic perception of information, as well as a propensity, on the part of the individual, to think spatially or verbally, i.e. in the Imager or Verbaliser mode of the CSA. The inclusion of school subject performance/attainment levels, in each of the four studies, was to enable comparisons to be made between the various measurements of field dependence – field independence, i.e. EST, SWT, NSWT and COIT, and GPCT, BASI/Short form IQ, CSA, and the characteristics of the different school subjects themselves. This enabled the exploration as to whether or not the characteristics of particular school subjects have an affinity to field dependence – field independence in terms of cognitive style and/or cognitive ability. This exploration was augmented in Study Three by the inclusion of GCSE subject grades for the sample members. In Chapter 14 (Comparisons of the Outcomes from the Four Studies), Witkin’s Group Embedded Figures Test (GEFT) was introduced to provide a comparison of reliability between the EST3, SWT2 and NSWT1, using the Cronbach Alpha statistic. The sample used for each of the pilot studies and studies consisted of Year 8, male and female, students of mixed ability, from Comprehensive and Middle Schools. The sample sizes for each of the two pilot studies were thirty, but varied for the four studies from 62 to 224 to increase the statistical validity and to decrease the standard error. Within each of the four studies, means; standard deviations; Multivariate Analysis of Variance (MANOVA); correlations (Pearson Product Moment) and Cronbach Alpha statistics were used to analyse the data. The EST3, SWT2 and NSWT1 were used in each of the four studies and produced a high level of construct validity across the four studies. A number of significant F ratios and correlations were obtained between the variables in each of the four studies. In particular, significant F ratios were obtained for EST3; Sex; Test Type (SWT2 and NSWT1), e.g. (F = 4.00; df 2, 56; p = 0.05 – Study 1); School Subjects and GCSE Subjects; and significant correlations between EST3 and Science (General), e.g. r = 0.4006; p = 0.05 – Study 1; Modern Language (French); COIT A and S; COIT B and S; and IQ within the corresponding that included these variables. Correlations (r) between EST3, SWT2 and NSWT1 within each study produced a number of high reliability values (See Chapter 14, Table 14.4 for full details). In addition, the Cronbach Alpha values for the EST3, SWT2 and NSWT1 gave a high reliability, i.e. 0.84263, 0.92799, 0.91946 for Study One; 0.90281, 0.93720, 0.88725 for Study Two; 0.89441, 0.94493, 0.92713 for Study Three; and 0.84942, 0.93779, 0.91099 for Study Four, respectively. The Cronbach Alpha value for the EST3 was greater than that for the GEFT, i.e. 0.93544 and 0.89234, respectively. Similar values were obtained for the SWT2 and NSWT1 when compared to the GEFT, i.e. 0.97891 and 0.96411, respectively

Motiivide semiootika: Kenneth Burke ja Deely-Tartu semiootika

Kõrvutades Kenneth Burke'i (1897–1993) ja tänapäeva semiootika võtmekontseptsioonie, uurib käesolev töö motiivi kriitilist rolli tähenduse arenemisel ja avastamisel inimkogemuses. Analüüsime Burke’i töid motiivi rollist inimese tähendusloomes, mida ta kirjeldas kui aktiivset, tahtelist protsessi. Tuginedes eriti Burke’i nn Motivorumi seeria teostele (A Grammar of Motives, A Rhetoric of Motives ja Symbolic of Motives), täiendame Burke'i kontseptuaalset skeemi semiootilise raamiga, tuletades selle John Deely üldsemiootika ja Tartu koolkonna teooria, mida esindab käesolevas eelkõige Kalevi Kull, ristumiskohast. Tulemuseks on analüütiline raamistik, mis jälgib motiivi vajalikkust alates bioloogilisele eesmärgile orienteeritusest kuni inimmõtte formaalsete piirideni. Kasutades Burke'i terministlike ekraanide mõistet, mida käsitletakse siin kui hierarhilise modelleerimise vormi, näitame, et tegelikkuse kohta järelduste tegemiseks peame selle kirjeldamisel kasutama valitud mõisteid, mis samas tähendab nendega kokkusobimatute võimaluste tagasilükkamist. Mõned tähendused tehakse võimalikuks, teised aga sõelutakse semiootiliselt välja. Nii võib tähenduse ja kogemuse osas saada väga erinevaid tulemusi– eriti kontekstides, mida juhivad nn jumalaterminid ehk terministlike ekraanide maksimaalsed vormid. Meie valikud, mis konkretiseerivad motiive, tekitavad kogemuse semiootilise filtri, misläbi kujuneme aja jooksul ise oma valikute ja motiivide märgiks. See on vastuolus modernismi paradigmaga, mis käsitleb motiveeritud järeldusi põhjuslike protsesside tagajärgedena. See vastuolu tuleneb modernismi semiootilisest vundamendist; me kasutame oma Burke'i-semiootilist raamistikku selle aluse, tema motiivide ja tulenevate tähenduste analüüsimiseks. Me väidame, et modernismi sisemised vastuolud põhjustavad realismi ja idealismi kollapsi, mille tulemuseks on kaks tavaliselt seisukohta: (1) naiivne realism, mida praktikas dikteerivad võimuinstitutsioonid, ja (2) naiivne relativism, mille dikteerib üksikagent. Alternatiivina pakume välja teleoloogilise meetodi, mis vaatleb kokkusobimatute tähendusvõimaluste erinevaid trajektoore, jälgib nende (entelehhiaalseid) lõpp-punkte ja näeb neid lõpp-punkte agendi valikute alusena – eriti suure olulisusega tähenduskontekstides. Küsimused on seega järgmised: „Milleni see valik viib? Millise kogemusliku maailma see avab või sulgeb? Milline motiiv selle aluseks on? Mille märgiks ma tahan olla?”By laying in parallel key concepts from Kenneth Burke (1897–1993) and contemporary semiotics, this work explores the critical role that motive plays in the development and discovery of meaning within human experience. We highlight Burke’s thought regarding the role of motive in human meaning-making, which he highlighted as an active, volitional process. Drawing especially from the so-called Motivorum works (A Grammar of Motives, A Rhetoric of Motives, and the nebulous Symbolic of Motives), we enrich Burke’s conceptual schema within an explicitly semiotic frame, derived from an intersection between John Deely’s general semiotics and the Tartu School approach, represented especially by Kalevi Kull. The result is an analytical framework that traces out the necessity of motive from biological goal-orientation to the formal boundaries of human thought. Employing Burke’s notion of terministic screens, treated here as a form of hierarchical modeling, we show that in order to make conclusions about reality, we must use some terms to describe it, which also entails rejecting other incompatible options. Some meanings are made possible, while others are semiotically screened out, which can lead to quite different results for meaning and experience – especially in contexts governed by God-terms, the maximal form of terministic screens. Our choices, which concretize motives, result in a semiotic filter for experience, and over time we become signs of our choices and motives. This runs counter to the Modern paradigm, which treats its motivated conclusions as deterministic entailments. That contradiction arises from Modernism’s semiotic foundation, and we employ our Burkean-semiotic framework to analyze this foundation, its underlying motives, and its meaning results. We argue that Modernism’s internal contradictions result in a collapsing of realism and idealism, with two commonly resulting stances: (1) naive realism, in practice dictated by institutions of power, and (2) naive relativism, dictated by the individual agent. Contrastively, we suggest a teleological method that looks at the various trajectories of incompatible meaning possibilities, traces out their (entelechial) end points, and uses those end points as the basis for which the agent will choose – especially in high-stakes meaning contexts. The questions asked are thus: “What does this choice lead to? What kind of experiential world will it open or close? What kind of motive underlies it? What do I want to be a sign of?”https://www.ester.ee/record=b548547

Robust expansion and hamiltonicity

This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments. In Chapters 3 and 4, we prove that every sufficiently large 3-connected $D$-regular graph on $n$ vertices with $D$ ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this. Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the $i$$^{th}$ largest degree in a sufficiently large graph $G$ on n vertices is at least a little larger than $n$/3 + $i$ for $i$ ≤ $n$/3, then $G$ contains the square of a Hamilton cycle

The art and architecture of mathematics education: a study in metaphors

This chapter presents the summary of a talk given at the Eighth European Summer University, held in Oslo in 2018. It attempts to show how art, literature, and history, can paint images of mathematics that are not only useful but relevant to learners as they can support their personal development as well as their appreciation of mathematics as a discipline. To achieve this goal, several metaphors about and of mathematics are explored