In this thesis we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups. This correspondence between Q-powered nilpotent groups and rational nilpotent Lie algebras was discovered by Anatoly Mal'cev in 1951. We show how the Mal'cev correspondence can be realized on a computer. We explore two possibilities for this purpose and compare them: the first one uses matrix embeddings and the second the Baker-Campbell-Hausdorff formula. Then, we describe a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection. Algorithms for collection lie at the heart of most methods dealing with polycyclically presented groups. The current state of the art is "collection from the left" as recently studied by Gebhardt, Leedham-Green/Soicher and Vaughan-Lee. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory. Further, we explore how the Mal'cev correspondence can be used to describe symbolically the collection process in polycyclically presented groups. In particular, we describe an algorithm that computes the collection functions for splittable polycyclic groups. This algorithm is based on work by du Sautoy. We apply it to the computation of pro-p-completions of polycyclic groups. Finally we describe a practical algorithm for testing polycyclicity of finitely generated rational matrix groups. Previously, not only did no such method exist but it was not clear whether this question was decidable at all. Most of the methods described in this thesis are implemented in the computer algebra system GAP and publicly available as part of the GAP packages Guarana and Polenta. Reports on the implementation including runtimes for some examples are given at the appropriate places.

Let G be a reductive algebraic group defined over an algebraically closed field of characteristic p. A subgroup H of G is called G-complete reducible whenever H is contained in a parabolic subgroup P of G, it is contained in some Levi subgroup of P. In this thesis, we present a pair of reductive subgroups H and M of G of type E_7 such that H<M and H is G-completely reducible but not M-completely reducible.

Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

Benson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibilty of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to the complex method of interpolation and to the real method of interpolation for the case q=1. Two new characterizations are presented of the domain space of the "derivation" operator in the context of orbital methods. Comparisons to the work of others are made, especially the unifying paper of M. Cwikel, N. Kalton, M. Milman and R. Rochberg, A United Theory of Commutator Estimates for a Class of Interpolation Methods, Advances in Mathematics 169 2002, 241-312. / by Eva, Kasprikova / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

Background and Objectives: In a previous randomized study, we reported the benefits of providing integrated care by the Joint Asia Diabetes Evaluation (JADE) Program consisting of comprehensive assessment (CA) and 3-4 monthly follow up (FU) assessments with personalized feedback reports and decision support on control of cardio-metabolic risk factors through reduced clinical inertia and improved self-care. Further, provision of additional peer support improved psychological well-being and all-cause hospitalization especially in those with negative emotions who tended to have co-morbidities. However, given the multicomponent nature of the program, the differential effects of peer support in different patient subgroups and independent effect of personalized FU reports had not been systematically evaluated.. In this thesis, I used a randomized and case-control design to examine the effects of providing regular FU reports and peer support on metabolic control, psychological health, and all-cause hospitalization in patients with diabetes. / Methods: Between February and December 2013, 1488 Chinese patients with diabetes aged 18-75 years underwent CA using the JADE portal and returned in 4 weeks in groups to receive their personalized CA report with explanation by nurses about their complications, risk factors and treatment targets. Amongst them, I selected 288 high risk patients defined as 1) HbA1c≥8%, 2) obesity (body mass index≥27.5 kg/m² and/or waist≥80cm (women) or ≥90cm (men), and/or 3) chronic kidney disease (CKD, eGFR<60ml/min/1.73m²) and offered them a telephone-based peer support program, to which 144 (50%) agreed (P+ group) and 144 (50%) refused (P- group). Within each group, they were also randomized to receive 2 JADE FU reports by mail after their clinic visits. These FU reports displayed their trends of ABC (HbA1c, BP, LDL-C) control and body weight with individualized reminders for self-care during a 12-month period. In the remaining patients (n=1200), half were randomized to receive 2 FU reports by mail (R+ group: n=600) while half received usual care (R- group: n=600). Amongst patients not offered peer support, 425 patients (50% received FU report) were matched to the P+ group (52% received FU report) on a 3:1 basis by age, gender, diabetes duration, and baseline HbA1c as a control group. The outcome measures were reduction in HbA1c and all-cause hospitalization at month 12. / Results: Amongst patients not offered peer support, after a median (IQR) follow-up period of 575 (519-646) days, the R+ group had greater reduction in HbA1c (mean [95% CI]: -0.24[-0.35,-0.14]% versus -0.15[-0.24,-0.06]%, p=0.030) with similar hospitalization rate and frequency compared with the R- group. Amongst patients offered peer support, the P+ group and P- group had similar baseline clinical and psychological-behavioral parameters. After 530 (463-575) days, the P+ group (n=138) tended to have greater reduction in HbA1c (-0.75[-0.97,-0.52]% versus -0.42[-0.68,-0.15]%, p=0.106) with significant improvements in mental health and quality of life than the P- group (n=131). In the case-control cohort for peer support,the P+ group tended to have greater reduction in HbA1c than the control group (n=425) (-0.75[-0.97,-0.52]% versus -0.49[-0.64, -0.35]%, p=0.119) with lower rate, frequency and length of hospitalization. On multivariable analysis of the entire cohort (n=1488), peer support (β coefficient [95% CI] -0.31[-0.56, -0.06], p=0.015) and receiving FU reports (-0.14[-0.25, -0.04], p=0.009) were independent predictors for reduction in HbA1c. Peer support (OR [95% CI] 0.36[0.16, 0.79], p=0.011) and CKD with FU reports (0.40[0.18, 0.88], p=0.022) were also associated with reduced hospitalization. / Conclusion: In this real-world quality improvement program, both peer support and regular personalized feedback report by mail were associated with reduced HbA1c in patients with diabetes. Peer support was associated with lower risk of hospitalization, while feedback report was associated with reduced hospitalization only in patients with diabetes and comorbid CKD. / 背景及目的：在一項隨機對照研究中，我們發現通過亞洲糖尿病評估計劃JADE提供整合了全面糖尿病併發症篩查（CA）、定期隨訪（FU）評估、個體化回饋報告及決策支持的綜合護理可通过降低臨床惰性，促進自我管理從而改善糖尿病風險因素的控制。在此基礎上，同伴支持可進一步改善患者的心理健康，降低住院率， 并且在有負面情緒的人群中作用尤其明顯。然而，由於綜合護理由多部分組成，個體的作用並未被系統評估。本文分別採用隨機對照和病例對照研究，評估個體化隨訪報告和同伴支持對糖尿病患者代謝控制、心理健康及住院率的作用。 / 研究方法：2013 年2 月至12 月，1488 位年齡18 至75 歲的糖尿病患者進行了CA，並于4 周後領取個體化CA 報告，由護士說明其併發症和危險因子的控制以及治療有無達標。其中，我邀請了288 位有以下高危因素的患者參加一項基於電話的同伴支持計劃：1）糖化血紅蛋白（HbA1c）≥8%；2）肥胖（體重指數≥27.5 kg/m² 和/或女性腰圍≥80cm/男性腰圍≥90cm；和/或3）慢性腎臟病（CKD，腎小球濾過率<60 ml/min/1.73m²）。其中，144 位（50%）同意（P+ 組）參與，144 位（50%）（P- 組）拒絕參與。每組再隨機抽取一半患者郵寄2 份JADEFU 報告。該報告顯示了患者HbA1c、血壓、低密度膽固醇和體重的控制情況，並附有針對自我管理的個體化建議。餘下的1200 位患者中，隨機抽取一半患者（R+ 組，n=600）郵寄2 份JADE FU 報告，另外一半患者常規護理（R- 組，n=600）。未被邀請參加同伴支持計劃的1200 位患者中，425 位（50%有FU 報告）按年齡、性別、病程和基礎HbA1c 與P+組（52%有FU 報告）匹配成為對照組。研究指標為12 個月後HbA1c 和住院率的改變。 / 研究結果：未提供同伴支援的患者中，經過575（519-646）日的隨訪，R+組HbA1c 降低更多（ 均值[95% 置信區間]: -0.24[-0.35,-0.14]% versus-0.15[-0.24,-0.06]%, p=0.030），但住院率與R-組相同。提供同伴支持的患者中，P+組和P-組研究開始時臨床、心理和行為指標皆相似。530（463-575）日後，與P-組比較，P+組有HbA1c 降低更多的趨勢（-0.75[-0.97,-0.52] % versus-0.42[-0.68,-0.15]%, p=0.106），住院率相似，但心理健康和生活品質均有明顯改善。在病例對照研究中，與對照組比較（n=425），P+組（n=142） HbA1c降低更多（-0.92[-1.25, -0.59]% versus -0.39[-0.58, -0.21], p=0.004），且住院率、住院次數和時間皆有明顯降低。採用多變量回歸分析，同伴支持（β 係數[95% 置信區間] -0.31[-0.56, -0.06], p=0.015）和FU 報告（-0.14[-0.25, -0.04], p=0.009）均是降低HbA1c 的獨立預測因子。同伴支持（0.36[0.16, 0.79], p=0.011） 和CKD伴FU 報告（0.40[0.18, 0.88], p=0.022）與住院风险降低明顯相關。 / 結論：在此項品質改進計畫中，同伴支持和定期郵寄個體化隨訪報告均与糖尿病患者的糖化血紅蛋白降低相關。同伴支持伴隨住院风险降低，但個體化隨訪報告僅在糖尿病伴慢性腎臟病患者中与住院风险降低相關。 / Yin, Junmei. / Thesis Ph.D. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 209-235). / Abstracts also in Chinese; appendix 4 in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

In this thesis, we have presented our discovery of true finite homomorphic images of various permutation and monomial progenitors, such as 2*7: D14, 2*7 : (7 : 2), 2*6 : S3 x 2, 2*8: S4, 2*72: (32:(2S4)), and 11*2 :m D10. We have given delightful symmetric presentations and very nice permutation representations of these images which include, the Mathieu groups M11, M12, the 4-fold cover of the Mathieu group M22, 2 x L2(8), and L2(13). Moreover, we have given constructions, by using the technique of double coset enumeration, for some of the images, including M11 and M12. We have given proofs, either by hand or computer-based, of the isomorphism type of each image. In addition, we use Iwasawa's Lemma to prove that L2(13) over A5, L2(8) over D14, L2(13) over D14, L2(27) over 2D14, and M11 over 2S4 are simple groups. All of the work presented in this thesis is original to the best of our knowledge.

A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).

Representations of SO(4,2;R) are constructed using 4 x 4 and 2 x 2 matrices with elements in H' ��� C . Using 2 x 2 matrix representations of C and H', the 4 x 4 representation is interpreted in terms of 16 x 16 real matrices. Finally, the known isomorphism between the conformal group and SO(4,2;R) is written explicitly in terms of the 4 x 4 representation. The 4 x 4 construction should generalize to matrices with elements in K' ��� K for K any normed division algebra over the reals and K' any split algebra over the reals. / Graduation date: 2013

This article describes the contribution of the British to the development of Northeastern Ontario since the beginning of the 20th century.