No description available.

No description available.

No description available.

No description available.

<p>Whether the growth series of a group is a rational function is investigated in this paper.Parry showed certain torus bundle groups of even trace exhibits rational growth, and thisresult has been extended by the author, Turbo Ho and Mark Pengitore. In this paper, bothresults are combined into a single proof used in [1], and the result is pushed further into thenegative case not covered in earlier works</p>

This study compares the scholastic achievement of 2,811 students enrolled in Fairfax County, Virginia, Public Schools for the 1983-1984 school year. Scholastic achievement of an experimental group of 1,068 students enrolled in combination or split/grade classes is compared with a control group of 1,743 students enrolled in regular graded classes. Five research questions were developed, three of which related directly to grade level student scholastic achievement by comparing test results for combination and regular grade classes, and two which attempted to identify any significance resulting from differences used by principals to select teachers and students for placement in combination classes. / Ed. D.

Gromov's theorem on finitely generated groups of polynomial growth is one of the cornerstones of geometric group theory. It has seen many applications and has immense importance within its realm. However, the early proofs of Gromov's theorem were based on deep results and were practically unsuitable for teaching. Recently, a new family of proofs have been published, potentially providing new ways to learn and teach Gromov's theorem. In this Master's thesis, we aim to provide an expository introduction to Gromov's theorem and its proofs.  We will first explore a brief theoretical overview of geometric group theory, which will include a proof of a theorem of Bass-Guivarc'h. We will then give a detailed account of an elementary proof of Gromov's theorem for groups of sub-polynomial growth, which was originally outlined by Tao in a blog post. The main contribution we make in this thesis is providing a significant amount of additional detail, bridging the gaps, and providing theoretical clarifications to all the steps in this proof. Finally, we will present some applications of Gromov's theorem. / Gromovs sats om ändligt genererade grupper med polynomiell tillväxt är en av grundpelarna inom geometrisk gruppteori. Satsen har sett många tillämpningar och har enorm betydelse inom sitt område. Dock baserades de tidiga bevisen för Gromovs sats på djupa resultat och var praktiskt taget olämpliga för undervisning. Nyligen har en ny familj av bevis publicerats, vilket potentiellt ger nya sätt att lära sig och undervisa Gromovs sats. I denna uppsats syftar vi till att ge en förklarande introduktion till Gromovs sats och dess bevis. Först ges en kort teoretisk översikt över geometrisk gruppteori, vilket kommer att inkludera ett bevis av en sats av Bass-Guivarc'h. Sedan ger vi en detaljerad redogörelse för ett elementärt bevis av Gromovs sats om grupper med sub-polynomiell tillväxt, vilket ursprungligen beskrevs av Tao i en bloggpost. Det huvudsakliga bidraget vi gör i denna uppsats är att tillhandahålla en betydande mängd ytterliga detaljer, och att ge teoretiska förtydliganden i alla steg i detta bevis. Slutligen presenterar vi några tillämpningar av Gromovs sats.

In 1911, mathematician Max Dehn posed three decision problems for finitely presented groups that have remained central to the study of combinatorial group theory. His work provided the foundation for geometric group theory, which aims to analyze groups using the topological and geometric properties of the spaces they act on. In this thesis, we study group actions on Cayley graphs and the Farey tree. We prove that a group has a solvable word problem if and only if its associated Cayley graph is constructible. Moreover, we prove that a group is finitely generated if and only if it acts geometrically on a proper path-connected metric space. As an example, we show that SL(2, Z) is finitely generated by proving that it acts geometrically on the Farey tree.

The purpose of this study is to examine young adults', 18-19 years old, attitudes towards corporal punishment and with the help of social constructionism, grid-group theory and relevant concepts analyze the results and compare it with previous research. The research questions introduce various factors such as gender, origin and personal experiences of corporal punishment that may affect the attitudes. In order to fulfill the purpose and answer the research questions, a quantitative research method has been selected. There was 118 participants in the survey. The study was conducted by means of surveys in high schools in Örebro. Social constructionism made us realize how the community, perceptions and expectations might influence attitudes. Grid-group theory gave us an insight into how different family structures may explain why attitudes toward corporal punishment of children differ between individuals. One of the conclusions of this study is that the long-term trend of decreasing positive attitudes toward corporal punishment seems to continue. Results show that all of 41, 2% of young adults in our study, with parents born outside the Nordic countries, have been subjected to corporal punishment. Personal experiences and parents' origin has been shown to have effects on attitudes towards corporal punishment. Different family structures and attitudes towards child rearing may be important to understand and take into consideration in social work. The study shows that corporal punishment still exists, which makes the topic worth of attention and we recommend it to be studied further.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: An integro-differential equation describing the dynamical behaviour of phytoplankton cells is considered in which the effects of cell division and aggregation are incorporated by coupling the coagulation-fragmentation equation with growth, and the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial-boundary value problem is shown to be well posed in a physically relevant Banach space using the theory of strongly continuous semigroups of operators, the theory of perturbation of positive semigroups and the semilinear abstract Cauchy problems theory. In particular, we provide sufficient conditions for honesty of the model. Finally, the results on the effects of the growth-fragmentation-coagulation on the overall evolution of the phytoplankton population are summarised. / AFRIKAANSE OPSOMMING: ’n Integro-differensiaalvergelyking wat die dinamiese ontwikkeling van fitoplanktonselle beskryf, word beskou. Die uitwerking van seldeling en -aggregasie is geïnkorporeer deur die vergelyking van koagulasie en fragmentasie met groeiaan die McKendrick-von Foerster hernuwingsmodel van ’n ouderdomsgestruktureerde populasie te koppel. Die teorie van sterk kontinue semigroepe van operatore, steuringsteorie van positiewe semigroepe en die teorie van semilineêre abstrakte Cauchy probleme word aangewend om, onder gepaste voorwaardes met betrekking tot die model se parameters, te bewys dat die geassosieerde beginwaarde-probleem met randvoorwaardes ‘goed gestel’ is in ’n fisies relevante Banach-ruimte. In die besonder word voldoende voorwaardes vir eerlikheid van die model verskaf. Ten slotte word ’n opsomming van die resultate met betrekking tot die gekombineerde uitwerking van groei-fragmentasie- koagulasie op die gesamentlike ontwikkeling van die fitoplanktonpopulasie verskaf.