Pietra ratio (Pietra index), also known as Robin Hood index, Schutz coefficient (Ricci-Schutz index) or half the relative mean deviation, is a good measure of statistical heterogeneity in the context of positive-valued data sets. In this thesis, two novel methods namely "adjusted jackknife empirical likelihood" and "extended jackknife empirical likelihood" are developed from the jackknife empirical likelihood method to obtain interval estimation of the Pietra ratio of a population. The performance of the two novel methods are compared with the jackknife empirical likelihood method, the normal approximation method and two bootstrap methods (the percentile bootstrap method and the bias corrected and accelerated bootstrap method). Simulation results indicate that under both symmetric and skewed distributions, especially when the sample is small, the extended jackknife empirical likelihood method gives the best performance among the six methods in terms of the coverage probabilities and interval lengths of the confidence interval of Pietra ratio; when the sample size is over 20, the adjusted jackknife empirical likelihood method performs better than the other methods, except the extended jackknife empirical likelihood method. Furthermore, several real data sets are used to illustrate the proposed methods.

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let $X_{\mathcal{F}}=\cup_{F\in\mathcal{F}}F$. A union closed family of sets is {\it separating} if for any two distinct elements in $\mathcal{F}$, there is a set in $\mathcal{F}$ containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family $\mathcal{F}$ is a poset with set inclusion as the partial order relation. A separating union closed family $\mathcal{F}$ is {\it irreducible} ({\it normalized}) if $|X_{\mathcal{F}}|$ is the minimum (maximum, resp.) with respect to the poset structure of $\mathcal{F}$. In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph $D$, a set $F$ of edges is defined to be a {\it directed cut} if there is a nontrivial partition $(X,Y)$ of $V(D)$ such that $F$ consists of all the directed edges from $X$ to $Y$. The maximum size of a directed cut in a given digraph $D$ is denoted by $\Lambda (D)$, and we let $\mathcal{D}(1,1)$ be the set of all digraphs $D$ such that $d^{+}(v)=1$ or $d^{-}(v)=1$ for every vertex $v$ in $D$. In this part of dissertation, we prove that $\Lambda (D) \geq \frac{3}{8}(|E(D)|-1)$ for any connected digraph $D\in\mathcal{D}(1,1)$, which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in $\mathcal{D}(1,1)$ and answer their another question.

In this work we discover for the first time a strong relationship between Geršgorin theory and the geometric multiplicities of eigenvalues. In fact, if λ is an eigenvalue of an n × n matrix A with geometric multiplicity k, then λ is in at least k Geršgorin discs of A. Moreover, construct the matrix C by replacing, in every row, the (k − 1) smallest off-diagonal entries in absolute value by 0, then λ is in at least k Geršgorin discs of C. We also state and prove many new applications and consequences of these results as well as we update an improve some important existing ones.

When an emerging outbreak occurs, stable parameter estimation and reliable projections of future incidence cases using limited (early) data can play an important role in optimal allocation of resources and in the development of effective public health intervention programs. However, the inverse parameter identification problem is ill-posed and cannot be solved with classical tools of computational mathematics. In this dissertation, various regularization methods are employed to incorporate stability in parameter estimation algorithms. The recovered parameters are then used to generate future incident curves as well as the carrying capacity of the epidemic and the turning point of the outbreak. For the nonlinear generalized Richards model of disease progression, we develop a novel iteratively regularized Gauss-Newton-type algorithm to reconstruct major characteristics of an emerging infection. This problem-oriented numerical scheme takes full advantage of a priori information available for our specific application in order to stabilize the iterative process. Another important aspect of our research is a reliable estimation of time-dependent transmission rate in a compartmental SEIR disease model. To that end, the ODE-constrained minimization problem is reduced to a linear Volterra integral equation of the first kind, and a combination of regularizing filters is employed to approximate the unknown transmission parameter in a stable manner. To justify our theoretical findings, extensive numerical experiments have been conducted with both synthetic and real data for various infectious diseases.

This dissertation reports on an investigation of transition-to-proof students' understanding of proof by contradiction. A plethora of research on the construction aspect of proof by contradiction is available and suggests that the method is one of the most difficult for students to construct and comprehend. However, there is little research on the students' comprehension of proofs and, in particular, proofs by contradiction. This study aims to fill this gap in the literature. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, this study proposes a preliminary genetic decomposition for how a student might construct the concept `proof by contradiction' and a series of five teaching interventions based on this preliminary genetic decomposition. Data was analyzed in two ways: (1) group analysis of the first two teaching interventions to consider students' initial conceptions of the proof method and (2) case study analysis of two individuals to consider how students' understanding developed over time. The genetic decomposition and teaching interventions were then revised based on the results of the data analysis. This study concludes with implications for teaching the concept of proof by contradiction and suggestions for further research on the topic.

Sodium ion (Na+) channels play an important role in excitable cells, as they are responsible for the initiation of action potentials. Understanding the electrical characteristics of sodium channels is essential in predicting their behavior under different physiological conditions. We investigated several Markov models for the human cardiac sodium channel (NaV1.5) to derive a minimal mathematical model that can describe the reported experimental data obtained using major voltage-clamp protocols. We obtained simulation results for current-voltage relationships, steady-state inactivation, the voltage dependence of normalized ion channel conductance; activation and deactivation, fast and slow inactivation and recovery from inactivation kinetics. Good agreement with the experimental data provides us with the mechanisms of the fast and slow inactivation of the human sodium channel and the coupling of its inactivation states to the closed and open states in the activation pathway.

碩士 / 國立師範大學 / 數學系 / 81 / Felix Klein 是 一位偉大的數學家 。他不但發現非歐幾何 有射影幾何 性質之外，也提出雙曲幾何模型，解決了非歐幾何公設系統的一致性問題 。1872年，他更發表"Erlanger Programm"，其中以變換群論幾何的觀點 ，影響了後來幾何學發展的方向。並且也開創了"自守函數"這一研究領域 ，讓後來的數學家繼續耕耘其上。此外，他也領導哥廷根學派，使得哥廷 根大學在二十世紀初期，成為世界數學的中心。事實上，Klein也是一位 關心數學教育的數學家。他在1900年以後，努力於德國中學數學教育改革 的活動，強調函數概念在數學教學上的重要性，並且主張將微積分和解析 幾何放入中學數學課程裡。1908年，國際數學教育委員會成力之後， Klein被推選為委員會的主席。在他領導下的數學教育委員會，確實地發 揮了國際數學教育資訊交流的功效。然而對於這樣一位在數學教育史上， 如此重要的人物，就筆者所掌握的資料而言，並無專論Klein在數學教育 方面的貢獻和成就。所以本論文的主要目標便是往這方面努力，以便給 Klein在數學教育史上，一個合適的定位。此外，由於Klein是一位偉大的 數學家，所以筆者在論文裡亦強調Klein的數學觀，如何影響到他的數學 教育思想。更重要的是: 筆者希望藉由本論文，能夠對關心數學教育的人 士有所啟發。

Thesis (MEd)--Stellenbosch University, 2012 / Includes bibliography / ENGLISH ABSTRACT: The use of information and communications technologies (ICTs) in the form of spreadsheet algebra programmes (SAPs) is important in the professional development of high school mathematics teachers. This is in line with The Namibian government‟s Vision 2030 in which ICT skills and competencies are regarded as core elements of living and participating in the 21st century. ICTs are also considered to be fundamental to the development of a dynamic knowledge-based economy (KBE) through the Education and Training Sector Improvement Program (ETSIP). ETSIP‟s aim is to embed ICT at all levels of the education system. It also aims to integrate the use of ICTs as tools in the delivery of curriculum and learning and in so doing, lead to a marked improvement in the quality of the learning and teaching process across all levels. Education has a key role in achieving Vision 2030. The aim of this research was to investigate mathematics teachers‟ perceptions of SAPs as curriculum materials in selected Namibian secondary (high) schools. This research adopted a qualitative methodology, which in this instance was a case study. The sample population consisted of five teachers from Okamu (pseudonym) secondary school in the Ohangwena Region of Namibia. Four of them had been teaching mathematics at different levels in the mentioned school for a period of four years, and one of them was teaching physical science. Three methods of data collection were used. The first two were semi-structured interviews and focus group interviews based on teachers‟ experiences using SAPs. The third method was an audio taped observation of a lesson taught by one of the teachers,. This research provides evidence about teachers‟ perceptions regarding time concerns and constraints with regards to the SAPs and the use of the SAPs. The teachers showed willingness and enthusiasm to use SAPs on linear and quadratic functions in their teaching. Some of the teachers became more aware of the epistemic dimensions associated with mathematical and algebraic symbols. Interview data reveal that the teachers had not considered these dimensions when teaching with the usual paper-and-pen format. The research also provides evidence of a teacher‟s early vision about the use of spreadsheets as an instrument to teach linear functions. This teacher did not consider any epistemic value for the instrumented spreadsheets techniques, or that they might contribute to a deeper understanding of the linear functions. His concern was focused more on getting the learners to acquire computer skills, such as learning how to use spreadsheets. It is recommended that in-service professional development about ICT integration into mathematics teaching be offered. This might help teachers to learn how their knowledge and skills could be used in the classroom more effectively in order to save time. It is also suggested that professional development programmes be designed to stimulate and promote teachers‟ willingness to develop an understanding of the characteristics of ICTs such as SAPs and their uses. Lastly, it is recommended that new SAPs be designed in order to deepen the understanding of algebra at the secondary level. / AFRIKAANSE OPSOMMING: Die gebruik van inligting- en kommunikasietegnologieë (IKT's) in die vorm van sigblad-algebra-programme (SAP's) is belangrik vir die professionele ontwikkeling van hoërskoolwiskunde-onderwysers. Dit is in pas met die Namibiese regering se visie vir 2030, Vision 2030, waarin IKT-vaardighede en -bevoegdhede beskou word as kernelemente van die lewe in en deelname aan die 21ste eeu. IKT's word ook beskou as grondliggend aan die ontwikkeling van ‟n dinamiese kennisekonomie (KE) deur middel van die Verbeteringsprogram vir die Onderwys- en Opleidingsektore (ETSIP). ETSIP het as oogmerk om IKT op alle vlakke van die onderwysstelsel vas te lê. Dit het ook ten doel om die gebruik van IKT's as hulpmiddele te integreer in die lewering van kurrikulum en leer en sodoende ‟n duidelike verbetering in die gehalte van die onderrig-en-leerproses oor alle vlakke heen tot gevolg te hê. Onderwys het ‟n sleutelrol te speel by die bereiking van Vision 2030. Die doel van hierdie navorsing was om wiskundeonderwysers se persepsies van SAP's as kurrikulummateriaal in geselekteerde Namibiese sekondêre (hoër-) skole te ondersoek. Hierdie navorsing het ‟n kwalitatiewe metode gevolg, in hierdie geval ‟n gevallestudie. Die proefgroep het bestaan uit vyf onderwysers van die sekondêre skool Okamu (skuilnaam) in die Ohangwena-streek van Namibië. Vier van hulle het reeds vier jaar lank wiskunde op verskillende vlakke in die betrokke skool gegee en een van hulle het fisiese wetenskap gegee. Drie metodes is ingespan om data in te samel. Die eerste twee was semigestruktureerde onderhoude en fokusgroeponderhoude gebaseer op onderwysers se ervaringe ten opsigte van die gebruik van SAP's. Die derde metode was ‟n klankopname van ‟n waarnemingsessie van ‟n les wat deur een van die onderwysers gegee is. Hierdie navorsing getuig van onderwysers se persepsies ten opsigte van tydskwessies en -beperkinge met betrekking tot die SAP's en die gebruik van die SAP's. Die onderwysers het ‟n gewilligheid en geesdrif geopenbaar om by lineêre en kwadratiese funksies SAP's in hul onderrig te gebruik. Party onderwysers het meer bewus geraak van die epistemiese dimensies in verband met wiskundige en algebraïese simbole. Onderhouddata onthul dat die onderwysers nie hierdie dimensies in ag geneem het toe hulle met die gewone papier-en-pen-formaat klasgegee het nie. Die navorsing bevestig ook ‟n onderwyser se aanvanklike insig oor die gebruik van sigblaaie as 'n instrument om lineêre funksies te onderrig. Hierdie onderwyser het nie die epistemiese waarde vir die geïnstrumenteerde sigbladtegnieke in ag geneem nie, of dat hulle sou kon bydra tot ‟n grondiger begrip van die lineêre funksies nie. Sy belangstelling was eerder daarop gefokus om die leerders sover te kry om rekenaarvaardighede, soos om sigblaaie te kan gebruik, te verwerf. Daar word aanbeveel dat indiens- professionele ontwikkeling oor IKT-integrasie in wiskundeonderrig aangebied word. Dit sal onderwysers moontlik help om te leer hoe hulle kennis en vaardighede meer doeltreffend in die klaskamer gebruik kan word om tyd te bespaar. Daar word ook voorgestel dat professionele ontwikkelingsprogramme ontwerp word, ter stimulering en bevordering van onderwysers se bereidwilligheid om ‟n begrip te ontwikkel van die kenmerke en gebruike van IKT's soos SAP's. Laastens word daar aanbeveel dat nuwe SAP's ontwerp word om die verstaan van algebra op sekondêre vlak te verdiep.

Pasaulyje yra plačiai diskutuojamas klausimu, kodėl mažai moterų dalyvauja tiksliųjų mokslų srityje. Lietuvoje ši tema nėra nagrinėta. Todėl pagrindinis šio darbo tikslas yra ištyrinėti Lietuvos mergaičių ir berniukų skirtumus mokantis matematikos. Darbą sudaro keturios pagrindinės dalys. Teorinėje dalyje apžvelgiama problema bei jos aktualumas. Tyrinėjama, kas gali lemti skirtingus mergaičių ir berniukų matematikos rezultatus, ir ar tiesa, kad berniukams matematika gali sektis geriau negu mergaitėms. Praktinėje dalyje nagrinėjami surinkti matematikos olimpiadų bei TIMSS tyrimo matematinio raštingumo testų rezultatai. Tyrinėjama, kuriai lyčiai geriau sekasi spręsti uždavinius olimpiadose, kokie uždaviniai geriau sekasi mergaitėms, o kokie berniukams. Nagrinėjamos berniukų ir mergaičių nuostatos matematikos atžvilgiu. Dar vieną svarbią dalį sudaro testo – tyrimo rezultatai. Testas atliktas ketvirtose ir aštuntose klasėse. Nagrinėjama, kokie uždaviniai geriau sekėsi merginoms, o kokie vaikinams. Analizuojama, ar pasitvirtino iškeltos hipotezės. Ketvirtoje dalyje pateikiamos darbo išvados ir rekomendacijos. Pagrindinė darbo išvada yra ta, kad mokykloje matematika sekasi vienodai gerai berniukams ir mergaitėms. Vaikinų sėkmę olimpiadose lemia tai, kad jiems geriau sekasi nestandartinių uždavinių sprendimas. / The academic faculties of the world universities in the fields of mathematics, engineering, and science are predominantly male. Therefore, there are many debates on the issue of gender differences on math achievement. In Lithuania this issue is also up for discussion. However, first of all, differences between girls and boys in learning mathematics should be investigated. This is the main purpose of the present work. The work is made of four parts. The first analysis is based on theoretical investigations. Factors that can cause gender differences in math perfomance are analysed. Then, the results of math olympiads and the results of TIMSS study math tests are investigated. Moreover, the results of tests made by Lithuanian fourth and eighth grade students are presented. Finally, the conclusions are made. The main conclusion of the work is that overall math perfomance of girls and boys in school is the same. However, research on math olympiads results show that boys during olympiades perform much better than girls. This is mainly because non-standard tasks seem to be performed much better by boys than by girls.

Šiame darbe apžvelgėme du egzaminavimo raštu metodus: be šaltinių, su šaltiniais. Išsiaiškinome, jų privalumus ir trūkumus. Remiantis moksliniais tyrimais išsiaiškinome, kad egzaminai su šaltiniais mažina stresinę įtampą, skatina studentus ne aklai įsiminti studijuojamą medžiagą, o mąstyti ir mokėti ją pritaikyti praktikoje. Lietuvoje populiaresnis egzaminavimo metodas yra be šaltinių, tačiau studentai mieliau ruošiasi egzaminams su šaltiniais. Atlikome tyrimą, kuriuo nagrinėjome kontrolinio ar egzamino įvertinimų ryšį su informacijos kiekiu studentų paruoštukėse. Pirmiausia, studentų paruoštukių turinį suskaitmeninome, t.y. pagalbinių medžiagų (paruoštukių) informacija užkodavome 0 ir 1. Tuomet atrinkome, kokia teorinė informacija galėjo būti naudinga sprendžiant kiekvieną kontrolinio ar egzamino uždavinį. Nagrinėjome kiekvieno uždavinio įvertinimų ryšį su informacijos kiekiu paruoštukėse, taip gavome trijų skirtingų koreliacijų rūšis: labai mažą, teigiamą didelę ir neigiamą didelę. Apibendrinant, galima daryti išvadą, jog pagalbinės medžiagos naudojimas egzaminų metu skatina studentus skirti egzamino pasiruošimui daugiau laiko, pasiruošimas egzaminui tampa nuoseklesnis, nes studentai anksčiau pradeda galvoti apie pagalbinės medžiagos turinį. Verta ir toliau tęsti pagalbinės medžiagos ir egzamino rezultatų ryšių tyrimus. Tai galėtų pagerinti mokymo kokybę. / Lina Jakubonienė THE ANALYSIS OF THE SUPPORTING MATERIAL IN MATHEMATICS EXAM. Supervisor: Dr. associate professor Ričardas Kudžma Vilnius University, the Faculty of Mathematics and Informatics, Department of Didactics of Mathematics and Informatics Vilnius, 2011 Aim of this work. Inspect the examination methods with the supporting material and without it. Explore the supporting material from the mathematical analysis exam, find out whether the amount of information in the supporting material has the capability to influence the test score results. Tasks for the investigation: 1. Examine the most popular written examination techniques. 2. Analyze the tasks for the test and the exam. 3. Figure out which theoretical knowledge is required to successfully solve the tasks for the tests and the exams. 4. Evaluate the supporting material sheets. 5. Ascertain whether the students had the correct theoretical information written down on their supporting material sheets and if they used it during the examination process. 6. Compare the student‘s test and exam results from three programs at the Faculty of Mathematics and Informatics of Vilnius University. Conclusions: The results about correlation between the information in the supporting material and exams score can be divided into three areas. The first one with the low correlation. This can be explained that we had not sufficient amount of data. Such was the most often case. Cases with big positive correlation can be explained that... [to full text]